Feedback is an essential composition operator in many classes of reactive and other systems. This paper studies feedback in the context of compositional theories with refinement. Such theories allow to reason about systems on a component-by-component basis, and to characterize substitutability as a refinement relation. Although compositional theories of feedback do exist, they are limited either to deterministic systems (functions) or input-receptive systems (total relations). In this work we propose a compositional theory of feedback which applies to non-deterministic and non-input-receptive systems (e.g., partial relations). To achieve this, we use the semantic frameworks of predicate and property transformers, and relations with fail and unknown values. We show how to define instantaneous feedback for stateless systems and feedback with unit delay for stateful systems. Both operations preserve the refinement relation, and both can be applied to non-deterministic and non-input-receptive systems.

Refinement calculus is a powerful and expressive tool for reasoning about sequential programs in a compositional manner. In this paper we present an extension of refinement calculus for reactive systems. Refinement calculus is based on monotonic predicate transformers, which transform sets of post-states into sets of pre-states. To model reactive systems, we introduce monotonic property transformers, which transform sets of output infinite sequences into sets of input infinite sequences. We show how to model in this semantics refinement, sequential composition, demonic choice, and other semantic properties of reactive systems. We also show how such transformers can be defined by various formalisms such as linear temporal logic formulas (suitable for specifications) and symbolic transition systems (suitable for implementations). Finally, we show how this framework generalizes previous work on relational interfaces to systems with infinite behaviors and liveness properties.

Abstract Invariant based programming is an approach where we start to construct a program by first identifying the basic situations (pre- and post-conditions as well as invariants) that could arise during the execution of the algorithm. These situations are identified before any code is written. After that, we identify the transitions between the situations, which will give us the flow of control in the program. Data refinement is a technique of building correct programs working on concrete data structures as refinements of more abstract programs working on abstract data types. We study in this paper data refinement for invariant based programs and we apply it to the construction of the classical Deutsch–Schorr–Waite graph marking algorithm. Our results are formalized and mechanically proved in the Isabelle/HOL theorem prover.

Abstract. We introduce a new algebraic model for program variables, suitable for reasoning about recursive procedures with parameters and local variables in a mechanical verification setting. We give a predicate transformer semantics to recursive procedures and prove refinement rules for introducing recursive procedure calls, procedure parameters, and local variables. We also prove, based on the refinement rules, Hoare total correctness rules for recursive procedures, and parameters. We introduce a special form of Hoare specification statement which alone is enough to fully specify a procedure. Moreover, we prove that this Hoare specification statement is equivalent to a refinement specification. We implemented this theory in the PVS theorem prover.

Pointers are an important programming concept. They are used explicitly or implicitly in many programming languages. In particular, the semantics of object-oriented programming languages rely on pointers. We introduce a semantics for pointer structures. Pointers are seen as indexes and pointer fields are functions from these indexes to values. Using this semantics we turn all pointer operations into simple assignments and then we use refinement calculus techniques to construct a pointer-manipulating program that checks whether or not a single linked list has a loop. We also introduce an induction principle on pointer structures in order to reduce complexity of the proofs.

Using a categorical model of abstract data types [2, 3, 13, 15], we show, following the Kozen's technique [10] and Tarjan's constructions for a deterministic automaton [16], that if two unambiguous regular expressions define the same regular language, then they represent two isomorphic abstract data types.