Countable nondeterminism and random assignment

Reviewer: D. John Cooke

This is an impressive paper detailing work first presented at ICALP8 in 1981 [1]. It includes two main results: the nonexistence of a “reasonable” semantics to describe computations that exhibit countable nondeterminism, and that proofs of total correctness in such a system require all the recursive ordinals. However, much other useful (and, indeed, necessary) material is also included, making the paper essentially self-contained. After a preliminary introduction to domain theory, the paper addresses the central issue of countable nondeterminism in programs. This is approached by considering the definition of a small programming language that incorporates the notion of a nondeterministic assignment x := __?__, the application of which results in x being given an arbitrary value from a countable set. An informal discussion of the language is followed by formal definitions that effectively constitute a concise introduction to four semantic definition systems: an operational semantics, two denotational semantics (one based on Egli-Milner orderings, and the other on Smyth orderings), and a weakest precondition semantics. Even though several simplifications are adopted in order to ease the presentation and comparison of the various systems, this part of the paper forms a readable, if very concentrated, presentation of the four chosen systems. This in itself makes the paper of value to those wishing to widen their vocabulary of definitional systems. The equivalence of all four definitions is proved and the authors then consider what properties might be desirable for a “reasonable” semantics. They conclude that it should be a compositional, continuous, correct (defined via operational equivalence and arbitrary contexts), and complete least fixed-point semantics. Several examples are given to illustrate how apparently reasonable semantics fail on at least one of these criteria; this culminates in a proof that, for a language with a countably nondeterministic assignment, such a semantic does not exist. The topological significance of this result for other systems is briefly discussed. The latter part of the paper is concerned with proof theory and the theory of recursive functions. A Hoare-style logic is extended to cope with the countable nondeterminism by means of a generalized deduction rule for the while construct; the resulting proof system is shown to be sound and complete for partial correctness considerations. On the other hand, proofs of termination require exactly all the recursive ordinals, and, as is to be expected, a comprehensive development of the associated theory is included in substantiating this claim. In a paper of this length and intensity, it is almost inevitable that typographical errors will occur; I found some, but very few. Even though the results included herein were announced five years ago, this fuller and more refined presentation contributes significantly to the general development, greater understanding, and overall rationalization of the theory of computing science. The paper is to be highly recommended.