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A parameterised Boolean equation system (PBES) is a set of equations thatdefines sets satisfying the equations as the least and/or greatestfixed-points. Thus this system is regarded as a declarative program definingpredicates, where a program execution returns whether a given ground atomicformula holds or not. The program execution corresponds to the membershipproblem of PBESs, which is however undecidable in general.  This paper proposes a subclass of PBESs which expresses universal-quantifiersfree formulas, and studies a technique to solve the problem on it. We use thefact that the membership problem is reduced to the problem whether a proofgraph exists. To check the latter problem, we introduce a so-called dependencyspace which is a graph containing all of the minimal proof graphs. Dependencyspaces are, however, infinite in general. Thus, we propose some conditions forequivalence relations to preserve the result of the membership problem, then weidentify two vertices as the same under the relation. In this sense, dependencyspaces possibly result in a finite graph. We show some examples having infinitedependency spaces which are reducible to finite graphs by equivalencerelations. We provide a procedure to construct finite dependency spaces andshow the soundness of the procedure. We also implement the procedure using anSMT solver and experiment on some examples including a downsized McCarthy 91function.

Reduced Dependency Spaces for Existential Parameterised Boolean Equation Systems