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A (Biased) Proof Complexity Survey for SAT Practitioners Jakob Nordstr om KTH Royal Institute of Technology Stockholm, Sweden 17th International Conference on Theory and Applications of Satisfiability Testing Vienna, Austria July 1417,

  1. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D 5 . x ∨ z Axiom Refutation ends when empty clause ⊥ derived x ∨ y Res (2 , 4) 6 . Can represent refutation as Res (1 , 6) 7 . x annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  2. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D Axiom Refutation ends when empty clause ⊥ 5 . x ∨ z derived x ∨ y Res (2 , 4) 6 . Can represent refutation as x 7 . Res (1 , 6) annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  3. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D Axiom Refutation ends when empty clause ⊥ 5 . x ∨ z derived x ∨ y Res (2 , 4) 6 . Can represent refutation as x 7 . Res (1 , 6) annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  4. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D 5 . x ∨ z Axiom Refutation ends when empty clause ⊥ derived x ∨ y Res (2 , 4) 6 . Can represent refutation as x 7 . Res (1 , 6) annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  5. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D 5 . x ∨ z Axiom Refutation ends when empty clause ⊥ derived x ∨ y Res (2 , 4) 6 . Can represent refutation as Res (1 , 6) 7 . x annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  6. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D 5 . x ∨ z Axiom Refutation ends when empty clause ⊥ derived x ∨ y Res (2 , 4) 6 . Can represent refutation as Res (1 , 6) 7 . x annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  7. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D 5 . x ∨ z Axiom Refutation ends when empty clause ⊥ derived x ∨ y Res (2 , 4) 6 . Can represent refutation as x 7 . Res (1 , 6) annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  8. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D 5 . x ∨ z Axiom Refutation ends when empty clause ⊥ derived x ∨ y Res (2 , 4) 6 . Can represent refutation as x 7 . Res (1 , 6) annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  9. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs The Resolution Proof System Goal: refute unsatisfiable CNF x ∨ y 1 . Axiom Start with clauses of formula (axioms) x ∨ y ∨ z 2 . Axiom Derive new clauses by resolution rule 3 . x ∨ z Axiom C ∨ x D ∨ x y ∨ z 4 . Axiom C ∨ D 5 . x ∨ z Axiom Refutation ends when empty clause ⊥ derived x ∨ y Res (2 , 4) 6 . Can represent refutation as x 7 . Res (1 , 6) annotated list or Res (3 , 5) DAG 8 . x Res (7 , 8) Tree-like resolution if DAG is tree 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 5/45

  10. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Resolution Size/Length Size/length = # clauses in refutation Most fundamental measure in proof complexity Lower bound on CDCL running time (can extract resolution proof from execution trace) Never worse than exp( O ( N )) Matching exp(Ω( N )) lower bounds known Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 6/45

  11. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Examples of Hard Formulas w.r.t Resolution Length (1/3) Pigeonhole principle (PHP) [Hak85] “ n + 1 pigeons don’t fit into n holes” Variables p i,j = “pigeon i goes into hole j ” p i, 1 ∨ p i, 2 ∨ · · · ∨ p i,n every pigeon i gets a hole no hole j gets two pigeons i � = i ′ p i,j ∨ p i ′ ,j Can also add “functionality” and “onto” axioms no pigeon i gets two holes j � = j ′ p i,j ∨ p i,j ′ p 1 ,j ∨ p 2 ,j ∨ · · · ∨ p n +1 ,j every hole j gets a pigeon Even onto functional PHP formula is hard for resolution √ � 3 � �� But only length lower bound exp Ω N in terms of formula size Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 7/45

  12. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Examples of Hard Formulas w.r.t Resolution Length (2/3) Tseitin formulas [Urq87] “Sum of degrees of vertices in graph is even” Variables = edges (in undirected graph of bounded degree) Label every vertex 0/1 so that sum of labels odd Write CNF requiring parity of edges around vertex = label � � �� Requires length exp Ω N on well-connected so-called expanders 1 ( x ∨ y ) ∧ ( x ∨ z ) y x ∧ ( x ∨ y ) ∧ ( y ∨ z ) ∧ ( x ∨ z ) ∧ ( y ∨ z ) z 0 0 Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 8/45

  13. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Examples of Hard Formulas w.r.t Resolution Length (3/3) Random k -CNF formulas [CS88] ∆ n randomly sampled k -clauses over n variables ( ∆ � 4 . 5 sufficient to get unsatisfiable 3 -CNF almost surely) � � �� Again lower bound exp Ω N And more. . . k -colourability [BCMM05] Independent sets and vertex covers [BIS07] Zero-one designs [Spe10, VS10, MN14] Et cetera. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 9/45

  14. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Examples of Hard Formulas w.r.t Resolution Length (3/3) Random k -CNF formulas [CS88] ∆ n randomly sampled k -clauses over n variables ( ∆ � 4 . 5 sufficient to get unsatisfiable 3 -CNF almost surely) � � �� Again lower bound exp Ω N And more. . . k -colourability [BCMM05] Independent sets and vertex covers [BIS07] Zero-one designs [Spe10, VS10, MN14] Et cetera. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 9/45

  15. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Resolution Width Width = size of largest clause in refutation (always ≤ N ) Width upper bound ⇒ length upper bound Proof: at most (2 · # variables ) width distinct clauses (This simple counting argument is essentially tight [ALN14]) Width lower bound ⇒ length lower bound Much less obvious. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 10/45

  16. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Resolution Width Width = size of largest clause in refutation (always ≤ N ) Width upper bound ⇒ length upper bound Proof: at most (2 · # variables ) width distinct clauses (This simple counting argument is essentially tight [ALN14]) Width lower bound ⇒ length lower bound Much less obvious. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 10/45

  17. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Resolution Width Width = size of largest clause in refutation (always ≤ N ) Width upper bound ⇒ length upper bound Proof: at most (2 · # variables ) width distinct clauses (This simple counting argument is essentially tight [ALN14]) Width lower bound ⇒ length lower bound Much less obvious. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 10/45

  18. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Width Lower Bounds Imply Length Lower Bounds Theorem ([BW01]) width 2 � � �� length ≥ exp Ω formula size N � √ N log N � Yields superpolynomial length bounds for width ω Almost all known lower bounds on length derivable via width For tree-like resolution have length ≥ 2 width [BW01] � √ N log N � General resolution: width up to O implies no length lower bounds — possible to tighten analysis? No! Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 11/45

  19. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Width Lower Bounds Imply Length Lower Bounds Theorem ([BW01]) width 2 � � �� length ≥ exp Ω formula size N � √ N log N � Yields superpolynomial length bounds for width ω Almost all known lower bounds on length derivable via width For tree-like resolution have length ≥ 2 width [BW01] � √ N log N � General resolution: width up to O implies no length lower bounds — possible to tighten analysis? No! Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 11/45

  20. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Width Lower Bounds Imply Length Lower Bounds Theorem ([BW01]) width 2 � � �� length ≥ exp Ω formula size N � √ N log N � Yields superpolynomial length bounds for width ω Almost all known lower bounds on length derivable via width For tree-like resolution have length ≥ 2 width [BW01] � √ N log N � General resolution: width up to O implies no length lower bounds — possible to tighten analysis? No! Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 11/45

  21. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Optimality of the Length-Width Lower Bound Ordering principles [St˚ a96, BG01] “Every (partially) ordered set { e 1 , . . . , e n } has minimal element” Variables x i,j = “ e i < e j ” x i,j ∨ x j,i anti-symmetry; not both e i < e j and e j < e i x i,j ∨ x j,k ∨ x i,k transitivity; e i < e j and e j < e k implies e i < e k � 1 ≤ i ≤ n, i � = j x i,j e j is not a minimal element Can also add “total order” axioms x i,j ∨ x j,i totality; either e i < e j or e j < e i Reuftable in resolution in length O ( N ) √ � 3 � Requires resolution width Ω N ( 3 -CNF version) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 12/45

  22. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Resolution Space x ∨ y 1 . Axiom Space = max # clauses in memory when performing refutation x ∨ y ∨ z 2 . Axiom Motivated by SAT solver memory usage 3 . x ∨ z Axiom (but also intrinsically interesting for proof complexity) y ∨ z 4 . Axiom Can be measured in different ways — 5 . x ∨ z Axiom focus here on most common measure clause space x ∨ y Res (2 , 4) 6 . Space at step t : # clauses at steps ≤ t x Res (1 , 6) 7 . used at steps ≥ t Res (3 , 5) 8 . x Example: Space at step 7 . . . Res (7 , 8) 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 13/45

  23. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Resolution Space x ∨ y 1 . Axiom Space = max # clauses in memory when performing refutation x ∨ y ∨ z 2 . Axiom Motivated by SAT solver memory usage 3 . x ∨ z Axiom (but also intrinsically interesting for proof complexity) y ∨ z 4 . Axiom Can be measured in different ways — 5 . x ∨ z Axiom focus here on most common measure clause space x ∨ y Res (2 , 4) 6 . Space at step t : # clauses at steps ≤ t x x Res (1 , 6) Res (1 , 6) 7 . 7 . used at steps ≥ t Res (3 , 5) 8 . x Example: Space at step 7 . . . Res (7 , 8) 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 13/45

  24. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Resolution Space x ∨ y 1 . Axiom Space = max # clauses in memory when performing refutation x ∨ y ∨ z 2 . Axiom Motivated by SAT solver memory usage 3 . x ∨ z Axiom (but also intrinsically interesting for proof complexity) y ∨ z 4 . Axiom Can be measured in different ways — 5 . x ∨ z Axiom focus here on most common measure clause space x ∨ y Res (2 , 4) 6 . Space at step t : # clauses at steps ≤ t x x Res (1 , 6) 7 . used at steps ≥ t Res (3 , 5) 8 . x Example: Space at step 7 . . . Res (7 , 8) 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 13/45

  25. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Resolution Space x ∨ y 1 . Axiom Space = max # clauses in memory when performing refutation x ∨ y ∨ z 2 . Axiom Motivated by SAT solver memory usage 3 . x ∨ z Axiom (but also intrinsically interesting for proof complexity) y ∨ z 4 . Axiom Can be measured in different ways — 5 . x ∨ z Axiom focus here on most common measure clause space x ∨ y Res (2 , 4) 6 . Space at step t : # clauses at steps ≤ t x x Res (1 , 6) 7 . used at steps ≥ t Res (3 , 5) 8 . x Example: Space at step 7 is 5 Res (7 , 8) 9 . ⊥ Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 13/45

  26. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Bounds on Resolution Space Space always at most N + O (1) [ET01] Lower bounds for Pigeonhole principle [ABRW02, ET01] Tseitin formulas [ABRW02, ET01] Random k -CNFs [BG03] Results always matching width bounds And proofs of very similar flavour. . . What is going on? Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 14/45

  27. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Bounds on Resolution Space Space always at most N + O (1) [ET01] Lower bounds for Pigeonhole principle [ABRW02, ET01] Tseitin formulas [ABRW02, ET01] Random k -CNFs [BG03] Results always matching width bounds And proofs of very similar flavour. . . What is going on? Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 14/45

  28. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Space vs. Width Theorem ([AD08]) space ≥ width + O (1) Are space and width asymptotically always the same? No! Pebbling formulas [BN08] Can be refuted in width O (1) May require space Ω( N/ log N ) A bit more involved to describe than previous benchmarks. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 15/45

  29. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Space vs. Width Theorem ([AD08]) space ≥ width + O (1) Are space and width asymptotically always the same? No! Pebbling formulas [BN08] Can be refuted in width O (1) May require space Ω( N/ log N ) A bit more involved to describe than previous benchmarks. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 15/45

  30. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Space vs. Width Theorem ([AD08]) space ≥ width + O (1) Are space and width asymptotically always the same? No! Pebbling formulas [BN08] Can be refuted in width O (1) May require space Ω( N/ log N ) A bit more involved to describe than previous benchmarks. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 15/45

  31. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Pebbling Formulas: Vanilla Version CNF formulas encoding so-called pebble games on DAGs 1 . u z 2 . v sources are true 3 . w truth propa- y x 4 . u ∨ v ∨ x gates upwards 5 . v ∨ w ∨ y but sink is false u v w 6 . x ∨ y ∨ z 7 . z Extensive literature on pebbling space and time-space trade-offs from 1970s and 80s Have been useful in proof complexity before in various contexts Hope that pebbling properties of DAG somehow carry over to resolution refutations of pebbling formulas. Except. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 16/45

  32. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Pebbling Formulas: Vanilla Version CNF formulas encoding so-called pebble games on DAGs 1 . u z 2 . v sources are true 3 . w truth propa- y x 4 . u ∨ v ∨ x gates upwards 5 . v ∨ w ∨ y but sink is false u v w 6 . x ∨ y ∨ z 7 . z Extensive literature on pebbling space and time-space trade-offs from 1970s and 80s Have been useful in proof complexity before in various contexts Hope that pebbling properties of DAG somehow carry over to resolution refutations of pebbling formulas. Except. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 16/45

  33. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Pebbling Formulas: Vanilla Version CNF formulas encoding so-called pebble games on DAGs 1 . u z 2 . v sources are true 3 . w truth propa- y x 4 . u ∨ v ∨ x gates upwards 5 . v ∨ w ∨ y but sink is false u v w 6 . x ∨ y ∨ z 7 . z Extensive literature on pebbling space and time-space trade-offs from 1970s and 80s Have been useful in proof complexity before in various contexts Hope that pebbling properties of DAG somehow carry over to resolution refutations of pebbling formulas. Except. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 16/45

  34. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Pebbling Formulas: Vanilla Version CNF formulas encoding so-called pebble games on DAGs 1 . u z 2 . v sources are true 3 . w truth propa- y x 4 . u ∨ v ∨ x gates upwards 5 . v ∨ w ∨ y but sink is false u v w 6 . x ∨ y ∨ z 7 . z Extensive literature on pebbling space and time-space trade-offs from 1970s and 80s Have been useful in proof complexity before in various contexts Hope that pebbling properties of DAG somehow carry over to resolution refutations of pebbling formulas. Except. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 16/45

  35. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Pebbling Formulas: Vanilla Version CNF formulas encoding so-called pebble games on DAGs 1 . u z 2 . v sources are true 3 . w truth propa- y x 4 . u ∨ v ∨ x gates upwards 5 . v ∨ w ∨ y but sink is false u v w 6 . x ∨ y ∨ z 7 . z Extensive literature on pebbling space and time-space trade-offs from 1970s and 80s Have been useful in proof complexity before in various contexts Hope that pebbling properties of DAG somehow carry over to resolution refutations of pebbling formulas. Except. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 16/45

  36. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Pebbling Formulas: Vanilla Version CNF formulas encoding so-called pebble games on DAGs 1 . u z 2 . v sources are true 3 . w truth propa- y x 4 . u ∨ v ∨ x gates upwards 5 . v ∨ w ∨ y but sink is false u v w 6 . x ∨ y ∨ z 7 . z Extensive literature on pebbling space and time-space trade-offs from 1970s and 80s Have been useful in proof complexity before in various contexts Hope that pebbling properties of DAG somehow carry over to resolution refutations of pebbling formulas. Except. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 16/45

  37. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Substituted Pebbling Formulas Won’t work — solved by unit propagation, so supereasy Make formula harder by substituting x 1 ⊕ x 2 for every variable x (also works for other Boolean functions with “right” properties): x ∨ y ⇓ ¬ ( x 1 ⊕ x 2 ) ∨ ( y 1 ⊕ y 2 ) ⇓ ( x 1 ∨ x 2 ∨ y 1 ∨ y 2 ) ∧ ( x 1 ∨ x 2 ∨ y 1 ∨ y 2 ) ∧ ( x 1 ∨ x 2 ∨ y 1 ∨ y 2 ) ∧ ( x 1 ∨ x 2 ∨ y 1 ∨ y 2 ) Now CNF formula inherits pebbling graph properties! Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 17/45

  38. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Space-Width Trade-offs Given a formula easy w.r.t. these complexity measures, can refutations be optimized for two or more measures? For space vs. width, the answer is a strong no Theorem ([Ben09]) There are formulas for which exist refutations in width O (1) exist refutations in space O (1) optimization of one measure causes (essentially) worst-case behaviour for other measure Holds for vanilla version of pebbling formulas Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 18/45

  39. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Length-Space Trade-offs Theorem ([BN11, BBI12, BNT13]) There are formulas for which exist refutations in short length exist refutations in small space optimization of one measure causes dramatic blow-up for other measure Holds for Substituted pebbling formulas over the right graphs Tseitin formulas over long, narrow rectangular grids So no meaningful simultaneous optimization possible for length and space in the worst case Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 19/45

  40. Resolution Preliminaries Connections Between Resolution and CDCL Length, Width and Space Stronger Proof Systems than Resolution Resolution Trade-offs Length-Width Trade-offs? What about length versus width? [BW01] transforms short refutation to narrow one, but blows up length exponentially Is this blow-up inherent? Or just an artifact of the proof? Open Problem Are there length-width trade-offs in resolution? Or is a narrow refutation never much longer than the shortest one? Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 20/45

  41. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Recap of Complexity Measures for Resolution Recall that N = size of formula Length # clauses in refutation at most exp( N ) Width Size of largest clause in refutation at most N Space Max # clauses one needs to remember when “verifying correctness of refutation” at most N (!) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 21/45

  42. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Proof Complexity Measures and CDCL Hardness Recall log( length ) � width � space Length Lower bound on running time for CDCL CDCL polynomially simulates resolution [PD11] But short proofs may be worst-case intractable to find [AR08] Width Searching in small width known heuristic in AI community Small width ⇒ CDCL solver will run fast [AFT11] Space In practice, memory consumption important bottleneck Space complexity gives lower bound on clause database size Plus assumes solver knows exactly which clauses to keep ⇒ in reality, probably (much) more memory needed Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 22/45

  43. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Proof Complexity Measures and CDCL Hardness Recall log( length ) � width � space Length Lower bound on running time for CDCL CDCL polynomially simulates resolution [PD11] But short proofs may be worst-case intractable to find [AR08] Width Searching in small width known heuristic in AI community Small width ⇒ CDCL solver will run fast [AFT11] Space In practice, memory consumption important bottleneck Space complexity gives lower bound on clause database size Plus assumes solver knows exactly which clauses to keep ⇒ in reality, probably (much) more memory needed Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 22/45

  44. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Proof Complexity Measures and CDCL Hardness Recall log( length ) � width � space Length Lower bound on running time for CDCL CDCL polynomially simulates resolution [PD11] But short proofs may be worst-case intractable to find [AR08] Width Searching in small width known heuristic in AI community Small width ⇒ CDCL solver will run fast [AFT11] Space In practice, memory consumption important bottleneck Space complexity gives lower bound on clause database size Plus assumes solver knows exactly which clauses to keep ⇒ in reality, probably (much) more memory needed Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 22/45

  45. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Proof Complexity Measures and CDCL Hardness Recall log( length ) � width � space Length Lower bound on running time for CDCL CDCL polynomially simulates resolution [PD11] But short proofs may be worst-case intractable to find [AR08] Width Searching in small width known heuristic in AI community Small width ⇒ CDCL solver will run fast [AFT11] Space In practice, memory consumption important bottleneck Space complexity gives lower bound on clause database size Plus assumes solver knows exactly which clauses to keep ⇒ in reality, probably (much) more memory needed Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 22/45

  46. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Relations Between Theoretical and Practical Hardness? 1 Are width or even space lower bounds relevant indicators of CDCL hardness? 2 Or is it true in practice that CDCL does essentially as well as resolution w.r.t. length/running time? 3 Can CDCL even do as well as resolution w.r.t. time and space simultaneously? Not mathematically well-defined questions. . . But perhaps still possible to perform experiments and draw interesting conclusions? Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 23/45

  47. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Relations Between Theoretical and Practical Hardness? 1 Are width or even space lower bounds relevant indicators of CDCL hardness? 2 Or is it true in practice that CDCL does essentially as well as resolution w.r.t. length/running time? 3 Can CDCL even do as well as resolution w.r.t. time and space simultaneously? Not mathematically well-defined questions. . . But perhaps still possible to perform experiments and draw interesting conclusions? Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 23/45

  48. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Practical Experimental Evaluation Proposed by [ABLM08] First(?) systematic attempt in [JMNˇ Z12] Length as a proxy for hardness seems too optimistic. . . So start by looking at width vs. space Run experiments on formulas with fixed complexity w.r.t. width (and length) but varying space complexity ∗ Is running time essentially the same? Or does it increase with increasing space? Experimental results Running times sometimes correlate well with space complexity But sometimes they really don’t. . . (*) Note: such formulas nontrivial to find; only know one construction Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 24/45

  49. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Practical Experimental Evaluation Proposed by [ABLM08] First(?) systematic attempt in [JMNˇ Z12] Length as a proxy for hardness seems too optimistic. . . So start by looking at width vs. space Run experiments on formulas with fixed complexity w.r.t. width (and length) but varying space complexity ∗ Is running time essentially the same? Or does it increase with increasing space? Experimental results Running times sometimes correlate well with space complexity But sometimes they really don’t. . . (*) Note: such formulas nontrivial to find; only know one construction Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 24/45

  50. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Practical Experimental Evaluation Proposed by [ABLM08] First(?) systematic attempt in [JMNˇ Z12] Length as a proxy for hardness seems too optimistic. . . So start by looking at width vs. space Run experiments on formulas with fixed complexity w.r.t. width (and length) but varying space complexity ∗ Is running time essentially the same? Or does it increase with increasing space? Experimental results Running times sometimes correlate well with space complexity But sometimes they really don’t. . . (*) Note: such formulas nontrivial to find; only know one construction Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 24/45

  51. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Practical Experimental Evaluation Proposed by [ABLM08] First(?) systematic attempt in [JMNˇ Z12] Length as a proxy for hardness seems too optimistic. . . So start by looking at width vs. space Run experiments on formulas with fixed complexity w.r.t. width (and length) but varying space complexity ∗ Is running time essentially the same? Or does it increase with increasing space? Experimental results Running times sometimes correlate well with space complexity But sometimes they really don’t. . . (*) Note: such formulas nontrivial to find; only know one construction Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 24/45

  52. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Practical Experimental Evaluation Proposed by [ABLM08] First(?) systematic attempt in [JMNˇ Z12] Length as a proxy for hardness seems too optimistic. . . So start by looking at width vs. space Run experiments on formulas with fixed complexity w.r.t. width (and length) but varying space complexity ∗ Is running time essentially the same? Or does it increase with increasing space? Experimental results Running times sometimes correlate well with space complexity But sometimes they really don’t. . . (*) Note: such formulas nontrivial to find; only know one construction Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 24/45

  53. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Example Results for Glucose Without Preprocessing or 3 glucose no-pre 3500 3000 2500 Time (s) 2000 1500 1000 pyr1seq bintree pyrofpyr 500 pyrseqsqrt pyramid gtb 0 0 1 2 3 4 5 6 Number of variables × 10 5 Looks nice. . . “Easy” formulas solved fast; “hard” take longer time Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 25/45

  54. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Example Results for Glucose with Preprocessing or 3 glucose pre 250 200 150 Time (s) 100 pyr1seq 50 bintree pyrofpyr pyrseqsqrt pyramid gtb 0 0 5 10 15 20 25 30 Number of variables × 10 5 Preprocessing makes formulas much easier, but this still looks nice Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 26/45

  55. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Some Lingeling Results (Without Preprocessing) maj 3 lingeling no-pre 3500 3000 2500 Time (s) 2000 1500 1000 pyr1seq bintree pyrofpyr 500 pyrseqsqrt pyramid gtb 0 0 1 2 3 4 5 6 Number of variables × 10 5 But sometimes we see pretty random behaviour. . . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 27/45

  56. Resolution Complexity Measures and CDCL Hardness Connections Between Resolution and CDCL Experimental Results Stronger Proof Systems than Resolution Future Directions? Practical Conclusions? No firm conclusions — both space and width seem relevant And sometimes other structural properties more important? More generally, CDCL performance on combinatorial benchmarks sometimes surprising; e.g.: For PHP, worse behaviour with heuristics than without For ordering principles, highly dependent on specific solver Sometimes “easy” formulas harder than “hard” ones?! [MN14] Open Problems Could explanations of above phenomena help us understand CDCL better? Could controlled experiments on easily scalable theoretical benchmarks yield other interesting insights? Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 28/45

  57. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Polynomial Calculus (or Actually PCR) Introduced in [CEI96]; below modified version from [ABRW02] Clauses interpreted as polynomial equations over finite field Any field in theory; GF(2) in practice Example: x ∨ y ∨ z gets translated to xyz = 0 (Think of 0 ≡ true and 1 ≡ false ) Derivation rules Boolean axioms Negation x 2 − x = 0 x + x = 1 Linear combination p = 0 q = 0 p = 0 Multiplication αp + βq = 0 xp = 0 Goal: Derive 1 = 0 ⇔ no common root ⇔ formula unsatisfiable Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 29/45

  58. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Polynomial Calculus (or Actually PCR) Introduced in [CEI96]; below modified version from [ABRW02] Clauses interpreted as polynomial equations over finite field Any field in theory; GF(2) in practice Example: x ∨ y ∨ z gets translated to xyz = 0 (Think of 0 ≡ true and 1 ≡ false ) Derivation rules Boolean axioms Negation x 2 − x = 0 x + x = 1 Linear combination p = 0 q = 0 p = 0 Multiplication αp + βq = 0 xp = 0 Goal: Derive 1 = 0 ⇔ no common root ⇔ formula unsatisfiable Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 29/45

  59. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Size, Degree and Space Write out all polynomials as sums of monomials W.l.o.g. all polynomials multilinear (because of Boolean axioms) Size — analogue of resolution length total # monomials in refutation (counted with repetitions) Can also define length measure — might be much smaller Degree — analogue of resolution width largest degree of monomial in refutation (Monomial) space — analogue of resolution (clause) space max # monomials in memory during refutation (with repetitions) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 30/45

  60. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Size, Degree and Space Write out all polynomials as sums of monomials W.l.o.g. all polynomials multilinear (because of Boolean axioms) Size — analogue of resolution length total # monomials in refutation (counted with repetitions) Can also define length measure — might be much smaller Degree — analogue of resolution width largest degree of monomial in refutation (Monomial) space — analogue of resolution (clause) space max # monomials in memory during refutation (with repetitions) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 30/45

  61. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Polynomial Calculus Simulates Resolution Polynomial calculus can simulate resolution proofs efficiently with respect to length/size, width/degree, and space simultaneously Can mimic resolution refutation step by step Hence worst-case upper bounds for resolution carry over Example: Resolution step: x ∨ y ∨ z y ∨ z x ∨ y simulated by polynomial calculus derivation: z + z − 1 = 0 yz = 0 yz + yz − y = 0 xyz = 0 xyz + xyz − xy = 0 xyz = 0 − xyz + xy = 0 xy = 0 Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 31/45

  62. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Polynomial Calculus Simulates Resolution Polynomial calculus can simulate resolution proofs efficiently with respect to length/size, width/degree, and space simultaneously Can mimic resolution refutation step by step Hence worst-case upper bounds for resolution carry over Example: Resolution step: x ∨ y ∨ z y ∨ z x ∨ y simulated by polynomial calculus derivation: z + z − 1 = 0 yz = 0 yz + yz − y = 0 xyz = 0 xyz + xyz − xy = 0 xyz = 0 − xyz + xy = 0 xy = 0 Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 31/45

  63. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Polynomial Calculus Simulates Resolution Polynomial calculus can simulate resolution proofs efficiently with respect to length/size, width/degree, and space simultaneously Can mimic resolution refutation step by step Hence worst-case upper bounds for resolution carry over Example: Resolution step: x ∨ y ∨ z y ∨ z x ∨ y simulated by polynomial calculus derivation: z + z − 1 = 0 yz = 0 yz + yz − y = 0 xyz = 0 xyz + xyz − xy = 0 xyz = 0 − xyz + xy = 0 xy = 0 Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 31/45

  64. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Polynomial Calculus Strictly Stronger than Resolution Polynomial calculus strictly stronger w.r.t. size and degree Tseitin formulas on expanders (just do Gaussian elimination) Onto functional pigeonhole principle [Rii93] Open Problem Show that polynomial calculus is strictly stronger than resolution w.r.t. space Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 32/45

  65. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Polynomial Calculus Strictly Stronger than Resolution Polynomial calculus strictly stronger w.r.t. size and degree Tseitin formulas on expanders (just do Gaussian elimination) Onto functional pigeonhole principle [Rii93] Open Problem Show that polynomial calculus is strictly stronger than resolution w.r.t. space Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 32/45

  66. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Size vs. Degree Degree upper bound ⇒ size upper bound [CEI96] Qualitatively similar to resolution bound A bit more involved argument Again essentially tight by [ALN14] Degree lower bound ⇒ size lower bound [IPS99] Precursor of [BW01] — can do same proof to get same bound Size-degree lower bound essentially optimal [GL10] Example: again ordering principle formulas Most size lower bounds for polynomial calculus derived via degree lower bounds (but machinery less developed) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 33/45

  67. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Examples of Hard Formulas w.r.t. Size (and Degree) Pigeonhole principle formulas Follows from [AR03] Earlier work on other encodings in [Raz98, IPS99] Tseitin formulas with “wrong modulus” Can define Tseitin-like formulas counting mod p for p � = 2 Hard if p � = characteristic of field [BGIP01] Random k -CNF formulas Hard in all characteristics except 2 [BI99] Lower bound for all characteristics in [AR03] Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 34/45

  68. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Bounds on Polynomial Calculus Space Lower bound for PHP with wide clauses [ABRW02] k -CNFs much trickier — sequence of lower bounds for Obfuscated 4 -CNF versions of PHP [FLN + 12] Random 4 -CNFs [BG13] Tseitin formulas on (some) expanders [FLM + 13] Open Problems Prove tight space lower bounds for Tseitin on any expander Prove any space lower bound on random 3 -CNFs Prove any space lower bound for any 3 -CNF!? Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 35/45

  69. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Space vs. Degree Open Problem (analogue of [AD08]) Is it true that space ≥ degree + O (1) ? Partial progress: if formula requires large resolution width, then XOR-substituted version requires large space [FLM + 13] Optimal separation of space and degree in [FLM + 13] by flavour of Tseitin formulas which can be refuted in degree O (1) require space Ω( N ) but separating formulas depend on characteristic of field Open Problem Prove space lower bounds for substituted pebbling formulas (would give space-degree separation independent of characteristic) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 36/45

  70. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Space vs. Degree Open Problem (analogue of [AD08]) Is it true that space ≥ degree + O (1) ? Partial progress: if formula requires large resolution width, then XOR-substituted version requires large space [FLM + 13] Optimal separation of space and degree in [FLM + 13] by flavour of Tseitin formulas which can be refuted in degree O (1) require space Ω( N ) but separating formulas depend on characteristic of field Open Problem Prove space lower bounds for substituted pebbling formulas (would give space-degree separation independent of characteristic) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 36/45

  71. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Trade-offs for Polynomial Calculus Strong, essentially optimal space-degree trade-offs [BNT13] Same vanilla pebbling formulas as for resolution Same parameters Strong size-space trade-offs [BNT13] Same formulas as for resolution Some loss in parameters Open Problem Are there size-degree trade-offs in polynomial calculus? Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 37/45

  72. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Algebraic SAT Solvers? Quite some excitement about Gr¨ obner basis approach to SAT solving after [CEI96] Promise of performance improvement failed to deliver Meanwhile: the CDCL revolution. . . Some current SAT solvers do Gaussian elimination, but this is only very limited form of polynomial calculus Is it harder to build good algebraic SAT solvers, or is it just that too little work has been done (or both)? Some shortcut seems to be needed — full Gr¨ obner basis computation does too much work Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 38/45

  73. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Algebraic SAT Solvers? Quite some excitement about Gr¨ obner basis approach to SAT solving after [CEI96] Promise of performance improvement failed to deliver Meanwhile: the CDCL revolution. . . Some current SAT solvers do Gaussian elimination, but this is only very limited form of polynomial calculus Is it harder to build good algebraic SAT solvers, or is it just that too little work has been done (or both)? Some shortcut seems to be needed — full Gr¨ obner basis computation does too much work Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 38/45

  74. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Cutting Planes Introduced in [CCT87] Clauses interpreted as linear inequalities over the reals with integer coefficients Example: x ∨ y ∨ z gets translated to x + y + (1 − z ) ≥ 1 (Now 1 ≡ true and 0 ≡ false again) Derivation rules � a i x i ≥ A Variable axioms Multiplication 0 ≤ x ≤ 1 � ca i x i ≥ cA � a i x i ≥ A � b i x i ≥ B � ca i x i ≥ A Addition Division � ( a i + b i ) x i ≥ A + B � a i x i ≥ ⌈ A/c ⌉ Goal: Derive 0 ≥ 1 ⇔ formula unsatisfiable Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 39/45

  75. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Cutting Planes Introduced in [CCT87] Clauses interpreted as linear inequalities over the reals with integer coefficients Example: x ∨ y ∨ z gets translated to x + y + (1 − z ) ≥ 1 (Now 1 ≡ true and 0 ≡ false again) Derivation rules � a i x i ≥ A Variable axioms Multiplication 0 ≤ x ≤ 1 � ca i x i ≥ cA � a i x i ≥ A � b i x i ≥ B � ca i x i ≥ A Addition Division � ( a i + b i ) x i ≥ A + B � a i x i ≥ ⌈ A/c ⌉ Goal: Derive 0 ≥ 1 ⇔ formula unsatisfiable Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 39/45

  76. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Size, Length and Space Length = total # lines/inequalities in refutation Size = sum also size of coefficients Space = max # lines in memory during refutation No (useful) analogue of width/degree Cutting planes simulates resolution efficiently w.r.t. length/size and space simultaneously is strictly stronger w.r.t. length/size — can refute PHP efficiently [CCT87] Open Problem Show cutting planes strictly stronger than resolution w.r.t. space Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 40/45

  77. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Size, Length and Space Length = total # lines/inequalities in refutation Size = sum also size of coefficients Space = max # lines in memory during refutation No (useful) analogue of width/degree Cutting planes simulates resolution efficiently w.r.t. length/size and space simultaneously is strictly stronger w.r.t. length/size — can refute PHP efficiently [CCT87] Open Problem Show cutting planes strictly stronger than resolution w.r.t. space Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 40/45

  78. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Hard Formulas w.r.t Cutting Planes Length Clique-coclique formulas [Pud97] “A graph with a k -clique is not ( k − 1) -colourable” Lower bound via interpolation and circuit complexity Open Problems Prove length lower bounds for cutting planes for Tseitin formulas for random k -CNFs for any formula using other technique than interpolation Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 41/45

  79. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Hard Formulas w.r.t Cutting Planes Length Clique-coclique formulas [Pud97] “A graph with a k -clique is not ( k − 1) -colourable” Lower bound via interpolation and circuit complexity Open Problems Prove length lower bounds for cutting planes for Tseitin formulas for random k -CNFs for any formula using other technique than interpolation Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 41/45

  80. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Hard Formulas w.r.t Cutting Planes Space? No space lower bounds known except conditional ones: Short cutting planes refutations of Tseitin formulas on expanders require large space [GP14] (But such short refutations probably don’t exist anyway) Short cutting planes refutations of (some) pebbling formulas require large space [HN12, GP14] (and such short refutations do exist; hard to see how exponential length could help bring down space) Above results obtained via communication complexity No (true) length-space trade-off results known (Although results above can also be phrased as trade-offs) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 42/45

  81. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Hard Formulas w.r.t Cutting Planes Space? No space lower bounds known except conditional ones: Short cutting planes refutations of Tseitin formulas on expanders require large space [GP14] (But such short refutations probably don’t exist anyway) Short cutting planes refutations of (some) pebbling formulas require large space [HN12, GP14] (and such short refutations do exist; hard to see how exponential length could help bring down space) Above results obtained via communication complexity No (true) length-space trade-off results known (Although results above can also be phrased as trade-offs) Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 42/45

  82. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Geometric SAT Solvers? Some work on pseudo-Boolean solvers using (subset of) cutting planes Seems hard to make competitive with CDCL on CNFs One key problem to recover cardinality constraints But. . . If cardinality constraints can be detected, then solvers can do really well (at least on combinatorial benchmarks) E.g., PHP formulas and also zero-one design formulas become easy [BBLM14] Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 43/45

  83. Resolution Polynomial Calculus Connections Between Resolution and CDCL Cutting Planes Stronger Proof Systems than Resolution And Beyond. . . Building SAT Solvers on Extended Resolution? Resolution + introduce new variables to name subformulas Without restrictions, corresponds to extended Frege Extremely strong — pretty much no lower bounds known In order to study extended resolution, would need to: Describe heuristics/rules actually used See if possible to reason about such restricted proof system Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 44/45

  84. Summing up Overview of resolution, polynomial calculus and cutting planes (More details in conference proceedings or survey [Nor13]) Resolution fairly well understood Polynomial calculus less so Cutting planes almost not at all Could there be interesting connections between proof complexity measures and hardness of SAT? How can we build efficient SAT solvers on stronger proof systems than resolution? Thank you for your attention! Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 45/45

  85. Summing up Overview of resolution, polynomial calculus and cutting planes (More details in conference proceedings or survey [Nor13]) Resolution fairly well understood Polynomial calculus less so Cutting planes almost not at all Could there be interesting connections between proof complexity measures and hardness of SAT? How can we build efficient SAT solvers on stronger proof systems than resolution? Thank you for your attention! Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 45/45

  86. References I [ABLM08] Carlos Ans´ otegui, Mar´ ıa Luisa Bonet, Jordi Levy, and Felip Many` a. Measuring the hardness of SAT instances. In Proceedings of the 23rd National Conference on Artificial Intelligence (AAAI ’08) , pages 222–228, July 2008. [ABRW02] Michael Alekhnovich, Eli Ben-Sasson, Alexander A. Razborov, and Avi Wigderson. Space complexity in propositional calculus. SIAM Journal on Computing , 31(4):1184–1211, 2002. Preliminary version appeared in STOC ’00 . [AD08] Albert Atserias and V´ ıctor Dalmau. A combinatorial characterization of resolution width. Journal of Computer and System Sciences , 74(3):323–334, May 2008. Preliminary version appeared in CCC ’03 . [AFT11] Albert Atserias, Johannes Klaus Fichte, and Marc Thurley. Clause-learning algorithms with many restarts and bounded-width resolution. Journal of Artificial Intelligence Research , 40:353–373, January 2011. Preliminary version appeared in SAT ’09 . Jakob Nordstr¨ om (KTH) A (Biased) Proof Complexity Survey SAT ’14 46/45

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