A unification of probabilistic choice within a design-based model of reversible computation

We see reversible computing as a generalisation of sequential computation obtained by revoking the law of the excluded miracle. Our execution language includes naked guarded commands and non-deterministic choice. Choices which lead to miraculous continuations invoke reverse computation, and non-deterministic choice plays the rôle of provisional choice within a backtracking context. We require probabilistic choice for symmetry breaking and sampling large search spaces, but must formulate it differently from previous approaches to obtain the required interactions between probabilistic choice and non-deterministic choice and between probabilistic choice and feasibility. Our formulation allows us to derive the post-distributions which characterise a program, and we use these to construct a relational model. We consider refinement as containment of convex closures within distribution space, qualified with additional conditions to avoid over-refinement. We link the non-probabilistic and probabilistic versions of the model with a Galois connection and show that classical designs are a retract of our probabilistic designs. We consider the interaction between probabilistic and non-deterministic choice and find the same initially counter-intuitive results that have been noted by other investigators. We provide an alternative formulation, within the same model, of oblivious non-determinism, which allows all non-deterministic choices to be moved to the start of a computation. We consider the interaction between probabilistic choice and feasibility that is required to match an operational interpretation in which infeasible commands provoke reverse execution, and we present a small case study to show how the interaction between probabilistic choice and feasibility can be exploited in a practical program. All programming structures described here are supported by our implementation platform, the Reversible Virtual Machine, whose development has accompanied our theoretical investigations

Calculational Verification of Reactive Programs with Reactive Relations and Kleene Algebra

Reactive programs are ubiquitous in modern applications, and so verification is highly desirable. We present a verification strategy for reactive programs with a large or infinite state space utilising algebraic laws for reactive relations. We define novel operators to characterise interactions and state updates, and an associated equational theory. With this we can calculate a reactive program’s denotational semantics, and thereby facilitate automated proof. Of note is our reasoning support for iterative programs with reactive invariants, which is supported by Kleene algebra. We illustrate our strategy by verifying a reactive buffer. Our laws and strategy are mechanised in Isabelle/UTP, which provides soundness guarantees, and practical verification support

Predicative Semantics of Loops

A predicative semantics is a mapping of programs to predicates. These predicates characterize sets of acceptable observations. The presence of time in the observations makes the obvious weakest fixed-point semantics of iterative constructs unacceptable. This paper proposes an alternative. We will see that this alternative semantics is monotone and implementable (feasible). Finally a programming theorem for iterative constructs is proposed, proved, and demonstrated. A novel aspect of this theorem is that it is not based on invariants. Keywords Predicative semantics, fixedpoint semantics, recursion, loops, refinement calculi. 0 FORMALIZATION 0.0 Specifications and refinement Define xnat as the set of all natural numbers (nat) joined with an additional object 1. We will suppose the following properties of 1: it is larger than any natural number; 1 + i = 1 \Gamma i = 1; for all natural numbers i; and 1 \Gamma 1 = 0. I will use a `batch' model for specifications borrowed, in most res..

Semantics of quasi-boolean expressions.

No abstract

Logic Group Preprint Series

We provide rules for calculating with invariants in process algebra with data, and illustrate these with examples. The new rules turn out to be equivalent to the well known Recursive Specification Principle which states that guarded recursive equations have at most one solution. In the setting with data this is reformulated as `every convergent linear process operator has at most one fixed point ' (CL-RSP). As a consequence, one can carry out verifications in well-known process algebras satisfying CL-RSP using invariants