Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

Abductive reasoning involves finding the missing premise of an “unsaturated” deductive inference, thereby selecting a possible _explanans_ for a conclusion based on a set of previously accepted premises. In this paper, we explore abductive reasoning from a structural proof-theory perspective. We present a hybrid sequent calculus for classical propositional logic that uses sequents and antisequents to define a procedure for identifying the set of analytic hypotheses that a rational agent would be expected to select as _explanans_ when presented with an (...) abductive problem. Specifically, we show that this set may not include the deductively minimal hypothesis due to the presence of redundant information. We also establish that the set of all analytic hypotheses exhausts all possible solutions to the given problem. Finally, we propose a deductive criterion for differentiating between the best _explanans_ candidates and other hypotheses. (shrink)

This paper presents Rasiowa–Sikorski deduction systems for logics \, \, \ and \. For each of the logics two systems are developed: an R–S system that can be supplemented with admissible cut rule, and a \-version of R–S system in which the non-admissible rule of cut is the only branching rule. The systems are presented in a Smullyan-like uniform notation, extended and adjusted to the aims of this paper. Completeness is proved by the use of abstract refutability properties which are (...) dual to consistency properties used by Fitting. Also the notion of admissibility of a rule in an R–S-system is analysed. (shrink)

We study an extension of first-degree entailment (FDE) by Dunn and Belnap with a non-contingency operator |$\blacktriangle \phi $| which is construed as ‘|$\phi $| has the same value in all accessible states’ or ‘all sources give the same information on the truth value of |$\phi $|’. We equip this logic dubbed |$\textbf {K}^\blacktriangle _{\textbf {FDE}}$| with frame semantics and show how the bi-valued models can be interpreted as interconnected networks of Belnapian databases with the |$\blacktriangle $| operator modelling search (...) for inconsistencies in the provided information. We construct an analytic cut system for the logic and show its soundness and completeness. We prove that |$\blacktriangle $| is not definable via the necessity modality |$\Box $| of |$\textbf {K}_{\textbf{FDE}}$|⁠. Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, |$\textbf {S4}$| and |$\textbf {S5}$| (among others) frames are definable. (shrink)

In this paper we devise non-distributive relatives of Exactly true logic by Pietz and Riveccio and its dual Non-falsity logic by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL’s and NFL’s entailment relations on the de Morgan lattice 4. We generalise these pre-orders and determine which distributive properties that hold on 4 are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For (...) these logics we also devise a truth table semantics which uses non-distributive lattice M3 as their lattice of truth values. We also provide analytic tableaux systems which work with sequents of the form \. We then prove correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL- and NFL-like logics built over a certain family of non-distributive modular lattices. (shrink)

ABSTRACT This paper reports on an empirical performance analysis of four modal theorem provers on benchmark suites of randomly generated formulae. The theorem provers tested are the Davis-Putnam-based procedure KSAT, the tableaux-based system KRZIS, the sequent-based Logics Workbench, and a translation approach combined with the first-order theorem prover SPASS. Our benchmark suites are sets of multi-modal formulae in a certain normal form randomly generated according to the scheme of Giunchiglia and Sebastiani [GS 96a, GS 96b]. We investigate the quality of (...) the random modal formulae and show that the scheme has some shortcomings, which may lead to mistaken conclusions. We propose improvements to the evaluation method and show that the translation approach provides a viable alternative to the other approaches. (shrink)

We describe a relational framework that uniformly supports formalization and automated reasoning in varied propositional modal logics. The proof system we propose is a relational variant of the classical Rasiowa-Sikorski proof system. We introduce a compact graph-based representation of formulae and proofs supporting an efficient implementation of the basic inference engine, as well as of a number of refinements. Completeness and soundness results are shown and a Prolog implementation is described.

Traditional abduction imposes as a precondition the restriction that the background information may not derive the goal data. In first-order logic such precondition is, in general, undecidable. To avoid such problem, we present a first-order cut-based abduction method, which has KE-tableaux as its underlying inference system. This inference system allows for the automation of non-analytic proofs in a tableau setting, which permits a generalization of traditional abduction that avoids the undecidable precondition problem. After demonstrating the correctness of the method, we (...) show how this method can be dynamically iterated in a process that leads to the construction of non-analytic first-order proofs and, in some terminating cases, to refutations as well. (shrink)

In this paper we study families of resource aware logics that explore resource restriction on rules; in particular, we study the use of controlled cut-rule and introduce three families of parameterised logics that arise from different ways of controlling the use of cut. We start with a formulation of classical logic in which cut is non-eliminable and then impose restrictions on the use of cut. Three Cut-and-Pay families of logics are presented, and it is shown that each family provides an (...) approximation process for full propositional classical logic when the control over the use of cut is progressively weakened. A sound and complete semantics is given for each component of each of the three families of approximated logics. One of these families is shown to possess the uniform substitution property, a new result for approximated reasoning. A tableau based decision procedure is presented for each element of the approximation families and the complexity of each decision procedure is studied. We show that there are families in which every component logic can be decided polynomially. (shrink)

In this paper we provide a detailed proof-theoretical analysis of a natural deduction system for classical propositional logic that (i) represents classical proofs in a more natural way than standard Gentzen-style natural deduction, (ii) admits of a simple normalization procedure such that normal proofs enjoy the Weak Subformula Property, (iii) provides the means to prove a Non-contamination Property of normal proofs that is not satisfied by normal proofs in the Gentzen tradition and is useful for applications, especially in formal argumentation, (...) (iv) naturally leads to defining a notion of depth of a proof, to the effect that, for every fixed natural k, normal k-depth deducibility is a tractable problem and converges to classical deducibility as k tends to infinity. (shrink)

In this paper we explore a generalization of traditional abduction which can simultaneously perform two different tasks: given an unprovable sequent Γ ⊢ G, find a sentence H such that Γ, H ⊢ G is provable ; given a provable sequent Γ ⊢ G, find a sentence H such that Γ ⊢ H and the proof of Γ, H ⊢ G is simpler than the proof of Γ ⊢ G . We argue that the two tasks should not be distinguished, (...) and present a general procedure for finding suitable hypotheses or lemmas. When the original sequent is provable, the abduced formula can be seen as a cut formula with respect to Gentzen's sequent calculus, so the abduction method is cut-based. Our method is based on the tableau-like system KE and we argue for its advantages over existing abduction methods based on traditional Smullyan-style Tableaux. (shrink)

Resolution and cut-free LK are the most popular propositional systems used for logical automated reasoning. The question whether or not resolution and cut-free LK have the same efficiency on the system of CNF formulas has been asked and studied since 1960 425–467). It was shown in Cook and Reckhow, J. Symbolic Logic 44 36–50 that tree resolution has super-polynomial speed-up over cut-free LK. Naturally, the current issue is whether or not resolution and cut-free LK expressed as directed acyclic graphs have (...) the same efficiency. In this paper, we introduce a new algorithm to eliminate atomic cuts and show that cut-free LK polynomially simulates resolution when the input formula is expressed as a k-CNF formula. As a corollary, we show that regular resolution does not polynomially simulate cut-free LK . We also show that cut-free LK polynomially simulates regular resolution. (shrink)