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Abstract-The time separation of events (TSE) problem is that of finding the maximum and minimum separation between the times of occurrence of two events in a concurrent system. It has applications in the performance analysis, optimization and verification of concurrent digital systems. This paper introduces an efficient polynomial-time algorithm to give exact bounds on TSE's for choice-free concurrent systems, whose operational semantics obey the max-causality rule. A choicefree concurrent system is modeled as a strongly-connected marked graph, where delays on operations are modeled as bounded intervals with unspecified distributions. While previous approaches handle acyclic systems only, or else require graph unfolding until a steady-state behavior is reached, the proposed approach directly identifies and evaluates the asymptotic steady-state behavior of a cyclic system via a graph-theoretical approach. As a result, the method has significantly lower computational complexity than previously-proposed solutions. A prototype CAD tool has been developed to demonstrate the feasibility and efficacy of our method. A set of experiments have been performed on the tool as well as two existing tools, with noticeable improvement on runtime and accuracy for several examples.

We present a novel method for state minimization of incompletely-specified finite state machines. Where classic methods simply minimize the number of states, ours directly addresses the implementation's logic complexity, and produces an exactly optimal implementation under input encoding. The method incorporates optimal "state mapping", i.e., the process of reducing the symbolic next-state relation which results from state splitting to an optimal conforming symbolic function. Further, it offers a number of convenient sites for applying heuristics to reduce time and space complexity, and is amenable to implementation based on implicit representations. Although our method currently makes use of an input encoding model, we believe it can be extended smoothly to encompass output encoding as well.1 Introduction State minimization is the problem of finding a machine realizing the input/output behaviour of a given FSM, with fewer internal states [11,17,12]. This is an important step in sequential synthesis: implementing unminimized FSM's often leads to considerably larger and/or slower implementations. However, it is well known that the classic formulation for state minimization expresses a heuristic -reducing the number of states only tends to decrease logic complexity. Early on, Hartmanis observed [9] that this heuristic sometimes fails; realizations having more states may be simpler to implement. Moreover, there may be many minimum-state realizations of a given FSM, and their logic complexity can vary significantly [18,16]. Hence, simply targeting any minimum-state solution is insufficient.The major contribution of this paper is a state minimization method which, in contrast to existing ones, directly targets logic complexity. In particular, we define and solve the optimal state minimization problem, that of finding for a given FSM a realization having minimum 2-level logic complexity over all realizations.Classic sequential synthesis comprises several steps: state minimization, state encoding, 2-level logic minimization, multi-level optimization and technology mapping. Each step has traditionally been treated as an isolated problem, which limits early steps most * This research was funded in part by NSF CAREER Award MIP-9501880 and by an Alfred P. Sloan Research Fellowship.severely. In [7], a key insight into optimal state encoding was presented: symbolic logic minimization can be performed concurrently with state encoding. More recent methods for optimal encoding have been developed [20,8] based on the same insight, but yielding even better results.We borrow this insight, and take it one step further: symbolic logic minimization is performed concurrently with both state minimization and state encoding. Our method is cast as a unique form of generalized prime implicant minimization [8]. Specifically, symbolic prime implicants are generated, and a binate covering problem is formed and solved, yielding a reduced machine and logic cover.Our paper offers a novel theoretical framework for formulating and solving...