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Non-deterministic Matrices and their Applications for Non-classical Logics

Final Report Summary - NDMANCL (Non-deterministic matrices and their applications for non-classical logics)

The project has successfully contributed to the development of the semantic framework of a new logical formalism, based on non-deterministic matrices (Nmatrices) and its applications in proof theory and reasoning under uncertainty. The theoretical results include refining and enlarging the framework and introducing new useful modifications which allow to capture much wider classes of logics. As envisaged in the beginning, these theoretical results found applications in proof theory of non-classical logics. In particular, they were exploited for the following:

(1) decidable semantic characterisation of crucial syntactic properties of calculi, such as cut-elimination, analyticity, invertibility of rules, etc.;
(2) systematic generation of analytic calculi (which are a prerequisite for efficient automated deduction methods) and non-deterministic semantics for a large class of logics given in terms of axiomatic (non-analytic) proof systems;
(3) modular construction of analytic calculi for thousands of paraconsistent logics, thereby revolving a long-standing open question and creating the theoretical basis for developing paraconsistent automated deduction methods.

More specifically, the following results have been achieved:

(1) The theory of canonical systems has been extended to a very general family of labelled calculi, which includes many previously studied calculi. LK, canonical signed calculi and labelled calculi of Baaz et al. are just a few cases in point. To define semantics for this family of canonical labelled calculi, a generalisation of Nmatrices called partial PNmatrices was introduced, in which empty cells are allowed in the truth-tables of logical connectives. The semantics is effective, in the sense that it naturally leads to a decision procedure for these calculi. It was then applied to provide simple decidable semantic criteria for crucial syntactic properties of these calculi, namely (strong) analyticity and cut-admissibility. Related publications (written together with M. Baaz and O. Lahav), are 'Effective finite-valued semantics for labelled calculi' (presented at IJCAR'12) and 'Finite-valued semantics for canonical labelled calculi' (accepted to Journal of Automated Reasoning).

(2) Another type of canonical systems, the single-conclusioned ones, was considered in 'Basic constructive connectives, determinism and matrix-based semantics' (with A. Ciabattoni and O. Lahav), where we applied non-deterministic Kripke-style semantics to characterise two syntactic properties: invertibility of rules and axiom-expansion. An alternative matrix-based formulation of such semantics was introduced, which provides an algorithm for checking these properties, and also some new insights into constructive connectives. The paper was presented by A. Zamansky at Tablieaux '11.

(3) The framework of Nmatrices was used to systematically extract cut-free sequent calculi for practically all paraconsistent C-systems studied in the literature, thus providing the theoretical basis for the implementation of efficient paraconsistent theorem provers. Related publications (with A. Avron and B. Konikowska) include 'Modular construction of cut-free sequent calculi for paraconsistent logics' (presented at LICS'12) and 'Cut-free sequent calculi for C-systems with generalised finite-valued semantics' (accepted to the Journal of Logic and Computation).

(4) The results from (3) were extended to obtain a fully automatic method for generating an analytic sequent calculus and an effective finite-valued semantics for logics formulated as a Hilbert-style calculus of a natural general form. This can be applied to a wide variety of useful logics, including the paraconsistent logics mentioned in (3), as well as other logics for which neither analytic calculi nor suitable semantics have so far been available. Related publications (with A. Ciabattoni, O. Lahav and L. Spendier) include 'Automated analysis of paraconsistent and other logics' (presented at LFCS'13), and 'Taming logics: An algorithmic approach' (under preparation). ParaLyzer, the Prolog implementation of the method is available at http://www.logic.at/people/lara/tinc/webparalyzer/paralyzer.html

(5) A method for constructing inconsistency-tolerant variants of logics was extended to logics based on Nmatrices, as well as other types of logics based on denotational semantics, both in monotonic and non-monotonic settings.

Related publications include 'A dissimilarity-based framework for generating inconsistency-tolerant logics' (with O. Arieli, accepted to Annals of Mathematics and Artificial Intelligence), 'Inconsistency-tolerance in knowledge-based systems by dissimilarities' (with O. Arieli, presented at FOiKS'12), and 'Preferential framework for trivialisation-resistant reasoning with inconsistent information' (presented at JELIA'12).

The project has significantly contributed to the recognition and adoption of the non-deterministic semantic paradigm by the scientific community. A. Zamansky has participated in numerous scientific conferences and workshops, and has presented her work at several occasions. She was invited to present a tutorial on the framework of Nmatrices at UNILOG'13, a major international event in the logical community. Moreover, she organised an international workshop on non-classical logics in Tel Aviv University, which further increased the scientific collaboration between the Austrian and Israeli groups. These occasions have served well both the transfer of her knowledge and her training.