Abstract
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.
Highlights
The central problem in proof complexity is to determine the size of the smallest proof for a given formula in a specified proof system, typically defined through a set of axioms and inference rules
The trace of the run of a SAT solver on an unsatisfiable formula can be interpreted as a proof of unsatisfiability, whereby each solver implicitly defines a proof system for unsatisfiable formulas
It follows from the above theorem that 2 f (n) is a lower bound for the number of modal resolution steps, N required to refute C, if 2 f (n) is superpolynomial we have proved a superpolynomial lower bound for N
Summary
The central problem in proof complexity is to determine the size of the smallest proof for a given formula in a specified proof system, typically defined through a set of axioms and inference rules. The most important proof complexity results on modal logics concern exponential lower bounds for modal Frege systems [40,42]. A game-based lower bound technique for modal resolution Most research effort in proof complexity is directed towards showing lower bounds for the proof size of specific families of formulas. We devise modal decision trees (Definition 25), which represent partial Kripke models These modal decision trees correspond to the partial models constructed during a game, and the size of modal decision trees is proportional to the number of modal resolution steps in Kmc-Res refutations (Proposition 4). In contrast to the case of propositional and QBF resolution, where the asymmetric Prover–Delayer game is known to precisely characterise tree-like resolution size [17,18], our game here does not provide a similar characterisation in the modal setting.
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