Computing a Complete Basis for Equalities Implied by a System of LRA Constraints 7/2/2016
Linear Arithmetic / Linear Programming Input: coefficients (∈ ℤ ) bound (∈ ℤ ) variables Basis for Equalities – Bromberger, Weidenbach 7/2/2016 2/15
Linear Arithmetic / Linear Programming Input: Goal: P NP Complexity: Example: Basis for Equalities – Bromberger, Weidenbach 7/2/2016 2/15
Fast Cube Tests for LIA constraint solving (Bromberger & Weidenbach, IJCAR 2016) Largest Cube Test • largest cube inside the set of real solutions Basis for Equalities – Bromberger, Weidenbach 7/2/2016 3/15
Fast Cube Tests for LIA constraint solving (Bromberger & Weidenbach, IJCAR 2016) Largest Cube Test • largest cube inside the set of real solutions • center point Basis for Equalities – Bromberger, Weidenbach 7/2/2016 3/15
Fast Cube Tests for LIA constraint solving (Bromberger & Weidenbach, IJCAR 2016) Largest Cube Test • largest cube inside the set of real solutions • center point → integer point • optimization LP (LRA) + evaluation Basis for Equalities – Bromberger, Weidenbach 7/2/2016 3/15
Equalities substitute Basis for Equalities – Bromberger, Weidenbach 7/2/2016 5/15
Equalities substitute Diophantine Equation Handler A Practical Approach to Satisfiability Modulo Linear Integer Arithmetic by A. Griggio. JSAT 2012 Basis for Equalities – Bromberger, Weidenbach 7/2/2016 5/15
Equalities substitute Diophantine Equation Handler A Practical Approach to Satisfiability Modulo Linear Integer Arithmetic by A. Griggio. JSAT 2012 Not Always Explicit! Basis for Equalities – Bromberger, Weidenbach 7/2/2016 5/15
Implied Equalities 𝑦 𝑧 = 2 2 explicit implicit Basis for Equalities – Bromberger, Weidenbach 7/2/2016 6/15
Equality Basis explicit implicit Basis for Equalities – Bromberger, Weidenbach 7/2/2016 7/15
Equality Basis Equality Basis: Implied Equalites: 1.set of All linear independent linear combinations equalities of the 2. maximal Equality Basis + Basis for Equalities – Bromberger, Weidenbach 7/2/2016 7/15
Equality Basis Equality Basis: Implied Equalites: 1.set of All linear independent linear combinations equalities of the 2. maximal Equality Basis − Basis for Equalities – Bromberger, Weidenbach 7/2/2016 7/15
Equality Basis Simplification: Eliminating Equalities via Substitution Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15
Equality Basis Simplification: Verifying Eliminating Implied Equalities: Equalities via via Substitution Substitution (Nelson-Oppen) ? Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15
Nelson-Oppen Combination pairs of equivalent variables 𝜏 ≔ { } , Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15
Nelson-Oppen Combination pairs of equivalent variables 𝜏 + normalizing semantic syntactic equivalence equivalence Find equivalent variables with DAGs! Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15
Nelson-Oppen Combination pairs of How do we find equalities? equivalent variables 𝜏 + normalizing semantic syntactic equivalence equivalence Find equivalent variables with DAGs! Basis for Equalities – Bromberger, Weidenbach 7/2/2016 8/15
Largest Cube Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 9/15
Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15
Equality Test Interior & Surface Interior Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15
Equality Test Interior & Surface Interior Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15
Equality Test sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15
Equality Test sat unsat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 10/15
Equality Explanation original Positive Linear Combinations Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15
Equality Explanation original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15
Equality Explanation strict original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15
Equality Explanation strict Multiply original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15
Equality Explanation strict Multiply original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15
Equality Explanation strict original Basis for Equalities – Bromberger, Weidenbach 7/2/2016 11/15
Equality Test Original System of Constraints actual implying strict inequalities inequalities inequalities Equality Explanation Minimal Conflict System for Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15
Equality Test Original System of Constraints actual implying strict inequalities inequalities inequalities Equality Explanation Minimal Conflict System for Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15
Equality Test Original System of Constraints strict non-strict inequalities inequalities System for Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15
Equality Test Original System of Constraints strict non-strict inequalities inequalities Satisfiable Assignment for Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15
Equality Test Original System of Constraints strict non-strict inequalities inequalities System for Equality Test Basis for Equalities – Bromberger, Weidenbach 7/2/2016 12/15
Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15
Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15
Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15
Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15
Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15
Computing an Equality Basis strict equalities original substitutions sat unsat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15
Computing an Equality Basis strict equalities original substitutions sat sat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 13/15
Computing an Equality Basis simplified equalities original substitutions sat sat Basis for Equalities – Bromberger, Weidenbach 7/2/2016 14/15
Computing an Equality Basis simplified equalities original substitutions sat sat ? Nelson- Oppen Basis for Equalities – Bromberger, Weidenbach 7/2/2016 14/15
Conclusions Finding Equalities: A Basis for Equalities: Applications: Thank you for your attention! Basis for Equalities – Bromberger, Weidenbach 7/2/2016 15/15
Inequality Tableau & Bounds Representation Representation Equalitiy Test substitution Equalitiy Basis Basis for Equalities – Bromberger, Weidenbach 7/2/2016 16/17
Inequality Tableau & Bounds Representation Representation Equalitiy Test pivoting substitution Equalitiy Basis Basis for Equalities – Bromberger, Weidenbach 7/2/2016 16/17
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