Towards the Compression of First-Order Resolution Proofs by Lowering - PowerPoint PPT Presentation

Towards the Compression of First-Order Resolution Proofs by Lowering Unit Clauses J. Gorzny 1 B. Woltzenlogel Paleo 2 1 University of Victoria 2 Vienna University of Technology 6 August 2015 Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 1 / 16

Our Goal Lifting propositional proof compression algorithms to first-order logic. This work: LowerUnits Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 2 / 16

Proof Compression Motivation The best, most efficient provers, do not generate the best, least redundant proofs. Many compression algorithms for propositional proofs; few for first-order proofs. Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 3 / 16

A Propositional Proof ¯ a,b a a, ¯ b , ¯ c a, ¯ b b ,c a, ¯ a,c c ¯ c c ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 4 / 16

LowerUnits Definition (Unit) A unit clause is a subproof with a conclusion clause (final clause) having exactly 1 literal Theorem A unit clause can always be lowered Compression is achieved by delaying resolution with unit clause subproofs. Two Traversals ↑ Collect units with more than one resolvent ↓ Delete units and reintroduce them at the bottom of the proof Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 5 / 16

Propositional Example ¯ a,b ¯ a,b a a a, ¯ a, ¯ b , ¯ b , ¯ c c a, ¯ a, ¯ b b ,c b b ,c a, ¯ a,c a, ¯ a,c c c ¯ c ¯ c c c ⊥ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 6 / 16

Propositional Example ¯ a,b ¯ a,b a a a, ¯ a, ¯ b , ¯ b , ¯ c c a, ¯ a, ¯ b b ,c b b ,c a, ¯ a,c a, ¯ a,c c c ¯ c ¯ c c c ⊥ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 6 / 16

Propositional Example ¯ a,b ¯ a,b a a a, ¯ a, ¯ b , ¯ b , ¯ c c a, ¯ a, ¯ b b ,c b b ,c a, ¯ a, ¯ a,c c a,c c ¯ c ¯ c c c a, ¯ ⊥ b Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 6 / 16

Propositional Example ¯ a,b a a, ¯ a, ¯ a, ¯ b , ¯ b , ¯ c c b ,c a, ¯ b b ,c a, ¯ b a, ¯ a,c c ¯ c c ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 6 / 16

Propositional Example ¯ a,b a a, ¯ a, ¯ a, ¯ b , ¯ b , ¯ c c b ,c a, ¯ b b ,c a, ¯ a,b b a, ¯ a,c c a ¯ c c ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 6 / 16

Propositional Example ¯ a,b a a, ¯ a, ¯ a, ¯ b , ¯ b , ¯ c b ,c c a, ¯ b b ,c a, ¯ a,b b a, ¯ a,c c ¯ a a ¯ c c ⊥ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 6 / 16

Propositional Example ¯ a,b a a, ¯ a, ¯ a, ¯ b , ¯ b , ¯ c b ,c c a, ¯ b b ,c a, ¯ a,b b a, ¯ a,c c ¯ a a ¯ c c ⊥ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 6 / 16

First-Order Change: Helpful Contractions η 2 : ⊢ p ( Y ) η 1 : p ( W ) ⊢ q ( Z ) η 3 : ⊢ q ( Z ) η 4 : p ( X ) , q ( Z ) ⊢ η 5 : p ( X ) ⊢ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 7 / 16

First-Order Change: Helpful Contractions η 2 : ⊢ p ( Y ) η 1 : p ( W ) ⊢ q ( Z ) η 3 : ⊢ q ( Z ) η 4 : p ( X ) , q ( Z ) ⊢ η 5 : p ( X ) ⊢ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 7 / 16

First-Order Change: Helpful Contractions η 2 : ⊢ p ( Y ) η 1 : p ( W ) ⊢ q ( Z ) η ′ η ′ 1 : p ( W ) ⊢ q ( Z ) 4 : p ( X ) , q ( Z ) ⊢ η 3 : ⊢ q ( Z ) η 4 : p ( X ) , q ( Z ) ⊢ η ′ 5 : p ( X ) , p ( Y ) ⊢ η 5 : p ( X ) ⊢ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 7 / 16

First-Order Change: Helpful Contractions η 2 : ⊢ p ( Y ) η 1 : p ( W ) ⊢ q ( Z ) η ′ η ′ 1 : p ( W ) ⊢ q ( Z ) 4 : p ( X ) , q ( Z ) ⊢ η 3 : ⊢ q ( Z ) η 4 : p ( X ) , q ( Z ) ⊢ η ′ 5 : p ( X ) , p ( Y ) ⊢ η 5 : p ( X ) ⊢ ⌊ η ′ 5 ⌋ : p ( U ) ⊢ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 7 / 16

First-Order Change: Helpful Contractions η ′ η ′ 1 : p ( W ) ⊢ q ( Z ) 4 : p ( X ) , q ( Z ) ⊢ η 2 : ⊢ p ( Y ) η 1 : p ( W ) ⊢ q ( Z ) η ′ η 3 : ⊢ q ( Z ) η 4 : p ( X ) , q ( Z ) ⊢ 5 : p ( X ) , p ( Y ) ⊢ ⌊ η ′ η ′ η 5 : p ( X ) ⊢ 5 ⌋ : p ( U ) ⊢ 2 : ⊢ p ( Y ) ⊥ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 7 / 16

First-Order Challenge: Pre-Deletion Check η 5 : q ( Y ) ⊢ p ( a ) η 4 : p ( X ) ⊢ η 3 : ⊢ p ( b ) , q ( Y ) η 1 : q ( Y ) ⊢ η 2 : ⊢ q ( Y ) ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 8 / 16

First-Order Challenge: Pre-Deletion Check η 5 : q ( Y ) ⊢ p ( a ) η 4 : p ( X ) ⊢ η 3 : ⊢ p ( b ) , q ( Y ) η 1 : q ( Y ) ⊢ η 2 : ⊢ q ( Y ) ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 8 / 16

First-Order Challenge: Pre-Deletion Check η 5 : q ( Y ) ⊢ p ( a ) η 4 : p ( X ) ⊢ η 3 : ⊢ p ( b ) , q ( Y ) η 1 : q ( Y ) ⊢ η 2 : ⊢ q ( Y ) ⊥ η ′ η ′ 5 : q ( Y ) ⊢ p ( a ) 3 : ⊢ p ( b ) , q ( Y ) η : ⊢ p ( a ) , p ( b ) Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 8 / 16

First-Order Challenge: Pre-Deletion Check η 5 : q ( Y ) ⊢ p ( a ) η 4 : p ( X ) ⊢ η 3 : ⊢ p ( b ) , q ( Y ) η 1 : q ( Y ) ⊢ η 2 : ⊢ q ( Y ) ⊥ η ′ η ′ 5 : q ( Y ) ⊢ p ( a ) 3 : ⊢ p ( b ) , q ( Y ) η : ⊢ p ( a ) , p ( b ) ⌊ η ⌋ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 8 / 16

First-Order Challenge: Pre-Deletion Check η 5 : q ( Y ) ⊢ p ( a ) η 4 : p ( X ) ⊢ η 3 : ⊢ p ( b ) , q ( Y ) η 1 : q ( Y ) ⊢ η 2 : ⊢ q ( Y ) ⊥ Definition (Pre-Deletion Property) η unit, l ∈ η , such that l is resolved with literals l 1 , . . . , l n in a proof ψ . η satisfies the pre-deletion unifiability property in ψ if l 1 , . . . , l n and l are unifiable. Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 8 / 16

First-Order Challenge: Post-Deletion Check η 1 : r ( Y ) , p ( X , q ( Y , b )) , p ( X , Y ) ⊢ η 2 : ⊢ p ( U , V ) η 4 : ⊢ r ( W ) η 3 : r ( V ) , p ( U , q ( V , b )) ⊢ η 5 : p ( U , q ( W , b )) ⊢ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 9 / 16

First-Order Challenge: Post-Deletion Check η 1 : r ( Y ) , p ( X , q ( Y , b )) , p ( X , Y ) ⊢ η 2 : ⊢ p ( U , V ) η 4 : ⊢ r ( W ) η 3 : r ( V ) , p ( U , q ( V , b )) ⊢ η 5 : p ( U , q ( W , b )) ⊢ ⊥ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 9 / 16

First-Order Challenge: Post-Deletion Check η 1 : r ( Y ) , p ( X , q ( Y , b )) , p ( X , Y ) ⊢ η 2 : ⊢ p ( U , V ) η 4 : ⊢ r ( W ) η 3 : r ( V ) , p ( U , q ( V , b )) ⊢ η 5 : p ( U , q ( W , b )) ⊢ ⊥ η ′ η ′ 4 : ⊢ r ( W ) 1 : r ( Y ) , p ( X , q ( Y , b )) , p ( X , Y ) ⊢ η ′ 5 : p ( X , q ( W , b )) , p ( X , W ) ⊢ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 9 / 16

First-Order Challenge: Post-Deletion Check η 1 : r ( Y ) , p ( X , q ( Y , b )) , p ( X , Y ) ⊢ η 2 : ⊢ p ( U , V ) η 4 : ⊢ r ( W ) η 3 : r ( V ) , p ( U , q ( V , b )) ⊢ η 5 : p ( U , q ( W , b )) ⊢ ⊥ η ′ η ′ 4 : ⊢ r ( W ) 1 : r ( Y ) , p ( X , q ( Y , b )) , p ( X , Y ) ⊢ η ′ 5 : p ( X , q ( W , b )) , p ( X , W ) ⊢ ⌊ η ′ 5 ⌋ Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 9 / 16

First-Order Challenge: Post-Deletion Check η 1 : r ( Y ) , p ( X , q ( Y , b )) , p ( X , Y ) ⊢ η 2 : ⊢ p ( U , V ) η 4 : ⊢ r ( W ) η 3 : r ( V ) , p ( U , q ( V , b )) ⊢ η 5 : p ( U , q ( W , b )) ⊢ ⊥ Definition (Post-Deletion Property) η unit, l ∈ η , such that l is resolved with literals l 1 , . . . , l n in a proof ψ . η satisfies the post-deletion unifiability property in ψ if l †↓ 1 , . . . , l †↓ n and l † are unifiable, where l † is the literal in ψ ′ = ψ \ { η } corresponding to l in ψ , and l †↓ is the descendant of l † in the roof of ψ ′ . Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 9 / 16

First-Order Lower Units Challenges Deletion changes literals Unit collection depends on whether contraction is possible after propagation down the proof Deletion of units require knowledge of proof after deletion, and deletion depends on what will be lowered. O ( n 2 ) solution to have full knowledge Difficult bookkeeping required for implementation Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 10 / 16

Greedy First-Order Lower Units - A Quicker Alternative Ignore post-deletion satisfaction Focus on pre-deletion satisfaction Greedy contraction Faster run-time (linear; one traversal) Easier to implement Doesn’t always compress (returns original proof sometimes) Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 11 / 16

Greedy First-Order Lower Units - A Quicker Alternative Ignore post-deletion satisfaction Focus on pre-deletion satisfaction Greedy contraction Faster run-time (linear; one traversal) Easier to implement Doesn’t always compress (returns original proof sometimes) Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 11 / 16

Greedy First-Order Lower Units - A Quicker Alternative Ignore post-deletion satisfaction Focus on pre-deletion satisfaction Greedy contraction Faster run-time (linear; one traversal) Easier to implement Doesn’t always compress (returns original proof sometimes) Gorzny, Woltzenlogel Paleo First-Order Lower Units CADE25 11 / 16