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Modeling of symbolic systems: Part I - Vector space representation of probabilistic finite state automata

Modeling of symbolic systems: Part I - Vector space representation of probabilistic finite state automata,Yicheng Wen,Asok Ray,Ishanu Chattopadhyay,Sh

Modeling of symbolic systems: Part I - Vector space representation of probabilistic finite state automata   (Citations: 1)
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Published in 2011.
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    • ...The main contribution of this paper lies in the construction of a family of inner products on the vector space of PFSA [13]...
    • ...The technical contents of this second part are built upon the vector space of PFSA, which is reported in the first part [13]...
    • ...where Np denotes a partition induced by Nerode equivalence [13]...
    • ...Q and all σ ∈ �}. Algorithm 1 in the first part [13] has been constructed in the context of the probabilistic Nerode equivalence Np such that a map e H : f A → P + f is surjective...
    • ...By Algorithm 1 in the first part [13], it follows that...
    • ...By use of the bijection H and its inverse F (see Algorithm 1 in the first part [13]), new vector addition and scalar multiplication operations are introduced on the quotient space A ...
    • ...Definition II.2 (Vector space A ) Let G1, G2 ∈ A and k ∈ R. Then, following the definitions of vector addition ⊕ and scalar multiplication ⊙ in the space of PFSA in the first part [13], • The addition operation + : A × A → A is defined as...
    • ...Proof: It follows from the definitions of vector addition ⊕ and scalar multiplication ⊙ in the space of PFSA in the first part [13] that (p1 ⊕ p2)(τ |x) = p1(τ |x)p2(τ |x)...
    • ...The following result is generated based on Algorithm 1 in the first part [13]...
    • ...Following the definition of a state transition probability matrix in the first part [13], a matrix representation of Eq. (18) is obtained as...
    • ...This second part of the two-part paper introduces a family of inner products on the vector space of PFSA that is constructed in the first part [13]...

    Yicheng Wenet al. Modeling of symbolic systems: Part II - Hilbert space construction for...

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