Algebraic simulations

A fruitful approach to the study of state-based systems consists in their mathematical formalization by means of models like Kripke structures, which allows to study their associated properties by means of simulations to relate them to other, possibly better-known systems [5, 7, 23]. This work tries to advance two main goals along those lines: the first, to generalize the notion of simulation between Kripke structures as much as possible, and the second, to provide general representability results showing that Kripke structures and generalized simulations can be represented in rewriting logic [28], an executable logical framework with good properties for representing many concurrent systems [28, 26]. These two goals are themselves motivated by pragmatic reasons. The reason for trying to advance the first goal is that simulations are essential for compositional reasoning. A cornerstone in such reasoning is the result that simulations reflect temporal logic properties, that is, if we have a simulation of Kripke structures H : A −→ B and a suitable temporal logic formula φ, then if aHb and B, b |= φ, we can conclude that A, a |= φ. Since this result is enormously powerful, there are strong reasons to generalize it: a more general notion of simulation will give it a wider applicability, even when the class of formulas φ for which it applies may have to be restricted. Advancing the second goal is also motivated by pragmatic reasons, namely: (i) executability, (ii) ease of specification, and (iii) ease of proof. The point about (i) and (ii) is that rewriting logic is a very flexible framework, so that concurrent systems can usually be specified quite easily and at a very high level; furthermore, such specifications can be used directly to execute a system, or to reason about it, which is point (iii). Indeed, both rewriting logic and its underlying equational logic can be very useful for formal reasoning, since often one needs to reason beyond the propositional level. For example, even when we use a model checker to prove that an infinite state system satisfies A, a |= φ by constructing a finite state abstraction simulation H : A −→ B and model checking that B, b |= φ for some b such that aHb, we are still left with verifying the correctness of H, which requires discharging proof obligations. More generally, any temporal logic deductive reasoning needs to include first-order and often inductive reasoning at the level of state predicates. This is precisely where rewriting and equational logics and their initial models supporting inductive reasoning are quite useful. In a previous paper [33] we have shown the usefulness of defining abstraction simulations equationally in rewriting logic, and of using tools such as Maude’s LTL model checker [18] and inductive theorem prover [12] to verify properties and prove abstractions correct. The conference paper [27] further generalized [33] by allowing not just the addition of equations E′ to a theory (Σ,E) for abstraction purposes, thus obtaining a subtheory inclusion (Σ,E) ⊆ (Σ′, E ∪E′),