Abstract
The problem of designing an unknown component that combined with the known part of a discrete event system satisfies the given overall specification arises in a number of applications. In this paper, we extend the known results for classical Finite State Machines (FSM) to Finite State Machines with Timeouts (TFSM). A TFSM is an FSM augmented with an input and an output timeout functions, prescribing the change of current state if no input is applied until a specified timeout expires and system delays needed to response to applied input, respectively. We represent the behavior of a TFSM by the corresponding regular language. The parallel composition of two TFSMs S and P is defined via composition of languages L(S) and L(P) intersected with L(MC), where MC is a maximal TFSM over input and output alphabets of the composition. The unknown component X is then designed as a solution to the equation where A and X are compared by ≤ with C. Here A is a context, C is a specification, and ≤ is the reduction relation which specifies that the behavior of a system to be designed is contained in that of the specification. Similar to classical FSMs, the equation is transformed to a language equation, the largest solution for which provides the largest solution to the solvable TFSM equation. After the largest solution is derived, a corresponding reduction can be extracted in order to provide the component with required properties. The application areas vary from testing in context to quality optimization of service compositions. Future work includes studying equation solvability criteria and properties of nondeterministic and partial specifications and solutions.
Highlights
The problem of designing an unknown component that combined with the known part of a discrete event system satisfies the given overall specification arises in a number of applications
We extend the known results for classical Finite State Machines (FSM) to Finite State Machines with Timeouts (TFSM)
A TFSM is an FSM augmented with an input and an output timeout functions, prescribing the change of current state if no input is applied until a specified timeout expires and system delays needed to response to applied input, respectively
Summary
Функционирование автомата всегда осуществляется в некоторой внешней среде, предполагаемой или заданной явно. Язык максимального автомата в инерционной среде будет иметь вид [(1)*I(1)*O(1)n]*(1)*, где n N представляет минимальное время реакции среды. Язык максимального автомата для инерционных систем будет иметь вид [(1)*I(1)m(1)*O]*(1)*, где m N представляет минимальное время реакции системы. Язык максимального автомата для систем в такой среде будет иметь вид [(1)*I(1) kO]*(1)*, где k N представляет максимальное время ожидания реакции системы. Наибольшее решение LSolutionTFSM, являющееся языком временного автомата и, следовательно, решением автоматного уравнения, есть наибольшее префикс-замкнутое подмножество языка LSolution L(MX), где MXесть максимальный автомат для неизвестной компоненты. Что в силу того, что языковые уравнения решаются одинаковым образом для любых коэффициентов-языков, описанный процесс построения наибольшего решения уравнения для временных автоматов является универсальным при любых ограничениях внешней среды для композиции, если данные ограничения могут быть выражены через поведение максимального временного автомата в заданной среде
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