Natural Deduction for Four-Valued both Regular and Monotonic Logics

The development of recursion theory motivated Kleene to create regular three-valued logics. Remove it taking his inspiration from the computer science, Fitting later continued to investigate regular three-valued logics and defined them as monotonic ones. Afterwards, Komendantskaya proved that there are four regular three-valued logics and in the three-valued case the set of regular logics coincides with the set of monotonic logics. Next, Tomova showed that in the four-valued case regularity and monotonicity do not coincide. She counted that there are 6400 four-valued regular logics, but only six of them are monotonic. The purpose of this paper is to create natural deduction systems for them. We also describe some functional properties of these logics

Coherent Integration of Databases by Abductive Logic Programming

We introduce an abductive method for a coherent integration of independent data-sources. The idea is to compute a list of data-facts that should be inserted to the amalgamated database or retracted from it in order to restore its consistency. This method is implemented by an abductive solver, called Asystem, that applies SLDNFA-resolution on a meta-theory that relates different, possibly contradicting, input databases. We also give a pure model-theoretic analysis of the possible ways to `recover' consistent data from an inconsistent database in terms of those models of the database that exhibit as minimal inconsistent information as reasonably possible. This allows us to characterize the `recovered databases' in terms of the `preferred' (i.e., most consistent) models of the theory. The outcome is an abductive-based application that is sound and complete with respect to a corresponding model-based, preferential semantics, and -- to the best of our knowledge -- is more expressive (thus more general) than any other implementation of coherent integration of databases

Generalized knowledge-based semantics for multi-valued logic programs

A generalized logic programming system is presented which uses bilattices as the underlying framework for the semantics of programs. The two orderings of the bilattice reflect the concepts of truth and knowledge. Programs are interpreted according to their knowledge content, resulting in a monotonic semantic operator even in the presence of negation. A special case, namely, logic programming based on the four-valued bilattice is carefully studied on its own right. In the four-valued case, a version of the Closed World Assumption is incorporated into the semantics. Soundness and Completeness results are given with and without the presence of the Closed World Assumption. The concepts studied in the four-valued case are then generalized to arbitrary bilattices. The resulting logic programming systems are well suited for representing incomplete or conflicting information. Depending on the choice of the underlying bilattice, the knowledge-based logic programming language can provide a general framework for other languages based on probabilistic logics, intuitionistic logics, modal logics based on the possible-worlds semantics, and other useful non-classical logics. A novel procedural semantics is given which extends SLDNF-resolution and can retrieve both negative and positive information about a particular goal in a uniform setting. The proposed procedural semantics is based on an AND-parallel computational model for logic programs. The concept of substitution unification is introduced and many of its properties are studied in the context of the proposed computational model. Some of these properties may be of independent interest, particularly in the implementation of parallel and distributed logic programs. Finally, soundness and completeness results are proved for the proposed logic programming system. It is further shown that for finite distributive bilattices (and, more generally, bilattices with the descending chain property), an alternate procedural semantics can be developed based on a small subset of special truth values which turn out to be the join irreducible elements of the knowledge part of the bilattice. The algebraic properties of these elements and their relevance to the corresponding logic programming system are extensively studied