Reduced Dependency Spaces for Existential Parameterised Boolean Equation Systems
نویسندگان
چکیده
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. Thus this system is regarded as a declarative program defining predicates, where a program execution returns whether a given ground atomic formula holds or not. The program execution corresponds to the membership problem of PBESs, which is however undecidable in general. This paper proposes a subclass of PBESs which expresses universal-quantifiers free formulas, and studies a technique to solve the problem on it. We use the fact that the membership problem is reduced to the problem whether a proof graph exists. To check the latter problem, we introduce a socalled dependency space which is a graph containing all of the minimal 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 infinite dependency spaces which are reducible to finite graphs by equivalence relations. 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 some examples including a downsized McCarthy 91 function.
منابع مشابه
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 ...
متن کاملAn Extension of Proof Graphs for Disjunctive Parameterised Boolean Equation Systems
A parameterised Boolean equation system (PBES) is a set of equations that defines sets as the least and/or greatest fixed-points that satisfy the equations. 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 correspond...
متن کاملEfficient Instantiation of Parameterised Boolean Equation Systems to Parity Games
Parameterised Boolean Equation Systems (PBESs) are sequences of Boolean fixed point equations with data variables, used for, e.g., verification of modal μ-calculus formulae for process algebraic specifications with data. Solving a PBES is usually done by instantiation to a Parity Game and then solving the game. Practical game solvers exist, but the instantiation step is the bottleneck. We enhan...
متن کاملParameterised boolean equation systems
Boolean equation system are a useful tool for verifying formulas from modal mu-calculus on transition systems (see [18] for an excellent treatment). We are interested in an extension of boolean equation systems with data. This allows to formulate and prove a substantially wider range of properties on much larger and even infinite state systems. In previous works [11, 15] it has been outlined ho...
متن کاملA Abstraction in Fixpoint Logic
ion in Fixpoint Logic SJOERD CRANEN, MACIEJ GAZDA, WIEGER WESSELINK and TIM A.C. WILLEMSE, Eindhoven University of Technology We present a theory of abstraction for the framework of parameterised Boolean equation systems, a firstorder fixpoint logic. Parameterised Boolean equation systems can be used to solve a variety of problems in verification. We study the capabilities of the abstraction th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1802.06496 شماره
صفحات -
تاریخ انتشار 2018