Simplified and Improved Separations Between Regular and General - PowerPoint PPT Presentation

Simplified and Improved Separations Between Regular and General Resolution by Lifting Marc Vinyals Technion Haifa, Israel joint work with Jan Elffers, Jan Johannsen, and Jakob Nordström

Background Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 1 / 17

Regular Resolution Res ’37 Resolution. [Blake] Axioms (CNF clauses) C ∨ x D ∨ x C ∨ D Contradiction Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 2 / 17

Regular Resolution Res ’37 Resolution. [Blake] ’62 Tree-like resolution. [DPLL] Tree Res Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 2 / 17

Regular Resolution Res ’37 Resolution. [Blake] ’62 Tree-like resolution. [DPLL] ’68 Regular resolution: do not resolve a variable twice on same path. [Tseitin] ◮ Tree-like resolution is regular wlog. Q Is regular resolution as powerful as general resolution? Reg Res No resolving over x C ∨ x D ∨ x C ∨ D Tree Res Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 2 / 17

Regular Resolution Res ’37 Resolution. [Blake] ’62 Tree-like resolution. [DPLL] ’68 Regular resolution: do not resolve a variable twice on same path. [Tseitin] ◮ Tree-like resolution is regular wlog. Q Is regular resolution as powerful as general resolution? Reg Res ◮ Formulas need exponentially long regular proofs. [Tseitin,Galil] ◮ If regular ≡ general, resolution needs exponentially long proofs. Tree Res Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 2 / 17

Regular Resolution Res ’37 Resolution. [Blake] ’62 Tree-like resolution. [DPLL] ’68 Regular resolution: do not resolve a variable twice on same path. [Tseitin] ◮ Tree-like resolution is regular wlog. Q Is regular resolution as powerful as general resolution? Reg Res ◮ Formulas need exponentially long regular proofs. [Tseitin,Galil] ◮ If regular ≡ general, resolution needs exponentially long proofs. ’87 Separation regular vs general (by a constant). [Huang, Yu] ’93 Separation regular vs general (superpolynomial). [Goerdt] ’02 Separation regular vs general (exponential). [AJPU] Tree Res ’11 Best separation to date: exp ( L / log 7 L loglog L ) . [Urquhart] Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 2 / 17

CDCL and Restarts CDCL Res ’96 CDCL: DPLL + Learning [MS; MMZZM] ◮ Also: VSIDS, Restarts. Reg Res ≡ DPLL Tree Res Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 3 / 17

CDCL and Restarts ≡ CDCL Res ’96 CDCL: DPLL + Learning [MS; MMZZM] ◮ Also: VSIDS, Restarts. No ’09 CDCL as powerful as resolution. [PD; AFT] ◮ Crucially uses restarts. restarts ◮ Restarts also seem very important in practice. Q Are restarts really needed? Reg Res ≡ DPLL Tree Res Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 3 / 17

CDCL and Restarts ≡ CDCL Res ’96 CDCL: DPLL + Learning [MS; MMZZM] ◮ Also: VSIDS, Restarts. No ’09 CDCL as powerful as resolution. [PD; AFT] ≃ Pool Res ◮ Crucially uses restarts. restarts ◮ Restarts also seem very important in practice. Q Are restarts really needed? Reg Res ’05 Pool resolution ≃ CDCL w/o restarts. [van Gelder] ◮ Pool res ≥ Regular res ⇒ Formulas that separate general and regular are good candidates to separate general and pool. ≡ DPLL Tree Res Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 3 / 17

CDCL and Restarts ≡ CDCL Res ’96 CDCL: DPLL + Learning [MS; MMZZM] ◮ Also: VSIDS, Restarts. No ’09 CDCL as powerful as resolution. [PD; AFT] ≃ Pool Res ◮ Crucially uses restarts. restarts ◮ Restarts also seem very important in practice. Q Are restarts really needed? Reg Res ’05 Pool resolution ≃ CDCL w/o restarts. [van Gelder] ◮ Pool res ≥ Regular res ⇒ Formulas that separate general and regular are good candidates to separate general and pool. ’14 All such formulas easy for pool resolution. [BBJ; BK] ◮ Also: formulas not good to run experiments with. ≡ DPLL Tree Res ◮ Need new formulas! Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 3 / 17

Proving Resolution Lower Bounds Largest clause in proof Size–Width Relation Resolution F requires width W ⇒ F requires length exp ( W 2 / n ) Tree-like resolution F requires width W ⇒ F requires length exp ( W ) Regular resolution ?? Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 4 / 17

Proving Resolution Lower Bounds Largest clause in proof Size–Width Relation Resolution F requires width W ⇒ F requires length exp ( W 2 / n ) Tree-like resolution F requires width W ⇒ F requires length exp ( W ) Regular resolution ?? Lifting Resolution F requires width W ⇒ T ( F ) requires length exp ( W ) Tree-like resolution F requires depth D ⇒ T ( F ) requires length exp ( D ) Regular resolution ?? Longest path in proof DAG Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 4 / 17

Results Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 5 / 17

Main Result (Informal) Theorem F requires large depth ⇒ T ( F ) requires long regular proofs. Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 6 / 17

Main Result (Informal) Theorem F requires large depth ⇒ T ( F ) requires long regular proofs. ◮ Simplifies separation between regular and general resolution. ◮ If F has narrow proofs, then T ( F ) still has short proofs. ◮ Obtain separation from F with small width and large depth, e.g. pebbling formulas. Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 6 / 17

Main Result (Informal) Theorem F requires large depth ⇒ T ( F ) requires long regular proofs. ◮ Simplifies separation between regular and general resolution. ◮ If F has narrow proofs, then T ( F ) still has short proofs. ◮ Obtain separation from F with small width and large depth, e.g. pebbling formulas. ◮ New family of “sparse stone formulas”. ◮ Improved separation: exp ( L / log 3 L loglog 5 L ) . ◮ Can use in experiments. Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 6 / 17

Lifting Usual Lifting ◮ Replace each original variable x i with a gadget g i ( y 1 i ,..., y k i ) . ( y 1 1 ⊕ y 2 1 ) ∨ ¬ ( y 1 2 ⊕ y 2 ◮ e.g. x 1 ∨ ¬ x 2 2 ) . → Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 7 / 17

Lifting Usual Lifting ◮ Replace each original variable x i with a gadget g i ( y 1 i ,..., y k i ) . ( y 1 1 ⊕ y 2 1 ) ∨ ¬ ( y 1 2 ⊕ y 2 ◮ e.g. x 1 ∨ ¬ x 2 2 ) . → Lifting with Reusing ◮ Share variables among gadgets. Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 7 / 17

Lifting Selector variables Main variables Lifting with Indexing i ) : if s j i = 1 , then g i ( ··· ) = r j ◮ Gadget g i ( s 1 i ,..., s m i ; r 1 i ,..., r m i . (Assume exactly one s i variable is 1 ) Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 8 / 17

Lifting Selector variables Main variables Lifting with Indexing i ) : if s j i = 1 , then g i ( ··· ) = r j ◮ Gadget g i ( s 1 i ,..., s m i ; r 1 i ,..., r m i . (Assume exactly one s i variable is 1 ) Lifting with Indexing and Reusing ◮ Share all main variables among all gadgets. Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 8 / 17

Lifting Selector variables Main variables Lifting with Indexing i ) : if s j i = 1 , then g i ( ··· ) = r j ◮ Gadget g i ( s 1 i ,..., s m i ; r 1 i ,..., r m i . (Assume exactly one s i variable is 1 ) r 2 Lifting with Indexing and Reusing Original ◮ Share all main variables among all gadgets. variables s 2 7 Lifting with Sparse Indexing and Reusing ◮ Fix a bipartite graph G ([ n ] ∪ [ m ] , E ) ; variable s j i exists iff ( i , j ) ∈ E . x 7 ◮ G is n disjoint stars ⇒ usual lifting. ◮ F is pebbling formula and G is complete graph K n , m ⇒ stone formula. ◮ F is pebbling formula and G is random graph ⇒ sparse stone formula. Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 8 / 17

Main Result Theorem (Dense) If F requires depth D , then L K ( F ) requires regular length ∼ exp ( D 2 / n ) . Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 9 / 17

Main Result Theorem (Dense) If F requires depth D , then L K ( F ) requires regular length ∼ exp ( D 2 / n ) . Theorem (Sparse) If F requires depth D , then L G ( F ) requires regular length ∼ exp ( D 3 / n 2 log 2 n ) . G is a random graph of degree d = log ( n / D ) . Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 9 / 17

Proof Marc Vinyals (Technion) Separations Between Regular and General Resolution by Lifting 10 / 17