Fixed-parameter tractability, definability, and model checking

In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For example, we establish the equivalence between the model-checking for existential first-order logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that model-checking for first-order formulas is fixed-parameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for every t >= 0 we prove an equivalence between model-checking for first-order formulas with t quantifier alternations and the parameterized halting problem for alternating Turing machines with t alternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows' W-hierarchy. On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixed-parameter tractable problems in terms of slicewise definability in finite variable least fixed-point logic, which is reminiscent of the Immerman-Vardi Theorem characterizing the class PTIME in terms of definability in least fixed-point logic.Comment: To appear in SIAM Journal on Computin

Descriptive Complexity Theories

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Descriptive Complexity Theories

In this article we review some of the main results of descriptive complexity theory in order to make the reader familiar with the nature of the investigations in this area. We start by presenting the characterization of automata recognizable languages by monadic second-order logic. Afterwards we explain the characterization of various logics by fixed-point logics. We assume familiarity with logic but try to keep knowledge of complexity theory to a minimum

Descriptive Complexity Theories

ABSTRACT: In this article we review some of the main results of descriptive complexity theory in order to make the reader familiar with the nature of the investigations in this area. We start by presenting the characterization of automata recognizable languages by monadic second-order logic. Afterwards we explain the characterization of various logics by fixed-point logics. We assume familiarity with logic but try to keep knowledge of complexity theory to a minimum. Keywords: Computational complexity theory, complexity classes, descriptive characterizations, monadic second-order logic, fixed-point logic, Turing machine. Complexity theory or more precisely, computational complexity theory (cf. Papadimitriou 1994), tries to classify problems according to the complexity of algorithms solving them. Of course, we can think of various ways of measuring the complexity of an algorithm, but the most important and relevant ones are time and space. That is, we think we have a type of machine capable, in principle, to carry out any algorithm, a so-called general purpose machine (or, general purpose computer), and we measure the complexity of an algorithm in terms of the time or the number of steps needed to carry out this algorithm. By space, we mean the amount of memory the algorithm uses. Time bounds yield (time) complexity classes consisting of all problems solvable by an algorithm keeping to the time bound. Similarly, space complexity classes are obtained. It turns out that these definitions are quite robust in the sense that, for reasonable time or space bounds, the corresponding complexity classes do not depend on the special type of machine model chosen. But are the resources time and space tied to the inherent mathematical complexity of a given problem? We can try to classify problems also in terms of the complexity of formal languages or logics that allow to express the problems (e.g., if problem P2 is expressible in second-order logic but not in first-order logic then it is more complex than a problem P1 expressible in first-order logic). We speak of descriptive complexity theory (cf. However, descriptive characterizations of complexity classes have been criticized for being merely simple translations of the original machine-based definitions of the classes into a logical formalism and therefore, for not providing new insights. Although it is certainly true that the logical characterizations are often very close to th

Model-Checking Problems as a Basis for Parameterized Intractability

Most parameterized complexity classes are defined in terms of a parameterized version of the Boolean satisfiability problem (the so-called weighted satisfiability problem). For example, Downey and Fellow's W-hierarchy is of this form. But there are also classes, for example, the A-hierarchy, that are more naturally characterised in terms of model-checking problems for certain fragments of first-order logic. Downey, Fellows, and Regan were the first to establish a connection between the two formalisms by giving a characterisation of the W-hierarchy in terms of first-order model-checking problems. We improve their result and then prove a similar correspondence between weighted satisfiability and model-checking problems for the A-hierarchy and the W^*-hierarchy. Thus we obtain very uniform characterisations of many of the most important parameterized complexity classes in both formalisms. Our results can be used to give new, simple proofs of some of the core results of structural parameterized complexity theory

Model-Checking Problems as a Basis for Parameterized Intractability

Most parameterized complexity classes are defined in terms of a parameterizedversion of the Boolean satisfiability problem (the so-called weightedsatisfiability problem). For example, Downey and Fellow's W-hierarchy is ofthis form. But there are also classes, for example, the A-hierarchy, that aremore naturally characterised in terms of model-checking problems for certainfragments of first-order logic. Downey, Fellows, and Regan were the first to establish a connection betweenthe two formalisms by giving a characterisation of the W-hierarchy in terms offirst-order model-checking problems. We improve their result and then prove asimilar correspondence between weighted satisfiability and model-checkingproblems for the A-hierarchy and the W^*-hierarchy. Thus we obtain very uniformcharacterisations of many of the most important parameterized complexityclasses in both formalisms. Our results can be used to give new, simple proofs of some of the coreresults of structural parameterized complexity theory.Comment: Changes in since v2: Metadata update