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Publication date
2011
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Informatics Engineering
and research's language is
English
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A classical result (often credited to Y. Medvedev) states that every language recognized by a finite automaton is the homomorphic image of a local language, over a much larger so-called local alphabet, namely the alphabet of the edges of the transition graph. Local languages are characterized by the value k=2 of the sliding window width in the McNaughton and Paperts infinite hierarchy of strictly locally testable languages (k-slt). We generalize Medvedevs result in a new direction, studying the relationship between the width and the alphabetic ratio telling how much larger the local alphabet is. We prove that every regular language is the image of a k-slt language on an alphabet of doubled size, where the width logarithmically depends on the automaton size, and we exhibit regular languages for which any smaller alphabetic ratio is insufficient. More generally, we express the trade-off between alphabetic ratio and width as a mathematical relation derived from a careful encoding of the states. At last we mention some directions for theoretical development and application.
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Finite automata whose computations can be reversed, at any point, by knowing the last k symbols read from the input, for a fixed k, are considered. These devices and their accepted languages are called k-reversible automata and k-reversible languages, respectively. The existence of k-reversible languages which are not (k-1)-reversible is known, for each k>1. This gives an infinite hierarchy of weakly irreversible languages, i.e., languages which are k-reversible for some k. Conditions characterizing the class of k-reversible languages, for each fixed k, and the class of weakly irreversible languages are obtained. From these conditions, a procedure that given a finite automaton decides if the accepted language is weakly or strongly (i.e., not weakly) irreversible is described. Furthermore, a construction which allows to transform any finite automaton which is not k-reversible, but which accepts a k-reversible language, into an equivalent k-reversible finite automaton, is presented.
In a previous work we introduced slice graphs as a way to specify both infinite languages of directed acyclic graphs (DAGs) and infinite languages of partial orders. Therein we focused on the study of Hasse diagram generators, i.e., slice graphs that generate only transitive reduced DAGs, and showed that they could be used to solve several problems related to the partial order behavior of p/t-nets. In the present work we show that both slice graphs and Hasse diagram generators are worth studying on their own. First, we prove that any slice graph SG can be effectively transformed into a Hasse diagram generator HG representing the same set of partial orders. Thus from an algorithmic standpoint we introduce a method of transitive reducing infinite families of DAGs specified by slice graphs. Second, we identify the class of saturated slice graphs. By using our transitive reduction algorithm, we prove that the class of partial order languages representable by saturated slice graphs is closed under union, intersection and even under a suitable notion of complementation (cut-width complementation). Furthermore partial order languages belonging to this class can be tested for inclusion and admit canonical representatives in terms of Hasse diagram generators. As an application of our results, we give stronger forms of some results in our previous work, and establish some unknown connections between the partial order behavior of $p/t$-nets and other well known formalisms for the specification of infinite families of partial orders, such as Mazurkiewicz trace languages and message sequence chart (MSC) languages.
Difference hierarchies were originally introduced by Hausdorff and they play an important role in descriptive set theory. In this survey paper, we study difference hierarchies of regular languages. The first sections describe standard techniques on difference hierarchies, mostly due to Hausdorff. We illustrate these techniques by giving decidability results on the difference hierarchies based on shuffle ideals, strongly cyclic regular languages and the polynomial closure of group languages.
We prove that the genus of a regular language is decidable. For this purpose, we use a graph-theoretical approach. We show that the original question is equivalent to the existence of a special kind of graph epimorphism - a directed emulator morphism -- onto the underlying graph of the minimal deterministic automaton for the regular language. We also prove that the class of directed emulators of genus less than or equal to $g$ is closed under minors. Decidability follows from the Robertson-Seymour theorem.
In structured prediction, a major challenge for models is to represent the interdependencies within their output structures. For the common case where outputs are structured as a sequence, linear-chain conditional random fields (CRFs) are a widely used model class which can learn local dependencies in output sequences. However, the CRFs Markov assumption makes it impossible for these models to capture nonlocal dependencies, and standard CRFs are unable to respect nonlocal constraints of the data (such as global arity constraints on output labels). We present a generalization of CRFs that can enforce a broad class of constraints, including nonlocal ones, by specifying the space of possible output structures as a regular language $mathcal{L}$. The resulting regular-constrained CRF (RegCCRF) has the same formal properties as a standard CRF, but assigns zero probability to all label sequences not in $mathcal{L}$. Notably, RegCCRFs can incorporate their constraints during training, while related models only enforce constraints during decoding. We prove that constrained training is never worse than constrained decoding, and show using synthetic data that it can be substantially better in practice. Additionally, we demonstrate a practical benefit on downstream tasks by incorporating a RegCCRF into a deep neural model for semantic role labeling, exceeding state-of-the-art results on a standard dataset.
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