Semantics and A utomation of Higher-Order Logic Some Remarks - PowerPoint PPT Presentation

Semantics and A utomation of Higher-Order Logic � Some Remarks � Christoph Benzm¤ uller Department of Computer Science, Saarland University Workshop on Logic, Proofs and Programs 17� 18 June 2004, Nancy

� First-order Logic c Benzm¤ uller 2004

� Higher-Order Logic ATP in FOL and HOL c Benzm¤ uller 2004

� Motivation for Talk ATP in FOL and HOL Is the situation really hopeless? Is it justi� able that the deduction cn064 2 PC 141Om�(Is)o Td�(dedu�/H 1�/ ir0 0entrti� ab)8182.22044 0 tes�(FOL)T361.73546 0so�(the)4031.73546 0strong(reall-27.3.2-331.34103546 0main(reall3.-9.4843177 0oTd�(in)3�q .05454 0 Td�(the)T031843177 0autn041I5.1587)Tj�0.53of1.82 Tm49he)T03184� rst-ord3 0 Td4.4927 c Benzm¤ uller 2004

� HOL: Classical Type Theory c Benzm¤ uller 2004

� HOL: Semantics c Benzm¤ uller 2004

Sidetrack: Logical Frameworks ATP in FOL and HOL Presentation by Marc Wagner Logical Frameworks c Benzm¤ uller 2004

� Exercise Sheet III c Benzm¤ uller 2004

� � � � HOL Semantics: Applications ATP in FOL and HOL Henkin semantics Mathematics Without Boolean extensionality Linguistics, intensional contexts �I believe c Benzm¤ uller 2004

� ATP in FOL and HOL HOL: Problems Problem uller 2004 Benzm¤ c

� � � � Abstract Consistency ATP in FOL and HOL Completeness proofs in HOL much harder than in FOL Direct semantical arguments are too complicate Abstract consistency proof c Benzm¤ uller 2004

� Abstract Consistency A c Benzm¤ uller 2004

� ND Calculi: Completeness ATP in FOL and HOL Excerpt from completeness proof . . . ? ? y y r β : Let A 2 � and � � A β be NK � -inconsistent. That is, � � A β ‘‘ F o . ? y By NK ( : I ) , we know � ‘‘ : A β . Since A β by NK ( Hyp ) �/R179 ET Q�1 0 0 RG�1 0 0 rg�q 10 0 0 10 0 0 cm BT�/R298 1 Tf�0 20.6626 -20.6626 -0 244.68 241.42 Tm�(N)Tj�0.73127 0 Td�(K)Tj�/R173 1 Tf�0.666999 0 Td�(()Tj�/R170 1 Tf�0.388999 0 Td�(� )Tj�/R173 1 Tf�0.615267 0 Td�())Tj�ET Q�0 G�0 g�q 10 0 0 10 0 0 cm BT�/R167 1 Tf�0 20.6626 -20.6626 -0 244.68 299.26 Tm�(.)Tj�0.620648 0 Td�(So)Tj�1.18335 0 Td�(,)Tj�0.550957 0 Td�(b)Tj�0.539577 0 Td�(y)Tj�ET Q�1 0 0 RG�1 0 0 rg�q 10 0 0 10 0 0 cm BT�/R298 1 Tf�0 20.6626 -20.6626 -0 244.68 375.1 Tm�(N)Tj�0.73127 0 Td�(K)Tj�/R173 1 Tf�0.672806 0 Td�(()Tj�/R182 1 Tf�0.388998 0 Td�(:)Tj�/R301 1 Tf�0.667 0 Td�(E)Tj�/R173 1 Tf�0.800959 0 Td�())Tj�ET Q�0 G�0 g�q 10 0 0 10 0 0 cm BT�/R167 1 Tf�0 20.6626 -20.6626 -0 244.68 456.22 Tm�(w)Tj�0.708383 0 Td�(e)Tj�0.835765 0 Td�(kno)Tj�1.59557 0 Td�(w)Tj�/R179 1 Tf�0.998764 0 Td�(�)Tj�/R182 1 Tf�0.998764 0 Td�(‘)Tj�0.109547 0 Td�(‘)Tj�/R292 1 Tf�0.887765 0 Td�(F)Tj�/R188 1 Tf�0 14.4637 -14.4637 -0 247.68 598.06 Tm�(o)Tj�/R167 1 Tf�0 20.6626 -20.6626 -0 244.68 612.58 Tm�(�/R179 /R179 1 Tf�1.94396 0 Td�(�)Tj�/R167 1 Tf�0.998764 0 Td�(is)Tj�ET Q�1 0 0 RG�1 0 0 rg�q 10 0 0 10 0 0 cm BT�/R298 1 Tf�0 20.6626 -20.6626 -0 273.84 113.26 Tm�(N)Tj�0.73127 0 Td�(K)Tj�/R185 1 Tf�0 14.4637 -14.4637 -0 276.96 142.18 Tm�(� )Tj�ET Q�0 G�0 g�q 10 0 0 10 0 0 cm BT�/R167 1 Tf�0 20.6626 -20.6626 -0 273.84 151.66 Tm�(-inconsistent.)Tj�/R182 1 Tf�-4.44282 -1.55063 Td�(r)Tj�/R304 1 Tf�0 14.4637 -14.4637 -0 309 72.34 Tm�(b)Tj�/R167 1 Tf�0 20.6626 -20.6626 -0 305.88 82.3 Tm�(:)Tj�1.49836 0 Td�(W)Tj�0.916963 0 Td�(e)Tj�0.829956 0 Td�(argue)Tj�2.83196 0 Td�(b)Tj�0.539577 0 Td�(y)Tj�0.771956 0 Td�(contr)Tj�2.21039 0 Td�(adiction.)Tj�4.05984 0 Td�(Assume)Tj�3.88776 0 Td�(that)Tj�1 Tf�0.667248942d�(A)Tj�/:)Tj�/R292 1 Tf�0.667 0 Td�(A)Tj�/R170 1 Tf�1.41854 0.563337 Tj�/: c Benzm¤ uller 2004

� Saturation and Cut ATP in FOL and HOL Saturation condition r sat is a challenge for machine-oriented calculi: � as hard as cut-elimination � therefore development of alternative, weaker conditions in [Benzm� llerBrownKohlhase-Draft03] which are c Benzm¤ uller 2004

Logik h� herer Stufe: Probleme ATP in FOL and HOL � Problem 3: The two crucial challenges for automation of HOL � treatment of equality and extensionality � instantiation of set variables are too hard to control successfully. Really? c Lecture IX � p.23 Benzm¤ uller 2004

� Extensional c Benzm¤ uller 2004

� c Benzm¤ uller 2004

� Extensional Resolution ATP in FOL and HOL Further small examples c Benzm¤ uller 2004

� Sidetrack: Lambda Cube ATP in FOL and HOL Presentation by Matthias Berg Lambda Cube See extra c Benzm¤ uller 2004

� Sidetrack: New Foundations ATP in FOL and HOL c Benzm¤ uller 2004

c resolution of � constr aints and negativ e equality liter als � Al resolution stil and � ization not alo on uni�cation constr aints � Some fur ther required ne xt Identi�cation uni�cation equality � Benzm ¤uler 204 Extensional in FOL and HOL 195 � 19: Extensional e [Benzm�ler-PhD-9] [Benzm�ler-CADE-9] slide ne imitiv pr = symbol logical w is ore; as Notation P aramodulation A TP R UE-Resolution v ef f or r ules f or e xtensional v alid; f actor w ed r ules f or = ; se

� � � � Difference Reduction ATP in FOL and HOL Extensional RUE-resolution [Benzm� ller-PhD-99] Difference reduction matrix calculus [Brown-PhD-04] Alcrules for extensional resolution Positive extensionality rules, but no paramodulation rule New: Resolution and factorization allowed on uni� cation constraints c Benzm¤ uller 2004

� � � Prover L EO ATP in FOL and HOL [Benzm� llerKohlhase-CADE-98] Extended set-of-support-architecture c Benzm¤ uller 2004