This paper studies parallel recursions. The trace specification language used in this paper incorporates sequentiality, nondeter-minism, reactiveness (including infinite traces), conjunctive parallelism and general recursion. The language is the minimum of its kind and thus provides a context in which we can study parallel recursions in general. In order to use Tarski's theorem to determine the fixpoints of recursions, we need to identify a well-founded partial order. A theorem of this paper shows that no appropriate order exists. Tarski's theorem alone is not enough to determine the fixpoints of parallel recursions. Instead of using Tarski's theorem directly, we reason about the fixpoints of terminating and nonterminating behaviours separately. Such reasoning is supported by the laws of a new composition called partition. We propose a fixpoint technique called the partitioned fixpoint, which is the least fixpoint of the nonterminating behaviours after the terminating behaviours reach their greatest fixpoint. The surprising result is that although a recursion may not be monotonic with regard to the lexical order, it must have the partitioned fixpoint, which equals the least lexical-order fixpoint. Since the partitioned fixpoint is well defined in any complete lattice, the results are applicable to various semantic models. Major existing fixpoint techniques simply become special cases of the partitioned fixpoint. For example, an Egli-Milner-monotonic recursion has its least Egli-Milner fixpoint, which can be shown to be the same as the partitioned fixpoint. The new technique is more general than the least Egli-Milner fixpoint in that the partitioned fixpoint can be determined even when a recursion is not Egli-Milner monotonic. Examples of non-monotonic recursions with fair-interleaving parallelism are studied. Their partitioned fixpoints are shown to be consistent with our intuitions.
展开▼
