. . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution Maria Luisa Bonet CS, UPC, Barcelona, Spain Jordi Levy IIIA, CSIC, Barcelona, Spain SAT 2020, July 7-9, Alghero, Italy Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution1 / 12
. . . . . . . . . . . . . . . . SAT Solvers Based on Proof Systems SAT solvers have 2 components: One tries to fjnd a satisfying assignment (solves SAT ) Another tries to fjnd a proof (solves TAUT ) Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution2 / 12
. . . . . . . . . . . . . . SAT Solvers Based on Proof Systems SAT solvers have 2 components: One tries to fjnd a satisfying assignment (solves SAT ) Another tries to fjnd a proof (solves TAUT ) Why base a SAT solver in a certain proof system? For most known proof systems there are hard to prove principles Most proof systems are diffjcult to automatize (able to fjnd a proof in polynomial time on the size of the smallest proof) Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution2 / 12
. SAT Solvers Based on Proof Systems . . . . . . . . . . SAT solvers have 2 components: . One tries to fjnd a satisfying assignment (solves SAT ) Another tries to fjnd a proof (solves TAUT ) Why base a SAT solver in a certain proof system? For most known proof systems there are hard to prove principles Most proof systems are diffjcult to automatize (able to fjnd a proof in polynomial time on the size of the smallest proof) Aim: Find a proof system slightly stronger that resolution Able to “effjciently” proof PHP A proof system “easy” to automatize Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution2 / 12
. . . . . . . . . . . . . . . Overview We analyze the relative strength of 3 proof systems All of them effjciently prove PHP and simulate Resolution Circular Proofs MaxSAT Resolution with Extension Weighted Dual-Rail Encoding Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution3 / 12
. . . . . . . . . . . . Circular Proofs [Atserias & Lauria] . Axiom A Cut A Split Axiom Cut Cut Cut x Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution4 / 12 . . . . . . . . . . . . x ∨ A ¬ x ∨ A x ∨ ¬ x x ∨ A ¬ x ∨ A x ∨ ¬ x ¬ x
. A . . . . . . . . . Circular Proofs [Atserias & Lauria] Axiom Cut . A Split Axiom F 1 Cut F 2 Cut F 3 Cut F 4 x Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution4 / 12 . . . . . . . . . . . x ∨ A ¬ x ∨ A x ∨ ¬ x x ∨ A ¬ x ∨ A x ∨ ¬ x ¬ x
. Cut . . . . Circular Proofs [Atserias & Lauria] Axiom A Cut A Split Axiom F 1 Cut F 2 F 3 . Cut F 4 x F 1 F 2 F 3 F 2 F 3 F 4 F 4 F 4 F 2 F 3 Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution4 / 12 x ∨ A ¬ x ∨ A x ∨ ¬ x x ∨ A ¬ x ∨ A x ∨ ¬ x ¬ x
. A . . . . . . . . . Circular Proofs [Atserias & Lauria] Axiom Cut . A Split Axiom F 1 Cut F 2 Cut F 3 Cut F 4 x Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution4 / 12 . . . . . . . . . . . . x ∨ A ¬ x ∨ A x ∨ ¬ x x ∨ A ¬ x ∨ A F 2 ≥ F 2 + F 4 F 1 ≥ F 2 + F 3 F 4 > 0 x ∨ ¬ x F 3 ≥ F 3 + F 4 ¬ x
. . . . . . . . . . . . . . MaxSAT Resolution with Extension [Larrosa & Rollon] MaxSAT Resolution Extension Clauses have a (positive or negative) weight associated Proceed replacing premises by consequences Clauses may be combined or decomposed: Fold Unfold Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution5 / 12 . . . . . . . . . . . . ( x ∨ A , u ) ( ¬ x ∨ B , u ) ( A ∨ B , u ) ( x ∨ A ∨ B , u ) ( ¬ x ∨ B ∨ A , u ) ( A , − u ) ( x ∨ A , u ) ( ¬ x ∨ A , u ) ( C , u ) ( C , v ) ( C , u + v ) ( C , u + v ) ( C , u ) ( C , v )
. . . . . . . . . . . . . . . (Weighted) Dual-Rail Encoding [Marques & Bonet et al.] MaxSAT Resolution Clauses have a positive weight associated Proceed replacing premises by consequences Clauses may be combined or decomposed Dual-Rail Encoding: Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution6 / 12 ( x ∨ A , u ) ( ¬ x ∨ B , u ) ( A ∨ B , u ) ( x ∨ A ∨ B , u ) ( ¬ x ∨ B ∨ A , u ) Change x i by p i , and ¬ x i by n i Add clauses ( ¬ p i ∨ ¬ n i , ∞ ) , ( p i , u i ) and ( n i , u i ) We can derive 1 + ∑ n i = 1 u i empty clauses
. . . . . . . . . . . . . . . Split or Extension Lemma The Split and Extension rules are equivalent in weighted proof Extension Fold Unfold Split Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution7 / 12 . . . . . ( A , u ) ( A , u ) ( A , − u ) ( x ∨ A , u ) ( ¬ x ∨ A , u ) ( x ∨ A , u ) ( ¬ x ∨ A , u ) ( A , u ) ( A , − u ) ( A , − u ) ( x ∨ A , u ) ( ¬ x ∨ A , u )
. . . . . . . . . . . . . . . MaxSAT Resolution Split replacing Extension Clauses have a (positive or negative) weight associated Proceed replacing premises by consequences Clauses may be combined or decomposed: Fold Unfold Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution8 / 12 . . . . . . . . . . . Generalizing Weighted Proofs Using R ( x ∨ A , u ) ( ¬ x ∨ B , u ) ( A ∨ B , u ) ( x ∨ A ∨ B , u ) ( ¬ x ∨ B ∨ A , u ) ( A , u ) ( x ∨ A , u ) ( ¬ x ∨ A , u ) ( C , u ) ( C , v ) ( C , u + v ) ( C , u + v ) ( C , u ) ( C , v )
. . . . . . . . . . . . . . . MaxSAT Resolution Split replacing Extension Clauses have a (positive or negative) weight associated Clauses may be combined or decomposed: Fold Unfold Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution8 / 12 . . . . . . . . . . . Generalizing Weighted Proofs Using R ( x ∨ A , u ) ( ¬ x ∨ B , u ) ( A ∨ B , u ) ( x ∨ A ∨ B , u ) R ( ¬ x ∨ B ∨ A , u ) ( A , u ) ( x ∨ A , u ) ( ¬ x ∨ A , u ) Proceed replacing premises by consequences in rules of R with same positive weight u > 0 ( C , u ) ( C , v ) ( C , u + v ) ( C , u + v ) ( C , u ) ( C , v )
. . . . . . . . . . . . . . . . MaxSAT Resolution or Symmetric Cut Lemma Split Split Symmetric Cut Maria Luisa Bonet, Jordi Levy . . . . . . . . . . . . . Equivalence Between Systems Stronger Than Resolution9 / 12 . . . . . . . . . . . Weighted proofs using R = { MaxSAT Resolution, Split } are poly-equivalent to proofs using R = { Symmetric Cut, Split } . ( x ∨ A , u ) ( ¬ x ∨ B , u ) ( x ∨ A ∨ ¬ b 1 , u ) ( x ∨ A ∨ b 1 , u ) ( ¬ x ∨ B , u ) ( x ∨ A ∨ ¬ b 1 , u ) ( x ∨ A ∨ b 1 ∨ ¬ b 2 , u ) ( x ∨ A ∨ b 1 ∨ b 2 , u ) ( ¬ x ∨ B , u ) ( s − 2 ) × Split ( x ∨ A ∨ B , u ) ( x ∨ A ∨ ¬ b 1 , u ) · · · ( x ∨ A ∨ b 1 ∨ · · · ∨ b s − 1 ∨ ¬ b s , u ) ( ¬ x ∨ B , u ) r × Split ( x ∨ A ∨ B , u ) ( ¬ x ∨ A ∨ B , u ) ( x ∨ A ∨ B , u ) ( ¬ x ∨ B ∨ A , u ) ( A ∨ B , u ) ( x ∨ A ∨ B , u ) ( ¬ x ∨ B ∨ A , u )
. Our Results . . . . . . . . . . Lemma . Weighted proofs and Circular proofs using the same set of inference rules are polynomially equivalent Theorem MaxSAT Resolution with Extension and Circular Resolution are polynomially equivalent Theorem polynomially simulates the Weighted Dual-Rail MaxSAT x 0-Split Maria Luisa Bonet, Jordi Levy Equivalence Between Systems Stronger Than Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 / 12 The weighted proof using R = { MaxSAT Resolution , 0-Split } ¬ x
. p 22 . . . . . PHP 3 2 p 11 p 21 p 31 p 11 p 12 p 12 p 32 . p 22 p 32 Split Split Cut Cut Cut Cut Cut Cut Cut Maria Luisa Bonet, Jordi Levy Equivalence Between Systems Stronger Than Resolution . . . . . . . . . . . . . . . . . . . . 11 / 12 . . . . . . . . . . . . . p 11 ∨ p 12 p 21 ∨ p 22 p 31 ∨ p 32 p 11 ∨ p 21 p 11 ∨ p 21 p 11 ∨ p 21
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