Equivalence Between Systems Stronger Than Resolution Maria Luisa - - PowerPoint PPT Presentation

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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

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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)

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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)

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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

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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

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Circular Proofs [Atserias & Lauria]

x ∨ ¬x

Axiom

x ∨ A ¬x ∨ A A

Cut

A x ∨ A ¬x ∨ A

Split Axiom Cut Cut Cut

x ∨ ¬x ¬x x

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Circular Proofs [Atserias & Lauria]

x ∨ ¬x

Axiom

x ∨ A ¬x ∨ A A

Cut

A x ∨ A ¬x ∨ A

Split Axiom F1 Cut F2 Cut F3 Cut F4

x ∨ ¬x ¬x x

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Circular Proofs [Atserias & Lauria]

x ∨ ¬x

Axiom

x ∨ A ¬x ∨ A A

Cut

A x ∨ A ¬x ∨ A

Split Axiom F1 Cut F2 Cut F3 Cut F4

x ∨ ¬x ¬x x F1 F2 F3 F2 F3 F4 F4 F4 F2 F3

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Circular Proofs [Atserias & Lauria]

x ∨ ¬x

Axiom

x ∨ A ¬x ∨ A A

Cut

A x ∨ A ¬x ∨ A

Split Axiom F1 Cut F2 Cut F3 Cut F4

x ∨ ¬x F1 ≥ F2 + F3 ¬x F3 ≥ F3 + F4 x F2 ≥ F2 + F4 F4 > 0

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MaxSAT Resolution with Extension [Larrosa & Rollon]

(x ∨ A, u) (¬x ∨ B, u) (A ∨ B, u) (x ∨ A ∨ B, u) (¬x ∨ B ∨ A, u)

MaxSAT Resolution

(A, −u) (x ∨ A, u) (¬x ∨ A, u)

Extension

Clauses have a (positive or negative) weight associated Proceed replacing premises by consequences Clauses may be combined or decomposed: (C, u) (C, v) (C, u + v)

Fold

(C, u + v) (C, u) (C, v)

Unfold

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(Weighted) Dual-Rail Encoding [Marques & Bonet et al.]

(x ∨ A, u) (¬x ∨ B, u) (A ∨ B, u) (x ∨ A ∨ B, u) (¬x ∨ B ∨ A, u)

MaxSAT Resolution

Clauses have a positive weight associated Proceed replacing premises by consequences Clauses may be combined or decomposed Dual-Rail Encoding: Change xi by pi, and ¬xi by ni Add clauses (¬pi ∨ ¬ni, ∞), (pi, ui) and (ni, ui) We can derive 1 + ∑n

i=1 ui empty clauses

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Split or Extension

Lemma The Split and Extension rules are equivalent in weighted proof (A, u)

Extension

(A, u) (A, −u) (x ∨ A, u) (¬x ∨ A, u)

Fold

(x ∨ A, u) (¬x ∨ A, u)

Unfold

(A, u) (A, −u)

Split

(A, −u) (x ∨ A, u) (¬x ∨ A, u)

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Generalizing Weighted Proofs Using R

(x ∨ A, u) (¬x ∨ B, u) (A ∨ B, u) (x ∨ A ∨ B, u) (¬x ∨ B ∨ A, u)

MaxSAT Resolution

(A, u) (x ∨ A, u) (¬x ∨ A, u)

Split replacing Extension

Clauses have a (positive or negative) weight associated Proceed replacing premises by consequences Clauses may be combined or decomposed: (C, u) (C, v) (C, u + v)

Fold

(C, u + v) (C, u) (C, v)

Unfold

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Generalizing Weighted Proofs Using R

(x ∨ A, u) (¬x ∨ B, u) (A ∨ B, u) (x ∨ A ∨ B, u) (¬x ∨ B ∨ A, u)

MaxSAT Resolution

(A, u) (x ∨ A, u) (¬x ∨ A, u)

Split replacing Extension

R

Clauses have a (positive or negative) weight associated Proceed replacing premises by consequences in rules of R with same positive weight u > 0 Clauses may be combined or decomposed: (C, u) (C, v) (C, u + v)

Fold

(C, u + v) (C, u) (C, v)

Unfold

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MaxSAT Resolution or Symmetric Cut

Lemma Weighted proofs using R = {MaxSAT Resolution, Split} are poly-equivalent to proofs using R = {Symmetric Cut, Split}.

(x ∨ A, u) (¬x ∨ B, u) Split (x ∨ A ∨ ¬b1, u) (x ∨ A ∨ b1, u) (¬x ∨ B, u) Split (x ∨ A ∨ ¬b1, u) (x ∨ A ∨ b1 ∨ ¬b2, u) (x ∨ A ∨ b1 ∨ b2, u) (¬x ∨ B, u) (s − 2) × Split (x ∨ A ∨ B, u) (x ∨ A ∨ ¬b1, u) · · · (x ∨ A ∨ b1 ∨ · · · ∨ bs−1 ∨ ¬bs, u) (¬x ∨ B, u) r × Split (x ∨ A ∨ B, u) (¬x ∨ A ∨ B, u) (x ∨ A ∨ B, u) (¬x ∨ B ∨ A, u) Symmetric Cut (A ∨ B, u) (x ∨ A ∨ B, u) (¬x ∨ B ∨ A, u)

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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 The weighted proof using R = {MaxSAT Resolution, 0-Split} polynomially simulates the Weighted Dual-Rail MaxSAT x ¬x

0-Split

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PHP3

2 p11 p21 p31 p11 ∨ p21 p11 ∨ p21 p11 ∨ p21 p11 p12 p11 ∨ p12 p12 p22 p32 p21 ∨ p22 p31 ∨ p32 p22 p32

Split Split Cut Cut Cut Cut Cut Cut Cut Maria Luisa Bonet, Jordi Levy Equivalence Between Systems Stronger Than Resolution 11 / 12

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PHP3

2 p11 p21 p31 p11 ∨ p21 p11 ∨ p21 p11 ∨ p21 p11 p12 p11 ∨ p12 p12

Split Split Cut Cut Cut

(p11 ∨ p12, 1), · · · (p11 ∨ p21, 1), · · · Unfold (p11 ∨ p12, 1), · · · (p11 ∨ p21, 1), · · · (p11, 1), (p11, −1) Cut (p21 ∨ p22, 1), · · · (p11 ∨ p21, 1), · · · (p12, 1), (p11, −1) Unfold (p21 ∨ p22, 1), · · · (p11 ∨ p21, 1), · · · (p12, 1), (p11, −1) (p12, 1), (p12, −1) Cut (p21 ∨ p22, 1), · · · (p11 ∨ p21, 1), · · · (p11, −1), (p12, −1) ( , 1)

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Conclusions and Further Work

Circular proofs and weighted proofs are equivalent and parameterizable on a set of rules R Both implement a sort of irreversible proofs where it makes sense to prove twice the same clause and where we can use clauses still not proved

Split makes Symmetric Cut complete, and allow us to

reverse its application

0-split is enough to prove PHP effjciently

Further work: Design a proof system for practical SAT solvers:

Use some sort of Split rule Use some sort of circularity or (negative) weights

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