Linear time and branching time semantics for recursion with merge

Abstract

Abstract. We consider two ways of assigning semantics to a class of statements built from a set of atomic actions (the 'alphabet'), by means of sequential composition, nondeterministic choice, recursion and merge (arbitrary interleaving). The first is linear time semantics (LT), stated in terms of trace theory; the semantic domain is the collection of all closed sets of finite and infinite words. The second is branching time semantics (BT), as introduced by De Bakker and Zucker; here the semantic domain is the metric completion of the collection of finite processes. For LT we prove the continuity of the operations (merge, sequential composition) in a direct, combinatorial way.Next, a connection between LT and BT is established by means of the operation trace which assigns to a process its set of traces. We show that the trace set of a process is closed and that trace is continuous. This requires the compactness of the semantic domains, ensured by the finiteness of the alphabet. Using trace, we then can carry over BT into LT.

Synchronous cooperation for explicit multi-threading

“…The following is a useful corollary from the proof of Theorem 1. This corollary implies that each closed term from T TA sc in which all subterms of the form p nt(r ) q occur in a subterm of the form s ( p 1 . .…”

“…Axiom SCf5 expresses that, in the case where some threads in the thread vector can fork off a thread, forking off threads takes place such that the threads forking off a thread and the threads being forked off keep pace with the other threads in the thread vector. The crucial axiom for reply conditionals 1 We write for the empty sequence, d for the sequence having d as sole element, and α β for the concatenation of finite sequences α and β. We assume the usual laws for concatenation of finite sequences.…”

Designing equivalent semantic models for process creation

“…(2) A denotational semantics based on a cpo structure on (certain) sets of so-called finite observations equipped with the order of reverse set inclusion [12,14]. (3) A branching time denotational semantics based on a process domain of the kind described in Section 2.3 [ 11]. The equivalence of the models in (1) and (2) has been established in [14], the equivalence of the model in (1) and the denotational metric model is proved in [13], and the relationship between the branching time model and (any of) the linear time models is settled in [ 11 ].…”

“…In Sections 3 and 4, the languages are uniform and the semantic models are of the so-called "linear time" variety (see, e.g., [11] or [ 40]), i.e., they consist of sets of (finite or infinite) sequences over a certain alphabet. The operational semantics is a uniform version of the Structured Operational Semantics (SOS) of Hennessy and Plotkin [29,38,39].…”

“…The mathematical structures used to model 2nus and 2nud are Plotkin's resumptions [37], presented in a fully metric framework as first described in [17] and subsequently extended and put in a category-theoretic perspective in [8]. We use the terminology of process domains P, satisfying certain (reflexive) domain equations of the form P== ;J/P(P) and we shall design the semantics of programs in 2nus and !tnud such that the meaning of a program is a process p E P. Processes are objects which have a branching structure, and the models for !tnus and !tnud are called branching time [11,40].…”