Theoroidal Maps as Algebraic Simulations

Abstract

Abstract. Computational systems are often represented by means of Kripke structures, and related using simulations. We propose rewriting logic as a flexible and executable framework in which to formally specify these mathematical models, and introduce a particular and elegant way of representing simulations in it: theoroidal maps. A categorical viewpoint is very natural in the study of these structures and we show how to organize Kripke structures in categories that afterwards are lifted to the rewriting logic's level. We illustrate the use of theoroidal maps with two applications: predicate abstraction and the study of fairness constraints.

A Categorical Approach to Simulations

“…In previous papers [13,11] we have studied the suitability of different kinds of simulations between transition systems and Kripke structures for the study of the relationships between formal models of concurrent systems. The range of available notions of simulations makes it very natural to adopt a categorical viewpoint in which Kripke structures become the objects of several categories while the morphisms are obtained from the corresponding notion of simulation.…”

“…On the one hand, we would like to finally prove or disprove the existence of limits in the Grothendieck categories. On the other hand, as briefly discussed in [11], rewriting logic theories representing Kripke structures can also be organized in categories: we plan to organize them in an institution and to study its relationship with I K .…”

“…Other institutions for temporal logics are discussed in [1], but their notions of signature morphism and of simulation are more restricted. Some of the ideas in this section were presented in [11]: we also include them here for the sake of completeness.…”

“…That is, we consider increasingly more general categories whose objects are Kripke structures, and whose morphisms are adequate simulations between them. There are several orthogonal dimensions along which simulations can be generalized as discussed in detail in [13,11]. We can extend them: (1) from functions to relations; (2) from strictly preserving state predicates to only doing so in a looser way; (3) from simulations in which one step is simulated by another to "stuttering" simulations in which several steps in one system can correspond to several steps in the other; and (4) from the case in which all systems we relate share the same set AP of atomic predicates to one in which systems with different atomic predicates can be related among each other.…”

Symbolic Model Checking of Infinite-State Systems Using Narrowing

“…In particular, it is related to, and complements, abstraction techniques for rewrite theories such as [39,33,23]. In fact, all the simulations we propose, especially the ones involving folding, can be viewed as suitable abstractions.…”