Resolution and the binary encoding of combinatorial principles - PowerPoint PPT Presentation

Resolution and the binary encoding of combinatorial principles Stefan Dantchev 2 Nicola Galesi 1 Barnaby Martin 2 1 Sapienza University Rome 2 Durham University Conference on Computational Complexity — July 20, 2019 Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Proof Complexity in Resolution over s -DNF A 2-DNF: (( v 1 ∧ ¬ v 2 ) ∨ ( v 2 ∧ v 3 ) ∨ ( ¬ v 1 ∧ v 3 )) Resolution (= Res ( 1 ) ) Res ( 2 ) C ∨ x ¬ x ∨ D C ∨ ( x ∧ y ) ( ¬ x ∨¬ y ) ∨ D Main Rule C ∨ D C ∨ D Refutations for CNF CNF Proof Size for UNSAT CNF: minimal number of s -DNFs to derive the empty clause � . Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Proof Complexity in Resolution over s -DNF The ∧ - introduction rule is 1 D 1 ∨ � D 2 ∨ � j ∈ J 1 l j j ∈ J 2 l j , D 1 ∨ D 2 ∨ � j ∈ J 1 ∪ J 2 l j provided that | J 1 ∪ J 2 | ≤ s . The cut (or resolution) rule is 2 D 1 ∨ � j ∈ J l j D 2 ∨ � j ∈ J ¬ l j , D 1 ∨ D 2 The two weakening rules are 3 D ∨ � j ∈ J 1 ∪ J 2 l j D and , D ∨ � j ∈ J l j D ∨ � j ∈ J 1 l j provided that | J | ≤ s . Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

We turn a Res ( s ) proof upside-down, i.e. reverse the edges of the underlying graph and negate the s -DNF on the vertices, we get a special kind of restricted branching s -program whose nodes are labelled by s -CNFs and at each node some s -disjunction is queried. 1 Querying a new s -disjunction, and branching on the answer, which can be depicted as follows. C ? � j ∈ J l j (1) ⊤ ւ ց ⊥ C ∧ � j ∈ J l j C ∧ � j ∈ J ¬ l j 2 Querying a known s -disjunction, and splitting it according to the answer: C∧ � j ∈ J 1 ∪ J 2 l j ? � j ∈ J 1 l j (2) ⊤ ւ ց ⊥ C ∧ � C ∧ � j ∈ J 1 l j j ∈ J 2 l j Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

3 There are two ways of forgetting information, C ∧ � j ∈ J 1 l j C 1 ∧ C 2 ↓ ↓ and , (3) C 1 C ∧ � j ∈ J 1 ∪ J 2 l j Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

k -clique principle G = ( V , E ) . We want to define a formula Clique k ( G ) satisfiable iff G contains a k -clique. x iv ≡ ” v is the i -th node in the clique”  � i ∈ [ k ] v ∈ V x i , v a node in each position  Clique k ( G ) = ¬ x i , v ∨ ¬ x i , u u � = v ∈ V , i ∈ [ k ] no two nodes in one position ¬ x i , u ∨ ¬ x j , v ( u , v ) �∈ E , i � = j ∈ [ k ] ”no-edges” are not in the clique  Fact Clique k ( G ) UNSAT iff G does not have a k-clique Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Binary Combinatorial Principles: What and Why k -Clique Principle: Simplified version G formed from k blocks V b of n nodes each: G = ( � b ∈ [ k ] V b , E ) Variables v i , q with i ∈ [ k ] , a ∈ [ n ] , with clauses � ¬ v i , a ∨ ¬ v j , b (( i , a ) , ( j , b )) �∈ E Clique n k ( G ) = � a ∈ [ n ] v i , a i ∈ [ k ] Fact Clique n k ( G ) UNSAT iff G does not have a k-clique Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

( 1 , 1 ) ( 3 , 1 ) ( 2 , 1 )  x 1 , 1   x 2 , 1  Clique n k ( G ) = x 3 , 1   ( ¬ x 1 , 1 ∨ ¬ x 3 , 1 )  Motivations(Informal): Clique n k captures the proof strength of adding to a proof system the ability to count up to k . [1,2] [1]=[Beyersorff Galesi Lauria Razborov 12] [2]=[Dantchev Martin Szeider 11] Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

k -Clique Principle (Binary Version): (Bit-)Variables: ω i , j , for i ∈ [ k ] , j ∈ [ log n ] Notation: � ω i , j if a j = 1 a j ω i , j = ¬ ω i , j if a j = 0 a log n v i , j ≡ ( ω a 1 i , log n ) , where ( j ) 2 = � i , 1 ∧ . . . ∧ ω a � 1 − a log n 1 − b log n � � ( ω 1 − a 1 i , log n ) ∨ ( ω 1 − b 1 Bin - Clique n k ( G ) = ∨ . . . ∨ ω ∨ . . . ∨ ω j , log n ) i , 1 j , 1 (( i , a ) , ( j , b )) �∈ E Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

preserve the combinatorial hardness of the unary principle; are less exposed to details of the encoding when attacked with a lower bound technique; give significative lower bounds. PHP m Bin - PHP m n : Unary encoding n : Binary encoding � n j = 1 v i , j i ∈ [ m ] i � = i ′ ∈ [ m ] � log n j = 1 ¬ ω i , j ∨ � log n j = 1 ¬ ω i ′ , j i , � = i ′ ∈ [ m ] , j ∈ [ n ] v i , j ∨ v i ′ , j Size-Width tradeoffs for Res: Size ( F ⊢ ) ≥ e Ω( ( w ( F ⊢ ) − w ( F )) 2 ) Vars ( F ) Space-Width Relations for Res: Space ( F ⊢ ) ≥ w ( F ⊢ ) − w ( F ) + 1 w ( PHP ) = n while w ( Bin - PHP ) = 2 logn Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Fact Res ( 1 ) proofs of Clique n → Res ( log n ) proofs of Bin - Clique n k ( G ) �− k ( G ) . a log n v i , a ≡ ( ω a 1 i , 1 ∧ . . . ∧ ω i , log n ) Fact Res ( 1 ) proofs of PHP m → Res ( log n ) proofs of Bin - PHP m n �− n Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Known results for k -Clique Principles in Res For any G there are O ( n k ) proofs in tree-Res (brute force) If G is the ( k − 1 ) -partite graph: Clique n k ( G ) has Reg-Res refutations of size O ( 2 k n 2 ) [1] Difficult to find G ’s without a k -clique making hard to refute Clique n k ( G ) . Known Lower Bounds: ( G ∼ G ( n , p ) , p = n − 2 ( 1 + ǫ ) k − 1 ) G ∼ G ( n , p ) tree-Res Reg-Res Res(1) Res(s) Clique n Ω( n k ) [1] Ω( n k ) [2] Open - Ω( 2 k ) [4] k ( G ) Open Bin - Clique n Ω( n k ) [3] Ω( n k ) , s = o ( log log n ) k ( G ) − − [1] = [Beyersdorff Galesi Lauria 13 ] [2] = [Atserias Bonacina de Rezende Lauria N¨ ordstrom Razborov 18] [3] = [Lauria Pudl´ ak R¨ odl Thapen 17 ] [4] = [Pang 19, ECCC] Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Results for Bin - PHP m n Theorem ǫ, δ > 0 . Any refutation of Bin - PHP m � 2 + ǫ n in Res ( s ) for s ≤ log n is of size 2 Ω( n 1 − δ ) . Theorem There are tree- Res ( 1 ) refutations of Bin - PHP m n of size 2 Θ( n ) . Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Lower Bound Proof (for Bin - Clique n k ( G ) ) Main Tools(for Binary Principles): Covering Number on s -DNFs [1] 1 Res ( s ) proofs with small CN efficiently simulated in Res ( s − 1 ) Bottlenecks (Random) restrictions for binary principles 2 Hardness properties of Bin - Clique n k ( G ) , when G ∼ G ( n , p ) [2,3,4] 3 Induction on s . 4 Base Case: known hardness on Res ( 1 ) [4]. [1]=[Segerlind Buss Impagliazzo 04] [2]=[Beyersdorff Galesi Lauria 13 ] [3]=[Atserias Bonacina de Rezende Lauria N¨ ordstrom Razborov 18] [4]=[Lauria Pudl´ ak R¨ odl Thapen 17] Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Covering number of a Res ( s ) proof A covering set for a s -DNF F is a set of literals L such that each term of F has at least a literal in L . The covering number cv ( F ) of a s -DNF F is the minimal size of a covering set for D . CN ( π ) = max F∈ π c ( F ) Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Small covering number vs simulations Lemma (Simulation Lemma) If F has a size N refutation π in Res ( s ) with CN ( π ) < d, then F has a Res ( s − 1 ) refutation of size at most 2 d + 2 N. Put π upside-down. Get a restricted branching s -program whose nodes are labelled by s -CNFs and at each node some s -disjunction � j ∈ [ s ] l j is queried. Example . . . C (4) ? � j ∈ [ s ] l j 1 ւ ց 0 C ∧ � j ∈ [ s ] l j C ∧ � j ∈ [ s ] ¬ l j Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Let cv ( C ) < d , witnessed by variable set { v 1 , . . . , v d } . Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles

Bottlenecks in Res ( s ) A c -bottleneck in a Res ( s ) proof is a s -DNF F whose cv ( F ) ≥ c . c ( s ) is the bottleneck number at Res ( s ) . Fact (Independence) If c = rs, r ≥ 1 and cv ( F ) ≥ c, then in F it is always possible to find r pairwise disjoint s-tuples of literals r ) such that the � T i ’s are T 1 = ( ℓ 1 1 , . . . , ℓ s 1 ) , . . . , T r = ( ℓ 1 r , . . . , ℓ s terms of F. Stefan Dantchev, Nicola Galesi , Barnaby Martin Resolution and the binary encoding of combinatorial principles