78 research outputs found

    Publication list of Zoltán Ésik

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    Deterministic Constrained Multilinear Detection

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    Revisiting the growth of polyregular functions: output languages, weighted automata and unary inputs

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    Polyregular functions are the class of string-to-string functions definable by pebble transducers (an extension of finite automata) or equivalently by MSO interpretations (a logical formalism). Their output length is bounded by a polynomial in the input length: a function computed by a kk-pebble transducer or by a kk-dimensional MSO interpretation has growth rate O(nk)O(n^k). Boja\'nczyk has recently shown that the converse holds for MSO interpretations, but not for pebble transducers. We give significantly simplified proofs of those two results, extending the former to first-order interpretations by reduction to an elementary property of N\mathbb{N}-weighted automata. For any kk, we also prove the stronger statement that there is some quadratic polyregular function whose output language differs from that of any kk-fold composition of macro tree transducers (and which therefore cannot be computed by any kk-pebble transducer). In the special case of unary input alphabets, we show that kk pebbles suffice to compute polyregular functions of growth O(nk)O(n^k). This is obtained as a corollary of a basis of simple word sequences whose ultimately periodic combinations generate all polyregular functions with unary input. Finally, we study polyregular and polyblind functions between unary alphabets (i.e. integer sequences), as well as their first-order subclasses.Comment: 27 pages, not submitted ye

    An Upper Bound on the Complexity of Recognizable Tree Languages

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    The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class ⅁(D_n(ÎŁ0_2))\Game (D\_n({\bf\Sigma}^0\_2)) for some natural number n≄1n\geq 1, where ⅁\Game is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space 2ω2^\omega into the class Δ1_2{\bf\Delta}^1\_2, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual Δ1_2{\bf\Delta}^1\_2

    Unambiguous Separators for Tropical Tree Automata

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    In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that f ? g, there exists effectively an unambiguous tropical automaton computing h such that f ? h ? g. This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different

    On the Sensitivity Conjecture for Read-k Formulas

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    Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and degree over R, are all polynomially related to one another. The sensitivity conjecture states that there is also a polynomial relationship between sensitivity and block sensitivity, thus supplying the "missing link". Since its introduction in 1991, the sensitivity conjecture has remained a challenging open question in the study of Boolean functions. One natural approach is to prove it for special classes of functions. For instance, the conjecture is known to be true for monotone functions, symmetric functions, and functions describing graph properties. In this paper, we consider the conjecture for Boolean functions computable by read-k formulas. A read-k formula is a tree in which each variable appears at most k times among the leaves and has Boolean gates at its internal nodes. We show that the sensitivity conjecture holds for read-once formulas with gates computing symmetric functions. We next consider regular formulas with OR and AND gates. A formula is regular if it is a leveled tree with all gates at a given level having the same fan-in and computing the same function. We prove the sensitivity conjecture for constant depth regular read-k formulas for constant k
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