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Algorithms for Weighted Finite Automata with Failure Transitions

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Implementation and Application of Automata (CIAA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10977))

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Abstract

In this paper we extend several weighted finite automata (WFA) algorithms to automata with failure transitions (\(\varphi \)-WFAs). Failure transitions, which are taken only when no immediate match is possible at a given state, are used to compactly represent automata and have many applications. Efficient algorithms to intersect two \(\varphi \)-WFAs, to remove failure transitions, to trim, and to compute (over \(\mathbb {R}_+\)) the shortest distance in a \(\varphi \)-WFA are presented.

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Notes

  1. 1.

    A metric \(\varDelta :\mathbb {K}\times \mathbb {K}\rightarrow \mathbb {R}_+\) satisfies (1) \(\varDelta (x,y) = \varDelta (y,x)\), (2) \(\varDelta (x,y) = 0\) iff \(x = y\), and (3) \(\varDelta (x,y) \le \varDelta (x,z) + \varDelta (y,z)\) for all \(x,y,z \in \mathbb {K}\).

  2. 2.

    The condition that a successful path cannot end in a \(\varphi \)-labeled transition simplifies the presentation without loss of generality since there is an equivalent \(\varphi \)-WFA with the final weights propagated to the \(\varphi \) sources.

  3. 3.

    Label multiplicity at state q is the maximum number of outgoing transitions in q sharing the same label.

  4. 4.

    Such as Dijkstra or Bellman-Ford with the appropriate queue disciplines on S.

  5. 5.

    Semiring K is k-closed if for all a in \(\mathbb {K}\), \(\bigoplus _{i=0}^{k+1} a^i = \bigoplus _{i=0}^{k} a^i\). It is k-closed for \(A\) if the weight a of each cycle in \(A\) verifies \(\bigoplus _{n=0}^{k+1} a^n = \bigoplus _{n=0}^{k} a^n\). The tropical semiring is 0-closed [17].

  6. 6.

    We could also add logic so that when line 12 of the shortest distance algorithm is executed for a disallowed transition then it is also always executed for any negative compensating transition in case \(\left| r[q+|Q|]\nu \right|< \epsilon < \left| r[q']\nu \right| \). This however is an unneeded precaution since with small enough \(\epsilon \) any discrepancy is insignificant compared to the floating-point precision of \(d[q'']\).

  7. 7.

    The real numbers can be defined axiomatically as a field with a complete, monotonic total order [24].

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  1. Google, New York, NY, USA

    Cyril Allauzen & Michael D. Riley

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  1. Cyril Allauzen
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  2. Michael D. Riley
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Correspondence to Cyril Allauzen .

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  1. University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada

    Cezar Câmpeanu

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Allauzen, C., Riley, M.D. (2018). Algorithms for Weighted Finite Automata with Failure Transitions. In: Câmpeanu, C. (eds) Implementation and Application of Automata. CIAA 2018. Lecture Notes in Computer Science(), vol 10977. Springer, Cham. https://doi.org/10.1007/978-3-319-94812-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-94812-6_5

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