A characterization of tree-like Resolution size

Abstract

We explain an asymmetric Prover-Delayer game which precisely characterizes proof size in tree-like Resolution. This game was previously described in a parameterized complexity context to show lower bounds for parameterized formulas [BGL13] and for the classical pigeonhole principle [BGL10]. The main point of this note is to show that the asymmetric game in fact characterizes tree-like Resolution proof size, i. e. in principle our proof method allows to always achieve the optimal lower bounds. This is in contrast with previous techniques described in the literature. We also provide a very intuitive information-theoretic interpretation of the game.

A Game Characterisation of Tree-like Q-resolution Size

“…A unified game-theoretic approach was recently established in [17]. Building on the classic game of Pudlák and Impagliazzo [38] for tree-like Resolution, the papers [14,16] devise an asymmetric Prover-Delayer game, which was shown in [15] to even characterise tree-like Resolution size. Thus, in contrast to the classic symmetric 2 Prover-Delayer game of [38], the asymmetric game in principle allows to always obtain the optimal lower bounds, 3 which was demonstrated in [14] for the pigeonhole principle.…”

“…Inspired by this asymmetric Prover-Delayer game of [14][15][16], we develop here a Prover-Delayer game which tightly characterises the proof size in tree-like Q-Resolution. The general idea behind this game is that a Delayer claims to know a satisfying assignment to a false formula, while a Prover asks for values of variables until eventually finding a contradiction.…”

“…Thus exhibiting clever Delayer strategies automatically gives lower bounds to the proof size, and in principle these bounds are guaranteed to be optimal. In comparison to the game of [14][15][16], our formulation here needs a somewhat more powerful Prover, who can forget information as well as freely set universal variables. This is necessary as the Prover needs to simulate more complex Q-Resolution proofs involving universal variables and rules for them absent in classical Resolution.…”

“…[15] for details. 3 For specific formulas this optimal lower bound might be very difficult to show via the game though.…”

Unified Characterisations of Resolution Hardness Measures

“…This can now be understood through hardness; by Lemma 9 we know that for unsatisfiable clause-set F holds cts(F ) = hd(F )+1, while in [27] it was shown that the optimal value of the above game plus one equals cts(F ), and thus hd(F ) is the optimal value of that game for F . We remark that thus the game of Pudlák and Impagliazzo does not characterise tree resolution size precisely; in [16,14] a modified (asymmetric) version of the game is introduced, which precisely characterises tree resolution size ( [15]). We present now the generalised hardness game, also handling satisfiable clause-sets.…”