Hall operators on the set of formations of finite groups

Let π be a nonempty set of primes and let F be a saturated formation of all finite soluble π-groups. It is constructed the saturated formation consisting of all finite π-soluble groups whose F-projectors contain a Hall π-subgroup

On Stone sublattices of the lattice of totally local Fitting classes

Totally local Fitting classes with a Stone lattice of totally local Fitting subclasses are described

Complete solution of a constrained tropical optimization problem with application to location analysis

We present a multidimensional optimization problem that is formulated and solved in the tropical mathematics setting. The problem consists of minimizing a nonlinear objective function defined on vectors over an idempotent semifield by means of a conjugate transposition operator, subject to constraints in the form of linear vector inequalities. A complete direct solution to the problem under fairly general assumptions is given in a compact vector form suitable for both further analysis and practical implementation. We apply the result to solve a multidimensional minimax single facility location problem with Chebyshev distance and with inequality constraints imposed on the feasible location area.Comment: 20 pages, 3 figure

Inference Rules in Nelson’s Logics, Admissibility and Weak Admissibility

© 2015, Springer Basel. Our paper aims to investigate inference rules for Nelson’s logics and to discuss possible ways to determine admissibility of inference rules in such logics. We will use the technique offered originally for intuitionistic logic and paraconsistent minimal Johannson’s logic. However, the adaptation is not an easy and evident task since Nelson’s logics do not enjoy replacement of equivalences rule. Therefore we consider and compare standard admissibility and weak admissibility. Our paper founds algorithms for recognizing weak admissibility and admissibility itself – for restricted cases, to show the problems arising in the course of study

Strong Negation in Well-Founded and Partial Stable Semantics for Logic Programs

Abstract. A formalism called partial equilibrium logic (PEL) has recently been proposed as a logical foundation for the well-founded semantics (WFS) of logic programs. In PEL one defines a class of minimal models, called partial equilibrium models, in a non-classical logic, HT 2. On logic programs partial equilibrium models coincide with Przymusinski’s partial stable (p-stable) models, so that PEL can be seen as a way to extend WFS and p-stable semantics to arbitrary propositional theories. We study several extensions of PEL with strong negation and compare these with previous systems extending WFS with explicit negation, notably WSFX [10] and p-stable models with “classical ” negation [11].

A Tableau Calculus for Equilibrium Entailment

We apply tableau methods to the problem of computing entailment in the nonmonotonic system of equilibrium logic, a generalisation of the inference relation associated with the stable model and answer set semantics for logic programs. We describe tableau calculi for the nonclassical logics underyling equilibrium entailment, namely hereand -there with strong negation and its strengthening classical logic with strong negation. A further tableau calculus is then presented for computing equilibrium entailment. This makes use of a new method for reducing the complexity of the tableau expansion rules, which we call signing-up

Strong cut-elimination systems for Hudelmaier’s depth-bounded sequent calculus for implicational logic

Inspired by the Curry-Howard correspondence, we study normalisation procedures in the depth-bounded intuitionistic sequent calculus of Hudelmaier (1988) for the implicational case, thus strengthening existing approaches to Cut-admissibility. We decorate proofs with proofterms and introduce various term-reduction systems representing proof transformations. In contrast to previous papers which gave different arguments for Cut-admissibility suggesting weakly normalising procedures for Cut-elimination, our main reduction system and all its variations are strongly normalising, with the variations corresponding to different optimisations, some of them with good properties such as confluence