Beata Konikowska

A Logic for Reasoning about Relative Similarity

A similarity relation is a reflexive and symmetric binary relation between objects. Similarity is relative: it depends on the set of properties of objects used in determining their similarity or dissimilarity. A multi-modal logical language for reasoning about relative similarities is presented. The modalities correspond semantically to the upper and lower approximations of a set of objects by similarity relations corresponding to all subsets of a given set of properties of objects. A complete deduction system for the language is presented.

Multi-valued Calculi for Logics Based on Non-determinism

Non-deterministic matrices (Nmatrices) are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one (the latter is new here). We use the Rasiowa-Sikorski (R-S) decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such structures with either of the above types of semantics. Later we demonstrate how these systems can be converted into cut-free ordinary Gentzen calculi which are also sound and complete for the corresponding non-deterministic semantics. As a by-product, we get new semantic characterizations for some well-known logics (like the logic CAR from [18, 28]).

Decomposition Proof Systems for Gödel-Dummett Logics

The main goal of the paper is to suggest some analytic proof systems for LC and its finite-valued counterparts which are suitable for proof-search. This goal is achieved through following the general Rasiowa-Sikorski methodology for constructing analytic proof systems for semantically-defined logics. All the systems presented here are terminating, contraction-free, and based on invertible rules, which have a local character and at most two premises.

Reducing model checking from multi-valued CTL* to CTL

A multi-valued version of CTL* (mv-CTL*), where both the propositions and the accessibility relation are multi-valued taking values in a finite quasi-boolean algebra, is considered. A general translation from mv-CTL* to CTL* model checking is defined. An application of the translation is shown for the most commonly used quasi-boolean algebras.

Cut-free Ordinary Sequent Calculi for Logics Having Generalized Finite-Valued Semantics

Abstract.The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general structure of proofs in the original calculus in a way ensuring preservation of the weak cut elimination theorem under the transformation. The described transformation metod is illustrated on several concrete examples of many-valued logics, including a new application to information sources logics.

Rough Sets and 3-Valued Logics

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics.We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix. Then we derive from these calculi sequent calculi with the same properties for the various “determinizations” of those two versions of the logic (including Łukasiewicz 3-valued logic). Finally, we show how to embed the four above-mentioned determinizations in extensions of the basic rough set logics obtained by adding to those logics a special two-valued “definedness” or “crispness” operator.

A Logic for Reasoning about Similarity

A similarity relation is a reflexive and symmetric, but in general not transitive binary relation between objects. Similarity can be regarded as a relative notion parametrised by the set of classification attributes used as a basis for determining similarity or dissimilairty of objects. In the paper we present a polymodal formal language for reasoning about such a relative notion of similarity. For each subset of a given set of attributes, we have two modalities, corresponding semantically to so-called upper and lower approximations of a set of objects with respect to that set of attributes; intuitively, the latter approximations could be described as the interior and completion of a set of objects with respect to the similarity relation generated by the considered set of attributes, respectively. Formulae of the language evaluate to sets of objects, and a formula is said to be true if it evaluates to the whole universe of the model. The language is given a sound and complete deduction system in Rasiowa-Sikorski style: it consists of fundamental sequences of formulae which represent axioms of the system, and decomposition rules for sequences of formulae which represent inference rules.

A three-valued logic for software specification and validation

Different calculi of partial or three-valued predicates have been used and studied by several authors in the context of software specification, development and validation. This paper offers a critical survey on the development of three-valued logics based on such calculi.

Modular Construction of Cut-free Sequent Calculi for Paraconsistent Logics

This paper makes a substantial step towards automatization of Para consistent reasoning by providing a general method for a systematic and modular generation of cut-free calculi for thousands of Para consistent logics known as Logics of Formal (In)consistency. The method relies on the use of non-deterministic semantics for these logics.

Model checking for multi-valued computation tree logics

A multi-valued version of CTL* (mv-CTL*), where both the propositions and the accessibility relation are multi-valued taking values in a finite quasi-Boolean algebra, is defined. A translation from mv-CTL* model checking to CTL* model checking is investigated. First, the case where the elements of quasi-Boolean algebras are totally ordered is considered. Secondly, it is shown how to design a translation algorithm for the two most commonly applied quasi-Boolean algebras. This construction suggests the way one can deal with more complex quasi-Boolean algebras if necessary.