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

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Superposition Modulo Linear Arithmetic – Sup(LA)

Max Planck Institute for Computer Science Saarbrücken

Ernst Althaus, Evgeny Kruglov, Christoph Weidenbach

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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.

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• Clause:

–

– a linear arithmetic constraint (LAC), defined as conjunction of atoms built over the rationals, the theory symbols: – are sequences of first-order atoms, only containing signature symbols from the free first-

• rder theory.

– All parts share universally quantified variables.

Notions

Sup(LA) calculus

|| Λ Γ → ∆

Λ ∩ Γ ⇒ ∆

∩ ∩ ∪

Λ

, , , , , + < > ≈ ≤ ≥ ∆ Γ,

1 2

, 4 3.5 0 || ( , ) ( , ) x y x y S x y S x y > − ≥ →

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Inference rules

• Resolution:

where is the unifier of and

• Factoring:

where is the unifier of and

Sup(LA) calculus

σ ) , , || , ( , || , ||

2 1 2 1 2 1 2 2 2 2 1 1 1 1

∆ ∆ → Γ Γ Λ Λ ∆ → Γ Λ ∆ → Γ Λ E E I σ ) , || ( , , ||

1 2 1

E E E I ∆ → Γ Λ ∆ → Γ Λ

1

E

σ

2

E ). (

2 1

σ σ E E =

1

E

σ

.

2

E

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Reduction rules

• Tautology Deletion:

where is a tautology or is unsatisfiable.

• Subsumption Deletion:

where The substitution :

– the standard subsumption matcher between the free parts of the clauses – a theory matcher mapping the variables solely occurring in first constraint to variables in the second one.

Λ ∃x

• 1

1 1 2 2 2 1 1 1

|| || || ∆ → Γ Λ ∆ → Γ Λ ∆ → Γ Λ R ∆ → Γ Λ || R

Sup(LA) calculus

∆ → Γ

1 2 1 2

, , σ σ Γ ⊆ Γ ∆ ⊆ ∆

σ δτ = δ τ

2 1 .

σ Λ ⇒ Λ

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LAC Implication Problem

• Recall the problem:
• is an affine transformation:

common variables variables solely occurring in variables solely occurring in

• With the substitution the constraint contains

parameter products (non-linear problem).

2 1δτ

Λ ⇒ Λ

τ

2 1 1 2 2 1

( ) ( ), ( ) ( ) \ ( ), ( ) \ ( ), x vars vars y dom vars vars z vars vars δ τ δ δ = Λ ∩ Λ = = Λ Λ = Λ Λ

• β

τ

• ֏
• +

+ z T x S y :

σ

1

Λ

τ

1δ

Λ

2

Λ

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LAC Implication Problem

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Non-closed Polyhedra Containment

• Decide whether the set

contains the set

• }

, | {

1

c x A c x A x ′ ′ < ′ ′ ′ ≤ ′ = Λ

• Polyhedra Containment Problem

} , | {

1

d x B d x B x ′ ′ < ′ ′ ′ ≤ ′ = Λ

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Polyhedra Containment

Polyhedra Containment Problem