[PPT] - A CDCL(LA) Solver SPASS-SATT A CDCL(LA) Solver Translation: fun PowerPoint Presentation

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SPASS-SATT A CDCL(LA) Solver

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SPASS-SATT A CDCL(LA) Solver

Translation: fun (=SPASS) sated (=SATT)

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SPASS-SATT A CDCL(LA) Solver

Translation: fun (=SPASS) sated (=SATT) being sick/tired of having fun…

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Quantifier-Free Linear Arithmetic 𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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Quantifier-Free Linear Arithmetic

Signature: Σ𝑀𝐵 ≔ {+, −, <, ≤, ≥, >, 0, 1, 2, … }

𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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Quantifier-Free Linear Arithmetic

Signature: Σ𝑀𝐵 ≔ {+, −, <, ≤, ≥, >, 0, 1, 2, … } Multiplication only as syntactic sugar! E.g.: 3 ⋅ 𝑦 ↦ 𝑦 + 𝑦 + 𝑦

𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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Quantifier-Free Linear Arithmetic

Goal: Quantifier-Free Linear Rational Arithmetic (QF_LRA)

⇒ rational solution, i.e., 𝑦, 𝑧, … ∈ ℚ

Quantifier-Free Linear Integer Arithmetic (QF_LIA)

⇒ integer solution, i.e., 𝑦, 𝑧, … ∈ ℤ

Signature: Σ𝑀𝐵 ≔ {+, −, <, ≤, ≥, >, 0, 1, 2, … } Multiplication only as syntactic sugar! E.g.: 3 ⋅ 𝑦 ↦ 𝑦 + 𝑦 + 𝑦

𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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CDCL(T) 𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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CDCL(T) 𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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CDCL(T) CDCL solver:

CDCL = conflict-driven clause-learning Decision procedure for propositional CNF formulas

SAT 𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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CDCL(T) CDCL solver:

CDCL = conflict-driven clause-learning Decision procedure for propositional CNF formulas

Theory solver:

Decision procedure for conjunctions of theory atoms e.g. Simplex for QF_LRA & Branch-and-Bound for QF_LIA

SAT

Theory

𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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CDCL(LA) 𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0)

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𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0) CDCL(LA)

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𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0) CDCL(LA) 𝐵 ⟺ 𝑦 > 0;

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𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0) 𝐵 𝐵 CDCL(LA) 𝐵 ⟺ 𝑦 > 0;

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𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0) 𝐵 𝐵 CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝑦 > 0 ∨ 𝑦 + 𝑧 > 0 ∧ 𝑦 < 0 ∨ 𝑦 + 𝑧 < 3 ∧ 𝑧 < 0 ∧ ¬(𝑦 > 0) 𝐵 𝐸 𝐷 𝐶 𝐹 𝐵 CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation 𝐹 ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation 𝐹 ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation 𝐹 ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation 𝐹 ¬𝐵 ⊥ ⊤ ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation 𝐹 ¬𝐵 ⊥ ⊤ ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation 𝐹 ¬𝐵 ⊥ ⊤ ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation 𝐹 ¬𝐵 𝐶 ⊥ ⊤ ⊤ ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Unit Propagation 𝐹 ¬𝐵 𝐶 ⊥ ⊤ ⊤ ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Decision 𝐹 ¬𝐵 𝐶 ⊥ ⊤ ⊤ ⊤ CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 Model: Decision 𝐹 ¬𝐵 𝐶 ⊥ ⊤ ⊤ ⊤ ⊤ 𝐷† CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0;

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Model: 𝐹 ¬𝐵 𝐶 𝐷† Theory Satisfiable? CDCL(LA) 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; 𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ ⊤

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Model: 𝐹 ¬𝐵 𝐶 𝐷† Theory Satisfiable? CDCL(LA) No! 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; 𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ ⊤

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¬𝐵 ⟺ 𝑦 ≤ 0; Model: 𝐹 ¬𝐵 𝐶 𝐷† Theory Satisfiable? CDCL(LA) 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; No! 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; 𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ ⊤

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Model: 𝐹 ¬𝐵 𝐶 𝐷† Conflict Analysis: CDCL(LA) ¬𝐵 ⟺ 𝑦 ≤ 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; 𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ ⊤

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Model: 𝐹 ¬𝐵 𝐶 𝐷† Conflict Analysis: CDCL(LA) (¬𝐹 ∧ 𝐵 ∧ ¬𝐶) ¬𝐵 ⟺ 𝑦 ≤ 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; 𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ ⊤

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Model: 𝐹 ¬𝐵 𝐶 𝐷† Conflict Analysis: CDCL(LA) (¬𝐹 ∧ 𝐵 ∧ ¬𝐶) ⊥ ⊥ ⊥ ¬𝐵 ⟺ 𝑦 ≤ 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; 𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ ⊤

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Model: 𝐹 ¬𝐵 𝐶 𝐷† Conflict Analysis: CDCL(LA) UNSAT! (¬𝐹 ∧ 𝐵 ∧ ¬𝐶) ⊥ ⊥ ⊥ ¬𝐵 ⟺ 𝑦 ≤ 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; 𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ ⊤

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SMT-COMP 2018

Solver Solved Score CPU time Score Solved CVC4 1586.833 69.006 1566 SPASS-SATT 1586.396 64.292 1590 Yices 2.6.0 1583.186 63.901 1567 veriT 1568.212 79.840 1527 SMTInterpol 1548.476 102.257 1521 MathSATn 1536.458 107.673 1461 z3-4.7.1n 1527.249 113.154 1435

pensmt2

1498.663 131.674 1329 Ctrl-Ergo 1450.082 172.097 1354 SMTRAT-Rat 1297.891 275.918 984 SMTRAT-MCSAT 1090.526 409.015 711 Solver Solved Score CPU time Score Solved SPASS-SATT 6587.626 72.048 6744 Ctrl-Ergo 6221.467 156.086 6259 MathSATn 6135.114 164.626 6528 SMTInterpol 5915.623 204.123 6286 CVC4 5891.019 194.986 6357 Yices 2.6.0 5867.976 209.452 6232 z3-4.7.1n 5733.374 224.539 6195 SMTRAT-Rat 4049.914 515.394 3112 veriT 3155.162 295.434 2734

QF_LIA (Main Track)

QF_LIA = quantifier-free linear integer arithmetic Benchmarks: 6947 Time limit: 1200s

QF_LRA (Main Track)

QF_LRA = quantifier-free linear rational arithmetic Benchmarks: 1649 Time limit: 1200s

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SAT and theory interaction:

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SAT and theory interaction:

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Theory solver extensions:

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SAT and theory interaction:

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Data-structure improvements: Theory solver extensions:

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SAT and theory interaction:

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Data-structure improvements: Theory solver extensions: Preprocessing:

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SAT and theory interaction:

weakened early pruning [Sebastiani07]
unate propagations and bound refinements [Dutertre06]
decision recommendations [Yices]

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Data-structure improvements:

priority queue for pivot selection [pretty much everyone]
integer coefficients instead of rational coefficients [veriT]
backup instead of recalculation [pretty much everyone]

Theory solver extensions:

unit cube test [Bromberger16]
bounding transformation [Bromberger18]
simple rounding and bound propagation [Schrijver86]

Preprocessing:

if-then-else (reconstruction, lifting, simplification, bounding) [CVC4]
pseudo-Boolean inequalities [CVC4]
small CNF transformation [Weidenbach01]

[…] invented by our team […] invented & published by someone else […] never published but implemented

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SAT and theory interaction:

weakened early pruning [Sebastiani07]
unate propagations and bound refinements [Dutertre06]
decision recommendations [Yices]

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Data-structure improvements:

priority queue for pivot selection [pretty much everyone]
integer coefficients instead of rational coefficients [veriT]
backup instead of recalculation [pretty much everyone]

Theory solver extensions:

unit cube test [Bromberger16]
bounding transformation [Bromberger18]
simple rounding and bound propagation [Schrijver86]

Preprocessing:

if-then-else (reconstruction, lifting, simplification, bounding) [CVC4]
pseudo-Boolean inequalities [CVC4]
small CNF transformation [Weidenbach01]

[…] invented by our team […] invented & published by someone else […] never published but implemented

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SAT and Theory Interaction

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SAT and Theory Interaction Bare minimum requirements:

theory check for complete model
return theory conflict for learning

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SAT and Theory Interaction Bare minimum requirements:

theory check for complete model
return theory conflict for learning

(Weakened) early pruning [Sebastiani07]

theory check for some partial models (⇒ early conflicts)
weaker check if full check too expensive

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SAT and Theory Interaction Bare minimum requirements:

theory check for complete model
return theory conflict for learning

(Weakened) early pruning [Sebastiani07]

theory check for some partial models (⇒ early conflicts)
weaker check if full check too expensive

Theory Propagation

unate propagations and bound refinements [Dutertre06]

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SAT heuristics based on theory knowledge

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SAT and Theory Interaction Bare minimum requirements:

theory check for complete model
return theory conflict for learning

(Weakened) early pruning [Sebastiani07]

theory check for some partial models (⇒ early conflicts)
weaker check if full check too expensive

Theory Propagation

unate propagations and bound refinements [Dutertre06]
decision recommendations [Yices]

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SAT heuristics based on theory knowledge

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SAT and Theory Interaction (Weakened) early pruning [Sebastiani07]

theory check for some partial models (⇒ early conflicts)
weaker check if full check too expensive
decision recommendations [Yices]

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¬𝐵 ⟺ 𝑦 ≤ 0; Model: 𝐹 ¬𝐵 𝐶 𝐷† Theory Satisfiable? Early Pruning 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; No! 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; 𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ ⊤

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ Early Pruning 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; Check for theory satisfiability before each decision!

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ Early Pruning 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; Check for theory satisfiability before each decision! Full theory check is too expensive? (NP for QF_LIA)

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𝐵 ∨ 𝐶 ∧ 𝐷 ∨ 𝐸 ∧ 𝐹 ∧ ¬ 𝐵 ⊥ ⊤ ⊤ ⊤ Weakened Early Pruning 𝐵 ⟺ 𝑦 > 0; 𝐹 ⟺ 𝑧 < 0; 𝐶 ⟺ 𝑦 + 𝑧 > 0; 𝐸 ⟺ 𝑦 + 𝑧 > 4; 𝐷 ⟺ 𝑦 < 0; Full theory check is too expensive? (NP for QF_LIA) Do a weaker check! (Check only for rational solutions) Check for theory satisfiability before each decision!

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Decision Recommendations

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Model: 𝐵 𝐶 𝐵 ⟺ 𝑦 ≥ 0; 𝐷 ⟺ 𝑧 ≥ 5; 𝐶 ⟺ 𝑧 ≥ 𝑦 + 1; 𝐷† How to select phase of decision literal?

r ¬𝐷†

SLIDE 56

Decision Recommendations

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Model: 𝐵 𝐶 Use rational assignment as heuristic

(Assignment is side effect of failed weakened early pruning)

𝐵 ⟺ 𝑦 ≥ 0; 𝐷 ⟺ 𝑧 ≥ 5; 𝐶 ⟺ 𝑧 ≥ 𝑦 + 1; Assignment: 𝑦 = 0, 𝑧 = 1 𝐷† How to select phase of decision literal?

r ¬𝐷†

SLIDE 57

Decision Recommendations

9/25

Model: 𝐵 𝐶 Use rational assignment as heuristic

(Assignment is side effect of failed weakened early pruning)

𝐵 ⟺ 𝑦 ≥ 0; 𝐷 ⟺ 𝑧 ≥ 5; 𝐶 ⟺ 𝑧 ≥ 𝑦 + 1; Assignment: 𝑦 = 0, 𝑧 = 1 𝐷† How to select phase of decision literal?

r ¬𝐷†

Goal: assignment should stay solution for model

SLIDE 58

Decision Recommendations

9/25

Model: 𝐵 𝐶 Use rational assignment as heuristic

(Assignment is side effect of failed weakened early pruning)

𝐵 ⟺ 𝑦 ≥ 0; 𝐷 ⟺ 𝑧 ≥ 5; 𝐶 ⟺ 𝑧 ≥ 𝑦 + 1; Assignment: 𝑦 = 0, 𝑧 = 1 𝐷† How to select phase of decision literal?

r ¬𝐷†

Goal: assignment should stay solution for model

(Why? Might reduce time spent on theory checking)

SLIDE 59

Decision Recommendations

9/25

Model: 𝐵 𝐶 Use rational assignment as heuristic

(Assignment is side effect of failed weakened early pruning)

𝐵 ⟺ 𝑦 ≥ 0; 𝐷 ⟺ 𝑧 ≥ 5; 𝐶 ⟺ 𝑧 ≥ 𝑦 + 1; Assignment: 𝑦 = 0, 𝑧 = 1 𝐷† How to select phase of decision literal?

r ¬𝐷†

Goal: assignment should stay solution for model

(Why? Might reduce time spent on theory checking)

𝐷† ⟺ 1 ≥ 5; ¬𝐷† ⟺ 1 < 5;

SLIDE 60

Decision Recommendations

9/25

Model: 𝐵 𝐶 Use rational assignment as heuristic

(Assignment is side effect of failed weakened early pruning)

𝐵 ⟺ 𝑦 ≥ 0; 𝐷 ⟺ 𝑧 ≥ 5; 𝐶 ⟺ 𝑧 ≥ 𝑦 + 1; Assignment: 𝑦 = 0, 𝑧 = 1 𝐷† How to select phase of decision literal?

r ¬𝐷†

Goal: assignment should stay solution for model ¬𝐷†

(Why? Might reduce time spent on theory checking)

𝐷† ⟺ 1 ≥ 5; ¬𝐷† ⟺ 1 < 5;

SLIDE 61

Decision Recommendations

11/25

additional instances: 129

QF_LIA (6947 problems)

twice as fast/slow: 389/58

SLIDE 62

Decision Recommendations

11/25

additional instances: 129

QF_LIA (6947 problems)

additional instances: 116

convert (319 problems)

twice as fast/slow: 389/58

SLIDE 63

Theory Solver Input:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛}

12/25

Goal:

QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 64

Theory Solver Input: Example:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛}

2𝑦2 ≤ 5𝑦1, 3𝑦2 ≥ 4𝑦1, 2𝑦2 ≤ −5𝑦1 + 15, 2𝑦2 ≥ −3𝑦1 + 4,

12/25

Goal:

QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 65

Theory Solver Input: Example:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛}

2𝑦2 ≤ 5𝑦1, 3𝑦2 ≥ 4𝑦1, 2𝑦2 ≤ −5𝑦1 + 15, 2𝑦2 ≥ −3𝑦1 + 4,

12/25

Goal:

QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 66

Theory Solver Input: Example:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛}

2𝑦2 ≤ 5𝑦1, 3𝑦2 ≥ 4𝑦1, 2𝑦2 ≤ −5𝑦1 + 15, 2𝑦2 ≥ −3𝑦1 + 4,

𝑦1, 𝑦2 ∈ ℚ QF_LRA

12/25

Goal:

QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 67

Theory Solver Input: Example:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛}

2𝑦2 ≤ 5𝑦1, 3𝑦2 ≥ 4𝑦1, 2𝑦2 ≤ −5𝑦1 + 15, 2𝑦2 ≥ −3𝑦1 + 4,

𝑦1, 𝑦2 ∈ ℚ QF_LRA

12/25

Goal:

QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 68

Theory Solver Example:

2𝑦2 ≤ 5𝑦1, 3𝑦2 ≥ 4𝑦1, 2𝑦2 ≤ −5𝑦1 + 15, 2𝑦2 ≥ −3𝑦1 + 4,

𝑦1, 𝑦2 ∈ ℤ

12/25

QF_LIA

Input: Goal:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 69

Theory Solver Example:

2𝑦2 ≤ 5𝑦1, 3𝑦2 ≥ 4𝑦1, 2𝑦2 ≤ −5𝑦1 + 15, 2𝑦2 ≥ −3𝑦1 + 4,

𝑦1, 𝑦2 ∈ ℤ

12/25

QF_LIA

Input: Goal:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 70

Theory Solver Example:

2𝑦2 ≤ 5𝑦1, 3𝑦2 ≥ 4𝑦1, 2𝑦2 ≤ −5𝑦1 + 15, 2𝑦2 ≥ −3𝑦1 + 4,

𝑦1, 𝑦2 ∈ ℤ

12/25

QF_LRA: dual simplex QF_LIA: branch and bound

Solver:

QF_LIA

Input: Goal:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 71

Theory Solver Example:

2𝑦2 ≤ 5𝑦1, 3𝑦2 ≥ 4𝑦1, 2𝑦2 ≤ −5𝑦1 + 15, 2𝑦2 ≥ −3𝑦1 + 4,

𝑦1, 𝑦2 ∈ ℤ

12/25

QF_LRA: dual simplex QF_LIA: branch and bound

Solver:

QF_LIA

Input: Goal:

𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} QF_LRA: 𝑦1, … , 𝑦𝑜 ∈ ℚ

r QF_LIA: 𝑦1, … , 𝑦𝑜 ∈ ℤ

SLIDE 72

1 2 1 2 1 1 2 1 −1 1 2 1 2 1 −1

Theory Solver Extensions for absolutely unbounded problems for partially unbounded problems

13/25

Unit Cube Test

(IJCAR 2016)

Bounding Transformation

(IJCAR 2018)

SLIDE 73

Unbounded Problems

14/25

SLIDE 74

Unbounded Problems Requirement: unbounded direction

14/25

SLIDE 75

Unbounded Problems Requirement: unbounded direction

ℎ ∈ ℚ𝑜 is bounded iff ∃𝑚, 𝑣 ∈ ℤ. ∀𝑦 ∈ ℚ𝑜. 𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} → 𝑚 ≤ ℎ𝑈𝑦 ≤ 𝑣

14/25

SLIDE 76

Unbounded Problems Requirement: unbounded direction

ℎ ∈ ℚ𝑜 is bounded iff ∃𝑚, 𝑣 ∈ ℤ. ∀𝑦 ∈ ℚ𝑜. 𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} → 𝑚 ≤ ℎ𝑈𝑦 ≤ 𝑣

ℎ′

14/25

SLIDE 77

Unbounded Problems Requirement: unbounded direction

ℎ ∈ ℚ𝑜 is bounded iff ∃𝑚, 𝑣 ∈ ℤ. ∀𝑦 ∈ ℚ𝑜. 𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} → 𝑚 ≤ ℎ𝑈𝑦 ≤ 𝑣

ℎ′

14/25

SLIDE 78

Unbounded Problems Requirement: unbounded direction

ℎ ∈ ℚ𝑜 is bounded iff ∃𝑚, 𝑣 ∈ ℤ. ∀𝑦 ∈ ℚ𝑜. 𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} → 𝑚 ≤ ℎ𝑈𝑦 ≤ 𝑣

ℎ′

14/25

SLIDE 79

Unbounded Problems Requirement: unbounded direction

ℎ ℎ

14/25

ℎ ∈ ℚ𝑜 is bounded iff ∃𝑚, 𝑣 ∈ ℤ. ∀𝑦 ∈ ℚ𝑜. 𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} → 𝑚 ≤ ℎ𝑈𝑦 ≤ 𝑣

SLIDE 80

Unbounded Problems Requirement: unbounded direction

ℎ ℎ

14/25

ℎ ∈ ℚ𝑜 is bounded iff ∃𝑚, 𝑣 ∈ ℤ. ∀𝑦 ∈ ℚ𝑜. 𝑏𝑗

𝑈𝑦 ≤ 𝑐𝑗

𝑗 = 1, … , 𝑛} → 𝑚 ≤ ℎ𝑈𝑦 ≤ 𝑣

SLIDE 81

Unbounded Problems partially unbounded

ℎ ℎ′

partially unbounded: both bounded and unbounded directions

14/25

SLIDE 82

Unbounded Problems absolutely unbounded

ℎ′

absolutely unbounded:

nly unbounded directions

partially unbounded

14/25

SLIDE 83

Overview: Unit Cube Test for absolutely unbounded problems

15/25

(IJCAR 2016)

SLIDE 84

Overview: Unit Cube Test for absolutely unbounded problems

15/25

(IJCAR 2016)

unit cube guarantees

integer solution

SLIDE 85

Overview: Unit Cube Test for absolutely unbounded problems

15/25

(IJCAR 2016)

unit cube guarantees

integer solution

computable in

polynomial time

SLIDE 86

Overview: Unit Cube Test for absolutely unbounded problems

15/25

(IJCAR 2016)

unit cube guarantees

integer solution

computable in

polynomial time

incremental

SLIDE 87

Overview: Unit Cube Test for absolutely unbounded problems

15/25

(IJCAR 2016)

unit cube guarantees

integer solution

computable in

polynomial time

incremental
not complete in general

SLIDE 88

Overview: Unit Cube Test for absolutely unbounded problems

15/25

(IJCAR 2016)

unit cube guarantees

integer solution

computable in

polynomial time

incremental
not complete in general
always succeeds on
abs. unbd. problems

SLIDE 89

Results: Unit Cube Test for absolutely unbounded problems

16/25

(IJCAR 2016)

additional instances: 56 more than twice as fast: 705

QF_LIA (6947 problems)

SLIDE 90

Overview: Bounding Transformation

17/25

1 2 1 2 1 1 2 1 −1 1 2 1 2 1 −1

for partially unbounded problems

(IJCAR 2018)

SLIDE 91

Overview: Bounding Transformation

17/25

1 2 1 2 1 1 2 1 −1 1 2 1 2 1 −1

transforms unbounded

into bounded problems for partially unbounded problems

(IJCAR 2018)

SLIDE 92

Overview: Bounding Transformation

17/25

1 2 1 2 1 1 2 1 −1 1 2 1 2 1 −1

transforms unbounded

into bounded problems

computable in

polynomial time for partially unbounded problems

(IJCAR 2018)

SLIDE 93

Overview: Bounding Transformation

17/25

1 2 1 2 1 1 2 1 −1 1 2 1 2 1 −1

transforms unbounded

into bounded problems

computable in

polynomial time

solution & conflict

conversion (polynomial time) for partially unbounded problems

(IJCAR 2018)

SLIDE 94

Overview: Bounding Transformation

17/25

1 2 1 2 1 1 2 1 −1 1 2 1 2 1 −1

transforms unbounded

into bounded problems

computable in

polynomial time

solution & conflict

conversion (polynomial time)

incremental

for partially unbounded problems

(IJCAR 2018)

SLIDE 95

Results: Bounding Transformation

18/25

1 2 1 2 1 1 2 1 −1 1 2 1 2 1 −1

for partially unbounded problems

(IJCAR 2018)

additional instances: 169 more than twice as fast: 167

QF_LIA (6947 problems)

SLIDE 96

Preprocessing:

if-then-else (reconstruction, lifting, simplification, bounding) [CVC4]
pseudo-Boolean inequalities [CVC4]
small CNF transformation [Weidenbach01]

21/25

[…] invented by our team […] invented & published by someone else […] never published but implemented

SLIDE 97

Preprocessing:

if-then-else (reconstruction, lifting, simplification, bounding) [CVC4]
pseudo-Boolean inequalities [CVC4]
small CNF transformation [Weidenbach01]

21/25

[…] invented by our team […] invented & published by someone else […] never published but implemented

additional instances:1776

QF_LIA (6947 problems)

SLIDE 98

Preprocessing:

if-then-else (reconstruction, lifting, simplification, bounding) [CVC4]
pseudo-Boolean inequalities [CVC4]
small CNF transformation [Weidenbach01]

21/25

[…] invented by our team […] invented & published by someone else […] never published but implemented

additional instances:1776

QF_LIA (6947 problems)

SLIDE 99

22/25

Modular Arithmetic

Type equation here.

SLIDE 100

22/25

Modular Arithmetic 2 ≡9 3 ⋅ 𝑦

Type equation here.

for 𝑦 ∈ ℤ

SLIDE 101

22/25

Modular Arithmetic 2 ≡9 3 ⋅ 𝑦

Type equation here.

UNSAT

for 𝑦 ∈ ℤ

SLIDE 102

22/25

Modular Arithmetic 2 ≡9 3 ⋅ 𝑦

Type equation here.

UNSAT Proof by case distinction:

for 𝑦 ∈ ℤ

SLIDE 103

22/25

Modular Arithmetic 2 ≡9 3 ⋅ 𝑦 0 ≡9 3 ⋅ (3 ⋅ 𝑙)

Type equation here.

𝑦 = 3 ⋅ 𝑙 for 𝑙 ∈ ℤ

UNSAT Proof by case distinction:

for 𝑦 ∈ ℤ

SLIDE 104

22/25

Modular Arithmetic 2 ≡9 3 ⋅ 𝑦 0 ≡9 3 ⋅ (3 ⋅ 𝑙)

Type equation here.

3 ≡9 3 ⋅ (3 ⋅ 𝑙 + 1) 𝑦 = 3 ⋅ 𝑙 for 𝑙 ∈ ℤ 𝑦 = 3 ⋅ 𝑙 + 1 for 𝑙 ∈ ℤ

UNSAT Proof by case distinction:

for 𝑦 ∈ ℤ

SLIDE 105

22/25

Modular Arithmetic 2 ≡9 3 ⋅ 𝑦 0 ≡9 3 ⋅ (3 ⋅ 𝑙)

Type equation here.

3 ≡9 3 ⋅ (3 ⋅ 𝑙 + 1) 6 ≡9 3 ⋅ (3 ⋅ 𝑙 + 2) 𝑦 = 3 ⋅ 𝑙 for 𝑙 ∈ ℤ 𝑦 = 3 ⋅ 𝑙 + 1 for 𝑙 ∈ ℤ 𝑦 = 3 ⋅ 𝑙 + 2 for 𝑙 ∈ ℤ

UNSAT Proof by case distinction:

for 𝑦 ∈ ℤ

SLIDE 106

23/25

2 ≡9 3 ⋅ 𝑦 for 𝑦 ∈ ℤ Modular Arithmetic via If-Then-Else

SLIDE 107

3 ⋅ 𝑦 − 18 3 ⋅ 𝑦 3 ⋅ 𝑦 − 9

23/25

2 = 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 0 ≤ 𝑦 < 9 ∧ 2 ≡9 3 ⋅ 𝑦 for 𝑦 ∈ ℤ Modular Arithmetic via If-Then-Else

SLIDE 108

𝑧 = 3 ⋅ 𝑦 − 18 3 ⋅ 𝑦 𝑧 = 3 ⋅ 𝑦 − 9

23/25

2 = 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 𝑧 ∧ ∧ 2 ≡9 3 ⋅ 𝑦 for 𝑦, 𝑧 ∈ ℤ Modular Arithmetic via If-Then-Else 0 ≤ 𝑦 < 9

SLIDE 109

23/25

( 3 ⋅ 𝑦 < 18 ∨ 𝑧 = 3 ⋅ 𝑦 − 18 ) ∧ ∧ (¬ 3 ⋅ 𝑦 < 18 ∨ 𝑧 = 3 ⋅ 𝑦 − 9 ) ∧ 2 ≡9 3 ⋅ 𝑦 3 ⋅ 𝑦 2 = 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑧 for 𝑦, 𝑧 ∈ ℤ Modular Arithmetic via If-Then-Else 0 ≤ 𝑦 < 9

SLIDE 110

𝑨 = 3 ⋅ 𝑦

23/25

2 = 𝑨 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑨 = 𝑧 ∧ ∧ 2 ≡9 3 ⋅ 𝑦 for 𝑦, 𝑧, 𝑨 ∈ ℤ Modular Arithmetic via If-Then-Else ∧ (¬ 3 ⋅ 𝑦 < 18 ∨ 𝑧 = 3 ⋅ 𝑦 − 9 ) ∧ ( 3 ⋅ 𝑦 < 18 ∨ 𝑧 = 3 ⋅ 𝑦 − 18 ) 0 ≤ 𝑦 < 9

SLIDE 111

23/25

2 ≡9 3 ⋅ 𝑦 2 = 𝑨 ∧ ( 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 3 ⋅ 𝑦 ) ∧ (¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 𝑧 ) ∧ for 𝑦, 𝑧, 𝑨 ∈ ℤ Modular Arithmetic via If-Then-Else ∧ (¬ 3 ⋅ 𝑦 < 18 ∨ 𝑧 = 3 ⋅ 𝑦 − 9 ) ∧ ( 3 ⋅ 𝑦 < 18 ∨ 𝑧 = 3 ⋅ 𝑦 − 18 )

SLIDE 112

23/25

2 ≡9 3 ⋅ 𝑦 2 = 𝑨 ∧ ( 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 3 ⋅ 𝑦 ) ∧ (¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 𝑧 ) ∧

two new variables
suboptimally connected

for 𝑦, 𝑧, 𝑨 ∈ ℤ Modular Arithmetic via If-Then-Else ∧ (¬ 3 ⋅ 𝑦 < 18 ∨ 𝑧 = 3 ⋅ 𝑦 − 9 ) ∧ ( 3 ⋅ 𝑦 < 18 ∨ 𝑧 = 3 ⋅ 𝑦 − 18 ) 0 ≤ 𝑦 < 9

SLIDE 113

3 ⋅ 𝑦 − 18 3 ⋅ 𝑦 3 ⋅ 𝑦 − 9

24/25

If-Then-Else: Shared Monomial Lifting 2 = 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 ∧ 2 ≡9 3 ⋅ 𝑦 for 𝑦 ∈ ℤ

SLIDE 114

3 ⋅ 𝑦 − 18 3 ⋅ 𝑦 3 ⋅ 𝑦 − 9

24/25

If-Then-Else: Shared Monomial Lifting 2 = 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 ∧ 2 ≡9 3 ⋅ 𝑦 for 𝑦 ∈ ℤ All share the monomial 3 ⋅ 𝑦 ! 0 ≤ 𝑦 < 9

SLIDE 115

−18 −9 2 = 3 ⋅ 𝑦 + 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 ∧ 2 ≡9 3 ⋅ 𝑦 for 𝑦 ∈ ℤ If-Then-Else: Shared Monomial Lifting

24/25

SLIDE 116

−18 −9 2 = 3 ⋅ 𝑦 + 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 ∧ 2 ≡9 3 ⋅ 𝑦 for 𝑦 ∈ ℤ If-Then-Else: Shared Monomial Lifting All divisible by −9 !

24/25

0 ≤ 𝑦 < 9

SLIDE 117

2 1 2 = 3 ⋅ 𝑦 − 9 ⋅ 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 ∧ 2 ≡9 3 ⋅ 𝑦 for 𝑦 ∈ ℤ If-Then-Else: Shared Monomial Lifting

24/25

0 ≤ 𝑦 < 9

SLIDE 118

𝑨 = 2 𝑨 = 0 𝑨 = 1 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 ∧ 2 ≡9 3 ⋅ 𝑦 ∧ 2 = 3 ⋅ 𝑦 − 9 ⋅ 𝑨 for 𝑦, 𝑨 ∈ ℤ If-Then-Else: Bounding

24/25

SLIDE 119

𝑨 = 2 𝑨 = 0 𝑨 = 1 𝑗𝑔 3 ⋅ 𝑦 < 9 𝑗𝑔 3 ⋅ 𝑦 < 18 ∧ 2 ≡9 3 ⋅ 𝑦 ∧ 2 = 3 ⋅ 𝑦 − 9 ⋅ 𝑨 0 ≤ 𝑨 ≤ 2 ∧ for 𝑦, 𝑨 ∈ ℤ If-Then-Else: Bounding

24/25

0 ≤ 𝑦 < 9

SLIDE 120

∧ 2 ≡9 3 ⋅ 𝑦 2 = 3 ⋅ 𝑦 − 9 ⋅ 𝑨 0 ≤ 𝑨 ≤ 2 ∧ for 𝑦, 𝑨 ∈ ℤ If-Then-Else: Preprocessing ¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 0 3 ⋅ 𝑦 < 9 ∨ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 1 ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 2 ∧ ∧ ∧

24/25

0 ≤ 𝑦 < 9

SLIDE 121

¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 0 ∧ 2 ≡9 3 ⋅ 𝑦 ∧ 2 = 3 ⋅ 𝑦 − 9 ⋅ 𝑨 0 ≤ 𝑨 ≤ 2 ∧ 3 ⋅ 𝑦 < 9 ∨ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 1 ∧ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 2 ∧ for 𝑦, 𝑨 ∈ ℤ If-Then-Else: Preprocessing

24/25

0 ≤ 𝑦 < 9

SLIDE 122

∧ 2 ≡9 3 ⋅ 𝑦 2 ≤ 3 ⋅ 𝑦 − 9 ⋅ 𝑨 0 ≤ 𝑨 ≤ 2 ∧ ∧ 2 ≥ 3 ⋅ 𝑦 − 9 ⋅ 𝑨 for 𝑦, 𝑨 ∈ ℤ If-Then-Else: Preprocessing ¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 0 ∧ 3 ⋅ 𝑦 < 9 ∨ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 1 ∧ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 2 ∧

24/25

0 ≤ 𝑦 < 9

SLIDE 123

∧ 2 ≡9 3 ⋅ 𝑦 2 3 ≤ 1 ⋅ 𝑦 − 3 ⋅ 𝑨 0 ≤ 𝑨 ≤ 2 ∧ ∧ 2 3 ≥ 1 ⋅ 𝑦 − 3 ⋅ 𝑨 for 𝑦, 𝑨 ∈ ℤ If-Then-Else: Preprocessing ¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 0 ∧ 3 ⋅ 𝑦 < 9 ∨ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 1 ∧ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 2 ∧

24/25

0 ≤ 𝑦 < 9

SLIDE 124

∧ 2 ≡9 3 ⋅ 𝑦 2 3 ≤ 1 ⋅ 𝑦 − 3 ⋅ 𝑨 0 ≤ 𝑨 ≤ 2 ∧ ∧ 2 3 ≥ 1 ⋅ 𝑦 − 3 ⋅ 𝑨 for 𝑦, 𝑨 ∈ ℤ If-Then-Else: Preprocessing ¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 0 ∧ 3 ⋅ 𝑦 < 9 ∨ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 1 ∧ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 2 ∧

24/25

0 ≤ 𝑦 < 9

SLIDE 125

∧ 2 ≡9 3 ⋅ 𝑦 1 ≤ 1 ⋅ 𝑦 − 3 ⋅ 𝑨 0 ≤ 𝑨 ≤ 2 ∧ ∧ 0 ≥ 1 ⋅ 𝑦 − 3 ⋅ 𝑨 for 𝑦, 𝑨 ∈ ℤ If-Then-Else: Preprocessing ¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 0 ∧ 3 ⋅ 𝑦 < 9 ∨ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 1 ∧ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 2 ∧

24/25

SLIDE 126

∧ 2 ≡9 3 ⋅ 𝑦 1 ≤ 1 ⋅ 𝑦 − 3 ⋅ 𝑨 0 ≤ 𝑨 ≤ 2 ∧ ∧ 0 ≥ 1 ⋅ 𝑦 − 3 ⋅ 𝑨 for 𝑦, 𝑨 ∈ ℤ 1 ≤ 0 If-Then-Else: Preprocessing ¬ 3 ⋅ 𝑦 < 9 ∨ 𝑨 = 0 ∧ 3 ⋅ 𝑦 < 9 ∨ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 1 ∧ ¬ 3 ⋅ 𝑦 < 18 ∨ 𝑨 = 2 ∧

24/25

0 ≤ 𝑦 < 9

SLIDE 127

25/25

If-Then-Else: Preprocessing additional instances:157

rings (294 problems)

Techniques: shared monomial lifting, ite bounding, (ite reconstruction)

SLIDE 128

25/25

If-Then-Else: Preprocessing additional instances:157

rings (294 problems)

additional instances: 1422

nec_smt (2800 problems)

Techniques: shared monomial lifting, ite bounding, (ite reconstruction) Techniques: constant-ite simplification, conjunctive-ite compression

SLIDE 129

SAT and theory interaction:

weakened early pruning [Sebastiani07]
unate propagations and bound refinements [Dutertre06]
decision recommendations [Yices]

Data-structure improvements:

priority queue for pivot selection [pretty much everyone]
integer coefficients instead of rational coefficients [veriT]
backup instead of recalculation [pretty much everyone]

Theory solver extensions:

unit cube test [Bromberger16]
bounding transformation [Bromberger18]
simple rounding and bound propagation [Schrijver86]

Preprocessing:

if-then-else (reconstruction, lifting, simplification, bounding) [CVC4]
pseudo-Boolean inequalities [CVC4]
small CNF transformation [Weidenbach01]

[…] invented by our team […] invented & published by someone else […] never published but implemented