Composability of Infinite-State Activity Automata

Abstract

Let ${\mathcal M}$ be a class of (possibly nondeterministic) language acceptors with a one-way input tape. A system (A; A ₁, ..., A _r ) of automata in ${\mathcal M}$, is composable if for every string w = a ₁ ... a _n of symbols accepted by A, there is an assignment of each symbol in w to one of the A _i ’s such that if w _i is the subsequence assigned to A _i , then w _i is accepted by A _i . For a nonnegative integer k, a k-lookahead delegator for (A; A ₁, ..., A _r ) is a deterministic machine D in ${\mathcal M}$ which, knowing (a) the current states of A, A ₁, ..., A _r and the accessible “local” information of each machine (e.g., the top of the stack if each machine is a pushdown automaton, whether a counter is zero on nonzero if each machine is a multicounter automaton, etc.), and (b) the k lookahead symbols to the right of the current input symbol being processed, can uniquely determine the A _i to assign the current symbol. Moreover, every string w accepted by A is also accepted by D, i.e., the subsequence of string w delegated by D to each A _i is accepted by A _i . Thus, k-lookahead delegation is a stronger requirement than composability, since the delegator D must be deterministic. A system that is composable may not have a k-delegator for any k. We look at the decidability of composability and existence of k-delegators for various classes of machines ${\mathcal M}$. Our results have applications to automated composition of e-services. When e-services are modeled by automata whose alphabet represents a set of activities or tasks to be performed (namely, activity automata), automated design is the problem of “delegating” activities of the composite e-service to existing e-services so that each word accepted by the composite e-service can be accepted by those e-services collectively with each accepting a subsequence of the word, under possibly some Presburger constraints on the numbers and types of activities that can be delegated to the different e-services. Our results generalize earlier ones (and resolve some open questions) concerning composability of deterministic finite automata as e-services to finite automata that are augmented with unbounded storage (e.g., counters and pushdown stacks) and finite automata with discrete clocks (i.e., discrete timed automata). We look at the decidability of composability and existence of k-delegators for various classes of machines ${\mathcal M}$. Our results have applications to automated composition of e-services. E-services provide a general framework for discovery, flexible interoperation, and dynamic composition of distributed and heterogeneous processes on the Internet. Automated composition allows a specified composite e-service to be implemented by composing existing e-services. When e-services are modeled by automata whose alphabet represents a set of activities or tasks to be performed (namely, activity automata), automated design is the problem of “delegating” activities of the composite e-service to existing e-services so that each word accepted by the composite e-service can be accepted by those e-services collectively with each accepting a subsequence of the word, under possibly some Presburger constraints on the numbers and types of activities that can be delegated to the different e-services. Our results generalize earlier ones (and resolve some open questions) concerning composability of deterministic finite automata as e-services to finite automata that are augmented with unbounded storage (e.g., counters and pushdown stacks) and finite automata with discrete clocks (i.e., discrete timed automata).

The research of Oscar H. Ibarra and Jianwen Su was supported in part by NSF Grants IIS-0101134 and CCR02-08595.