A natural and established way to restrict the constraint satisfaction problem is to fix the relations that can be used to pose constraints; such a family of relations is called a constraint language. In this article, we study arc consistency, a heavily investigated inference method, and three extensions thereof from the perspective of constraint languages. We conduct a comparison of the studied methods on the basis of which constraint languages they solve, and we present new polynomial-time tractability results for singleton arc consistency, the most powerful method studied.
Abstract. We extend first-order logic with counting by a new operator that allows it to formalise a limited form of recursion which can be evaluated in logarithmic space. The resulting logic LREC has a data complexity in LOGSPACE, and it defines LOGSPACEcomplete problems like deterministic reachability and Boolean formula evaluation. We prove that LREC is strictly more expressive than deterministic transitive closure logic with counting and incomparable in expressive power with symmetric transitive closure logic STC and transitive closure logic (with or without counting). LREC is strictly contained in fixed-point logic with counting FP+C. We also study an extension LREC= of LREC that has nicer closure properties and is more expressive than both LREC and STC, but is still contained in FP+C and has a data complexity in LOGSPACE.Our main results are that LREC captures LOGSPACE on the class of directed trees and that LREC= captures LOGSPACE on the class of interval graphs.
The question of whether there is a logic that captures polynomial time is one of the main open problems in descriptive complexity theory and database theory. In 2010 Grohe showed that fixed-point logic with counting captures polynomial time on all classes of graphs with excluded minors. We now consider classes of graphs with excluded induced subgraphs. For such graph classes, an effective graph decomposition, called modular decomposition, was introduced by Gallai in 1976. The graphs that are non-decomposable with respect to modular decomposition are called prime. We present a tool, the Modular Decomposition Theorem, that reduces (definable) canonization of a graph class C to (definable) canonization of the class of prime graphs of C that are colored with binary relations on a linearly ordered set. By an application of the Modular Decomposition Theorem, we show that fixed-point logic with counting captures polynomial time on the class of permutation graphs. Within the proof of the Modular Decomposition Theorem, we show that the modular decomposition of a graph is definable in symmetric transitive closure logic with counting. We obtain that the modular decomposition tree is computable in logarithmic space. It follows that cograph recognition and cograph canonization is computable in logarithmic space. CC Creative Commons24:2 B. Grußien Vol. 15:1 does not capture PTIME. In order to obtain a candidate for a logic capturing PTIME on all structures, Immerman proposed in 1987 to add to FP the ability to count [Imm87a]. Although the resulting logic, fixed-point logic with counting (FP+C), is not strong enough to capture PTIME on all finite structures [CFI92], it does so on many interesting classes of structures: FP+C captures PTIME, for example, on planar graphs [Gro98], all classes of graphs of bounded treewidth [GM99] and on K 5 -minor free graphs [Gro08]. Note that all these classes can be defined by a list of forbidden minors. In fact, Grohe showed in 2010 that FP+C captures PTIME on all graph classes with excluded minors [Gro10b]. This leads to the question whether a similar result can be obtained for graph classes that are characterized by excluded induced subgraphs, i.e., graph classes that are closed under taking induced subgraphs. For FP+C such a general result is not possible: Capturing PTIME on the class of chordal graphs, comparability graphs or co-comparability graphs is as hard as capturing PTIME on the class of all graphs for any "reasonable" logic [Gro10a, Lau11]. Yet, this gives us reason to consider the three mentioned graph classes and their subclasses more closely. So far, there are results showing that FP+C captures PTIME on the class of chordal line graphs [Gro10a] and on the class of interval graphs (chordal co-comparability graphs) [Lau10].We add to these results the following results: FP+C captures PTIME on the class of permutation graphs (comparability co-comparability graphs) (see Section 5) and on the class of chordal comparability graphs (see [Gru17a]). Both results are based on modular decom...
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