Existence of reduction hierarchies

Abstract

In the automata-theoretic approach to verification, we model programs and specifications by automata on infinite words. Correctness of a program with respect to a specification can then be reduced to the language-containment problem. In a concurrent setting, the program is typically a parallel composition of many coordinating processes, and the language-containment problem that corresponds to verification is

$$(\dag )\mathcal{L}(P_1 ) \cap \mathcal{L}(P_2 ) \cap ... \cap \mathcal{L}(P_n ) \subseteq \mathcal{L}(T),$$

where P ₁, P ₂, ..., P _n are automata that model the underlying coordinating processes, and T is the task they should perform. In 1994, Kurshan suggested the heuristic of Reduction Hierarchies for circumventing the exponential blow-up introduced by conventional methods that solve the problem (†). In the reduction-hierarchy heuristic, we solve the problem (†) by solving a sequence of easier problems, which involve only automata of tractable sizes. Complexity-theoretic conjectures (NP ≠ PSPACE) imply that there are settings in which the heuristic cannot circumvent the exponential blow-up. In this paper, we demonstrate the strength of the heuristic, study its properties, characterize settings in which it performs effectively, and suggest a method for searching for reduction hierarchies. In particular, we prove, independently of the NP ≠ PSPACE question, that reduction hierarchies of tractable sizes do not always exist.

Supported in part by the ONR YIP award N00014-95-1-0520, by the NSF CAREER award CCR-9501708, by the NSF grant CCR-9504469, by the AFOSR contract F49620-93-1-0056, by the ARO MURI grant DAAH-04-96-1-0341, by the ARPA grant NAG2-892, and by the SRC contract 95-DC-324.036.