Language acceptability of finite automata based on theory of semi‐tensor product of matrices

Yue, Jumei; Yan, Yongyi; Chen, Zengqiang

doi:10.1002/asjc.2190

Cited by 9 publications

(5 citation statements)

References 23 publications

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“…It is easy to understand that sum operation

\sum

is the inverse operation of Ψ, that is,

\sum normalΨ false(M false) = \sum_{i \in normalΨ false(M false)} i = M

. We report the following equation [13]:

f false(x_{i}, e false) = normalΨ false({\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{i} double-struck⋉ u false(t false) false) .

According to Equations () and (), we have

\begin{matrix} X false(t false) & = ⋃_{x} false{f false(x, e false) (|, x \in X_{0}) false} \\ = normalΨ false({\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{p_{1}} double-struck⋉ u false(t false) false) \cup \dots \cup normalΨ false({\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{p_{k}} double-struck⋉ u false(t false) false) \\ = normalΨ false({\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{p_{1}} double-struck⋉ u false(t false) + \dots + {\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{p_{K}} double-struck⋉ u false(t false) false) \\ = normalΨ false({\overset{F}{true}}^{t} double-struck⋉ false(δ_{n}^{p_{1}} + \dots + δ_{n}^{p_{K}} f... \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

“…In particular, structure complexity of critical observers is one of the most difficult issues that must be overcome to make theoretical results applicable to real applications. The matrix approach was introduced to the field of FSMs by Xu [7], with the help of the theory of semi-tensor product which has been applied to many fields successfully [8][9][10][11][12][13][14][15][16][17][18][19]. Xu further investigated the reachability and observability problems of FSMs [20].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new look at the critical observability of finite state machines from an algebraic viewpoint

Yan

Deng

Chen

2021

Asian Journal of Control

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The traditional critical observability of FSMs refers to the possibility of detecting if the current state of an FSM belongs to a critical set for any input. In practical applications, the input is always of finite length, and we may only care about some critical states at a given time step or the inverse problem that how to find all possible inputs moving an FSM to a critical state of interest. Thus, we propose two new concepts about critical observability to cope with these two cases.Using the algebraic model of FSMs proposed in recent years, we propose several algebraic criteria for these types of critical observability. Moreover, the results are also applicable to large-scale networks of FSMs.

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“…It is easy to understand that sum operation

\sum

is the inverse operation of Ψ, that is,

\sum normalΨ false(M false) = \sum_{i \in normalΨ false(M false)} i = M

. We report the following equation [13]:

f false(x_{i}, e false) = normalΨ false({\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{i} double-struck⋉ u false(t false) false) .

According to Equations () and (), we have

\begin{matrix} X false(t false) & = ⋃_{x} false{f false(x, e false) (|, x \in X_{0}) false} \\ = normalΨ false({\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{p_{1}} double-struck⋉ u false(t false) false) \cup \dots \cup normalΨ false({\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{p_{k}} double-struck⋉ u false(t false) false) \\ = normalΨ false({\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{p_{1}} double-struck⋉ u false(t false) + \dots + {\overset{F}{true}}^{t} double-struck⋉ δ_{n}^{p_{K}} double-struck⋉ u false(t false) false) \\ = normalΨ false({\overset{F}{true}}^{t} double-struck⋉ false(δ_{n}^{p_{1}} + \dots + δ_{n}^{p_{K}} f... \end{matrix}

…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A new look at the critical observability of finite state machines from an algebraic viewpoint

Yan

Deng

Chen

2021

Asian Journal of Control

Self Cite

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“…Also, Deterministic refers to the distinctiveness of the computation. The finite automata are called deterministic finite automata if the machine reads an input string one symbol at a time [6,8]. Tree automata are state machines.…”

Section: Preliminariesmentioning

confidence: 99%

“…Aydin et al [1] had done their work on Automata-based model counting for string constraints. Most recently, Yue et al [8] developed the language acceptability of finite automata based on theory of semi-tensor product of matrices. Dobronravov et al [3] introduced the length of the shortest strings accepted by two-way finite automata.…”

Section: Introductionmentioning

confidence: 99%

Acceptable Strings in an Automaton

Dutta,

Borah

2023

jnanabha

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This paper is a presentation and discussion of two proofs of a theorem. The theorem is a statement about a specific type of sequence of inputs where all multiples of fixed denominations are accepted as inputs of an automaton. The theorem has interesting implications for accepted strings in a nite automaton in general setting. Here, we have examined different methods for proving the theorem. We present one analytic proof by using automata theory with graph theoretic concepts and another proof by utilizing the ordered partition of number theory. We also illustrate the results with the help of examples.

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“…In the last decade, semi-tensor product (STP) of matrices [15] has been introduced to the study of DESs, and several interesting results have been established [16][17][18][19][20][21][22][23]. Xu et al [19] proposed a matrix method for the reachability analysis of deterministic finite automata.…”

Section: Introductionmentioning

confidence: 99%

Matrix approach to I‐detectability of partially observed discrete event systems

Dou

2021

Asian Journal of Control

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I‐detectability is an important issue in partially observed discrete event systems (DESs). In this paper, a Boolean semi‐tensor product (BSTP) approach is proposed to investigate the I‐detectability problem of partially observed DESs. First, two concepts of I‐detectability, that is, strong I‐detectability and weak I‐detectability, are recalled for partially observed DESs. Second, in order to estimate the initial state of DESs after finite observation events, an observer is constructed and converted to an algebraic form based on BSTP. Third, based on the algebraic form of observer, two necessary and sufficient conditions are presented for strong I‐detectability and weak I‐detectability of partially observed DESs, respectively. Finally, two examples are given to illustrate the obtained results.

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Language acceptability of finite automata based on theory of semi‐tensor product of matrices

Cited by 9 publications

References 23 publications

A new look at the critical observability of finite state machines from an algebraic viewpoint

A new look at the critical observability of finite state machines from an algebraic viewpoint

Acceptable Strings in an Automaton

Matrix approach to I‐detectability of partially observed discrete event systems

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