Lower bounds for width-restricted with Lemmas The Pigeonhole - PowerPoint PPT Presentation

Width-restricted clause learning Jan Johannsen Resolution Trees Lower bounds for width-restricted with Lemmas The Pigeonhole clause learning Principle The Ordering Principle Small Width Jan Johannsen Formulas Institut f¨ ur Informatik LMU M¨ unchen Banff, 04. 10. 2011 partially based on joint work with Sam Buss, Jan Hoffmann & Eli Ben-Sasson

Width-restricted Outline clause learning Jan Johannsen Resolution Trees with Lemmas The Pigeonhole Principle Resolution Trees with Lemmas The Ordering Principle Small Width Lower Bound for the Pigeonhole Principle Formulas Lower Bound for the Ordering Principle Lower Bound for Small Width Formulas

Width-restricted Resolution clause learning Jan Johannsen Resolution Trees Clause: disjunction a 1 ∨ . . . ∨ a k of literals a i = x or a i = ¯ x . with Lemmas The Pigeonhole Principle The width of C is w ( C ) := k . The Ordering Principle Small Width Formulas Formula (in CNF): conjunction C 1 ∧ . . . ∧ C m of clauses. Resolution rule If C , D are clauses with x ∈ C and ¯ x ∈ D , then Res x ( C , D ) := ( C \ x ) ∨ ( D \ ¯ x )

Width-restricted Resolution proofs clause learning Jan Johannsen Definition Resolution Trees with Lemmas A Resolution derivation R of clause C from formula F The Pigeonhole is a dag labelled with clauses s.t. Principle ◮ there is exactly one sink labelled C The Ordering Principle ◮ If v has 2 predecessors u and u ′ , then Small Width Formulas C v = Res x ( C u , C u ′ ) for some variable x ◮ if v is a source, then C v ∈ F The width of R is the maximal width of a clause in R If the dag is a tree, we call R tree-like A Resolution refutation of F is a derivation of the empty clause ✷ from F .

Width-restricted DLL and Tree Resolution clause learning Jan Johannsen Resolution Trees with Lemmas Algorithm DLL (Davis, Logemann, Loveland 1962) The Pigeonhole Principle DLL ( F , α ) The Ordering Principle test if α | = F or ✷ ∈ F α Small Width Formulas pick variable x in F α recursively solve DLL ( F , α [ x := 0]) and DLL ( F , α [ x := 1]) Theorem If unsatisfiable formula F is refuted by DLL in s steps, then F has a tree-like resolution refutation R of size s.

Width-restricted Clause Learning clause learning Jan Johannsen Resolution Trees with Lemmas In the case ✷ ∈ F α : (conflict) The Pigeonhole α ′ ⊆ α ◮ find Principle implying conflict (conflict analysis) The Ordering � Principle ◮ add clause a to F (learning) Small Width α ′ ( a )=0 Formulas Learning too many clauses memory explosion ❀ ❀ Heuristic to decide which clauses to learn. We show: Learning only short clauses does not help!

Width-restricted Resolution Trees with Lemmas clause learning Jan Johannsen Resolution Trees A Resolution tree with lemmas ( RTL ) for formula F with Lemmas is an ordered binary tree labelled with clauses s.t. The Pigeonhole Principle ◮ C root = ✷ The Ordering Principle ◮ if v has 2 children u and u ′ , then Small Width Formulas C v = Res x ( C u , C u ′ ) for some variable x ◮ if v has 1 child u , then C v ⊇ C u ◮ if v is a leaf, then C v ∈ F or C v = C u for some u ≺ v (lemma) ≺ is the post-order on trees.

Width-restricted Clause learning and RTL clause learning Jan Johannsen Resolution Trees Theorem (Buss, Hoffmann, JJ) with Lemmas The Pigeonhole If unsatisfiable formula F is refuted by DLL+CL in s steps, Principle then F has an RTL-refutation R of size s · n O (1) . The Ordering Principle Small Width Moreover, the lemmas used in R are among the clauses Formulas learned by the algorithm. In fact, the paper defines a subsystem WRTI < RTL for which also the converse holds. Here: lower bounds for RTL ( k ): A refutation R in RTL is in RTL ( k ), if every lemma C used in R is of width w ( C ) ≤ k .

Width-restricted The Pigeonhole Principle clause learning Jan Johannsen Resolution Trees with Lemmas The Pigeonhole . . . says: There is no injective map [ n + 1] → [ n ] Principle The Ordering Principle Small Width Formulas The formula PHP n : ◮ variables x i , j for i ≤ n + 1 and j ≤ n ◮ pigeon clauses x i , 1 ∨ . . . ∨ x i , n for every i for i < i ′ ◮ hole clauses ¯ x i , j ∨ ¯ x i ′ , j

Width-restricted Complexity of the Pigeonhole Principle clause learning Jan Johannsen Resolution Trees with Lemmas Theorem (Haken 1985) The Pigeonhole Principle Resolution proofs of PHP n require size 2 Ω( n ) . The Ordering Principle Small Width Formulas Theorem (Buss, Pitassi 1997) There are regular resolution proofs of PHP n of size n 3 2 n . Theorem (Iwama, Miyazaki 1999) Tree-like resolution proofs of PHP n require size 2 Ω( n log n ) .

Width-restricted The lower bound clause learning Jan Johannsen Resolution Trees with Lemmas The Pigeonhole Goal: solving PHP n takes long when learning Principle only short clauses. The Ordering Principle Small Width Formulas To this end: lower bound for RTL ( k )-refutations of PHP n : Theorem Every RTL ( n / 2) -refutation of PHP n is of size 2 Ω( n log n ) .

Width-restricted Matching restrictions clause learning Jan Johannsen A restriction ρ is a partial truth assignment. Resolution Trees with Lemmas F ⌈ ρ for ρ applied to F . Notation: The Pigeonhole Principle The Ordering Property: Let R be a derivation of C from F . Principle There is a derivation R ′ of C ⌈ ρ from F ⌈ ρ of size | R ′ | ≤ | R | . Small Width Formulas We denote R ′ by R ⌈ ρ . defined by { ( i 1 , j 1 ) , . . . , ( i k , j k ) } : Matching restriction:  if ( i , j ) ∈ ρ 1   if ( i , j ′ ) ∈ ρ or ( i ′ , j ) ∈ ρ ρ ( x i , j ) = 0  undefined otherwise.  PHP n ⌈ ρ ≡ PHP n −| ρ | . Property:

Width-restricted Proof of the lower bound clause learning Jan Johannsen ◮ Let R be a refutation of PHP n Resolution Trees with Lemmas The Pigeonhole Principle ◮ Find first C with w ( C ) ≤ k The Ordering Principle ◮ Subtree R C is tree-like Small Width derivation of C Formulas ◮ Pick ρ with C ⌈ ρ = 0 ◮ R C ⌈ ρ is refutation of PHP n ⌈ ρ ◮ ρ matching restriction → PHP n ⌈ ρ = PHP n −| ρ | ◮ lower bound by Iwama/Miyazaki Main Lemma: For C in R with w ( C ) ≤ k , there is a matching restriction ρ with C ⌈ ρ = 0 and | ρ | ≤ k

Width-restricted The Ordering Principle clause learning Jan Johannsen Resolution Trees with Lemmas . . . says: An ordering of [ n ] has a maximum The Pigeonhole Principle The Ordering The formula Ord n : Principle Small Width ◮ variables x i , j for i , j ≤ n and i � = j Formulas ◮ totality clauses x i , j ∨ x j , i for all i , j ◮ asymmetry clauses x i , j ∨ ¯ ¯ for all i , j x j , i ◮ transitivity clauses ¯ x i , j ∨ ¯ x j , k ∨ ¯ x k , i for all i , j , k ◮ maximum clauses � j � = i x i , j for all i

Width-restricted Complexity of the Ordering Principle clause learning Jan Johannsen Resolution Trees with Lemmas The Pigeonhole Principle Theorem (St˚ almarck 1997) The Ordering Principle There are regular resolution proofs of Ord n of size O ( n 3 ) . Small Width Formulas Theorem (Bonet, Galesi 1999) Tree-like resolution proofs of Ord n require size 2 Ω( n ) .

Width-restricted Ordering restrictions clause learning Jan Johannsen Resolution Trees with Lemmas Ordering restriction: defined by S ⊆ [ n ] The Pigeonhole and an ordering ≺ on S . Principle The Ordering  1 if i , j ∈ S and i ≺ j Principle   Small Width  0 if i , j ∈ S and j ≺ i  Formulas    σ ( x i , j ) = if i ∈ S and j / ∈ S x s , j  x i , s if i / ∈ S and j ∈ S      otherwise, x i , j  where s ∈ S is fixed. Ord n ⌈ σ ≡ Ord n −| S | +1 . Property:

Width-restricted Cyclic clauses clause learning Jan Johannsen For clause C , the graph G ( C ) has edges Resolution Trees with Lemmas The Pigeonhole x i , j ∈ C for x i , j ∈ C ( i , j ) for ¯ and ( j , i ) Principle The Ordering Principle Small Width Definition: C is cyclic , if G ( C ) contains a cycle. Formulas Lemma: A cyclic clause C has a tree-like resolution derivation from Ord n of size O ( w ( C )).

Width-restricted The main lemmas clause learning Jan Johannsen Resolution Trees with Lemmas Lemma The Pigeonhole If there is an RTL ( k ) -refutation of Ord n of size s, then there Principle is another one using no cyclic lemmas of size O ( sk ) . The Ordering Principle Proof: Replace each cyclic lemma by its derivation Small Width Formulas of size O ( k ). Lemma If C is acyclic with w ( C ) ≤ k, then there is an ordering restriction σ with | σ | ≤ 2 k such that C ⌈ σ = 0 . Proof: For C acyclic G ( C ) is a dag obtain σ as a topological ordering of G ( C ). ❀

Width-restricted The lower bound clause learning Jan Johannsen Theorem Resolution Trees with Lemmas For k < n / 4 , every RTL ( k ) -refutation of Ord n The Pigeonhole is of size 2 Ω( n ) . Principle The Ordering ◮ Let R be a refutation of Ord n Principle Small Width ◮ Remove cyclic lemmas Formulas ◮ Find first C with w ( C ) ≤ k ◮ Subtree R C is tree-like derivation of C ◮ Pick σ with C ⌈ σ = 0 ◮ R C ⌈ σ is refutation of Ord n ⌈ σ ◮ Ord n ⌈ σ = Ord n −| σ | +1 ◮ lower bound by Bonet/Galesi