A Proof Complexity View of Pseudo-Boolean Solving Marc Vinyals Tata Institute of Fundamental Research Mumbai, India Joint work with Jan Elffers, Jesús Giráldez-Cru, Stephan Gocht, and Jakob Nordström Theory and Practice of Satisfiability Solving Workshop August 28 2018, Casa Matemática Oaxaca, Mexico
Background Theory Experiments The Power of CDCL Solvers ◮ Current SAT solvers use CDCL algorithm ◮ Replace heuristics by nondeterminism → CDCL proof system Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 1 / 30
Background Theory Experiments The Power of CDCL Solvers ◮ Current SAT solvers use CDCL algorithm ◮ Replace heuristics by nondeterminism → CDCL proof system ◮ All CDCL proofs are resolution proofs ◮ Lower bound for resolution length ⇒ lower bound for CDCL run time *(Ignoring preprocessing) Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 1 / 30
Background Theory Experiments The Power of CDCL Solvers ◮ Current SAT solvers use CDCL algorithm ◮ Replace heuristics by nondeterminism → CDCL proof system ◮ All CDCL proofs are resolution proofs ◮ Lower bound for resolution length ⇒ lower bound for CDCL run time *(Ignoring preprocessing) And the opposite direction? Theorem [Pipatsrisawat, Darwiche ’09; Atserias, Fichte, Thurley ’09] CDCL ≡ poly Resolution ◮ CDCL can simulate any resolution proof ◮ Not true for DPLL: limited to tree-like Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 1 / 30
Background Theory Experiments More Powerful Solvers Resolution is a weak proof system ◮ e.g. cannot count ◮ x 1 + ··· + x n = n / 2 needs exponentially many clauses Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 2 / 30
Background Theory Experiments More Powerful Solvers Resolution is a weak proof system ◮ e.g. cannot count ◮ x 1 + ··· + x n = n / 2 needs exponentially many clauses Pseudo-Boolean constraints more expressive x 1 + ··· + x n ≥ n / 2 x 1 + ··· + x n ≥ n / 2 Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 2 / 30
Background Theory Experiments More Powerful Solvers Resolution is a weak proof system ◮ e.g. cannot count ◮ x 1 + ··· + x n = n / 2 needs exponentially many clauses Pseudo-Boolean constraints more expressive x 1 + ··· + x n ≥ n / 2 x 1 + ··· + x n ≥ n / 2 Build solvers with native pseudo-Boolean constraints? ◮ Can generalize CDCL, even if tricky ◮ Not as successful as SAT solvers Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 2 / 30
Background Theory Experiments What do we do Question What limits pseudo-Boolean solvers? Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 3 / 30
Background Theory Experiments What do we do Question What limits pseudo-Boolean solvers? Theoretical Barriers ◮ Study proof systems arising from pseudo-Boolean solvers Implementation ◮ Evaluate solvers on theoretically easy formulas Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 3 / 30
Background Theory Experiments What do we do Question What limits pseudo-Boolean solvers? Theoretical Barriers ◮ Study proof systems arising from pseudo-Boolean solvers Implementation ◮ Evaluate solvers on theoretically easy formulas Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 3 / 30
Background Theory Experiments Cutting Planes All pseudo-Boolean proofs are cutting planes proofs Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 4 / 30
Background Theory Experiments Cutting Planes All pseudo-Boolean proofs are cutting planes proofs Work with linear pseudo-Boolean inequalities x ∨ y → x + y ≥ 1 ≡ x + ( 1 − y ) ≥ 1 y = 1 − y degree Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 4 / 30
Background Theory Experiments Cutting Planes All pseudo-Boolean proofs are cutting planes proofs Work with linear pseudo-Boolean inequalities x ∨ y → x + y ≥ 1 ≡ x + ( 1 − y ) ≥ 1 y = 1 − y degree Rules Variable axioms Addition Division � � � a i x i ≥ a b i x i ≥ b a i x i ≥ a x ≥ 0 − x ≥ − 1 � � ( α a i + β b i ) x i ≥ α a + β b ( a i / k ) x i ≥ ⌈ a / k ⌉ Goal: derive 0 ≥ 1 Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 4 / 30
Background Theory Experiments Addition in Practice Addition � � a i x i ≥ a b i x i ≥ b � ( α a i + β b i ) x i ≥ α a + β b ◮ Unbounded choices ◮ Need a reason to add inequalities Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 5 / 30
Background Theory Experiments Division in Practice Division � a i x i ≥ a � ( a i / k ) x i ≥ ⌈ a / k ⌉ ◮ Too expensive Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 6 / 30
Background Theory Experiments Weaker Rules What is the bare minimum to simulate resolution? x ∨ y ∨ z x ∨ y y ∨ z Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 7 / 30
Background Theory Experiments Weaker Rules What is the bare minimum to simulate resolution? x ∨ y ∨ z x ∨ y x + y + z ≥ 1 x + y ≥ 1 y ∨ z x + x + 2 y + z ≥ 2 Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 7 / 30
Background Theory Experiments Weaker Rules What is the bare minimum to simulate resolution? x ∨ y ∨ z x ∨ y x + y + z ≥ 1 x + y ≥ 1 y ∨ z ▲ + 2 y + z ≥ 1 ◮ Addition only if some variable cancels Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 7 / 30
Background Theory Experiments Weaker Rules What is the bare minimum to simulate resolution? x ∨ y ∨ z x ∨ y x + y + z ≥ 1 x + y ≥ 1 y ∨ z 2 y + z ≥ 1 y + z ≥ 1 ◮ Addition only if some variable cancels ◮ Division brings coefficients down to degree Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 7 / 30
Background Theory Experiments Addition in Practice Addition � � a i x i ≥ a b i x i ≥ b � ( α a i + β b i ) x i ≥ α a + β b ◮ Unbounded choices ◮ Need a reason to add inequalities Cancelling Addition ◮ Some variable cancels: α a i + β b i = 0 ◮ aka. Generalized Resolution Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 8 / 30
Background Theory Experiments Division in Practice Division � a i x i ≥ a � ( a i / k ) x i ≥ ⌈ a / k ⌉ ◮ Too expensive Saturation � a i x i ≥ a � min ( a , a i ) x i ≥ a Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 9 / 30
Background Theory Experiments Proof Systems Power of subsystems of CP? CP saturation CP division general addition general addition CP saturation CP division cancelling addition cancelling addition Resolution Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 10 / 30
Background Theory Experiments Proof Systems Cancelling addition is a CP saturation CP division particular case of addition general addition general addition CP saturation CP division cancelling addition cancelling addition Resolution A B : B simulates A (with only polynomial loss) Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 11 / 30
Background Theory Experiments Proof Systems All subsystems simulate CP saturation CP division resolution general addition general addition ◮ Trivial over CNF inputs ◮ Also holds over linear pseudo-Boolean inputs CP saturation CP division cancelling addition cancelling addition Resolution A B : B simulates A (with only polynomial loss) Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 11 / 30
Background Theory Experiments Proof Systems Repeated divisions CP saturation CP division simulate saturation general addition general addition † ◮ Polynomial simulation only if polynomial coefficients CP saturation CP division cancelling addition cancelling addition Resolution A B : B simulates A (with only polynomial loss) † : known only for polynomial-size coefficients Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 11 / 30
Background Theory Experiments Proof Systems CP stronger than resolution CP saturation CP division general addition general addition † CP saturation CP division ◮ Pigeonhole principle cancelling addition cancelling addition ◮ Subset cardinality have proofs of size ◮ polynomial in CP Resolution ◮ exponential in resolution A B : B simulates A (with only polynomial loss) A B : B cannot simulate A (separation) † : known only for polynomial-size coefficients Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 11 / 30
Background Theory Experiments Bad News Theorem On CNF inputs all subsystems as weak as resolution ◮ No subsystem is implicationally complete ◮ Solvers very sensitive to input encoding Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 12 / 30
▲ Background Theory Experiments Cancelling Addition ≡ Resolution Observation [Hooker ’88] Over CNF inputs CP with cancelling addition ≡ resolution. Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 13 / 30
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