Short Proofs Are Narrow (Well, Sort of), But Are They Tight? Jakob - PowerPoint PPT Presentation

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Some Notation and Terminology ◮ Literal a : variable x or its negation x ◮ Clause C = a 1 ∨ . . . ∨ a k : set of literals At most k literals: k -clause ◮ CNF formula F = C 1 ∧ . . . ∧ C m : set of clauses k -CNF formula: CNF formula consisting of k -clauses ◮ Vars ( · ) : set of variables in clause or formula Lit ( · ) : set of literals in clause or formula ◮ F � D : semantical implication, α ( F ) true ⇒ α ( D ) true for all truth value assignments α ◮ [ n ] = { 1 , 2 , . . . , n } Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 17 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Resolution Proof System Resolution derivation π : F ⊢ A of clause A from F : Sequence of clauses π = { D 1 , . . . , D s } such that D s = A and each line D i , 1 ≤ i ≤ s , is either ◮ a clause C ∈ F (an axiom) ◮ a resolvent derived from clauses D j , D k in π (with j , k < i ) by the resolution rule B ∨ x C ∨ x B ∨ C resolving on the variable x Resolution refutation of CNF formula F : Derivation of empty clause 0 (clause with no literals) from F Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 18 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Example Resolution Refutation F = ( x ∨ z ) ∧ ( z ∨ y ) ∧ ( x ∨ y ∨ u ) ∧ ( y ∨ u ) ∧ ( u ∨ v ) ∧ ( x ∨ v ) ∧ ( u ∨ w ) ∧ ( x ∨ u ∨ w ) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) 2 . z ∨ y Axiom 10 . x ∨ y Res ( 3 , 4 ) x ∨ y ∨ u x ∨ u Res ( 5 , 6 ) 3 . Axiom 11 . 4 . y ∨ u Axiom 12 . x ∨ u Res ( 7 , 8 ) u ∨ v 5 . Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 19 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Resolution Sound and Complete Resolution is sound and implicationally complete. Sound If there is a resolution derivation π : F ⊢ A then F � A Complete If F � A then there is a resolution derivation π : F ⊢ A ′ for some A ′ ⊆ A . In particular, F is unsatisfiable ⇔ ∃ resolution refutation of F Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 20 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Completeness of Resolution: Proof by Example Decision tree: x 0 1 y u 0 1 0 1 z u v w 0 1 0 1 0 1 0 1 x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w Resulting resolution refutation: 0 x x x ∨ y x ∨ y x ∨ u x ∨ u x ∨ z y ∨ z x ∨ y ∨ u y ∨ u u ∨ v x ∨ v u ∨ w x ∨ u ∨ w Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 21 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Derivation Graph and Tree-Like Derivations Derivation graph G π of a resolution derivation π : directed acyclic graph (DAG) with ◮ vertices: clauses of the derivations ◮ edges: from B ∨ x and C ∨ x to B ∨ C for each application of the resolution rule A resolution derivation π is tree-like if G π is a tree (We can make copies of axiom clauses to make G π into a tree) Example Our example resolution proof is tree-like. (The derivation graph is on the previous slide.) Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 22 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Length  1 . x ∨ z  ◮ Length L ( F ) of CNF formula F is  z ∨ y 2 .    # clauses in it  3 . x ∨ y ∨ u     ◮ Length of derivation π : F ⊢ A is y ∨ u 4 .     5 . u ∨ v # clauses in π (with repetitions)    6 . x ∨ v   ◮ Length of deriving A from F is   7 . u ∨ w   Length 8 . x ∨ u ∨ w � � L ( F ⊢ A ) = min L ( π ) 15  x ∨ y 9 .  π : F ⊢ A    10 . x ∨ y    where minimum taken over all x ∨ u  11 .    derivations of A 12 . x ∨ u     13 . x  ◮ Length of deriving A from F in    14 . x  tree-like resolution is L T ( F ⊢ A )   15 . 0 (min of all tree-like derivations) Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 23 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Exponential Lower Bound for Proof Length Theorem (Haken 1985) � � ∞ There is a family of unsatisfiable CNF formulas F n n = 1 of size � � polynomial in n such that L ( F n ⊢ 0 ) = exp Ω( n ) . Also known: general resolution is exponentially stronger than tree-like resolution (Bonet et al. 1998, Ben-Sasson et al. 1999) Resolution widely used in practice anyway because of nice properties for proof search algorithms (but is probably not automatizable) Theoretical point of view: we want to understand resolution Gain insights and develop techniques that perhaps can be used to attack more powerful proof systems Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 24 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Weakening In proofs, sometimes convenient to add a derivation rule for weakening B B ∨ C (for arbitrary clauses B , C ). Proposition Any resolution refutation π : F ⊢ 0 using weakening can be transformed into a refutation π ′ : F ⊢ 0 without weakening in at most the same length. Proof. Easy proof by induction over the resolution refutation. Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 25 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Restriction Restriction ρ : partial truth value assignment Represented as set of literals ρ = { a 1 , . . . , a m } set to true by ρ For a clause C , the ρ -restriction of C is � if ρ ∩ Lit ( C ) � = ∅ 1 C | ρ = C \ { a | a ∈ ρ } otherwise where 1 denotes the trivially true clause For a formula F , define F | ρ = � C ∈ F C | ρ For a derivation π = { D 1 , . . . , D s } , define π | ρ = { D 1 | ρ , . . . , D s | ρ } (with all trivial clauses 1 removed) Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 26 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Example Restriction π = π | x = 1 . x ∨ z Axiom in F 1 . — 2 . z ∨ y Axiom in F 2 . z ∨ y Axiom in F | x 3 . x ∨ y ∨ u Axiom in F 3 . — 4 . y ∨ u Axiom in F 4 . y ∨ u Axiom in F | x u ∨ v u ∨ v Axiom in F | x 5 . Axiom in F 5 . 6 . x ∨ v Axiom in F 6 . v Axiom in F | x u ∨ w u ∨ w Axiom in F | x 7 . Axiom in F 7 . 8 . x ∨ u ∨ w Axiom in F 8 . u ∨ w Axiom in F | x 9 . x ∨ y Res ( 1 , 2 ) 9 . — x ∨ y 10 . Res ( 3 , 4 ) 10 . — 11 . x ∨ u Res ( 5 , 6 ) 11 . u Res ( 5 , 6 ) x ∨ u Res ( 7 , 8 ) Res ( 7 , 8 ) 12 . 12 . u 13 . x Res ( 9 , 10 ) 13 . — 14 . x Res ( 11 , 12 ) 14 . 0 Res ( 11 , 12 ) 15 . 0 Res ( 13 , 14 ) Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 27 / 63

Propositional Proof Systems and Unsatisfiable CNFs Introduction Resolution Basics Resolution Proof Length Resolution Width Two Useful Tools Restrictions Preserve Resolution Derivations Proposition If π : F ⊢ A is a resolution derivation and ρ is a restriction on Vars ( F ) , then π | ρ is a derivation of A | ρ from F | ρ , possibly using weakening. Proof. Easy proof by induction over the resolution derivation. In particular, if π : F ⊢ 0 then π | ρ can be transformed into a resolution refutation of F | ρ without weakening in at most the same length as π . Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 28 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length Width 1 . x ∨ z 2 . z ∨ y ◮ Width W ( C ) of clause C is | C | , 3 . x ∨ y ∨ u y ∨ u i.e., # literals 4 . 5 . u ∨ v ◮ Width of formula F or derivation x ∨ v 6 . π is width of the widest clause in 7 . u ∨ w the formula / derivation 8 . x ∨ u ∨ w ◮ Width of deriving A from F is 9 . x ∨ y 10 . x ∨ y � � W ( F ⊢ A ) = min W ( π ) x ∨ u 11 . π : F ⊢ A 12 . x ∨ u 13 . x (No difference between tree-like and 14 . x general resolution) 15 . 0 � � Always W ( F ⊢ 0 ) ≤ � Vars ( F ) � �� � � Width 3 Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 29 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length Width and Length A narrow resolution proof is necessarily short. � w is an upper bound on � 2 · | Vars ( F ) | For a proof in width w , the number of possible clauses. Ben-Sasson & Wigderson proved (sort of) that the converse also holds. If there is a short resolution refutation of F , then there is a resolution refutation in small width as well. Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 30 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length Technical Lemma 1 Lemma � � � � If W F | x ⊢ A ≤ w then W F ⊢ A ∨ x ≤ w + 1 (possibly by use of the weakening rule). Proof. ◮ Suppose π = { D 1 , . . . , D s } derives A from F | x in width W ( π ) ≤ w . ◮ Add the literal x to all clauses in π . ◮ Claim: this yields a legal derivation π ′ from F (possibly with weakening). ◮ If so, obviously W ( π ′ ) ≤ w + 1, and last line is A ∨ x . Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 31 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length Proof of Technical Lemma 1 (continued) Proof of claim. Need to show that each D i ∨ x ∈ π ′ can be derived from previous clauses by resolution and/or weakening. Let F x = { C ∈ F | x ∈ Lit ( C ) } be the set of all clauses of F containing the literal x . Three cases: 1. D i ∈ F x | x : This means that D i ∨ x ∈ F , which is OK. 2. D i ∈ F | x \ F x | x : This means that D i ∈ F , so D i ∨ x can be derived by weakening. 3. D i derived from D j , D k ∈ π by resolution: By induction D j ∨ x and D k ∨ x ∈ π ′ derivable; resolve to get D i ∨ x . Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 32 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length Technical Lemma 2 Lemma If � � ◮ W F | x ⊢ 0 ≤ w − 1 and � � ◮ W F | x ⊢ 0 ≤ w then � � ◮ W F ⊢ 0 ≤ max { w , W ( F ) } . Proof. ◮ Derive x in width ≤ w by Technical Lemma 1. ◮ Resolve x with all clauses C ∈ F containing literal x to get F | x in width ≤ W ( F ) . ◮ Derive 0 from F | x in width ≤ w (by assumption). Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 33 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length Warm-Up: Tree-Like Resolution Theorem (Ben-Sasson & Wigderson 1999) For tree-like resolution, the width of refuting a CNF formula F is bounded from above by W ( F ⊢ 0 ) ≤ W ( F ) + log 2 L T ( F ⊢ 0 ) . Corollary For tree-like resolution, the length of refuting a CNF formula F is bounded from below by L T ( F ⊢ 0 ) ≥ 2 ( W ( F ⊢ 0 ) − W ( F )) . Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 34 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length Proof for Tree-Like Resolution (1 / 2) Proof by nested induction over b and # variables n that L T ( F ⊢ 0 ) ≤ 2 b ⇒ W ( F ⊢ 0 ) ≤ W ( F ) + b Base cases: b = 0 ⇒ proof of length 1 ⇒ empty clause 0 ∈ F n = 1 ⇒ formula over 1 variable, i.e., x ∧ x ⇒ ∃ proof of width 1 Induction step: Suppose for formula F with n variables that π is tree-like refutation in length ≤ 2 b Last step in refutation π : F ⊢ 0 is x x for some x 0 Let π x and π x be the tree-like subderivations of x and x , respectively Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 35 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length Proof for Tree-Like Resolution (2 / 2) 0 Since L ( π ) = L ( π x ) + L ( π x ) + 1 ≤ 2 b x x (true since π is tree-like), one of π x and π x has length ≤ 2 b − 1 π x π x Suppose w.l.o.g. L ( π x ) ≤ 2 b − 1 π x | x is a refutation of F | x in length ≤ 2 b − 1 � � � � ⇒ by induction W F | x ⊢ 0 ≤ W F | x + b − 1 ≤ W ( F ) + b − 1 π x | x is a refutation in length ≤ 2 b of F | x with ≤ n − 1 variables � � � � ⇒ by induction W F | x ⊢ 0 ≤ W F | x + b ≤ W ( F ) + b � � Technical Lemma 2: W F | x ⊢ 0 ≤ W ( F ) + b − 1 and � � � � F | x ⊢ 0 ≤ W ( F ) + b ⇒ W F ⊢ 0 ≤ W ( F ) + b W (But construction leads to exponential blow-up in length, so short proofs are not narrow after all) Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 36 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length The General Case Theorem (Ben-Sasson & Wigderson 1999) The width of refuting a CNF formula F over n variables in general resolution is bounded from above by �� � W ( F ⊢ 0 ) ≤ W ( F ) + O n log L ( F ⊢ 0 ) . Note: 2 n + 1 − 1 maximal possible proof length, so bound is � W ( F ⊢ 0 ) � W ( F ) + log ( max possible ) · log L ( F ⊢ 0 ) This bound on width in terms of length is essentially optimal (Bonet & Galesi 1999). Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 37 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length The General Case: Corollary Corollary For general resolution, the length of refuting a CNF formula F over n variables is bounded from below by � � ( W ( F ⊢ 0 ) − W ( F )) 2 �� � � F ⊢ 0 ≥ exp L Ω . n Has been used to simplify many length lower bound proofs in resolution (and to prove a couple of new ones) � √ n � Need W ( F ⊢ 0 ) − W ( F ) = ω to get non-trivial bounds Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 38 / 63

Introduction Definition of Width Resolution Two Technical Lemmas Resolution Width Width is Upper-Bounded by Length (Not a) Proof of the General Case Proof for tree-like resolution breaks down in general case 0 Not true that L ( π ) = L ( π x ) + L ( π x ) + 1 x x Subderivations π x and π x may share clauses! Instead π x π x ◮ Look at very wide clauses in π ◮ Eliminate many of them by applying restriction setting commonly occurring literal to true ◮ More complicated inductive argument (still exponential blow-up in length) Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 39 / 63

Resolution Space Combinatorial Characterization of Width Space is Greater than Width Open Questions Part II Resolution Width and Space Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 40 / 63

Resolution Space Combinatorial Characterization of Width Space is Greater than Width Open Questions Outline of Part II: Resolution Width and Space Resolution Space Definition of Space Some Basic Properties Combinatorial Characterization of Width Boolean Existential Pebble Game Existential Pebble Game Characterizes Resolution Width Space is Greater than Width Open Questions Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 41 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Introducing Space ◮ Results on width lead to question: Can other complexity measures yield interesting insights as well? ◮ Esteban & Tor´ an (1999) introduced proof space (maximal # clauses in memory while verifying proof) ◮ Many lower bounds for space proven All turned out to match width bounds! Coincidence? ◮ Atserias & Dalmau (2003): space ≥ width − constant for k -CNF formulas The subject of the 2nd part of this talk Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 42 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Resolution Derivation (Revisited) Sequence of sets of clauses, or clause configurations, { C 0 , . . . , C τ } such that C 0 = ∅ and C t follows from C t − 1 by: Download C t = C t − 1 ∪ { C } for clause C ∈ F (axiom) Erasure C t = C t − 1 \ { C } for clause C ∈ C t − 1 Inference C t = C t − 1 ∪ { C ∨ D } for clause C ∨ D inferred by resolution rule from C ∨ x , D ∨ x ∈ C t − 1 Resolution derivation π : F ⊢ D of clause D from F : Derivation { C 0 , . . . , C τ } such that C τ = { D } Resolution refutation of F : Derivation π : F ⊢ 0 of empty clause 0 from F Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 43 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom       Empty start configuration       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ z     Download axiom x ∨ z       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ z     Download axiom x ∨ z       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ z z ∨ y     Download axiom z ∨ y       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ z z ∨ y     Download axiom z ∨ y       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ z z ∨ y   Infer x ∨ y from     x ∨ z and z ∨ y     Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ z z ∨ y   Infer x ∨ y from   x ∨ y   x ∨ z and z ∨ y     Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ z z ∨ y   Infer x ∨ y from   x ∨ y   x ∨ z and z ∨ y     Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ z z ∨ y     x ∨ y Erase clause x ∨ z       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   z ∨ y x ∨ y     Erase clause x ∨ z       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   z ∨ y x ∨ y     Erase clause z ∨ y       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y     Erase clause z ∨ y       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y ∨ u     Download axiom x ∨ y ∨ u       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y ∨ u     Download axiom x ∨ y ∨ u       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y ∨ u     y ∨ u Download axiom y ∨ u       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y ∨ u     y ∨ u Download axiom y ∨ u       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y ∨ u   Infer x ∨ y from   y ∨ u   x ∨ y ∨ u and y ∨ u     Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y ∨ u   Infer x ∨ y from   y ∨ u   x ∨ y ∨ u and y ∨ u   x ∨ y   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y ∨ u   Infer x ∨ y from   y ∨ u   x ∨ y ∨ u and y ∨ u   x ∨ y   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y ∨ u     y ∨ u Erase clause x ∨ y ∨ u     x ∨ y   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y y ∨ u     x ∨ y Erase clause x ∨ y ∨ u       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y y ∨ u     x ∨ y Erase clause y ∨ u       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y     Erase clause y ∨ u       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y   Infer x from     x ∨ y and x ∨ y     Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y   Infer x from   x   x ∨ y and x ∨ y     Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y   Infer x from   x   x ∨ y and x ∨ y     Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x ∨ y     Erase clause x ∨ y x       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x     Erase clause x ∨ y       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x ∨ y x     Erase clause x ∨ y       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x     Erase clause x ∨ y       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x u ∨ v     Download axiom u ∨ v       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x u ∨ v     Download axiom u ∨ v       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x u ∨ v     x ∨ v Download axiom x ∨ v       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x u ∨ v     x ∨ v Download axiom x ∨ v       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x u ∨ v   Infer x ∨ u from   x ∨ v     u ∨ v and x ∨ v   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x u ∨ v   Infer x ∨ u from   x ∨ v     u ∨ v and x ∨ v x ∨ u   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x u ∨ v   Infer x ∨ u from   x ∨ v     u ∨ v and x ∨ v x ∨ u   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x u ∨ v     x ∨ v Erase clause u ∨ v     x ∨ u   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ v     x ∨ u Erase clause u ∨ v       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ v     x ∨ u Erase clause x ∨ v       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     Erase clause x ∨ v       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     u ∨ w Download axiom u ∨ w       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     u ∨ w Download axiom u ∨ w       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     u ∨ w Download axiom x ∨ u ∨ w     x ∨ u ∨ w   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     u ∨ w Download axiom x ∨ u ∨ w     x ∨ u ∨ w   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u   Infer x ∨ u from   u ∨ w     u ∨ w and x ∨ u ∨ w x ∨ u ∨ w   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u   Infer x ∨ u from   u ∨ w     u ∨ w and x ∨ u ∨ w x ∨ u ∨ w   x ∨ u Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u   Infer x ∨ u from   u ∨ w     u ∨ w and x ∨ u ∨ w x ∨ u ∨ w   x ∨ u Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     u ∨ w Erase clause u ∨ w     x ∨ u ∨ w   x ∨ u Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     x ∨ u ∨ w Erase clause u ∨ w     x ∨ u   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     x ∨ u ∨ w Erase clause x ∨ u ∨ w     x ∨ u   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     x ∨ u Erase clause x ∨ u ∨ w       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u   Infer x from   x ∨ u     x ∨ u and x ∨ u   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u   Infer x from   x ∨ u     x ∨ u and x ∨ u x   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u   Infer x from   x ∨ u     x ∨ u and x ∨ u x   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     x ∨ u Erase clause x ∨ u     x   Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63

Resolution Space Combinatorial Characterization of Width Definition of Space Space is Greater than Width Some Basic Properties Open Questions Example (Our Favourite Resolution Refutation Again) 1 . x ∨ z Axiom 9 . x ∨ y Res ( 1 , 2 ) z ∨ y x ∨ y 2 . Axiom 10 . Res ( 3 , 4 ) 3 . x ∨ y ∨ u Axiom 11 . x ∨ u Res ( 5 , 6 ) y ∨ u x ∨ u Res ( 7 , 8 ) 4 . Axiom 12 . 5 . u ∨ v Axiom 13 . x Res ( 9 , 10 ) 6 . x ∨ v Axiom 14 . x Res ( 11 , 12 ) u ∨ w 7 . Axiom 15 . 0 Res ( 13 , 14 ) 8 . x ∨ u ∨ w Axiom   x x ∨ u     Erase clause x ∨ u x       Short Proofs Are Narrow (Well, Sort of), But Are They Tight? TCS PhD Student Seminar April 3rd, 2006 44 / 63