Logic-Independent Premise Selection for Automated Theorem Proving - - PowerPoint PPT Presentation

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Logic-Independent Premise Selection for Automated Theorem Proving - - PowerPoint PPT Presentation

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29 March 2017 Logic-Independent Premise Selection | 1 FACULTY OF COMPUTER SCIENCE Logic-Independent Premise Selection for Automated Theorem Proving Eugen Kuksa AITP 2017 Obergurgl 29 March 2017 Logic-Independent Premise Selection | 2 FACULTY

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FACULTY OF COMPUTER SCIENCE

29 March 2017 Logic-Independent Premise Selection | 1

Logic-Independent Premise Selection for Automated Theorem Proving

Eugen Kuksa

AITP 2017 Obergurgl

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Outline

  • 1. Introduction
  • 2. Theoretical Foundation: Entailment Relations
  • 3. Case Study
  • 4. There’s More: Signatures and Logic Translations.
  • 5. Conclusion
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Introduction

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Automated Theorem Proving

  • A theorem proving problem consists of Axioms and a conjecture.
  • An automated theorem prover (ATP) runs an algorithm to find

a proof.

  • A typical ATP is efficient on small problems.
  • Large problems lead to combinatorial explosion.
  • ATP reach their time or memory limit.
  • Return with no result.
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Example

  • Suggested Upper Merged Ontology (SUMO)
  • Formalised in FOF and THF in the TPTP library
  • Problems with (tens of) thousands of axioms
  • Pick CSR119^3 (THF): ≈5000 Axioms
  • Higher-Order Prover Leo-II runs into a timeout (60 seconds)
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Solution: Premise Selection

  • Reduce the set of axioms for the proving task
  • Proving time decreases or proving even becomes possible at all
  • The SUMO example CSR119^3 passes in less than a second
  • The axiom set was reduced to 390 out of over 5000 axioms by

SInE

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Logic Dependence

Problem:

  • There are many premise selection algorithms
  • Implemented only for FOF or some higher order logics
  • . . . even though some are described logic-independently

Solution:

  • Lift the algorithms to logic-independence
  • Run them in an abstract notion of ‘logic’
  • Transfer results to the concrete logic
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Tool Support

Problem: Many provers operate on one logic/syntax only

  • FaCT, Pellet: Description logic with OWL
  • Darwin, E-Prover, Geo-III, SPASS, Vampire: First-order logic

with TPTP/FOF

  • Leo-II, Satallax, Isabelle: Higher-order logic with TPTP/THF
  • Isabelle/HOL’s own logic
  • . . .

Solution:

  • Lift the algorithms to logic-independence
  • Run them in an abstract notion of ‘logic’
  • Transfer results to the concrete logic
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Theoretical Foundation: Entailment Relations

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Entailment relation

An entailment relation with symbols (Sen, Sym ⊢, symbols) consists

  • f
  • A set of sentences Sen
  • A set of symbols Sym
  • A relation ⊢ ⊆ P(Sen) × Sen which is
  • reflexive (Axioms are theorems)
  • transitive (We may use lemmas)
  • monotonic (We may use premise selection)
  • A function symbols : Sen → P(Sym) giving the symbols that
  • ccur in a sentence
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Case Study

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Implementation: Ontohub

  • Web application: https://ontohub.org
  • Version controlled repository for
  • ntologies/specifications/theories
  • Version control (git)
  • Integrated editor for small files
  • Analyses theories
  • Has interfaces with ATPs
  • Back-end: Hets
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Implementation: Ontohub cont’d

  • Supports different logics
  • Propositional Logic
  • OWL
  • FOL / TPTP-FOF
  • FOL + Induction
  • CASL
  • Modal Logic
  • Common Logic
  • HOL / TPTP-THF
  • Isabelle/HOL
  • . . .
  • Brings tool support
  • FaCT, Pellet
  • CVC4, Darwin, E-Prover, Geo-III, SPASS, Vampire
  • Leo-II, Satallax, Isabelle
  • . . .
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Premise Selection: The Algorithm ‘SInE’

  • Developed by Kryštof Hoder
  • Fully automatic with a few user-defined parameters
  • Operates on syntax
  • Selects recursively the axioms that share a symbol with the

conjecture or an already selected axiom

  • The shared symbol that allows to select an axiom must hold

more conditions

  • Selection stops after n recursion steps
  • We implemented SInE in Ontohub
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Data Flow

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Experiments: Setup

We applied our implementation of SInE to

  • All 2078 problems of the MPTP2078 (FOF)
  • A subset (501 problems) of a formalisation into THF0 of the

Automath formalization of Landau’s ‘Grundlagen der Analysis’

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Results: FOF

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Results: THF0

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There’s More: Signatures and Logic Translations.

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Special handling of THF

Problem:

  • THF (among other logics) is typed
  • Symbols must be declared with a formula before their first use
  • Such ‘signature-defining delarations’ must not be removed

Solution:

  • Preserve the needed ‘signature-defining declarations’ after the

premise selection

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Theoretical Foundation: Entailment Systems

An entailment system with symbols (Sign, Sen, Sym, ⊢, symbols) consists of

  • a category Sign of signatures and signature morphisms
  • a functor Sen : Sign → Set giving the set of sentences over a

signature

  • a faithful functor Sym : Sign → Set giving the set of symbols of

a signature

  • for each Σ ∈ |Sign| a relation ⊢Σ ⊆ P(Sen(Σ)) × Sen(Σ) which
  • is reflexive, transitive, monotonic
  • and satisfies ⊢-translation: Given a signature morphism

σ : Σ1 → Σ2, we have Γ ⊢Σ1 ϕ ⇒ Sen(σ)(Γ) ⊢Σ2 Sen(σ)(ϕ)

  • a natural transformation symbols : Sen → P ◦ Sym giving the

symbols of a sentence

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Tool Support for Logics

Problem:

  • Some logics don’t have direct tool support, e.g. CASL, Common

Logic, modal logic

  • People need to formalise the theory in tool-supported logics
  • . . . or cannot use ATP (with premise selection)

Solution:

  • Run premise selection in the desired logic
  • Translate the modified theory to logic with tool support
  • Run the prover on the translation
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Theoretical Foundation: Entailment relation morphism

An entailment relation morphism α : (SenS, SymS, ⊢S, symbolsS) → (SenT , SymT , ⊢T , symbolsT ) is a function α : SenS → SenT such that for all Γ ⊆ SenS, ϕ ∈ SenS : Γ ⊢S ϕ implies α(Γ) ⊢T α(ϕ) α is called conservative if for all Γ ⊆ SenS, ϕ ∈ SenS : Γ ⊢S ϕ if and only if α(Γ) ⊢T α(ϕ)

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Theoretical Foundation: Theoroidal entailment relation morphism

A conservative theoroidal entailment relation morphism (α, ∆) contains

  • a function α : (SenS, SymS, ⊢S, symbolsS)

→ (SenT , SymT , ⊢T , symbolsT )

  • a base set of sentences ∆ ⊆ SenT

that hold for all Γ ⊆ SenS, ϕ ∈ SenS : Γ ⊢S ϕ if and only if ∆∪α(Γ) ⊢T α(ϕ)

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Hets

  • Evaluation component of Ontohub
  • Actually analyses theories
  • Translates theories
  • Interfaces with provers
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Data Flow

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Conclusion

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Conclusion and Future Work

Conclusion

  • Premise selection improves proving performance significantly
  • Entailment relation morphisms allow its use with different logics
  • And different reasoning tools
  • SInE in Ontohub is only a proof of concept

Future Work

  • Develop more premise selection algorithms and deploy them to

Ontohub

  • Learn from found proofs and disproofs (after premise selection)
  • Use modular structure (signature morphisms)
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Thank you for listening! Questions?