Weighted Automata and Logics for Infinite Nested Words

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 B\"uchi'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.Comment: LATA 2014, 12 page

Multi-weighted Automata Models and Quantitative Logics

Recently, multi-priced timed automata have received much attention for real-time systems. These automata extend priced timed automata by featuring several price parameters. This permits to compute objectives like the optimal ratio between rewards and costs. Arising from the model of timed automata, the multi-weighted setting has also attracted much notice for classical nondeterministic automata. The present thesis develops multi-weighted MSO-logics on finite, infinite and timed words which are expressively equivalent to multi-weighted automata, and studies decision problems for them. In addition, a Nivat-like theorem for weighted timed automata is proved; this theorem establishes a connection between quantitative and qualitative behaviors of timed automata. Moreover, a logical characterization of timed pushdown automata is given

Pebble Weighted Automata and Weighted Logics

34 pagesInternational audienceWe introduce new classes of weighted automata on words. Equipped with pebbles, they go beyond the class of recognizable formal power series: they capture weighted first-order logic enriched with a quantitative version of transitive closure. In contrast to previous work, this calculus allows for unrestricted use of existential and universal quantifications over positions of the input word. We actually consider both two-way and one-way pebble weighted automata. The latter class constrains the head of the automaton to walk left-to-right, resetting it each time a pebble is dropped. Such automata have already been considered in the Boolean setting, in the context of data words. Our main result states that two-way pebble weighted automata, one- way pebble weighted automata, and our weighted logic are expressively equivalent. We also give new logical characterizations of standard recognizable series

Maximal Partition Logic: Towards a Logical Characterization of Copyless Cost Register Automata

It is highly desirable for a computational model to have a logic characterization like in the seminal work from Buchi that connects MSO with finite automata. For example, weighted automata are the quantitative extension of finite automata for computing functions over words and they can be naturally characterized by a subframent of weighted logic introduced by Droste and Gastin. Recently, cost register automata (CRA) were introduced by Alur et al. as an alternative model for weighted automata. In hope of finding decidable subclasses of weighted automata, they proposed to restrict their model with the so-called copyless restriction. Unfortunately, copyless CRA do not enjoy good closure properties and, therefore, a logical characterization of this class seems to be unlikely. In this paper, we introduce a new logic called maximal partition logic (MPL) for studying the expressiveness of copyless CRA. In contrast from the previous approaches (i.e. weighted logics), MPL is based on a new set of "regular" quantifiers that partition a word into maximal subwords, compute the output of a subformula over each subword separately, and then aggregate these outputs with a semiring operation. We study the expressiveness of MPL and compare it with weighted logics. Furthermore, we show that MPL is equally expressive to a natural subclass of copyless CRA. This shows the first logical characterization of copyless CRA and it gives a better understanding of the copyless restriction in weighted automata

A Unifying Survey on Weighted Logics and Weighted Automata: Core Weighted Logic: Minimal and Versatile Specification of Quantitative Properties

International audienceLogical formalisms equivalent to weighted automata have been the topic of numerous research papers in the recent years. It started with the seminal result by Droste and Gastin on weighted logics over semir-ings for words. It has been extended in two dimensions by many authors. First, the weight domain has been extended to valuation monoids, valuation structures, etc., to capture more quantitative properties. Along another dimension, different structures such as ranked or unranked trees, nested words, Mazurkiewiz traces, etc., have been considered. The long and involved proofs of equivalences in all these papers are implicitely based on the same core arguments. This article provides a meta-theorem which unifies these different approaches. Towards this, we first introduce a core weighted logic with a minimal number of features and a simplified syntax. Then, we define a new semantics for weighted automata and weighted logics in two phases—an abstract semantics based on multisets of weight structures (independent of particular weight domains) followed by a concrete semantics. We show at the level of the abstract semantics that weighted automata and core weighted logic have the same expressive power. We show how previous results can be recovered from our result by logical reasoning. In this paper, we prove the meta-theorem for words, ranked and unranked trees, showing the robustness of our approach

Weighted Automata and Logics on Hierarchical Structures and Graphs

Formal language theory, originally developed to model and study our natural spoken languages, is nowadays also put to use in many other fields. These include, but are not limited to, the definition and visualization of programming languages and the examination and verification of algorithms and systems. Formal languages are instrumental in proving the correct behavior of automated systems, e.g., to avoid that a flight guidance system navigates two airplanes too close to each other. This vast field of applications is built upon a very well investigated and coherent theoretical basis. It is the goal of this dissertation to add to this theoretical foundation and to explore ways to make formal languages and their models more expressive. More specifically, we are interested in models that are able to model quantitative features of the behavior of systems. To this end, we define and characterize weighted automata over structures with hierarchical information and over graphs. In particular, we study infinite nested words, operator precedence languages, and finite and infinite graphs. We show Büchi-like results connecting weighted automata and weighted monadic second order (MSO) logic for the respective classes of weighted languages over these structures. As special cases, we obtain Büchi-type equivalence results known from the recent literature for weighted automata and weighted logics on words, trees, pictures, and nested words. Establishing such a general result for graphs has been an open problem for weighted logics for some time. We conjecture that our techniques can be applied to derive similar equivalence results in other contexts like traces, texts, and distributed systems

Weighted Logics for Nested Words and Algebraic Formal Power Series

Nested words, a model for recursive programs proposed by Alur and Madhusudan, have recently gained much interest. In this paper we introduce quantitative extensions and study nested word series which assign to nested words elements of a semiring. We show that regular nested word series coincide with series definable in weighted logics as introduced by Droste and Gastin. For this we establish a connection between nested words and the free bisemigroup. Applying our result, we obtain characterizations of algebraic formal power series in terms of weighted logics. This generalizes results of Lautemann, Schwentick and Therien on context-free languages

Model Checking One-clock Priced Timed Automata

We consider the model of priced (a.k.a. weighted) timed automata, an extension of timed automata with cost information on both locations and transitions, and we study various model-checking problems for that model based on extensions of classical temporal logics with cost constraints on modalities. We prove that, under the assumption that the model has only one clock, model-checking this class of models against the logic WCTL, CTL with cost-constrained modalities, is PSPACE-complete (while it has been shown undecidable as soon as the model has three clocks). We also prove that model-checking WMTL, LTL with cost-constrained modalities, is decidable only if there is a single clock in the model and a single stopwatch cost variable (i.e., whose slopes lie in {0,1}).Comment: 28 page