Parameterized Proof Complexity

Abstract

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Relativization makes contradictions harder for Resolution

“…The system of parameterized Resolution [12] seeks to refute the parameterized contradictions of PCon. Given (F , k), where F is a CNF in variables x 1 , .…”

“…The second part of the paper is in the area of Parameterized proof complexity, a program initiated in [12], which generally aims to gain evidence that W[i] is different from FPT. Typically, i is so that the former is W [2], though-in the journal version [13] of [12]-this has been W[SAT] and-in the note [20]-W [1] was entertained.…”

“…The second part of the paper is in the area of Parameterized proof complexity, a program initiated in [12], which generally aims to gain evidence that W[i] is different from FPT. Typically, i is so that the former is W [2], though-in the journal version [13] of [12]-this has been W[SAT] and-in the note [20]-W [1] was entertained. In the W [2] context, parameterized refutation systems aim at refuting parameterized contradictions which are pairs (F , k) in which F is a propositional CNF with no satisfying assignment of weight ≤ k. Several parameterized (hereafter often abbreviated as "p-") proof systems are discussed in [12,3,6].…”

“…In the W [2] context, parameterized refutation systems aim at refuting parameterized contradictions which are pairs (F , k) in which F is a propositional CNF with no satisfying assignment of weight ≤ k. Several parameterized (hereafter often abbreviated as "p-") proof systems are discussed in [12,3,6]. The lower bounds in [12], [3] and [6] amount to proving that the systems p-tree-Resolution, p-Resolution and p-bounded-depth Frege, respectively, are not fpt-bounded. Indeed, this is witnessed by the Pigeonhole principle, and so holds even when one considers parameterized contradictions (F , k) where F is itself an actual contradiction.…”

“…Such parameterized contradictions are termed "strong" in [6], in which the authors suggest these might be the only parameterized contradictions worth considering, as general lower bounds-even in pbounded-depth Frege-are trivial (see [6]). We sympathize with this outlook, but remind that there are alternative parameterized refutation systems built from embedding (see [12,13]) for which no good lower bounds are known even for general parameterized contradictions.…”