A Proof Complexity View of Pseudo-Boolean Solving Marc Vinyals Tata - - PowerPoint PPT Presentation
A Proof Complexity View of Pseudo-Boolean Solving Marc Vinyals Tata Institute of Fundamental Research Mumbai, India Joint work with Jan Elffers, Jess Girldez-Cru, Stephan Gocht, and Jakob Nordstrm Theory and Practice of Satisfiability
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
◮ Current SAT solvers use CDCL algorithm ◮ Replace heuristics by nondeterminism → CDCL proof system
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Background Theory Experiments
◮ 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)
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Background Theory Experiments
◮ 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
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Background Theory Experiments
Resolution is a weak proof system
◮ e.g. cannot count ◮ x1 + ··· + xn = n/2 needs exponentially many clauses
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Background Theory Experiments
Resolution is a weak proof system
◮ e.g. cannot count ◮ x1 + ··· + xn = n/2 needs exponentially many clauses
Pseudo-Boolean constraints more expressive
x1 + ··· + xn ≥ n/2 x1 + ··· + xn ≥ n/2
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Background Theory Experiments
Resolution is a weak proof system
◮ e.g. cannot count ◮ x1 + ··· + xn = n/2 needs exponentially many clauses
Pseudo-Boolean constraints more expressive
x1 + ··· + xn ≥ n/2 x1 + ··· + xn ≥ n/2
Build solvers with native pseudo-Boolean constraints?
◮ Can generalize CDCL, even if tricky ◮ Not as successful as SAT solvers
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Background Theory Experiments
Question
What limits pseudo-Boolean solvers?
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Background Theory Experiments
Question
What limits pseudo-Boolean solvers? Theoretical Barriers
◮ Study proof systems arising from pseudo-Boolean solvers
Implementation
◮ Evaluate solvers on theoretically easy formulas
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Background Theory Experiments
Question
What limits pseudo-Boolean solvers? Theoretical Barriers
◮ Study proof systems arising from pseudo-Boolean solvers
Implementation
◮ Evaluate solvers on theoretically easy formulas
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Background Theory Experiments
All pseudo-Boolean proofs are cutting planes proofs
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Background Theory Experiments
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
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Background Theory Experiments
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
x ≥ 0 −x ≥ −1
Addition
Division
Goal: derive 0 ≥ 1
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Background Theory Experiments
Addition
◮ Unbounded choices ◮ Need a reason to add inequalities
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Background Theory Experiments
Division
◮ Too expensive
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Background Theory Experiments
What is the bare minimum to simulate resolution?
x ∨ y ∨ z x ∨ y y ∨ z
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Background Theory Experiments
What is the bare minimum to simulate resolution?
x ∨ y ∨ z x ∨ y y ∨ z x + y + z ≥ 1 x + y ≥ 1 x + x + 2y + z ≥ 2
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Background Theory Experiments
What is the bare minimum to simulate resolution?
x ∨ y ∨ z x ∨ y y ∨ z x + y + z ≥ 1 x + y ≥ 1 ▲ + 2y + z ≥ 1
◮ Addition only if some variable cancels
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Background Theory Experiments
What is the bare minimum to simulate resolution?
x ∨ y ∨ z x ∨ y y ∨ z x + y + z ≥ 1 x + y ≥ 1 2y + z ≥ 1 y + z ≥ 1
◮ Addition only if some variable cancels ◮ Division brings coefficients down to degree
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Background Theory Experiments
Addition
◮ Unbounded choices ◮ Need a reason to add inequalities
Cancelling Addition
◮ Some variable cancels: αai + βbi = 0 ◮ aka. Generalized Resolution
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Background Theory Experiments
Division
◮ Too expensive
Saturation
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition Power of subsystems of CP?
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
A B: B simulates A (with only polynomial loss)
Cancelling addition is a particular case of addition
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
A B: B simulates A (with only polynomial loss)
All subsystems simulate resolution
◮ Trivial over CNF inputs ◮ Also holds over linear
pseudo-Boolean inputs
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
†
A B: B simulates A (with only polynomial loss) †: known only for polynomial-size coefficients
Repeated divisions simulate saturation
◮ Polynomial simulation
coefficients
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
†
A B: B simulates A (with only polynomial loss) A B: B cannot simulate A (separation) †: known only for polynomial-size coefficients
CP stronger than resolution
◮ Pigeonhole principle ◮ Subset cardinality
have proofs of size
◮ polynomial in CP ◮ exponential in resolution
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Background Theory Experiments
Theorem
On CNF inputs all subsystems as weak as resolution
◮ No subsystem is implicationally complete ◮ Solvers very sensitive to input encoding
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Background Theory Experiments
Observation [Hooker ’88]
Over CNF inputs CP with cancelling addition ≡ resolution.
▲
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Background Theory Experiments
Observation [Hooker ’88]
Over CNF inputs CP with cancelling addition ≡ resolution. Proof Sketch
◮ Start with clauses (degree 1) ◮ Add two clauses → a clause
x +
x +
▲ + 1 +
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Background Theory Experiments
Observation [Hooker ’88]
Over CNF inputs CP with cancelling addition ≡ resolution. Proof Sketch
◮ Start with clauses (degree 1) ◮ Add two clauses → a clause
x +
x +
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Background Theory Experiments
Observation [Hooker ’88]
Over CNF inputs CP with cancelling addition ≡ resolution. Proof Sketch
◮ Start with clauses (degree 1) ◮ Add two clauses → a clause
x +
x +
≡ x ∨ C x ∨ D C ∨ D
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
†
A B: B simulates A (with only polynomial loss) A B: B cannot simulate A (separation) †: known only for polynomial-size coefficients
Cancellation ≡ Resolution
◮ Over CNF inputs [Hooker ’88] ◮ Pigeonhole principle ◮ Subset cardinality
have proofs of size
◮ polynomial in CP ◮ exponential in resolution
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
†
A B: B simulates A (with only polynomial loss) A B: B cannot simulate A (separation) †: known only for polynomial-size coefficients
Cancellation ≡ Resolution
◮ Over CNF inputs [Hooker ’88] ◮ Pigeonhole principle ◮ Subset cardinality
have proofs of size
◮ polynomial in CP ◮ exponential in CP
with cancelling addition and any rounding
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Background Theory Experiments
Theorem
Over CNF inputs CP with saturation ≡ CP with cancelling addition.
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Background Theory Experiments
Theorem
Over CNF inputs CP with saturation ≡ CP with cancelling addition.
Corollary
Over CNF inputs CP with saturation ≡ resolution.
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
†
A B: B simulates A (with only polynomial loss) A B: B cannot simulate A (separation) †: known only for polynomial-size coefficients
Saturation ≡ Resolution
◮ Over CNF inputs ◮ Pigeonhole principle ◮ Subset cardinality
have proofs of size
◮ polynomial in CP ◮ exponential in CP
with cancelling addition
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
†
A B: B simulates A (with only polynomial loss) A B: B cannot simulate A (separation) †: known only for polynomial-size coefficients
Saturation ≡ Resolution
◮ Over CNF inputs ◮ Pigeonhole principle ◮ Subset cardinality
have proofs of size
◮ polynomial in CP ◮ exponential in CP
with cancelling addition
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Background Theory Experiments
Lemma
If a formula defines an empty polytope over then have polynomial size proof in CP with cancelling addition.
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
†
A B: B simulates A (with only polynomial loss) A B: B cannot simulate A (separation) †: known only for polynomial-size coefficients
Pseudo-Boolean versions of
◮ Pigeonhole principle ◮ Subset cardinality
have proof of size
◮ polynomial in all CP
subsystems
◮ exponential in resolution
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Background Theory Experiments
Resolution CP saturation cancelling addition CP division cancelling addition CP saturation general addition CP division general addition
†
A B: B simulates A (with only polynomial loss) A B: B cannot simulate A (separation) †: known only for polynomial-size coefficients
Pseudo-Boolean versions of
◮ Pigeonhole principle ◮ Subset cardinality
have proof of size
◮ polynomial in all CP
subsystems
◮ exponential in resolution
CNF version exponential ⇒ Cannot recover encoding ⇒ Subsystems are incomplete
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Background Theory Experiments
Question
What limits pseudo-Boolean solvers? Theoretical Barriers
◮ Study proof systems arising from pseudo-Boolean solvers ◮ Cancelling addition and saturation not enough
Implementation
◮ Evaluate solvers on theoretically easy formulas
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Background Theory Experiments
Question
What limits pseudo-Boolean solvers? Theoretical Barriers
◮ Study proof systems arising from pseudo-Boolean solvers ◮ Cancelling addition and saturation not enough
Implementation
◮ Evaluate solvers on theoretically easy formulas
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Background Theory Experiments
Craft combinatorial formulas easy for CP
◮ Build proofs in different subsystems ◮ Choice of parameters for different levels of hardness
1
Easy for resolution
2
Hard for resolution, infeasible LP
3
Feasible LP, easy for saturation
4
Require division?
◮ Easy for CP, even tree-like
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Background Theory Experiments
Solvers from PB evaluation 2016 with different techniques
◮ Open-WBO
◮ Translate into CNF ◮ ≃ Resolution Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 22 / 30
Background Theory Experiments
Solvers from PB evaluation 2016 with different techniques
◮ Open-WBO
◮ Translate into CNF ◮ ≃ Resolution
◮ Sat4j
◮ Linear inequalities ◮ ≃ CP saturation cancelling addition
◮ RoundingSat
◮ Linear inequalities ◮ CP division cancelling addition Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 22 / 30
Background Theory Experiments
Experimental Observation
PB solvers not good at proof search.
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Background Theory Experiments
Experimental Observation
PB solvers not good at proof search.
◮ Sometimes exponentially faster than CDCL
◮ e.g. when LP close to infeasible Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 23 / 30
Background Theory Experiments
0.01 0.1 1 10 100 1000 500 1000 1500 2000 2500 3000 3500 4000 Wall clock time (s) matrix dimension RoundingSat Open-WBO Sat4jCP ◮ No rational solutions ◮ Exponentially hard for resolution ⇒ Open-WBO times out ◮ Cutting planes solvers run fast
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Background Theory Experiments
Experimental Observation
PB solvers not good at proof search.
◮ Sometimes exponentially faster than CDCL
◮ e.g. when LP close to infeasible
◮ Often not good at “truly Boolean” reasoning
◮ in particular when division useful Marc Vinyals (TIFR) A Proof Complexity View of Pseudo-Boolean Solving 25 / 30
Background Theory Experiments
0.01 0.1 1 10 100 1000 100 200 300 400 500 600 700 800 900 Wall clock time (s) #vertices RoundingSat Open-WBO Sat4jCP ◮ Provably hard for resolution ⇒ Open-WBO times out ◮ Conjecture hard for CP with saturation ⇒ Sat4j times out ◮ RoundingSat works best ⇒ division necessary?
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Background Theory Experiments
Experimental Observation
PB solvers not good at proof search.
◮ Sometimes exponentially faster than CDCL
◮ e.g. when LP close to infeasible
◮ Often not good at “truly Boolean” reasoning
◮ in particular when division useful
◮ Sometimes even not good for infeasible LPs
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Background Theory Experiments
0.01 0.1 1 10 100 1000 50 100 150 200 Wall clock time (s) #columns RoundingSat Open-WBO Sat4jCP ◮ No rational solutions ◮ Open-WBO runs fast ◮ Cutting planes solvers fairly bad
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Background Theory Experiments
Question
What limits pseudo-Boolean solvers? Theoretical Barriers
◮ Study proof systems arising from pseudo-Boolean solvers ◮ Cancelling addition and saturation not enough
Implementation
◮ Evaluate solvers on theoretically easy formulas ◮ Need to improve on proof search
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Remarks
◮ Classified subsystems of Cutting Planes ◮ On CNF Subsystems ≡ Resolution → Sensitive to encoding ◮ Solvers not good at proof search
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Remarks
◮ Classified subsystems of Cutting Planes ◮ On CNF Subsystems ≡ Resolution → Sensitive to encoding ◮ Solvers not good at proof search
Open problems
◮ Is division needed? Separation on PB inputs? ◮ Better search heuristics
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Remarks
◮ Classified subsystems of Cutting Planes ◮ On CNF Subsystems ≡ Resolution → Sensitive to encoding ◮ Solvers not good at proof search
Open problems
◮ Is division needed? Separation on PB inputs? ◮ Better search heuristics
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