Reduced dependency spaces for existential parameterised

A parameterised Boolean equation system (PBES) is a set of equations that defines sets satisfying the equations as the least and/or greatest fixed-points. This system is regarded as a declarative program defining functions that take a datum and returns a Boolean value. The membership problem of PBESs is a problem to decide whether a given element is in the defined set or not, which corresponds to an execution of the program. It is known that the problem, is undecidable in general, is reduced to the existence of proof graphs. This paper proposes a subclass of PBESs which expresses ∀-quantifiers free formulas, and studies a technique to solve the problem on it. To check the existence of a proof graph, we introduce a dependency space which is a graph containing all of the proof graphs. Dependency spaces are, however, infinite in general. Thus, we propose some conditions for equivalence relations to preserve the result of the membership problem, then we identify two vertices as the same under the relation. In this sense, dependency spaces possibly result in a finite graph. We show some examples having no finite dependency space except for reduced one. We provide a procedure to construct finite dependency spaces and show the soundness of the procedure. We also implement the procedure using an SMT solver and experiment on a downsized McCarthy 91 function.