The Proof-Search Problem (between bdd-width resolution and - PowerPoint PPT Presentation

The Proof-Search Problem (between bdd-width resolution and bdd-degree semi-algebraic proofs) Albert Atserias Universitat Polit` ecnica de Catalunya Barcelona, Spain

Satisfiability Example : 15 variables and 40 clauses x 1 ∨ x 2 ∨ x 6 x 1 ∨ x 3 ∨ x 7 x 1 ∨ x 4 ∨ x 8 x 1 ∨ x 5 ∨ x 9 x 2 ∨ x 3 ∨ x 10 x 2 ∨ x 4 ∨ x 11 x 2 ∨ x 5 ∨ x 12 x 3 ∨ x 4 ∨ x 13 x 3 ∨ x 5 ∨ x 14 x 4 ∨ x 5 ∨ x 15 x 6 ∨ x 7 ∨ x 10 x 6 ∨ x 8 ∨ x 11 x 6 ∨ x 9 ∨ x 12 x 7 ∨ x 8 ∨ x 13 x 7 ∨ x 9 ∨ x 14 x 8 ∨ x 9 ∨ x 15 x 10 ∨ x 11 ∨ x 13 x 10 ∨ x 12 ∨ x 14 x 11 ∨ x 12 ∨ x 15 x 13 ∨ x 14 ∨ x 15 x 1 ∨ x 2 ∨ x 6 x 1 ∨ x 3 ∨ x 7 x 1 ∨ x 4 ∨ x 8 x 1 ∨ x 5 ∨ x 9 x 2 ∨ x 3 ∨ x 10 x 2 ∨ x 4 ∨ x 11 x 2 ∨ x 5 ∨ x 12 x 3 ∨ x 4 ∨ x 13 x 3 ∨ x 5 ∨ x 14 x 4 ∨ x 5 ∨ x 15 x 6 ∨ x 7 ∨ x 10 x 6 ∨ x 8 ∨ x 11 x 6 ∨ x 9 ∨ x 12 x 7 ∨ x 8 ∨ x 13 x 7 ∨ x 9 ∨ x 14 x 8 ∨ x 9 ∨ x 15 x 10 ∨ x 11 ∨ x 13 x 10 ∨ x 12 ∨ x 14 x 11 ∨ x 12 ∨ x 15 x 13 ∨ x 14 ∨ x 15

Satisfiability Example : R (3 , 3) ≤ 6 In every party of six, either three of them are mutual friends, or three of them are mutual strangers.

Part I PROPOSITIONAL PROOF COMPLEXITY

Proof systems Definition : A proof system for A ⊆ Σ ∗ is a binary relation R ⊆ Σ ∗ × Σ ∗ s.t.: • x ∈ A ⇒ ∃ y ∈ Σ ∗ (( x , y ) ∈ R ), • x �∈ A ⇒ ∀ y ∈ Σ ∗ (( x , y ) �∈ R ), and ? • ( x , y ) ∈ R decidable in time poly ( | x | + | y | ).

Proof systems Terminology : • If ( x , y ) ∈ R , then y is an R -proof that x ∈ A ,

Proof systems Terminology : • If ( x , y ) ∈ R , then y is an R -proof that x ∈ A , • For x in A , let c R ( x ) = min {| y | : y is an R -proof that x ∈ A } .

Proof systems Terminology : • If ( x , y ) ∈ R , then y is an R -proof that x ∈ A , • For x in A , let c R ( x ) = min {| y | : y is an R -proof that x ∈ A } . Definition : A proof system R for A is polynomially-bounded if c R ( x ) ≤ poly ( | x | ) , for x ∈ A .

Polynomial simulation Definition : Given proof systems R 1 and R 2 for A , R 1 ≤ p R 2 if there exist f computable in polynomial-time such that: ( x , y ) ∈ R 1 ⇒ ( x , f ( y )) ∈ R 2 .

Resolution and Frege Proof Systems Cut rule (Resolution) : A ∨ C B ∨ C . A ∨ B

Resolution and Frege Proof Systems Cut rule (Resolution) : A ∨ C B ∨ C . A ∨ B Rest of rules of inference (Frege) : A ∨ C B ∨ D A A ∨ B ∨ ( C ∧ D ) . A ∨ B A ∨ A

Resolution and Frege Proof Systems Cut rule (Resolution) : A ∨ C B ∨ C . A ∨ B Rest of rules of inference (Frege) : A ∨ C B ∨ D A A ∨ B ∨ ( C ∧ D ) . A ∨ B A ∨ A Proof that C 1 ∧ . . . ∧ C m ∈ UNSAT : ❄ C 1 , . . . , C m , F 1 , . . . , F i , . . . , F j , . . . , F k , . . . , ∅

Hierarchy of proof systems ✛ NC 1 -Frege ✲ Frege (arbitrary formulas) ✻ ✻ TC 0 -Frege ✻ ■ ❅ ❅ AC 0 -Frege ❅ ❅ ✻ . ❅ . . ❅ ❅ ❅ Σ 3 -Frege ❅ ✻ ✟ ✯ ✟✟✟✟✟✟✟ Σ 2 -Frege ✻ ✛ ✲ Σ 1 -Frege Resolution (clauses only)

Hierarchy of proof systems ✛ NC 1 -Frege ✲ Frege (arbitrary formulas) ✻ ✻ TC 0 -Frege ✻ ■ ❅ ❅ AC 0 -Frege ❅ ❅ ✻ . ❅ . . ❅ ❅ ❅ Σ 3 -Frege ❅ ✻ Cutting planes ✟ ✯ ✟✟✟✟✟✟✟ ✻ Σ 2 -Frege ✻ ✛ ✲ Σ 1 -Frege Resolution (clauses only)

Hierarchy of proof systems ✛ NC 1 -Frege ✲ Frege (arbitrary formulas) ✻ ✻ TC 0 -Frege ✻ ■ ❅ ❅ AC 0 -Frege ❅ ❅ ✻ . ❅ . . ❅ ❅ ❅ Σ 3 -Frege ❅ ✻ Cutting planes ✟ ✯ ✟✟✟✟✟✟✟ ✻ Σ 2 -Frege ✻ ✛ ✲ Σ 1 -Frege Resolution (clauses only)

Hierarchy of proof systems ✛ NC 1 -Frege ✲ Frege (arbitrary formulas) ✻ ✻ TC 0 -Frege ✻ ■ ❅ ❅ NO poly bounded. AC 0 -Frege ❄ ❅ (unconditional) ❅ ✻ . ❅ . . ❅ ❅ ❅ Σ 3 -Frege ❅ ✻ Cutting planes ✟ ✯ ✟✟✟✟✟✟✟ ✻ Σ 2 -Frege ✻ ✛ ✲ Σ 1 -Frege Resolution (clauses only)

Proof search Definition : The proof search problem for a proof system R for A is: Given x ∈ A , find some y ∈ Σ ∗ (any y ∈ Σ ∗ ) such that ( x , y ) ∈ R .

Proof search Definition : The proof search problem for a proof system R for A is: Given x ∈ A , find some y ∈ Σ ∗ (any y ∈ Σ ∗ ) such that ( x , y ) ∈ R . Definition [Bonet-Pitassi-Raz]: A proof system R for A is automatizable if the proof search problem for R is solvable in time poly ( | x | + c R ( x )).

An easier task Definition The weak proof search problem for a proof system R for A is: Given x ∈ Σ ∗ and a size parameter s ∈ N , if c P ( x ) ≤ s , say YES, if c P ( x ) = ∞ , say NO.

An easier task Definition The weak proof search problem for a proof system R for A is: Given x ∈ Σ ∗ and a size parameter s ∈ N , if c P ( x ) ≤ s , say YES, if c P ( x ) = ∞ , say NO. Definition [Razborov] [Pudlak] A proof system R for A is weakly automatizable if the weak proof search problem for R is solvable in time poly ( | x | + s ).

Some known results Theorems [Bonet-Pitassi-Raz] [Alekhnovich-Razborov] 1. Weak automatizability of Frege is crypto-hard. 2. Automatizability of Resolution is W[P]-hard.

Status of the question ✛ NC 1 -Frege ✲ Frege ✻ ✻ NO weakly autom. TC 0 -Frege ✻ (crypto-hardness) ✻ ■ ❅ ❅ AC 0 -Frege ❅ ❅ ✻ . ❅ . . ❅ ❅ ❅ Σ 3 -Frege ❅ ✻ Cutting planes ✟ ✯ ✟✟✟✟✟✟✟ ✻ Σ 2 -Frege ✻ ✛ ✲ NO autom. Σ 1 -Frege Resolution (W[P]-hardness)

Part II MEAN-PAYOFF STOCHASTIC GAMES

Mean-payoff games 0 1 −2 8 −1 −2 1 0 1 4 −2 3 0 −1 −2 2 2 0 −1 2 4 Box: player max. Diamond: player min. Circle: random (nature).

Mean-payoff stochastic games A mean-payoff stochastic game is given by: • Game graph G = ( V , E ): finite directed graph. • Partition: V = V max ∪ V min ∪ V avg . • Weights on edges: w : E → Z .

Mean-payoff stochastic games A mean-payoff stochastic game is given by: • Game graph G = ( V , E ): finite directed graph. • Partition: V = V max ∪ V min ∪ V avg . • Weights on edges: w : E → Z . Goals of players: lim t →∞ 1 � t � � max / min E i =0 w ( v i − 1 , v i ) t (simplifying issues: lim vs. lim sup or lim inf, measurability, etc.).

Four types of games Mean-payoff stochastic games [Shapley 1953]: No restrictions. Simple stochastic games [Condon]: All weights are 0 except at one +1-sink and one − 1-sink. Mean-payoff games [Ehrenfeucht-Mycielski]: There are no random nodes. Parity games [Emerson-Jutla]: There are no random nodes and all weights outgoing node i are ( − 1) i · ( | V | + 1) i .

Complexity of the games Definition The MPSG-problem is: Given a game graph, does player max have a strategy securing value ≥ 0?

Complexity of the games Definition The MPSG-problem is: Given a game graph, does player max have a strategy securing value ≥ 0? Theorem [C, EM, EJ, Zwick-Paterson] 1. PG ≤ p m MPG ≤ p m SSG ≤ p m MPSG. 2. All four versions are in NP ∩ co-NP.

Complexity of the games Definition The MPSG-problem is: Given a game graph, does player max have a strategy securing value ≥ 0? Theorem [C, EM, EJ, Zwick-Paterson] 1. PG ≤ p m MPG ≤ p m SSG ≤ p m MPSG. 2. All four versions are in NP ∩ co-NP. Open problems Membership in P is unknown. Any kind of hardness is unknown.

Back to the proof-search problem Theorem [A.-Maneva] There is a polynomial time algorithm MPG instance G �→ CNF formula F so that: 1. If max wins G , then F is satisfiable. 2. If min wins G , then F has poly-size Σ 2 -refutation.

Status of the question ✛ NC 1 -Frege ✲ Frege ✻ ✻ NO weakly autom. TC 0 -Frege ✻ (crypto-hardness) ✻ ❅ ■ ❅ AC 0 -Frege ❅ ❅ ✻ . ❅ . . ❅ NO weakly autom. ❅ ✻ (MPG-hardness) ❅ Σ 3 -Frege ✟ ✟✟✟✟✟✟✟✟ ❅ ✻ Cutting planes ✯ ✟ ✟✟✟✟✟✟✟ ✻ Σ 2 -Frege ✻ ✛ ✲ Σ 1 -Frege Resolution

Status of the question ✛ NC 1 -Frege ✲ Frege ✻ ✻ NO weakly autom. TC 0 -Frege ✻ (crypto-hardness) ✻ ■ ❅ ❅ AC 0 -Frege ❅ NO weakly autom. ❅ ✻ ✻ . ❅ (SSG-hardness) . . ✟ ❅ ✟✟✟✟✟✟✟✟ NO weakly autom. ❅ ✻ (MPG-hardness) ❅ Σ 3 -Frege ✟ ✟✟✟✟✟✟✟✟ ❅ ✻ Cutting planes ✟ ✯ ✟✟✟✟✟✟✟ ✻ Σ 2 -Frege ✻ ✛ ✲ Σ 1 -Frege Resolution

Status of the question ✛ NC 1 -Frege ✲ Frege ✻ ✻ NO weakly autom. TC 0 -Frege ✻ (crypto-hardness) ✻ ■ ❅ ❅ AC 0 -Frege ❅ NO weakly autom. ❅ ✻ ✻ . ❅ (SSG-hardness) . . ✟ ❅ ✟✟✟✟✟✟✟✟ NO weakly autom. ❅ ✻ (MPG-hardness) ❅ Σ 3 -Frege ✟ ✟✟✟✟✟✟✟✟ ❅ ✻ Cutting planes ✯ ✟ ✟✟✟✟✟✟✟ ✻ Σ 2 -Frege ✻ ✛ ✲ NO weakly autom. Σ 1 -Frege Resolution ✻ (PG-hardness)

Part III BOUNDED-WIDTH RESOLUTION

Bounded-width resolution Definition 1. The width of a clause is its number of literals. 2. The width of a refutation is the width of its widest clause.