Zenon Modulo: When Achilles Outruns the Tortoise using Deduction Modulo November 18, 2013 David Delahaye David.Delahaye@cnam.fr Cnam / Inria, CPR / Deducteam, Paris, France GDR GPL, GT LTP , LaBRI, Bordeaux, France
Proof Search in Axiomatic Theories Extending Zenon to Deduction Modulo David Delahaye 1 Introduction Current Trends Principles of Deduction Modulo ◮ Axiomatic theories (Peano arithmetic, set theory, etc.); Overview of the Zenon ATP ◮ Decidable fragments (Presburger arithmetic, arrays, etc.); Deduction Modulo ◮ Applications of formal methods in industrial settings. for Zenon Zenon Modulo over the TPTP Library A Backend for Place of the Axioms? Zenon Modulo References for ◮ Leave axioms wandering among the hypotheses? Zenon Modulo Deduction Modulo ◮ Induce a combinatorial explosion in the proof search space; for BWare Conclusion ◮ Do not bear meaning usable by automated theorem provers. Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
Proof Search in Axiomatic Theories Extending Zenon to Deduction Modulo David Delahaye 1 Introduction A Solution Principles of Deduction Modulo ◮ A cutting-edge combination between: Overview of the Zenon ATP ◮ First order automated theorem proving method (resolution); Deduction Modulo ◮ Theory-specific decision procedures (SMT approach). for Zenon Zenon Modulo over the TPTP Library Drawbacks A Backend for Zenon Modulo References for ◮ Specific decision procedure for each given theory; Zenon Modulo ◮ Decidability constraint over the theories; Deduction Modulo for BWare ◮ Lack of automatability and genericity. Conclusion Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
Proof Search in Axiomatic Theories Extending Zenon to Deduction Modulo David Delahaye Use of Deduction Modulo 1 Introduction Principles of ◮ Transform axioms into rewrite rules; Deduction Modulo Overview of the ◮ Turn proof search among the axioms into computations; Zenon ATP ◮ Avoid unnecessary blowups in the proof search; Deduction Modulo for Zenon ◮ Shrink the size of proofs (record only meaningful steps). Zenon Modulo over the TPTP Library A Backend for Zenon Modulo This Talk References for Zenon Modulo ◮ Introduce the principles of deduction modulo; Deduction Modulo for BWare ◮ Present the results of an experiment with Zenon; Conclusion ◮ Give an overview of the BWare project. Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
Principles of Deduction Modulo Extending Zenon to Inclusion Deduction Modulo David Delahaye Introduction ∀ a ∀ b (( a ⊆ b ) ⇔ ( ∀ x ( x ∈ a ⇒ x ∈ b ))) Principles of 2 Deduction Modulo Overview of the Zenon ATP Proof in Sequent Calculus Deduction Modulo for Zenon Zenon Modulo over the TPTP Library A Backend for Ax Zenon Modulo . . . , x ∈ A ⊢ A ⊆ A , x ∈ A References for ⇒ R Zenon Modulo . . . ⊢ A ⊆ A , x ∈ A ⇒ x ∈ A ∀ R Deduction Modulo Ax . . . ⊢ A ⊆ A , ∀ x ( x ∈ A ⇒ x ∈ A ) . . . , A ⊆ A ⊢ A ⊆ A for BWare ⇒ L Conclusion . . . , ( ∀ x ( x ∈ A ⇒ x ∈ A )) ⇒ A ⊆ A ⊢ A ⊆ A ∧ L A ⊆ A ⇔ ( ∀ x ( x ∈ A ⇒ x ∈ A )) ⊢ A ⊆ A ∀ L × 2 ∀ a ∀ b (( a ⊆ b ) ⇔ ( ∀ x ( x ∈ a ⇒ x ∈ b ))) ⊢ A ⊆ A Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
Principles of Deduction Modulo Extending Zenon to Inclusion Deduction Modulo David Delahaye Introduction ∀ a ∀ b (( a ⊆ b ) − → ( ∀ x ( x ∈ a ⇒ x ∈ b ))) Principles of 2 Deduction Modulo Overview of the Zenon ATP Rewrite Rule Deduction Modulo for Zenon Zenon Modulo over the TPTP Library ( a ⊆ b ) − → ( ∀ x ( x ∈ a ⇒ x ∈ b )) A Backend for Zenon Modulo References for Zenon Modulo Proof in Deduction Modulo Deduction Modulo for BWare Conclusion Ax x ∈ A ⊢ x ∈ A ⇒ R ⊢ x ∈ A ⇒ x ∈ A ∀ R , A ⊆ A − →∀ x ( x ∈ A ⇒ x ∈ A ) ⊢ A ⊆ A Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
From Axioms to Rewrite Rules Extending Zenon to Deduction Modulo David Delahaye Difficulties Introduction 3 Principles of ◮ Confluence and termination of the rewrite system; Deduction Modulo Overview of the ◮ Preservation of the consistency; Zenon ATP ◮ Preservation of the cut-free completeness; Deduction Modulo for Zenon ◮ Automation of the transformation. Zenon Modulo over the TPTP Library A Backend for Zenon Modulo An Example References for Zenon Modulo ◮ Axiom A ⇔ ( A ⇒ B ) ; Deduction Modulo for BWare ◮ Transformed into A − → A ⇒ B ; Conclusion ◮ We want to prove: B . Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
From Axioms to Rewrite Rules Extending Zenon to Deduction Modulo An Example (Continued) David Delahaye Introduction ◮ In sequent calculus, we have a cut-free proof: 3 Principles of Deduction Modulo Overview of the Zenon ATP ∼ Π Deduction Modulo A ⇒ ( A ⇒ B ) , A ⊢ B , B for Zenon Π ⇒ R Zenon Modulo over A ⇒ ( A ⇒ B ) ⊢ B , A ⇒ B A ⇒ ( A ⇒ B ) , A ⊢ B the TPTP Library ⇒ L A Backend for A ⇒ ( A ⇒ B ) , ( A ⇒ B ) ⇒ A ⊢ B Zenon Modulo ⇔ L A ⇔ ( A ⇒ B ) ⊢ B References for Zenon Modulo Where Π is: Deduction Modulo for BWare ax ax Conclusion A ⊢ B , A A , B ⊢ B ⇒ L ax A ⊢ B , A A , A ⇒ B ⊢ B ⇒ L A ⇒ ( A ⇒ B ) , A ⊢ B Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
From Axioms to Rewrite Rules Extending Zenon to Deduction Modulo David Delahaye An Example (Continued) Introduction 3 Principles of ◮ In deduction modulo, we have to cut A to get a proof: Deduction Modulo Overview of the Zenon ATP Π Deduction Modulo for Zenon Π A ⊢ B ⇒ R , A − → A ⇒ B Zenon Modulo over A ⊢ B ⊢ A cut the TPTP Library ⊢ B A Backend for Zenon Modulo Where Π is: References for Zenon Modulo ax ax Deduction Modulo A ⊢ A A , B ⊢ B ⇒ L , A − for BWare ax → A ⇒ B Conclusion A ⊢ A A , A ⊢ B cut A ⊢ B Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
The Zenon Automated Theorem Prover Extending Zenon to Features of Zenon Deduction Modulo David Delahaye ◮ First order logic with equality; Introduction ◮ Tableau-based proof search method; Principles of Deduction Modulo ◮ Extensible by adding new deductive rules; 4 Overview of the Zenon ATP ◮ Certifying, 3 outputs: Coq, Isabelle, Dedukti; Deduction Modulo for Zenon ◮ Used by other systems: Focalize, TLA. Zenon Modulo over the TPTP Library A Backend for Zenon Zenon Modulo References for Zenon Modulo ◮ Reference: Deduction Modulo for BWare R. Bonichon, D. Delahaye, D. Doligez. Zenon: An Extensible Automated Theorem Conclusion Prover Producing Checkable Proofs. LPAR (2007). ◮ Freely available (BSD license); ◮ Developed by D. Doligez; Cnam / Inria ◮ Download: http://focal.inria.fr/zenon/ CPR / Deducteam GDR GPL, GT LTP 20
The Zenon Automated Theorem Prover Extending Zenon to The Tableau Method Deduction Modulo David Delahaye ◮ We start from the negation of the goal (no clausal form); Introduction ◮ We apply the rules in a top-down fashion; Principles of Deduction Modulo ◮ We build a tree whose each branch must be closed; 4 Overview of the Zenon ATP ◮ When the tree is closed, we have a proof of the goal. Deduction Modulo for Zenon Zenon Modulo over the TPTP Library Closure and Cut Rules A Backend for Zenon Modulo References for Zenon Modulo Deduction Modulo ⊥ ⊙ ⊥ ¬⊤ ⊙ ¬⊤ cut for BWare P | ¬ P ⊙ ⊙ Conclusion ¬ R r ( t , t ) ⊙ r R s ( a , b ) ¬ R s ( b , a ) ⊙ s P ¬ P ⊙ ⊙ ⊙ ⊙ Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
The Zenon Automated Theorem Prover Extending Zenon to Deduction Modulo Analytic Rules David Delahaye Introduction Principles of Deduction Modulo ¬ ( P ⇔ Q ) 4 Overview of the P ⇔ Q ¬¬ P α ¬¬ Zenon ATP β ⇔ β ¬⇔ P ¬ P , ¬ Q | P , Q ¬ P , Q | P , ¬ Q Deduction Modulo for Zenon Zenon Modulo over ¬ ( P ∨ Q ) α ¬∨ ¬ ( P ⇒ Q ) α ¬⇒ P ∧ Q α ∧ the TPTP Library P , Q A Backend for ¬ P , ¬ Q P , ¬ Q Zenon Modulo References for ¬ ( P ∧ Q ) β ¬∧ P ∨ Q β ∨ P ⇒ Q β ⇒ Zenon Modulo Deduction Modulo P | Q ¬ P | Q ¬ P | ¬ Q for BWare Conclusion ∃ x P ( x ) ¬∀ x P ( x ) δ ∃ δ ¬∀ P ( ǫ ( x ) . P ( x )) ¬ P ( ǫ ( x ) . ¬ P ( x )) Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
The Zenon Automated Theorem Prover Extending Zenon to Deduction Modulo David Delahaye γ -Rules Introduction Principles of Deduction Modulo ∀ x P ( x ) γ ∀ M ¬∃ x P ( x ) γ ¬∃ M 4 Overview of the Zenon ATP P ( X ) ¬ P ( X ) Deduction Modulo for Zenon ∀ x P ( x ) γ ∀ inst ¬∃ x P ( x ) γ ¬∃ inst Zenon Modulo over the TPTP Library P ( t ) ¬ P ( t ) A Backend for Zenon Modulo References for Zenon Modulo Deduction Modulo Relational Rules for BWare Conclusion ◮ Equality, reflexive, symmetric, transitive rules; ◮ Are not involved in the computation of superdeduction rules. Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
The Zenon Automated Theorem Prover Extending Zenon to Deduction Modulo David Delahaye Introduction Example of Proof Search Principles of Deduction Modulo 4 Overview of the Zenon ATP Deduction Modulo ∀ x ( P ( x ) ∨ Q ( x )) , ¬ P ( a ) , ¬ Q ( a ) for Zenon Zenon Modulo over the TPTP Library A Backend for Zenon Modulo References for Zenon Modulo Deduction Modulo for BWare Conclusion Cnam / Inria CPR / Deducteam GDR GPL, GT LTP 20
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