Proof Complexity of Modal Resolution

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.

Chapter 7. Proof Complexity and SAT Solving

This chapter gives an overview of proof complexity and connections to SAT solving, focusing on proof systems such as resolution, Nullstellensatz, polynomial calculus, and cutting planes (corresponding to conflict-driven clause learning, algebraic approaches using linear algebra or Gröbner bases, and pseudo-Boolean solving, respectively). There is also a discussion of extended resolution (which is closely related to DRAT proof logging) and Frege and extended Frege systems more generally. An ample supply of references for further reading is provided, including for some topics omitted in this chapter.

ON NON-MONOTONOUS PROPERTIES OF SOME CLASSICAL AND NONCLASSICAL PROPOSITIONAL PROOF SYSTEMS

We investigate the relations between the proof lines of non-minimal tautologies and its minimal tautologies for the Frege systems, the sequent systems with cut rule and the systems of natural deductions of classical and nonclassical logics. We show that for these systems there are sequences of tautologies ψn, every one of which has unique minimal tautologies φn such that for each n the minimal proof lines of φn are an order more than the minimal proof lines of ψn.

Investigation of Monotonous Properties for Frege Systems

In this paper, we investigate the relations between the Frege proof lines of minimal tautologies and the results of substitutions in them. We show that there is a sequence of tautologies ψn, each of which has a unique minimal tautology ϕn, such that for every n the Frege proof lines of ϕn are an order more than the Frege proof lines for ψn.