Probabilistic Choice, Reversibility, Loops, and Miracles

We consider an addition of probabilistic choice to Abrial’s Generalised Substitution Language (GSL) in a form that accommodates the backtracking interpretation of nondeterministic choice. Our formulation is introduced as an extension of the Prospective Values formalism we have developed to describe the results from a backtracking search. Significant features are that probabilistic choice is governed by feasibility, and nontermination is strict. The former property allows us to use probabilistic choice to generate search heuristics. In this paper we are particularly interested in iteration. By demonstrating sub-conjunctivity and monotonicity properties of expectations we give the basis for a fixed point semantics of iterative constructs, and we consider the practical proof treatment of probabilistic loops. We discuss loop invariants, loops with probabilistic behaviour, and probabilistic termination in the context of our formalism, which is strict with respect to nontermination. The formal constructs described are incorporated in a reversal virtual machine (RVM).