A proof-theoretic analysis of the classical propositional matrix method

The matrix method, due to Bibel and Andrews, is a proof procedure designed for automated theorem-proving. We show that underlying this method is a fully structured combinatorial model of conventional classical proof theory. © 2012 The Author, 2012. Published by Oxford University Press

Automated proof search in non-classical logics : efficient matrix proof methods for modal and intuitionistic logics

In this thesis we develop efficient methods for automated proof search within an important class of mathematical logics. The logics considered are the varying, cumulative and constant domain versions of the first-order modal logics K, K4, D, D4, T, S4 and S5, and first-order intuitionistic logic. The use of these non-classical logics is commonplace within Computing Science and Artificial Intelligence in applications in which efficient machine assisted proof search is essential. Traditional techniques for the design of efficient proof methods for classical logic prove to be of limited use in this context due to their dependence on properties of classical logic not shared by most of the logics under consideration. One major contribution of this thesis is to reformulate and abstract some of these classical techniques to facilitate their application to a wider class of mathematical logics. We begin with Bibel's Connection Calculus: a matrix proof method for classical logic comparable in efficiency with most machine orientated proof methods for that logic. We reformulate this method to support its decomposition into a collection of individual techniques for improving the efficiency of proof search within a standard cut-free sequent calculus for classical logic. Each technique is presented as a means of alleviating a particular form of redundancy manifest within sequent-based proof search. One important result that arises from this anaylsis is an appreciation of the role of unification as a tool for removing certain proof-theoretic complexities of specific sequent rules; in the case of classical logic: the interaction of the quantifier rules. All of the non-classical logics under consideration admit complete sequent calculi. We anaylse the search spaces induced by these sequent proof systems and apply the techniques identified previously to remove specific redundancies found therein. Significantly, our proof-theoretic analysis of the role of unification renders it useful even within the propositional fragments of modal and intuitionistic logic

Theories of truth based on four-valued infectious logics

Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems

Elimination of Cuts in First-order Finite-valued Logics

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information

On the Complexity of Reconstructing Chemical Reaction Networks

The analysis of the structure of chemical reaction networks is crucial for a better understanding of chemical processes. Such networks are well described as hypergraphs. However, due to the available methods, analyses regarding network properties are typically made on standard graphs derived from the full hypergraph description, e.g.\ on the so-called species and reaction graphs. However, a reconstruction of the underlying hypergraph from these graphs is not necessarily unique. In this paper, we address the problem of reconstructing a hypergraph from its species and reaction graph and show NP-completeness of the problem in its Boolean formulation. Furthermore we study the problem empirically on random and real world instances in order to investigate its computational limits in practice

Proof Theory of Finite-valued Logics

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics

Type-driven semantic interpretation and feature dependencies in R-LFG

Once one has enriched LFG's formal machinery with the linear logic mechanisms needed for semantic interpretation as proposed by Dalrymple et. al., it is natural to ask whether these make any existing components of LFG redundant. As Dalrymple and her colleagues note, LFG's f-structure completeness and coherence constraints fall out as a by-product of the linear logic machinery they propose for semantic interpretation, thus making those f-structure mechanisms redundant. Given that linear logic machinery or something like it is independently needed for semantic interpretation, it seems reasonable to explore the extent to which it is capable of handling feature structure constraints as well. R-LFG represents the extreme position that all linguistically required feature structure dependencies can be captured by the resource-accounting machinery of a linear or similiar logic independently needed for semantic interpretation, making LFG's unification machinery redundant. The goal is to show that LFG linguistic analyses can be expressed as clearly and perspicuously using the smaller set of mechanisms of R-LFG as they can using the much larger set of unification-based mechanisms in LFG: if this is the case then we will have shown that positing these extra f-structure mechanisms is not linguistically warranted.Comment: 30 pages, to appear in the the ``Glue Language'' volume edited by Dalrymple, uses tree-dvips, ipa, epic, eepic, fullnam

Relative Entailment Among Probabilistic Implications

We study a natural variant of the implicational fragment of propositional logic. Its formulas are pairs of conjunctions of positive literals, related together by an implicational-like connective; the semantics of this sort of implication is defined in terms of a threshold on a conditional probability of the consequent, given the antecedent: we are dealing with what the data analysis community calls confidence of partial implications or association rules. Existing studies of redundancy among these partial implications have characterized so far only entailment from one premise and entailment from two premises, both in the stand-alone case and in the case of presence of additional classical implications (this is what we call "relative entailment"). By exploiting a previously noted alternative view of the entailment in terms of linear programming duality, we characterize exactly the cases of entailment from arbitrary numbers of premises, again both in the stand-alone case and in the case of presence of additional classical implications. As a result, we obtain decision algorithms of better complexity; additionally, for each potential case of entailment, we identify a critical confidence threshold and show that it is, actually, intrinsic to each set of premises and antecedent of the conclusion