A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a @math . A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each @math in one of three ways: it grows forward [ @math , backward [ @math ], or just occurs once [ @math ]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth @math iff it is a zigzag word of depth @math . As corollaries, we obtain that the morphic words with growth @math are exactly the ultimately periodic words, and the morphic words with growth @math are exactly the multilinear words.
The class of multilinear words appears in @cite_3 as the infinite words determined by one-way stack automata, and also in @cite_0 (as the reducts of the prime'' stream @math ). In @cite_11 , prediction of periodic words and multilinear words is studied in an automata-theoretic setting.
@cite_0 @cite_3 @cite_11
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We characterize the infinite words determined by one-way stack automata. An infinite language L determines an infinite word alpha if every string in L is a prefix of alpha. If L is regular or context-free, it is known that alpha must be ultimately periodic. We extend this result to the class of languages recognized by one-way nondeterministic checking stack automata (1-NCSA). We then consider stronger classes of stack automata and show that they determine a class of infinite words which we call multilinear. We show that every multilinear word can be written in a form which is amenable to parsing. Finally, we consider the class of one-way multihead deterministic finite automata (1:multi-DFA). We show that every multilinear word can be determined by some 1:multi-DFA, but that there exist infinite words determined by 1:multi-DFA which are not multilinear. We propose the model of nested words for representation of data with both a linear ordering and a hierarchically nested matching of items. Examples of data with such dual linear-hierarchical structure include executions of structured programs, annotated linguistic data, and HTML XML documents. Nested words generalize both words and ordered trees, and allow both word and tree operations. We define nested word automata—finite-state acceptors for nested words, and show that the resulting class of regular languages of nested words has all the appealing theoretical properties that the classical regular word languages enjoys: deterministic nested word automata are as expressive as their nondeterministic counterparts; the class is closed under union, intersection, complementation, concatenation, Kleene-*, prefixes, and language homomorphisms; membership, emptiness, language inclusion, and language equivalence are all decidable; and definability in monadic second order logic corresponds exactly to finite-state recognizability. We also consider regular languages of infinite nested words and show that the closure properties, MSO-characterization, and decidability of decision problems carry over. The linear encodings of nested words give the class of visibly pushdown languages of words, and this class lies between balanced languages and deterministic context-free languages. We argue that for algorithmic verification of structured programs, instead of viewing the program as a context-free language over words, one should view it as a regular language of nested words (or equivalently, a visibly pushdown language), and this would allow model checking of many properties (such as stack inspection, pre-post conditions) that are not expressible in existing specification logics. We also study the relationship between ordered trees and nested words, and the corresponding automata: while the analysis complexity of nested word automata is the same as that of classical tree automata, they combine both bottom-up and top-down traversals, and enjoy expressiveness and succinctness benefits over tree automata. In the classic problem of sequence prediction, a predictor receives a sequence of values from an emitter and tries to guess the next value before it appears. The predictor masters the emitter if there is a point after which all of the predictor’s guesses are correct. In this paper we consider the case in which the predictor is an automaton and the emitted values are drawn from a finite set; i.e., the emitted sequence is an infinite word. We examine the predictive capabilities of finite automata, pushdown automata, stack automata (a generalization of pushdown automata), and multihead finite automata. We relate our predicting automata to purely periodic words, ultimately periodic words, and multilinear words, describing novel prediction algorithms for mastering these sequences. Abstract Motivated by the success of powerful while expensive techniques to recognize words in a holistic way (, 2013; , 2014; , 2016) object proposals techniques emerge as an alternative to the traditional text detectors. In this paper we introduce a novel object proposals method that is specifically designed for text. We rely on a similarity based region grouping algorithm that generates a hierarchy of word hypotheses. Over the nodes of this hierarchy it is possible to apply a holistic word recognition method in an efficient way. Our experiments demonstrate that the presented method is superior in its ability of producing good quality word proposals when compared with class-independent algorithms. We show impressive recall rates with a few thousand proposals in different standard benchmarks, including focused or incidental text datasets, and multi-language scenarios. Moreover, the combination of our object proposals with existing whole-word recognizers (, 2014; , 2016) shows competitive performance in end-to-end word spotting, and, in some benchmarks, outperforms previously published results. Concretely, in the challenging ICDAR2015 Incidental Text dataset, we overcome in more than 10 F -score the best-performing method in the last ICDAR Robust Reading Competition (Karatzas, 2015). Source code of the complete end-to-end system is available at https: github.com lluisgomez TextProposals .