Tree resolution proofs of the weak pigeon-hole principle

Information Processing Letters

Galesi

Lauria

2010

In this note we show that the asymmetric Prover-Delayer game developed in Beyersdorff et al. (2010) [2] for Parameterized Resolution is also applicable to other tree-like proof systems. In particular, we use this asymmetric Prover-Delayer game to show a lower bound of the form 2 Ω(n log n) for the pigeonhole principle in tree-like Resolution. This gives a new and simpler proof of the same lower bound established by Iwama and Miyazaki (1999) [7] and Dantchev and Riis (2001) [5].

Section: And There Is No I ∈ [M]supporting

confidence: 64%

Section: Introductionmentioning

confidence: 99%

A lower bound for the pigeonhole principle in tree-like Resolution by asymmetric Prover–Delayer games

Information Processing Letters

Galesi

Lauria

2010

“…in the dag-like model. This is in contrast to the propositional proof complexity of PHP, which is known to be 2 (n log n) in tree-like [32], but only 2 (n) in dag-like propositional resolution [38].…”

Section: A Game-based Lower Bound Technique For Modal Resolutionmentioning

confidence: 69%

Proof Complexity of Modal Resolution

Sigley

2021

J Autom Reasoning

We investigate the proof complexity of modal resolution systems developed by Nalon and Dixon (J Algorithms 62(3–4):117–134, 2007) and Nalon et al. (in: Automated reasoning with analytic Tableaux and related methods—24th international conference, (TABLEAUX’15), pp 185–200, 2015), which form the basis of modal theorem proving (Nalon et al., in: Proceedings of the twenty-sixth international joint conference on artificial intelligence (IJCAI’17), pp 4919–4923, 2017). We complement these calculi by a new tighter variant and show that proofs can be efficiently translated between all these variants, meaning that the calculi are equivalent from a proof complexity perspective. We then develop the first lower bound technique for modal resolution using Prover–Delayer games, which can be used to establish “genuine” modal lower bounds for size of dag-like modal resolution proofs. We illustrate the technique by devising a new modal pigeonhole principle, which we demonstrate to require exponential-size proofs in modal resolution. Finally, we compare modal resolution to the modal Frege systems of Hrubeš (Ann Pure Appl Log 157(2–3):194–205, 2009) and obtain a “genuinely” modal separation.

“…Its complexity in Resolution was first determined by Haken's seminal exponential lower bound [Hak85]. However, in tree-like Resolution the complexity of PHP is indeed 2 θ(n log n) as shown independently by Iwama and Miyazaki [IM99] and Dantchev and Riis [DR01]. The paper [BGL10] provides an elegant proof of this optimal n!…”

Section: Advantage Of the Asymmetric Prover-delayer Gamementioning

confidence: 92%

“…This connection with clause space complexity limits the strength of the symmetric method, since there are formulas for which the above lower bound is not tight (e.g. the classical pigeonhole principle [IM99,DR01,BGL10]). This is so because the clause space complexity of a formula F is s if and only if any proof tree for F contains a complete binary tree of height s. The gap between the size of such a minor and the size of the proof tree is exactly what the symmetric game fails to analyse.…”

Section: C∨x D∨¬x C∨dmentioning

confidence: 99%

A characterization of tree-like Resolution size

Information Processing Letters

Galesi

Lauria

2013

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.