9,792 research outputs found
Continuity of Functional Transducers: A Profinite Study of Rational Functions
A word-to-word function is continuous for a class of languages~
if its inverse maps _languages to~. This notion
provides a basis for an algebraic study of transducers, and was integral to the
characterization of the sequential transducers computable in some circuit
complexity classes.
Here, we report on the decidability of continuity for functional transducers
and some standard classes of regular languages. To this end, we develop a
robust theory rooted in the standard profinite analysis of regular languages.
Since previous algebraic studies of transducers have focused on the sole
structure of the underlying input automaton, we also compare the two algebraic
approaches. We focus on two questions: When are the automaton structure and the
continuity properties related, and when does continuity propagate to
superclasses
Rationality of semialgebraic functions
Let X be an algebraic subset of Rⁿ, and ƒ: X → R a semialgebraic function. We prove that if ƒ is continuous rational on each curve C ⊂ X then: 1) ƒ is arc-analytic, 2) ƒ is continuous rational on X. As a consequence we obtain a characterization of hereditarily rational functions recently studied by J. Kollár and K. Nowak
On univoque Pisot numbers
We study Pisot numbers which are univoque, i.e., such that
there exists only one representation of 1 as , with . We prove in particular that there
exists a smallest univoque Pisot number, which has degree 14. Furthermore we
give the smallest limit point of the set of univoque Pisot numbers.Comment: Accepted by Mathematics of COmputatio
Calibrating Generative Models: The Probabilistic Chomsky-Schützenberger Hierarchy
A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning
Multiplicative measures on free groups
We introduce a family of atomic measures on free groups generated by
no-return random walks. These measures are shown to be very convenient for
comparing "relative sizes" of subgroups, context-free and regular subsets
(that, subsets generated by finite automata) of free groups. Many asymptotic
characteristics of subsets and subgroups are naturally expressed as analytic
properties of related generating functions. We introduce an hierarchy of
asymptotic behaviour "at infinity" of subsets in the free groups, more
sensitive than the traditionally used asymptotic density, and apply it to
normal subgroups and regular subsets.Comment: LaTeX, requires amssymb.sty; 31 pp Version 3: more detail in Example
2 and Tauberian theorem
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