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First Order Logic: Resolution-based Automated Theorem Proving - - PowerPoint PPT Presentation

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First Order Logic: Resolution-based Automated Theorem Proving Valentin Goranko DTU Informatics September 2010 V Goranko Resolution for first-order logic with equality Additional rules for equational reasoning are needed. Paramodulation:

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V Goranko

First Order Logic: Resolution-based Automated Theorem Proving

Valentin Goranko DTU Informatics September 2010

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V Goranko

Resolution for first-order logic with equality

Additional rules for equational reasoning are needed.

  • Paramodulation:

Par : C ∨ s1 = s2, D ∨ Q(t, t1, . . . , tn) (C ∨ D ∨ Q(s2, t1, . . . , tn))σ (σ = MGU(s1, t))

  • Superposition.
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V Goranko

First-order resolution: some optimizations

  • Tautology deletion.
  • Subsumption resolution.
  • Clause ordering.
  • Indexing techniques.
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V Goranko

First-order resolution: some strategies

  • Unit resolution.
  • Set-of-support resolution.
  • Hyper-resolution.
  • Linear resolution.
  • Selective Linear Definite clause (SLD)-resolution.
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V Goranko

Resolution-based Logic Programming: PROLOG

  • PROLOG: Language for Logic Programming based on

SLD-resolution plus backtracking.

  • Because of the depth-first search strategy, the pure PROLOG

does not always terminate, even when a derivation exists, and hence is incomplete.

  • To remedy that problem, PROLOG employs some non-logical

features, such as cut and fail, implementing ‘Negation as Failure’.

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V Goranko

Resolution-based automated reasoning in FOL: some tool implementations

  • OTTER, Prover9, Mace (Argonne National Laboratory)
  • SPASS (Max Planck Institute for Informatics, Saarbr¨

ucken)

  • Vampire (University of Manchester)
  • E equational theorem prover (Technical University of Munich)
  • SETHEO, based on model generation, and E-SETHEO

(Technical University of Munich)

  • CADE ATP System Competition (CASC).
  • Thousand Problems for Theorem Provers (TPTP) library:

www.tptp.org

  • See more references and links on the course website.
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V Goranko

SPASS

  • SPASS: automated theorem prover for first-order logic with

equality.

  • Website: http://www.spass-prover.org/
  • Downloadable and online versions. WebSPASS
  • Easy user interface. Tutorial.
  • The last problem in Assignment 1 uses SPASS.
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V Goranko

Automated reasoning in AI

  • Automated reasoning: major area of AI
  • Logic-based knowledge representation
  • Automated deduction. Automated theorem provers: mostly

for first-order logic.

  • Interactive theorem proving. Automated proof assistants for

first-order, non-classical, and higher-order logics: HOL, Isabelle, Coq, Mizar, etc.