Reasoning algebraically about loops

Deriving real-time action systems with multiple time bands using algebraic reasoning

“…The following lemmas allow one to fold/unfold infinite and finite iterations and to prove iterative definitions via induction [6,22,36]. For an interval predicate p, we let p + = p ; p * and p ω+ = p ; p ω denote the finite and possibly infinite positive iterations of p, respectively.…”

“…We let p V denote the framed interval predicate ∅: [p] V , i.e., a framed interval predicate with an empty frame. The abstract syntax of actions is given below, which we note differs from the standard syntax of Back et al [4,6,7].…”

“…Note that action A ∞ can be treated as a special case of A ω , namely A ω V A * A ∞ . The syntax of an action system consists of an initialisation followed by a potentially infinite loop that executes a non-deterministic choice over a set of guarded statements [6].…”

“…Back and von Wright present an algebraic definition that allows termination of the action system [6]. The methods in this paper apply to reactive systems, which we assume are nonterminating, but can be extended to reason about potentially terminating behaviour in a straightforward manner.…”

“…In this paper, we remove this bottleneck by defining a semantics for action systems using algebraic operators in the style of Back and von Wright [6]. However, unlike Back and von Wright, we address real-time issues by basing our semantics on an algebra for interval predicates [22,35,36].…”