problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P ̸ = NP). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.

Abstract. In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post’s lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P. 2000 ACM Subject Classification:

vollmerATthi.uni-hannover.de Abstract. In a seminal paper from 1985, Sistla and Clarke showed that the model-checking problem for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper systematically studies the model-checking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the model-checking problem is tractable (in P) or intractable (NP-hard). We then focus on the tractable cases, showing that they all are NL-complete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators. 1

Ekin, Foldes, Hammer, and Hellerstein showed that a set of Boolean functions is characterizable by a (possibly infinite) set of Boolean equations iff it is closed under permutation of variables, addition of dummy variables, and identification of variables. Subsequently, Pippenger introduced constraint characterizations, which generalize previous characterizations of clones. He showed that a set of finite functions can be characterized by a set of constraints iff it is characterizable by Boolean equations. He also developed a Galois theory for sets of finite functions characterizable by constraints. In this paper we generalize Pippenger's results by considering sets of finite functions, such as the unate functions, that are closed under permutation of variables and addition of dummy variables but not necessarily under identification of variables. We introduce the notion of a generalized constraint, and consider the question of which sets of functions can be characterized by...

The H-Coloring problem can be expressed as a particular case of the Constraint Satisfaction Problem (CSP) whose computational complexity has been intensively studied under various approaches inthe last several years. We show that the dichotomy theorem proved by P.Hell and J.Nesetril [12] for the complexity of the H-Coloring problem for undirected graphs can be obtained using general methods for studying CSP, and that the criterion distinguishing the tractablecases of the H-Coloring problem agrees with that conjectured in [5]for the complexity of the general CSP.

Abstract. In this paper we consider the complexity of several problems involving finite algebraic structures. Given finite algebras A and B, these problems ask the following. (1) Do A and B satisfy precisely the same identities? (2) Do they satisfy the same quasi-identities? (3) Do A and B have the same set of term operations? In addition to the general case in which we allow arbitrary (finite) algebras, we consider each of these problems under the restrictions that all operations are unary and that A and B have cardinality two. We briefly discuss the relationship of these problems to algebraic specification theory.

Abstract. Reiter’s default logic formalizes nonmonotonic reasoning using default assumptions. The semantics of a given instance of default logic is based on a fixpoint equation defining an extension. Three different reasoning problems arise in the context of default logic, namely the existence of an extension, the presence of a given formula in an extension, and the occurrence of a formula in all extensions. Since the end of 1980s, several complexity results have been published concerning these default reasoning problems for different syntactic classes of formulas. We derive in this paper a complete classification of default logic reasoning problems by means of universal algebra tools using Post’s clone lattice. In particular we prove a trichotomy theorem for the existence of an extension, classifying this problem to be either polynomial, NP-complete, or Σ2P-complete, depending on the set of underlying Boolean connectives. We also prove similar trichotomy theorems for the two other algorithmic problems in connection with default logic reasoning. 1

Abstract. We introduce a new general polynomial-time constructionthe fibre construction- which reduces any constraint satisfaction problem CSP(H) to the constraint satisfaction problem CSP(P), where P is any subprojective relational system. As a consequence we get a new proof (not using universal algebra) that CSP(P) is NP-complete for any subprojective (and thus also projective) relational system. The fibre construction allows us to prove the NP-completeness part of the conjectured Dichotomy Classification of CSPs, previously obtained by algebraic methods. We show that this conjectured Dichotomy Classification is equivalent to the dichotomy of whether or not the template is subprojective. This approach is flexible enough to yield NP-completeness of coloring problems with large girth and bounded degree restrictions thus reducing the Feder-Hell-Huang and Kostočka-Neˇsetˇril-Smolíková problems to the Dichotomy Classification of coloring problems. 1. Introduction and Previous

In this paper, we propose a computational classification of finite characteristic numbers

The class composition C ◦ K of Boolean clones, being the set of composite functions f(g1,..., gn) with f ∈ C, g1,..., gn ∈ K, is investigated. This composition C ◦ K is either the join C ∨ K in the Post Lattice or it is not a clone, and all pairs of clones C, K are classified accordingly. Factorizations of the clone Ω of all Boolean functions as a composition of minimal clones are described and seen to correspond to normal form representations of Boolean functions. The median normal form, arising from the factorization of Ω with the clone SM of self-dual monotone functions as the leftmost composition factor, is compared in terms of complexity with the well-known DNF, CNF, and Zhegalkin (Reed–Muller) polynomial representations, and it is shown to provide a more efficient normal form representation. 1 Introduction and