This article surveys various complexity and expressiveness results on different forms of logic programming. The main focus is on decidable forms of logic programming, in particular, propositional logic programming and datalog, but we also mention general logic programming with function symbols. Next to classical results on plain logic programming (pure Horn clause programs), more recent results on various important extensions of logic programming are surveyed. These include logic programming with different forms of negation, disjunctive logic programming, logic programming with equality, and constraint logic programming.

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Complexity and expressive power of logic programming

We consider disjunctive Datalog, a powerful database query language based on disjunctive logic programming. Briefly, disjunctive Datalog is a variant of Datalog where disjunctions may appear in the rule heads; advanced versions also allow for negation in the bodies which can be handled according to a semantics for negation in disjunctive logic programming. In particular, we investigate three different semantics for disjunctive Datalog: the minimal model semantics the perfect model semantics, and the stable model semantics. For each of these semantics, the expressive power and complexity are studied. We show that the possibility variants of these semantics express the same set of queries. In fact, they precisely capture the complexity class ΣP2. Thus, unless the Polynomial Hierarchy collapses, disjunctive Datalog is more expressive that normal logic programming with negation. These results are not only of theoretical interest; we demonstrate that problems relevant in practice such as computing the optimal tour value in the Traveling Salesman Problem and eigenvector computations can be handled in disjunctive Datalog, but not Datalog with negation (unless the Polynomial Hierarchy collapses). In addition, we study modularity properties of disjunctive Datalog and investigate syntactic restrictions of the formalisms.

Disjunctive datalog

For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

This report documents the program and the outcomes of Dagstuhl Seminar 15511 “The Graph Isomorphism Problem”. The goal of the seminar was to bring together researchers working on the numerous topics closely related to the Isomorphism Problem to foster their collaboration. To this end the participants of the seminar included researchers working on the theoretical and practical aspects of isomorphism ranging from the fields of algorithmic group theory, finite model theory, combinatorial optimization to algorithmics. A highlight of the conference was the presentation of a new quasi-polynomial time algorithm for the Graph Isomorphism Problem, providing the first improvement since 1983. Seminar December 13–18, 2015 – http://www.dagstuhl.de/15511 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems, G.2.1 Combinatorics, G.2.2 Graph Theory

The Graph Isomorphism Problem

It is shown that BPP can be simulated in subexponential time for infinitely many input lengths unless exponential time collapses to the second level of the polynomial-time hierarchy, has polynomial-size circuits, and has publishable proofs (EXPTIME=MA). It is also shown that BPP is contained in subexponential time unless exponential time has publishable proofs for infinitely many input lengths. In addition, it is shown that BPP can be simulated in subexponential time for infinitely many input lengths unless there exist unary languages in MA/P. The proofs are based on the recent characterization of the power of multiprover interactive protocols and on random self-reducibility via low degree polynomials. They exhibit an interplay between Boolean circuit simulation, interactive proofs and classical complexity classes. An important feature of this proof is that it does not relativize.<<ETX>>

BPP has Subexponential Time Simulation unless EXPTIME has Pubishable Proofs.

This paper is concerned with three basic questions about sparse sets: (1) With respect to what types of reductions might NP have hard or complete sparse sets? (2) If a set A reduces to a sparse set, does it follow that A is reducible to some sparse set that is "simple" relative to A? (3) With respect to what types of reductions might NP have hard or complete sets of low instance complexity, and, relatedly, what is the structure of the class of sets with low instance complexity? .pp With respect to the first and third questions, intuitively one would expect that even with respect to flexible reductions NP is unlikely to have complete sets whose information content is low. With respect to the second question, one might intuitively feel that the structure imposed on a set by the fact that it reduces to a sparse set makes it plausible that we can indeed find a simple sparse set that can masquerade as the original sparse set. These two intuitions are in many ways certified by the current literature, and by the results of this paper.

Reductions to sets of low information content

Highly regular combinatorial objects can be represented advantageously by some kind of description shorter than their full standard encoding For instance, graphs exhibiting enough regularities can be described using encodings substantially shorter than the full adjacency matrix Anatural scheme for such succinct representations is by means of boolean circuits computing, as a boolean function, the values of individual bits of the binary encoding of the object The complexity of many algorithmic problems changes drastically when this succinct representation is used to present the input Two powerful lemmas quantifying exactly this increase of complexity are presented These are applied to show that previous results in the area can be interpreted assufficient conditions for completeness in the logarithmic time and polynomial time counting hierarchies

The complexity of algorithmic problems on succinct instances

The authors study one-word-decreasing self-reducible sets, which are the usual self-reducible sets with the peculiarity that the self-reducibility machine makes at most one query to a word lexicographically smaller than the input. It is first shown that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one-word-decreasing self-reducible. Using this fact it is proved that, for any class K chosen from a certain set of complexity classes, it holds that (1) if there is a sparse polynomial-time bounded-truth-table-hard set for K, then K=P, and (2) if there is a sparse strongly nondeterministic bounded-truth-table-hard set for K, then K contained in NP intersection co-NP. The main result also shows that the same facts hold for the class PSPACE. >

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On one query, self-reducible sets

The nonuniform complexity of the graph isomorphism (GI) and graph automorphism (GA) problems is studied, and the implications of different types of polynomial-time reducibilities from these problems to sparse sets are considered. It is shown that if GI (or GA) is bounded truth-table or conjunctively reducible to a sparse set, then it is in P; if it is supposed that it is truth-table reducible without restrictions to a sparse set (or, equivalently, that it belongs to P/poly), then the problem is low for MA, the class of sets with publishable proofs. With respect to nondeterministic reductions, it is shown that if the considered problems are btt strong nondeterministically reducible to a sparse set, then they are in co-NP. Some of these results are proved using graph constructions that show properties of the GI and GA problems. >

Antoni Lozano

Papers

Reductions to sets of low information content

The complexity of algorithmic problems on succinct instances

On one query, self-reducible sets

On the nonuniform complexity of the graph isomorphism problem

On the nonuniform complexity of the graph isomorphism problem