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An inner product space on irreducible and synchronizable probabilistic finite state automata

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Abstract

Probabilistic finite state automata (PFSA) have found their applications in diverse systems. This paper presents the construction of an inner-product space structure on a class of PFSA over the real field via an algebraic approach. The vector space is constructed in a stationary setting, which eliminates the need for an initial state in the specification of PFSA. This algebraic model formulation avoids any reference to the related notion of probability measures induced by a PFSA. A formal language-theoretic and symbolic modeling approach is adopted. Specifically, semantic models are constructed in the symbolic domain in an algebraic setting. Applicability of the theoretical formulation has been demonstrated on experimental data for robot motion recognition in a laboratory environment.

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Abbreviations

\({\mathcal{A}}\) :

is the set of all irreducible FSA [Definition 2.4];

\({\bar{\mathcal{A}}}\) :

is the set of all FSA [Definition 2.2];

\({\mathcal{A}_s}\) :

is the set of all irreducible and synchronizable FSA [Definition 3.3];

\({\mathcal{B}^+}\) :

is the set of all PFSA such that the probability map \({\tilde{\pi}}\) has only strictly positive entries and the underlying FSA is irreducible [Definition 4.1];

\({\mathcal{B}^+_s}\) :

is the subset of synchronizable PFSA in \({\mathcal{B}^+}\) [Definition 4.1];

\({\mathcal{C}^+}\) :

is the quotient set \({\mathcal{B}^+/\equiv}\) of all PFSA in \({\mathcal{B}^+}\) [Definition 4.7];

\({\mathcal{C}^+_s}\) :

is the quotient set \({\mathcal{B}^+_s/\equiv}\) of all PFSA in \({\mathcal{B}^+_s}\) [Definition 4.7];

\({\lfloor G \rfloor_K}\) :

is the lift of an FSA G relatively to a PFSA K [Definition 4.8];

\({\bar{K}}\) :

is the underlying FSA of the PFSA K [Definition 2.3];

\({\widehat{K}}\) :

is the minimal representation of the underlying FSA of the PFSA K [Theorem 4.5];

\({\lceil K \rceil}\) :

is the equivalence class of \({K \in \mathcal{B}^+}\) under the relation \({\equiv}\) [Definition 4.7];

\({\wp}\) :

is the stationary-probability distribution of the states [Sect. 5.3];

Q :

is the set of states of the FSA or PFSA [Definition 2.2];

T :

is the relabeling map between two FSA [Definition 3.1];

δ :

is the state transition map for FSA [Definition 2.2];

δ*:

is the extended state transition map of FSA [Sect. 2];

μ :

is the merging map for FSA [Definition 3.2];

π :

is the state-to-state transition probability map of PFSA [Sect. 5.3];

\({\tilde{\pi}}\) :

is the probability map of a PFSA [Definition 2.3];

\({\tilde{\pi}*}\) :

is the extended probability map of a PFSA [Sect. 2];

Σ:

is a finite alphabet of cardinality |Σ| [Sect. 2];

Σ*:

is the collection of all finite-length words made from Σ [Sect. 2];

\({\varpi}\) :

weight for the inner product \({\langle \cdot, \cdot \rangle}\) [Subsect. 5.3];

\({\mathfrak{S}}\) :

is the state relabeling equivalence for FSA [Definition 3.1];

\({\equiv}\) :

is the algebraic equivalence over PFSA [Definition 4.6];

\({\vartriangleleft}\) :

is the state splitting operation for FSA [Definition 3.2];

\({\vartriangleright}\) :

is the state merging operation for FSA [Definition 3.2];

\({\asymp}\) :

is the state relabeling operation for PFSA [Definition 4.2];

\({\preceq}\) :

is the state merging operation for PFSA [Definition 4.2];

\({\vee}\) :

is the join of two FSA [Definition 2.1] [Theorem B.1];

\({\curlyvee}\) :

is the join composition of two PFSA [Definition 2.1] [Definition 4.9];

\({\oplus}\) :

is the vector addition operation on the space \({\mathcal{B}^+_s}\) [Definition 5.2];

\({\odot}\) :

is the scalar multiplication operation on the space \({\mathcal{B}^+_s}\) [Definition 5.7];

 + :

is the vector addition operation on the space \({\mathcal{C}^+_s}\) [Definition 5.4];

\({\cdot}\) :

is the scalar multiplication operation on the space \({\mathcal{C}^+_s}\) [Definition 5.9];

\({\langle\langle \bullet, \bullet \rangle\rangle}\) :

is an inner product on the space \({\mathcal{B}^+_s}\) [Definition 5.12];

\({\langle \bullet, \bullet \rangle}\) :

is an inner product on the space \({\mathcal{C}^+_s}\) [Definition 5.14].

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  1. The Pennsylvania State University, University Park, PA, 16802, USA

    Patrick Adenis, Yicheng Wen & Asok Ray

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  1. Patrick Adenis
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Correspondence to Asok Ray.

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This work has been supported in part by the US Office of Naval Research under Grant No. N00014-09-1-0688, and by the US Army Research Laboratory and the US Army Research Office under Grant No. W911NF-07-1-0376. Any opinions, findings and conclusions or recommendations in this publication are those of the authors and do not reflect the views of the sponsoring agencies.

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Adenis, P., Wen, Y. & Ray, A. An inner product space on irreducible and synchronizable probabilistic finite state automata. Math. Control Signals Syst. 23, 281–310 (2012). https://doi.org/10.1007/s00498-012-0075-1

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