Parameterized complexity of decision problems in non-classical logics

Abstract

Parameterized complexity is a branch of a computational complexity. The pioneers of this new and promising research field are Downey and Fellows. They suggest to examine the structural properties of a given problem and restrict the instance by a parameter. In this thesis we investigate the parameterized complexity of various problems in default logic and in temporal logics. In the first section of Chapter 3 we introduce a dynamic programming algorithm which decides whether a given default theory has a consistent stable extension in fpt-time and enumerates all generating defaults that lead to a stable extension with a pre-computation step that is linear in the input theory and triple exponential in the tree-width followed by a linear delay to output solutions. In the second part of this chapter we lift the notion of backdoors to the field of default logics. We consider two problems, first we are interested to detect a backdoor and then to evaluate it for the target formulae classes HORN, KROM, POSITIVE-UNITand MONOTONE. In Chapter 4, we investigate the parameterized complexity of problems in various tem- poral logics. In the first section we introduce several graph-like structures for formula representation and the corresponding notion of tree-width and path-width. To obtain the fixed parameter tractability of different fragments, we generalize the prominent Courcelle’s Theorem to work for infinite signatures. In this section, we also consider Boolean operator fragments in the sense of Post’s lattice. In the second part of Chapter 4 we introduce the notion of backdoors for the glob- ally fragment of linear temporal logic. Again, our problems of interest are to detect a backdoor and to evaluate it, this time, for the target formulae classes HORN and KROM.