An Algebraic Approach to Temporal Logic

. The sequential calculus is an algebraic calculus, intended for reasoning about phenomena with a duration and their sequencing. It can be specialized to various domains used for reasoning about programs and systems, including Tarski's calculus of binary relations, Kleene's regular expressions, Hoare's CSP and Dijkstra's regularity calculus. In this paper we use the sequential calculus as a tool for algebraizing temporal logics. We show that temporal operators are definable in terms of sequencing and we show how a specific logic may be selected by introducing additional axioms. All axioms of the complete proof system for discrete linear temporal logic (given in [9]) are obtained as theorems of sequential algebra. Our work embeds temporal logic into an algebra naturally equipped with sequencing constructs, and in which recursion is definable. This could be a first step towards a design calculus for transforming temporal specifications by stepwise refinement into executable programs. Thi..

Temporal Algebra

This paper is tutorial in style and there are no difficult technical results. To the experts in temporal logics, we hope to convey the simplicity and beauty of algebraic reasoning as opposed to the machine-orientedness of logical deduction. To those familiar with the calculational approach to programming, we want to show that their methods extend easily and smoothly to temporal reasoning. For anybody else, this text may serve as a gentle introduction to both areas. 1. Introductio

Formal Derivation of CSP Programs From Temporal Specifications

. The algebra of relations has been very successful for reasoning about possibly non-deterministic programs, provided their behaviour can be fully characterized by just their initial and final states. We use a slight generalization, called sequential algebra, to extend the scope of relation-algebraic methods to reactive systems, where the behaviour between initiation and termination is also important. To illustrate this approach, we integrate Communicating Sequential Processes and linear temporal logic in sequential algebra and show that the associated calculus permits the formal derivation of CSP programs from temporal specifications. 1 Introduction CSP is a process language for describing concurrent agents that cooperate via synchronous communication [12]. It is the conceptual core of the occam programming language. The theory of CSP and occam is anchored in denotational semantics [3, 21, 11], which has been used to establish algebraic identities between processes [14]. The collecti..

Towards a Design Calculus for CSP

The algebra of relations has been very successful for reasoning about possibly non-deterministic programs, provided their behaviour can be fully characterized by just their initial and final states. We use a slight generalization, called sequential algebra, to extend the scope of relation-algebraic methods to reactive systems, where the behaviour between initiation and termination is also important. The sequential calculus is clearly differentiated from the relational one by the absence of a general converse operation. As a consequence, the past can be distinguished from the future, so that we can define the temporal operators and mix them freely with executable programs. We use a subset of CSP in this paper, but the sequential calculus can also be instantiated to different theories of programming. In several examples we demonstrate the use of the calculus for specification, derivation and verification. 1 Introduction The main theme of this paper is the marriage of sequential and tempo..

A Relational Model for Temporal Logic

We use Tarski's relational calculus to construct a model of linear temporal logic. Both discrete and dense time are covered and we obtain denotational domains for a large variety of reactive systems. 1 Introduction The relational calculus has been very successful for modelling possibly nondeterministic systems provided they can be fully described in terms of their initial and final states. In contrast, the linear temporal logic (LTL) of Manna and Pnueli was designed for describing properties of reactive systems, where the behaviour between initiation and termination is also important. Unlike the calculus of relations, LTL has no operator for sequential composition. Therefore the standard model of LTL is unsuitable as a semantic domain for programs: LTL formulae describe properties of executions, but they are not themselves programs. Program constructs and temporal connectives cannot be mixed. This is very different from the relational approach, where programs are relations. The contri..