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A unifying survey on weighted logics and weighted automata

Core weighted logic: minimal and versatile specification of quantitative properties

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Abstract

Logical formalisms equivalent to weighted automata have been the topic of numerous research papers in the recent years. It started with the seminal result by Droste and Gastin on weighted logics over semirings for words. It has been extended in two dimensions by many authors. First, the weight domain has been extended to valuation monoids, valuation structures, etc. to capture more quantitative properties. Along another dimension, different structures such as ranked or unranked trees, nested words, Mazurkiewicz traces, etc. have been considered. The long and involved proofs of equivalences in all these papers are implicitly based on the same core arguments. This article provides a meta-theorem which unifies these different approaches. Towards this, we first revisit weighted automata by defining a new semantics for them in two phases—an abstract semantics based on multisets of weight structures (independent of particular weight domains) followed by a concrete semantics. Then, we introduce a core weighted logic with a minimal number of features and a simplified syntax, and lift the new semantics to this logic. We show at the level of the abstract semantics that weighted automata and core weighted logic have the same expressive power. Finally, we show how previous results can be recovered from our result by logical reasoning. In this paper, we prove the meta-theorem for words, ranked and unranked trees, showing the robustness of our approach.

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Notes

  1. This valuation does not satisfy \(\mathsf {Sq}(s)=s\).

  2. We do not require that \(\mathsf {Val}(r)=r\) for all \(r\in U\) as in Droste and Perevoshchikov (2013).

  3. Actually, (3) holds for arbitrary multisets \(A_i,B_j\in \mathbb {N} \langle R \rangle \).

  4. As already noticed in Droste and Meinecke (2012), a simple projection does not work. Indeed, it would result in transition labels that are multisets of weights, which is not possible since our theorem requires the same set of weights for the automaton and the formula.

  5. We could also use an unambiguous automaton for \(\mathcal A_\varphi \).

  6. These formulæ are call strongly \(\wedge \)-restricted in Droste and Meinecke (2012), in contrast with a slightly less restrained fragment of \(\wedge \)-restricted formulæ.

  7. In Droste and Meinecke (2012), the regularity is defined with respect to some weighted automata. Here, we prefer to use a purely logical definition. However, using Corollary 11-2., those two definitions are equivalent.

  8. The product in the semiring of weight sequences is based on the concatenation of sequences. There is no natural counterpart for trees.

  9. Equivalently, we may assume that transitions are disjoint: if \((L,a,q),(L',a,q)\in \Delta \) then \(L=L'\) or \(L\cap L'=\emptyset \). In this case, a run can be defined as a simpler Q-tree \(\rho \) with \(\mathrm {dom}(\rho )=\mathrm {dom}(t)\) and such that for all \(u\in \mathrm {dom}(t)\) there is a (unique) transition \((L,t(u),\rho (u))\in \Delta \) with \(\rho (u\cdot 1)\ldots \rho (u\cdot \mathsf {ar}(u))\in L\).

  10. An even simpler formula can be obtained in the simpler setting of ranked trees.

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Authors and Affiliations

  1. LSV ENS Cachan, CNRS Université Paris-Saclay, 94235, Cachan, France

    Paul Gastin

  2. Aix-Marseille Université, LIF, CNRS, Marseille, France

    Benjamin Monmege

Authors
  1. Paul Gastin
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  2. Benjamin Monmege
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Correspondence to Benjamin Monmege.

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The authors declare that they have no conflict of interest.

Additional information

Communicated by M. Droste, Z. Esik and K. Larsen.

Part of the research leading to these results was achieved when the second author was at Université libre de Bruxelles (Belgium), and has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under Grant Agreement n601148 (CASSTING).

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Gastin, P., Monmege, B. A unifying survey on weighted logics and weighted automata. Soft Comput 22, 1047–1065 (2018). https://doi.org/10.1007/s00500-015-1952-6

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