We consider forkable regular expressions, which enrich regular expressions with a fork operator, to establish a formal basis for static and dynamic analysis of the communication behavior of concurrent programs. We define a novel compositional semantics for forkable expressions, establish their fundamental properties, and define derivatives for them as a basis for the generation of automata, for matching, and for language containment tests. Forkable expressions may give rise to non-regular languages, in general, but we identify sufficient conditions on expressions that guarantee finiteness of the automata construction via derivatives.
The latter connection is made precise by Garg and Ragunath @cite_6 , who study concurrent regular expressions (CRE), which are shuffle expressions extended with synchronous composition. They show that the class of CRE languages is equal to the class of Petri net languages. The proof requires the presence of synchronous composition. Forkable expressions do not support synchronous composition, but they are equivalent to , which are also defined by Garg and Ragunath and shown to be strictly less powerful than CREs.
@cite_6
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Abstract We define algebraic systems called concurrent regular expressions which provide a modular description of languages of Petri nets. Concurrent regular expressions are extension of regular expressions with four operators - interleaving, interleaving closure, synchronous composition and renaming. This alternative characterization of Petri net languages gives us a flexible way of specifying concurrent systems. Concurrent regular expressions are modular and, hence, easier to use for specification. The proof of equivalence also provides a natural decomposition method for Petri nets. We propose the model of nested words for representation of data with both a linear ordering and a hierarchically nested matching of items. Examples of data with such dual linear-hierarchical structure include executions of structured programs, annotated linguistic data, and HTML XML documents. Nested words generalize both words and ordered trees, and allow both word and tree operations. We define nested word automata—finite-state acceptors for nested words, and show that the resulting class of regular languages of nested words has all the appealing theoretical properties that the classical regular word languages enjoys: deterministic nested word automata are as expressive as their nondeterministic counterparts; the class is closed under union, intersection, complementation, concatenation, Kleene-*, prefixes, and language homomorphisms; membership, emptiness, language inclusion, and language equivalence are all decidable; and definability in monadic second order logic corresponds exactly to finite-state recognizability. We also consider regular languages of infinite nested words and show that the closure properties, MSO-characterization, and decidability of decision problems carry over. The linear encodings of nested words give the class of visibly pushdown languages of words, and this class lies between balanced languages and deterministic context-free languages. We argue that for algorithmic verification of structured programs, instead of viewing the program as a context-free language over words, one should view it as a regular language of nested words (or equivalently, a visibly pushdown language), and this would allow model checking of many properties (such as stack inspection, pre-post conditions) that are not expressible in existing specification logics. We also study the relationship between ordered trees and nested words, and the corresponding automata: while the analysis complexity of nested word automata is the same as that of classical tree automata, they combine both bottom-up and top-down traversals, and enjoy expressiveness and succinctness benefits over tree automata. Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schutzenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmanski to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semiring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schutzenberger's and Ochmanski's theorems. Abstract A hallmark of the Esterel language is the combination of perfect synchrony with total orthogonality and powerful constructs for preemption, suspension and trap handling. It is desirable to make this kind of expressiveness available for the description of hybrid systems, that is, systems whose evolution is understood in terms of segment-wise continuous functions over the real time axis. Our approach consists of modifying Esterel concepts, most notably by replacing the discrete time frame by a continuously advancing one. We are then able to state a semantics made up of transitions with closed execution intervals of non-zero length. By an instant we understand an execution interval within this framework. Hybrid signals may change their value during such an instant, non-hybrid ones, that is, classical signals immediately settle to a specific state and keep it the whole time. Time consumption still has to be specified explicitly, namely in that instants reflect jumps among control flow locations defined by pause statements; all other statements take no time in the sense that arbitrarily many of them may be sequentially executed regardless of the instant's duration. A transfer of perfect synchrony from the discrete to the continuous is in this way accomplished. We also consider an example, which is from the automotive domain, traces, bisimilarity and compositionality.