No abstract

A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fixed set C of Boolean functions. We consider the problem of determining whether two given constraint satisfaction instances are equivalent and prove a Dichotomy Theorem by showing that for all sets C of allowed constraints, this problem is either polynomial-time solvable or coNPcomplete, and we give a simple criterion to determine which case holds. A more general problem addressed in this paper is the isomorphism problem, the problem of determining whether there exists a renaming of the variables that makes two given constraint satisfaction instances equivalent in the above sense. We prove that this problem is coNP-hard if the corresponding equivalence problem is coNP-hard, and polynomial-time many-one reducible to the graph isomorphism problem in all other cases.

We introduce the probabilistic class SBP. This class emerges from BPP by keeping the promise of a probability gap but decreasing the probability limit from 1/2 to exponentially small values. We show that SBP is in the polynomial-time hierarchy, between MA and AM on the one hand and between BPP and BPP path on the other hand. We provide evidence that SBP does not coincide with these and other known complexity classes. In particular, in a suitable relativized world SBP is not contained in Σ P 2 . In the same world, BPP path is not contained in Σ P 2 , which solves an open question raised by Han, Hemaspaandra, and Thierauf. We study the question of whether SBP has many-one complete sets. We relate this question to the existence of uniform enumerations and construct an oracle relative to which SBP and AM do not have many-one complete sets. We introduce the operator SB· and prove that, for any class C with certain properties, BP · ∃ · C contains every class defined by applying an operator sequence over {U·, ∃·, BP·, SB·} to C.

Abstract. Any Boolean function can be defined by a Boolean circuit, provided we may use sufficiently strong functions in its gates. On the other hand, it depends on these gate functions, what Boolean functions can be defined: Each set B of gate functions defines the class of Boolean functions that can be defined by circuits over B. Although these classes are known since the 1920s, their computational complexity was never investigated. In this paper we will study how difficult it is to decide for a Boolean function f and a class B, whether f is in B. Moreover we will provide such a decision algorithm with additional information: How difficult is it to decide whether or not f is in B, provided we already know a circuit for f , but with gates from another class A? Given such a circuit, we know that f is in A. Is the problem harder if we do not have a concrete representation for f , but still know that it is from A? For nearly all possible combinations, we show that this is not the case,and that the problem is either in P or coNP-complete.

Abstract. In this paper we will look at restricted versions of the evaluation problem, the model checking problem, the equivalence problem, and the counting problem for quantified propositional formulas, both with and without bound on the number of quantifier alternations. The restrictions are such that we consider formulas in conjunctive normal-form with restricted types of clauses (e.g., positive, Horn, linear, etc.). For each of these algorithmic goals we will obtain full complexity classifications, exhibiting on the one hand severe syntactic restrictions of the original problems that are still computationally hard, and on the other hand non-trivial subcases that admit efficient solution algorithms. Generalizing these results to non Boolean domains, we obtain a number of hardnes results for quantified constraints over arbitrary finite universes.