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

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First-order Logic

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Higher-Order Logic

ATP in FOL and HOL

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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)3q .05454 0 Td(the)T031843177 0autn041I5.1587)Tj0.53of1.82 Tm49he)T03184 rst-ord3 0 Td4.4927

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HOL: Classical Type Theory

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HOL: Semantics

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Sidetrack: Logical Frameworks

ATP in FOL and HOL

Presentation by Marc Wagner Logical Frameworks

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Exercise Sheet III

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HOL Semantics: Applications

ATP in FOL and HOL

Henkin semantics

• Mathematics

Without Boolean extensionality

• Linguistics, intensional contexts
• I believe

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HOL: Problems

ATP in FOL and HOL

Problem

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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

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Abstract Consistency

A

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ND Calculi: Completeness

ATP in FOL and HOL

Excerpt from completeness proof . . . r β: Let A 2 and A ? y

β be NK -inconsistent. That is, A

? y

β ‘‘ Fo.

By NK(: I ), we know ‘‘ : A ? y

β. SinceA β by

NK(Hyp) /R179 ET Q1 0 0 RG1 0 0 rgq 10 0 0 10 0 0 cm BT/R298 1 Tf0 20.6626 -20.6626 -0 244.68 241.42 Tm(N)Tj0.73127 0 Td(K)Tj/R173 1 Tf0.666999 0 Td(()Tj/R170 1 Tf0.388999 0 Td( )Tj/R173 1 Tf0.615267 0 Td())TjET Q0 G0 gq 10 0 0 10 0 0 cm BT/R167 1 Tf0 20.6626 -20.6626 -0 244.68 299.26 Tm(.)Tj0.620648 0 Td(So)Tj1.18335 0 Td(,)Tj0.550957 0 Td(b)Tj0.539577 0 Td(y)TjET Q1 0 0 RG1 0 0 rgq 10 0 0 10 0 0 cm BT/R298 1 Tf0 20.6626 -20.6626 -0 244.68 375.1 Tm(N)Tj0.73127 0 Td(K)Tj/R173 1 Tf0.672806 0 Td(()Tj/R182 1 Tf0.388998 0 Td(:)Tj/R301 1 Tf0.667 0 Td(E)Tj/R173 1 Tf0.800959 0 Td())TjET Q0 G0 gq 10 0 0 10 0 0 cm BT/R167 1 Tf0 20.6626 -20.6626 -0 244.68 456.22 Tm(w)Tj0.708383 0 Td(e)Tj0.835765 0 Td(kno)Tj1.59557 0 Td(w)Tj/R179 1 Tf0.998764 0 Td()Tj/R182 1 Tf0.998764 0 Td(‘)Tj0.109547 0 Td(‘)Tj/R292 1 Tf0.887765 0 Td(F)Tj/R188 1 Tf0 14.4637 -14.4637 -0 247.68 598.06 Tm(o)Tj/R167 1 Tf0 20.6626 -20.6626 -0 244.68 612.58 Tm(/R179 /R179 1 Tf1.94396 0 Td()Tj/R167 1 Tf0.998764 0 Td(is)TjET Q1 0 0 RG1 0 0 rgq 10 0 0 10 0 0 cm BT/R298 1 Tf0 20.6626 -20.6626 -0 273.84 113.26 Tm(N)Tj0.73127 0 Td(K)Tj/R185 1 Tf0 14.4637 -14.4637 -0 276.96 142.18 Tm( )TjET Q0 G0 gq 10 0 0 10 0 0 cm BT/R167 1 Tf0 20.6626 -20.6626 -0 273.84 151.66 Tm(-inconsistent.)Tj/R182 1 Tf-4.44282 -1.55063 Td(r)Tj/R304 1 Tf0 14.4637 -14.4637 -0 309 72.34 Tm(b)Tj/R167 1 Tf0 20.6626 -20.6626 -0 305.88 82.3 Tm(:)Tj1.49836 0 Td(W)Tj0.916963 0 Td(e)Tj0.829956 0 Td(argue)Tj2.83196 0 Td(b)Tj0.539577 0 Td(y)Tj0.771956 0 Td(contr)Tj2.21039 0 Td(adiction.)Tj4.05984 0 Td(Assume)Tj3.88776 0 Td(that)Tj1 Tf0.667248942d(A)Tj/:)Tj/R292 1 Tf0.667 0 Td(A)Tj/R170 1 Tf1.41854 0.563337 Tj/:

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Saturation and Cut

ATP in FOL and HOL

Saturation condition rsat is a challenge for machine-oriented calculi:

as hard as cut-elimination therefore development of alternative, weaker conditions in

[Benzm llerBrownKohlhase-Draft03] which are

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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?

Lecture IX p.23

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Extensional

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Extensional Resolution

ATP in FOL and HOL

Further small examples

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Sidetrack: Lambda Cube

ATP in FOL and HOL

Presentation by Matthias Berg Lambda Cube See extra

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Sidetrack: New Foundations

ATP in FOL and HOL

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Extensional P aramodulation

A TP in FOL and HOL

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• 19:

Extensional R UE-Resolution [Benzmler-CADE-9] [Benzmler-PhD-9]

• Notation

as v ef

• re;

ne w is logical symbol = f or pr imitiv e equality

• Identication
• f

unication constr aints and negativ e equality liter als

• Al

r ules f or e xtensional resolution stil v alid; resolution and f actor ization not alo w ed

• n

unication constr aints

• Some

fur ther r ules required f or = ; se ne xt slide

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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

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Prover LEO

ATP in FOL and HOL

• [Benzm llerKohlhase-CADE-98]
• Extended set-of-support-architecture