Quantitative generalizations of classical languages, which assign to each word a real number instead of a Boolean value, have applications in modeling resource-constrained computation. We use weighted automata (finite automata with transition weights) to define several natural classes of quantitative languages over finite and infinite words; in particular, the real value of an infinite run is computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We define the classical decision problems of automata theory (emptiness, universality, language inclusion, and language equivalence) in the quantitative setting and study their computational complexity. As the decidability of the language-inclusion problem remains open for some classes of weighted automata, we introduce a notion of quantitative simulation that is decidable and implies language inclusion. We also give a complete characterization of the expressive power of the various classes of weighted automata. In particular, we show that most classes of weighted automata cannot be determinized.

/pdf/quantitative-languages-1qr7kbwj4u.pdf

Quantitative languages

Quantitative generalizations of classical languages, which assign to each word a real number instead of a boolean value, have applications in modeling resource-constrained computation. We use weighted automata (finite automata with transition weights) to define several natural classes of quantitative languages over finite and infinite words; in particular, the real value of an infinite run is computed as the maximum, limsup, liminf, limit average, or discounted sum of the transition weights. We define the classical decision problems of automata theory (emptiness, universality, language inclusion, and language equivalence) in the quantitative setting and study their computational complexity. As the decidability of language inclusion remains open for some classes of weighted automata, we introduce a notion of quantitative simulation that is decidable and implies language inclusion. We also give a complete characterization of the expressive power of the various classes of weighted automata. In particular, we show that most classes of weighted automata cannot be determinized.

Quantitative Languages

We survey some important results for weighted tree automata and weighted tree transducers over finite ranked trees and semirings as weight structure. In particular, we address closure properties of the class of recognizable tree series, results on the support of such tree series, the determinization of weighted tree automata, pumping lemmata and decidability results, and finite algebraic characterizations of recognizable tree series. We discuss the equivalence between recognizable tree series and equational, rational, and MSO-definable tree series, and we present a comparison of several other models of recognizability. For weighted tree transducers we show composition and decomposition results, an inclusion diagram of some fundamental classes of tree series transformations, and hierarchies obtained by composing weighted tree transducers. We also discuss other models of weighted tree transducers.

Weighted Tree Automata and Tree Transducers

We show that it is decidable in time complexity \(2^{2^{2^{O{(n)}}}}\) whether the language accepted by an n-state non-deterministic automaton is of star height one, which is the first ever complexity result for the star height one problem. To achieve this, we introduce distance desert automata as a joint generalization of distance automata and desert automata, and show the decidability of its limitedness problem by solving the underlying Burnside problem.

/pdf/distance-desert-automata-and-the-star-height-problem-2n1m63c3iw.pdf

Distance desert automata and the star height problem

Weighted automata are a generalisation of non-deterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for non-deterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of state-based systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on Set (the category of sets and functions) characterise weighted bisimilarity, while coalgebras on Vect (the category of vector spaces and linear maps) characterise weighted language equivalence. Relying on the second characterisation, we show three different procedures for computing weighted language equivalence. The first one consists in a generalisation of the usual partition refinement algorithm for ordinary automata. The second one is the backward version of the first one. The third procedure relies on a syntactic representation of rational weighted languages.

/pdf/a-coalgebraic-perspective-on-linear-weighted-automata-5d2g1xbl7y.pdf

A coalgebraic perspective on linear weighted automata

In the paper, we generalize an algorithm and some related results by Mohri [25] for determinization of weighted finite automata (WFA) over the tropical semiring We present the underlying mathematical concepts of his algorithm in a precise way for arbitrary semirings We define a class of semirings in which we can show that the twins property is sufficient for the termination of the algorithm We also introduce single-valued WFA and give a partial correction of a claim by Mohri [25] by showing several characterizations of single-valued WFA, eg, the formal power series computed by a single-valued WFA is subsequential iff it has bounded variation Also, it is decidable in polynomial time whether a given WFA over the tropical semiring is single-valued

On the Determinization of Weighted Automata.

This paper solves the unambiguity and the sequentiality problem for polynomially ambiguous min-plus automata. This result is proved through a decidable algebraic characterization involving so-called metatransitions and an application of results from the structure theory of finite semigroups. It is noteworthy that the equivalence problem is known to be undecidable for polynomially ambiguous automata.

/pdf/deciding-unambiguity-and-sequentiality-of-polynomially-21p071gn5b.pdf

Deciding Unambiguity and Sequentiality of Polynomially Ambiguous Min-Plus Automata

Since Buchi’s work in 1960 [17], automata play an important role in logic. Numerous different notions of automata provide decision and complexity results in various kinds of logic. Often, one develops a method to translate some given formula ϕ into an appropriate finite automaton A such that L(ϕ) = L(A). Such a translation reduces the model checking problem and the satisfiability problem in some logic to the word problem and the emptiness problem for finite automata. Moreover, such a translation provides algorithms to solve the model checking and the satisfiability problems on a computer. Consequently, one is interested in the decidability and the complexity of the word and emptiness problems of automata.

Alternating tree automata and parity games

We give a positive solution to the so-called finite substitution problem which was open for more than 10 years [11]: given recognizable languages K and L, decide whether there exists a finite substitution σ such that σ(K) = L. For this, we introduce a new model of weighted automata and show the decidability of its limitedness problem by solving the underlying Burnside problem.

Desert automata and the finite substitution problem

/pdf/distance-desert-automata-and-the-star-height-one-problem-1jbrdid0zz.pdf

Daniel Kirsten

Papers

On the Determinization of Weighted Automata.

Deciding Unambiguity and Sequentiality of Polynomially Ambiguous Min-Plus Automata

Alternating tree automata and parity games

Desert automata and the finite substitution problem

Distance Desert Automata and the Star Height One Problem