A main task in document transformation and information retrieval is locating subtrees satisfying some pattern. Therefore, unary queries, i.e., queries that map a tree to a set of its nodes, play an important role in the context of structured document databases. We want to understand how the natural and well-studied computation model of tree automata can be used to compute such queries. We dene a query automaton (QA) as a deterministic two-way nite automaton over trees that has the ability to select nodes depending on the state and the label at those nodes. We study QAs over ranked as well as over unranked trees. Unranked trees dier from ranked ones in that there is no bound on the number of children of nodes. We characterize the expressiveness of the dierent formalisms as the unary queries denable in monadic second-order logic (MSO). Surprisingly, in contrast to the ranked case, special stay transitions had to be added to QAs over unranked trees to capture MSO. We es...

Let U be a strictly increasing sequence of integers. By a greedy algorithm, every nonnegative integer has a greedy U-representation. The successor function maps the greedy U-representation of N onto the greedy U-representation of N+1. We characterize the sequences U such that the successor function associated to U is a left, resp. a right sequential function. We also show that the odometer associated to U is continuous if and only if the successor function is right sequential.

Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.

Abstract It is easy to get an upper bound for the state complexity of shuffle on trajectories that generalizes the bound for unrestricted shuffle. We establish improved bounds for slender trajectories. For trajectories with USL index 1 (or 1-thin trajectories) we obtain an asymptotically tight lower bound when the state complexity of the trajectory grows with respect to the state complexity of the component languages. Some estimations are improved by considering nondeterministic state complexity. 1 Introduction The notion of shuffle on trajectories was introduced as an extension of the existing notions of shuffle by Mateescu et al. [5] to provide an abstraction of parallel composition of words, an important operation in parallel computation.

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For a non-negative integer k, we say that a language L is k-poly-slender if the number of words of length n in L is of order O(n k ). We give a precise characterization of the k- poly-slender context-free languages. The well-known characterization of the k-poly-slender regular languages is an immediate consequence of ours. Keywords: context-free language, poly-slender language, Dyck loop Introduction An infinite sequence (#L (n)) n0 can be associated in a natural way to a language L: #L (n) is the number of words of length n in L. The idea is by no means new; for instance, in the first ICALP, Berstel [3] considered the notion of the population function of a language L which associates, to every n, the number of words of length at most n in L. The notion of the number of words of the same length is certainly very basic one in language theory and this is why some results have been proved several times. We recall briefly in the following the history of such results. When #L (n) is ...

Abstract. For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the n-th word of a genealogically ordered language L onto the (n+1)-th word of L. We show that, if the ratio of the number of elements of length n +1overthenumber of elements of length n of the language has a limit β>1, then the amortized cost of the successor function is equal to β/(β − 1). From this, we deduce the value of the amortized cost for several classes of numeration systems (integer base systems, canonical numeration systems associated with a Parry number, abstract numeration systems built on a rational language, and rational base numeration systems). 1

We prove that the function that maps a word of a rational language onto its successor for the radix order in this language is a finite union of co-sequential functions.

multiplication by a constant

. I survey some of the connections between formal languages and number theory. Topics discussed include applications of representation in base k, representation by sums of Fibonacci numbers, automatic sequences, transcendence in finite characteristic, automatic real numbers, fixed points of homomorphisms, automaticity, and k-regular sequences. Key words. finite automata, automatic sequences, transcendence, automaticity AMS(MOS) subject classifications. Primary 11B85, Secondary 11A63 11A55 11J81 1. Introduction. In this paper, I survey some interesting connections between number theory and the theory of formal languages. This is a very large and rapidly growing area, and I focus on a few areas that interest me, rather than attempting to be comprehensive. (An earlier survey of this area, written in French, is [1].) I also give a number of open questions. Number theory deals with the properties of integers, and formal language theory deals with the properties of strings. At the interse...