Weighted Automata and Weighted Logics

Abstract

Abstract. Weighted automata are used to describe quantitative properties in various areas such as probabilistic systems, image compression, speech-to-text processing. The behaviour of such an automaton is a mapping, called a formal power series, assigning to each word a weight in some semiring. We generalize Büchi's and Elgot's fundamental theorems to this quantitative setting. We introduce a weighted version of MSO logic and prove that, for commutative semirings, the behaviours of weighted automata are precisely the formal power series definable with our weighted logic. We also consider weighted first-order logic and show that aperiodic series coincide with the first-order definable ones, if the semiring is locally finite, commutative and has some aperiodicity property.

Generalizing input-driven languages: Theoretical and practical benefits

“…Since the size of G's nonterminal alphabet is bounded, the above procedure eventually terminates when no new rules are added to P. 20 We have seen that a fundamental issue to state the properties of most abstract machines is their determinizability: in the cases examined so far we have realized that the positive basic result holding for RL extends to the various versions of structured CFL, though at the expenses of more intricate constructions and size complexity of the deterministic versions obtained from the nondeterministic ones, but not to the general CF family. Having OPL been born just with the motivation of supporting deterministic parsing, and being they structured as well, it is not surprising to find that for any nondeterministic OPA with s states an equivalent deterministic one can be built with 2 O(s 2 ) states, as it happens for the analogous construction for VPL: in [37] besides giving a detailed construction for the above result, it is also noticed that the construction of an OPA from an OPG is such that, if the OPG is in FNF, then the obtained automaton is already deterministic since the grammar has no repeated rhs.…”

Descriptive Complexity for counting complexity classes

“…We introduce here the logical framework that we use for studying counting complexity classes. This framework is based on the framework of Weighted Logics (WL) [9] that has been used in the context of weighted automata for studying functions from words (or trees) to semirings. We propose here to use the framework of WL over any relational structure and to restrict the semiring to natural numbers.…”

“…In this section, we discuss some previous frameworks proposed in the literature and how they differ from our approach. We start by discussing the connection between QSO and weighted logics (WL) [9]. As it was previously discussed, QSO is a fragment of WL.…”

“…In this paper, we propose to study the descriptive complexity of counting complexity classes in terms of Weighted Logics (WL) [9], a general logical framework that combines Boolean formulae (e.g. in FO or SO) with operations over a fixed semiring (e.g.…”

Quantitative simulations by matrices

“…Remark 2.4 (size of state space Q) In the above definition, differently from the usual definitions of weighted automata (e.g. [15]), we allow Q to be infinite. This is because in Theorem 2.11, such an infinite-state automaton arises as a final coalgebra that captures the trace semantics of weighted automata.…”