Emil Jeřábek

Admissible Rules of Modal Logics

We construct explicit bases of admissible rules for a representative class of normal modal logics (including the systems K4, GL, S4, Grz, and GL:3), by extending the methods of S. Ghilardi and R. Iemhoff. We also investigate the notion of admissible multiple conclusion rules.

Dual weak pigeonhole principle, Boolean complexity, and derandomization

Abstract We study the extension (introduced as BT by Krajicek in Fund. Math. 170 (2001) 123) of the theory S21 by instances of the dual (onto) weak pigeonhole principle for p-time functions, dWPHP(PV)x2x. We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkies witnessing theorem for S21+dWPHP(PV). We construct a propositional proof system WF (based on a reformulation of Extended Frege in terms of Boolean circuits), which captures the ∀Π1b-consequences of S21+dWPHP(PV). We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of Buss et al. (Comput. Complexity 6 (1996/1997) 256). We prove that dWPHP(PV) is (over S21) equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan–Wigderson construction (derandomization of probabilistic p-time algorithms) in a conservative extension of S21+dWPHP(PV).

Admissible Rules of Łukasiewicz Logic

We investigate admissible rules of Łukasiewicz multi-valued propositional logic. We show that admissibility of multiple-conclusion rules in Łukasiewicz logic, as well as validity of universal sentences in free MV-algebras, is decidable (in PSPACE).

Bases of Admissible Rules of Łukasiewicz Logic

We construct explicit bases of single-conclusion and multiple-conclusion admissible rules of propositional Łukasiewicz logic, and we prove that every formula has an admissibly saturated approximation. We also show that Łukasiewicz logic has no finite basis of admissible rules.

Complexity of admissible rules

We investigate the computational complexity of deciding whether a given inference rule is admissible for some modal and superintuitionistic logics. We state a broad condition under which the admissibility problem is coNEXP-hard. We also show that admissibility in several well-known systems (including GL, S4, and IPC) is in coNE, thus obtaining a sharp complexity estimate for admissibility in these systems.

The strength of sharply bounded induction

We prove that the sharply bounded arithmetic T02 in a language containing the function symbol ⌊x /2y⌋ (often denoted by MSP) is equivalent to PV1. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Proof Complexity of the Cut-free Calculus of Structures

We investigate the proof complexity of analytic subsystems of the deep inference proof system SKSg (the calculus of structures). Exploiting the fact that the cut rule (i↑) of SKSg corresponds to the ¬-left rule in the sequent calculus, we establish that the ‘analyticsystem KSg+c↑ has essentially the same complexity as the monotone Gentzen calculus MLK. In particular, KSg+c↑ quasipolynomially simulates SKSg, and admits polynomial-size proofs of some variants of the pigeonhole principle.

Substitution Frege and extended Frege proof systems in non-classical logics

Abstract We investigate the substitution Frege ( SF ) proof system and its relationship to extended Frege ( EF ) in the context of modal and superintuitionistic (si) propositional logics. We show that EF is p-equivalent to tree-like SF , and we develop a “normal form” for SF -proofs. We establish connections between SF for a logic L , and EF for certain bimodal expansions of L . We then turn attention to specific families of modal and si logics. We prove p-equivalence of EF and SF for all extensions of KB , all tabular logics, all logics of finite depth and width, and typical examples of logics of finite width and infinite depth. In most cases, we actually show an equivalence with the usual EF system for classical logic with respect to a naturally defined translation. On the other hand, we establish exponential speed-up of SF over EF for all modal and si logics of infinite branching, extending recent lower bounds by P. Hrubes. We develop a model-theoretical characterization of maximal logics of infinite branching to prove this result.

Integer factoring and modular square roots

Buresh-Oppenheim proved that the NP search problem to find nontrivial factors of integers of a special form belongs to Papadimitrious class PPA, and is probabilistically reducible to a problem in PPP. In this paper, we use ideas from bounded arithmetic to extend these results to arbitrary integers. We show that general integer factoring is reducible in randomized polynomial time to a PPA problem and to the problem WeakPigeon ? PPP . Both reductions can be derandomized under the assumption of the generalized Riemann hypothesis. We also show (unconditionally) that PPA contains some related problems, such as square root computation modulo n, and finding quadratic nonresidues modulo n.

Frege systems for extensible modal logics

Abstract By a well-known result of Cook and Reckhow [S.A. Cook, R.A. Reckhow, The relative efficiency of propositional proof systems, Journal of Symbolic Logic 44 (1) (1979) 36–50; R.A. Reckhow, On the lengths of proofs in the propositional calculus, Ph.D. Thesis, Department of Computer Science, University of Toronto, 1976], all Frege systems for the classical propositional calculus (CPC) are polynomially equivalent. Mints and Kojevnikov [G. Mints, A. Kojevnikov, Intuitionistic Frege systems are polynomially equivalent, Zapiski Nauchnyh Seminarov POMI 316 (2004) 129–146] have recently shown p-equivalence of Frege systems for the intuitionistic propositional calculus (IPC) in the standard language, building on a description of admissible rules of IPC by Iemhoff [R. Iemhoff, On the admissible rules of intuitionistic propositional logic, Journal of Symbolic Logic 66 (1) (2001) 281–294]. We prove a similar result for an infinite family of normal modal logics, including K 4 , G L , S 4 , and S 4 Grz .