Superposition Modulo Linear Arithmetic Sup(LA) Ernst Althaus, - PowerPoint PPT Presentation

Superposition Modulo Linear Arithmetic – Sup(LA) Ernst Althaus, Evgeny Kruglov, Christoph Weidenbach Max Planck Institute for Computer Science Saarbrücken 1

Overview • Motivation – Building arithmetic into Automated Theorem Proving will constitute a milestone in Automated Reasoning. – Verification of linear hybrid systems, program analysis, protocol analysis, etc. – New decidability results. • Task – Integrate LA into the SUP calculus in a modular fashion. – Extend the technology of redundancy detection in the free first-order theory to the combination of the free theory and Linear Arithmetic. • Challenge – Many theoretical questions have been solved (Hierarchic Theorem Proving by Bachmair, Ganzinger, Waldmann), but there was no answer to redundancy detection in the combination of theories. 2

Notions Sup(LA) calculus • Clause: Λ Γ → ∆ || ∩ ∩ ∪ Γ ⇒ Λ ∩ ∆ – – Λ a linear arithmetic constraint (LAC), defined as conjunction of atoms built over the rationals, the theory symbols: + < > ≈ ≤ ≥ , , , , , – Γ , ∆ are sequences of first-order atoms, only containing signature symbols from the free first- order theory. – All parts share universally quantified variables. > − ≥ → x y , 4 x 3.5 y 0 || S x y ( , ) S ( , ) x y 1 2 3

Inference rules Sup(LA) calculus Λ Γ → ∆ Λ Γ → ∆ || , E || , E 1 1 1 1 2 2 2 2 I • Resolution: Λ Λ Γ Γ → ∆ ∆ σ ( , || , , ) 1 2 1 2 1 2 σ = σ σ E E ( E E ). where is the unifier of and 1 2 1 2 Λ Γ → ∆ || , E , E • Factoring: 1 2 I Λ Γ → ∆ σ ( || , E ) 1 σ E E . where is the unifier of and 2 1 4

Reduction rules Sup(LA) calculus Λ || Γ → ∆ • Tautology Deletion: R � Γ → ∆ ∃ x Λ where is a tautology or is unsatisfiable. Λ Γ → ∆ Λ Γ → ∆ || || • Subsumption Deletion: 1 1 1 2 2 2 R Λ Γ → ∆ || 1 1 1 Λ ⇒ Λ where σ Γ σ ⊆ Γ ∆ σ ⊆ ∆ 1 . , , 1 2 1 2 2 σ = δτ The substitution : δ – the standard subsumption matcher between the free parts of the clauses τ – a theory matcher mapping the variables solely occurring in first constraint to variables in the second one. 5

LAC Implication Problem Λ ⇒ Λ • Recall the problem: 1 δτ 2 � � � � τ • is an affine transformation: ֏ τ + z + β : y S x T � = Λ ∩ Λ δ x vars ( ) vars ( ), common variables 2 1 � Λ 1 δ = τ = Λ δ Λ ( ) ( ) \ ( ), variables solely occurring in y dom vars vars 1 2 � Λ variables solely occurring in = Λ Λ δ z vars ( ) \ vars ( ), 2 2 1 τ Λ σ • With the substitution the constraint contains 1 parameter products (non-linear problem). 6

LAC Implication Problem 7

Non-closed Polyhedra Containment Polyhedra Containment Problem • Decide whether the set � � � � � ′ ′ ′ ′ ′ ′ Λ = ≤ < { x | A x c , A x c } 1 contains the set � � � � � ′ ′ ′ ′ ′ ′ Λ = ≤ < { x | B x d , B x d } 1 • 8

Polyhedra Containment Problem Polyhedra Containment 9