Separation Logic with Linearly Compositional Inductive Predicates and Set Data Constraints

Abstract

We identify difference-bound set constraints (DBS), an analogy of difference-bound arithmetic constraints for sets. DBS can express not only set constraints but also arithmetic constraints over set elements. We integrate DBS into separation logic with linearly compositional inductive predicates, obtaining a logic thereof where set data constraints of linear data structures can be specified. We show that the satisfiability of this logic is decidable. A crucial step of the decision procedure is to compute the transitive closure of DBS-definable set relations, to capture which we propose an extension of quantified set constraints with Presburger Arithmetic (RQSPA). The satisfiability of RQSPA is then shown to be decidable by harnessing advanced automata-theoretic techniques.

Automated Reasoning with Restricted Intensional Sets

“…Tools such as SMT solvers deal with this kind of problems (see, e.g., [Deharbe et al 2011] and [Bjørner et al 2013]), although in general they are complete only in quite restricted cases [Ge and de Moura 2009]. Recently, a language admitting some forms of quantified formulas over sets was proven to be decidable in the context of separation logic [Gao et al 2019].…”