Countable nondeterminism and random assignment

Abstract

Abstract. Four semantics for a small programming language involving unbounded (but countable) nondeterminism are provided. These comprise an operational semantics, two state transformation semantics based on the Egli-Milner and Smyth orders, respectively, and a weakest precondition semantics. Their equivalence is proved. A Hoare-like proof system for total correctness is also introduced and its soundness and completeness in an appropriate sense are shown. Finally, the recursion theoretic complexity of the notions introduced is studied. Admission of countable nondeterminism results in a lack of continuity of various semantic functions, and this is shown to be necessary for any semantics satisfying appropriate conditions. In proofs of total correctness, one resorts to the use of (countable) ordinals, and it is shown that all recursive ordinals are needed.

Reasoning about Loops in Total and General Correctness

“…We are grateful to Walter Guttmann for feedback on the work-in-progress extended abstract which preceded this paper, to Roland Backhouse for directing us to Apt and Plotkin's Transfer Lemma in [2], and to the referees of the original review draft of this paper whose insightful comments we have endeavoured to address in this final version. The second author's research was supported by Australian Research Council (ARC) Discovery Grant DP0558408, Analysing and generating fault-tolerant real-time systems and the EPSRC-funded Trustworthy Ambient Systems (TrAmS) Platform Project.…”

Fairness in parallel programs: the transformational approach

“…and assign an arbitrary nonnegative integer to z [3]. await-statements S =await B then 81 end are used to achieve synchronization in the context of parallel composition.…”

“…In [24] a correctness proof of a parallel program S proceeds in two steps. First, one has to find appropriate correctness proofs for the sequential components of S. This is done using the proof rules of [24] extended by the following ones taken from [3] and [16]: The last two premises of the rule guarantee divergence freedom. Here however, owing to the presence of random assignments, it is, in general, not sufficient to let t be an integer expression and a an integer variable.…”

“…Here however, owing to the presence of random assignments, it is, in general, not sufficient to let t be an integer expression and a an integer variable. Instead, we shall need expressions involving infinite ordinals and variables ranging over infinite ordinals as in [1], [3], [7], and [21] (for details see the sample Siero below).…”

“…For example, if one component of the program Szero, say Sy, finds a zero of f, there are, upon termination, infinitely many values possible for the variable x of the other unsuccessful component Sx. It is well known that reasoning about unbounded nondeterminism leads to various complications-various semantic functions lose their continuity [ 11], and the standard technique of proving loop termination with integer-valued bound functions does not work any more (see, e.g., [3,21]). …”