Relating two width measures for resolution proofs

A clause is a disjunction of literals. We use set notation; a set of literals denotes the clause that is their disjunction. The empty clause is denoted by . The width of a clause C, denoted |C|, is the number of literals in it. The width of a CNF formula F , denoted width(F ), is the maximum width of any clause in F . For a partial assignment ρ, C|ρ denotes the restriction of the clause C by applying the partial assignment to all literals in C in the domain of ρ, and F |ρ denotes the conjunction of the clauses C|ρ for each C ∈ F . The resolution rule infers a clause C ∪D from clauses C ∪ {x}, D ∪ {¬x}. A resolution proof π deriving a clause C from a set of initial clauses C1, C2, . . . , Cm (called the axioms) is a sequence of clauses D1, D2, . . . , Dt where Dt is the clause C, and each Di is either an axiom, or is obtained from some Dj , Dk for j, k < i by resolution. The width of π is maxi{|Di|}, and its size is t. By width(F ` C) we denote the minimum width of a resolution proof π deriving C from F . The width measure has proven to be extremely useful in obtaining lower bounds on the total size of resolution proofs; see [BW01]. A clause C ′ is a weakening of a clause C if C ⊆ C ′, that is, every literal in C also appears in C ′. We may also allow weakening steps in a resolution proof; in the sequence above, Di may be obtained from Dj for some j < i by weakening. Weakening steps are redundant and can be eliminated. The measure width(F ` ) remains unchanged whether or not weakening is allowed. If π is restricted so that each derived clause Di is used at most once subsequently (that is, the underlying graph structure is a tree), then we have a tree-like resolution proof. Making a proof tree-like by duplicating sub-derivations as needed can increase its size significantly, but does not affect width. So without loss of generality we consider only tree-like proofs.