Automated Deduction Modulo November 8, 2013 David Delahaye David.Delahaye@cnam.fr Cnam / Inria, Paris, France PSATTT’13, École polytechnique, Palaiseau, France
Proof Search in Axiomatic Theories Automated Deduction Modulo David Delahaye Current Trends 1 Introduction Deduction Modulo & Superdeduction ◮ Axiomatic theories (Peano arithmetic, set theory, etc.); Superdeduction ◮ Decidable fragments (Presburger arithmetic, arrays, etc.); for Zenon Superdeduction for ◮ Applications of formal methods in industrial settings. the B Method Super Zenon for First Order Theories Deduction Modulo Place of the Axioms? for Zenon Zenon Modulo over ◮ Leave axioms wandering among the hypotheses? the TPTP Library A Backend for ◮ Induce a combinatorial explosion in the proof search space; Zenon Modulo Deduction Modulo ◮ Do not bear meaning usable by automated theorem provers. for BWare Conclusion Cnam / Inria PSATTT’13 25
Proof Search in Axiomatic Theories Automated Deduction Modulo David Delahaye 1 Introduction A Solution Deduction Modulo & Superdeduction ◮ A cutting-edge combination between: Superdeduction for Zenon ◮ First order automated theorem proving method (resolution); Superdeduction for ◮ Theory-specific decision procedures (SMT approach). the B Method Super Zenon for First Order Theories Drawbacks Deduction Modulo for Zenon ◮ Specific decision procedure for each given theory; Zenon Modulo over the TPTP Library ◮ Decidability constraint over the theories; A Backend for Zenon Modulo ◮ Lack of automatability and genericity. Deduction Modulo for BWare Conclusion Cnam / Inria PSATTT’13 25
Proof Search in Axiomatic Theories Automated Deduction Modulo David Delahaye Use of Deduction Modulo 1 Introduction Deduction Modulo ◮ Transform axioms into rewrite rules; & Superdeduction ◮ Turn proof search among the axioms into computations; Superdeduction for Zenon ◮ Avoid unnecessary blowups in the proof search; Superdeduction for the B Method ◮ Shrink the size of proofs (record only meaningful steps). Super Zenon for First Order Theories Deduction Modulo for Zenon This Talk Zenon Modulo over the TPTP Library ◮ Introduce deduction modulo (and superdeduction); A Backend for Zenon Modulo ◮ Present the experiments in automated deduction; Deduction Modulo for BWare ◮ Describe the applications in industrial settings. Conclusion Cnam / Inria PSATTT’13 25
Deduction Modulo & Superdeduction Automated Inclusion Deduction Modulo David Delahaye Introduction ∀ a ∀ b (( a ⊆ b ) ⇔ ( ∀ x ( x ∈ a ⇒ x ∈ b ))) Deduction Modulo 2 & Superdeduction Superdeduction for Zenon Proof in Sequent Calculus Superdeduction for the B Method Super Zenon for First Order Theories Deduction Modulo for Zenon Ax . . . , x ∈ A ⊢ A ⊆ A , x ∈ A Zenon Modulo over ⇒ R the TPTP Library . . . ⊢ A ⊆ A , x ∈ A ⇒ x ∈ A A Backend for ∀ R Ax Zenon Modulo . . . ⊢ A ⊆ A , ∀ x ( x ∈ A ⇒ x ∈ A ) . . . , A ⊆ A ⊢ A ⊆ A ⇒ L Deduction Modulo . . . , ( ∀ x ( x ∈ A ⇒ x ∈ A )) ⇒ A ⊆ A ⊢ A ⊆ A for BWare ∧ L Conclusion A ⊆ A ⇔ ( ∀ x ( x ∈ A ⇒ x ∈ A )) ⊢ A ⊆ A ∀ L × 2 ∀ a ∀ b (( a ⊆ b ) ⇔ ( ∀ x ( x ∈ a ⇒ x ∈ b ))) ⊢ A ⊆ A Cnam / Inria PSATTT’13 25
Deduction Modulo & Superdeduction Automated Inclusion Deduction Modulo David Delahaye Introduction ∀ a ∀ b (( a ⊆ b ) − → ( ∀ x ( x ∈ a ⇒ x ∈ b ))) Deduction Modulo 2 & Superdeduction Superdeduction for Zenon Rewrite Rule Superdeduction for the B Method Super Zenon for First Order Theories ( a ⊆ b ) − → ( ∀ x ( x ∈ a ⇒ x ∈ b )) Deduction Modulo for Zenon Zenon Modulo over Proof in Deduction Modulo the TPTP Library A Backend for Zenon Modulo Deduction Modulo Ax for BWare x ∈ A ⊢ x ∈ A Conclusion ⇒ R ⊢ x ∈ A ⇒ x ∈ A ∀ R , A ⊆ A − → ∀ x ( x ∈ A ⇒ x ∈ A ) ⊢ A ⊆ A Cnam / Inria PSATTT’13 25
Deduction Modulo & Superdeduction Automated Deduction Modulo David Delahaye Introduction Inclusion Deduction Modulo 2 & Superdeduction Superdeduction ∀ a ∀ b (( a ⊆ b ) − → ( ∀ x ( x ∈ a ⇒ x ∈ b ))) for Zenon Superdeduction for the B Method Super Zenon for First Order Theories Computation of the Superdeduction Rule Deduction Modulo for Zenon Zenon Modulo over the TPTP Library Γ ⊢ ∀ x ( x ∈ a ⇒ x ∈ b ) , ∆ A Backend for Zenon Modulo Γ ⊢ a ⊆ b , ∆ Deduction Modulo for BWare Conclusion Cnam / Inria PSATTT’13 25
Deduction Modulo & Superdeduction Automated Deduction Modulo David Delahaye Inclusion Introduction Deduction Modulo 2 & Superdeduction ∀ a ∀ b (( a ⊆ b ) − → ( ∀ x ( x ∈ a ⇒ x ∈ b ))) Superdeduction for Zenon Superdeduction for the B Method Computation of the Superdeduction Rule Super Zenon for First Order Theories Deduction Modulo for Zenon Γ , x ∈ a ⊢ x ∈ b , ∆ Zenon Modulo over the TPTP Library ⇒ R Γ ⊢ x ∈ a ⇒ x ∈ b , ∆ A Backend for ∀ R , x �∈ Γ , ∆ Zenon Modulo Γ ⊢ ∀ x ( x ∈ a ⇒ x ∈ b ) , ∆ Deduction Modulo for BWare Γ ⊢ a ⊆ b , ∆ Conclusion Cnam / Inria PSATTT’13 25
Deduction Modulo & Superdeduction Automated Inclusion Deduction Modulo David Delahaye Introduction ∀ a ∀ b (( a ⊆ b ) − → ( ∀ x ( x ∈ a ⇒ x ∈ b ))) Deduction Modulo 2 & Superdeduction Superdeduction for Zenon Computation of the Superdeduction Rule Superdeduction for the B Method Super Zenon for First Order Theories Γ , x ∈ a ⊢ x ∈ b , ∆ IncR , x �∈ Γ , ∆ Deduction Modulo for Zenon Γ ⊢ a ⊆ b , ∆ Zenon Modulo over the TPTP Library A Backend for Zenon Modulo Proof in Superdeduction Deduction Modulo for BWare Conclusion Ax x ∈ A ⊢ x ∈ A IncR Cnam / Inria ⊢ A ⊆ A PSATTT’13 25
From Axioms to Rewrite Rules Automated Deduction Modulo David Delahaye Difficulties Introduction Deduction Modulo 3 ◮ Confluence and termination of the rewrite system; & Superdeduction ◮ Preservation of the consistency; Superdeduction for Zenon ◮ Preservation of the cut-free completeness; Superdeduction for the B Method ◮ Automation of the transformation. Super Zenon for First Order Theories Deduction Modulo for Zenon An Example Zenon Modulo over the TPTP Library ◮ Axiom A ⇔ ( A ⇒ B ) ; A Backend for Zenon Modulo ◮ Transformed into A − → A ⇒ B ; Deduction Modulo for BWare ◮ We want to prove: B . Conclusion Cnam / Inria PSATTT’13 25
From Axioms to Rewrite Rules Automated Deduction Modulo An Example (Continued) David Delahaye ◮ In sequent calculus, we have a cut-free proof: Introduction Deduction Modulo 3 & Superdeduction Superdeduction for Zenon ∼ Π Superdeduction for A ⇒ ( A ⇒ B ) , A ⊢ B , B the B Method Π ⇒ R Super Zenon for A ⇒ ( A ⇒ B ) ⊢ B , A ⇒ B A ⇒ ( A ⇒ B ) , A ⊢ B First Order Theories ⇒ L Deduction Modulo A ⇒ ( A ⇒ B ) , ( A ⇒ B ) ⇒ A ⊢ B for Zenon ⇔ L A ⇔ ( A ⇒ B ) ⊢ B Zenon Modulo over the TPTP Library Where Π is: A Backend for Zenon Modulo ax ax Deduction Modulo for BWare A ⊢ B , A A , B ⊢ B ⇒ L ax Conclusion A ⊢ B , A A , A ⇒ B ⊢ B ⇒ L A ⇒ ( A ⇒ B ) , A ⊢ B Cnam / Inria PSATTT’13 25
From Axioms to Rewrite Rules Automated Deduction Modulo David Delahaye An Example (Continued) Introduction Deduction Modulo 3 ◮ In deduction modulo, we have to cut A to get a proof: & Superdeduction Superdeduction for Zenon Π Superdeduction for the B Method Π A ⊢ B ⇒ R , A − → A ⇒ B Super Zenon for A ⊢ B ⊢ A cut First Order Theories ⊢ B Deduction Modulo for Zenon Where Π is: Zenon Modulo over the TPTP Library ax ax A Backend for A ⊢ A A , B ⊢ B ⇒ L , A − Zenon Modulo ax → A ⇒ B Deduction Modulo A ⊢ A A , A ⊢ B cut for BWare A ⊢ B Conclusion Cnam / Inria PSATTT’13 25
Some References for Deduction Modulo Automated Deduction Modulo David Delahaye Seminal Papers Introduction Deduction Modulo 4 ◮ Deduction Modulo: & Superdeduction Superdeduction G. Dowek, T. Hardin, C. Kirchner. Theorem Proving Modulo . JAR (2003). for Zenon ◮ Superdeduction: Superdeduction for the B Method P . Brauner, C. Houtmann, C. Kirchner. Principles of Superdeduction. LICS (2007). Super Zenon for First Order Theories Deduction Modulo Theories Modulo for Zenon Zenon Modulo over the TPTP Library ◮ Arithmetic: A Backend for Zenon Modulo G. Dowek, B. Werner. Arithmetic as a Theory Modulo. RTA (2005). Deduction Modulo ◮ Set Theory: for BWare Conclusion G. Dowek, A. Miquel. Cut Elimination for Zermelo Set Theory. Draft (2007). Cnam / Inria PSATTT’13 25
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