Некоторые вопросы теории сложности вычислений с точки зрения элементарной теории моделей

L ogic P roof Íåêîòîðûå âîïðîñû òåîðèè ñëîæíîñòè âû÷èñëåíèé ñ òî÷êè çðåíèÿ ýëåìåíòàðíîé òåîðèè ìîäåëåé  ïðåäñòàâëåííîì èññëåäîâàíèè ðàññìàòðèâàåòñÿ îäíà èç èçâåñòíûõ ïðîáëåì òåîðèè ñëîæíîñòè âû÷èñëåíèé: êàêîâî ñîîòíîøåíèå êëàññîâ ñëîæíîñòè N P è co − N P ? Äëÿ îòâåòà íà ýòîò âîïðîñ áûëî ïåðåîñìûëåííî è ñîîòâåòñòâóþùèì îáðàçîì ïåðåôîðìóëèðîâàíî èçâåñòíîå ôóíäàìåíòàëüíîå ïîíÿòèå - ìîäåëüíàÿ ïîëíîòà èññëåäóåìîé òåîðèè, ðàçäåëà ìàòåìàòèêè ” Òåîðèÿ ìîäåëåé ” . Öåëüþ ïåðåôîðìó- ëèðîâêè ýòîãî ôóíäàìåíòàëüíîãî ïîíÿòèÿ ÿâëÿëîñü, îïèñàòü ñîîòíîøåíèå êëàññîâ ñëîæíîñòè N P è co − N P , ñ òåîðåòèêî - ìîäåëüíîé òî÷êè çðåíèÿ. Èçâåñòíûé ôàêò: èåðàðõèÿ ñâîéñòâ â ëþáîé ìîäå- ëè ìîäåëüíî ïîëíîé òåîðèè îáðûâàåòÿ íà ïåðâîì óðîâíå. Ýòà êëþ÷åâàÿ èäåÿ áûëà ïîëîæåíà â îñíîâó äëÿ ïëîäîòâîðíîãî èññëåäîâàíèÿ ñîîòíîøåíèÿ êëàññîâ ñëîæíîñòè N P è co − N P . Èçâåñòíûé ôàêò: ñóùåñòâóåò òàêîé îðàêóë A ïðè êîòîðîì êëàññ ñëîæíîñòè N P ( A ) îòëè÷àåòñÿ îò êëàññà ñëîæíîñòè co − N P ( A ) . Ðàçðàáîòàâ ñîîòâåòñòâóþùèì îáðàçîì îðàêóëüíûå âû÷èñëåíèÿ è ôîðìàëèçîâàâ èõ, â êëàñ- ñå ïðèìèòèâíî ðåêóðñèâíûõ àëãîðèòìîâ, à çàòåì, èñïîëüçóÿ òåîðåòèêî - ìîäåëüíîå ñîîòíîøåíèå ìåæ- äó óêàçàííûìè êëàññàìè, óäàëîñü ñâÿçàòü ñîîòíîøåíèå êëàññîâ ñëîæíîñòè âû÷èñëåíèé N P è co − N P ñ ñîîòíîøåíèåì êëàññîâ ñëîæíîñòè N P ( A ) è co − N P ( A ) , ÷òî çàòåì ïîçâîëèëî óñòàíîâèòü, ÷òî êëàññ ñëîæíîñòè N P íå ÿâëÿåòñÿ áóëåâîé àëãåáðîé. Ïðè ôîðàëèçàöèè îðàêóëüíûõ âû÷èñëåíèé â êëàññå ïðèìèòèâíî ðåêóðñèâíûõ àëãîðèòìîâ áûëè äîêàçàíû ðÿä èíòåðåñíûõ òåîðåì, îäíà èç êîòîðûõ åñòü àíàëîã òåîðåìû î íåïîäâèæíîé òî÷êå, êîòîðàÿ èñïîëüçîâàëàñü â êëþ÷åâîé òåîðåìå ïîçâîëèâøåé óñòà- íîâèòü, ÷òî êëàññ ñëîæíîñòè N P íå ÿâëÿåòñÿ áóëåâîé àëãåáðîé. Ïðî÷òÿ ïðåäñòàâëåííîå èññëåäîâàíèå, ìîæíî ïîíÿòü ïî÷åìó ýôôåêò ðåëÿòèâèçàöèè ïðåïÿòñòâóåò ïîëó÷åíèþ âûñîêèõ íèæíèõ îöåíîê èëè îòäåëåíèþ îäíîãî êëàññà ñëîæíîñòè îò äðóãîãî êëàññà ñëîæíîñòè âû÷èñëåíèé ìåòîäàìè "Äèñêðåòíîé ìàòåìàòèêè" 1 .  ñëàáûõ ìîäåëÿõ ñðåäñòâà äîêàçàòåëüñòâà íå ïîçâîëÿþò äîêàçàòü, íàïðèìåð, óòâåð- æäåíèå P̸ = N P , à â ñèëüíûõ ìîäåëÿõ ñðåäñòâà äîêàçàòåëüñòâà ïîäâåðæåíû ðåëÿòèâèçàöèè. Åñëè èìååòñÿ äîêàçàòåëüñòâî óòâåðæäåíèå P̸ = N P , òî äëÿ ëþáîãî îðàêóëà A ìû ïîëó÷èì äîêàçàòåëüñòâî óòâåðæäåíèå P ( A ) ̸ = N P ( A ) , îäíàêî èìååòñÿ òàêîé îðàêóë B , ÷òî âåðíî P ( B ) = N P ( B ) . Åñëè æå èìååòñÿ äîêàçàòåëüñòâî óòâåðæäåíèå P = N P , òî äëÿ ëþáîãî îðàêóëà A ìû ïîëó÷èì äîêàçàòåëüñòâî óòâåðæäåíèå P ( A ) = N P ( A ) , îäíàêî èìååòñÿ òàêîé îðàêóë C , ÷òî âåðíî P ( C ) ̸ = N P ( C ) . Îïèñàííûå òðóäíîñòè óñïåøíî ïðåîäîëåíû â ïðåäñòàâëåííîì èññëåäîâàíèè. Ïðåäñòàâëåííîå èññëåäîâàíèå ÿâëÿåòñÿ îðèãèíàëüíûì, à ìíîãèå âàæíûå ïîíÿòèå, êîòîðûå èñïîëü- çóþòñÿ â ýòîì èññëåäîâàíèè, âî âñåõ èçâåñòíûõ ìíå èññëåäîâàíèÿõ, íå âñòðå÷àëèñü . Ââåäåíèå Îäíîé èç ñàìûõ èíòåðåñíûõ ïðîáëåì ñëîæíîñòè âû÷èñëåíèé, ÿâëÿåòñÿ ïðîáëåìà, ñîñòîÿùàÿ â òîì, ÷òî ñóùåñòâóåò èëè íå ñóùåñòâóåò ïîëèíîìèàëüíûé àëãîðèòì, âûïîëíÿÿ êîòîðûé íà ïðîèçâîëüíîé ôîðìóëå èñ÷èñëåíèÿ âûñêàçûâàíèé, ìîæíî ïîëó÷èòü îòâåò âûïîëíèìà èëè íå âûïîëíèìà ðàññìàòðèâàåìàÿ ôîð- ìóëà èñ÷èñëåíèÿ âûñêàçûâàíèé. Ýòà ïðîáëåìà, êîòîðóþ êàê ïðàâèëî îáîçíà÷àþò SAT , èíòåðåñíà òåì, ÷òî îíà ïðîñòî ôîðìóëèðóåòñÿ è ôîðìèðóåò ÷óâñòâî íà ñêîðûé îòâåò íà ýòó ïðîáëåìó. À ïîëîæèòåëüíûé ðåøåíèå ýòîé ïðîáëåìû äàåò âîçìîæíîñòü ïîñòðîåíèÿ âåñüìà ýôôåêòèâíûõ àëãîðèòìîâ äëÿ ìíîãèõ ïðàê- òè÷åñêèõ çàäà÷. Ýòîé ïðîáëåìå ïîñâÿùåíû äåñÿòêè òûñÿ÷ íàó÷íûõ è âåñüìà èíòåðåñíûõ èññëåäîâàíèé, ïîçâîëèâøèå áîëåå ãëóáîêî âûÿñíèòü ñóùíîñòü ôóíäàìåíòàëüíîãî ïîíÿòèÿ àëãîðèòìà, íî íà ïîñòàâëåí- íûé âîïðîñ äî ñèõ ïîð íåò îòâåòà. Îäíàêî, îòâåò â îïðåäåëåííîì ñìûñëå áûë ïîëó÷åí â [13], â êîòîðîì 1 Ïîä ìåòîäàìè Äèñêðåòíîé ìàòåìàòèêè, ÿ ïîíèìàþ òå äîêàçàòåëüñòâà, êîòîîðûå ìîæíî âûðàçèòü â ñòàíäàðòíîé ìîäåëè àðèôìåòèêè, íàïðèìåð, äîêàçàòåëüñòâî Consis íåëüçÿ âûðàçèòü â ñòàíäàðòíîé ìîäåëè àðèôìåòèêè, õîòÿ ýòî ïðåäëîæåíèå èñòèíî â ýòîé ìîäåëè, Íå âñå óòâåðæäåíèÿ, êîòîðûå ïîäâåðãàþòñÿ ðåëÿòèâèçàöèè, ìîæíî äîêàçàòü ìåòîäàìè Äèñêðåòíîé ìàòåìàòèêè. È ýòî ïðîäåìîíñòðèðîâàíî â äàííîì èññëåäîâàíèè. 1 áûëà íà îïðåäåëåííîì óðîâíå, ïîíÿòà ñóùíîñòü ïðîáëåìû SAT .  óïîìÿíóòîì èññëåäîâàíèè, êîòîðîå áûëî îðèãèíàëüíûì è íåîæèäàííûì äëÿ ìåíÿ, áûë ïîñòðîåí îðàêóë A , òàêîé, ÷òî êëàññ ïîëèíîìèàëü- íûõ îðàêóëüíûõ àëãîðèòìîâ, èñïîëüçóþùèõ â êà÷åñòâå îðàêóëà A , íå ñìîæåò ðàñïîçàòü òàêîé ÿçûê êàê L ( A ) = { α : ∃ x ( | α | = | x | ∧ x ∈ A ) } , íî ñìîæåò î÷åâèäíî, ðàñïîçíàòü íåäåòåðìèíèðîâàííûé ïîëèíîìèàëü- íûé îðàêóëüíûé àëãîðèòì ñ òåì æå îðàêóëîì. Ýòîò âûäàþùèéñÿ ðåçóëüòàò ñôîðìèðîâàë ó ìåíÿ ïîäõîä, ðåàëèçóÿ êîòîðûé, ìîæíî áûëî áû ïîëó÷èòü îòâåò íà âûøåïîñòàâëåííûé âîïðîñ. ×òî äëÿ ýòîãî áûëî íåîáõîäèìî ïîëó÷èòü? Ïåðâîå . Ôîðìàëèçîâàòü(ñèíòàêñè÷åñêè îïèñàòü) âû÷èñëåíèÿ ïðèìèòèâíî ðåêóðñèâíûõ ñëîâàðíûõ ôóíêöèé è îðàêóëüíûõ ïðèìèòèâíî ðåêóðñèâíûõ ñëîâàðíûõ ôóíêöèé. Ñì. Part I è Part II. Âòîðîå . Ñèíòàêñè÷åñêè îïèñàòü êëàññ ïîëèíîìèàëüíî âû÷èñëèìûõ ñëîâàðíûõ ôóíêöèé è ïîëèíî- ìèàëüíî âû÷èñëèìûõ ñëîâàðíûõ ôóíêöèé, ïðè âû÷èñëåíèè êîòîðûõ, ìîæåò èñïîëüçîâàòüñÿ îðàêóë, ò.å. ïîëèíîìèàëüíûé îðàêóëüíûé àëãîðèòì. Ñì. Part III. Òðåòüå . Ñâÿçàòü ôîðìàëèçîâàííîå âû÷èñëåíèå ïðèìèòèâíî ðåêóðñèâíûõ ñëîâàðíûõ ôóíêöèé ñ ôîð- ìàëèçîâàííûì âû÷èñëåíèåì îðàêóëüíûõ ïðèìèòèâíî ðåêóðñèâíûõ ôóíêöèé. Ñì. Part IV è Part V. ×åòâåðòîå . Âûðàçèòü ñèíòàêòè÷åñêèå ñâîéñòâà èññëåäóåìûõ ÿçûêîâ, íàïðèìåð SAT , èñïîëüçóÿ äëÿ ýòîãî ïîíÿòèÿ è ìåòîäû, êîòîðûå ðàçâèòû â "Òåîðèè ìîäåëåé". Ñì. Part VI. Ðåàëèçîâàâ âñå ÷åòûðå ñëó÷àÿ, óäàëîñü îòâåòèòü íà âîïðîñ N P =? co − N P . Part I Èñ÷èñëåíèå ðàâåíñòâ äëÿ âû÷èñëåíèÿ çàìêíóòûõ òåðìîâ Ïóñòü àëôàâèòû L 1 , . . . , L 6 òàêîâû, ÷òî: L 1 = { S, I, Z , δ , Length , . . . . . . . . . , Concat , D , } , L 2 = { Λ , x } , L 3 = { R, J } , L 4 = { = , | , , } , L 5 = { [ , ] , ( , ) } , L 6 = { U } , L = i =5 S i =1 L i , L ( U ) = L ∪ L 6 .  àëôàâèòå L ( U ) áóäåò ôîðìàëèçîâàíî âû÷èñëåíèå çàìêíóòûõ òåðìîâ äëÿ êëàññà îðàêóëüíûõ ïðèìèòèâíî ðåêóðñèâíûõ ñëîâàðíûõ ôóíêöèé[1-4][5 ñ. 204]. Äëÿ ïîëíîòû ïðèâåä¼ì íåñêîëüêî äîñòàòî÷íî òðàäèöèîííûõ îïðåäåëåíèé. Îïðåäåëåíèå ôóíêòîðà è åãî ìåñòíîñòè: 2 1) Ñëîâà âèäà: Z , δ , Length , S | S, S || S, . . . , S || , . . . , | | {z } m − ðàç S - îäíîìåñòíûå ôóíêòîðû. Îäíîìåñòíûé ôóíêòîð S | , . . . , | | {z } k − ðàç S áóäåì îáîçíà÷àòü êàê S k . 2) Ñëîâà âèäà: I | , . . . , | | {z } m − ðàç , | , . . . , | | {z } n − ðàç - n - ìåñòíûé ôóíêòîð, êîòîðûé áóäåì îáîçíà÷àòü òðàäèöèîííî I n m , ïðè 1 ≤ m ≤ n . 3) Ñëîâî U - îäíîìåñòíûé ôóíêòîð.  äàëüíåéøåì ôóíêòîð U áóäåì íàçûâàòü òàêæå íåîïðåäåëÿåìûì ôóíêöèîíàëüíûì ñèìâîëîì èëè îðàêóëüíûì ñèìâîëîì. 4) Ñëîâà âèäà: . , Concat , D - äâóõìåñòíûå ôóíêòîðû. 5) Åñëè Φ - k - ìåñòíûé ôóíêòîð, Ψ 1 , . . . , Ψ k - n - ìåñòíûå ôóíêòîðû, òî ñëîâî [ J ΦΨ 1 , . . . , Ψ k ] - n - ôóíêòîð. Ýòîò ôóíêòîð áóäåì íàçûâàòü ôóíêòîðîì ñóïåðïîçèöèè. Ââåä¼ì ñëåäóþùåå âàæíîå ïîíÿòèå: à) Λ - àðãóìåíòíîå ñëîâî, êîòîðîå íàçûâàåòñÿ ïóñòûì ñëîâîì; á) åñëè α - àðãóìåíòíîå, òî ñëîâî S k ( α ) - àðãóìåíòíîå ñëîâî, êîòîðîå áóäåì îáîçíà÷àòü êàê αa k . Àðãóìåíòíîå ñëîâî α íàçûâàåòñÿ k - àëôàâèòíûì, åñëè ýòî ñëîâî íå ñîäåðæèò ôóíêòîðîâ S l , ïðè l > k . Ìíîæåñòâî B àðãóìåíòíûõ ñëîâ íàçûâàåòñÿ k - àëôàâèòíûì, åñëè êàæäîå ñëîâî α ∈ B ÿâëÿåòñÿ k - àëôàâèòíûì. Ïóñòü k > 1 . ×èñëî âñåõ k - àëôàâèòíûõ ñëîâ, äëèíà êîòîðûõ l > 0 , ðàâíî k l . ×èñëî âñåõ k - àëôàâèòíûõ ñëîâ, äëèíà êîòîðûõ íå áîëüøå l , ðàâíî k l +1 − 1 k − 1 . Àðãóìåíòíûå ñëîâà, íå ñîäåðæàùèå ôóíêòîð S k , ïðè k > 1 , íàçûâàþòñÿ íàòóðàëüíûìè ÷èñëàìè. 6 ) Åñëè α - àðãóìåíòíîå ñëîâî, Φ 1 , . . . , Φ m - 2 -õ ìåñòíûå ôóíêòîðû, òîãäà ñëîâî [ Rα Φ 1 , · · · , Φ m ] - 1 - ìåñòíûé ôóíêòîð. 7 ) Åñëè Φ - k ≥ 1 - ìåñòíûé ôóíêòîð, Ψ 1 , . . . , Ψ m - k +2 - ìåñòíûå ôóíêòîðû, òîãäà ñëîâî [ R ΦΨ 1 , . . . , Ψ m ] - k +1 - ìåñòíûé ôóíêòîð. Ôóíêòîðû ïóíêòîâ 6 è 7 áóäåì íàçûâàòü ðåêóðñèâíûìè ôóíêòîðàìè, à ôóíêòî- ðû Φ , Φ 1 , . . . Ψ m íàçûâàþòñÿ ñîñòàâëÿþùèìè ôóíêòîðàìè, ÷èñëî m íàçûâàåòñÿ ñòåïåíüþ ðàçâåòâë¼ííîñòè ðàññìàòðèâàåìîãî ðåêóðñèâíîãî ôóíêòîðà. Ôóíêòîðû ïóíêòîâ 1-4 áóäóò íàçûâàòüñÿ èñõîäíûìè ôóíêòîðàìè. Ñëîâà âèäà x | , . . . , | | {z } l − ðàç x - ïåðåìåííûå. Îáîçíà÷èì ýòè ñëîâà òðàäèöèîííî â âèäå x l . Ïîíÿòèå òåðìà. 1) Âñÿêîå àðãóìåíòíîå ñëîâî è âñÿêàÿ ïåðåìåííàÿ - òåðì. 2) Åñëè t 1 , . . . , t k - òåðìû, Ψ - k - ìåñòíûé ôóíêòîð, òîãäà ñëîâî Ψ( t 1 , . . . , t k ) - òåðì. Äëÿ ëþáîãî ïîäìíîæåñòâà A àðãóìåíòíûõ ñëîâ, ââåä¼ì ñëåäóþùèå ðàâåíñòâà(îïðåäåëÿþùèå ðàâåí- ñòâà), â êà÷åñòâå àêñèîì îïðåäåëÿåìîé ôîðìàëèçàöèè äëÿ âû÷èñëåíèÿ îðàêóëüíûõ çàìêíóòûõ òåðìîâ: 3 1) T = T , ãäå T - ïðîèçâîëüíûé òåðì, 2) Z ( x 1 ) = Λ , 3) I n m ( x 1 , . . . , x n ) = x m , 4) δ (Λ) = Λ , 5) δ ( x 1 a k ) = x 1 , 6) Length (Λ) = Λ , 7) Length ( x 1 a k ) = Length ( x 1 ) a 1 , 8) x 1 . . . . . . . . . Λ = x 1 , 9) x 1 . . . . . . . . . x 2 a k = δ ( x 1 . . . . . . . . . x 2 ) , 10) Concat ( x 1 , Λ) = x 1 , 11) Concat ( x 1 , x 2 a k ) = Concat ( x 1 , x 2 ) a k , 12) D ( x 1 , Λ) = Λ , 13) D ( x 1 , x 2 a k ) = Concat ( x 1 , D ( x 1 , x 2 )) , 14) [ J ΦΨ 1 , . . . , Ψ k ]( x ) = Φ(Ψ 1 ( x ) , . . . , Ψ k ( x )) , 15) [ Rα Φ 1 , . . . , Φ m ](Λ) = α , 16) [ Rα Φ 1 , . . . , Φ m ]( xa k ) = Φ k ( x, [ Rα Φ 1 , . . . , Φ m ]( x )) , ïðè k ⩽ m , 17) [ Rα Φ 1 , . . . , Φ m ]( xa k ) = [ Rα Φ 1 , . . . , Φ m ]( x ) , ïðè k > m , 18) [ R ΦΨ 1 , . . . , Ψ m ]( x, Λ) = Φ( x ) , 19) [ R ΦΨ 1 , . . . , Ψ m ]( x, ya k ) = Ψ k ( x, y, [ R ΦΨ 1 , . . . , Ψ m ]( x, y )) , ïðè k ⩽ m , 20) [ R ΦΨ 1 , . . . , Ψ m ]( x, ya k ) = [ R ΦΨ 1 , . . . , Ψ m ]( x, y )) , ïðè k > m . Ïóñòü A - íåêîòîðîå ìíîæåñòâî àðãóìåíòíûõ ñëîâ. Àêñèîìû èíòåðïðåòàöèè íåîïðåäåëÿåìîãî ôóíê- öèîíàëüíîãî ñèìâîëà U a ) U ( α ) = Λ , åñëè α ∈ A , b ) U ( α ) = a 1 , åñëè α / ∈ A . Ðàâåíñòâà a ) è b ) íàçûâàþòñÿ àêñèîìàìè èíòåðïðåòàöèè, êîòîðûå ñîîòâåòñòâóþò ìíîæåñòâó A , ìíî- æåñòâî àðãóìåíòíûõ ñëîâ A íàçûâàåòñÿ èíòåðïðåòàöèîííûì ìíîæåñòâîì. Ïðàâèëà âûâîäà Èñ÷èñëåíèå ðàâåíñòâ äëÿ çàìêíóòûõ òåðìîâ Sb : T 1 = Q 1 , T 2 = Q 2 [ T 2 ] x T 1 = [ Q 2 ] x Q 1 , Cut 1 : T 1 = T 2 , T 2 = T 3 T 1 = T 3 , Cut 2 : T 1 = T 2 , T 3 = T 2 T 1 = T 3 .  ïðàâèëå Sb ïåðåìåííàÿ x - ñîáñòâåííàÿ ïåðåìåííàÿ ýòîãî ïðàâèëà. 4 Çàìå÷àíèå. Äëÿ âû÷èñëåíèÿ çíà÷åíèé çàìêíóòûõ òåðìîâ, äîñòàòî÷íî íàðÿäó ñ ïðàâèëîì Sb èñïîëü- çîâàòü òîëüêî ïðàâèëî Cut 1 . Äëÿ äîêàçàòåëüñòâà ðàâåíñòâ çàìêíóòûõ òåðìîâ äîáàâëÿåòñÿ ïðàâèëî Cut 2 . Ìîæíî îáîéòèñü áåç ïðà- âèëà Cut 2 , çàìåíèâ ïðàâèëî Sb íà ïðàâèëî T 1 = Q 1 , T 2 = Q 2 [ T 2 ] x Q 1 = [ Q 2 ] x T 1 . Îïðåäåëåíèå âûâîäà. Ïîñëåäîâàòåëüíîñòü ðàâåíñòâ T 1 = Q 1 , . . . , T n = Q n ÿâëÿåòñÿ âûâîäîì(äîêàçàòåëüñòâîì ðàâåíñòâà T n = Q n ), åñëè äëÿ êàæäîãî i = 1 , 2 , . . . , n , ðàâåíñòâî T i = Q i ÿâëÿåòñÿ ëèáî àêñèîìîé, ëèáî àêñèîìîé èíòåðïðåòàöèè U ( α ) = Λ , U ( α ) = a 1 , êîòîðûå ñîîòâåòñòâóþò ìíîæåñòâó A , ëèáî ïîëó÷åíî èç ïðåäûäóùèõ ðàâåíñòâ ïî îäíîìó èç ïðàâèë âûâîäà. Åñëè âûâîä P òàêîâ, ÷òî îí ñîäåðæèò èíòåðïðåòàöèîííûå àêñèîìû U ( α ) = Λ èëè U ( β ) = a 1 , òîãäà áóäåì ãîâîðèòü, ÷òî ñëîâà α , β áûëè èñïîëüçîâàíû â ýòîì âûâîäå, ïðè÷¼ì ñëîâî α èñïîëüçîâàíî ïîëîæè- òåëüíî, ñëîâî β èñïîëüçîâàíî îòðèöàòåëüíî. Ñ âûâîäîì P , ïðè èíòåðïðåòàöèîííîì ìíîæåñòâå A , ñâÿæåì ïàðó ìíîæåñòâ: ( A + ) P - ìíîæåñòâî âñåõ ïîëîæèòåëüíî îïðîøåííûõ ñëîâ â âûâîäå P , ( A − ) P - ìíîæåñòâî âñåõ îòðèöàòåëüíî îïðîøåííûõ ñëîâ â âûâîäå P . Ïîñëåäîâàòåëüíîñòü ðàâåíñòâ t 1 = q 1 , . . . , t n = q n - êâàçèâûâîä, åñëè êàæäîå ðàâåíñòâî â ýòîé ïîñëåäî- âàòåëüíîñòè åñòü ëèáî âûâîäèìîå ðàâåíñòâî, ëèáî ïîëó÷åíî èç ïðåäûäóùèõ ðàâåíñòâ ïî îäíîìó èç ïðàâèë âûâîäà. Çàìå÷àíèå. Èäåÿ ïðèâåä¼ííîãî èñ÷èñëåíèÿ ðàâåíñòâ äëÿ âû÷èñëåíèÿ çàìêíóòûõ îðàêóëüíûõ òåðìîâ áûëà çàèìñòâîâàíà â ðàáîòàõ [1-6] èç êîòîðûõ îñíîâîïîëàãàþùåé ÿâëÿåòñÿ[3]. Äëèíà äîêàçàòåëüñòâà P - ÷èñëî ðàâåíñòâ â äîêàçàòåëüñòâå P . Ýòî ÷èñëî îáîçíà÷àåòñÿ êàê l P . Ïîëíàÿ äëèíà äîêàçàòåëüñòâà P - äëèíà ñëîâà, ïîëó÷åííîãî ñîåäèíåíèåì âñåõ ðàâåíñòâ, âõîäÿùèõ â äîêàçàòåëüñòâî P , ðàçäåë¼ííûõ ñèìâîëîì çàïÿòàÿ. Ýòî ÷èñëî îáîçíà÷àåòñÿ F l P . Îáîçíà÷èì ïîëó÷åííîå èñ÷èñëåíèå çàìêíóòûõ òåðìîâ êàê CalcEq , CalcEq U â àëôàâèòå L , L ( U ) ñîîòâåòñòâåííî. Îáîçíà÷åíèå ⊢ t = r - ðàâåíñòâî t = r âûâîäèìî â èñ÷èñëåíèè CalcEq U , ïðè ëþáîé èíòåðïðåòàöèè ôóíêòîðà U . Îáîçíà÷åíèå A ⊢ t = r - ðàâåíñòâî t = r âûâîäèìî â èñ÷èñëåíèè CalcEq U , ñ àêñèîìàìè èíòåðïðåòàöèè ñîîòâåòñòâóþùèìè ìíîæåñòâó A . Ôóíêòîð Φ , òåðì t íàçûâàþòñÿ n > 0 àëôàâèòíûì, åñëè ïðåäñòàâëåííûå ñëîâà íå ñîäåðæàò ôóíêòîðîâ S l , ïðè l > n . Âñå èñõîäíûå ôóíêòîðû ÿâëÿþòñÿ n àëôàâèòíûìè äëÿ êàæäîãî n . Äëÿ êàæäîãî n ≥ 1 è ëþáîãî àðãóìåíòíîãî ñëîâà α ìîæíî ïîñòðîèòü n - ìåñòíûé ôóíêòîð, îáîçíà÷à- 5 åìûé êàê Const n α , ÷òî âåðíî ⊢ Const n α ( x 1 , . . . , x n ) = α . Òåîðåìà 1.1 . Ìîæíî ñîñòàâèòü òàêîé àëãîðèòì, ñëåäóÿ êîòîðîìó äëÿ êàæäîãî òåðìà t ìîæíî ïî- ñòðîèòü òàêîé ôóíêòîð Φ t , ÷òî ⊢ Φ t ( y ) = t , ãäå y - ñïèñîê ïåðåìåííûõ, ñîäåðæàùèé ïåðåìåííûå òåðìà t . Äîêàçàòåëüñòâî è ïîëíóþ ôîðìóëèðîâêó ýòîé òåîðåìû ñì.â [2, ñòð. 62],[4, ñòð. 446]. Òåîðåìà 1.2 . Äëÿ ëþáîãî çàìêíóòîãî n àëôàâèòíîãî òåðìà t , äëÿ çàäàííîé èíòåðïðåòàöèè A ôóíê- òîðà U , ñóùåñòâóåò åäèíñòâåííîå òàêîå àðãóìåíòíîå ñëîâî α òîãî æå àëôàâèòà, ÷òî A ⊢ t = α . Äîêàçà- òåëüñòâî ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà, çàòåì èíäóêöèåé ïî ïîñòðîåíèþ òåðìà t . Äîêàçàòåëüñòâî P ðàâåíñòâà âèäà t = α , ãäå t - çàìêíóòûé òåðì, α - íåêîòîðîå àðãóìåíòíîå cëîâî, áóäåì íàçûâàòü âû÷èñëåíèåì òåðìà t . Çàìå÷àíèå. Äëÿ ëþáîãî òåðìà t ( x ) , äëÿ ëþáîé ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α , ñóùåñòâóåò òàêîå íàòóðàëüíîå ÷èñëî k , ÷òî äëÿ ëþáîãî èíòåðïðåòàöèîííîãî ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ìîæíî ïîñòðîèòü òàêîå âû÷èñëåíèå òåðìà t ( x ) íà α , â êîòîðîì áóäåò èñïîëüçîâàíî íå áîëüøå k àðãóìåíòíûõ ñëîâ. Òåîðåìà 1.3 . Ïóñòü äàí n - ìåñòíûé ôóíêòîð [ J ΦΨ 1 , . . . , Ψ k ] , α 1 , . . . , α n - íåêîòîðàÿ ïîñëåäîâàòåëü- íîñòü àðãóìåíòíûõ ñëîâ. Ïóñòü P Ψ i ,α - âû÷èñëåíèå ôóíêòîðà Ψ i íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , ñ ðåçóëüòàòîì âû÷èñëåíèÿ β i , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , 1 ≤ i ≤ k . Ïóñòü P Φ ,β - âû÷èñëåíèå ôóíêòîðà Φ íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ ñëîâ β 1 , . . . , β k , ñ ðåçóëüòàòîì âû÷èñëå- íèÿ γ , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , òîãäà ìîæíî ïîñòðîèòü âû÷èñëåíèå P [ J ΦΨ 1 ,..., Ψ k ] ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , äëÿ êîòîðîãî âåðíî: ( A + ) P [ J ΦΨ 1 ,..., Ψ k ] ,α = k S i =1 ( A + ) P Ψ i ,α S ( A + ) P Φ ,β , ( A − ) P [ J ΦΨ 1 ,..., Ψ k ] ,α = k S i =1 ( A − ) P Ψ i ,α S ( A − ) P Φ ,β . Äîêàçàòåëüñòâî. Ñîñòàâèì ñëåäóþùóþ ïîñëåäîâàòåëüíîñòü ðàâåíñòâ: P Ψ 1 ,α , . . . , P Ψ k ,α , [ J ΦΨ 1 , . . . , Ψ k ]( x 1 , . . . , x n ) = Φ(Ψ 1 ( x ) , . . . , Ψ k ( x )) , . . . , [ J ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n ) = Φ(Ψ 1 ( α ) , . . . , Ψ k ( α )) , Φ( x 1 , . . . , x n ) = Φ( x 1 , . . . , x n ) , . . . , Φ(Ψ 1 ( α ) , . . . , Ψ k ( α )) = Φ( β 1 , . . . , β n ) , P Φ , Φ(Ψ 1 ( α ) , . . . , Ψ k ( α )) = γ, [ J ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n ) = γ - âû÷èñ- ëåíèå ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] íà ïîñëåäîâàòåëüíîñòè α 1 , . . . , α n , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , ñ óêàçàííûì ìíîæåñòâîì ïîëîæèòåëüíî è îòðèöàòåëüíî îïðîøåííûõ ñëîâ. Òåîðåìà 1.4 . Ïóñòü äàí n + 1 ( n ≥ 1 ) - ìåñòíûé ôóíêòîð [ R ΦΨ 1 , . . . , Ψ k ] , α 1 , . . . , α n , βa i - íåêîòîðàÿ ïîñëåäîâàòåëüíîñòü àðãóìåíòíûõ ñëîâ. Ïóñòü P Φ - âû÷èñëåíèå ôóíêòîðà Φ íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , â èíòåðïðå- òàöèîííîì ìíîæåñòâå A . Ïóñòü P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,α n ,β - âû÷èñëåíèå ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ k ] íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ 6 ñëîâ α 1 , . . . , α n , β , ñ ðåçóëüòàòîì âû÷èñëåíèÿ γ , â èíòåðïðåòàöèîííîì ìíîæåñòâå A . Ïóñòü P Ψ i - âû÷èñëåíèå ôóíêòîðà Ψ i íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , β, γ , ñ ðå- çóëüòàòîì âû÷èñëåíèÿ η , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , òîãäà ìîæíî ïîñòðîèòü âû÷èñëåíèå P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,α n ,βa i ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ k ] íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , βa i , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , äëÿ êîòîðîãî âåðíî: 1. ( A + ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn, Λ = ( A + ) P Φ ; α 1 ,...,α n , ( A − ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn, Λ = ( A − ) P Φ ; α 1 ,...,α n 2. ( A + ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,βa i = ( A + ) P Ψ i , ,α 1 ,...,α n ,β,γ S ( A + ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,β , ( A − ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,βa i = ( A − ) P Ψ i ,α 1 ,...,α n ,β,γ S ( A − ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,β Äîêàçàòåëüñòâî. Ïóíêò (1) - î÷åâèäåí. Ïóíêò (2). Ñîñòàâèì ñëåäóþùóþ ïîñëåäîâàòåëüíîñòü ðà- âåíñòâ: P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,α n ,β , [ R ΦΨ 1 , . . . , Ψ k ]( x 1 , . . . , x n , za i ) = Ψ( x 1 , . . . , x n , z, [ R ΦΨ 1 , . . . , Ψ k ]( x 1 , . . . , x n , z )) , . . . , [ R ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n , βa i ) = Ψ i ( α 1 , . . . , α n , β, [ R ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n , β )) , Ψ i ( x 1 , . . . , x n , z, u ) = Ψ i ( x 1 , . . . , x n , z, u ) , . . . , Ψ i ( α 1 , . . . , α n , β, [ R ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n , β )) = Ψ i ( α 1 , . . . , α n , β, γ ) , P Ψ i ,α , Ψ i ( α 1 , . . . , α n , β, [ R ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n , β )) = η , [ R ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n , βa i ) = η - âû÷èñëåíèå ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ k ] íà ïîñëåäîâàòåëüíîñòè àðãó- ìåíòíûõ ñëîâ α 1 , . . . , α n , βa i , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , ñ óêàçàííûì ìíîæåñòâîì ïîëîæèòåëüíî è îòðèöàòåëüíî îïðîøåííûõ ñëîâ. Òåîðåìà 1.5 . Ïóñòü äàí ôóíêòîð [ Rα Ψ 1 , . . . , Ψ k ] , α 1 , . . . , α n , βa i - íåêîòîðàÿ ïîñëåäîâàòåëüíîñòü àð- ãóìåíòíûõ ñëîâ. Ïóñòü P [ Rα Ψ 1 ,..., Ψ k ] ,β - âû÷èñëåíèå ôóíêòîðà [ Rα Ψ 1 , . . . , Ψ k ] íà àðãóìåíòíîì ñëîâå β , ñ ðåçóëüòàòîì âû÷èñëåíèÿ γ , â èíòåðïðåòàöèîííîì ìíîæåñòâå A . Ïóñòü P Ψ i ,β,γ - âû÷èñëåíèå ôóíêòîðà Ψ i íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ ñëîâ β, γ , ñ ðåçóëüòàòîì âû÷èñëåíèÿ η , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , òîãäà ìîæíî ïîñòðîèòü âû÷èñëåíèå P [ Rα Ψ 1 ,..., Ψ k ] ,βa i ôóíêòîðà [ Rα Ψ 1 , . . . , Ψ k ] íà àðãóìåíòíîì ñëîâå βa i , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , äëÿ êîòîðîãî âåðíî: 1. ( A + ) P [ Rα Ψ 1 ,..., Ψ k ] , Λ = ∅ , ( A − ) P [ Rα Ψ 1 ,..., Ψ k ] , Λ = ∅ 7 2. ( A + ) P [ Rα Ψ 1 ,..., Ψ k ] ,βa i = ( A + ) P Ψ i ,β,γ S ( A + ) P [ Rα Ψ 1 ,..., Ψ k ] ,β , ( A − ) P [ Rα Ψ 1 ,..., Ψ k ] ,βa i = ( A − ) P Ψ i ,β,γ S ( A − ) P [ Rα Ψ 1 ,..., Ψ k ] ,β Äîêàçàòåëüñòâî. Ïóíêò (1) - î÷åâèäåí. Ïóíêò (2). Ñîñòàâèì ñëåäóþùóþ ïîñëåäîâàòåëüíîñòü ðà- âåíñòâ: P [ Rα Ψ 1 ,..., Ψ k ] ,β , [ Rα Ψ 1 , . . . , Ψ k ]( za i ) = Ψ i ( z, [ Rα Ψ 1 , . . . , Ψ k ]( z )) [ Rα Ψ 1 , . . . , Ψ k ]( βa i ) = Ψ i ( β, [ Rα Ψ 1 , . . . , Ψ k ]( β )) , Ψ i ( z, u ) = Ψ i ( z, u ) , . . . , Ψ i ( β, [ Rα Ψ 1 , . . . , Ψ k ]( β )) = Ψ i ( β, γ ) , P Ψ i ; β,γ , Ψ i ( β, [ R ΦΨ 1 , . . . , Ψ k ]( β )) = η , [ Rα Ψ 1 , . . . , Ψ k ]( βa i ) = η - âû÷èñëåíèå ôóíêòîðà [ Rα Ψ 1 , . . . , Ψ k ] íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ βa i , â èíòåðïðåòàöèîííîì ìíîæåñòâå A , ñ óêàçàííûì ìíîæåñòâîì ïîëîæèòåëüíî è îòðèöàòåëüíî îïðî- øåííûõ ñëîâ. Ó÷èòûâàÿ, ÷òî Length (Λ) = Λ , Length ( αa i ) = S 1 ( α ) . | Λ | = Λ , | αa i | = | α | + 1 = S ( | α | ) , | x | - ôóíêöèÿ äëèíû ñëîâà x , òî âûðàæåíèå âèäà Length ( t ) , áóäåì îáîçíà÷àòü â âèäå | t | . Î÷åâèäíî, ÷òî àðãóìåíòíîå ñëîâî α ÿâëÿåòñÿ íàòóðàëüíûì ÷èñëîì, òîãäà è òîëüêî òîãäà, êîãäà ⊢ | α | = α , Length ( α ) - íàòóðàëüíîå ÷èñëî. Òåðì, ñîäåðæàùèé òîëüêî ôóíêòîðû âèäà: Concat , D , à òàêæå íàòóðàëüíûå ÷èñëà, íàçûâàåòñÿ ñëî- âàðíûì ìíîãî÷ëåíîì. Ñëîâàðíûå ìíîãî÷ëåíû áóäóò îáîçíà÷àòüñÿ êàê P ( x ) . Èç ñâîéñòâ: | Concat ( x, y ) | = | Concat ( | x | , y ) | = | Concat ( x, | y | ) | = Concat ( | x | , | y | ) = | x | + | y | , | D ( x, y ) | = D ( | x | , y ) = | D ( x, | y | ) | = D ( | x | , | y | ) = | x | · | y | , ïîëó÷èì | P ( x 1 , . . . , x n ) | = P ( | x 1 | , . . . , | x n | ) , ∀ αβγ [ Concat ( | α | , | β | ) = γ ∨ D ( | α | , β ) = γ ] , òîãäà γ - íàòóðàëüíîå ÷èñëî. Äëÿ ëþáîãî ñëîâàðíîãî ìíîãî÷ëåíà P ( x ) , ìîæíî ïîñòðîèòü ìíîãî÷ëåí ñ íàòóðàëüíûìè êîýôôèöèåí- òàìè P ∗ ( x ) , ÷òî âåðíî ðàâåíñòâî P ∗ ( | x | ) = | P ( x ) | . Ñîñòàâèì 3 ≤ n − ìåñòíûé ôóíêòîð âèäà [ J ConcatI n 1 [ J ConcatI n 2 . . . [ J ConcatI n n − 1 I n n ] . . . ] . Äëÿ ýòîãî ôóíêòîðà â èñ÷èñëåíèè CalcEq âûâîäèìî ðàâåíñòâî [ J ConcatI n 1 [ J ConcatI n 2 . . . [ J ConcatI n n − 1 I n n ] . . . ]( x 1 . . . x n ) = Concat ( x 1 , Concat ( x 2 , . . . Concat ( x n − 1 , x n ) . . . )) . Ïóñòü Concat n ⇌ [ J ConcatI n 1 [ J ConcatI n 2 . . . [ J ConcatI n n − 1 I n n ] . . . ] , ïðè n ≥ 3 , òîãäà ⊢ Concat n ( x 1 , . . . x n ) = Concat ( x 1 , Concat ( x 2 , . . . Concat ( x n − 1 , x n ) . . . )) . Ïðè n = 2 , Concat 2 ⇌ Concat , ïðè n = 1 , Concat 1 ⇌ I 1 1 è âûâîäèìû ðàâåíñòâà: ⊢ Concat 2 ( x 1 , x 2 ) = Concat ( x 1 , x 2 ) , ⊢ Concat 1 ( x 1 ) = I 1 1 ( x 1 ) = x 1 . 8 ⊢ Concat n +1 ( x 1 , . . . x n , x n +1 ) = Concat ( x 1 , Concacat n ( x 2 , . . . x n +1 )) . Îïðåäåëåíèå. Ïóñòü D - íåêîòîðîå ìíîæåñòâî n - àëôàâèòíûõ ôóíêòîðîâ. Äëÿ êàæäîãî ìíîæå- ñòâà A àðãóìåíòíûõ ñëîâ, îïðåäåëèì ïîíÿòèå ñòàíäàðòíîé ñëîâàðíîé ìîäåëè, êîòîðóþ îáîçíà÷èì êàê WordM n, A , D . Íîñèòåëåì ýòîé ìîäåëè ÿâëÿþòñÿ âñå n - àëôàâèòíûå àðãóìåíòíûå ñëîâà. Äëÿ êàæäîãî k ≥ 1 - ìåñòíîãî ôóíêòîðà Φ ∈ D îïðåäåëèì îïåðàöèþ, îáîçíà÷àåìóþ f Φ è îïðåäåëÿåìóþ êàê ∀ α ∀ β [ f Φ ( α ) = β ⇐⇒ A ⊢ Φ( α ) = β ] . Åñëè ìíîæåñòâî n - àëôàâèòíûõ ôóíêòîðîâ D ñîâïàäàåò ñ ìíîæåñòâîì âñåõ n - àëôàâèòíûõ ïðè- ìèòèâíî ðåêóðñèâíûõ ôóíêòîðîâ, ñòàíäàðòíóþ ìîäåëü áóäåì îáîçíà÷àòü êàê WordM n, A , WordM n , â àëôàâèòå L ( U ) , L ñîîòâåòñòâåííî èëè åùå ïðîùå WordM A , WordM . Çàìå÷àíèå . Îòìåòèì, ÷òî ìíîæåñòâî âñåõ îïåðàöèé ìîäåëè WordM n ñîâïàäàåò ñ êëàññîì ñëîâàðíûõ ôóíêöèé P r (Σ) , ãäå Σ - àëôàâèò, ñîñòîÿùèé èç n ðàçëè÷íûõ ñèìâîëîâ[1, p.220, Denition 3]. Ðàññìîòðèì ñëåäóþùèå îïðåäåëÿþùèå ðàâåíñòâà: Exp ( x, Λ) = a 1 , Exp ( x, ya i ) = D ( x, Exp ( x, y )) . Î÷åâèäíî, ÷òî | Exp ( x, y )) | = Exp ( | x | , y )) = Exp ( | x | , | y | )) . Ìîæíî ïîñòðîèòü òàêîé 2- õ ìåñòíûé ôóíê- òîð, äëÿ êîòîðîãî áóäóò âåðíû ýòè îïðåäåëÿþùèå ñîîòíîøåíèÿ. Äëÿ êàæäîãî íàòóðàëüíîãî ÷èñëà k > 1 âûïèøåì ñëåäóþùèå îïðåäåëÿþùèå ðàâåíñòâà: exp k (Λ) = a 1 , exp k ( αa i ) = D ( a 1 , . . . , a 1 | {z } k − ðàç , exp k ( α )) . Èìååòñÿ ïðèìèòèâíî ðåêóðñèâíûé ñëîâàðíûé ôóíêòîð, óäîâëåòâîðÿþùèé ýòèì îïðåäåëÿþùèì ðàâåí- ñòâàì. Îáîçíà÷èì åãî êàê exp k . Äëÿ ôóíêòîðà exp k âåðíî WordM | = ∀ x [ | exp k ( x ) | = k | x | ] , ∀ αβ [ exp k ( α ) = β ] , òîãäà β - íàòóðàëüíîé ÷èñëî. Ïðè k > 1 exp k ( α ) - ÷èñëî k - àëôàâèòíûõ ñëîâ äëèíà êîòîðûõ ðàâíà äëèíå ñëîâà α , exp k ( αa 1 ) − 1 k − 1 - ÷èñëî k - àëôàâèòíûõ ñëîâ ïðåäøåñòâóþùèõ â ëåêñèêîãðàôè÷åñêîì óïîðÿäî÷èâàíèè ñëîâó | αa 1 | . Çàìå÷àíèå . Äëÿ êàæäîãî íàòóðàëüíîãî ÷èñëà k âåðíî WordM | = ∀ x [ | exp k ( x ) | = Exp ( k, x )) Òåîðåìà 1.6 . Èìååòñÿ àëãîðèòì, âûïîëíÿÿ êîòîðûé, ïî ïðîèçâîëüíîé ôîðìóëå A ( x 1 , . . . , x n ) èñ÷èñ- ëåíèÿ âûñêàçûâàíèé, â êîòîðîì ýëåìåíòàðíûå âûñêàçûâàíèÿ ÿâëÿþòÿ âûñêàçûâàíèÿ âèäà r = q , ãäå r, q - òåðìû àëôàâèòà L ( U ) , ìîæíî ìîæíî ïîñòðîèòü n - ìåñòíûé ôóíêòîð Φ A , òàêîé, ÷òî ∀ α [ WordM A | = A ( α ) ⇔ A ⊢ Φ A ( α ) = Λ] ( WordM A | = ∀ x [ A ( x ) ≡ Φ A ( x ) = Λ] )[4]. 9 Äëÿ êàæäîãî èñõîäíîãî ôóíêòîðà: I n m , S k , Z , δ , Length , . . . . . . . . . , Concat , D âåðíî ⊢ | I n m ( x 1 , . . . , x n ) | = I n m ( | x 1 | , . . . , | x n | ) , ⊢ | S k ( x ) | = S 1 ( | x | ) , ⊢ | Z ( x ) | = Z ( | x | ) , ⊢ | δ ( x ) | = δ ( | x | ) , ⊢ | Length ( x ) | = Length ( | x | ) , ⊢ | x 1 . . . . . . . . . x 2 | = | x 1 | . . . . . . . . . | x 2 | , ⊢ | Concat ( x 1 , x 2 ) | = Concat ( | x 1 | , | x 2 | ) , ⊢ | D ( x 1 , x 2 ) | = D ( | x 1 | , | x 2 | ) . Part II Ïðîñòîå âû÷èñëåíèå ôóíêòîðà Ñ êàæäûì n - ìåñòíûì ôóíêòîðîì Φ è ïîñëåäîâàòåëüíîñòüþ àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , â äàëüíåé- øåì îáîçíà÷àåìóþ êàê α , ñâÿæåì ïðîñòîå(êàíîíè÷åñêîå) âû÷èñëåíèå ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , îáîçíà÷àåìîå êàê P Φ; α 1 ,...α n . Ýòî ïðîñòîå âû÷èñëåíèå ïîñòðîèì èíäóê- öèåé ïî ïîñòðîåíèþ ôóíêòîðà Φ , à âíóòðè ýòîé èíäóêöèè, äëÿ ðåêóðñèâíîãî ôóíêòîðà èíäóêöèåé ïî ïîñòðîåíèþ àðãóìåíòíîãî ñëîâà. Ñ êàæäûì ïðîñòûì âû÷èñëåíèåì óêàæåì ìíîæåñòâà ( A + ) P , Φ; α 1 ,...,α n è ( A − ) P , Φ; α 1 ,...,α n è äëèíó âû÷èñëåíèÿ l P , Φ ( α ) - ÷èñëî ðàâåíòñòâ â âûâîäå P Φ; α 1 ,...α n . Äëÿ èñõîäíûõ ôóíêòîðîâ: S k , Z , I n m , δ , Length , . . . . . . . . . , Concat , D : Äëÿ ôóíêòîðà S k : 1. S k ( x ) = S k ( x ) , 2. S k ( α ) = S k ( α ) . ( A + ) P , S k ; α = ∅ , ( A − ) P , S k ; α = ∅ . Äëèíà âû÷èñëåíèÿ l P , S k ( α ) = 2 . Äëÿ ôóíêòîðà Z : 1. Z ( x 1 ) = Λ , 2. Z ( α ) = Λ . Äëèíà âû÷èñëåíèÿ l P , Z ( α ) = 2 . ( A + ) P , Z ; α = ∅ , ( A − ) P , Z ; α = ∅ . 10 Äëÿ ôóíêòîðà U : U ( α ) = Λ , åñëè α ∈ A , èíà÷å, U ( α ) = a 1 . Äëèíà âû÷èñëåíèÿ l P , U ( α ) = 1 . ( A + ) P , U ; α = { α } , ( A − ) P , U ; α = ∅ , åñëè α ∈ A , ( A + ) P , U ; α = ∅ , ( A − ) P , U ; α = { α } , åñëè α / ∈ A Äëÿ ôóíêòîðà I n m : 1. I n m ( x 1 , . . . , x n ) = x m , 2. I n m ( α 1 , x 2 , . . . , x n ) = x m , . . . , , n + 1 . I n m ( α 1 , . . . , α n ) = α m . Äëèíà âû÷èñëåíèÿ l P , I n m ( α 1 , . . . , α n ) = n + 1 . ( A + ) P , I n m ; α 1 ,...,α n = ∅ , ( A − ) P , I n m ; α 1 ,...,α n = ∅ . Äëÿ ôóíêòîðà δ : 1. δ (Λ) = Λ , 2. δ ( x 1 a i ) = x 1 , 3. δ ( αa i ) = α . Äëèíà âû÷èñëåíèÿ çàäà¼òñÿ îïðåäåëÿþùèìè ðàâåíñòâàìè: l P δ (Λ) = 1 , l P δ ( α ) = 2 , ïðè α ̸ = Λ , ( A + ) P , δ = ∅ , ( A − ) P , δ = ∅ . Äëÿ ôóíêòîðà Length : 1. | Λ | = Λ , 1. | x 1 a k | = | x 1 | a 1 , 2. | αa k | = | α | a 1 , [Ïóñòü P Length ; α - ïðîñòîå âû÷èñëåíèå ôóíêòîðà Length íà àðãóìåíòíîì ñëîâå α , äàëåå âûïèñûâàåì ýòî ïðîñòîå âû÷èñëåíèå P Length ; α , â êîíöå ýòîãî âûâîäà ñòîèò ðàâåíñòâî | α | = γ , ïðîäîëæàåì] P Length ,α , 3. S 1 ( x 1 ) = S 1 ( x 1 ) , 4. S 1 ( | α | ) = S 1 ( γ ) , 11 5. | αa k | = γa 1 . Äëèíà âû÷èñëåíèÿ çàäà¼òñÿ îïðåäåëÿþùèìè ðàâåíñòâàìè: l P , Length (Λ) = 1 , l P , Length ( αa k ) = l P , Length ( α ) + 5 . l P , Length ( α ) = 1 + 5 · | α | , ( A + ) P , Length = ∅ , ( A − ) P , Length = ∅ . Äëÿ ôóíêòîðà . . . . . . . . . : 1. x 1 . . . . . . . . . Λ = x 1 , 2. α . . . . . . . . . Λ = α , 1. x 1 . . . . . . . . . x 2 a k = δ ( x 1 . . . . . . . . . x 2 ) , 2. α . . . . . . . . . x 2 a k = δ ( α . . . . . . . . . x 2 ) , 3. α . . . . . . . . . βa k = δ ( α . . . . . . . . . β ) , [Ïóñòü P . . . . . . . . . ; α,β - ïðîñòîå âû÷èñëåíèå ôóíêòîðà . . . . . . . . . íà àðãóìåíòíûõ ñëîâàõ α, β , äàëåå âûïèñûâàåì ýòî ïðîñòîå âû÷èñëåíèå P . . . . . . . . . ; α,β , â êîíöå ýòîãî âûâîäà ñòîèò ðàâåíñòâî α . . . . . . . . . β = γ , ïðîäîëæàåì] P . . . . . . . . . ; α,β , 4. δ ( z ) = δ ( z ) , 5. δ ( α . . . . . . . . . β ) = δ ( γ ) . [ Ïóñòü P δ ; γ - ïðîñòîå âû÷èñëåíèå ôóíêòîðà δ íà àãðóìåíòíîì ñëîâå γ , â êîíöå ýòîãî âûâîäà ñòîèò ðàâåíñòâî âèäà δ ( γ ) = η , ïðîäîëæàåì] P δ ,γ , 6. δ ( α . . . . . . . . . β ) = η , 7. α . . . . . . . . . βa k = η , Äëèíà âû÷èñëåíèÿ çàäà¼òñÿ îïðåäåëÿþùèìè ðàâåíñòâàìè: l P . . . . . . . . . ( α, Λ) = 2 , l P . . . . . . . . . ( α, βa k ) = l P . . . . . . . . . ( α, β ) + l P δ ( α . . . . . . . . . β ) + 7 . l P . . . . . . . . . ( α,β ) =                    2 + 9 · | β | , if | α | > | β | ≥ 0; | α | + 8 · | β | , if 1 ≤ | α | ≤ | β | ; 2 + 8 · | β | , if | α | = Λ . ( A + ) P , . . . . . . . . . = ∅ , 12 ( A − ) P , . . . . . . . . . = ∅ . Äëÿ ôóíêòîðà Concat : 1. Concat ( x 1 , Λ) = x 1 , 2. Concat ( α, Λ) = α , 1. Concat ( x 1 , x 2 a k ) = Concat ( x 1 , x 2 ) a k , 2. Concat ( α, x 2 a k ) = Concat ( α, x 2 ) a k , 3. Concat ( α, βa k ) = Concat ( α, β ) a k , [ Ïóñòü P Concat ; α,β - ïðîñòîå âû÷èñëåíèå ôóíêòîðà Concat íà àãðóìåíòíûõ ñëîâàõ α è β , â êîíöå ýòîãî âûâîäà ñòîèò ðàâåíñòâî âèäà Concat ( α, β ) = γ , ïðîäîëæàåì] P Concat ; α,β , 4. S k ( x 1 ) = S k ( x 1 ) , 5. S k ( Concat ( α, β )) = S k ( γ ) , 6. Concat ( α, βa k ) = S k ( γ ) . Äëèíà âû÷èñëåíèÿ çàäà¼òñÿ îïðåäåëÿþùèìè ðàâåíñòâàìè: l P Concat ( α, Λ) = 2 , l P Concat ( α, βa k ) = l P Concat ( α, β ) + 6 , l P Concat ( α, β ) = 2 + 6 · | β | , ( A + ) P , Concat = ∅ , ( A − ) P , Concat = ∅ . Äëÿ ôóíêòîðà D : 1. D ( x 1 , Λ) = Λ , 2. D ( α, Λ) = Λ , 1. D ( x 1 , x 2 a k ) = Concat ( x 1 , D ( x 1 , x 2 )) , 2. D ( α, x 2 a k ) = Concat ( α, D ( α, x 2 )) , 3. D ( α, βa k ) = Concat ( α, D ( α, β )) , [ Ïóñòü P D ; α,β - ïðîñòîå âû÷èñëåíèå ôóíêòîðà D íà àãðóìåíòíîì ñëîâå α è ñëîâå β , â êîíöå ýòîãî âûâîäà ñòîèò ðàâåíñòâî âèäà D ( α, β ) = γ , ïðîäîëæàåì] P D ,α,β , 4. Concat ( x 1 , x 2 ) = Concat ( x 1 , x 2 ) , 5. Concat ( α, x 2 ) = Concat ( α, x 2 ) , 6. Concat ( α, D ( α, β )) = Concat ( α, γ ) , 13 [ Ïóñòü P Concat ; α,γ - ïðîñòîå âû÷èñëåíèå ôóíêòîðà Concat íà àãðóìåíòíîì ñëîâå α , γ , â êîíöå ýòîãî âûâîäà ñòîèò ðàâåíñòâî âèäà Concat ( α, γ ) = η , ïðîäîëæàåì]. P Concat ,α,γ , 7. Concat ( α, D ( α, β )) = η , 8. D ( α, βa k ) = η . Äëèíà âû÷èñëåíèÿ çàäà¼òñÿ îïðåäåëÿþùèìè ðàâåíñòâàìè: l P D ( α, Λ) = 2 , l P D ( α, βa k ) = l P D ( α, β ) + l P Concat ( α, D ( α, β )) + 8 , l P D ( α, β ) = 2 + 10 · | β | + 3 · | α | · | β | ( | β | . . . . . . . . . 1) , ( A + ) P , D = ∅ , ( A − ) P , D = ∅ . Äëÿ ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] : Ïóñòü P Ψ 1 ; α - ïðîñòîé âûâîä ôóíêòîðà Ψ 1 íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α, . . . , P Ψ k ; α - ïðîñòîé âûâîä ôóíêòîðà Ψ k íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α . Ñîñòàâèì ïîñëåäîâàòåëüíîñòü ðàâåíñòâ: P Ψ 1 ; α , . . . , P Ψ k ; α , [  êîíöå êàæäîãî âûâîäà P Ψ i íàõîäèòñÿ ðàâåíñòâî âèäà Ψ i ( α ) = γ i , ïðîäîëæàåì] s + 1 . Φ( x 1 , . . . , x k ) = Φ( x 1 , . . . , x k ) , s + 2 . Φ(Ψ 1 ( α ) , x 2 , . . . , x k ) = Φ( γ 1 , x 2 , . . . , x k ) , . . . , s + k + 1 . Φ(Ψ 1 ( α ) , . . . , Ψ k ( α )) = Φ( γ 1 , . . . , γ k ) , [Ïóñòü P Φ; γ - ïðîñòîé âûâîä ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ γ , â êîíöå ýòîãî âûâîäà íàõîäèòñÿ ðàâåíñòâî Φ( γ 1 , . . . , γ k ) = η , ïðîäîëæàåì] P Φ; γ , s + k + r + 2 . Φ(Ψ 1 ( α ) , . . . , Ψ 1 ( α )) = η , s + k + r + 3 . [ J ΦΨ 1 , . . . , Ψ k ]( x 1 , . . . , x n ) = Φ(Ψ 1 ( x 1 , . . . , x n ) , . . . , Ψ k ( x 1 , . . . , x n )) , s + k + r + 4 . [ J ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , x n ) = Φ(Ψ 1 ( α 1 , . . . , x n ) , . . . , Ψ k ( α 1 , . . . , x n )) , . . . , s + k + r + n + 3 . [ J ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n ) = Φ(Ψ 1 ( α 1 , . . . , α n ) , . . . , Ψ k ( α 1 , . . . , α n )) , s + k + r + n + 4 . [ J ΦΨ 1 , . . . , Ψ k ]( α 1 , . . . , α n ) = η - ïîëó÷åííàÿ ïîñëåäîâàòåëüíîñòü ðàâåíñòâ - ïðîñòîå âû÷èñëåíèå ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α . Äëèíà âû÷èñëåíèÿ çàäà¼òñÿ îïðåäåëÿþùèìè ðàâåíñòâàìè: l P [ J ΦΨ 1 ,..., Ψ k ] ( α ) = l P Ψ 1 ( α )+ , . . . , + l P Ψ k ( α ) + l P Φ (Ψ 1 ( α ) , . . . , Ψ k ( α )) + n + k + 4 . 14 ( A + ) P [ J ΦΨ 1 ,..., Ψ k ] ; α 1 ,...,α n = k S i =1 ( A + ) P Ψ i ; α 1 ,...,α n S ( A + ) P Φ ; γ 1 ,...,γ k , ( A − ) P [ J ΦΨ 1 ,..., Ψ k ] = k S i =1 ( A − ) P Ψ i ; α 1 ,...,α n S ( A − ) P Φ ; γ 1 ,...,γ k (ñì. òåîðåìó 1.3.). Äëÿ n ≥ 2 - ìåñòíîãî ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ m ] : Ñîñòàâèì ïîñëåäîâàòåëüíîñòü ðàâåíñòâ: 1. [ R ΦΨ 1 , . . . , Ψ m ]( x 1 , . . . , x n , Λ) = Φ( x 1 , . . . , x n ) , 2. [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , x n , Λ) = Φ( α 1 , . . . , x n ) , . . . , n + 1 . [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , α n , Λ) = Φ( α 1 , . . . , α n ) , [Ïóñòü P Φ; α - ïðîñòîé âûâîä ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α , â êîíöå ýòîãî âûâîäà íàõîäèòñÿ ðàâåíñòâî Φ( α 1 , . . . , α n ) = γ , ïðîäîëæàåì] P Φ; α , n + r + 2 . [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , α n , Λ) = γ , [Ïóñòü P [ R ΦΨ 1 ,..., Ψ m ]; α 1 ,...,α n +1 - ïðîñòîé âûâîä ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ m ] íà ïîñëåäîâàòåëüíîñòè àðãó- ìåíòíûõ ñëîâ α 1 , . . . , α n +1 , â êîíöå ýòîãî âûâîäà íàõîäèòñÿ ðàâåíñòâî [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , α n , α n +1 ) = β , ïðîäîëæàåì] P [ R ΦΨ 1 ,..., Ψ m ]; α 1 ,...,α n +1 , [Ïóñòü P Ψ k ; α 1 ,...,α n +1 ,β - ïðîñòîé âûâîä ôóíêòîðà Ψ k ] íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n +1 , β , â êîíöå ýòîãî âûâîäà íàõîäèòñÿ ðàâåíñòâî [Ψ k ( α 1 , . . . , α n , α n +1 , β ) = θ , ïðîäîëæàåì] P Ψ k ; α 1 ,...,α n +1 ,β , s + t + 1 . [ R ΦΨ 1 , . . . , Ψ m ]( x 1 , . . . , x n , x n +1 a k ) = Ψ k ( x 1 , . . . , x n , x n +1 , [ R ΦΨ 1 , . . . , Ψ m ]( x 1 , . . . , x n , x n +1 )) , s + t +2 . [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , x n , x n +1 a k ) = Ψ k ( α 1 , . . . , x n , x n +1 , [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , x n , x n +1 )) , . . . , n + s + t +2 . [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , α n , α n +1 a k ) = Ψ k ( α 1 , . . . , α n , α n +1 , [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , α n , α n +1 )) , n + s + t + 3 . Ψ k ( x 1 , . . . , x n +2 ) = Ψ k ( x 1 , . . . , x n +2 ) , n + s + t + 4 . Ψ k ( α 1 , . . . , x n +2 ) = Ψ k ( α 1 , . . . , x n +2 ) , . . . , 2 n + s + t + 4 . Ψ k ( α 1 , . . . , α n +1 , x n +2 ) = Ψ k ( α 1 , . . . , α n +1 , x n +2 ) , 2 n + s + t + 5 . Ψ k ( α 1 , . . . , α n +1 , [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , α n , α n +1 )) = Ψ k ( α 1 , . . . , α n +1 , β ) , 2 n + s + t + 6 . Ψ k ( α 1 , . . . , α n +1 , [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , α n , α n +1 )) = θ , 2 n + s + t + 7 . [ R ΦΨ 1 , . . . , Ψ m ]( α 1 , . . . , α n , α n +1 a k ) = θ - ýòà ïîñëåäîâàòåëüíîñòü ðàâåíñòâ ÿâëÿåòñÿ ïðîñòûì âûâîäîì ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ m ] íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n +1 . Äëèíà âû÷èñëåíèÿ çàäà¼òñÿ îïðåäåëÿþùèìè ðàâåíñòâàìè: l P [ R ΦΨ 1 ,..., Ψ m ] ( α, Λ) = l P Φ ( α ) + n + 2 , 15 l P [ R ΦΨ 1 ,..., Ψ m ] ( α, α n +1 a k ) = l P [ R ΦΨ 1 ,..., Ψ m ] ( α, α n +1 ) + l P Ψ k ( α, α n +1 , [ R ΦΨ 1 , . . . , Ψ m ]( α, α n +1 )) + 2 n + 7 1. ( A + ) P [ R ΦΨ 1 ,..., Ψ m ] ,α 1 ,...,αn, Λ = ( A + ) P Φ ; α 1 ,...,α n , ( A − ) P [ R ΦΨ 1 ,..., Ψ m ] ,α 1 ,...,αn, Λ = ( A − ) P Φ ; α 1 ,...,α n 2. ( A + ) P [ R ΦΨ 1 ,..., Ψ m ] ,α 1 ,...,αn,βa i = ( A + ) P Ψ i ; α 1 ,...,α n ,β,γ S ( A + ) P [ R ΦΨ 1 ,..., Ψ m ] ,α 1 ,...,αn,β , ( A − ) P [ R ΦΨ 1 ,..., Ψ m ] ,α 1 ,...,αn,βa i = ( A − ) P Ψ i ; α 1 ,...,α n ,β,γ S ( A − ) P [ R ΦΨ 1 ,..., Ψ m ] ,α 1 ,...,αn,β (ñì. òåîðåìó 1.4.). Ñëó÷àé, êîãäà ôóíêòîð ðåêóðñèè èìååò âèä [ Rα Φ 1 , . . . , Φ m ] , ðàçáèðàåòñÿ àíàëîãè÷íî. Äëÿ íåãî îïðåäåëÿþùèå ðàâåíñòâà äëèíû ïðîñòîãî âû÷èñëåíèÿ òàêîâû: l P [ Rα Ψ 1 ,..., Ψ m ] (Λ) = 1 , l P [ Rα Ψ 1 ,..., Ψ m ] ( αa k ) = l P [ Rα Ψ 1 ,..., Ψ k ] ( α ) + l P Ψ k ( α, [ Rα Ψ 1 , . . . , Ψ m ]( α )) + 7 1. ( A + ) P [ Rα Ψ 1 ,..., Ψ m ] , Λ = ∅ , ( A − ) P [ Rα Ψ 1 ,..., Ψ m ] , Λ = ∅ 2. ( A + ) P [ Rα Ψ 1 ,..., Ψ m ] ,βa i = ( A + ) P Ψ i ,β,γ S ( A + ) [ P [ R ΦΨ 1 ,..., Ψ m ] ,β , (ñì. òåîðåìó 1.5), Ñâîéñòâà ïðîñòîãî âû÷èñëåíèÿ ôóíêòîðîâ: 1 . Ïîñëåäíåå ðàâåíñòâî ïðîñòîãî âû÷èñëåíèÿ n − ìåñòíîãî ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àð- ãóìåíòíûõ ñëîâ α , ÿâëÿåòñÿ ðàâåíñòâî âèäà Φ( α ) = β , ãäå β - àðãóìåíòíîå ñëîâî, êîòîðîå íàçûâàåòñÿ ðåçóëüòàòîì ïðîñòîãî âû÷èñëåíèÿ n − ìåñòíîãî ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α . 2 . Ïðîñòîå âû÷èñëåíèå n − ìåñòíîãî ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α ñîñòîèò òîëüêî èç òåõ ôóíêòîðîâ, êîòîðûå ÿâëÿþòñÿ ïîäôóíêòîðàìè ôóíêòîðà Φ . Ïðè ýòîì èñïîëüçóþòñÿ òîëüêî ïðàâèëà âûâîäà Sb èëè Sub 1 . 3 . Âñå îïðàøèâàåìûå ñëîâà, ïðè ïðîñòîì âû÷èñëåíèè ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ m ] íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α , ñîñòîÿò èç îïðàøèâàåìûõ ñëîâ, êîòîðûå âõîäÿò â ïðîñòîå âû÷èñëåíèå ôóíêòîðà Ψ 1 íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α è ò.ä. èç îïðàøèâàåìûõ ñëîâ, êîòîðûå âõîäÿò â ïðîñòîå âû÷èñëåíèå ôóíêòîðà Ψ m íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α , èç îïðàøèâàåìûõ ñëîâ, êîòîðûå âõîäÿò â ïðîñòîå âû÷èñëåíèå ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ γ 1 , . . . , γ m , ãäå γ i - ðåçóëüòàò âû÷èñëåíèÿ ôóíêòîðà Ψ i íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α . 4 . Âñå îïðàøèâàåìûå ñëîâà, ïðè ïðîñòîì âû÷èñëåíèè ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ m ] íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α, βa k , ñîñòîÿò èç îïðàøèâàåìûõ ñëîâ, êîòîðûå âõîäÿò â ïðîñòîå âû÷èñëåíèå ôóíêòî- ðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α , èç îïðàøèâàåìûõ ñëîâ ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ m ] ïðè ïðîñòîì âû÷èñëåíèè íà ïîñëåäîâàòåëíîñòè àðãóìåíòíûõ ñëîâ α, β (ïðåäûäóùèé øàã), èç îïðàøèâàåìûõ ñëîâ, êîòîðûå âõîäÿò â ïðîñòîå âû÷èñëåíèå ôóíêòîðà Ψ k íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α, β, γ ãäå γ - ðåçóëüòàò âû÷èñëåíèÿ ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ m ] íà ïîñëåäîâàòååëüíîñòè ñëîâ α, β . 5 . Ëþáîé n - ìåñòíûé ôóíêòîð Φ ìîæíî èíòåðïðåòèðîâàòü êàê íåêîòîðûé àëãîðèòì, âûïîëíÿÿ êî- 16 òîðûé, ìîæíî âû÷èñëèòü çíà÷åíèå ýòîãî ôóíêòîðà íà çàäàííîé ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n . Ïðîñòîå âû÷èñëåíèå ýòîãî ôóíêòîðà íà çàäàííîé ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ ÿâëÿ- åòñÿ ðåàëèçàöèåé ýòîãî àëãîðèòìà. Äëÿ êàæäîãî èñõîäíîãî ôóíêòîðà: I n m , S k , Z , δ , Length , . . . . . . . . . , Concat , D âåðíî l P I n m ( x 1 , . . . , x n ) = l P I n m ( | x 1 | , . . . , | x n | ) , l P Z ( x ) = l P Z ( | x | ) , l P S k ( x ) = l P S k ( | x | ) , l P δ ( x ) = l P δ ( | x | ) , l P Length ( x ) = l P Length ( | x | ) , l P . . . . . . . . . ( x 1 , x 2 ) = l P . . . . . . . . . ( | x 1 | , | x 2 | ) , l P Concat ( x 1 , x 2 ) = l P Concat ( | x 1 | , | x 2 | ) , l P D ( x 1 , x 2 ) = l P D ( | x 1 | , | x 2 | ) . Çàìå÷àíèå . Ïóñòü Φ - n - ìåñòíûé ôóíêòîð àëôàâèòà L , ñîñòàâëåííûé èç ôóíêòîðîâ I n m , Z , δ , Length , . . . . . . . . . , Concat , D ñ ïîìîùüþ îïåðàòîðà ñóïåðïîçèöèè J , òîãäà äëÿ ýòîãî ôóíêòîðà âåðíî ⊢ | Φ( x 1 , . . . , x n ) | = Φ( | x 1 | , . . . , | x n | ) . Part III Ôóíêòîð îãðàíè÷åííîé ðåêóðñèè, PPr ôóíêòîðû Ïóñòü Φ - ïðîèçâîëüíûé n - ìåñòíûé ôóíêòîð. Ñîñòàâèì ôóêíêòîð [ J Φ I n +1 1 , . . . I n +1 n ] - ââåäåíèå n + 1 ôèêòèâíîé ïåðåìåííîé, ýòîò ôóíêòîð îáîçíà÷èì êàê [ J Φ n +1 ] . Ðàâåíñòâî âèäà x . . . . . . . . . y = Λ áóäåì îáîçíà÷àòü êàê x ⩽ y . Ó÷èòûâàÿ ñâîéñòâî x . . . . . . . . . y = Λ ⇐⇒ | x | . . . . . . . . . | y | = Λ , ôîðìóëó âèäà x ⩽ y , áóäåì çàïèñûâàòü òàêæå â âèäå | x | ⩽ | y | . x . . . . . . . . . y =          Λ , åñëè | x | ≤ | y | ; z, èíà÷å , ãäå z - òàêîå ñëîâî, êîòîðîå ÿâëÿåòñÿ íà÷àëîì ñëîâà x è äëèíà êîòîðîãî ðàâíà | x | − | y | . Äëÿ ëþáîãî ñëîâàðíîãî ìíîãî÷ëåíà P ( y ) , ó÷èòûâàÿ, ÷òî | P ( y ) | = P ( | y | ) , âåðíî: | x | ⩽ | P ( y ) | ⇐⇒ | x | ⩽ P ( | y | ) . Îáîçíà÷èì äâóõìåñòíûé ôóíêòîð J [ . . . . . . . . . I 2 1 J [ . . . . . . . . . I 2 1 I 2 2 ]] êàê min . Äëÿ ýòîãî ôóíêòîðà àëôàâèòà L , â èñ- ÷èñëåíèè CalcEq âûâîäèìî ðàâåíñòâî min ( x 1 , x 2 ) = x 1 . . . . . . . . . ( x 1 . . . . . . . . . x 2 ) . Ñâîéñòâà: ⊢ min ( x 1 , x 2 ) = min ( x 1 , | x 2 | ) , ⊢ | min ( x 1 , x 2 ) | = min ( | x 1 | , | x 2 | ) 17 min ( x, y ) =          x, åñëè | x | ≤ | y | ; z, èíà÷å , ãäå z - òàêîå ñëîâî, êîòîðîå ÿâëÿåòñÿ íà÷àëîì ñëîâà x è äëèíà êîòî- ðîãî ðàâíà | y | . ∀ αβ WordM | = | min ( α, β ) | ≤ | β | . Ïóñòü P - n - ìåñòíûé ìíîãî÷ëåííûé ôóíêòîð, Φ - n - ìåñòíûé ôóíêòîð, Ψ - n + 1 - ìåñòíûé ôóíêòîð. Ñîñòàâèì ôóíêòîðû: [ J min Φ , P ] , [ J min Ψ[ J P n +1 ]] - ôóíêòîðû îãðàíè÷åíèÿ, ñîîòâåòñòâåííî áåç ââåäåíèÿ ôèêòèâíîé ïåðåìåííîé è ñ âåäåíèåì ôèêòèâíîé ïåðåìåííîé. Ýòè ôóíêòîðû îáîçíà÷èì êàê Bound (Φ , P ) , Bound (Ψ , P ) . Ïóñòü Φ - n - ìåñòíûé ôóíêòîð, Ψ 1 , . . . , Ψ k - n +2 - ìåñòíûå ôóíêòîðû, P , P 1 - ñîîòâåòñòâåííî n , n +1 - ìåñòíûå ìíîãî÷ëåííûå ôóíêòîðû. Ñîñòàâèì ôóíêòîð [ R Bound (Φ , P ) , Bound (Ψ 1 , P 1 ) , . . . , Bound (Ψ k , P 1 )] - ôóíêòîð îãðàíè÷åííîé ðåêóðñèè. Äëÿ êàæäîãî ôóíêòîðà îãðàíè÷åííîé ðåêóðñèè Γ ⇌ [ R Bound (Φ , P ) , Bound (Ψ 1 , P 1 ) , . . . , Bound (Ψ k , P 1 )] âûâîäèìû ðàâåíñòâà: ⊢ Γ( x 1 , . . . , x n , Λ) = min (Φ( x 1 , . . . , x n ) , P ( x 1 , . . . , x n )) , ⊢ Γ( x 1 , . . . , x n , S k ( x n +1 )) = min (Ψ k ( x 1 , . . . , x n , x n +1 , Γ( x 1 , . . . , x n +1 )) , P 1 ( x 1 , . . . , x n , x n +1 )) , äëÿ ëþáîãî ìíîæåñòâà àðãóìåíòíûõ ñëîâ A âåðíî: ∀ α, ∀ β ̸ = Λ A ⊢ | Γ( α, Λ) | ⩽ | P ( α ) | , A ⊢ | Γ( α, β ) | ⩽ | P 1 ( α, β ) | . Èíäóêòèâíî îïðåäåëèì ìíîæåñòâî ôóíêòîðîâ, îáîçíà÷àåìîå êàê PPr ( U ) : 1) Cëîâî âèäà U - ïîëèíîìèàëüíàÿ ïðîãðàììà; 2) Ñëîâà âèäà Z , δ , Length , S k , I m n , . . . . . . . . . , Concat , D - ïîëèíîìèàëüíûå ïðîãðàììû; 3) Åñëè Φ - k - ìåñòíûé ôóíêòîð, Ψ 1 , . . . , Ψ k - n - ìåñòíûå ôóíêòîðû è ÿâëÿþòñÿ ïîëèíîìèàëüíû- ìè ïðîãðàììàìè, òî ôóíêòîð [ J ΦΨ 1 , . . . , Ψ k ] - ïîëèíîìèàëüíàÿ ïðîãðàììà, ò.å. ïðèíàäëåæèò ìíîæåñòâó PPr ( U ) ; 4) Åñëè Φ - n - ìåñòíûé ôóíêòîð, Ψ 1 , . . . , Ψ k - n +2 - ìåñòíûå ôóíêòîðû è ÿâëÿþòñÿ ïîëèíîìèàëüíûìè ïðîãðàììàìè, P , P 1 - ñîîòâåòñòâåííî n , n + 1 - ìåñòíûå ìíîãî÷ëåííûå ôóíêòîðû, òî ôóíêòîð [ R Bound (Φ , P ) , Bound (Ψ 1 , P 1 ) , . . . , Bound (Ψ k , P 1 )] - ïîëèíîìèëüíàÿ ïðîãðàììà. Ìíîæåñòâî ôóíêòîðîâ îïðåäåë¼ííûõ ñîãëàñíî ïóíêòàì 2-4, áóäåì òàêæå íàçûâàòü ïîëèíîìèàëüíûìè ïðîãðàììàìè, íî â àëôàâèòå L . Ýòî ìíîæåñòâî ôóíêòîðîâ áóäåì îáîçíà÷àòü êàê PPr . Çàìå÷àíèå . Ìíîæåñòâî âñåõ îïåðàöèé ñòàíäàðòíîé ñëîâàðíîé ìîäåëè WordM n, PPr ñîâïàäàåò ñ êëàñ- ñîì ôóíêöèè E 2 (Σ) [1, p.220. Denition 7], ãäå Σ - àëôàâèò, ñîñòîÿùèé èç n ≥ 2 ðàçëè÷íûõ ñèìâîëîâ. 18 Åñëè èç êîíòåêñòà áóäåò ïîíÿòíî èëè áóäåò íå âàæíî â êàêîì àëôàâèòå L èëè L ( U ) , ðàññìàòðèâàåòñÿ ìíîæåñòâî PPr èëè PPr ( U ) , òîãäà áóäåì ïðîñòî îáîçíà÷àòü PPr . Òåîðåìà 3.1 . Ïóñòü Φ - n - ìåñòíàÿ ïîëèíîìèàëüíàÿ ïðîãðàììà, ò.å. Φ ∈ PPr , òîãäà ñóùåñòâóåò(ìîæíî ïîñòðîèòü), òàêîé ñëîâàðíûé ìíîãî÷ëåí P ( x ) òîé æå ìåñòíîñòè, ÷òî äëÿ ëþáîãî íàáîðà àðãóìåíòíûõ ñëîâ α âåðíî: a ) ∀ A WordM A | = | Φ( α ) | < | P ( α ) | . b ) ∀ A WordM A | = l P Φ ( α ) < | P ( α ) | ; c ) ∀ A WordM A | = F l P Φ ( α ) < | P ( α ) | . Äîêàçàòåëüñòâî . Äîêàçàòåëüñòâî ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà PPr , èñïîëüçóÿ ïðè ýòîì ñâîéñòâà ïðîñòîãî âû÷èñëåíèÿ, ñòð. 10 - 15. Ïóñòü Φ ∈ PPr ( U ) - n - ìåñòíûé ôóíêòîð, òîãäà ñóùåñòâóåò òàêîé ñëîâàðíûé ìíîãî÷ëåí P ( x ) , ÷òî äëÿ ïðîèçâîëüíîãî ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ëþáûõ àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , äëèíà âñåõ îïðà- øèâàåìûõ ñëîâ â ïðîñòîì âû÷èñëåíèè ôóíêòîðà Φ íà α 1 , . . . , α n è ÷èñëî îïðîøåííûõ ñëîâ îãðàíè÷åíû | P ( α 1 , . . . , α n ) | . Çàìå÷àíèå . Ïóñòü MT - îðàêóëüíàÿ ìàøèíà Òüþðèíãà ñ âõîäíûì àëôàâèòîì A = { a 1 , . . . , a k } ( k ≥ 2 ), è îðàêóëüíûì ìíîæåñòâîì B , âðåìÿ ðàáîòû êîòîðîé îãðàíè÷åíî íåêîòîðûì ìíîãî÷ëåíîì P ( x 1 , . . . , x n ) ñ íàòóðàëüíûìè êîýôôèöèåíòàìè. Ïóñòü f MT ( x 1 , . . . , x n ) - ñëîâàðíàÿ ôóíêöèÿ, êîòîðàÿ, ïîðîæäåíà ðàñ- ñìàòðèâàåìîé îðàêóëüíîé MT . Òîãäà ìîæíî ïîñòðîèòü òàêîé ôóíêòîð Φ ∈ PPr ( U ) òîé æå ìåñòíîñòè ìíîæåñòâîì âõîäíûõ ñëîâ êîòîðîãî, ÿâëÿåòñÿ ìíîæåñòâî àðãóìåíòíûõ ñëîâ { S 1 (Λ) , . . . , S k (Λ) } , ÷òî âåðíî ∀ α 1 , . . . , α k , β B ⊢ Φ( α 1 . . . , α k ) = β ⇐⇒ f MT ( α 1 , . . . , α k ) = β [1 p. 224. Theorem 6] 2 [5, Òåîðåìà 1 ñòð 228] 3 . Çàìå÷àíèå . Ïóñòü Φ ∈ PPr ( U ) - n - ìåñòíûé ôóíêòîð, ìíîæåñòâîì âõîäíûõ ñëîâ êîòîðîãî, ÿâëÿåòñÿ ìíîæåñòâî àðãóìåíòíûõ ñëîâ { S 1 (Λ) , . . . , S k (Λ) } ( k ≥ 2 ). Ïóñòü B - èíòåïðåòàöèÿ îðàêóëüíîãî ñèìâîëà U . Òîãäà ìîæíî ïîñòðîèòü îðàêóëüíóþ ìàøèíó Òüþðèíãà MT - ñ âõîäíûì àëôàâèòîì A = { a 1 , . . . , a k } è îðàêóëüíûì ìíîæåñòâîì B , âðåìÿ ðàáîòû êîòîðîé îãðàíè÷åíî íåêîòîðîì ìíîãî÷ëåíîì P ( x 1 , . . . , x n ) ñ íàòóðàëüíûìè êîýôôèöèåíòàìè, ÷òî äëÿ ñëîâàðíîé ôóíêöèè f MT ( x 1 , . . . , x n ) , êîòîðàÿ ïîðîæäåíà ðàñ- ñìàòðèâàåìîé îðàêóëüíîé MT âåðíî ∀ α 1 , . . . , α k , β B ⊢ Φ( α 1 . . . , α k ) = β ⇐⇒ f MT ( α 1 , . . . , α k ) = β [[1 p. 224. Theorem 7][5, Òåîðåìà 1 ñòð. 230]. Ïócòü A - èíòåïðåòàöèÿ îðàêóëüíîãî ñèìâîëà U . Èíäóêòèâíî îïðåäåëèì ìíîæåñòâî ôóíêòîðîâ, îáî- çíà÷àåìîå êàê PPr ( A ) : 2 Ýòà òåîðåìà ëåãêî ïåðåíîñèòñÿ â ñëó÷àå êîãäà ðàññìàòðèâàåìÿ ìàøèíà Òüþðèíãà, ÿâëÿåòñÿ îðàêóëüíîé 3 Âñå óêàçàííûå íà ñòàíèöàõ 212-215 ñëîâàðíûå ôóíêöèè ÿâëÿþòñÿ PPr ôóíêöèÿìè, àëôàâèòà L , ðàññìàòðèâàåìàÿ òåîðåìà ëåãêî ïåðåíîñèòñÿ íà îðàêóëüíóþ ìàøèíó Òüþðèíãà 19 1) Cëîâî âèäà U - PPr ( A ) ïðîãðàììà; 2) Ñëîâà âèäà Z , δ , Length , S k , I m n , . . . . . . . . . , Concat , D - PPr ( A ) ïðîãðàììû; 3) Åñëè Φ - k - ìåñòíûé ôóíêòîð, Ψ 1 , . . . , Ψ k - n - ìåñòíûå ôóíêòîðû è ÿâëÿþòñÿ PPr ( A ) ïðîãðàììàìè, òî ôóíêòîð [ J ΦΨ 1 , . . . , Ψ k ] - PPr ( A ) ïðîãðàììà, 4) Åñëè Φ - n - ìåñòíûé ôóíêòîð, Ψ 1 , . . . , Ψ k - n +2 - ìåñòíûå ôóíêòîðû è ÿâëÿþòñÿ PPr ( A ) ïðîãðàììà- ìè, P - n +1 - ñëîâàðíûé ìíîãî÷ëåí, òîãäà, åñëè âåðíî WordM A | = ∀ x, y {| [ R ΦΨ 1 , . . . , Ψ k ]( x, y ) | ≤ | P ( x, y ) |} , òîãäà ôóíêòîð [ R ΦΨ 1 , . . . , Ψ k ] - PPr ( A ) - ïðîãðàììà. Çàìå÷àíèå . Åñëè Φ ∈ PPr ( A ) , òîãäà ìîæíî ïîñòðîèòü òàêîé ñëîâàðíûé ìíîãî÷ëåí P ( x ) , ÷òî âåðíî WordM A | = ∀ x {| Φ( x ) | ≤ | P ( x ) |} . Çàìå÷àíèå . Äëÿ ëþáîãî ôóíêòîðà Φ ∈ PPr ( U ) , äëÿ ëþáîãî îðàêóëà A , âåðíî Φ ∈ PPr ( A ) . Çàìå÷àíèå . Ïócòü A - èíòåïðåòàöèÿ îðàêóëüíîãî ñèìâîëà U . Äëÿ ëþáîãî ôóíêòîðà Φ ∈ PPr ( A ) , ìîæíî ïîñòðîèòü òàêîé ôóíêòîð Ψ ∈ PPr ( U ) , ÷òî âåðíî WordM A | = ∀ x [Φ( x ) = Ψ( x )] . Äîêàçàòåëüñòâî . Äîêàçàòåëüñòâî âåä¼òñÿ èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà Φ , à âíóòðè ýòîé èí- äóêöèè, èíäóêöèåé ïî ïîñòðîåíèþ àðãóìåíòíîãî ñëîâà. Çàìå÷àíèå . Îòìåòèì, ÷òî äëÿ îòíîøåíèÿ exp k ( x ) = y ìîæíî ñîñòàâèòü ôóíêòîð EXP k , ïðèíàäëå- æàùèé PPr , òàêîé, ÷òî WordM | = ∀ xy [ exp k ( x ) = y ⇔⊢ EXP k ( x, y ) = Λ] . Part IV Ôóíêöèîíàëüíûå ñëîâà è èõ ñâîéñòâà Ñîñòàâèì ñëåäóþùèé ñëîâàðíûé òåðì Concat ( | α | , Concat ( a 2 , Concat ( α, Concat ( β, Concat ( a 2 , a 2 ))))) . Ïóñòü 1 - îáîçíà÷åíèå àðãóìåíòíîãî ñëîâà S 1 (Λ) , 2 - îáîçíà÷åíèå àðãóìåíòíîãî ñëîâà S 2 (Λ) , òîãäà ñëîâàð- íûé òåðì Concat ( | α | , Concat ( a 2 , Concat ( α, Concat ( β, Concat ( a 2 , a 2 ))))) äëÿ íàãëÿäíîñòè áóäåì îáîçíà- ÷àòü â âèäå 1 , . . . , 1 | {z } | α | - ðàç 2 αβ 22 . Ïóñòü c - òàêîé ôóíêòîð, äëÿ êîòîðîãî â èñ÷èñëåíèè CalcEq âûâîäèìî ðàâåíñòâî c ( x, y ) = 1 , . . . , 1 | {z } | x | - ðàç 2 xy 22 = | x | 2 xy 22 . Ïóñòü äàíà ïðîèçâîëüíàÿ ïîñëåäîâàòåëüíîñòü ïàð àðãóìåíòíûõ ñëîâ ( α 1 , γ 1 ) . . . , ( α n , γ n ) (íå îáÿçàòåëü- íî ðàçëè÷íûõ ïàð). Ýòó ïîñëåäîâàòåëüíîñòü áóäåì íàçûâàòü ôóíêöèîíàëüíîé, åñëè âûïîëíÿþòñÿ óñëîâèÿ: 1. ∀ i [ γ i = Λ ∨ γ i = a 1 ] , 2. ∀ i, j [ α i = α j → γ i = γ j ] . 20 Ââåä¼ì ïîíÿòèå, êîòîðîå â äàëüíåéøåì áóäåò èìåòü âàæíîå çíà÷åíèå. Îïðåäåëåíèå . 1. Λ - ôóíêöèîíàëüíîå ñëîâî. 2. Åñëè ïîñëåäîâàòåëüíîñòü ïàð ( α 1 , γ 1 ) . . . , ( α n , γ n ) ÿâëÿåòñÿ ôóíêöèîíàëüíîé, òî ñëîâî âèäà Concat ( c ( α 1 , γ 1 ) , Concat ( c ( α 2 , γ 2 ) , . . . , Concat ( c ( α k , γ k ) , Λ)) , . . . , ) - ôóíêöèîíàëüíîå ñëîâî, ïðè 1 ≤ k ≤ n . Íàãëÿäíî ôóíêöèîíàëüíîå ñëîâî ìîæíî çàïèñàòü â âèäå | α 1 | 2 α 1 γ 1 22 , . . . , | α k | 2 α k γ k 22 . Ñëîâà ïîñëåäîâàòåëüíîñòè α 1 , . . . , α k áóäåì íàçûâàòü îáëàñòüþ îïðåäåëåíèÿ ðàññìàòðèâàåìîãî ôóíê- öèîíàëüíîãî ñëîâà, à ñëîâà ïîñëåäîâàòåëüíîñòè γ 1 , . . . , γ k ñîîòâåòñâóþùèìè çíà÷åíèÿìè. Çàìå÷àíèå . Ëþáîå ôóíêöèîíàëüíîå ñëîâî θ áóäåì èíòåðïðåòèðîâàòü êàê ñëîâî ñîãëàñíî åãî îïðåäå- ëåíèÿ è êàê ôóíêöèþ ñ òåì æå èìåíåì. Îáëàñòü îïðåäåëåíèÿ è ìíîæåñòâî çíà÷åíèé ôóíêöèè θ - îáëàñòü îïðåäåëåíèÿ è ìíîæåñòâî çíà÷åíèé ôóíêöèîíàëüíîãî ñëîâà θ , ïðè÷¼ì θ ( α ) = Λ , òîãäà è òîëüêî òîãäà, êîãäà ñëîâî c ( α, Λ) - ïîäñëîâî ñëîâà θ è θ ( α ) = a 1 , òîãäà è òîëüêî òîãäà, êîãäà ñëîâî c ( α, a 1 ) - ïîäñëîâî ñëîâà θ . Îáëàñòü îïðåäåëåíèÿ ôóíêöèîíàëüíîãî ñëîâà θ áóäåì îáîçíà÷àòü êàê dom ( θ ) . Ïóñòü θ - íåêîòîðîå ôóíêöèîíàëüíîå ñëîâî. Äëÿ ýòîãî ôóíêöèîíàëüíîãî ñëîâà ïîñòðîèì ìíîæåñòâî àðãóìåíòíûõ ñëîâ, îïðåäåëÿåìîå êàê A θ = { α : α ∈ dom ( θ ) è θ ( α ) = Λ } . Äëÿ ëþáîãî òåðìà t ( x ) , äëÿ ëþáîé ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α , äëÿ ëþáîãî ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ìîæíî ïîñòðîèòü òàêîå ôóêöèîíàëüíîå ñëîâî θ α, A ,t , ñîãëàñîâàííîå ñ ìíîæåñòâîì A , ÷òî äëÿ ëþáîãî àðãóìåíòíîãî ñëîâà β âåðíî A ⊢ t ( α ) = β ⇔ A θ α, A ,t ⊢ t ( α ) = β . Äëÿ ýòîãî äîñòàòî÷íî ïîñòðîèòü âû÷èñëåíèå çàìêíóòîãî òåðìà t ( α ) íà ìíîæåñòâå A , ñîáðàòü âñå îïðîøåííûå ñëîâà â ýòîì âû÷èñëåíèè è ñîñòàâèòü ïî ïîëó÷åííûì îïðîøåííûì ñëîâàì, ñîîòâåòñòâóþùåå ôóíêöèîíàëüíîå ñëîâî. Ðàçóìååòñÿ, òàê ñîñòàâëåííîå ôóíêöèîíàëüíîå ñëîâî çàâèñèò îò ïîñòðîåííîãî âû÷èñëåíèÿ òåðìà t ( α ) , íî ïðè ýòîì áóäåò âûïîëåíî ñâîéñòâî: äëÿ ëþáîãî ôóíêöèîíàëüíîãî ñëîâà θ ⊇ θ α, A ,t , âåðíî A ⊢ t ( α ) = β ⇔ A θ ⊢ t ( α ) = β . Ýòî ñâîéñòâî âåðíî äëÿ ëþáîãî áåñêâàíòîðíîãî ïðåäëîæåíèÿ Φ (ïðåäëîæåíèå, ñîñòàâëåííîé ñ ïîìîùüþ ëîãè÷åñêèõ ñâÿçîê èç ðàâåíñòâ çàìêíóòûõ òåðìîâ): WordM A | = Φ ⇔ WordM A θ | = Φ . Ïóñòü Fw - òàêîé ôóíêòîð àëôàâèòà L , äëÿ êîòîðîãî âåðíî: 1. ∀ α [ ⊢ Fw ( α ) = Λ ∨ ⊢ Fw ( α ) = a 1 ] ; 2. ⊢ Fw ( α ) = Λ ⇔ α - ôóíêöèîíàëüíîå ñëîâî; 3. Ôóíêòîð Fw - PPr ôóíêòîð. Ââåä¼ì áèíàðíîå îòíîøåíèå x ∈ dom ( θ ) : x ∈ dom ( θ ) ⇐⇒ WordM | = Fw ( θ ) ∧ ( | x | 2 x 22 èëè | x | 2 x 122 åñòü ïîäñëîâî ñëîâà θ ) . 21 Îòíåøåíèå x ∈ dom ( θ ) ïðèíàäëåæèò PPr àëôàâèòà L , ãäå Fw ( x ) ⇋ Fw ( x ) = Λ Ââåä¼ì áèíàðíîå îòíîøåíèå θ ⊆ θ 1 : θ ⊆ θ 1 ⇐⇒ WordM | = Fw ( θ ) ∧ Fw ( θ 1 ) ∧ ∀ x ∈ dom ( θ )[ θ ( x ) = θ 1 ( x )] . Îòíåøåíèå θ ⊆ θ 1 ïðèíàäëåæèò PPr àëôàâèòà L . Ââåä¼ì áèíàðíîå îòíîøåíèå ≈ : WordM | = θ ≈ θ 1 ⇐⇒ Fw ( θ ) ∧ Fw ( θ 1 ) ∧ dom ( θ ) = dom ( θ 1 ) ∧ ∀ x ∈ dom ( θ ) θ ( x ) = θ 1 ( x ) . Îòíîøåíèå x ≈ y ïðèíàäëåæèò PPr àëôàâèòà L . Çàìå÷àíèå . Åñëè θ ≈ θ 1 , òîãäà Concat ( θ, θ 1 ) ≈ Concat ( θ 1 , θ ) ∧ Concat ( θ, θ 1 ) ≈ θ ∧ θ ⊆ θ 1 ∧ θ 1 ⊆ θ . Ïóñòü G - åñòü òàêîé äâóõ ìåñòíûé ôóíêòîð, àëôàâèòà L , êîòîðûé óäîâëåòâîðÿåò ñëåäóùèì óñëîâèÿì: 1. Åñëè θ - ôóíêöèîíàëüíîå ñëîâî, α ∈ dom ( θ ) è θ ( α ) = γ , òîãäà ⊢ G ( θ, α ) = γ . 2. Åñëè θ - ôóíêöèîíàëüíîå ñëîâî, α / ∈ dom ( θ ) , òîãäà ⊢ G ( θ, α ) = a 1 . 3. Åñëè θ - íå ÿâëÿåòñÿ ôóíêöèîíàëüíûì ñëîâîì, òîãäà ∀ α ⊢ G ( θ, α ) = a 1 . 4. Ôóíêòîð G - PPr ôóíêòîð. Ôóíêòîð G îáëàäàåò ñâîéñòâàìè: 1. Äëÿ ëþáûõ ôóíêöèîíàëüíûõ ñëîâ θ, θ 1 , òàêèõ, ÷òî θ ⊆ θ 1 , âåðíî ∀ α ∈ dom ( θ ) ⊢ [ G ( θ, α ) = G ( θ 1 , α )] . 2. Îòíîøåíèå âèäà x ⊂ U ⇌ [ Fw ( x ) = Λ]& ∀ z ∈ dom ( x )[( G ( x, z ) = Λ ⇒ U ( z ) = Λ)&( G ( x, z ) = a 1 ⇒ U ( z ) = a 1 )] ïðèíàäëåæèò PPr ò.å. ñóùåñòâóåò òàêîé îäíîìåñòíûé PPr ôóíêòîð φ àëôàâèòà L ( U ) , ÷òî äëÿ ëþáîãî èíòåðïðåòàöèîííîãî ìíîæåñòâà àðãóìåíòíûõ ñëîâ A âåðíî WordM A | = ∀ x [ x ⊂ U ⇔ φ ( x ) = Λ] . 3. ∀ A WordM A | = ∀ α ( G ( | α | 2 α U ( α )22 , α ) = U ( α )) . Ôóíêöèîíàëüíîå ñëîâî θ ñîãëàñóåòñÿ ñ ìíîæåñòâîì A ( θ ⊂ A ), åñëè ∀ x ∈ dom ( θ )[ θ ( x ) = Λ ⇐⇒ x ∈ A ] , ( WordM A | = θ ⊂ U ). Çàìå÷àíèå. Äëÿ ëþáûõ ôóíêöèîíàëüíûõ ñëîâ θ 1 , θ 2 , êîòîðûå ñîãëàñîâàíû ñ ìíîæåñòâîì A , ñóùå- ñòâóåò ôóíêöèîíàëüíîå ñëîâî θ , ñîãëàñîâàííîå ñ ìíîæåñòâîì A è θ 1 ⊆ θ , θ 2 ⊆ θ , íàïðèìåð θ 1 ∪ θ 2 ( Concat ( θ 1 , θ 2 ) ). Îïðåäåëåíèå. Ïóñòü äàí ôóíêòîð Φ , ïîñëåäîâàòåëüíîñòü àðãóìåíòíûõ ñëîâ α è èíòåðïðåòàöèîííîå ìíîæåñòâî A . Ïóñòü P - ïðîñòîå âû÷èñëåíèå ôóíêòîðà Φ , íà ïîñëåäîâàòåëüíîñòè α , â èíòåðïðåòàöèè A . Òîãäà, èñïîëüçóÿ ïðîñòîå âû÷èñëåíèå P , ñîñòàâèì ôóíêöèîíàëüíîå ñëîâî: 1. Âûïèøåì âñå ñëîâà èç ìíîæåñòâà ( A + ) P Φ ,α . Ïóñòü ýòî áóäóò ñëîâà β 1 , . . . , β k , ðàñïîëîæèì èõ, íà- ïðèìåð, â ëåêñèêîãðàôè÷åñêîì ïîðÿäêå. 22 2. Âûïèøåì âñå ñëîâà èç ìíîæåñòâà ( A − ) P Φ ,α . Ïóñòü ýòî áóäóò ñëîâà γ 1 , . . . , γ s , ðàñïîëîæèì èõ, òàêæå â ëåêñèêîãðàôè÷åñêîì ïîðÿäêå. 3. Ñîñòàâèì ôóíêöèîíàëüíîå ñëîâî Concat ( c ( β 1 , Λ) , . . . , Concat ( c ( β k , Λ) , Concat ( c ( γ 1 , a 1 ) , . . . , Concat ( c ( γ s − 1 , a 1 ) , c ( γ s , a 1 ))) , . . . , ) . Òàê ñîñòàâ- ëåííîå ôóíêöèîíàëüíîå ñëîâî íàçîâ¼ì ôóíêöèîíàëüíûì ñëîâîì ñîñòàâëåííûì ñîãëàñíî ïðîñòîãî âû÷èñ- ëåíèÿ P ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè α , â èíòåðïðåòàöèè A . Îáîçíà÷èì òàêîå ôóíêöèîíàëüíîå ñëîâî êàê θ SimpleFw , Φ ,α, A . Îïðåäåëåíèå . Òåðìû âèäà | x | 2 x U ( x )22 | z | 2 z U ( z )22 , . . . , | v | 2 v U ( v )22 áóäåì íàçûâàòü ôóíêöèîíàëüíû- ìè òåðìàìè àëôàâèòà L ( U ) . Ìíîæåñòâî òàê ïîñòðîåííûõ ôóíêöèîíàëüíûõ òåðìîâ áóäåì îáîçíà÷àòü êàê F term U , à êîíêðåòíûé ôóíêöèîíàëüíûé òåðì ýòîãî ìíîæåñòâà, êàê f term ( x, z, . . . , v ) . Îïðåäåëåíèå . Òåðìû âèäà | x | 2 x G ( y , x )22 | z | 2 z G ( y , z )22 , . . . , | v | 2 v G ( y , v )22 áóäåì íàçûâàòü ôóíêöèî- íàëüíûìè òåðìàìè àëôàâèòà L . Ìíîæåñòâî òàê ïîñòðîåííûõ ôóíêöèîíàëüíûõ òåðìîâ áóäåì îáîçíà÷àòü êàê F term, à êîíêðåòíûé ôóíêöèîíàëüíûé òåðì ýòîãî ìíîæåñòâà, êàê f ∗ term ( y , x, z, . . . , v ) . Ñâîéñòâà . 1. ∀ A âåðíî WordM A | = f ∗ term ( f term ( x, z, . . . , v ) , x, z, . . . , v ) = f term ( x, z, . . . , v ) - êàê ñëîâà. 2. WordM | = f ∗ term ( f ∗ term ( x, z, . . . , v ) , x, z, . . . , v ) = f ∗ term ( x, z, . . . , v ) - êàê ñëîâà. 3. Äëÿ ëþáîãî ôóíêöèîíàëüíîãî ñëîâà θ âåðíî ∀ x, z, . . . , v ∈ dom ( θ ) WordM | = f ∗ term ( θ, x, z, . . . , v ) ⊆ θ . Îïðåäåëåíèå ôóíêòîðà äëÿ ïîñòðîåíèÿ ôóíêöèîíàëüíîãî ñëîâà Äëÿ êàæäîãî n - ìåñòíîãî ôóíêòîðà Φ , îïðåäåëèì ñâçÿííóþ ñ ýòèì ôóíêòîðîì, ôóíêòîð ïîñòðîåíèÿ ôóíêöèîíàëüíîãî ñëîâà. Îïðåäåëåíèå ïðîâåä¼ì èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà Φ . 1. Äëÿ èñõîäíûõ ôóíêòîðîâ: S k , Z , δ , Length , . , Concat , D , I n k , U : Θ S k = Z , Θ Z = Z , Θ δ = Z , Θ Length = [ J ZI 2 2 ] , Θ . = [ J ZI 2 2 ] , Θ Concat = [ J ZI 2 2 ] , Θ D = [ J ZI 2 2 ] , Θ I n k = [ J ZI n k ] , Θ U = [ J cI 1 1 U ] . Äëÿ ýòèõ ôóíêòîðîâ â èñ÷èñëåíèè CalcEq âûâîäèìû ðàâåíñòâà: Θ S k ( x 1 ) = Λ , Θ Z ( x 1 ) = Λ , Θ δ = Λ , Θ Length ( x 1 ) = Λ , Θ . ( x 1 , x 2 ) = Λ , Θ Concat ( x 1 , x 2 ) = Λ , Θ D ( x 1 , x 2 ) = Λ , Θ I n k ( x 1 , . . . , x n ) = Λ , ⊢ Θ U ( x 1 ) = c ( x 1 , U ( x 1 )) - â èñ÷èëåíèè CalcEq U è âåðíî ∀ α ∀ A A ⊢ G (Θ U ( α ) , α ) = U ( α ) . 2. Ïóñòü Φ - k - ìåñòíûé ôóíêòîð( k ≥ 2 ), Ψ 1 , . . . , Ψ k - n - ìåñòíûå ôóíêòîðû, ñîñòàâèì ôóíêòîð ñóïåðïîçèöèè [ J ΦΨ 1 , . . . , Ψ k ] . Äëÿ ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] ïîñòðîèì ôóíêòîð Θ [ J ΦΨ 1 ,..., Ψ k ] . Ïóñòü äëÿ ôóíêòîðà Φ ïîñòðîåí ôóíêòîð Θ Φ , äëÿ ôóíêòîðà Ψ 1 ïîñòðîåí ôóíêòîð Θ Ψ 1 , è ò.ä. äëÿ ôóíêòîðà Ψ k ïîñòðîåí ôóíêòîð Θ Ψ k , òîãäà 23 Θ [ J ΦΨ 1 ,..., Ψ k ] ⇌ [ J Concat k +1 [ J Θ Φ Ψ 1 . . . Ψ k ] , Θ Ψ 1 . . . Θ Ψ k ] ïðè k ≥ 2 . Ïîëó÷åííûé ôóíêòîð ÿâëÿåòñÿ n - ìåñòíûì è äëÿ íåãî âåðíû ñëåäóþùèå äîêàçóåìûå ðàâåíñòâà ⊢ [ J Concat k +1 [ J Θ Φ Ψ 1 . . . Ψ k ] , Θ Ψ 1 . . . Θ Ψ k ]( x 1 , . . . , x n ) = Concacat k +1 ([ J Θ Φ Ψ 1 . . . Ψ k ]( x 1 , . . . , x n ) , Θ Ψ 1 ( x 1 , . . . , x n ) . . . Θ Ψ k ( x 1 , . . . , x n )) ⊢ Concat k +1 ([ J Θ Φ Ψ 1 . . . Ψ k ]( x 1 , . . . , x n ) , Θ Ψ 1 ( x 1 , . . . , x n ) . . . Θ Ψ k ( x 1 , . . . , x n )) = Concat k +1 (Θ Φ (Ψ 1 ( x 1 . . . , x n ) . . . Ψ k ( x 1 , . . . , x n )) , Θ Ψ 1 ( x 1 , . . . , x n ) . . . Θ Ψ k ( x 1 , . . . , x n )) . Èòàê, èìååì ⊢ Θ [ J ΦΨ 1 ,..., Ψ k ] ( x 1 , . . . , x n ) = [ J Concat k +1 [ J Θ Φ Ψ 1 . . . Ψ k ] , Θ Ψ 1 . . . Θ Ψ k ]( x 1 , . . . , x n ) = Concat k +1 (Θ Φ (Ψ 1 ( x 1 , . . . , x n ) . . . Ψ k ( x 1 , . . . , x n )) , Θ Ψ 1 ( x 1 , . . . , x n ) . . . Θ Ψ k ( x 1 , . . . , x n )) , ⊢ Θ [ J ΦΨ 1 ,..., Ψ k ] ( x 1 , . . . , x n ) = Concat k +1 (Θ Φ (Ψ 1 ( x 1 , . . . , x n ) . . . Ψ k ( x 1 , . . . , x n )) , Θ Ψ 1 ( x 1 , . . . , x n ) . . . Θ Ψ k ( x 1 , . . . , x n )) , ⊢ Θ [ J ΦΨ 1 ,..., Ψ k ] ( x ) = Concat ([ J Θ Φ Ψ 1 , . . . , Ψ k ]( x ) , ( Concat (Θ Ψ 1 ( x ) , . . . , Concat (Θ Ψ k − 1 ( x ) , Θ Ψ k ( x ))) , . . . , ) . Ïóñòü Φ - k - ìåñòíûé ôóíêòîð( k = 1 ), òîãäà Θ [ J ΦΨ 1 ] ⇌ [ J Concat [ J Θ Φ Ψ 1 ]Θ Ψ 1 ] , òîãäà Θ [ J ΦΨ 1 ] ( x 1 , . . . , x n ) ⇌ [ J Concat [ J Θ Φ Ψ 1 ]Θ Ψ 1 ]( x 1 , . . . , x n ) = Concat (Θ Φ (Ψ 1 ( x 1 , . . . , x n )) , Θ Ψ ( x 1 , . . . , x n )) 3. Äëÿ ôóíêòîðà âèäà [ Rα Ψ 1 , . . . , Ψ k ] è ôóíêòîðà âèäà [ R ΦΨ 1 , . . . , Ψ k ] . Ïóñòü äëÿ ôóíêòîðà Φ ïî- ñòðîåí ôóíêòîð Θ Φ , äëÿ ôóíêòîðà Ψ 1 ïîñòðîåí ôóíêòîð Θ Ψ 1 , è ò.ä. äëÿ ôóíêòîðà Ψ k ïîñòðîåí ôóíêòîð Θ Ψ k , òîãäà äëÿ ôóíêòîðà Θ [ R ΦΨ 1 ,..., Ψ k ] â èñ÷èëåíèè CalcEq U èìååò ìåñòî (îïðåäåëÿþùåå ðàâåíñòâî)(ñì. òåîðåìó 1.1) 4 : ⊢ Θ [ Rα Ψ 1 ,..., Ψ k ] (Λ) = Λ . ⊢ Θ [ Rα Ψ 1 ,..., Ψ k ] ( x 1 a i ) = Concat (Θ Ψ i ( x 1 , [ R ΦΨ 1 , . . . , Ψ k ]( x 1 )) , Θ [ Rα Ψ 1 ,..., Ψ k ] ( x 1 )) , ïðè i ≤ k . ⊢ Θ [ Rα Ψ 1 ,..., Ψ k ] ( x 1 a i ) = Θ [ Rα Ψ 1 ,..., Ψ k ] ( x 1 ) , ïðè i > k ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = Θ Φ ( x ) . ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, x n +1 a i ) = Concat (Θ Ψ i ( x, x n +1 , [ R ΦΨ 1 , . . . , Ψ k ]( x, x n +1 )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, x n +1 )) , ïðè i ≤ k . ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, x n +1 a i ) = Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, x n +1 ) , ïðè i > k . Òåîðåìà 4.1 . Ïóñòü Φ - ôóíêòîð àëôàâìòà L , òîãäà â èñ÷èñëåíèè CalEq âåðíî ⊢ Θ Φ ( x ) = Λ . Äîêàçàòåëüñòâî ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåèþ ôóíêòîðà Φ , à âíóòðè ýòîé èíäóêöèè, èíäóêöèåé ïî ïîñòðîåíèþ àðãóìåíòíîãî ñëîâà. Òåîðåìà 4.2 . Ïóñòü Φ - ïðîèçâîëüíûé ôóíêòîð, êîòîðûé ïðèíàäëåæèò PPr , òîãäà ôóíêòîð Θ Φ ïðè- íàäëåæèò PPr . Äîêàçàòåëüñòâî ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåíèþ PPr ôóíêòîðà Φ , à âíóòðè ýòîé èíäóêöèè, èíäóêöèåé ïî ïîñòðîåíèþ àðãóìåíòíîãî ñëîâà. 4 ñì. Ïðèëîæåíèå 24 Òåîðåìà 4.3 . Ïóñòü Φ - ïðîèçâîëüíûé n − ìåñòíûé ôóíêòîð. Äëÿ ëþáîãî èíòåðïðåòàöèîííîãî ìíî- æåñòâà A , ëþáîé ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α, β , âåðíî: Åñëè A ⊢ Θ Φ ( α ) = β , òîãäà ñëîâî β - ôóíêöèîíàëüíîå ñëîâî è θ SimpleFw , Φ ,α, A ⊆ β ⊂ A . Äîêàçàòåëüñòâî . Äîêàçàòåëüñòâî ïðîâåä¼ì èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà. Áàçà èíäóêöèè . Äëÿ èñõîäíûõ ôóíêòîðîâ S k , Z , δ , Length , . , Concat , D , I n k äîêàçàòåëüñòâî ïîëó- ÷àåòñÿ íåïîñðåäñòâåííî. Äëÿ ôóíêòîðà U , ïîëó÷èì: A ⊢ Θ U ( α ) = β , òîãäà è òîëüêî òîãäà, êîãäà ( β = | α | 2 α 22& α ∈ A ) ∨ ( β = | α | 2 αa 1 22& α / ∈ A ) , òîãäà β = θ SimpleFw , U ,α, A è β ⊂ A . Èíäóêöèîííîå ïðåäïîëîæåíèå 1. Ïóñòü òåîðåìà âåðíà äëÿ ôóíêòîðà Φ , ôóíêòîðîâ Ψ 1 , . . . , Ψ k . Äîêàæåì, ÷òî òåîðåìà âåðíà äëÿ ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] . Ïî èíäóêöèîííîìó ïðåäïîëîæåíèþ èìååì: åñëè A ⊢ Θ Ψ 1 ( α ) = γ 1 , . . . , A ⊢ Θ Ψ k ( α ) = γ k , òîãäà γ i - ôóíêöèîíàëüíûå ñëîâà è θ SimpleFw , Ψ 1 ,α, A ⊆ γ 1 ⊂ A , . . . , θ SimpleFw , Ψ k ,α, A ⊆ γ k ⊂ A . Ôóíêöèîíàëüíîå ñëîâî θ SimpleFw , Ψ i ,α, A , ñîñòàâëåíî ñîãëàñíî ìíîæåñòâàì ( A + ) P Ψ i ,α , ( A − ) P Ψ i ,α . òîãäà k S i =1 θ SimpleFw , Ψ i ,α, A ⊆ Concat (Θ Ψ 1 ( α ) , . . . , Concat (Θ Ψ k − 1 ( α ) , Θ Ψ k ( α )) , . . . , ) . Èìååì A ⊢ Ψ i ( α ) = β i . Åñëè A ⊢ Θ Φ ( β 1 , . . . , β k ) = η , òîãäà ïî èíäóêöèîííîìó ïðåäïîëîæåíèþ, η - ôóíêöèîíàëüíîå ñëîâî è θ SimpleFw , Φ ,β 1 ,...,β k , A ⊆ η ⊂ A . Ôóíêöèîíàëüíîå ñëîâî θ SimpleFw , Φ ,β, A ñîñòàâëåíî ñîãëàñíî ìíîæåñòâàì ( A + ) P Φ ,β , ( A − ) P Φ ,β , òîãäà θ SimpleFw , Φ ,β 1 ,...,β k , A ⊆ Concat (Θ Φ ( β 1 , . . . , β k ) , Concat (Θ Ψ 1 ( α ) , . . . , Concat (Θ Ψ k − 1 ( α ) , Θ Ψ k ( α )) , . . . , ) . Ôóíê- öèîíàëüíîå ñëîâî θ SimpleFw , [ J ΦΨ 1 ,..., Ψ k ] ,α, A ñîñòàâëåíî ñîãëàñíî ìíîæåñòâàì ( A + ) P [ J ΦΨ 1 ,..., Ψ k ] ,α = k S i =1 ( A + ) P Ψ i ,α S ( A + ) P Φ ,β , ( A − ) P [ J ΦΨ 1 ,..., Ψ k ] ,α = k S i =1 ( A − ) P Ψ i ,α S ( A − ) P Φ ,β , òîãäà, ñîãëàñíî îïðåäåëÿþùåìó ðàâåíñòâó äëÿ Θ [ J ΦΨ 1 ,..., Ψ k ] , ïîëó÷èì θ SimpleFw , [ J ΦΨ 1 ,..., Ψ k ] ,α, A ⊆ Θ [ J ΦΨ 1 ,..., Ψ k ] ( α ) ⊂ A . Èíäóêöèîííîå ïðåäïîëîæåíèå 2. Ïóñòü òåîðåìà âåðíà äëÿ ôóíêòîðà Φ , ôóíêòîðîâ Ψ 1 , . . . , Ψ k . Äîêàæåì, ÷òî òåîðåìà âåðíà äëÿ ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ k ] . Ñîãëàñíî îïðåäåëÿþùèì ðàâåíñòâàì äëÿ ôóíêòîðà Θ [ R ΦΨ 1 ,..., Ψ k ] , èìååì: ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = Θ Φ ( x ) . ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, za i ) = Concat (Θ Ψ i ( x, z, [ R ΦΨ 1 , . . . , Ψ k ]( x, z )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) . Áàçà èíäóêöèè. Ïî èíäóêöèîííîìó ïðåäïîëîæåíèþ èìååì: A ⊢ Θ Φ ( α ) = β - ôóíêöèîíàëüíîå ñëîâî è θ SimpleFw , Φ ,α, A ⊆ β ⊂ A . Ôóíêöèîíàëüíîå ñëîâî θ SimpleFw , Φ ,α, A ïîñòðîåíî ïî ìíîæåñòâàì: ( A + ) P Φ ,α , ( A − ) P Φ ,α . Ôóíêöèîíàëüíîå ñëîâî θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α, Λ , A ïîñòðî- åíî òàêæå ïî ìíîæåñòâàì: ( A + ) P Φ ,α , ( A − ) P Φ ,α , ó÷èòûâàÿ îïðåäåëÿþùåå ðàâåíñòâî Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, Λ) = Θ Φ ( α ) , ïîëó÷èì θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α, Λ , A ⊆ β è β ⊂ A . 25 Èíäóêöèîííûé ïåðåõîä . Âîçüì¼ì ïîñëåäîâàòåëüíîñòü àðãóìåíòíûõ ñëîâ α 1 , . . . , α n .βa i . Ïî èíäóêöèîííîìó ïåðåõîäó èìååì: Åñëè A ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β ) = γ , òîãäà γ - ôóíêöèîíàëüíîå ñëîâî è θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α,β, A ⊆ γ ⊂ A . Ïóñòü A ⊢ [ R ΦΨ 1 , . . . , Ψ k ]( α, β ) = η . Åñëè A ⊢ Θ Ψ i ( α, β, η ) = ξ i , òîãäà ξ i - ôóíêöèîíàëüíîå ñëîâî è θ SimpleFw , Ψ i ,α,β,η, A ⊆ ξ i ⊂ A . Ñîãëàñíî îïðåäåëåíèÿ ôóíêöèîíàëüíîãî ñëîâà θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α,β, A îíî ïîñòðîåíî ïî ìíîæå- ñòâàì: ( A + ) P [ R ΦΨ 1 ,..., Ψ k ] ,α,β , ( A − ) P [ R ΦΨ 1 ,..., Ψ k ] ,α,β , òîãäà θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α,β, A ⊆ Concat (Θ Ψ i ( α, β, [ R ΦΨ 1 , . . . , Ψ k ]( α, β )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β )) . Ôóíêöèîíàëüíîå ñëîâî θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α,βa i , A ñòðîèòñÿ ïî ìíîæåñòâàì: ( A + ) P [ R ΦΨ 1 ,..., Ψ k ] ,α,βa i = ( A + ) P Ψ i , ,α,β,η S ( A + ) P [ R ΦΨ 1 ,..., Ψ k ] ,α,β , ( A − ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,βa i = ( A − ) P Ψ i , ,α,β,η S ( A − ) P [ R ΦΨ 1 ,..., Ψ k ] ,α,β , òîãäà θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α,βa i , A = θ SimpleFw , Ψ i ,α,β,η, A S θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α,β, A , òîãäà θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α,βa i , A ⊆ Concat (Θ Ψ i ( α, β, [ R ΦΨ 1 , . . . , Ψ k ]( α, β )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β )) . Ó÷èòûâàÿ îïðå- äåëÿþùåå ðàâåíñòâî Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, βa i ) = Concat (Θ Ψ i ( α, β, [ R ΦΨ 1 , . . . , Ψ k ]( α, β )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β )) , ïîëó÷èì θ SimpleFw , [ R ΦΨ 1 ,..., Ψ k ] ,α,βa i , A ⊆ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, βa i ) ⊂ A . Îñòàëüíûå àêñèîìû ðåêóðñèè(15,16,17,20) ðàññìàòðèâàþòñÿ àíàëîãè÷íî. Òåîðåìà 4.4 Ïóñòü Φ - ïðîèçâîëüíûé n − ìåñòíûé ôóíêòîð àëôàâèòà L ( U ) . Äëÿ ëþáîãî èíòåðïðå- òàöèîííîãî ìíîæåñòâà A , ëþáîé ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α , âåðíî: WordM A | = Θ Φ ( α ) ≈ Θ Θ Φ ( α ) 5 . Äîêàçàòåëüñòâî ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà Φ , íîñèò ÷èñòî ñèíòàêñè÷åñêèé õà- ðàêòåð. Íà ñîäåðæàòåëüíîì óðîâíå ýòî îòíîøåíèå ÿâëÿåòñÿ î÷åâèäíûì. Íåôîðìàëüíî, Θ Φ - íåïîäâèæíàÿ òî÷êà îïåðàòîðà Θ . Çàìå÷àíèå . Ïóñòü Φ - ïðîèçâîëüíûé n − ìåñòíûé ôóíêòîð àëôàâèòà L ( U ) . Äëÿ ëþáîãî èíòåðïðå- òàöèîííîãî ìíîæåñòâà A âåðíî WordM A | = ∀ x [ θ SimpleFw , Φ ,x, A ≈ Θ Φ ( x )] . Part V Ïðåîáðàçîâàíèå âûðàæåíèé àëôàâèòà L ( U ) , â âûðàæåíèÿ àëôàâèòà L Èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà, ïîñòðîèì ïðåîáðàçîâàíèå, îáîçíà÷àåìîå êàê ∗ , ôóíêòîðà àëôàâèòà L ( U ) â ôóíêòîð àëôàâèòà L . Äëÿ èñõîäíûõ ôóíêòîðîâ: 5 ñì. Ïðèëîæåíèå 26 1. ( S k ) ∗ = [ J S k I 2 2 ] , 2. ( Z ) ∗ = [ J ZI 2 2 ] , 3. ( δ ) ∗ = [ J δ I 2 2 ] , 4. ( U ) ∗ = G , 5. ( Length ) ∗ = [ J LengthI 3 2 I 3 3 ] , 6. ( . ) ∗ = [ J . I 3 2 I 3 3 ] , 7. ( Concat ) ∗ = [ J ConcatI 3 2 I 3 3 ] , 8. ( D ) ∗ = [ J DI 3 2 I 3 3 ] , 9. ( I n k ) ∗ = [ J I n k I n +1 2 , . . . , I n +1 n +1 ] , 1. ([ J ΦΨ 1 , . . . , Ψ k ]) ∗ = [ J (Φ) ∗ I n + 1 1 (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ] , 2. ([ Rα Φ 1 , . . . , Φ m ]) ∗ = [ R Const 1 α (Φ 1 ) ∗ , . . . , (Φ m ) ∗ ] , 3. ([ R ΦΨ 1 , . . . , Ψ m ]) ∗ = [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ m ) ∗ ] . Çàìå÷àíèå. Åñëè ôóíêòîð Φ - ôóíêòîð àëôàâèòà L , òîãäà ïåðâûé àðãóìåíò ôóíêòîðà (Φ) ∗ - ôèê- òèâíàÿ ïåðåìåííàÿ: ⊢ Φ( x ) = (Φ) ∗ ( y , x ) . Òåîðåìà 5.1 . Äëÿ ëþáîãî n - ìåñòíîãî ôóíêòîðà Φ àëôàâèòà L ( U ) ∀ A , ∀ α ∀ θ ⊇ θ SimpleFw , Φ ,α, A âåðíî A ⊢ [Φ( α ) = (Φ) ∗ ( θ, α )] ( WordM A | = [Φ( α ) = (Φ) ∗ ( θ, α )] ). Äîêàçàòåëüñòâî. Äîêàçàòåëüñòâî ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà, âíóòðè ýòîé èí- äóêöèè äëÿ ðåêóðñèâíîãî ôóíêòîðà, äîêàçàòåëüñòâî ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåíèþ àðãóìåíòíîãî ñëîâà. Áàçà èíäóêöèè. Èñõîäíûå ôóíêòîðû Äëÿ èñõîäíûõ ôóíêòîðîâ: S k , Z , δ , Length , . , Concat , D , I n k ìîæíî óáåäèòüñÿ íåïîñðåäñòâåííî, âû- ïèñàâ óêàçàííûé ôóíêòîð ϕ è ôóíêòîð ( ϕ ) ∗ . Äîêàæåì òåîðåìó äëÿ ôóíêòîðà U . Ñîãëàñíî îïðåäåëåíèÿ ( U ) ∗ = G , íåîáõîäèìî ïîêàçàòü ∀ A ∀ α ∀ θ ⊇ θ SimpleFw , U ,α, A A ⊢ [ U ( α ) = G ( θ, α )] . Ïóñòü α ∈ A , òîãäà U ( α ) = Λ - àêñèîìà è ÿâëÿåòñÿ ïðîñòûì âû÷èñëåíèåì ôóíêòîà U íà ñëîâå α .  êà÷åñòâå ôóíêöèîíàëüíîãî ñëîâà âîçüì¼ì ñëîâî θ SimpleFw , U ,α, A = c ( α, Λ) = | α | 2 α 22 , òîãäà ñî- ãëàñíî îïðåäåëåíèÿ ôóíêòîðà G , ïîëó÷èì ∀ θ ⊇ θ SimpleFw , U ,α, A ⊢ G ( θ, α ) = Λ . Ïóñòü P θ,α - (íàïðè- ìåð)ïðîñòîé âûâîä ôóíêòîðà G íà ïîñëåäîâàòåëüíîñòè ñëîâ θ, α , òîãäà ïîñëåäîâàòåëüíîñòü ðàâåíñòâ U ( α ) = Λ , P θ,α , U ( α ) = G ( θ, α ) - âûâîä ðàâåíñòâà U ( α ) = G ( θ, α ) , ïðè èíòåðïðåòàöèè ôóíêöèîíàëüíîãî ñèìâîëà U ìíîæåñòâîì A . Àíàëîãè÷íî: ïóñòü α ̸ ∈ A , òîãäà U ( α ) = a 1 - àêñèîìà è ÿâëÿåòñÿ ïðîñòûì âû÷èñëåíèåì ôóíê- 27 òîðà U íà ñëîâå α .  êà÷åñòâå ôóíêöèîíàëüíîãî ñëîâà âîçüì¼ì ñëîâî θ SimpleFw , U ,α, A = c ( α, a 1 ) = | α | 2 αa 1 22 ( | α | 2 α 122 ), òîãäà ñîãëàñíî îïðåäåëåíèÿ ôóíêòîðà G , ïîëó÷èì ∀ θ ⊇ θ SimpleFw , U ,α, A , ⊢ G ( θ, α ) = a 1 ( ⊢ G ( θ, α ) = 1 ). Ïóñòü P θ,α - ïðîñòîé âûâîä ôóíêòîðà G íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ θ, α , òîãäà ïîñëåäîâàòåëüíîñòü ðàâåíñòâ U ( α ) = a 1 , P θ,α , U ( α ) = G ( θ, α ) - âûâîä ðàâåíñòâà U ( α ) = G ( θ, α ) , ïðè èíòåðïðåòàöèè ôóíêöèîíàëüíîãî ñèìâîëà U ìíîæåñòâîì A . a ). Èíäóêöèîííîå ïðåäïîëîæåíèå . Ïóñòü òåîðåìà âåðíà äëÿ ôóíêòîðîâ: Φ , Ψ 1 , . . . , Ψ k , äîêàæåì òåîðåìó äëÿ ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] . Îáîçíà÷èì f ⇋ [ J ΦΨ 1 , . . . , Ψ k ] . Èìååì ( a ): äëÿ ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , äëÿ ôóíêòîðà Ψ 1 , âåðíî ∀ θ ⊇ θ SimpleFw , Ψ 1 ,α, A , A ⊢ Ψ 1 ( α ) = (Ψ 1 ) ∗ ( θ, α ) , . . . , äëÿ ôóíêòîðà Ψ k , âåðíî ∀ θ ⊇ θ SimpleFw , Ψ k ,α, A , A ⊢ Ψ k ( α ) = (Ψ k ) ∗ ( θ, α ) ; äëÿ ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ β 1 , . . . , β k , äëÿ ôóíêòîðà Φ , âåðíî ∀ θ ⊇ θ SimpleFw , Φ ,β, A , A ⊢ Φ( β ) = (Φ) ∗ ( θ, β ) . Ôóíêöèîíàëüíîå ñëîâî θ SimpleFw , f ,α, A , ñîãëàñíî åãî îïðåäåëåíèÿ, ñîñòàâëåíî ïî ìíîæåñòâàì: ( A + ) P [ J ΦΨ 1 ,..., Ψ k ] ,α = k S i =1 ( A + ) P Ψ i ,α S ( A + ) P Φ ,γ , ( A − ) P [ J ΦΨ 1 ,..., Ψ k ] ,α = k S i =1 ( A − ) P Ψ i ,α S ( A − ) P Φ ,γ ], òîãäà θ SimpleFw , f ,α, A ⊇ θ SimpleFw , Ψ i ,α, A è θ SimpleFw , f ,α, A ⊇ θ SimpleFw , Φ ,β, A . Ïóñòü θ f ⊇ θ SimpleFw , f ,α, A . Ñëåäóþùàÿ ïîñëåäîâàòåëüíîñòü ðàâåíñòâ: 1. (Ψ 1 ) ∗ ( θ f , α ) = γ 1 , . . . , (Ψ k ) ∗ ( θ f , α ) = γ k - âû÷èñëÿåì, 2. (Φ) ∗ ( θ f , γ 1 , . . . , γ k ) = η - âû÷èñëÿåì, 3. [ J (Φ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . , (Ψ) ∗ k ]( θ f , α ) = (Φ) ∗ ( θ f , (Ψ 1 ) ∗ ( θ f , α ) , . . . , (Ψ) ∗ k ( θ f , α )) - ïî÷òè àêñèîìà, 4. (Φ) ∗ ( y , x 1 , . . . , x k ) = (Φ) ∗ ( y , x 1 , . . . , x k ) - àêñèîìà, 5. (Φ) ∗ ( θ f , (Ψ 1 ) ∗ ( θ f , α ) , . . . , (Ψ) ∗ k ( θ f , α )) = (Φ) ∗ ( θ f , γ 1 , . . . , γ k ) - èç 1,4, 6. (Φ) ∗ ( θ f , (Ψ 1 ) ∗ ( θ f , α ) , . . . , (Ψ) ∗ k ( θ f , α )) = η - èç 2,5, 7. [ J (Φ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . , (Ψ) ∗ k ]( θ f , α ) = η - èç 3,6, [Ðàâåíñòâà 1-7 äîêàçûâàþòÿ â èñ÷èñëåíèè CalcEq ] 8. [ J ΦΨ 1 , . . . , Ψ k ]( α ) = Φ(Ψ 1 ( α ) , . . . , Ψ k ( α )) - ïî÷òè àêñèîìà, 9. Ψ 1 ( α ) = (Ψ 1 ) ∗ ( θ f , α ) , . . . , Ψ k ( α ) = (Ψ k ) ∗ ( θ f , α ) - èíäóêöèîíîå ïðåäïîëîæåíèå, 10. Φ( x 1 , . . . , x k ) = Φ( x 1 , . . . , x k ) - àêñèîìà, 11. Φ(Ψ 1 ( α ) , . . . , Ψ 1 ( α )) = Φ((Ψ 1 ) ∗ ( θ f , α ) , . . . , (Ψ k ) ∗ ( θ f , α )) - èç 9,10, 12. Φ((Ψ 1 ) ∗ ( θ f , α ) , . . . , (Ψ k ) ∗ ( θ f , α )) = Φ( γ 1 , . . . , γ k ) - èç 1,10, 13. Φ( γ 1 , . . . , γ k ) = (Φ) ∗ ( θ f , γ 1 , . . . , γ k ) - èíäóêöèîííîå ïðåäïîëîæåíèå,(ïîëàãàåì â ( a ) β i = γ i ) 14. Φ( γ 1 , . . . , γ k ) = η - èç 2,13, 28 15. Φ((Ψ 1 ) ∗ ( θ f , α ) , . . . , Ψ k ) ∗ ( θ f , α )) = η - èç 12,14, 16. Φ(Ψ 1 ( α ) , . . . , Ψ 1 ( α )) = η - èç 11,15, 17. [ J ΦΨ 1 , . . . , Ψ k ]( α ) = η - èç 8,16, 18. [ J ΦΨ 1 , . . . , Ψ k ]( α ) = [ J (Φ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . , (Ψ) ∗ k ]( θ f , α ) - èç 7,17 19. [ J ΦΨ 1 , . . . , Ψ k ]( α ) = ([ J ΦΨ 1 , . . . , Ψ k ]) ∗ ( θ f , α ) - èç 18 - êâàçèâûîä ïðè èòåðïðåòàöèîííîì ìíîæåñòâå A , ò.å. â èñ÷èñëåíèè CalcEq A . b ). Èíäóêöèîííîå ïðåäïîëîæåíèå . Ïóñòü òåîðåìà âåðíà äëÿ ôóíêòîðîâ: Φ , Ψ 1 , . . . , Ψ k , äîêàæåì òåîðåìó äëÿ ôóíêòîðà f ⇋ [ R ΦΨ 1 , . . . , Ψ k ] . Ïî èíäóêöèîííîìó ïðåäïîëîæåíèþ èìååì: äëÿ ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , äëÿ ôóíêòîðà Φ , âåðíî ∀ θ ⊇ θ SimpleFw , Φ ,α, A , è A ⊢ Φ( α ) = (Φ) ∗ ( θ, α ) ; äëÿ ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , β, γ , äëÿ ôóíê- òîðà Ψ 1 , âåðíî ∀ θ ⊇ θ SimpleFw , Ψ 1 ,α,β,γ, A , A ⊢ Ψ 1 ( α ) = (Ψ 1 ) ∗ ( θ, α, β, γ ) , . . . , äëÿ ôóíêòîðà Ψ k , âåðíî ∀ θ ⊇ θ SimpleFw , Ψ k ,α,β,γ, A , A ⊢ Ψ k ( α ) = (Ψ k ) ∗ ( θ, α, β, γ ) . Äàëåå äîêàçàòåëüñòâî áóäåò ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåíèåþ àðãóìåíòíîãî ñëîâà. Áàçà èíäóêöèè. Äîêàæåì, ÷òî äëÿ ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , âåðíî ∀ θ f ⊇ θ SimpleFw , f ,α, Λ A ⊢ [ R ΦΨ 1 , . . . , Ψ k ]( α, Λ) = ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( θ f , Λ) . Cîãëàñíî îïðåäåëåíèÿ ïðåîáðàçîâàíèÿ" ∗ "èìååì: ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ = [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ] , ñëåäîâà- òåëüíî, íåîáõîäèìî äîêàçàòü ∀ θ f ⊇ θ SimpleFw , f ,α, Λ A ⊢ [ R ΦΨ 1 , . . . , Ψ k ]( α, Λ) = [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , Λ) . Ñëåäóþùàÿ ïîñëåäîâàòåëüíîñòü ðàâåíñòâ: 1. [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( y , x, Λ) = (Φ) ∗ ( y , x ) - àêñèîìà, 2. [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, Λ) = (Φ) ∗ ( θ f , α ) - èç 1; [Ðàâåíñòâà 1,2 äîêàçûâàþòÿ â èñ÷èñëåíèè CalcEq ] [Ó÷èòûâàÿ, ÷òî θ f ⊇ θ SimpleFw , f ,α, Λ ⊇ θ SimpleFw , Φ ,α, A , ïîëó÷èì ] 3. [ R ΦΨ 1 , . . . , Ψ k ]( α, Λ) = Φ( α ) - ïî÷òè àêñèîìà, 4. Φ( α ) = (Φ) ∗ ( θ f , α ) - èíäóêöèîííîå ïðåäïîëîæåíèå, 5. [ R ΦΨ 1 , . . . , Ψ k ]( α, Λ) = (Φ) ∗ ( θ f , α ) - èç 3,4, 6. [ R ΦΨ 1 , . . . , Ψ k ]( α, Λ) = [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , , α, Λ) -èç 2,5 7. [ R ΦΨ 1 , . . . , Ψ k ]( α, Λ) = ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( θ f , , α, Λ) èç 6 - êâàçèâûâîä ïðè èíòåðïðåòàöèîííîì ìíîæå- ñòâå A . Èíäóêöèîííûé ïåðåõîä. Ïî èíäóêöèîííîìó ïðåäïîëîæåíèþ èìååì: äëÿ ìíîæåñòâà àðãóìåíòíûõ 29 ñëîâ A , äëÿ ëþáîé ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , β , âåðíî ∀ θ f ⊇ θ SimpleFw , f ,α,β A ⊢ [ R ΦΨ 1 , . . . , Ψ k ]( α, β ) = [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , , α, β ) ; á) Äëÿ ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , β, γ , âåðíî: ∀ θ ⊇ θ SimpleFw , Ψ i ,α,β,γ, A , A ⊢ Ψ i ( α, β, γ ) = (Ψ i ) ∗ ( θ, α, β, γ ) . Òðåáóåòñÿ äîêàçàòü, ÷òî äëÿ ôóíêòîðà f , äëÿ ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , βa i , âåðíî ∀ θ f ⊇ θ SimpleFw , f ,α,βa i A ⊢ [ R ΦΨ 1 , . . . , Ψ k ]( α, βa i ) = [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , , α, βa i ) . Ñîñòàâèì ôóíêöèîíàëüíîå ñëîâî ñîãëàñíî ìíîæåñòâàì: ( A + ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,βa i = ( A + ) P Ψ i ,α 1 ,...,α n ,β,γ S ( A + ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,β , ( A − ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,βa i = ( A − ) P Ψ i ,α 1 ,...,α n ,β,γ S ( A − ) P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,αn,β , ãäå P [ R ΦΨ 1 ,..., Ψ k ] ,α 1 ,...,α n ,βa i - ïðîñòîå âû÷èñëåííèå ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ k ] íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , βa i , P Ψ i - ïðîñòîå âû÷èñëåíèå ôóíêòîðà Ψ i íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , β, γ , ïîëó÷èì θ SimpleFw , f ,α,βa i , òîãäà θ SimpleFw , f ,α,βa i ⊇ θ SimpleFw , Ψ i ,α,β,γ, A , θ SimpleFw , f ,α,βa i ⊇ θ SimpleFw , f ,α,β . Âîçüì¼ì θ f ⊇ θ SimpleFw , f ,α,βa i Ñëåäóþùàÿ ïîñëåäîâàòåëüíîñòü ðàâåíñòâ: 1. [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, β ) = γ - âû÷èñëÿåì, 2. (Ψ i ) ∗ ( θ f , α, β, γ ) = η i - âû÷èñëÿåì, 3. [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, βa i ) = (Ψ i ) ∗ ( θ f , α, β, [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, β )) - ïî÷òè àêñèîìà, 4. (Ψ i ) ∗ ( y , x, z, u ) = (Ψ i ) ∗ ( y , x, z, u ) - àêñèîìà, 5. (Ψ i ) ∗ ( θ f , α, β, [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, β )) = (Ψ i ) ∗ ( θ f , α, β, γ )) - èç 1,4, 6. (Ψ i ) ∗ ( θ f , α, β, [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, β )) = η i - èç 2,5, 7. [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, βa i ) = η i - èç 3,6, [Ðàâåíñòâà 1-7 äîêàçûâàþòÿ â èñ÷èñëåíèè CalcEq ] 8. [ R ΦΨ 1 , . . . , Ψ k ]( α, βa i ) = Ψ i ( α, β, [ R ΦΨ 1 , . . . , Ψ k ]( α, β )) - ïî÷òè àêñèîìà, 9. [ R ΦΨ 1 , . . . , Ψ k ]( α, β ) = [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , , α, β ) - èíäóêöèîííîå ïðåäïîëîæåíèå, 10. Ψ i ( x, u, v ) = Ψ i ( x, u, v ) - àêñèîìà, 11. Ψ i ( α, β, [ R ΦΨ 1 , . . . , Ψ k ]( α, β )) = Ψ i ( α, β, [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, β )) - èç 9,10, 12. Ψ i ( α, β, [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , α, β )) = Ψ i ( α, β, γ ) - èç 1,10, 13. Ψ i ( α, β, γ ) = (Ψ i ) ∗ ( θ f , α, β, γ ) - èíäóêöèîíîå ïðåäïîëîæåíèå, 14. Ψ i ( α, β, [ R ΦΨ 1 , . . . , Ψ k ]( α, β )) = Ψ i ( α, β, γ ) - èç 11,12, 15. Ψ i ( α, β, [ R ΦΨ 1 , . . . , Ψ k ]( α, β )) = (Ψ i ) ∗ ( θ f , α, β, γ ) - èç 13,14, 30 16. Ψ i ( α, β, [ R ΦΨ 1 , . . . , Ψ k ]( α, β )) = η i - èç 2,15, 17. [ R ΦΨ 1 , . . . , Ψ k ]( α, βa i ) = η i - èç 8,16, 18. [ R ΦΨ 1 , . . . , Ψ k ]( α, βa i ) = [ R (Φ) ∗ (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( θ f , , α, βa i ) - èç 7,17 19. [ R ΦΨ 1 , . . . , Ψ k ]( α, βa i ) = ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( θ f , , α, βa i ) - èç 18 - êâàçèâûâîä ïðè èòåðïðåòàöèîííîì ìíîæåñòâå A , ò.å. â èñ÷èñëåíèè CalcEq A . Îñòàëüíûå àêñèîìû ðåêóðñèè(15,16,17,20) ðàññìàòðèâàþòñÿ àíàëîãè÷íî. Ñëåäñòâèå 5.2 . 1. Äëÿ ëþáîãî n - ìåñòíîãî ôóíêòîðà Φ àëôàâèòà L ( U ) ∀ A , ∀ α âåðíî A ⊢ [Φ( α ) = Φ ∗ (Θ Φ ( α ) , α )] ( WordM A | = [Φ( α ) = Φ ∗ (Θ Φ ( α ) , α )]) . 2. Äëÿ ëþáîãî n - ìåñòíîãî ôóíêòîðà Φ àëôàâèòà L ( U ) ∀ A , ∀ α ∀ θ ⊇ Θ Φ ( α ) , âåðíî A ⊢ [Φ( α ) = Φ ∗ ( θ, α )] ( WordM A | = [Φ( α ) = Φ ∗ ( θ, α )]) , 3. Äëÿ ëþáîãî n - ìåñòíîãî ôóíêòîðà Φ àëôàâèòà L ( U ) ∀ A , ∀ α, β ∀ θ ⊇ Θ Φ ( α ) , âåðíî A ⊢ Φ( α ) = β ⇔⊢ Φ ∗ ( θ, α ) = β ] ( WordM A | = Φ( α ) = β ⇔ WordM | = Φ ∗ ( θ, α ) = β ) . 4. Äëÿ ëþáîãî n - ìåñòíîãî ôóíêòîðà Φ àëôàâèòà L ( U ) ∀ α âåðíî WordM A | = (Θ Φ ) ∗ (Θ Φ ( α ) , α ) = Θ Φ ( α ) - êàê ðàâåíñòâî ñëîâ(íàèìåíüøàÿ íåïîäâèæíàÿ òî÷êà: ∀ θ ⊇ Θ Φ ( α )[(Θ Φ ) ∗ ( θ, α ) ⊆ θ ] { ∀ θ ⊇ Θ Φ ( α )[Θ Φ ( α ) = (Θ Φ ) ∗ ( θ, α )] }, ïðè÷¼ì, åñëè Θ Φ ( α ) ̸ = Λ , òîãäà dom (Θ Φ ( α )) = dom ((Θ Φ ) ∗ ( θ, α )) äëÿ ëþáîãî ëþáîãî ôóíêöèîíàëüíîãî ñëîâà θ . Ñì. òåîðåìó 4.4 . Çàìå÷àíèå . Ïóñòü Φ - ïðîèçâîëüíûé n - ìåñòíûé ôóíêòîð àëôàâèòà L ( U ) . Ïðîñòîå âû÷èñëåíèå ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α ìîæíî ðàçëîæèòü íà äâà âû÷èñëåíèÿ: ïðîñòîå âû÷èñëåíèå ôóíêòîðà Θ Φ àëôàâèòà L ( U ) íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α è çàòåì ïðîñòîå âû÷èñëåíèå ôóíêòîðà (Φ) ∗ àëôàâèòà L íà ïîñëåäîâàòåëüíîñòè àðãóìåíòûõ ñëîâ Θ Φ ( α ) , α . Ïðè÷¼ì îá- ëàñòü îïðåäåëåíèÿ ôóíêöèîíàëüíîãî ñëîâà Θ Φ ( α ) ñîñòîèò èç òåõ è òîëüêî òåõ àðãóìåíòíûõ ñëîâ, êîòîðûå áûëè èñïîëüçîâàíû â ïðîñòîì âû÷èñëåíèè ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α è ïðè ëþáîì ðàñøèðåíèè ôóíêöèîíàëüíîãî ñëîâà Θ Φ ( α ) ⊆ θ , íåîáÿçàòåëüíî ñîãëàñîâàííîãî ñ îðàêóëüíûì ìíîæåñòâîì A , ðåçóëüòàò ïðîñòîãî âû÷èñëåíèÿ ôóíêòîðà (Φ) ∗ íà ïîñëåäîâàòåëüíîñòè Θ Φ ( α ) , α ñîâïàä¼ò ñ ðåçóëüòàòîì ïðîñòîãî âû÷èñëåíèÿ ýòîãî æå ôóíêòîðà (Φ) ∗ íà ïîñëåäîâàòåëüíîñòè θ, α (àíàëîã "Ïðèí- öèïà èñïîëüçîâàíèÿ"(" Use Principle ") îðàêóëüíûõ âû÷èñëåíèé, íàïðèìåð íà ìàøèíàõ Òüþðèíãà ) è êàê ðàíåå îòìå÷àëîñü (Θ Φ ) ∗ ( θ, α ) ⊆ θ - " Use Principle "ñûãðàåò â äàëüíåéøåì âàæíóþ ðîëü ïðè ïåðå- íåñåíèè(ðàñïðîñòðàíåíèè) ýòîãî ôóíäàìåíòàëüíîãî ïîíÿòèÿ, ñâÿçàííîå ñ âû÷èñëåíèÿìè â ñòàíäàðòíîé ìîäåëè, íà íåñòàíäàðòíûå ìîäåëè. Åñëè â ïðîñòîì âû÷èñëåíèè ôóíêòîðà Φ íà ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α êàæäóþ èí- òåðïðåòàöèîííóþ àêñèîìó âèäà U ( α ) = Λ , çàìåíèòü íà ðàâåíñòâî âèäà G ( | α | 2 α 22 , α ) = Λ (çàìåíèòü íà 31 ðàâåíñòâî âèäà G ( θ, α ) = Λ , ãäå | α | 2 α 22 ⊂ θ ), àêñèîìó âèäà U ( α ) = a 1 , çàìåíèòü íà ðàâåíñòâî âèäà G ( | α | 2 αa 1 22 , α ) = a 1 (çàìåíèòü íà ðàâåíñòâî âèäà G ( θ, α ) = a ! , ãäå | α | 2 αa 1 22 ⊂ θ ), òî ïîëó÷åííàÿ ïîñëå- äîâàòåëüíîñòü ðàâåíñòâ áóäåò êâàçèâûâîäîì, êîòîðûé íå áóäåò ñîäåðæàòü èíòåðïðåòàöèîííûõ àêñèîì, ïðè÷¼ì ýòîò êâàçèâûâîä ìîæíî ëåãêî ïðåîáðàçîâàòü â âûâîä àëôàâèòà L , çàìåíèâ óêàçàííûå ðàâåíñòâà íà èõ, íàïðèìåð, ïðîñòûå âûâîäû. Òåîðåìà 5.3 . Äëÿ ïðîèçâîëüíîãî n - ìåñòíîãî ôóíêòîðà Φ àëôàâèòà L ( U ) , äëÿ ïðîèçâîëüíûõ àðãó- ìåíòíûõ ñëîâ α , β, γ , åñëè WordM A | = (Θ Φ ) ∗ ( β, α ) = γ , òîãäà γ - ôóíêöèîíàëüíîå ñëîâî. Äîêàçàòåëüñòâî ïðîâîäèòñÿ èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà Φ . Òåîðåìà 5.4 . Äëÿ ïðîèçâîëüíîãî n - ìåñòíîãî ôóíêòîðà Φ àëôàâèòà L ( U ) , äëÿ ïðîèçâîëüíîãî ìíî- æåñòâà àðãóìåíòíûõ ñëîâ A , âåðíî WordM A | = ∀ α ∀ β [Θ Φ ( α ) = β = ⇒ (Θ Φ ) ∗ ( β, α ) = β ∧ β ⊂ A ] . Äîêàçàòåëüñòâî . Äëÿ ïðîèçâîëüíûõ àðãóìåíòíûõ ñëîâ α, β èìååì WordM A | = Θ Φ ( α ) = β ⇐⇒ (Θ Φ ) ∗ (Θ Θ Φ ( α ) , α ) = β , ó÷èòûâàÿ òåîðåìó 4.4, ïîëó÷èì WordM A | = (Θ Φ ) ∗ ( β, α ) = β ∧ β ⊂ A . Òåîðåìà 5.5 . Ïóñòü äëÿ n - ìåñòíîãî ôóíêòîðà Φ àëôàâèòà L ( U ) , äëÿ àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , äëÿ ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , äëÿ ôóíêöèîíàëüíîãî ñëîâà β âåðíî WordM A | = (Θ Φ ) ∗ ( β, α 1 , . . . , α n ) ⊆ β ∧ β ⊂ A , òîãäà WordM A | = Θ Φ ( α 1 , . . . , α n ) ⊆ β . Äîêàçàòåëüñòâî . Äîêàçàòåëüñòâî ïðîâåä¼ì èíäóêöèåé ïî ïîñòðîåíèþ ôóíêòîðà Φ . Áàçà èíäóêöèè . Φ - èñõîäíûé ôóíêòîð, íàïðèìåð U , òîãäà Θ U = [ J cI 1 1 U ] , òîãäà (Θ U ) ∗ = [ J c ∗ I 2 1 [ J I 1 1 I 2 2 ] G ] , òîãäà WordM A | = (Θ U ) ∗ ( β, α ) = c ∗ ( β, α, G ( β, α )) = | α | 2 α G ( β, α )22 .( ⊢ c ∗ ( x, y, z ) = c ( y, z ) , ñì Çàìå÷àíèå ñòð.27). Ïóñòü WordM A | = (Θ U ) ∗ ( β, α ) ⊆ β è β ⊂ A , òîãäà | α | 2 α G ( β, α )22 ⊆ β . Ðàçáåð¼ì ñëó÷àè: a ) G ( β, α ) = Λ , òîãäà β ( α ) = Λ , òîãäà α ∈ A , òîãäà U ( α ) = Λ , òîãäà Θ U ( α ) = | α | 2 α 22 , ó÷èòûâàÿ, ÷òî (Θ U ) ∗ ( β, α ) = | α | 2 α G ( β, α )22 = | α | 2 α 22 ⊆ β , òîãäà Θ U ( α ) ⊆ β ; b ) G ( β, α ) = 1 , òîãäà β ( α ) = 1 , òîãäà α / ∈ A , òîãäà U ( α ) = 1 , òîãäà Θ U ( α ) = | α | 2 α 122 , ó÷èòûâàÿ, ÷òî (Θ U ) ∗ ( β, α ) = | α | 2 α G ( β, α )22 = | α | 2 α 122 ⊆ β , òîãäà Θ U ( α ) ⊆ β . Äëÿ îñòàëüíûõ èñõîäíûõ ôóíêòîðîâ äîêàçàòåëüñòâî äîñòàòî÷íî íàãëÿäíîå. Èíäóêöèîííîå ïðåäïîëîæåíèå 1). Ïóñòü òåîðåìà âåðíà äëÿ k - ìåñòíîãî ôóíêòîðà Φ , äëÿ n - ìåñòíûõ ôóíêòîðîâ Ψ 1 , . . . , Ψ k . Äîêàæåì òåîðåìó äëÿ ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] . Èìååì ⊢ Θ [ J ΦΨ 1 ,..., Ψ k ] ( x ) = Concat ([ J Θ Φ Ψ 1 , . . . , Ψ k ]( x ) , ( Concat (Θ Ψ 1 ( x ) , . . . , Concat (Θ Ψ k − 1 ( x ) , Θ Ψ k ( x ))) , . . . , ) , òîãäà ⊢ (Θ [ J ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x ) = Concat ([ J (Θ Φ ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( y , x ) , ( Concat ((Θ Ψ 1 ) ∗ ( y , x ) , . . . , Concat ((Θ Ψ k − 1 ) ∗ ( y , x ) , (Θ Ψ k ) ∗ ( y , x ))) , . . . , ) 6 , òîãäà 6 äëÿ óòî÷íåíèÿ, ñì. Ïðèëîæåíèå 32 WordM A | = (Θ [ J ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α ) = Concat ([ J (Θ Φ ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( β, α ) , ( Concat ((Θ Ψ 1 ) ∗ ( β, α ) , . . . , Concat ((Θ Ψ k − 1 ) ∗ ( β, α ) , (Θ Ψ k ) ∗ ( β, α ))) , . . . , ) . Ïóñòü (Θ [ J ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α ) ⊆ β ⊂ A , òîãäà [ J (Θ Φ ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . , (Ψ k ) ∗ ]( β, α ) = (Θ Φ ) ∗ ( β, (Ψ 1 ) ∗ ( β, α ) , . . . , (Ψ k ) ∗ ( β, α )) ⊆ β è (Θ Ψ 1 ) ∗ ( β, α ) ⊆ β, . . . , (Θ Ψ k ) ∗ ( β, α ) ⊆ β . Ïî èíäóêöèîííîìó ïðåäïîëîæåíèþ ïîëó÷èì Θ Φ ((Ψ 1 ) ∗ ( β, α ) , . . . , (Ψ k ) ∗ ( β, α )) ⊆ β , à òàêæå Θ Ψ 1 ( α ) ⊆ β, . . . , Θ Ψ k ( α ) ⊆ β , òîãäà Ψ 1 ( α ) = (Ψ 1 ) ∗ ( β, α ) , . . . , Ψ k ( α ) = (Ψ k ) ∗ ( β, α ) , òîãäà Θ Φ ((Ψ 1 ) ∗ ( β, α ) , . . . , (Ψ k ) ∗ ( β, α )) = Θ Φ (Ψ 1 ( α ) , . . . Ψ k ( α )) , òîãäà Θ Φ (Ψ 1 ( α ) , . . . Ψ k ( α )) ⊆ β , òîãäà Concat ([ J Θ Φ Ψ 1 , . . . , Ψ k ]( x ) , ( Concat (Θ Ψ 1 ( x ) , . . . , Concat (Θ Ψ k − 1 ( x ) , Θ Ψ k ( x ))) , . . . , ) ⊆ β , òîãäà Θ [ J ΦΨ 1 ,..., Ψ k ] ( α ) ⊆ β . Èíäóêöèîííîå ïðåäïîëîæåíèå 2). Ïóñòü òåîðåìà âåðíà äëÿ n - ìåñòíîãî ôóíêòîðà Φ , äëÿ n + 2 - ìåñòíûõ ôóíêòîðîâ Ψ 1 , . . . , Ψ k . Äîêàæåì òåîðåìó äëÿ n + 1 - ìåñòíîãî ôóíêòîðà [ R ΦΨ 1 , . . . , Ψ k ] . Èìååì: ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = Θ Φ ( x ) . ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, x n +1 a i ) = Concat (Θ Ψ i ( x, x n +1 , [ R ΦΨ 1 , . . . , Ψ k ]( x, x n +1 )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, x n +1 )) , òî- ãäà ⊢ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, Λ) = (Θ Φ ) ∗ ( y , x ) , ⊢ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, x n +1 a i ) = Concat ((Θ Ψ i ) ∗ ( y , x, x n +1 , ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, x n +1 )) , (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, x n +1 )) 7 , òîãäà c ) WordM A | = (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α, Λ) = (Θ Φ ) ∗ ( β, α ) , d ) WordM A | = (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α, γa i ) = = Concat ((Θ Ψ i ) ∗ ( β, α, γ, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( β, α, γ )) , (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α, γ )) . Ñëó÷àé ( c ) . Ïóñòü (Θ Φ ) ∗ ( β, α ) ⊆ β ⊂ A . Ïî èíëóêöèîííîìó ïðåäïîëîæåíèþ ïîëó÷èì WordM A | = Θ Φ ( α ) ⊆ β , òîãäà Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) ⊆ β . Ñëó÷àé ( d ) . Ïóñòü (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α, γa i ) ⊆ β ⊂ A , òîãäà Concat ((Θ Ψ i ) ∗ ( β, α, γ, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( β, α, γ )) , (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α, γ )) ⊆ β , òîãäà (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α, γ ) ⊆ β , òîãäà ïî èíäêöèîííîìó ïðåäïîëîæåíèþ, ïîëó÷èì Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, γ ) ⊆ β , òîãäà Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, γ ) = δ ⇐⇒ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α, γ )) = δ è δ ⊆ β . Äàëåå ó÷èòûâàÿ [ R ΦΨ 1 , . . . , Ψ k ])( α, γ )) = η ⇐⇒ ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ (Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, γ ) , α, γ )) = η è Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, γ ) ⊆ β , ïîëó÷èì [ R ΦΨ 1 , . . . , Ψ k ])( α, γ )) = η ⇐⇒ ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( β, α, γ ) , α, γ )) = η , òîãäà Concat ((Θ Ψ i ) ∗ ( β, α, γ, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( β, α, γ )) , (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( β, α, γ )) = = Concat ((Θ Ψ i ) ∗ ( β, α, γ, [ R ΦΨ 1 , . . . , Ψ k ]( α, γ )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, γ )) . Èìååì (Θ Ψ i ) ∗ ( β, α, γ, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( β, α, γ )) ⊆ β , òîãäà (Θ Ψ i ) ∗ ( β, α, γ, [ R ΦΨ 1 , . . . , Ψ k ]( α, γ )) ⊆ β , òîãäà ïî èíäóêöèîííîìó ïðåäïîëîæåíèþ, ïîëó÷èì Θ Ψ i ( α, γ, [ R ΦΨ 1 , . . . , Ψ k ]( α, γ )) ⊆ β , òîãäà 7 ñì. Ïðèëîæåíèå 33 Concat (Θ Ψ i ( α, γ, [ R ΦΨ 1 , . . . , Ψ k ]( α, γ )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, γ )) ⊆ β , òîãäà Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, γa i ) ⊆ β è β ⊂ A . Îñòàëüíûå àêñèîìû ðåêóðñèè(15,16,17,20) ðàññìàòðèâàþòñÿ àíàëîãè÷íî, òîãäà ∀ x ∀ y { Fw ( y ) ⇒ [[(Θ Φ ) ∗ ( y, x ) ⊆ y ∧ y ⊂ U ] ⇒ Θ Φ ( x ) ⊆ y ] } ∈ Th ( U ) (ñì. ñòð.39). Ñëåäñòâèå 5.6 . Äëÿ ïðîèçâîëüíîãî n - ìåñòíîãî ôóíêòîðà Φ , äëÿ ïðîèçâîëüíîé ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α 1 , . . . , α n , äëÿ ïðîèçâîëüíîãî ìíîæåñòâà àðãóìåíòíûõ ñëîâ A , ñóùåñòâóåò åäèíñòâåííîå ôóíêöèîíàëüíîå ñëîâî β , òàêîå, ÷òî âåðíî WordM A | = Θ Φ ( α 1 , . . . , α n ) = β ⇐⇒ WordM A | = (Θ Φ ) ∗ ( β, α 1 , . . . , α n ) = β ∧ β ⊂ U , òîãäà ∀ x ∃ ! y [(Θ Φ ) ∗ ( y, x ) = y ∧ y ⊂ U ] ∈ Th ( U ) , ∀ x ∃ ! y { [Θ Φ ( x ) = y ≡ [(Θ Φ ) ∗ ( y, x ) = y ∧ y ⊂ U ] } ∈ Th ( U ) . Ñëåäñòâèå 5.7 . Ïóñòü äëÿ n - ìåñòíîãî ôóíêòîðà Φ , äëÿ ïîñëåäîâàòåëüíîñòè àðãóìåíòíûõ ñëîâ α , äëÿ ôóíêöèîíàëüíîãî ñëîâà β âåðíî WordM | = (Θ Φ ) ∗ ( β, α ) = β , òîãäà ìîæíî ïîñòðîèòü òàêîå èíòåðïðåòàöè- îííîå ìíîæåñòâî B îðàêóëüíîãî ñèìâîëà U , ÷òî âåðíî WordM B | = [(Θ Φ ) ∗ ( β, α ) = β ∧ β ⊂ U ∧ Θ Φ ( α ) = β ] . Îãðàíè÷åííûå ôîðìóëû . Óíèâåðñàëüíîå ôóíêöèîíàëüíîå ñëîâî Îïðåäåëèì ïîíÿòèå îãðàíè÷åííîé ôîðìóëû Ψ è ñîïðîâîæäàþùåå ýòî ïîíÿòèå, ìíîæåñòâà, îáîçíà÷à- åìûå êàê Bwp Ψ , V wp Ψ . 1) Âñÿêàÿ áåñêâàíòîðíàÿ ôîðìóëà A ÿâëÿåòñÿ îãðàíè÷åííîé ôîðìóëîé, Bwp A = ∅ , V wp A = ∅ ; 2) Ïóñòü A - îãðàíè÷åííàÿ ôîðìóëà è z / ∈ Bwp A S V wp A , P ( x 1 , . . . , x n ) - ñëîâàðíûé ìíîãî÷ëåí, ïåðåìåííûe êîòîðîãî, íå ïðèíàäëåæàò ìíîæåñòâó Bwp A S V wp A , òîãäà ôîðìóëà B ⇌ ∃ z [ | z | ⩽ | P ( x 1 , . . . , x n ) | ∧ A ] èëè B ⇌ ∀ z [ | z | ⩽ | P ( x 1 , . . . , x n ) | ⊃ A ] - îãðàíè÷åííàÿ ôîðìóëà, Bwp B = { z } S Bwp A , V wp B = { x 1 , . . . , x n } S V wp A Òàêóþ ôîðìóëó áóäåì îáîçíà÷àòü â âèäå ∃ | P ( x 1 ,...,x n ) | z A èëè ∀ | P ( x 1 ,...,x n ) | z A . Îãðàíè÷åííàÿ ôîðìóëà A íàçûâàåòñÿ ∃ ( ∀ ) îãðàíè÷åííîé ôîðìóëîé ñ ïàðàìåòðàìè èëè áåç íèõ, åñëè îíà èìååò âèä ∃ | P 1 ( x ) | z 1 , . . . ∃ | P k ( x ) | z k B ( z 1 , . . . z k , x ; y 1 , . . . , y k ) ( ∀ | P 1 ( x ) | z 1 , . . . ∀ | P k ( x ) | z k B ( z 1 , . . . z k , x ; y 1 , . . . , y k ) ), ãäå B ( z 1 , . . . z k , x ; y 1 , . . . , y k ) - áåñêâàíòîðíàÿ ôîðìóëà. Çàìå÷àíèå . ∀ Φ ∀ α ∀ P ( x ) : 1 . Word A | = ∃ | P ( α ) | u Φ( α, u ) = Λ ⇔ ∃ | P ( α ) | u (Φ) ∗ (Θ Φ ( α, u ) , α, u ) = Λ , 2 . Word A | = ∀ | P ( α ) | u Φ( α, u ) = Λ ⇔ ∀ | P ( α ) | u (Φ) ∗ (Θ Φ ( α, u ) , α, u ) = Λ , ∀ β : 3 . Word A | = ∀ | P ( α ) | u Θ Φ ( α, u ) ⊆ β ⇒ {∃ | P ( α ) | u Φ( α, u ) = Λ ⇔ ∀ θ [ β ⊆ θ ⇒ ∃ | P ( α ) | u (Φ) ∗ ( θ, α, u ) = Λ] } , 4 . Word A | = ∀ | P ( α ) | u Θ Φ ( α, u ) ⊆ β ⇒ {∀ | P ( α ) | u Φ( α, u ) = Λ ⇔ ∀ θ [ β ⊆ θ ⇒ ∀ | P ( α ) | u (Φ) ∗ ( θ, α, u ) = Λ] } . Ðàññìîòðèì ñëåäóþùóþ ñëîâàðíóþ ôóíêöèþ: Order( α ) - òàêîå ñëîâî β , ÷òî ÷èñëî ñëîâ ïðåäøåñòâóþ- ùèõ ñëîâó β â ëåêñèêîãðàôè÷åñêîì óïîðÿäî÷èâàíèè, ðàâíî | α | .  [6 ñ. 217] äîêàçàíî, ÷òî ýòà ñëîâàðíàÿ 34 ôóíêöèÿ ÿâëÿåòñÿ ñëîâàðíîé ïðèìèòèâíî ðåêóðñèâíîé ôóíêöèåé, òîãäà äëÿ ñëîâàðíîé ôóíêöèè Order ñóùåñòâóåò òàêîé îäíîìåñòíûé ôóíêòîð Order àëôàâèòà L , ÷òî âåðíî: ∀ α, β [Order( α ) = β ⇔ ⊢ Order ( α ) = β ] . Âûïèøåì îïðåäåëÿþùèå ðàâåíñòâà äëÿ ôóíêòîðà Order : 1). Order (Λ) = Λ ; 2). Order ( S i ( α )) = R ( Order ( α )) , ãäå a ). R (Λ) = a 1 ; b ). R ( S i ( α )) = S i +1 ( α ) , ãäå 1 ≤ i < p ;  c). R ( S p ( α )) = S 1 ( R ( α )) . Çàìå÷àíèå. Äëÿ êàæäîãî ìíîæåñòâà p - àëôàâèòíûõ ñëîâ, áóäåò ñâîé p - àëôàâèòíûé ôóíêòîð Order p . Èç êîíòåêñòà áóäåò ïîíÿòíî êàêîé p - àëôàâèòíûé ôóíòîð èìååòñÿ ââèäó. Ôóíêòîð Order îáëà- äàåò ñâîéñòâîì: 1. WordM | = ∀ αβ [ | α | = | β | ⊃ Order ( α ) = Order ( β )] ; 2. ïðè k ≥ 2 ∧ p ≥ 2 âåðíî WordM | = | Order p ( k ) | < k ; 3. Äëÿ êàæäîãî p ≥ 2 âåðíî WordM | = Order p ( p n +1 − 1 p − 1 ) = 1 , . . . , 1 | {z } n +1 − ðàç ; 4. Äëÿ êàæäîãî p ≥ 2 âåðíî WordM | = ∀ x [ | x | ≥ 2 ⇒ Order p ( p | x | ) = ( p − 1) , . . . , ( p − 1) | {z } | x |− 1 − ðàç p ] ; 5. Ñîãëàñíî îïðåäåëÿþùèì ðàâåíñòâàì, êîòîðûå ïðèâåäåíû â [6 ñ. 217], ñëåäóåò, ÷òî ýòîò ôóíêòîð ïðèíàäëåæèò PPr , ò.å. Order ∈ PPr . Èç (3) ïîëó÷èì WordM | = Order p ( p | P ( x ) | +1 − 1 p − 1 ) = 1 , . . . , 1 | {z } | P ( x ) | +1 − ðàç . Åñëè äëÿ àðãóìåíòíûõ ñëîâ α, β âåðíî WordM | = Order ( α ) = β , òî àðãóìåíòíîå ñëîâî | α | - íàòóðàëü- íîå ÷èñëî, áóäåì íàçûâàòü ã¼äåëåâûì íîìåðîì àðãóìåíòíîãî ñëîâà β è îáîçíà÷àòü ⌜ β ⌝ = | α | . Î÷åâèäíî, ÷òî äëÿ ëþáîãî àðãóìåíòíîãî ñëîâà β ñóùåñòâóò òàêîå íàòóðàëüíîå ÷èñëî α , ÷òî âåðíî ⌜ β ⌝ = α ( WordM | = Order ( α ) = β ), òîãäà WordM | = ⌜ ( p − 1) , . . . , ( p − 1) | {z } | x |− 1 − ðàç p ⌝ = p | x | , ïðè | x | ≥ 2 . Ñ êàæäûì n ≥ 1 - ìåñòíûì ôóíêòîðîì Φ ñâÿæåì òàêîé ôóíêòîð, îáîçíà÷àåìûé êàê Θ Φ , ∀ ∀ ∀ , óäîâëåòâî- ðÿþùèé ñëåäóþùèì îïðåäåëÿþùèì ðàâåíñòâàì: Θ Φ , ∀ ∀ ∀ ( x, Λ) = Θ Φ ( x, Λ) ; Θ Φ , ∀ ∀ ∀ ( x, S k ( x n )) = Concat (Θ Φ , ∀ ∀ ∀ ( x, x n ) , Θ Φ ( x, Order ( S k ( x n )))) . Âåðíî: 1. Åñëè Φ ∈ PPr ⇒ Θ Φ , ∀ ∀ ∀ ∈ PPr ; 2. WordM A | = ∀ α, β, x [ | α | = | β | ⇒ Θ Φ , ∀ ∀ ∀ ( x, α ) = Θ Φ , ∀ ∀ ∀ ( x, β )] ; 35 3. WordM A | = ∀ x ∀ | z | y [Θ Φ ( x, y ) ⊆ Θ Φ , ∀ ∀ ∀ ( x, p | z | +1 − 1 p − 1 )] . Îïðåäåëèì óíèâåðñàëüíîå ôóíêöèîíàëüíîå ñëîâî, îáîçíà÷àåìîå êàê Θ U . Îïðåäåëÿþùèå ðàâåíñòâà: Θ U (Λ) = Λ , Θ U ( αa i ) = Concat (Θ U ( α ) , c ( Order ( α ) , U ( Order ( α )))) . Ñîãëàñíî îïðåäåëåíèÿ, ôóíêòîð Θ U ïðèíàäëå- æèò PPr àëôàâèòà L ( U ) , ó÷èòûâàÿ Concat ( u, c ( v, w )) ⇋ u | v | 2 vw 22 , ïîëó÷èì áîëåå íàãëÿäíóþ çàïèñü âèäà Θ U ( αa i ) = Θ U ( α ) | Order ( α ) | Order ( α ) U ( Order ( α ))22 . Èìååì: (Θ U ) ∗ ( y , Λ) = Λ , (Θ U ) ∗ ( y , αa i ) = Concat ((Θ U ) ∗ ( y , α ) , c ( Order ( α, G ( y , Order ( α )))) , ∀ αβ ∀ θ ⊇ Θ U ( α ) WordM A | = [Θ U ( α ) = β ⇐⇒ (Θ U ) ∗ ( θ, α ) = β ] , â ÷àñòíîñòè WordM A | = ∀ x ∀ y [Θ U ( x ) = y ⇐⇒ (Θ U ) ∗ (Θ U ( x ) , x ) = y ] . Çàìå÷àíèå . Äëÿ êàæäîãî p ≥ 2 , îïðåäåëÿåòñÿ ñâî¼ p - àëôàâèòíîå óíèâåðñàëüíîå ôóíêöèîíàëüíîå ñëîâî Θ U . Óòâåðæäåíèå 5.7 . Ïóñòü β - òàêîå ôóíêöèîíàëüíîå ñëîâî, ÷òî äëÿ íåêîòîðîãî ñëîâà α âåðíî WordM A | = (Θ U ) ∗ ( β, α ) ⊆ β ⊂ A , òîãäà WordM A | = Θ U ( α ) ⊆ β . Èìååì ∀ x ∀ y { Fw ( y ) ⇒ [[(Θ U ) ∗ ( y, x ) ⊆ y ∧ y ⊂ U ] ⇒ Θ U ( x ) ⊆ y ] } ∈ Th ( U ) . Âåðíî: 1. WordM A | = ∀ α, β [ | α | = | β | ⇐⇒ Θ U ( α ) = Θ U ( β )] ; 2. WordM A | = ∀ α, β [ | α | ≤ | β | ⇐⇒ Θ U ( α ) ⊆ Θ U ( β )] . 3. WordM A | = ∀ x {| x | ≤ | y | ⇒ [Θ U ( p | y | +1 − 1 p − 1 )]( x ) = Λ ⇐⇒ x ∈ A } , ãäå A - ìíîæåñòâî p ≥ 2 - àëôàâèòíûõ àðãóìåíòíûõ ñëîâ; 4. WordM A | = ∀ α [Θ U ( α ) = Θ Θ U ( α )] , ïðè èñïîëüçîâàíèè ïðàâèëà Ãóäñòåéíà, ìîæíî äîêàçàòü, ÷òî ⊢ Θ U ( x ) = Θ Θ U ( x ) . Ïóñòü çàäàí n + m - ìåñòíûé( n ≥ 1 , m ≥ 1 ) p - àëôàâèòíûé ôóíêòîð Φ ∈ PPr , èíòåðïðåòàöèîííîå p - àëôàâèòíîå ìíîæåñòâî àðãóìåíòíûõ ñëîâ A . Äëÿ ýòîãî ôóíêòîðà ñóùåñòâóåò òàêîé ñëîâàðíûé ìíîãî÷ëåí P ( x 1 , . . . , x n ; y 1 , . . . , y m ) , ÷òî äëÿ ∀ α 1 , . . . , α n ; β 1 , . . . , β m , â ïðîñòîì âû÷èñëåíèè ôóíêòîðà Φ íà α, β , âñå îïðîøåííûå ñëîâà èìåþò äëèíó, íå ïðåâîñõîäÿùóþ | P ( α, β ) | , òîãäà Θ Φ ( α, β ) ⊆ Θ U ( p | P ( α,β ) | +1 − 1 p − 1 ) . Äàëåå, ïóñòü | γ 1 | ≤ | P 1 ( α ) | , . . . , | γ m | ≤ | P m ( α ) | , òîãäà â ïðîñòîì âû÷èñëåíèè ôóíêòîðà Φ íà α, γ , âñå îïðîøåííûå ñëîâà èìåþò äëèíó, íå ïðåâîñõîäÿùóþ | P ( α, P 1 ( α ) , . . . , P m ( α )) | , òîãäà ïîëó÷èì 36 Θ Φ ( α, γ ) ⊆ Θ U ( p | P ( α, P 1 ( α ) ,..., P m ( α )) | +1 − 1 p − 1 ) . Äàëåå, ðàññìîòðèì ôîðìóëó âèäà ∃ | P 1 ( α ) | z [Φ( α, z ) = Λ] , òîãäà äëÿ ëþáîãî ñëîâà γ , òàêîãî, ÷òî | γ | ≤ | P 1 ( α ) | , âåðíî Θ Φ ( α, γ ) ⊆ Θ U ( p | P ( α, P 1 ( α )) | +1 − 1 p − 1 ) , òîãäà WordM A | = {∃ | P 1 ( α ) | z Φ( α, z ) = Λ ⇔ ∃ | P 1 ( α ) | z (Φ) ∗ (Θ U ( p | P ( α, P 1 ( α )) | +1 − 1 p − 1 ) , α, z ) = Λ } . Àíàëîãè÷íî ðàññóæ- äàÿ, ïîëó÷èì WordM A | = {∃ | P 2 ( β ) | z 2 ∃ | P 1 ( β ) | z 1 Φ( β, z 1 , z 2 ) = Λ ⇔ ∃ | P 2 ( β ) | z 2 ∃ | P 1 ( β ) | z 1 (Φ) ∗ (Θ U ( p | P ( β, P 1 ( β ) , P 2 ( β )) | +1 − 1 p − 1 ) , β, z 1 , z 2 ) = Λ } . Óòâåðæäåíèå 5.8 Ïóñòü n + m - ìåñòíûé( n ≥ 1 , m ≥ 1 ). p - àëôàâèòíûé ôóíêòîð Φ ∈ PPr , èíòåð- ïðåòàöèîííîå p - àëôàâèòíîå ìíîæåñòâî àðãóìåíòíûõ ñëîâ A è ñëîâàðíûå ìíîãî÷ëåíû P 1 ( x ) , . . . , P m ( x ) , òîãäà ìîæíî ïîñòðîèòü òàêîé ñëîâàðíûé ìíîãî÷ëåí P ( x ) , ÷òî WordM A | = {∃ | P 1 ( x ) | z 1 . . . ∃ | P m ( x ) | z m Φ( x, z 1 , . . . z m ) = Λ ⇔ ∃ | P 1 ( x ) | z 1 , . . . , ∃ | P m ( x ) | z m (Φ) ∗ (Θ U ( p | P ( x ) | +1 − 1 p − 1 ) , x, z 1 , . . . , z m ) = Λ } . Àíàëîãè÷íî. Óòâåðæäåíèå 5.9 Ïóñòü n + m - ìåñòíûé( n ≥ 1 , m ≥ 1 ), p - àëôàâèòíûé ôóíêòîð Φ ∈ PPr , èíòåð- ïðåòàöèîííîå p - àëôàâèòíîå ìíîæåñòâî àðãóìåíòíûõ ñëîâ A è ñëîâàðíûå ìíîãî÷ëåíû P 1 ( x ) , . . . , P m ( x ) , òîãäà ìîæíî ïîñòðîèòü òàêîé ñëîâàðíûé ìíîãî÷ëåí P ( x ) , ÷òî WordM A | = {∀ | P 1 ( x ) | z 1 . . . ∀ | P m ( x ) | z m Φ( x, z 1 , . . . z m ) = Λ ⇔ ∀ | P 1 ( x ) | z 1 , . . . , ∀ | P m ( x ) | z m (Φ) ∗ (Θ U ( p | P ( x ) | +1 − 1 p − 1 ) , x, z 1 , . . . , z m ) = Λ } . Part VI Ñëîæíîñòíûå êëàññû è ýëåìåíòàðíàÿ òåîðèÿ ìîäåëåé Ìíîæåñòâî B n -îê àðãóìåíòíûõ ñëîâ, ïðè çàäàííîé èíòåðïðåòàöèè îðàêóëüíîãî ñèìâîëà U ìíî- æåñòâîì àðãóìåíòíûõ ñëîâ A , íàçûâàåòñÿ ïîëèíîìèàëüíûì, åñëè ñóùåñòâóåò(ìîæíî ïîñòðîèòü) òàêóþ áåñêâàíòîðíóþ ôîðìóëó B , êîòîðàÿ ïîñòðîåíà èç ôóíêòîðîâ, ïðèíàäëåæàùèõ PPr , ÷òî âåðíî ∀ ∀ ∀ α [ WordM A | = B ( α ) ⇐⇒ α ∈ B ] . Êëàññ âñåõ ïîëèíîìèàëüíûõ ìíîæåñòâ îòíîñèòåëüíî ìíîæåñòâà A - àðãóìåíòíûõ ñëîâ áóäåì îáîçíà- ÷àòü P A ( U ) . Ýòîò êëàññ ñëîâàðíûõ ìíîæåñòâ çàìêíóò îòíîñèòåëüíî áóëåâûõ îïåðàöèé: ïåðåñå÷åíèÿ, îáú- åäèíåíèÿ, äîïîëíåíèÿ è íàâåøèâàíèÿ îãðàíè÷åííîãî êâàíòîðà ñóùåñòâîâàíèÿ è âñåîáùíîñòè( ∃ , ∀ ) ïî ïîä- ñëîâàì. Ìíîæåñòâî B n -îê àðãóìåíòíûõ ñëîâ, íàçûâàåòñÿ ìíîæåñòâîì òèïà P , îòíîñèòåëüíî íåêîòîðîãî ìíî- 37 æåñòâà àðãóìåíòíûõ ñëîâ A , åñëè äëÿ íåêîòîðîé ∃ îãðàíè÷åííîé ôîðìóëû B ( x ) , áåñêâàíòîðíàÿ ôîðìóëà êîòîðîé, ïîñòðîåíà èç ôóíêòîðîâ ïðèíàäëåæèò PPr , âåðíî ∀ α { α ∈ B ⇐⇒ WordM A | = B ( α ) } . Êëàññ âñåõ ñëîâàðíûõ ìíîæåñòâ òèïà P áóäåì îáîçíà÷àòü ïîñðåäñòâîì N P A ( U ) . Ýòîò êëàññ ñëîâàð- íûõ ìíîæåñòâ çàìêíóòû îòíîñèòåëüíî îïåðàöèé: ïåðåñå÷åíèÿ, îáúåäèíåíèÿ è íàâåøèâàíèÿ îãðàíè÷åííîãî êâàíòîðà ñóùåñòâîâàíèÿ ïî äëèíàì ñëîâ. Ïóñòü co − N P A ( U ) = { C : C ∈ N P A ( U } . Ìíîæåñòâî, ïðèíàäëåæàùèå êëàññó co − N P A ( U ) áóäåì íàçûâàòü ìíîæåñòâîì òèïà Q [7]. Ïóñòü äàíà ôîðìóëà âèäà ∃ | P ( x | z Φ( x, z ) = Λ , ãäå ôóíêòîð Φ - PPr - ôóíêòîð. Ñ ïîìîùüþ ôóíêòîðà Order ìîæíî ïîñòðîèòü òàêîé ïðèìèòèâíî ðåêóðñèâíûé ôóíêòîð Ψ , ÷òî âåðíî WordM A | = ∀ x [ ∃ | P ( x ) | z Φ( x, z ) = Λ ≡ Ψ( x ) = Λ] . Àíàëîãè÷íî èìååì WordM A | = ∀ x [ ∀ | P ( x ) | z Φ( x, z ) = Λ ≡ Ψ( x ) = Λ] . Èçâåñòíûå ñîîòíîøåíèÿ ìåæäó ââåäåííûìè êëàññàìè: a ). Ñóùåñòâóåò òàêîé îðàêóë A , ÷òî P A ( U ) = N P A ( U ) ; b). Ñóùåñòâóåò òàêîé îðàêóë B , ÷òî P B ( U ) ̸ = N P B ( U ) ; c). Ñóùåñòâóåò òàêîé îðàêóë C , ÷òî P C ( U ) ̸ = N P C ( U ) è N P C ( U ) = co − N P C ( U ) ; d). Ñóùåñòâóåò òàêîé îðàêóë D , ÷òî N P D ( U ) ̸ = co − N P D ( U ) ; e). Ñóùåñòâóåò òàêîé îðàêóë E , ÷òî P E ( U ) = N P E ( U ) T co − N P E ( U ) ; f). Ñóùåñòâóåò òàêîé îðàêóë F , ÷òî P F ( U ) = N P F ( U ) T co − N P F ( U ) è N P F ( U ) = co − N P F ( U ) ; g). Ñóùåñòâóåò òàêîé îðàêóë G , ÷òî P G ( U ) = N P G ( U ) T co − N P G ( U ) è N P G ( U ) ̸ = co − N P G ( U ) ; h). Ñóùåñòâóåò òàêîé îðàêóë H , ÷òî P H ( U ) ̸ = N P H ( U ) T co − N P H ( U ) è N P H ( U ) = co − N P H ( U ) i). Ñóùåñòâóåò òàêîé îðàêóë I , ÷òî P I ( U ) ̸ = N P I ( U ) T co − N P I ( U ) è N P I ( U ) ̸ = co − N P I ( U ) . Êàæäîå óêàçàííîå ñîîòíîøåèå â íåðåëÿòèâèçîâàííîì âàðèàíòå - ïðîáëåìà. Îñíîâíûå ïîíÿòèÿ è ðàññìàòðèâàåìûå òåîðåìû â ýòîì ðàçäåëå, çàèìñòâîâàíû èç [8-13] è ïðåîáðàçîâàíû ñîîòâåòñòâóþùèì îáðàçîì. Ïóñòü F ( U ) - íåêîòîðîå ìíîæåñòâî ôóíêòîðîâ, ñîäåðæàùåå èñõîäíûå ôóíêòîðû. ßçûê ïåðâîãî ïîðÿäêà, îïðåäåëÿåìûé çàäàííûì ìíîæåñòâîì ôóíêòîðîâ, è îáîçíà÷àåìûé êàê L F ( U ) , ñîñòîèò èç ôóíêöèîíàëüíûõ ñèìâîëîâ f Φ , äëÿ êàæäîãî ôóíêòîðà Φ ∈ F ( U ) , ìåñòíîñòü êîòîðîãî ðàâíà ìåñòíîñòè ôóíêòîðà Φ , êîíñòàíòíîãî ñèìâîëà Λ , îñíîâíîãî ïðåäèêàòíîãî ñèìâîëà ≤ . Çàìå÷àíèå . Êàê ïðàâèëî ôóíêöèîíàëüíûé ñèìâîë f Φ áóäåì îáîçíà÷àòü êàê Φ è èíòåðïðåòèðîâàòü åãî êàê ôóíêöèþ. ßçûê L F ( U ) íàçûâàåòñÿ k - àëôàâèòíûì, åñëè ìíîæåñòâî ôóíêòîðîâ F ( U ) - k - àëôàâèòíîå. 38 Åñëè ìíîæåñòâî ôóíêòîðîâ F ( U ) ñîñîòîèò èç âñåãî ìíîæåñòâà ñëîâàðíûõ ïðèìèòèâíî ðåêóðñèâíûõ ôóíêòîðîâ, òî ÿçûê L F ( U ) áóäåò îáîçíà÷àòüñÿ êàê L ( U ) ( L ). Äëÿ êàæäîãî ôèêñèðîâàííîãî ìíîæåñòâà p ≥ 2 -àëôàâèòíûõ àðãóìåíòíûõ ñëîâ îïðåäåëèì ñëåäóþùèå òåîðèè: Th = { A : WordM | = A - ïðåäëîæåíèå ÿçûêà L } + ∀ xy ( x ≤ y ≡ | x | . | y | = Λ) . Òåîðèÿ Th - ïîëíàÿ òåîðèÿ â ÿçûêå L . Òåîðèþ â ÿçûêå L ( U ) , îáîçíà÷àåìóþ êàê Th ( U ) : Th ( U ) = { A : Äëÿ ëþáîãî ìíîæåñòâà p ≥ 2 − àëôàâèòíûõ àðãóìåíòíûõ ñëîâ A WordM A | = A , A - ïðåäëîæåíèå ÿçûêà L ( U ) } + ∀ xy ( x ≤ y ≡ | x | . | y | = Λ) . Th ( A ) = { A : WordM A | = A , A - ïðåäëîæåíèå ÿçûêà L A } + ∀ xy ( x ≤ y ≡ | x | . | y | = Λ) Òåîðèÿ Th ( A ) - ïîëíàÿ òåîðèÿ â ÿçûêå L A . Èìååò ìåñòî: Th ⊂ Th ( U ) ⊂ Th ( A ) . Òåîðåìà 6.1 . Ïóñòü A ′ | = Th . Ïóñòü u : A ′ → { Λ , 1 } , òîãäà ìîäåëü A ′ ÿçûêà L ìîæíî îáîãàòèòü äî òàêîé ìîäåëè A ÿçûêà L ( U ) , ÷òî A ′ ⊆ A , A | = Th ( U ) , ∀ b ∈ A ′ u ( b ) = U A ( b ) è äëÿ ëþáîé ôîðìóëû A ( x ) ÿçûêà L , ∀ a ∈ A ′ , åñëè A ′ | = A ( a ) , òîãäà A | = A ( a ) . Äîêàçàòåëüñòâî . Ñîñòàâèì òåîðèþ Th ( A ′ A ′ ) (ñì.[9, ñòð. 130]). Ïðîèíòåðïðåòèðóåì îðàêóëüíóþ ôóíê- öèþ U : U ( a ) =          Λ , if u ( a ) = Λ; S 1 (Λ) , if u ( a ) = 1 . äëÿ êàæäîãî ýëåìåíòà a ∈ A ′ ( S 1 (Λ) ⇌ a 1 ⇌ 1 ) Äàëåå ñîñòàâèì òåîðèþ Th ( A ′ A ′ ) + Th ( U ) + { U ( c a ) = b : a ∈ A ′ } . Ýòà òåîðèÿ ëèáî íåïðîòèâåðî÷èâà èëè òàêîâîé íå ÿâëÿåòñÿ. Ïóñòü òåîðèÿ Th ( A ′ A ′ ) + Th ( U ) + { U ( c a ) = b : a ∈ A ′ } - ïðîòèâîðå÷èâà, òîãäà Th ( U ) ⊢ V i ≤ k U ( c a i ) = b i ⊃ ¬ V j ≤ m B j ( c a 1 , . . . , c a k ) , òîãäà Th ( U ) ⊢ V i ≤ k U ( y i ) = b i ⊃ ¬ V j ≤ m B j ( y 1 , . . . , y k ) . Èìååì: òåîðèÿ Th - ïîëíàÿ òåîðèÿ â ÿçûêå L , A ′ | = Th , è A ′ | = V j ≤ m B j ( a 1 , . . . , a k ) , òîãäà ñóùåñòâó- þò òàêèå ðàçëè÷íûå àðãóìåíòíûå ñëîâà α 1 , . . . , α k , ÷òî WordM | = V j ≤ m B j ( α 1 , . . . , α k ) , òîãäà Th ( U ) ⊢ V i ≤ k U ( α i ) = b i ⊃ ¬ V j ≤ m B j ( α 1 , . . . , α k ) . Ïðîèíòåïðåòèðóåì îðàêóë U : U ( α i ) = b i , äëÿ äðóãèõ àðãóìåíò- íûõ ñëîâ β ïîëîæèì U ( β ) = Λ . Îáîçíà÷èì ïîëó÷åííóþ èíòåïðåòàöèÿ êàê A , òîãäà WordM A | = Th ( U ) è WordM A | = V i ≤ k U ( α i ) = b i , òîãäà WordM A | = ¬ V j ≤ m B j ( α 1 , . . . , α k ) . Ôîðìóëà V j ≤ m B j ( α 1 , . . . , α k ) åñòü ôîðìóëà ÿçûêà L , òîãäà WordM | = ¬ V j ≤ m B j ( α 1 , . . . , α k ) - ïðîòèâîðå÷èå. Ïóñòü A | = Th ( A ′ A ′ ) + Th ( U ) + { U ( a ) = b : a ∈ A ′ } , òîãäà ∀ b ∈ A ′ u ( b ) = U A ( b ) è äëÿ ëþáîé ôîðìóëû A ( x ) ÿçûêà L , äëÿ ∀ a ∈ A ′ , åñëè A ′ | = A ( a ) , òîãäà A | = A ( a ) . Ââåä¼ì ñëåäóþùåå âàæíîå ïîíÿòèå. 39 Îïðåäåëåíèå. Ïóñòü A - ìîäåëü ÿçûêà L ( U ) è a ∈ A . Ïîëèíîìèàëüíîé ñðåçêîé, îïðåäåëÿåìîé íàáî- ðîì ýëåìåíòîâ a ∈ A , íàçûâàåòñÿ òàêàÿ ìîäåëü(àëãåáðàè÷åñêàÿ ñèñòåìà), îáîçíà÷àåìàÿ êàê A a , íîñèòåëåì êîòîðîé ÿâëÿåòñÿ ìíîæåñòâî A a = { b : äëÿ íåêîòîðîãî ñëîâàðíîãî ìíîãî÷ëåíà P ( x ) A | = | b | ≤ | P ( a ) |} , à ñèãíàòóðà ñîñòîèò èç âñåõ òåõ ôóíêöèé f Φ , äëÿ êîòîðûõ ôóíêòîð Φ ∈ PPr . Èìååò ìåñòî: 1. A a ⊆ A [9 c.36]. 2. ∀ a [ A a | = A ( a ) ⇐⇒ A | = A ( a )] , ãäå A ( x ) - îãðàíè÷åííàÿ ôîðìóëà ñèãíàòóðû PPr . Ïîñðåäñòâîì ∆ A - áóäåì îáîçíà÷òü äèàãðàììó ìîäåëè A [9. c. 86], â ÷àñòíîñòè ∆ A a - äèàãðàììà ïîëè- íîìèàëüíîé ñðåçêè A a . Çàìå÷àíèå . Ïóñòü A ′ - îáåäíåíèå ìîäåëè A ÿçûêà L ( U ) äî ÿçûêà L . èìååò ìåñòî ∆ A ′ ⊆ ∆ A , â ÷àñòíîñòè ∆ A ′ a ⊆ ∆ A a , ó÷èòûâàÿ ñëåäñòâèå 5.2, ñòð. 27, ïî äèàãðàììå ∆ A ′ , ìîæíî âîññòàíîâèòü äèàãðàììó ∆ A , ïðè ýòîì íîáõîäèìî çíàòü ãðàôèê îðàêóëà U , â ìîäåëè A . Äèàãðàììà ∆ A ′ ñîäåðæèò òîëüêî ñëåäû îðàêóëüíûõ âû÷èñëåíèé, ïîëíàÿ èíôîðìàöèÿ îá îðàêóëüíûõ âû÷èñëåíèõ ñîäåðæèòñÿ â äèàãðàììå ∆ A , íàïðèìåð, åñëè A | = Θ Φ ( b ) = c , òîãäà A ′ | = (Θ Φ ) ∗ ( c.b ) = c , åñëè [Θ Φ ( b ) = c ] ∈ ∆ A a , òîãäà [(Θ Φ ) ∗ ( c.b ) = c ] ∈ ∆ A ′ a . Óòâåðæäåíèå 6.2 . Äëÿ ïðîèçâîëüíîãî n + 1 ( n ≥ 1 ) - ìåñòíîãî ôóíêòîðà Φ , äëÿ ïðîèçâîëüíîé ìîäåëè A ÿçûêà L ( U ) , òàêîé, ÷òî A | = Th ( A ) , äëÿ ïðîèçâîëüíîãî íàáîðà a ∈ A , äëÿ ïðîèçâîëüíîãî ñëîâàðíîãî ìíîãî÷ëåíà P ( x ) , ñóùåñòâóåò íàèìåíüøèé ïî äëèíå ôóíêöèîíàëüíûé ýëåìåíò b , òàêîé, ÷òî ∀ | P ( a ) | u Θ Φ ( a , u ) ⊆ b . Ïðèìå÷àíèå . Òàêîé ôóíêöèîíàëüíûé ýëåìåíò íå åäèíñòâåííûé, íî äëÿ ëþáûõ òàêèõ ôóíêöèîíàëü- íûõ ýëåìåíòîâ b , c âåðíî | b | = | c | ∧ dom ( b ) = dom ( c ) ∧ ∀ x [ x ∈ dom ( b ) ⇒ b ( x ) = c ( x )] . Óòâåðæäåíèå 6.3 . Äëÿ ïðîèçâîëüíîãî n +1 - ìåñòíîãî ôóíêòîðà Φ , äëÿ ïðîèçâîëüíîé ìîäåëè A ÿçûêà L ( U ) , òàêîé, ÷òî A | = Th ( A ) , äëÿ ïðîèçâîëüíîãî íàáîðà a ∈ A , äëÿ ïðîèçâîëüíîãî ñëîâàðíîãî ìíîãî÷ëåíà P ( x ) , ∀ b ∈ A a , äëÿ ïðîèçâîëüíîãî ôóíêöèîíàëüíîãî ýëåìåíòà c , òàêîãî, ÷òî ∀ | P ( b ) | u Θ Φ ( b , u ) ⊆ c , âåðíî: 1. A | = ∀ | P ( b ) | u Φ( b , u ) = Λ ⇐⇒ A | = ∀ | P ( b ) | u (Φ) ∗ ( c , b , u ) = Λ . 2. A | = ∃ | P ( b ) | u Φ( b , u ) = Λ ⇐⇒ A | = ∃ | P ( b ) | u (Φ) ∗ ( c , b , u ) = Λ . Óòâåðæäåíèå 6.4 . Ëþáîå ðàñøèðåíèå ïîëèíîìèàëüíé ñðåçêè A a ⊆ M | = Th ( A ) , ïîðîæäàåò ñëåäóþ- ùèå ñîîòíîøåíèÿ, äëÿ ëþáîãî ôóíêòîðà Φ ∈ PPr è ëþáîãî ñëîâàðíîãî ìíîãî÷ëåíà P ( x ) : 1. A a ⊆ M a ; 2. ∀ b ∈ A a âåðíî Θ Φ ( b ) A = Θ Φ ( b ) M ; 3. ∀ b , c ∈ A a , åñëè A | = (Φ) ∗ (Θ Φ ( b ) , b ) = c , òîãäà ∀ f ∈ M , òàêîãî ÷òî Θ Φ ( b ) ⊆ f , âåðíî M | = (Φ) ∗ ( f , b ) = c - "Ïðèíöèì èñïîëüçîâàíèÿ" (" Use Principle ") ïðè ðàñøèðåíèè ìîäåëåé; 4. Ïóñòü b ∈ A a . Ïóñòü c A - íàèìåíüøèé ïî äëèíå ôóíêöèîíàëüíûé ýëåìåíò, òàêîé, ÷òî 40 A | = ∀ | P ( b ) | u Θ Φ ( b , u ) ⊆ c A . Ïóñòü d M - íàèìåíüøèé ïî äëèíå ôóíêöèîíàëüíûé ýëåìåíò, òàêîé, ÷òî M | = ∀ | P ( b ) | u Θ Φ ( b , u ) ⊆ d M . Òîãäà: a ). dom ( c A ) ⊂ A a ∧ dom ( d M ) ⊂ M a ∧ c A ⊆ d M ; b ). Åñëè | e | ≤ | P ( b ) | è A | = Φ( b , e ) = h , òîãäà A | = (Φ) ∗ ( c A , b , e ) = h è M | = (Φ) ∗ ( d M , b , e ) = h ; c ). Åñëè A | = ∀ | P ( b ) | u (Φ) ∗ ( c A , b , u ) = Λ , òîãäà äëÿ êàæäîãî | f | ≤ | P ( b ) | ïîëó÷èì A | = (Φ) ∗ ( c A , b , f ) = Λ , òîãäà M | = (Φ) ∗ ( d M , b , f ) = Λ , òîãäà M | = ∀ u ( | u | ≤ | P ( b ) | ∧ u ∈ A a ⊃ (Φ) ∗ ( d M , b , u ) = Λ . Èìååò ìåñòî òàêæå, åñëè M | = ∀ u ( | u | ≤ | P ( b ) | ∧ u ∈ A a ⊃ (Φ) ∗ ( d M , b , u ) = Λ) , òîãäà A | = ∀ | P ( b ) | u (Φ) ∗ ( c A , b , u ) = Λ . "Ïðèíöèì èñïîëüçîâàíèÿ"(" Use Principle ") ïðè ðàñøèðåíèè ìîäåëåé. Òåîðåìà 6.5. Ïóñòü A - ïðîèçâîëüíûé îðàêóë. N P ( A ) = co − N P ( A ) , òîãäà è òîëüêî òîãäà, êîãäà äëÿ ëþáîé îãðàíè÷åííîé ∃ ôîðìóëû A ( x ) ñèãíàòóðû PPr ( U ) , ñóùåñòâóåò îãðàíè÷åííàÿ ∀ ôîðìóëà B ( x ) òîé æå ñèãíàòóðû, ÷òî Th ( A ) ⊢ ∀ x [ A ( x ) ≡ B ( x )] . Òåîðåìà 6.6. Ïóñòü A - ïðîèçâîëüíûé îðàêóë, A ( x ) - îãðàíè÷åííàÿ ∀ ôîðìóëà ñèãíàòóðû PPr ÿçûêà L ( U ) . Ñëåäóþùèå óñëîâèÿ ðàâíîñèëüíû: 1. Äëÿ ëþáîé ìîäåëè A | = Th ( A ) , äëÿ ëþáûõ a ∈ A , äëÿ ëþáûõ b ∈ A a , ëþáîé ìîäåëè M | = Th ( A ) , òàêîé, ÷òî A a ⊆ M , åñëè A | = A ( b ) , òîãäà M | = A ( b ) . 2. Äëÿ ôîðìóëû A ( x ) ñóùåñòâóåò òàêàÿ îãðàíè÷åííàÿ ∃ ôîðìóëà B ( x ) ÿçûêà ÿçûêà L ( U ) , ñèãíàòóðû PPr , ÷òî Th ( A ) | = ∀ x [ A ( x ) ≡ B ( x )] . Äîêàçàòåëüñòâî . Äîêàæåì, ÷òî èç (1) ñëåäóåò (2). Èäåÿ äîêàçàòåëüñòâà çàèìñòâîâàíà â [8 ñ. 156], [9 ñ.133-134]. Åñëè ôîðìóëà A ( x ) òàêîâà, ÷òî Th ( A ) | = ∀ x A ( x ) , òîãäà Th ( A ) | = ∀ ( x [ A ( x ) ≡ B ( x )] , ãäå B ( x ) - îãðàíè÷åííàÿ ∃ ôîðìóëà, ñèãíàòóðû PPr òàêàÿ, ÷òî Th ( A ) | = ∀ x B ( x ) . Ïóñòü ôîðìóëà A ( x ) òàêîâà, ÷òî Th ( A ) / | = ∀ x A ( x ) ( a ). Ñîñòàâèì ìíîæåñòâî ïðåäëîæåíèé Γ ( c ) = { Θ( c ) : Th ( A ) | = ¬ A ( x ) ⊃ Θ( x ) } , ãäå Θ( x ) - îãðàíè÷åííàÿ ∀ ôîðìóëà, ñèãíàòóðû PPr . Èç ( a ) ïîëó÷èì, ÷òî Th ( A ) + Γ ( c ) - ñîâìåñòíî. Äîêàæåì Th ( A ) + Γ ( c ) | = ¬ A ( c ) . Ïóñòü A | = Th ( A ) + Γ ( c ) , c ∈ A . Ñîñòàâèì ∆ A c - äèàãðàììà ìîäåëè A c . Ìíîæåñòâî ïðåäëîæåíèé Th ( A ) + ∆ A c + ¬ A ( c ) - ñîâìåñòíî èëè òàêîâûì íå ÿâëÿåòñÿ. Åñëè Th ( A ) + ∆ A c + ¬ A ( c ) - íåñîâìåñòíî, òîãäà Th ( A ) + ∆ A c | = A ( c ) , òîãäà Th ( A ) + V i ≤ k Λ i ( c, d ) | = A ( c ) , ãäå Λ i ( c, d ) ∈ ∆ A c , òîãäà Th ( A ) + ¬ A ( c ) | = ¬ V i ≤ k Λ i ( c, d ) , òîãäà Th ( A ) | = ¬ A ( c ) ⊃ ¬ V i ≤ k Λ i ( c, d ) , òîãäà Th ( A ) | = ∀ x ∀ y [ ¬ A ( x ) ⊃ ¬ V i ≤ k Λ i ( x, y )] , òîãäà Th ( A ) | = ∀ x [ ¬ A ( x ) ⊃ ∀ y ¬ V i ≤ k Λ i ( x, y )] ( b ). Äëÿ d , ñóùåñòâåò òàêîé ñëîâàðíûé ìíîãî÷ëåí ˜ P ( x ) , ÷òî | d | ≤ | ˜ P ( c ) | . Èç ( b ) ïîëó÷èì 41 Th ( A ) | = ∀ x [ ¬ A ( x ) ⊃ ∀ | ˜ P ( x ) | y ¬ V i ≤ k Λ i ( x, y )] , òîãäà ∀ | ˜ P ( c ) | y ¬ V i ≤ k Λ i ( c, y ) ∈ Γ ( c ) , òîãäà A | = ¬ V i ≤ k Λ i ( c, d ) - ïðîòèâîðå÷èå, ñëåäîâàòåëüíî ìíîæåñòâî ïðåäëîæåíèé Th ( A ) + ∆ A c + ¬ A ( c ) - ñîâìåñòíî. Ïóñòü M | = Th ( A ) + ∆ A c + ¬ A ( c ) , òîãäà ïîëó÷èì A c ⊆ M è M | = ¬ A ( c ) , òîãäà A | = ¬ A ( c ) , ñëåäîâà- òåëüíî Th ( A ) + Γ ( c ) | = ¬ A ( c ) , òîãäà Th ( A ) | = V j ≤ m Ω j ( c ) ⊃ ¬ A ( c ) , ãäå Ω j ( c ) ∈ Γ ( c ) , òîãäà Th ( A ) | = V j ≤ m Ω j ( x ) ⊃ ¬ A ( x ) . Èìååì Th ( A ) | = ¬ A ( x ) ⊃ V j ≤ m Ω j ( x ) , òîãäà Th ( A ) | = ¬ A ( x ) ≡ V j ≤ m Ω j ( x ) , òîãäà Th ( A ) | = ¬ V j ≤ m Ω j ( x ) ≡ A ( x ) . Äîêàæåì, ÷òî èç (2) ñëåäóåò (1). Ïóñòü A | = Th ( A ) , a ∈ A , b ∈ A a , M | = Th ( A ) , òàêàÿ, ÷òî A a ⊆ M è A | = A ( b ) . Èìååì: äëÿ ôîðìóëû A ( x ) ñóùåñòâóåò òàêàÿ îãðàíè÷åííàÿ ∃ ôîðìóëà B ( x ) ÿçûêà L ( U ) , ñèãíàòóðû PPr , ÷òî Th ( A ) | = ∀ x [ A ( x ) ≡ B ( x )] , ó÷èòûâàÿ, A | = Th ( A ) è M | = Th ( A ) , ïîëó÷èì A | = ∀ x [ A ( x ) ≡ B ( x )] è M | = ∀ x [ A ( x ) ≡ B ( x )] òîãäà A | = A ( b ) ≡ B ( b ) è M | = A ( b ) ≡ B ( b ) , òîãäà A a | = A ( b ) ≡ B ( b ) è M a | = A ( b ) ≡ B ( b ) , ó÷èòûâàÿ A | = A ( b ) , ïîëó÷èì A | = B ( b ) , òîãäà A a | = B ( b ) , ó÷èòâàÿ, ÷òî B ( x ) - îãðàíè÷åííàÿ ∃ ôîðìóëà ñèãíàòóðû PPr , A a ⊆ M è b ∈ M a , ïîëó÷èì M | = B ( b ) , òîãäà M | = A ( b ) . Çàìå÷àíèå . Àíàëîãè÷íî òåîðåìå 6.6, äîêàçûâàåòñÿ òåîðåìà äëÿ îãðàíè÷åííîé ∀ ôîðìóëû A ( x ) ñèã- íàòóðû PPr äëÿ òåîðèè Th â ÿçûêå L . Çàìå÷àíèå . Àíàëîãè÷íî òåîðåìå 6.6, äîêàçûâàåòñÿ òåîðåìà äëÿ îãðàíè÷åííîé ∀ ôîðìóëû A ( x ) â ÿçûêå L ( U ) ñèãíàòóðû PPr ( U ) äëÿ òåîðèè Th ( U ) . Òåîðåìà 6.7 . Äëÿ ëþáîãî îðàêóëà A , ëþáîé îãðàíè÷åííîé ∀ ôîðìóëû A ( x ) ÿçûêà L ( U ) ñèãíàòóðû PPr , ñëåäóþùèå óñëîâèÿ ðàâíîñèëüíû: 1). Äëÿ ëþáîé ìîäåëè A | = Th ( A ) , ëþáûõ ýëåìåíòîâ a ∈ A , ëþáûõ ýëåìåíòîâ b ∈ A a , åñëè A a | = A ( b ) , òîãäà äëÿ ëþáîé ìîäåëè B | = Th ( A ) òàêîé, ÷òî B ⊇ A a , âåðíî B | = A ( b ) . 2). Äëÿ ëþáîé ìîäåëè A | = Th ( A ) , ëþáûõ ýëåìåíòîâ a ∈ A , ëþáûõ ýëåìåíòîâ b ∈ A a , åñëè A a | = A ( b ) , òîãäà Th ( A ) + ∆ A a ⊢ A ( b ) . 3). Äëÿ ôîðìóëû A ( x ) ñóùåñòâóåò îãðàíè÷åííàÿ ∃ ôîðìóëà B ( x ) ñèãíàòóðû PPr , ÷òî Th ( A ) ⊢ ∀ x [ A ( x ) ≡ B ( x )] (ðàâíîñèëüíî WordM A | = ∀ x [ A ( x ) ≡ B ( x )] ). Äîêàçàòåëüñòâî . Äîêàçàòåëüñòâî (1) ⇔ (3) - ñì. òåîðåìó 6.6. Äîêàæåì, ÷òî èç (1) ñëåäóåò (2). Ïóñòü A | = Th ( A ) , a ∈ A , b ∈ A a è A a | = A ( b ) . Ïðåäïîëîæèì, ÷òî Th ( A ) + ∆ A a / ⊢ A ( b ) , òîãäà òåîðèÿ Th ( A ) + ∆ A a + ¬ A ( b ) - íåïðîòèâîðå÷èâà. Ïóñòü B | = Th ( A ) + ∆ A a + ¬ A ( b ) , òîãäà B | = Th ( A ) , B ⊇ A a , B | = ¬ A ( b ) è B | = A ( b ) - ïðîòèâîðå÷èå, ñëåäîâàòåëüíî Th ( A ) + ∆ A a ⊢ A ( b ) . 42 Äîêàæåì, ÷òî èç (2) ñëåäóåò (1). Ïóñòü A | = Th ( A ) , a ∈ A , b ∈ A a , A a | = A ( b ) , B | = Th ( A ) è B ⊇ A a . Èç A a | = A ( b ) , ïîëó÷èì Th ( A ) + ∆ A a ⊢ A ( b ) . Èç B | = Th ( A ) è B ⊇ A a ïîëó÷èì B | = Th ( A ) + ∆ A a , òîãäà B | = A ( b ) . Çàìå÷àíèå . Àíàëîãè÷íî òåîðåìå 6.7, äîêàçûâàåòñÿ òåîðåìà äëÿ îãðàíè÷åííîé ∀ ôîðìóëû A ( x ) ñèã- íàòóðû PPr äëÿ òåîðèè Th â ÿçûêå L . Çàìå÷àíèå . Àíàëîãè÷íî òåîðåìå 6.7, äîêàçûâàåòñÿ òåîðåìà äëÿ îãðàíè÷åííîé ∀ ôîðìóëû A ( x ) â ÿçûêå L ( U ) ñèãíàòóðû PPr ( U ) äëÿ òåîðèè Th ( U ) . Çàìå÷àíèå . Èñïîëüçóÿ óòâåðæäåíèå 6.3 è òåîðåìó 6.7, ìîæíî äîñòàòî÷íî ïðîñòî äîêàçàòü òåîðåìó 4.5[12 p.469]. Òåîðåìà 6.8 Ïóñòü äëÿ òåîðèè Th âûïîëíÿåòñÿ âòîðîé ïóíêò òåîðåìû 6.7 â ÿçûêå L , äëÿ ëþáîé îãðàíè÷åííîé ∀ ôîðìóëû ñèãíàòóðû PPr . Ïóñòü A | = Th ( U ) , a ∈ A , b ∈ A a . Ïóñòü äëÿ ôîðìóëû ∀ | P ( x ) | y Φ( x, y ) = Λ âåðíî A | = ∀ | P ( b ) | y Φ( b , y ) = Λ , ãäå Φ - ôóíêòîð ñèãíàòóðû PPr ( U ) , òîãäà Th ( U ) + ∆ A ⊢ ∀ | P ( b ) | y Φ( b , y ) = Λ . Äîêàçàòåëüñòâî . Ôîðìóëà ∀ x [ ∀ | P ( x ) | z Φ( x, z ) = Λ ≡ ∀ | P ( x ) | u (Φ) ∗ ( Θ U ( exp p ( ˜ | P ( x )) | ) , x, u )) = Λ] , ïðèíàä- ëåæèò òåîðèè Th ( U ) , ãäå ˜ P ( x ) - ïîäõîäÿùèé ñëîâàðíûé ìíîãî÷ëåí(ñì. Óòâåðæäåíèå 5.9), òîãäà Th ( U ) ⊢ ∀ x [ ∀ | P ( x ) | z Φ( x, z ) = Λ ≡ ∀ | P ( x ) | u (Φ) ∗ (Θ U ( exp p ( ˜ | P ( x ) | )) , x, u ) = Λ] , òîãäà Th ( U ) ⊢ [ ∀ | P ( b ) | z Φ( b , z ) = Λ ≡ ∀ | P ( b ) | u (Φ) ∗ (Θ U ( exp p ( ˜ | P ( b ) | )) , b , u ) = Λ] ( a ) Âû÷èñëèì: ˜ | P ( b ) | = d 1 , exp p ( d 1 ) = d 2 , Θ U ( d 2 ) = d 3 , òîãäà A ′ | = ∀ | P ( b ) | u (Φ) ∗ ( d 3 , b , u ) = Λ , ãäå A ′ - îáåäíåíèå ìîäåëè A ÿçûêà L ( U ) äî ìîäåëè ÿçûêà L , òîãäà Th + ∆ A ′ a ; d 2 ⊢ ∀ | P ( b ) | u (Φ) ∗ ( d 3 , b , u ) = Λ , ó÷èòûâàÿ ðàâåíñòâî Θ U ( d 2 ) = d 3 , ïîëó÷èì Th + ∆ A ′ a ; d 2 + { Θ U ( d 2 )= d 3 } ⊢ ∀ | P ( b ) | u (Φ) ∗ (Θ U ( d 2 ) , b , u ) = Λ , ó÷èòûâàÿ (Θ U ( d 2 ) = d 3 ) ∈ ∆ A a ; d 2 , ïîëó÷èì Th + ∆ A a ; d 2 ⊢ ∀ | P ( b ) | u (Φ) ∗ (Θ U ( d 2 ) , b , u ) = Λ . Ó÷èòûâàÿ Th ⊢ ∀ x ∀ y [ exp p ( x ) = y ≡ EXP p ( x, y ) = Λ] è Th ⊢ ∀ z ∀ v ∀ x [ A ( z, v ) ∧ EXP p ( x, v ) = Λ ⊃ A ( z, exp p ( x ))] , ïîëó÷èì Th + ∆ A a ; d 2 ⊢ ∀ | P ( b ) | u (Φ) ∗ (Θ U ( exp p ( d 1 )) , b , u ) = Λ , ó÷èòûâàÿ ˜ | P ( b ) | = d 1 , ïîëó÷èì Th + ∆ A a ; d 2 ⊢ ∀ | P ( b ) | u (Φ) ∗ (Θ U ( exp p ( ˜ | P ( b ) | )) , b , u ) = Λ] , ó÷èòûâàÿ( a ), ïîëó÷èì Th ( U ) + ∆ A a ; d 2 ⊢ ∀ | P ( b ) | y Φ( b , y ) = Λ , òîãäà Th ( U ) + ∆ A ⊢ ∀ | P ( b ) | y Φ( b , y ) = Λ . Çàìå÷àíèå . Ýòó òåîðåìó ìîæíî ðàñïðîñòðàíèòü íà ëþáóþ îãðíè÷åííóþ ∀ ôîðìóëó ñèãíàòóðû PPr ( U ) . Òåîðåìà 6.9 . Åñëè äëÿ òåîðèè Th äëÿ ëþáîé îãðàíè÷åíîé ∀ - ôîðìóëû ÿçûêà L ñèãíàòóðû PPr âûïîëíÿåòñÿ ïåðâûé ïóíêò òåîðåìû 6.7, òîãäà äëÿ ëþáîãî îðàêóëà A , äëÿ ëþáîé îãðàíè÷åíîé ∀ - ôîðìóëû ÿçûêà L ( U ) ñèãíàòóðû PPr ( U ) , äëÿ òåîðèè Th ( A ) òàêæå âûïîëíÿåòñÿ ïåðâûé ïóíêò ýòîé òåîðåìû. 43 Äîêàçàòåëüñòâî. Ïóñòü Φ - ïðîèçâîëüíûé n + 1 - ìåñòíûé ôóíêòîð, ñèãíàòóðû PPr ( U ) ÿçûêà L ( U ) . Ïóñòü P ( x 1 , . . . , x n ) - ïðîèçâîëüíûé ñëîâàðíûé ìíîãî÷ëåí. Äîêàæåì òåîðåìó äëÿ ôîðìóëû âèäà ∀ | P ( x 1 ,...,x n ) | z Φ( x 1 , . . . , x n , z ) = Λ . Ïóñòü A | = Th ( A ) , a ∈ A , b ∈ A a , A | = ∀ | P ( b ) | z [Φ( b , z ) = Λ] , äîêàæåì, ÷òî ∀ M | = Th ( A ) , òàêîé, ÷òî M ⊇ A a , âåðíî M | = ∀ | P ( b ) | z [Φ( b , z ) = Λ] . Ïóñòü A ′ - îáåäíåíèå ìîäåëè A ÿçûêà L ( U ) äî ìîäåëè ÿçûêà L . Ñîñòàâèì òåîðèþ Th ( A ′ A ) 8 (ñì.[9, ñòð. 130]), äàëåå ñîñòàâèì òåðèþ Th ( A ′ A ) + ∆ M ′ a , ãäå M ′ a - îáåäíåíèå ìîäåëè M a äî ÿçûêà L . Ýòà òåîðèÿ ïðîòèâîðå÷èâà èëè òàêîâîé íå ÿâëÿåòñÿ. Ïðåäïîëîæèì, ÷òî òåîðèÿ Th ( A ′ A ) + ∆ M ′ a - ïðîòèâîðå÷èâà, òîãäà Th ⊢ V A i ( e, f ) ⊃ ¬ V B j ( e, h ) , ãäå e ∈ A a , A i ( e, f ) ∈ Th ( A ′ A ) , B j ( e, h ) ∈ ∆ M ′ a , òîãäà Th ⊢ V A i ( e, f ) ⊃ ¬ V B j ( e, x ) , òîãäà Th ⊢ V A i ( e, f ) ⊃ ∀ x ¬ V B j ( e, x ) , òîãäà A ′ | = ∀ x ¬ V B j ( e, x ) . Äëÿ h ìîæíî ïîñòðîèòü òàêîé ñëîâàðíûé ìíîãî÷ëåí Q ( x ) , ÷òî A ′ | = | h | ≤ | Q ( a ) | , òîãäà A ′ | = ∀ | Q ( a ) | x ¬ V B j ( e, x ) , òîãäà, ñîãëàñíî ïóíêòà 1 òåîðåìû 6.7 äëÿ ÿçûêà L , ó÷èòûâàÿ A ′ a ⊆ M ′ a , ïîëó÷èì M ′ a | = ∀ | Q ( a ) | x ¬ V B j ( e, x ) - ïðîòèâîðå÷èå, ñëåäîâàòåëüíî òåîðèÿ Th ( A ′ A )+∆ M ′ a - íåïðîòèâîðå÷èâà. Îòìåòèì, ÷òî â ìîäåëÿõ A ′ è M ′ èìåþòñÿ ñëåäû îðàêóëüíûõ âû÷èñëåíèé îðàêóëà U A è îðàêóëà U M . Ïóñòü N | = Th ( A ′ A ) + ∆ M ′ a â ÿçûêå L . Èìååì: A ′ ⊆ N è M ′ a ⊆ N Ïîñòðîèì èíòåðïðåòàöèþ îðàêóëüíîãî ñèìâîëà U : U ( a ) =                    U A ( a ) , if a ∈ A ; U M ( a ) , if a ∈ M a ; Λ , other. Îáîçíà÷èì ïîëó÷åííóþ èíòåðïðåòàöèþ êàê B . Ñîãëàñíî òåîðåìå 6.1, ïîëó÷èì N 1 | = Th ( U ) + Th ( A ′ A ) + ∆ M ′ a . Èìååì: 1. A ′ ⊆ N 1 ′ , M ′ a ⊆ N 1 ′ . 2. Äëÿ èíòåïðåòàöèè A A îðàêóëüíîãî ñèìâîëà U â ìîäåëè A è äëÿ èíòåïðåòàöèè B N 1 îðàêóëüíîãî ñèìâîëà U â ìîäåëè N 1 âåðíî A A ⊆ B N 1 ( ∀ b ∈ A U A ( b ) = U N 1 ( b ) ). 3. Äëÿ èíòåïðåòàöèè A M a îðàêóëüíîãî ñèìâîëà U â ìîäåëè M a è äëÿ èíòåïðåòàöèè B N 1 îðàêóëüíîãî ñèìâîëà U â ìîäåëè N 1 âåðíî A M a ⊆ B N 1 ( ∀ b ∈ M a U M a ( b ) = U N 1 ( b ) ). 4. Äëÿ ëþáîãî ôóíêòîðà Φ ÿçûêà L âåðíî ∀ b ∈ A ∀ c ∈ A A | = Φ( b ) = c ⇐⇒ N 1 | = Φ( b ) = c , ïðè ýòîì èñïîëüçóåòñÿ (1), òîãäà âåðíî ∀ b ∈ A ∀ c ∈ A ∀ d ∈ A A | = (Θ Φ ) ∗ ( c, b ) = d ⇐⇒ N 1 | = Θ ∗ Φ ( c, b ) = d . 5. Äëÿ ëþáîãî ôóíêòîðà Φ ÿçûêà L ( U ) âåðíî ∀ b ∈ A ∀ c ∈ A A | = Θ Φ ( b ) = c ⇐⇒ N 1 | = Θ Φ ( b ) = c , ïðè 8 Ìîæíî âçÿòü òåîðèþ Th + ∆ A ′ 44 ýòîì èñïîëüçóåòñÿ (2,4), òåîðåìà 4.3, òåîðåìà 4.4, òåîðåìà 5.1 è òåîðåìà 5.5: A | = Θ Φ ( b ) = c A ⇐⇒ A | = (Θ Φ ) ∗ ( c A .b ) = c A ∧ c A ⊂ A A , òîãäà N 1 | = (Θ Φ ) ∗ ( c A .b ) = c A ∧ c A ⊂ B N 1 , òîãäà N 1 | = Θ Φ ( b ) = d N 1 ⊂ c A , òîãäà N 1 | = (Θ Φ ) ∗ ( c A .b ) = Θ Φ ( b ) = d N 1 , òîãäà N 1 | = Θ Φ ( b ) = d N 1 = c A 6. Äëÿ ëþáîãî ôóíêòîðà Φ ÿçûêà L ( U ) âåðíî ∀ b ∈ A ∀ c ∈ A A | = Φ( b ) = c ⇐⇒ N 1 | = Φ( b ) = ñ, ïðè ýòîì èñïîëüçóåòñÿ (4,5) è òåîðåìà 5.1. Òàêèì îáðàçîì ïîëó÷èì A ⊆ N 1 , òîãäà N 1 | = ∆ A , òîãäà N 1 | = Th ( U )+∆ A , ó÷èòûâàÿ òåîðåìó 6.8, ïîëó÷èì N 1 | = ∀ | P ( b ) | z [Φ( b , z ) = Λ] . 7. Äëÿ ëþáîãî ôóíêòîðà Φ ÿçûêà L ñèãíàòóðû PPr âåðíî ∀ b ∈ M a ∀ c ∈ M a M a | = Φ( b ) = c ⇐⇒ N 1 | = Φ( b ) = c , ïðè ýòîì èñàïîëüçóåòñÿ (1), òîãäà âåðíî ∀ b ∈ M a ∀ c ∈ M a ∀ d ∈ M a M a | = (Θ Φ ) ∗ ( c, b ) = d ⇐⇒ N 1 | = Θ ∗ Φ ( c.b ) = d 8. Äëÿ ëþáîãî ôóíêòîðà Φ ÿçûêà L ( U ) , ñèãíàòóðû PPr ( U ) âåðíî ∀ b ∈ M a ∀ c ∈ M a M a | = Θ Φ ( b ) = c ⇐⇒ N 1 | = Θ Φ ( b ) = c , ïðè ýòîì èñïîëüçóåòñÿ (3,7), òåîðåìà 4.3, òåîðåìà 4.4, òåîðåìà 5.1 è òåîðåìà 5.5. 9. Äëÿ ëþáîãî ôóíêòîðà Φ ÿçûêà L ( U ) , ñèãíàòóðû PPr ( U ) âåðíî ∀ b ∈ M a ∀ c ∈ M a M a | = Φ( b ) = c ⇐⇒ N 1 | = Φ( b ) = c , ïðè ýòîì èñïîëüçóåòñÿ (7,8) è òåîðåìà 5.1. Òàêèì îáðàçîì ïîëó÷èì M a ⊆ N 1 , ó÷èòûâàÿ (6), ïîëó÷èì M a | = ∀ | P ( b ) | y Φ( b , y ) = Λ , òîãäà M | = ∀ | P ( b ) | z [Φ( b , z ) = Λ] . Ïðîäîëæàåì . Ïóñòü A ( z, x 1 , . . . , x n ) - ïðîèçâîëüíàÿ áåñêâàíòîðíàÿ ôîðìóëà ñèãíàòóðû PPr . Äëÿ ýòîé ôîðìóëû ìîæíî ïîñòðîèòü òàêîé n + 1 - ìåñòíûé ôóíêòîð Φ A (òåîðåìà 1.6), ÷òî Th ( U ) ⊢ ∀ x, ∀ z [ A ( z, x ) ≡ Φ A ( z, x ) = Λ] , òîãäà äëÿ ëþáîãî ñëîâàðíîãî ìíîãî÷ëåíà P ( x ) âåðíî Th ( U ) ⊢ ∀ x [ ∃ | P ( x ) | z A ( z, x ) ≡ ∃ | P ( x ) | u [Φ A ( u, x ) = Λ]] , à òàêæå Th ( U ) ⊢ ∀ x [ ∀ | P ( x ) | z A ( z, x ) ≡ ∀ | P ( x ) | u [Φ A ( u, x ) = Λ]] , òîãäà Th ( U ) ⊢ ∀ | P ( b ) | z A ( z, b ) ≡ ∀ | P ( b ) | u [Φ A ( u, b ) = Λ] . Ïóñòü A | = ∀ | P ( b ) | z A ( z, b ) , òîãäà A | = ∀ | P ( b ) | u [Φ A ( u, b ) = Λ] , òîãäà M | = ∀ | P ( b ) | u [Φ A ( u, b ) = Λ] , òîãäà M | = ∀ | P ( b ) | z A ( z, b ) = Λ . Äëÿ ôîðìóëû, êîòîðàÿ èìååò äâà è áîëåå îãðàíè÷åííûõ êâàíòîðîâ ∀ , äîêàçàòåëüñòâî ïðîâîäèòñÿ àíà- ëîãè÷íî. Êîíåö äîêàçàòåëüñòâà òåîðåìû . Òåîðåìà 6.10. Ñóùåñòâóåò òàêàÿ èíòåðïðåòàöèÿ A ôóíêòîðà U , ÷òî N P ( A ) ̸ = co − N P ( A ) , òîãäà äëÿ òåîðèè Th ( A ) íå âûïîëíÿåòñÿ òðåòèé ïóíêò òåîðåìû 6.7. Äîêàçàòåëüñòâî. Ðàññìîòðèì ôîðìóëó âèäà ∃ | x | y [ | x | = | y | ∧ U ( y ) = Λ] . Äëÿ ýòîé ôîðìóëû ìîæíî ïîñòðîèòü òàêóþ n - àëôàâèòíóþ èíòåðïðåòàöèþ A ôóíêòîðà U , ÷òî ïðè n ≥ 2 , äëÿ ëþáîé îãðàíè÷åííîé ∀ ôîðìóëû A ( x, z ) , ñèãíàòóðû PPr , âåðíî WordM A ̸ | = ∃ z ∀ x [ ∃ | x | y [ | x | = | y | ∧ U ( y ) = Λ] ≡ A ( x, z )] . 45 Êîíñòðóêöèÿ ïîñòðîåíèÿ îðàêóëà A èìååòñÿ â[13, p. 437]. Çàìå÷àíèå . Äëÿ èñ÷èñëåíèÿ CalcEq U ïîñòðîèòü îðàêóë A , âåñüìà ïðîñòî. Ñëåäñòâèå . N P̸ = co − N P . Äîêàçàòåëüñòâî. Èñïîëüçóåì òåîðåìû 6.5 - 6.10. P.S. ß ðàñïîëàãàþ äîêàçàòåëüñòâîì ñëåäóþùåãî ñîîòíîøåíèÿ êëàññîâ ñëîæíîñòè P =?( N P T co −N P ) , ïîäõîä ðåøåíèÿ ê êîòîðîìó, äàæå íå îáîçíà÷åí â ðàçäåëå "Äèñêðåòíàÿ ìàòåìàòèêà". The latest version of the presented study can be seen on the page https : // logicproof . org Ïóñòü PF n - êëàññ ôóíêöèé, êîòîðûå âû÷èñëèìû íà äåòåðìèíèðîâàííûõ ìàøèíàõ Òüþðèíãà ñ âõîä- íûì àëôàâèòîì { a 1 . . . . , a n } ( n ≥ 2 ) çà âðåìÿ ïîëèíîìèàëüíî çàâèñÿùåå îò äëèíû âõîäà. Èìååì PF n = PPr , ãäå PPr - n - àëôàâèòíûå ôóíêòîðû. Ïóñòü N = { Λ } S { m : S 1 ( S 1 ( , . . . , ( S 1 | {z } m − ðàç (Λ) , . . . , ) , m ≥ 1 } . Ïóñòü A = { a 1 , . . . , a n } , ãäå n ≥ 2 . Âñÿêàÿ ôóíêöèÿ f : N k → N , ïîðîæäàåò ñëîâàðíóþ ôóíêöèþ f w : ( A ∗ ) k → A ∗ ïî ïðàâèëó f w ( α 1 , . . . , α k ) = ⌜ f ( ⌜ α 1 ⌝ , . . . , ⌜ α k ⌝ ) ⌝ − 1 . Âñÿêàÿ ñëîâàðíàÿ ôóíêöèÿ f w : ( A ∗ ) k → A ∗ ïîðîæäàåò ôóíêöèþ f : N k → N , ïî ïðàâèëó f ( m 1 , . . . , m k ) = ⌜ f w ( ⌜ m 1 ⌝ − 1 , . . . , ⌜ m k ⌝ − 1 ) ⌝ . Ïóñòü P p - íàèìåíüøèé êëàññ ôóíêöèé, ñîäåðæàùèé ôóíêöèè: S 1 ( x ) , x + y , x . y , [ x/ 2] , [ √ x ] , x 2 , x | y | è çàìêíóòûé îòíîñèòåëüíî îïåðàöèè ñóïåðïîçèöèè è îãðàíè÷åííîé ðåêóðñèè âèäà f ( x, Λ) = g ( x ) , f ( x, p · y + i ) = h i ( x, y, f ( x, y )) , ãäå p ≥ 2 , 0 ≤ i < p ∀ x, y [ f ( x, y ) ≤ j ( x, y )] . Òåîðåìà ( A. Cobham[6]). P p = PF p . Òîãäà P p = PPr , ãäå PPr - ñîñòîèò èç p - àëôàâèòíûõ PPr ôóíêòîðîâ( p ≥ 2 ). Ïóñòü PF Space n - êëàññ ôóíêöèé, êîòîðûå âû÷èñëèìû íà äåòåðìèíèðîâàííûõ ìàøèíàõ Òüþðèíãà ñ âõîäíûì àëôàâèòîì { a 1 . . . . , a n } ( n ≥ 2 ) ñ ïîëèíîìèàëüíî îãðàíè÷åííîé ïàìÿòüþ, çàâèñÿùåé îò äëèíû âõîäà. Ïóñòü P Space - íàèìåíüøèé êëàññ ôóíêöèé, ñîäåðæàùèé ôóíêöèè: S 1 ( x ) , x + y , x . y , [ x/ 2] , [ √ x ] , x 2 , x | y | è çàìêíóòûé îòíîñèòåëüíî îïåðàöèè ñóïåðïîçèöèè è îãðàíè÷åííîé ðåêóðñèè âèäà f ( x, Λ) = g ( x ) , f ( x, y + 1) = h ( x, y, f ( x, y )) , 46 ∀ x, y [ f ( x, y ) ≤ j ( x, y )] . Èìååì PF Space n = P Space w . Èç ïîëó÷åííûõ ðåçóëüòàòîâ ïðåäñòàâëåííûõ â äàííîì èññëåäîâàíèè, ñëåäóåò PF n ⊊ PF Space n , ∀ p ≥ 2 âåðíî P p ⊊ P Space . 47 Ëèòåðàòóðà 1. F. W. v. Henke, K. Indermark, G. Rose, K. Weihrauch. On Primitive Recursive Wordfunctions. Computing, vol 15, 1975, p. 217-234. 2. Øàíèí Í.À. Âñòóïèòåëüíàÿ ñòàòüÿ. Î ðåêóðñèâíîì ìàòåìàòè÷åñêîì àíàëèçå è èñ÷èñëåíèè àðèô- ìåòè÷åñêèõ ðàâåíñòâ Ð.Ë. Ãóäñòåéíà -  êí.: Ð.Ë. Ãóäñòåéí. Ðåêóðñèâíûé ìàòåìàòè÷åñêèé àíàëèç. Èçäà- òåëüñòâî Ì."Íàóêà 1970. ñ. 7-75. 3. R.L. Goodstein. Recursive number theory. Amstrdam, 1957. p. 187. 4. Êàððè Õ.Á. Ôîðìàëèçàöèÿ ðåêóðñèâíîé àðèôìåòèêè -  êí.: Ð.Ë. Ãóäñòåéí. Ðåêóðñèâíûé ìàòåìà- òè÷åñêèé àíàëèç. Èçäàòåëüñòâî Ì."Íàóêà"ñòð. 437-469. 5. Ìàëüöåâ À. È. Àëãîðèòìû è ðåêóðñèâíûå ôóíêöèè. Ì., 1986. ñ. 368. 6. A. Cobham The intrinsic computational diculty of functions. Proc. of the 1964 International Congress for Logic, Methology, and the Philosophy of Sciens, North Holand Publishing Co., Amsterdam, p. 24-30. 7. L, J. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science vol 3 1977, p.1-22. 8. Åðøîâ Þ.Ë. Ïàëþòèí Å.À. Ìàòåìàòè÷åñêàÿ ëîãèêà. Ì., ÔÈÇÌÀÒËÈÒ, 2011.ñ. 356. 9. Ã. Êåéñëåð, ×ýí. ×.×. Òåîðèÿ ìîäåëåé, Ìîñêâà, Èçä-âî ÌÈÐ, 1977, ñ. 614. 10. Ñïðàâî÷íàÿ êíèãà ïî ìàòåìàòè÷åñêîé ëîãèêå, ÷. 1. Òåîðèÿ ìîäåëåé, Ìîñêâà "Íàóêà 1982, ñ. 391. 11. J. Donald Monk. Mathematical Logic. Springer - Verlag, New York, 1976. 12. Book R.V., Long T.J., Selman A.L. Quantitative relativization of complexity classes. SIAM J. Comput. vol 13 No 3 August 1984, p. 461-487. 13. Baker T, Gill J. Solovay R. Relativization of the P =? N P Question. SIAM J. Comput. vol 4 December 1971, p. 431-442. 48 Ïðèëîæåíèå Ñîñòàâèì k ≥ 3 ìåñòíûé ôóíêòîð âèäà [ J ConcatI k 1 [ J ConcatI k 2 . . . [ J ConcatI k k − 1 I k k ] . . . ] . Äëÿ ýòîãî ôóíêòîðà â èñ÷èñëåíèè CalcEq âûâîäèìî ðàâåíñòâî [ J ConcatI k 1 [ J ConcatI k 2 . . . [ J ConcatI k k − 1 I k k ] . . . ]( x 1 . . . x k ) = Concat ( x 1 , Concat ( x 2 , . . . Concat ( x k − 1 , x k ) . . . )) . Ïóñòü Concat k ⇌ [ J ConcatI k 1 [ J ConcatI k 2 . . . [ J ConcatI k k − 1 I k k ] . . . ] , ïðè k ≥ 3 , òîãäà ⊢ Concat k ( x 1 , . . . x k ) = Concat ( x 1 , Concat ( x 2 , . . . Concat ( x k − 1 , x k ) . . . )) . Ïðè k = 2 , ïîëó÷èì Concat 2 ⇌ Concat , ⊢ Concat 2 ( x 1 , x 2 ) = Concat ( x 1 , x 2 ) ïðè k = 1 Concat 1 ⇌ I 1 1 , ⊢ Concat 1 ( x ) = x . Èìååì ([ J ConcatI k k − 1 I k k ]) ∗ = [ J ( Concat ) ∗ I k +1 1 ( I k k − 1 ) ∗ ( I k k ) ∗ ] = [ J [ J ConcatI 3 2 I 3 3 ] I k +1 1 ( I k k − 1 ) ∗ ( I k k ) ∗ ] = [ J Concat ( I k k − 1 ) ∗ ( I k k ) ∗ ] = [ J ConcatI k +1 k I k +1 k +1 ] , òîãäà ( Concat k ) ∗ = ([ J ConcatI k 1 [ J ConcatI k 2 . . . [ J ConcatI k k − 1 I k k ] . . . ] ∗ = [ J ConcatI k +1 2 [ J ConcatI k +1 3 . . . [ J ConcatI k +1 k I k +1 k +1 ] . . . ] , òîãäà ⊢ ( Concat k ) ∗ ( x 1 , x 2 , . . . , x k +1 ) = Concat k ( x 2 , . . . , x k +1 ) Èìååì (Θ [ J ΦΨ 1 ,..., Ψ k ] ) ∗ = ( J [ Concat k +1 [ J Θ Φ Ψ 1 , . . . Ψ k ]Θ Ψ 1 , . . . , Θ Ψ k ]) ∗ , òîãäà ( J [ Concat k +1 [ J Θ Φ Ψ 1 , . . . Ψ k ]Θ Ψ 1 , . . . , Θ Ψ k ]) ∗ = [ J ( Concat k +1 ) ∗ I n +1 1 ([ J Θ Φ Ψ 1 , . . . Ψ k ]) ∗ (Θ Ψ 1 ) ∗ , . . . , (Θ Ψ k ) ∗ ] . Äàëåå ([ J Θ Φ Ψ 1 , . . . Ψ k ]) ∗ = [ J (Θ Φ ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . (Ψ k ) ∗ ] , òîãäà ⊢ ([ J Θ Φ Ψ 1 , . . . Ψ k ]) ∗ ( x 1 , x 2 , . . . , x n +1 ) = [ J (Θ Φ ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . (Ψ k ) ∗ ]( x 1 , x 2 , . . . , x n +1 ) , òîãäà ⊢ [ J (Θ Φ ) ∗ I n +1 1 (Ψ 1 ) ∗ , . . . (Ψ k ) ∗ ]( x 1 , x 2 , . . . , x n +1 ) = (Θ Φ ) ∗ ( x 1 , (Ψ 1 ) ∗ ( x 1 , x 2 , . . . , x n ) , . . . (Ψ k ) ∗ ( x 1 , x 2 , . . . , x n )) , òîãäà ⊢ ([ J Θ Φ Ψ 1 , . . . Ψ k ]) ∗ ( x 1 , x 2 , . . . , x n +1 ) = (Θ Φ ) ∗ ( x 1 , (Ψ 1 ) ∗ ( x 1 , x 2 , . . . , x n ) , . . . (Ψ k ) ∗ ( x 1 , x 2 , . . . , x n )) , òî- ãäà ⊢ (Θ [ J ΦΨ 1 ,..., Ψ k ] ) ∗ ( x 1 , x 2 , . . . , x n +1 ) = ( Concat k +1 ) ∗ ( x 1 , ([ J Θ Φ Ψ 1 , . . . Ψ k ]) ∗ ( x 1 , x 2 , . . . , x n +1 ) , (Θ Ψ 1 ) ∗ ( x 1 , x 2 , . . . , x n +1 ) , (Θ Ψ k ) ∗ ( x 1 , x 2 , . . . , x n +1 )) , òîãäà ⊢ (Θ [ J ΦΨ 1 ,..., Ψ k ] ) ∗ ( x 1 , x 2 , . . . , x n +1 ) = Concat k +1 (([ J Θ Φ Ψ 1 , . . . Ψ k ]) ∗ ( x 1 , x 2 , . . . , x n +1 ) , (Θ Ψ 1 ) ∗ ( x 1 , x 2 , . . . , x n +1 ) , . . . , (Θ Ψ k ) ∗ ( x 1 , x 2 , . . . , x n +1 )) , òîãäà ⊢ (Θ [ J ΦΨ 1 ,..., Ψ k ] ) ∗ ( x 1 , x 2 , . . . , x n +1 ) = Concat k +1 ((Θ Φ ) ∗ ( x 1 , (Ψ 1 ) ∗ ( x 1 , x 2 , . . . , x n +1 ) , . . . (Ψ k ) ∗ ( x 1 , x 2 , . . . , x n +1 )) , (Θ Ψ 1 ) ∗ ( x 1 , x 2 , . . . , x n +1 ) , . . . , (Θ Ψ k ) ∗ ( x 1 , x 2 , . . . , x n +1 )) , òîãäà ⊢ (Θ [ J ΦΨ 1 ,..., Ψ k ] ) ∗ ( x 1 , x 2 , . . . , x n +1 ) = Concat ((Θ Φ ) ∗ ( x 1 , (Ψ 1 ) ∗ ( x 1 , x 2 , . . . , x n +1 ) , . . . (Ψ k ) ∗ ( x 1 , x 2 , . . . , x n +1 )) , Concat ((Θ Ψ 1 ) ∗ ( x 1 , x 2 , . . . , x n +1 ) , . . . , Concat ((Θ Ψ k − 1 ) ∗ ( x 1 , x 2 , . . . , x n +1 ) , . . . , (Θ Ψ k ) ∗ ( x 1 , x 2 , . . . , x n +1 )) , . . . , ) . Äàíû Φ - n ≥ 1 - ìåñòíûé ôóíêòîð, Ψ 1 , . . . , Ψ k - ( n + 2) - ìåñòíûå ôóíêòîðû. Ñîñòàâèì ôóíêòîð [ R ΦΨ 1 , . . . , Ψ k ] - ( n + 1) - ìåñòíûé. Ïî ýòîìó ôóíêòîðó ïîñòðîèì ôóíêòîð Θ [ R ΦΨ 1 ,..., Ψ k ] . Ïóñòü x ⇋ x 1 , . . . , x n , λ ⇋ [ J [ R ΦΨ 1 , . . . , Ψ k ] I n +2 1 , . . . , I n +2 n +1 ] . Èìååì ⊢ λ ( x, z, u ) = [ J [ R ΦΨ 1 , . . . , Ψ k ] I n +2 1 , . . . , I n +2 n +1 ]( x, z, u ) = [ R ΦΨ 1 , . . . , Ψ k ]( x, z ) . 49 Ïóñòü ˜ Ψ i ⇋ [ J Concat [ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ] , ˜ Ψ i - ( n + 2) - ìåñòíûé ôóíòîð. Èìååì: ⊢ ˜ Ψ i ( x, z, u ) ⇋ [ J Concat [ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ]( x, z, u ) = Concat ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]( x, z, u ) , I n +2 n +2 ( x, z, u )) = Concat (Θ Ψ i ( x, z, λ ( x, z, u )) , u ) = Concat (Θ Ψ i ( x, z, [ R ΦΨ 1 , . . . , Ψ k ]( x, z )) , u ) . Èòàê, ⊢ ˜ Ψ i ( x, z, u ) = Concat (Θ Ψ i ( x, z, [ R ΦΨ 1 , . . . , Ψ k ]( x, z )) , u ) Ïóñòü Θ [ R ΦΨ 1 ,..., Ψ k ] ⇋ [ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ] . Îïðåäåëÿþùèå ðàâåíñòâà: ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = [ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ]( x, Λ) = Θ Φ ( x ) ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = [ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ]( x, S i ( z )) = ˜ Ψ i ( x, z, [ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ]( x, z )) = ˜ Ψ i ( x, z, Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) = Concat (Θ Ψ i ( x, z, [ R ΦΨ 1 , . . . , Ψ k ]( x, z )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) . Èòàê, èìååì ñëåäóþùèå îïðåäåëÿþùèå ðàâåíñòâà äëÿ ôóíòîðà Θ [ R ΦΨ 1 ,..., Ψ k ] : ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = Θ Φ ( x ) , ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( y )) = Concat (Θ Ψ i ( x, z, [ R ΦΨ 1 , . . . , Ψ k ]( x, z )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) , ãäå i ≤ k , ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( y )) = Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, y ) , ãäå i > k . Äàëåå ( λ ) ∗ ⇋ ([ J [ R ΦΨ 1 , . . . , Ψ k ] I n +2 1 , . . . , I n +2 n +1 ]) ∗ . ([ J [ R ΦΨ 1 , . . . , Ψ k ] I n +2 1 , . . . , I n +2 n +1 ]) ∗ = [ J ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ I n +3 1 ( I n +2 1 ) ∗ , . . . , ( I n +2 n +1 ) ∗ ] Èìååì: ⊢ ( λ ) ∗ ( y , x, z, u ) = [ J ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ I n +3 1 ( I n +2 1 ) ∗ , . . . , ( I n +2 n +1 ) ∗ ]( y , x, z, u ) = ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( I n +3 1 ( y , x, z, u ) , ( I n +2 1 ) ∗ ( y , x, z, u ) , . . . , ( I n +2 n +1 ) ∗ ( y , x, z, u )) = ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z ) Èòàê, ⊢ ( λ ) ∗ ( y , x, z, u ) = ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z ) . ( ˜ Ψ i ) ∗ ⇋ ([ J Concat [ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ]) ∗ = [ J ( Concat ) ∗ I n +3 1 ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ ( I n +2 n +2 ) ∗ ] = [ J [ J ConcatI 3 2 I 3 3 ] I n +3 1 ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ ( I n +2 n +2 ) ∗ ] . ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ = [ J (Θ Ψ i ) ∗ I n +3 1 ( I n +2 1 ) ∗ , . . . , ( I n +2 n +1 ) ∗ ( λ ) ∗ ] , ⊢ ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ ( y , x, z, u ) = [ J (Θ Ψ i ) ∗ I n +3 1 ( I n +2 1 ) ∗ , . . . , ( I n +2 n +1 ) ∗ ( λ ) ∗ ]( y , x, z, u ) = (Θ Ψ i ) ∗ ( I n +3 1 ( y , x, z, u ) , ( I n +2 1 ) ∗ ( y , x, z, u ) , . . . , ( I n +2 n +1 ) ∗ ( y , x, z, u ) , ( λ ) ∗ ( y , x, z, u )) = (Θ Ψ i ) ∗ ( y , x, z, ( λ ) ∗ ( y , x, z, u )) = (Θ Ψ i ) ∗ ( y , x, z, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z )) . Èòàê, ⊢ ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ ( y , x, z, u ) = (Θ Ψ i ) ∗ ( y , x, z, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z )) . ⊢ ( ˜ Ψ i ) ∗ ( y , x, z, u ) = [ J ( Concat ) ∗ I n +3 1 ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ ( I n +2 n +2 ]) ∗ ]( y , x, z, u ) = [ J [ J ConcatI 3 2 I 3 3 ] I n +3 1 ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ ( I n +2 n +2 ) ∗ ]( y , x, z, u ) = [ J ConcatI 3 2 I 3 3 ]( I n +3 1 ( y , x, z, u ) , ([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ ( y , x, z, u ) , ( I n +2 n +2 ) ∗ ( y , x, z, u )) = Concat (([ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]) ∗ ( y , x, z, u ) , ( I n +2 n +2 ) ∗ ( y , x, z, u )) = Concat ((Θ Ψ i ) ∗ ( y , x, z, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z )) , u ) . Èòàê, ⊢ ( ˜ Ψ i ) ∗ ( y , x, z, u ) = Concat ((Θ Ψ i ) ∗ ( y , x, z, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z )) , u ) . 50 (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ⇋ ([ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ = [ R (Θ Φ ) ∗ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ] ⊢ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, Λ) = ([ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ ( y , x, Λ) = [ R (Θ Φ ) ∗ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( y , x, Λ) = (Θ Φ ) ∗ ( y , x ) ⊢ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, S i ( z )) = ([ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ ( y , x, S i ( z )) = [ R (Θ Φ ) ∗ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( y , x, S i ( z )) = ( ˜ Ψ i ) ∗ ( y , x, z, [ R (Θ Φ ) ∗ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( y , x, z )) = ( ˜ Ψ i ) ∗ ( y , x, z, ([ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k )]) ∗ ( y , x, z )) = Concat ((Θ Ψ i ) ∗ ( y , x, z, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z )) , ([ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k )]) ∗ ( y , x, z )) = Concat ((Θ Ψ i ) ∗ ( y , x, z, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z )) , (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, z )) . Èòàê, ⊢ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, S i ( z )) = Concat ((Θ Ψ i ) ∗ ( y , x, z, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z )) , (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, z )) . Òàêèì îáðàçîì ïîëó÷èì : ⊢ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, Λ) = (Θ Φ ) ∗ ( y , x ) , ⊢ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, S i ( z )) = Concat ((Θ Ψ i ) ∗ ( y , x, z, ([ R ΦΨ 1 , . . . , Ψ k ]) ∗ ( y , x, z )) , (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, z )) , ïðè i ≤ k . ⊢ (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, S i ( z )) = (Θ [ R ΦΨ 1 ,..., Ψ k ] ) ∗ ( y , x, z ) , ïðè i > k . Ïóñòü Ψ 1 , . . . Ψ k - 2 - ìåñòíûå ôóíòîðû, α - íåêîòðîå p − àëôàâèòíîå àðãóìåíòíîå ñëîâî. Ñîñòàâèì ôóíêòîð [ Rα Ψ 1 , . . . , Ψ k ] Ïîñòðîèì ôóíêòîð Θ [ Rα Ψ 1 ,..., Ψ k ] . Ïóñòü γ ⇋ [ J [ Rα Ψ 1 , . . . , Ψ k ] I 2 1 ] . Èìååì ⊢ γ ( x, z ) = [ J [ Rα Ψ 1 , . . . , Ψ k ] I 2 1 ]( x, z ) = [ Rα Ψ 1 , . . . , Ψ k ]( x ) . Ïóñòü ˜ Ψ i ⇋ [ J Concat [ J Θ Ψ i I 2 1 γ ] I 2 2 ] . Èìååì: ⊢ ˜ Ψ i ( x, z ) ⇋ [ J Concat [ J Θ Ψ i I 2 1 γ ] I 2 2 ]( x, z ) = Concat ([ J Θ Ψ i I 2 1 γ ]( x, z ) , I 2 2 ( x, z )) = Concat (Θ Ψ i ( x, γ ( x, z )) , z ) = Concat (Θ Ψ i ( x, [ Rα Ψ 1 , . . . , Ψ k ]( x )) , z ) . Èòàê, ⊢ ˜ Ψ i ( x, z ) = Concat (Θ Ψ i ( x, [ Rα Ψ 1 , . . . , Ψ k ]( x )) , z ) . Ïóñòü Θ [ Rα Ψ 1 ,..., Ψ k ] ⇋ [ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ] , òîãäà ⊢ Θ [ Rα Ψ 1 ,..., Ψ k ] (Λ) = [ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ](Λ) = Λ , ⊢ Θ [ Rα Ψ 1 ,..., Ψ k ] ( S k ( x )) = [ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ]( S i ( x )) = ˜ Ψ i ( x, [ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ]( x )) = ˜ Ψ i ( x, Θ [ Rα Ψ 1 ,..., Ψ k ] ( x )) = Concat (Θ Ψ i ( x, [ Rα Ψ 1 , . . . , Ψ k ]( x )) , Θ [ Rα Ψ 1 ,..., Ψ k ] ( x )) . Èòàê, èìååì ñëåäóþùèå îïðåäåëÿþùèå ðàâåíñòâà äëÿ ôóíòîðà Θ [ Rα Ψ 1 ,..., Ψ k ] : ⊢ Θ [ Rα Ψ 1 ,..., Ψ k ] (Λ) = Λ ⊢ Θ [ Rα Ψ 1 ,..., Ψ k ] ( S i ( x )) = Concat (Θ Ψ i ( x, [ Rα Ψ 1 , . . . , Ψ k ]( x )) , Θ [ Rα Ψ 1 ,..., Ψ k ] ( x )) , ãäå i ≤ k . ⊢ Θ [ Rα Ψ 1 ,..., Ψ k ] ( S i ( x )) = Θ [ Rα Ψ 1 ,..., Ψ k ] ( x ) , ãäå i > k . Âûïèøåì îïðåäåëÿþùèå ñîîòíîøåíèÿ äëÿ ôóíêòîðà (Θ [ Rα Ψ 1 ,..., Ψ k ] ) ∗ ⇋ ([ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ : ([ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ = ([ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ] , ⊢ ([ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ ( x, y ) = [ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, y ) , 51 ⊢ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, Λ) = Const 1 Λ ( x ) = Λ ⊢ [ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, S i ( y )) = ( ˜ Ψ i ) ∗ ( x, y, [ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, y )) , òîãäà ⊢ [ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, S i ( y )) = ( ˜ Ψ i ) ∗ ( x, y, ([ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ ( x, y )) , ⊢ ([ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ ( x, S i ( y )) = ( ˜ Ψ i ) ∗ ( x, y, ([ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ]) ∗ ( x, y )) , ⊢ (Θ [ Rα Ψ 1 ,..., Ψ k ] ) ∗ ( x, S i ( y )) = ( ˜ Ψ i ) ∗ ( x, y, (Θ [ Rα Ψ 1 ,..., Ψ k ] ) ∗ ( x, y )) , Äàëåå ( γ ) ∗ ⇋ ([ J [ Rα Ψ 1 , . . . , Ψ k ] I 2 1 ]) ∗ , ([ J [ Rα Ψ 1 , . . . , Ψ k ] I 2 1 ]) ∗ = [ J ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ I 3 1 ( I 2 1 ) ∗ ] ( I 2 1 ) ∗ = [ J I 2 1 I 3 2 I 3 3 ] ⊢ ( I 2 1 ) ∗ ( x, y, z ) = [ J I 2 1 I 3 2 I 3 3 ]( x, y, z ) = I 2 1 ( I 3 2 ( x, y, z ) , I 3 3 ( x, y, z )) = y ⊢ ([ J [ Rα Ψ 1 , . . . , Ψ k ] I 2 1 ]) ∗ ( x, y, z ) = [ J ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ I 3 1 ( I 2 1 ) ∗ ]( x, y, z ) ⊢ [ J ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ I 3 1 ( I 2 1 ) ∗ ]( x, y, z ) = ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( I 3 1 ( x, y, z ) , ( I 2 1 ) ∗ ( x, y, z )) ⊢ ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( I 3 1 ( x, y, z ) , ( I 2 1 ) ∗ ( x, y, z )) = ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( x, y ) ⊢ ( γ ) ∗ ( x, y, z ) = ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( x, y ) . ( ˜ Ψ i ) ∗ ⇋ ([ J Concat [ J Θ Ψ i I 2 1 γ ] I 2 2 ]) ∗ , ([ J Concat [ J Θ Ψ i I 2 1 γ ] I 2 2 ]) ∗ = [ J ( Concat ) ∗ I 3 1 ([ J Θ Ψ i I 2 1 γ ]) ∗ ( I 2 2 ) ∗ ] , ( Concat ) ∗ = [ J ConcatI 3 2 I 3 2 ] , ([ J Θ Ψ i I 2 1 γ ]) ∗ = [ J (Θ Ψ i ) ∗ I 3 1 ( I 2 1 ) ∗ ( γ ) ∗ ] , ( I 2 1 ) ∗ = [ J I 2 1 I 3 2 I 3 3 ] ( I 2 2 ) ∗ = [ J I 2 2 I 3 2 I 3 3 ] Èìååì: ⊢ ( I 2 1 ) ∗ ( x, y, z ) = [ J I 2 1 I 3 2 I 3 3 ]( x, y, z ) = y , ⊢ ( I 2 2 ) ∗ ( x, y, z ) = [ J I 2 2 I 3 2 I 3 3 ]( x, y, z ) = z ⊢ ( Concat ) ∗ ( x, y, z ) = [ J ConcatI 3 2 I 3 2 ]( x, y, z ) = Concat ( y, z ) , ⊢ ( ˜ Ψ i ) ∗ ( x, y, z ) = ([ J Concat [ J Θ Ψ i I 2 1 γ ] I 2 2 ]) ∗ ( x, y, z ) = [ J ( Concat ) ∗ I 3 1 ([ J Θ Ψ i I 2 1 γ ]) ∗ ( I 2 2 ) ∗ ]( x, y, z ) = , ⊢ ([ J Θ Ψ i I 2 1 γ ]) ∗ ( x, y, z ) = [ J (Θ Ψ i ) ∗ I 3 1 ( I 2 1 ) ∗ ( γ ) ∗ ]( x, y, z ) = (Θ Ψ i ) ∗ ( I 3 1 ( x, y, z ) , ( I 2 1 ) ∗ ( x, y, z ) , ( γ ) ∗ ( x, y, z )) , ⊢ (Θ Ψ i ) ∗ ( I 3 1 ( x, y, z ) , ( I 2 1 ) ∗ ( x, y, z ) , ( γ ) ∗ ( x, y, z )) = (Θ Ψ i ) ∗ ( x, y, ( γ ) ∗ ( x, y, z )) ⊢ [ J ( Concat ) ∗ I 3 1 ([ J Θ Ψ i I 2 1 γ ]) ∗ ( I 2 2 ) ∗ ]( x, y, z ) = ( Concat ) ∗ ( I 3 1 ( x, y, z ) , ([ J Θ Ψ i I 2 1 γ ]) ∗ ( x, y, z ) , ( I 2 2 ) ∗ ( x, y, z )) ⊢ ( Concat ) ∗ ( I 3 1 ( x, y, z ) , ([ J Θ Ψ i I 2 1 γ ]) ∗ ( x, y, z ) , ( I 2 2 ) ∗ ( x, y, z )) = Concat (([ J Θ Ψ i I 2 1 γ ]) ∗ ( x, y, z ) , z ) ⊢ Concat (([ J Θ Ψ i I 2 1 γ ]) ∗ ( x, y, z ) , z ) = Concat ((Θ Ψ i ) ∗ ( x, y, ( γ ) ∗ ( x, y, z )) , z ) ⊢ Concat ((Θ Ψ i ) ∗ ( x, y, ( γ ) ∗ ( x, y, z )) , z ) = Concat ((Θ Ψ i ) ∗ ( x, y, ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( x, y )) , z ) . 52 Òàêèì îáðàçîì ïîëó÷èì ⊢ ( ˜ Ψ i ) ∗ ( x, y, z ) = Concat ((Θ Ψ i ) ∗ ( x, y, ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( x, y )) , z ) , òîãäà ⊢ ( ˜ Ψ i ) ∗ ( x, y, [ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, y )) = Concat ((Θ Ψ i ) ∗ ( x, y, ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( x, y )) , [ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, y )) , òîãäà ⊢ [ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, S i ( y )) = Concat ((Θ Ψ i ) ∗ ( x, y, ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( x, y )) , [ R Const 1 Λ ( ˜ Ψ 1 ) ∗ , . . . , ( ˜ Ψ k ) ∗ ]( x, y )) . Òàêèì îáðàçîì ïîëó÷èì ⊢ (Θ [ Rα Ψ 1 ,..., Ψ k ] ) ∗ ( x, Λ) = Λ ⊢ (Θ [ Rα Ψ 1 ,..., Ψ k ] ) ∗ ( x, S i ( y )) = Concat ((Θ Ψ i ) ∗ ( x, y, ([ Rα Ψ 1 , . . . , Ψ k ]) ∗ ( x, y )) , (Θ [ Rα Ψ 1 ,..., Ψ k ] ) ∗ ( x, y )) , ïðè i ≤ k . ⊢ (Θ [ Rα Ψ 1 ,..., Ψ k ] ) ∗ ( x, S i ( y )) = (Θ [ Rα Ψ 1 ,..., Ψ k ] ) ∗ ( x, y ) , ïðè i > k . Äëÿ ëþáîãî ôóíêòîðà Φ äëÿ ëþáîãî èíòåïðåòàöèîííîãî ìíîæåñòâà A äîêàæåì WordM A | = ∀ α Θ Φ ( α ) ≈ Θ Θ Φ ( α ) . Âûïèøåì çíà÷åíèå îïåðàòîðà Θ : äëÿ èñõîäíûõ ôóíêòîðîâ: S k , Z , δ , Length , . , Concat , D , I n k , U : Θ S k = Z , Θ Z = Z , Θ δ = Z , Θ Length = [ J ZI 2 2 ] , Θ . = [ J ZI 2 2 ] , Θ Concat = [ J ZI 2 2 ] , Θ D = [ J ZI 2 2 ] , Θ I n k = [ J ZI n k ] , Θ U = [ J cI 1 1 U ] . äëÿ ôóíêòîðà [ J ΦΨ 1 , . . . , Ψ k ] Θ [ J ΦΨ 1 ,..., Ψ k ] ⇌ [ J Concat k +1 [ J Θ Φ Ψ 1 . . . Ψ k ] , Θ Ψ 1 . . . Θ Ψ k ] . äëÿ ôóíêòîðà [ Rα Ψ 1 , . . . , Ψ k ] Θ [ Rα Ψ 1 ,..., Ψ k ] ⇋ [ R Λ ˜ Ψ 1 , . . . , ˜ Ψ k ] , äëÿ ôóíêòîðà Θ [ R ΦΨ 1 ,..., Ψ k ] Θ [ R ΦΨ 1 ,..., Ψ k ] ⇋ [ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ] . Äëÿ ëþáîãî ôóíêòîðà Ψ àëôàâèòà L âåðíî ∀ α ⊢ Θ Ψ ( α ) = Λ . Ïðè èñïîëüçîâàíèè ïðàâèëà Ãóäñòåéíà âåðíî ⊢ Θ Ψ ( x ) = Λ . Äîêàæåì ⊢ Θ U ( x ) = Θ Θ U ( x ) : Θ U = [ J cI 1 1 U ] , Θ Θ U = Θ [ J cI 1 1 U ] = [ J Concat 3 [ J Θ c I 1 1 U ]Θ I 1 1 Θ U ] , ó÷èòûâàÿ [ J Θ c I 1 1 U ] = [ J ZI 1 1 ] , Θ I 1 1 = [ J ZI 1 1 ] , ïîëó÷èì Θ Θ U = Θ U , òîãäà ⊢ Θ U ( x ) = Θ Θ U ( x ) . Èíäóêöèîííîå ïðåäïîëîæåíèå : a . Ïóñòü äëÿ ôóíêòîðà Φ âåðíî WordM A | = ∀ x [Θ Φ ( x ) ≈ Θ Θ Φ ( x )] . 53 b . Ïóñòü äëÿ ôóíêòîðîâ Ψ 1 . . . , Ψ k âåðíî WordM A | = ∀ y 1 , . . . , ∀ y n [Θ Ψ i ( y 1 , . . . , y n ) ≈ Θ Θ Ψ i ( y 1 , . . . , y n )] Äîêàæåì WordM A | = ∀ y 1 , . . . ∀ y n Θ [ J ΦΨ 1 ,..., Ψ k ] ( y 1 , . . . , y n ) ≈ Θ Θ [ J ΦΨ 1 ,..., Ψ k ] ( y 1 , . . . , y n ) . Èìååì Θ [ J ΦΨ 1 ,..., Ψ k ] = [ J Concat k +1 [ J Θ Φ Ψ 1 , . . . , Ψ k ]Θ Ψ 1 , . . . , Θ Ψ k ] , òîãäà ⊢ Θ [ J ΦΨ 1 ,..., Ψ k ] ( y ) = [ J Concat k +1 [ J Θ Φ Ψ 1 , . . . , Ψ k ]Θ Ψ 1 , . . . , Θ Ψ k ]( y ) = Concat k +1 (Θ Φ (Ψ 1 ( y ) , . . . , Ψ k ( y )) , Θ Ψ 1 ( y ) , . . . Θ Ψ k ( y )) ( A ). Âû÷èñëèì Θ [ J Θ Φ Ψ 1 ... Ψ k ] : Θ [ J Θ Φ Ψ 1 ... Ψ k ] = [ J Concat k +1 [ J Θ Θ Φ Ψ 1 . . . Ψ k ]Θ Ψ 1 . . . Θ Ψ k ] , òîãäà ⊢ Θ [ J Θ Φ Ψ 1 ... Ψ k ] ( y ) = [ J Concat k +1 [ J Θ Θ Φ Ψ 1 . . . Ψ k ]Θ Ψ 1 . . . Θ Ψ k ]( y ) = Concat k +1 (Θ Θ Φ (Ψ 1 ( y ) , . . . , Ψ k ( y )) , Θ Ψ 1 ( y ) , . . . Θ Ψ k ( y )) . Ó÷èòûâàÿ èíäóêöèîííîå ïðåäïîëîæåíèå WordM A | = ∀ x [Θ Φ ( x ) ≈ Θ Θ Φ ( x )] , ïîëó÷èì WordM A | = ∀ y [Θ [ J Θ Φ Ψ 1 ... Ψ k ] ( y ) ≈ Concat k +1 (Θ Φ (Ψ 1 ( y ) , . . . , Ψ k ( y )) , Θ Ψ 1 ( y ) , . . . Θ Ψ k ( y ))] . Ó÷èòûâàÿ èíäóêöèîííîå ïðåäïîëîæåíèå WordM A | = ∀ y 1 , . . . , ∀ y n [Θ Ψ i ( y 1 , . . . , y n ) ≈ Θ Θ Ψ i ( y 1 , . . . , y n )] , ïîëó÷èì WordM A | = ∀ y [ Concat (Θ [ J Θ Φ Ψ 1 ... Ψ k ] ( y ) , Θ Θ Ψ 1 ( y )) ≈ Concat k +1 (Θ Φ (Ψ 1 ( y ) , . . . , Ψ k ( y )) , Θ Ψ 1 ( y ) , . . . Θ Ψ k ( y ))] , WordM A | = ∀ y [ Concat 3 (Θ [ J Θ Φ Ψ 1 ... Ψ k ] ( y ) , Θ Θ Ψ 1 ( y ) , Θ Θ Ψ 2 ( y )) ≈ Concat k +1 (Θ Φ (Ψ 1 ( y ) , . . . , Ψ k ( y )) , Θ Ψ 1 ( y ) , . . . Θ Ψ k ( y ))] ,..., WordM A | = ∀ y [ Concat k +1 (Θ [ J Θ Φ Ψ 1 ... Ψ k ] ( y ) , Θ Θ Ψ 1 ( y ) , Θ Θ Ψ 2 ( y ) , . . . , Θ Θ Ψ k ( y )) ≈ Concat k +1 (Θ Φ (Ψ 1 ( y ) , . . . , Ψ k ( y )) , Θ Ψ 1 ( y ) , . . . Θ Ψ k ( y ))] , ó÷èòûâàÿ ( A ), ïîëó÷èì WordM A | = ∀ y [ Concat k +1 (Θ [ J Θ Φ Ψ 1 ... Ψ k ] ( y ) , Θ Θ Ψ 1 ( y ) , Θ Θ Ψ 2 ( y ) , . . . , Θ Θ Ψ k ( y )) ≈ Θ [ J ΦΨ 1 ,..., Ψ k ] ( y ) ( B ) Âû÷èñëèì Θ Θ [ J ΦΨ 1 ,..., Ψ k ] : Θ Θ [ J ΦΨ 1 ,..., Ψ k ] = Θ [ J Concat k +1 [ J Θ Φ Ψ 1 ,..., Ψ k ]Θ Ψ 1 ,..., Θ Ψ k ] Θ [ J Concat k +1 [ J Θ Φ Ψ 1 ,..., Ψ k ]Θ Ψ 1 ,..., Θ Ψ k ] = [ J Concat k +2 [ J Θ Concat k + 1 [ J Θ Φ Ψ 1 , . . . , Ψ k ]Θ Ψ 1 , . . . , Θ Ψ k ]Θ [ J Θ Φ Ψ 1 ,..., Ψ k ] Θ Θ Ψ 1 . . . , Θ Θ Ψ k ] , ó÷èòûâàÿ Θ Concat k + 1 [ J Θ Φ Ψ 1 , . . . , Ψ k ]Θ Ψ 1 , . . . , Θ Ψ k ] = [ J ZI n 1 ] , ïîëó÷èì Θ Θ [ J ΦΨ 1 ,..., Ψ k ] = [ J Concat k +1 Θ [ J Θ Φ Ψ 1 ,..., Ψ k ] Θ Θ Ψ 1 . . . , Θ Θ Ψ k ] , òîãäà ⊢ Θ Θ [ J ΦΨ 1 ,..., Ψ k ] ( y ) = [ J Concat k +1 Θ [ J Θ Φ Ψ 1 ,..., Ψ k ] Θ Θ Ψ 1 . . . , Θ Θ Ψ k ]( y ) = Concat k +1 (Θ [ J Θ Φ Ψ 1 ,..., Ψ k ] ( y ) , Θ Θ Ψ 1 ( y ) , . . . Θ Θ Ψ k ( y )) , ó÷èòûâàÿ ( B )ïîëó÷èì WordM A | = ∀ y [Θ Θ [ J ΦΨ 1 ,..., Ψ k ] ( y ) ≈ Θ [ J ΦΨ 1 ,..., Ψ k ] ( y ) Âû÷èñëèì Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ⇋ Θ [ R Θ Φ ˜ Ψ 1 ,..., ˜ Ψ k ] . Èìååì ˜ Ψ i ⇋ [ J Concat [ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ] . Âû÷èñëèì Θ [ J Θ Ψ i I n +2 1 ,..., I n +2 n +1 λ ] : Θ [ J Θ Ψ i I n +2 1 ,..., I n +2 n +1 λ ] = [ J Concat n +3 [ J Θ Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]Θ I n +2 1 , . . . , Θ I n +2 n +1 Θ λ ] , ó÷èòûâàÿ Θ I n +2 i = [ J ZI n +1 1 ] 54 ïîëó÷èì Θ [ J Θ Ψ i I n +2 1 ,..., I n +2 n +1 λ ] = [ J Concat [ J Θ Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]Θ λ ] . Âû÷èñëèì Θ ˜ Ψ i . Θ ˜ Ψ i ⇋ Θ [ J Concat [ J Θ Ψ i I n +2 1 ,..., I n +2 n +1 λ ] I n +2 n +2 ] = [ J Concat 3 [ J Θ Concat [ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ]Θ [ J Θ Ψ i I n +2 1 ,..., I n +2 n +1 λ ] Θ I n +2 n +2 ] , ó÷èòûâàÿ [ J Θ Concat [ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ] = [ J ZI n +2 1 ] è Θ I n +2 ò +2 = [ J ZI n +2 1 ] ïîëó÷èì [ J Concat 3 [ J Θ Concat [ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ]Θ [ J Θ Ψ i I n +2 1 ,..., I n +2 n +1 λ ] Θ I n +2 n +2 ] = Θ [ J Θ Ψ i I n +2 1 ,..., I n +2 n +1 λ ] , òîãäà Θ ˜ Ψ i = [ J Concat [ J Θ Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]Θ λ ] . Èìååì: Θ [ R ΦΨ 1 ,..., Ψ k ] = [ R Θ Φ ˜ Ψ 1 , . . . , Ψ k ] Θ [ R Θ Φ ˜ Ψ 1 ,..., ˜ Ψ k ] = [ R Θ Θ Φ ˜ ˜ Ψ 1 , . . . , ˜ ˜ Ψ k ] . ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = [ R Θ Φ ˜ Ψ 1 , . . . , ˜ Ψ k ]( x, Λ) = Θ Φ ( x ) ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = ˜ Ψ 1 ( x, z, Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = [ J Concat [ J Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ]( x, S i ( z )) ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = Concat (Θ Ψ i ( x, z, λ ( x, z )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) ⊢ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = [ R Θ Θ Φ ˜ ˜ Ψ 1 , . . . , ˜ ˜ Ψ k ]( x, Λ) = Θ Θ Φ ( x ) ⊢ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = ˜ ˜ Ψ i ( x, z, Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) Èìååì ˜ ˜ Ψ i ⇋ [ J Concat [ J Θ ˜ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ] , òîãäà ⊢ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = [ J Concat [ J Θ ˜ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ] I n +2 n +2 ]( x, z, Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) , òîãäà ⊢ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = Concat ([ J Θ ˜ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]( x, z, Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) , I n +2 n +2 ( x, z, Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z ))) ⊢ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = Concat (Θ ˜ Ψ i ( x, z, λ ( x, z )) , Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) . Èìååì Θ ˜ Ψ i = [ J Concat [ J Θ Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]Θ λ ] , òîãäà ⊢ Θ ˜ Ψ i (( x, z, u ) = [ J Concat [ J Θ Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]Θ λ ]( x, z, u ) ⊢ Θ ˜ Ψ i ( x, z, u ) = Concat ([ J Θ Θ Ψ i I n +2 1 , . . . , I n +2 n +1 λ ]( x, z, u ) , Θ λ ( x, z, u )) ⊢ Θ ˜ Ψ i ( x, z, u ) = Concat ( J Θ Θ Ψ i ( x, z, λ ( x, z )) , Θ λ , ( x, z, u )) Èìååì Θ λ = Θ [ J [ R ΦΨ 1 ,..., Ψ k ]] I n +2 1 ,..., I n +2 n +1 ] = [ J Concat n +2 [ J Θ [ R ΦΨ 1 ,..., Ψ k ] I n +2 1 , . . . , I n +2 n +1 ]Θ I n +2 1 , . . . , Θ I n +2 n +1 ] , òîãäà Θ λ = [ J Θ [ R ΦΨ 1 ,..., Ψ k ] I n +2 1 , . . . , I n +2 n +1 ] , òîãäà ⊢ Θ λ ( x, z, u ) = [ J Θ [ R ΦΨ 1 ,..., Ψ k ] I n +2 1 , . . . , I n +2 n +1 ]( x, z, u ) , òîãäà ⊢ Θ λ ( x, z, u ) = Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z ) , òîãäà ⊢ Θ ˜ Ψ i ( x, z, u ) = Concat (Θ Θ Ψ i ( x, z, λ ( x, z )) , Θ [ R ΦΨ 1 ,..., Ψ k ] , ( x, z )) , òîãäà ⊢ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = Concat ( Concat (Θ Θ Ψ i ( x, z, λ ( x, z )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) , Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) . Ïîäâåä¼ì èòîãè: 55 ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = Θ Φ ( x ) ⊢ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, Λ) = Θ Θ Φ ( x ) ⊢ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = Concat (Θ Ψ i ( x, z, λ ( x, z )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) ⊢ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, S i ( z )) = Concat ( Concat (Θ Θ Ψ i ( x, z, λ ( x, z )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) , Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )) . Ïðåäïîëîæèì, ÷òî âåðíî: ∀ α WordM A | = Θ Φ ( α ) ≈ Θ Θ Φ ( α ) ∀ α ∀ β ∀ γ WordM A | = Θ Ψ i ( α, β, γ ) ≈ Θ Θ Ψ i ( α, β, γ ) ∀ α ∀ β WordM A | = Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β ) ≈ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β ) , òîãäà ïîëó÷èì WordM A | = Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, S i ( β )) = Concat ( Concat (Θ Θ Ψ i ( α, β, λ ( α, β )) , Θ [ R ΦΨ i,..., Ψ k ] ( α, β )) , Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β )) = Concat (Θ Θ Ψ i ( α, β, λ ( α, β )) , Concat (Θ [ R ΦΨ i,..., Ψ k ] ( α, β ) , Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β ))) , èñïîëüçóÿ èíäóêöèîííîå ïðåä- ïîëîæåíèå WordM A | = Θ [ R ΦΨ i,..., Ψ k ] ( α, β ) ≈ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β )) , ïîëó÷èì WordM A | = Concat (Θ [ R ΦΨ i,..., Ψ k ] ( α, β ) , Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β )) ≈ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β ) , òîãäà WordM A | = Concat (Θ Θ Ψ i ( α, β, λ ( α, β )) , Concat (Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β ) , Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β ))) ≈ Concat (Θ Ψ i ( α, β, λ ( α, β )) , Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, β )) , òîãäà WordM A | = Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, S i ( β )) ≈ Θ [ R ΦΨ 1 ,..., Ψ k ] ( α, S i ( β )) , òîãäà WordM A | = ∀ x ∀ z [Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z ) ≈ Θ Θ [ R ΦΨ 1 ,..., Ψ k ] ( x, z )] . Èñïîëüçóÿ èíäóêöèþ ïî ïîñòðîåíèþ ôóíêòîðîâ è èíäóêöèþ ïî ïîñòðîåíèþ àðãóìåíòíîãî ñëîâà, ïî- ëó÷èì: äëÿ ëþáîãî ôóíêòîðà Φ âåðíî WordM A | = ∀ x [Θ Φ ( x ) ≈ Θ Θ Φ ( x ) . 56