2,545 research outputs found
Abstract numeration systems on bounded languages and multiplication by a constant
A set of integers is -recognizable in an abstract numeration system if
the language made up of the representations of its elements is accepted by a
finite automaton. For abstract numeration systems built over bounded languages
with at least three letters, we show that multiplication by an integer
does not preserve -recognizability, meaning that there always
exists a -recognizable set such that is not
-recognizable. The main tool is a bijection between the representation of an
integer over a bounded language and its decomposition as a sum of binomial
coefficients with certain properties, the so-called combinatorial numeration
system
Automatic Equivalence Structures of Polynomial Growth
In this paper we study the class EqP of automatic equivalence structures of the form ?=(D, E) where the domain D is a regular language of polynomial growth and E is an equivalence relation on D. Our goal is to investigate the following two foundational problems (in the theory of automatic structures) aimed for the class EqP. The first is to find algebraic characterizations of structures from EqP, and the second is to investigate the isomorphism problem for the class EqP. We provide full solutions to these two problems. First, we produce a characterization of structures from EqP through multivariate polynomials. Second, we present two contrasting results. On the one hand, we prove that the isomorphism problem for structures from the class EqP is undecidable. On the other hand, we prove that the isomorphism problem is decidable for structures from EqP with domains of quadratic growth
The Almost Equivalence by Asymptotic Probabilities for Regular Languages and Its Computational Complexities
We introduce p-equivalence by asymptotic probabilities, which is a weak
almost-equivalence based on zero-one laws in finite model theory. In this
paper, we consider the computational complexities of p-equivalence problems for
regular languages and provide the following details. First, we give an
robustness of p-equivalence and a logical characterization for p-equivalence.
The characterization is useful to generate some algorithms for p-equivalence
problems by coupling with standard results from descriptive complexity. Second,
we give the computational complexities for the p-equivalence problems by the
logical characterization. The computational complexities are the same as for
the (fully) equivalence problems. Finally, we apply the proofs for
p-equivalence to some generalized equivalences.Comment: In Proceedings GandALF 2016, arXiv:1609.0364
Finding the growth rate of a regular language in polynomial time
We give an O(n^3+n^2 t) time algorithm to determine whether an NFA with n
states and t transitions accepts a language of polynomial or exponential
growth. We also show that given a DFA accepting a language of polynomial
growth, we can determine the order of polynomial growth in quadratic time
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