Weighted Recognizability over Infinite Alphabets

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...

Information & Computation, 2019

We investigate weighted tree automata over Cauchy tree valuation monoids, a new type of weight structure which includes all commutative semirings and, in addition, average and discounted computations of weights for trees. We define rational tree series over these weight structures, and we prove Kleene's classical theorem for this setting: a tree series over a Cauchy tree valuation monoid is recognizable by a weighted tree automaton if and only if it is rational. The proof works via direct automata-theoretic constructions. Along the way, our results yield a new characterization of weighted tree automata over commutative semirings by rational expressions.

Modern Applications of Automata Theory, 2012

In many contexts such as validation of XML data, software model checking and parametrized verification, the systems studied are naturally abstracted as finite state automata, but whose input alphabet is infinite. While use of such automata for verification requires that the non-emptiness problem be decidable, ensuring this is non-trivial, since the space of configurations of such automata is infinite. We describe some recent attempts in this direction, and suggest that an entire theoretical framework awaits development. 1.1. Motivation The theory of finite state automata over (finite) words is an arena that is rich in concepts and results, offering interesting connections between computability theory, algebra, logic and complexity theory. Moreover, finite state automata provide an excellent abstraction for many real world applications, such as string matching in lexical analysis [1, 2], model checking finite state systems [3] etc. Considering that finite state machines have only bounded memory, it is a priori reasonable that their input alphabet is finite. If the input alphabet were infinite, it is hardly clear how such a machine can tell infinitely many elements apart. And yet, there are many good reasons to consider mechanisms that achieve precisely this. Abstract considerations first: consider the set of all finite sequences of natural numbers (given in binary) separated by hashes. A word of this language, for example, is 100#11#1101#100#10101. Now consider the subset L containing all sequences with some number repeating in it. It is easily seen that L is not regular, it is not even context-free. The problem with L has little to do with the representation of the input sequence. If we were given a bound on the numbers occurring in any sequence, we could easily build a finite state automaton recognizing L. The difficulty arises precisely because we don't have such a bound or because we have 'unbounded data'. It is not difficult to find instances of languages like L occurring naturally in the computing world. For example consider the sequences of all nonces used in a security protocol run. Ideally this language should be L. The question

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

Theoretical Computer Science, 2006

We define a weighted monadic second order logic for trees where the weights are taken from a commutative semiring. We prove that a restricted version of this logic characterizes the class of formal tree series which are accepted by weighted bottom-up finite state tree automata. The restriction on the logic can be dropped if additionally the semiring is locally finite. This generalizes corresponding classical results of Thatcher, Wright, and Doner for tree languages and it extends recent results of Droste and Gastin [Weighted automata and weighted logics, informal power series on words to formal tree series.

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.

Information Sciences, 2010

We consider weighted finite automata over strong bimonoids, where these weight structures can be considered as semirings which might lack distributivity. Then, in general, the well-known run semantics, initial algebra semantics, and transition semantics of an automaton are different. We prove an algebraic characterization for the initial algebra semantics in terms of stable finitely generated submonoids. Moreover, for a given weighted finite automaton we construct the Nerode automaton and Myhill automaton, both being crisp-deterministic, which are equivalent to the original automaton with respect to the initial algebra semantics, resp., the transition semantics. We prove necessary and sufficient conditions under which the Nerode automaton and the Myhill automaton are finite, and we provide efficient algorithms for their construction. Also, for a given weighted finite automaton, we show sufficient conditions under which a given weighted finite automaton can be determinized preserving its run semantics. and weighted tree automata over strong bimonoids begun in respectively . As usual, there are three different definitions of the behavior of such automata: the run semantics , the initial algebra semantics , and the transition semantics . In the run semantics, the weight of a word w is computed by summing up the weights of all successful runs of A on w where the weight of a run is the product of the weights of the involved transitions. For the other two semantics, we associate with each input symbol an n × n-matrix of transition weights assuming that A has n states. In both semantics, we multiply the vector of initial weights with the matrices corresponding to the symbols of w and with the final weight vector, but the sequence of these multiplications differs in the two semantics. In the case of semirings, these three semantics coincide for every weighted finite automaton. In contrast to this, we show by an easy example that over the unit interval with bounded sum these three semantics in general differ.

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.

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.

SIAM Journal on Computing, 2000

We study the problem of constructing the deterministic equivalent of a nondeterministic weighted finite-state automaton (WFA). Determinization of WFAs has important applications in automatic speech recognition (ASR). We provide the first polynomial-time algorithm to test for the twins property, which determines if a WFA admits a deterministic equivalent. We also give upper bounds on the size of the deterministic equivalent;