A Unifying Survey on Weighted Logics and Weighted Automata

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. FiPart 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). P. Gastin LSV, ENS Cachan, CNRS, Université Paris-Saclay, 94235 Cachan, France [email protected] B. Monmege Aix-Marseille Université, LIF, CNRS, France [email protected] nally, 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.