Jules Desharnais

Kleene algebra with domain

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra by simple equational axioms for a domain and a codomain operation. KAD considerably augments the expressiveness of Kleene algebra, in particular for the specification and analysis of programs and state transition systems. We develop the basic calculus, present the most interesting models and discuss some related theories. We demonstrate applicability by two examples: algebraic reconstructions of Noethericity and propositional Hoare logic based on equational reasoning.

Relational heuristics for the design of deterministic programs

SummaryIt has been shown that the design of deterministic programs can be formulated as the resolution of relational equations. Because relational calculi are not sufficiently structured, there are no algorithmic solutions to relational equations. In this paper, we formulate some heuristic solutions to these equations.

Integration of sequential scenarios

We give a formal relation-based definition of scenarios and we show how different scenarios can be integrated to obtain a more global view of user-system interactions. We restrict ourselves to the sequential case, meaning that we suppose that there is only one user (thus, the scenarios we wish to integrate cannot occur concurrently). Our view of scenarios is state-based, rather than event-based, like most of the other approaches, and can be grafted to the well-established specification language Z. Also, the end product of scenario integration, the specification of the functional aspects of the system, is given as a relation; this specification can be refined using independently developed methods. Our formal description is coupled with a diagram-based, transition-system like, presentation of scenarios, which is better suited to communication between clients and specifiers.

Embedding a demonic semilattice in a relation algebra

We present a refinement ordering between binary relations, viewed as programs or specifications. This ordering induces a complete join semilattice that can be embedded in a relation algebra. This embedding then allows an easy proof of many properties of the refinement semilattice, by making use of the well-known corresponding properties of relation algebras. The operations of the refinement semilattice corresponding to join and composition in the embedding algebra are, respectively, demonic join and demonic composition. The weakest prespecification and postspecification operators of Hoare and He, defined over a relation algebra, also have corresponding operators in the semilattice.

Defining and applying measures of distance between specifications

Echoing Louis Pasteurs quote, we submit the premise that it is advantageous to define measures of distance between requirements specifications because such measures open up a wide range of possibilities both in theory and in practice. The authors present a mathematical basis for measuring distances between specifications and show how their measures of distance can be used to address concrete problems that arise in the practice of software engineering.

Internal axioms for domain semirings

New axioms for domain operations on semirings and Kleene algebras are proposed. They generalise the relational notion of domain-the set of all states that are related to some other state-to a wide range of models. They are internal since the algebras of state spaces are induced by the domain axioms. They are simpler and conceptually more appealing than previous two-sorted external approaches in which the domain algebra is determined through typing. They lead to a simple and natural algebraic approach to modal logics based on equational reasoning. The axiomatisations have been developed in a new style of computer-enhanced mathematics by automated theorem proving, and the approach itself is suitable for automated systems analysis and verification. This is demonstrated by a fully automated proof of a modal correspondence result for Lobs formula that has applications in termination analysis.

Characterizing determinacy in Kleene algebras

Abstract Elements of Kleene algebras can be used, among other ways, as abstractions of the input–output semantics of nondeterministic programs or as models for the association of pointers with their target objects. One is interested in a notion of determinacy, in the first case, to distinguish deterministic programs and, in the second case, since it does not make sense for a pointer to point to two different objects. We discuss several candidate notions of determinacy and clarify their relationship. Some characterizations that are equivalent when the underlying Kleene algebra is an (abstract) relation algebra are not equivalent for general Kleene algebras.

Termination in Modal Kleene Algebra

Modal Kleene algebras (MKAs) are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. The paper formalizes and compares different notions of termination, including Lob’s formula, in MKA. It studies exhaustive iteration and gives calculational proofs of two fundamental termination-dependent statements from rewriting theory: the well-founded union theorem by Bachmair and Dershowitz and Newman’s lemma. These results are also of general interest for the termination analysis of programs and state transition systems.

Refinement and demonic semantics

In Chapter 8, it was shown how functional programs can be regarded as elements of a relation algebra. In this chapter, we consider imperative programs, which we view as computing an input-output relation on a set of states. We are interested here in programs that are meant to terminate, not in reactive programs. Our programming language is Dijkstra’s language of guarded commands [Dijkstra 1976], which allows the expression of nondeterminism, thus making a relational approach very natural.

Kleene under a Demonic Star

In relational semantics, the input-output semantics of a program is a relation on its set of states. We generalize this in considering elements of Kleene algebras as semantical values. In a nondeterministic context, the demonic semantics is calculated by considering the worst behavior of the program. In this paper, we concentrate on while loops. Calculating the semantics of a loop is diffcult, but showing the correctness of any candidate abstraction is much easier. For deterministic programs, Mills has described a checking method known as the while statement verification rule. A corresponding programming theorem for nondeterministic iterative constructs is proposed, proved and applied to an example. This theorem can be considered as a generalization of the while statement verification rule to nondeterministic loops.