On deterministic weighted automata

Highlights

• Two classes of deterministic weighted automata are considered: purely sequential automata and sequential automata.

• The class of semirings over which all weighted automata admit purely sequential equivalents is fully characterised.

• A partial characterisation is obtained for sequential automata as well.

Abstract

Two families of input-deterministic weighted automata over semirings are considered: purely sequential automata, in which terminal weights of states are either zero or unity, and sequential automata, in which states can have arbitrary terminal weights. The class of semirings over which all weighted automata admit purely sequential equivalents is fully characterised. A similar characterisation is proved for sequential automata under an assumption that all elements of the underlying semiring have finitely many multiplicative left inverses, which is in particular true for all commutative semirings and all division semirings.

Introduction

Classical nondeterministic finite automata admit a well known extension, in which each transition is weighted (typically) by an element of some semiring [7], [22]. Such automata – usually known as weighted automata – realise formal power series instead of recognising languages, and they have been studied extensively both from the theoretical point of view and in connection with their practical applications. The reader might consult [7] for an overview of some of the most important research directions.

For applications such as natural language processing [16], [20], input-determinism of weighted automata often turns out to be crucial. This basically means that the automaton has precisely one state with nonzero initial weight and there is at most one transition for each input symbol leading from each state. However, it is well known that not all weighted automata can be determinised [19], [21]. The research has thus focused mainly on providing sufficient conditions – both on the automaton and the underlying semiring – under which a weighted automaton admits a deterministic equivalent, and on devising efficient determinisation algorithms for automata satisfying such conditions [1], [15], [20], [21].

We shall focus here on a slightly different question: over which semirings all weighted automata can be determinised? This in fact amounts to the study of deterministic weighted automata from a negative point of view, as we shall see that the class of such semirings is fairly constrained.

More precisely, we shall deal with this question for two classes of deterministic weighted automata: for purely sequential weighted automata, in which terminal weights of states might only be chosen as zero or unity of the underlying semiring, and for sequential weighted automata, in which terminal weights can be arbitrary (the term “deterministic weighted automata” usually refers to the latter [21]). This terminology follows Lombardy and Sakarovitch [19]; it may differ significantly in other sources (in particular, purely sequential automata are often called sequential, while sequential automata are called subsequential [20]).

We shall prove that weighted automata over S always admit purely sequential equivalents if and only if S is a locally finite division semiring. Moreover, local finiteness of S is known to be sufficient to guarantee that all weighted automata over S have sequential equivalents [19]. We shall prove that if S has no element with infinitely many multiplicative left inverses, then this is also a necessary condition. In particular, if S is commutative or a division semiring, then weighted automata over S can always be sequentialised if and only if S is locally finite.

Finally, let us mention that there is a branch of research motivated by quantitative formal verification dealing with weighted – or quantitative – automata over various structures beyond semirings [2], [3], [4], [5], [6], [8], [9], [10], [11], [18]. We shall nevertheless confine ourselves to the classical setting of semirings in this article, making theoretical analysis more tractable. Possible extensions of the results presented herein to structures more general than semirings are left for further research.