Aperiodic Weighted Automata and Weighted First-Order Logic

We extend the disambiguation construction presented by Mohri and Riley [1] in two ways. First we change the underlying structure of their automata from words to trees and second we show that these results hold not only for the tropical semiring but also the arctic one.

International Journal of Algebra and Computation, 2013

This paper addresses the problem of the validity of weighted automata in which the presence of ε-circuits results in infinite summations. Earlier works either rule out such automata or characterize the semirings in which these infinite sums are all well-defined. By means of a topological approach, we take here a definition of validity that is strong enough to insure that in any kind of semirings, any closure algorithm will succeed on every valid weighted automaton and turn it into an equivalent proper automaton. This definition is stable with respect to natural transformations of automata. The classical closure algorithms, in particular algorithms based on the computation of the star of the matrix of ε-transitions, cannot be used to decide validity. This decision problem remains open for general topological semirings. We present a closure algorithm that yields a decision procedure for the validity of automata in the case where the weights are taken in ℚ or ℝ, two cases that had neve...

Leibniz International Proceedings in Informatics (LIPIcs), 2018

We present three pumping lemmas for three classes of functions definable by fragments of weighted automata over the min-plus semiring and the semiring of natural numbers. As a corollary we show that the hierarchy of functions definable by unambiguous, finitely-ambiguous, polynomially-ambiguous weighted automata, and the full class of weighted automata is strict for the min-plus semiring. 2012 ACM Subject Classification Theory of computation → Models of computation

Lecture Notes in Computer Science, 2014

Nested words introduced by Alur and Madhusudan are used to capture structures with both linear and hierarchical order, e.g. XML documents, without losing valuable closure properties. Furthermore, Alur and Madhusudan introduced automata and equivalent logics for both finite and infinite nested words, thus extending Büchi's theorem to nested words. Recently, average and discounted computations of weights in quantitative systems found much interest. Here, we will introduce and investigate weighted automata models and weighted MSO logics for infinite nested words. As weight structures we consider valuation monoids which incorporate average and discounted computations of weights as well as the classical semirings. We show that under suitable assumptions, two resp. three fragments of our weighted logics can be transformed into each other. Moreover, we show that the logic fragments have the same expressive power as weighted nested word automata.

Electronic Proceedings in Theoretical Computer Science

In this paper we deal with three models of weighted automata that take weights in the field of real numbers. The first of these models are classical weighted finite automata, the second one are crispdeterministic weighted automata, and the third one are weighted automata over a vector space. We explore the interrelationships between weighted automata over a vector space and other two models.

Electronic Proceedings in Theoretical Computer Science, 2016

We introduce a weighted linear dynamic logic (weighted LDL for short) and show the expressive equivalence of its formulas to weighted rational expressions. This adds a new characterization for recognizable series to the fundamental Schützenberger theorem. Surprisingly, the equivalence does not require any restriction to our weighted LDL. Our results hold over arbitrary (resp. totally complete) semirings for finite (resp. infinite) words. As a consequence, the equivalence problem for weighted LDL formulas over fields is decidable in doubly exponential time. In contrast to classical logics, we show that our weighted LDL is expressively incomparable to weighted LTL for finite words. We determine a fragment of the weighted LTL such that series over finite and infinite words definable by LTL formulas in this fragment are definable also by weighted LDL formulas.

Theory of Computing Systems, 2011

We define a weighted monadic second order logic for unranked trees and the concept of weighted unranked tree automata, and we investigate the expressive power of these two concepts. We show that weighted tree automata and a syntactically restricted weighted MSO-logic have the same expressive power in case the semiring is commutative or in case we deal only with ranked trees, but, surprisingly, not in general. This demonstrates a crucial difference between the theories of ranked trees and unranked trees in the weighted case.

2020

Recently, weighted ω-pushdown automata have been introduced by Droste, Ésik, Kuich. This new type of automaton has access to a stack and models quantitative aspects of infinite words. Here, we consider a simple version of those automata. The simple ω-pushdown automata do not use -transitions and have a very restricted stack access. In previous work, we could show this automaton model to be expressively equivalent to context-free ω-languages in the unweighted case. Furthermore, semiring-weighted simple ω-pushdown automata recognize all ω-algebraic series. Here, we consider ω-valuation monoids as weight structures. As a first result, we prove that for this weight structure and for simple ω-pushdown automata, Büchi-acceptance and Muller-acceptance are expressively equivalent. In our second result, we derive a Nivat theorem for these automata stating that the behaviors of weighted ω-pushdown automata are precisely the projections of very simple ω-series restricted to ω-context-free lang...

Theoretical Computer Science, 2018

Lecture Notes in Computer Science, 2017

Weighted automata (WA) are an important formalism to describe quantitative properties. Obtaining equivalent deterministic machines is a longstanding research problem. In this paper we consider WA with a set semantics, meaning that the semantics is given by the set of weights of accepting runs. We focus on multi-sequential WA that are defined as finite unions of sequential WA. The problem we address is to minimize the size of this union. We call this minimum the degree of sequentiality of (the relation realized by) the WA. For a given positive integer k, we provide multiple characterizations of relations realized by a union of k sequential WA over an infinitary finitely generated group: a Lipschitz-like machine independent property, a pattern on the automaton (a new twinning property) and a subclass of cost register automata. When possible, we effectively translate a WA into an equivalent union of k sequential WA. We also provide a decision procedure for our twinning property for commutative computable groups thus allowing to compute the degree of sequentiality. Last, we show that these results also hold for word transducers and that the associated decision problem is Pspace-complete.