A fixpoint theory for non-monotonic parallelism
Theoretical Computer Science
This paper studies parallel recursion. The trace speciÿcation language used in this paper incorporates sequentially, nondeterminism, reactiveness (including inÿnite traces), three forms of parallelism (including conjunctive, fair-interleaving and synchronous parallelism) and general recursion. In order to use Tarski's theorem to determine the ÿxpoints of recursions, we need to identify a well-founded partial order. Several orders are considered, including a new order called the lexical order,
... ich tends to simulate the execution of a recursion in a similar manner as the Egli-Milner order. A theorem of this paper shows that no appropriate order exists for the language. Tarski's theorem alone is not enough to determine the ÿxpoints of parallel recursions. Instead of using Tarski's theorem directly, we reason about the ÿxpoints of terminating and nonterminating behaviours separately. Such reasoning is supported by the laws of a new composition called partition. We propose a ÿxpoint technique called the partitioned ÿxpoint, which is the least ÿxpoint of the nonterminating behaviours after the terminating behaviours reach their greatest ÿxpoint. The surprising result is that although a recursion may not be lexical-order monotonic, it must have the partitioned ÿxpoint, which is equal to the least lexical-order ÿxpoint. Since the partitioned ÿxpoint is well deÿned in any complete lattice, the results are applicable to various semantic models. Existing ÿxpoint techniques simply become special cases of the partitioned ÿxpoint. For example, an Egli-Milner-monotonic recursion has its least Egli-Milner ÿxpoint, which can be shown to be the same as the partitioned ÿxpoint. The new technique is more general than the least Egli-Milner ÿxpoint in that the partitioned ÿxpoint can be determined even when a recursion is not Egli-Milner monotonic. Examples of non-monotonic recursions are studied. Their partitioned ÿxpoints are shown to be consistent with our intuition.