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V. W. Marek

Publications and source records attributed to V. W. Marek.

6 recordsLinked to original sources
arXiv2021

Pairwise Comparisons Matrix Decomposition into Approximation and Orthogonal Component Using Lie Theory

This paper examines the use of Lie group and Lie Algebra theory to construct the geometry of pairwise comparisons matrices. The Hadamard product (also known as coordinatewise, coordinate-wise, elementwise, or element-wise product) is analyzed in the context of inconsistency and inaccuracy by the decomposition method. The two designed components are the approximation and orthogonal components. The decomposition constitutes the theoretical foundation for the multiplicative pairwise comparisons. Keywords: approximate reasoning, subjectivity, inconsistency, consistency-driven, pairwise comparison, matrix Lie group, Lie algebra, approximation, orthogonality, decomposition.

math.GM↗
arXiv2020

On the use of group theory to generalize elements of pairwise comparisons matrix: a cautionary note

This paper examines the constricted use of group theory in the studies of pairwise comparisons. The presented approach is based on the application of the famous Levi Theorems of 1942 and 1943 for orderable groups. The theoretical foundation for multiplicative (ratio) pairwise comparisons has been provided. Counterexamples have been provided to support the theory. In our opinion, the scientific community must be made aware of the limitations of using the group theory in pairwise comparisons. Groups, which are not torsion free, cannot be used for ratios by Levi's theorems.

math.HO↗
arXiv2013

Index sets for Finite Normal Predicate Logic Programs

_e is the effective list of all finite predicate logic programs. is the list of recursive trees. We modify constructions of Marek, Nerode, and Remmel [25] to construct recursive functions f and g such that for all indices e, (i) there is a one-to-one degree preserving correspondence between the set of stable models of Q_e and the set of infinite paths through T_{f(e)} and (ii) there is a one-to-one degree preserving correspondence between the set of infinite paths through T_e and the set of stable models of Q_{g(e)}. We use these two recursive functions to reduce the problem of finding the complexity of the index set I_P for various properties P of normal finite predicate logic programs to the problem of computing index sets for primitive recursive trees for which there is a large variety of results [6], [8], [16], [17], [18], [19]. We use our correspondences to determine the complexity of the index sets of all programs and of certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having at least one stable model, (iii) having exactly c stable models for any given positive integer c, (iv) having only finitely many stable models, or (vi) having infinitely many stable models.

cs.LO↗
arXiv2010

An Application of Proof-Theory in Answer Set Programming

We apply proof-theoretic techniques in answer Set Programming. The main results include: 1. A characterization of continuity properties of Gelfond-Lifschitz operator for logic program. 2. A propositional characterization of stable models of logic programs (without referring to loop formulas.

cs.AI↗
arXiv2006

Logic programs with monotone abstract constraint atoms

We introduce and study logic programs whose clauses are built out of monotone constraint atoms. We show that the operational concept of the one-step provability operator generalizes to programs with monotone constraint atoms, but the generalization involves nondeterminism. Our main results demonstrate that our formalism is a common generalization of (1) normal logic programming with its semantics of models, supported models and stable models, (2) logic programming with weight atoms (lparse programs) with the semantics of stable models, as defined by Niemela, Simons and Soininen, and (3) of disjunctive logic programming with the possible-model semantics of Sakama and Inoue.

cs.AI↗