Toshiro Araki

Some decision problems related to the reachability problem for Petri nets

Abstract In this paper, we show that (1) the question to decide whether a given Petri net is consistent, Mo-reversible or live is reduced to the reachability problem in a unified manner, (2) the reachability problem for Petri nets is equivalent to the equality problem and the inclusion problem for the sets of all firing sequences of two Petri nets, (3) the equality problem for the sets of firing sequences of two Petri nets with only two unbounded places under homomorphism is undecidable, (4) the coverability and reachability problems are undecidable for generalized Petri nets in which a distinguished transition has priority over the other transitions, and (5) the reachability problem is undecidable for generalized Petri nets in which some transitions can reset a certain place to zero marking.

Decidable problems on the strong connectivity of Petri net reachability sets

Abstract For a Petri net N and a marking M, let RN(M) be the set of markings reachable from M and let CN(M) be the set of markings M′ such that M′ ϵ RN(M) and M ϵ RN(M′)CN(M) is a strongly connected component of RN(M) to which M belongs. If RN(M) = CN (M), then N is said to be M-reversible, and if N is M-reversible for every marking M, then N is said to be reversible. In this paper the following results are presented. (1) CN(M) is semilinear and therefore it is decidable whether (i) M′ ϵ CN(M), (ii) CN(M) is a finite set, (iii) CN(M) ⊆ CN′(M′), and (iv) given two markings M and M′ such that M ⩾ M′, there is a nonnegative integer k such that M + k(M − M′) ϵ RN(M′). (2) It is decidable whether (i) N is M-reversible or not, and (ii) N is reversible or not. (3) Given a Petri net N and a marking M, we can construct an M-reversible Petri net N′ such that CN(M) = RN′(M). (4) The equality problem for the sets of all firing sequences of an M-reversible Petri net N and an M′-reversible Petri net N′ is decidable. And some related problems are discussed.

Flow languages equal recursively enumerable languages

SummaryRecently, A.C. Shaw introduced a new class of expressions called flow expressions, and conjectured that the formal descriptive power of flow expressions lies somewhat below context-sensitive grammers. In this paper, we give a negative answer for his conjecture, that is, we show that all recursively enumerable languages may be denoted by flow expressions.

Relations of flow languages to Petri net languages

Abstract A.C. Shaw has introduced the concept of the flow expression which is an extended regular expression and state several results with some conjectures. His conjecture that the flow expressions define context-sensitive languages has been answered negatively by two of the present authors, that is, the flow languages are recursively enumerable. In this paper, it is shown that flow languages for some restricted class of flow expressions are equivalent to Petri net languages. Some related decision problems for flow expressions are also discussed.

Concurrent C: A programming language for distributed multiprocessor systems

Recent advances in hardware technology have made the construction of multiprocessor systems economically feasible. This paper describes a new programming language (Concurrent C) suitable for distributed systems which are networks of loosely connected processors, each with its own local storage. Concurrent C is the extended version of the programming language C, incorporating features for parallel processing and interprocess communications.

Descriptive power of synchronized shuffle expressions

As a model to describe the concurrent system with a synchronization mechanism, Holenderski has proposed the synchronized shuffle expression (SSE). SSE is a shuffle expression obtained by adding a synchronized shuffle operator with its closure and projection operator. This paper considers various subclasses of SSE. It is shown that the class of languages represented by the subclass excluding the projection operator is the same as the class of recursively enumerable languages. The hierarchical relations among the subclasses are described.

Descriptive powers of synchronized shuffle grammars and synchronized production systems

As models to describe the concurrent systems with synchronized mechanism, synchronized shuffle grammar and synchronized production system (SPS) have been proposed. The synchronized shuffle grammar is defined by the triplet composed of a shuffle grammar G, synchronizing language LC and a homeomorphism h. The language generated by is defined as L(G) = h (L(G) LC). As a subclass of the synchronized shuffle grammar, this paper considers a class syncRSG (sync ERSG), where G is an (extended) regular shuffle grammar and Lc is an arbitrary regular language. SPS can also be defined as a pair of extended context-free grammar and a state-transition machine. This paper considers as its subclass, the context-free SPS which is a pair of a context-free grammar and a finite automaton. It is shown that there exist some subclass.es in syncRSG (sync ERSG) and context-free SPS, which generate the same class of formal languages. It is also shown that there exists a certain hierarchical relation, and the full descriptive powers of syncRSG (syncERSG) and context-free SPS are the equivalent to that of Petri net.

Real‐time shuffle stack automaton

As models to describe concurrent systems, Petri net, shuffle stack automaton (SSA), flow expression, event expression, and shuffle grammar have been proposed. SSA is an automaton which is obtained by adding to the finite automaton a stack with shuffle function. Its descriptive power has already been discussed. This paper discusses the real-time shuffle stack automaton (RSSA, which is an SSA without λ-transition in regard to the input), which is a subclass of SSA. It is shown that RSSA has the same descriptive power, independently of the accepting mode (empty-stack acceptance or final-state acceptance), and of whether or not a terminating symbol is provided at the end of the input string. It is also shown that the descriptive power of RSSA is incomparable to that of the push-down automaton, but is properly contained in that of the linear bounded automaton. If a language is bounded and if the set obtained by applying Parikh mapping to that language is semi-linear, the language is accepted by RSSA. Another property of the language accepted by RSSA is that if it is a bounded context-free language, it is accepted by RSSA.

Specification of schedulers with algebraic specification techniques

The specification of a class of single resource schedulers is presented. A class of these schedulers is regarded as a parameterized data type whose formal parameter is a scheduling strategy. The specification is given by using the concept of parameterization in algebraic techniques. The specification of each scheduler instance can be directly obtained only by replacing the formal parameter with a concrete scheduling strategy. The assignment of an actual parameter to a formal parameter is given by a mechanism called extended parameter morphisms. This specification technique is applicable to many other objects.