900 research outputs found
On a zeta function associated with automata and codes
AbstractThe zeta function of a finite automaton A is exp{∑n=1∞anznn}, where an is the number of bi-infinite paths in A labelled by a bi-infinite word of period n. It reflects the properties of A: aperiodicity, nil-simplicity, existence of a zero. The results are applied to codes
Sofic-Dyck shifts
We define the class of sofic-Dyck shifts which extends the class of
Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck
shifts are shifts of sequences whose finite factors form unambiguous
context-free languages. We show that they correspond exactly to the class of
shifts of sequences whose sets of factors are visibly pushdown languages. We
give an expression of the zeta function of a sofic-Dyck shift
Cyclic languages and strongly cyclic languages
International audienceWe prove that cyclic languages are the boolean closure of languages called strongly cyclic languages. The result is used to give another proof of the rationality of the zeta function of rational cyclic languages
Dynamically affine maps in positive characteristic
We study fixed points of iterates of dynamically affine maps (a
generalisation of Latt\`es maps) over algebraically closed fields of positive
characteristic . We present and study certain hypotheses that imply a
dichotomy for the Artin-Mazur zeta function of the dynamical system: it is
either rational or non-holonomic, depending on specific characteristics of the
map. We also study the algebraicity of the so-called tame zeta function, the
generating function for periodic points of order coprime to . We then verify
these hypotheses for dynamically affine maps on the projective line,
generalising previous work of Bridy, and, in arbitrary dimension, for maps on
Kummer varieties arising from multiplication by integers on abelian varieties.Comment: Lois van der Meijden co-authored Appendix B. 31 p
Multiband linear cellular automata and endomorphisms of algebraic vector groups
We propose a correspondence between certain multiband linear cellular
automata - models of computation widely used in the description of physical
phenomena - and endomorphisms of certain algebraic unipotent groups over finite
fields. The correspondence is based on the construction of a universal element
specialising to a normal generator for any finite field. We use this
correspondence to deduce new results concerning the temporal dynamics of such
automata, using our prior, purely algebraic, study of the endomorphism ring of
vector groups. These produce 'for free' a formula for the number of fixed
points of the -iterate in terms of the -adic valuation of , a
dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic
formula for the number of periodic orbits. Since multiband linear cellular
automata simulate higher order linear automata (in which states depend on
finitely many prior temporal states, not just the direct predecessor), the
results apply equally well to that class.Comment: 11 page
The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
Only Human: a book review of The Turing Guide
This is a review of The Turing Guide (2017), written by Jack Copeland, Jonathan Bowen, Mark Sprevak, Robin Wilson, and others. The review includes a new sociological approach to the problem of computability in physics
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