# A Decision Method for The Recognizability of Sets Defined by Number Systems

@article{Honkala1986ADM, title={A Decision Method for The Recognizability of Sets Defined by Number Systems}, author={Juha Honkala}, journal={RAIRO Theor. Informatics Appl.}, year={1986}, volume={20}, pages={395-403} }

— We show that it is decidable whether or not a k-recognizable set is recognizable. Consequently, it is decidable whether or not the set defined by a number System is recognizable. Résumé. Nous montrons qu'il est décidable si un ensemble k-reconnaissable est reconnaissabîe. En conséquence, il est décidable si Vensemble défini par un système de numération est reconnaissabîe.

## 51 Citations

A Decision Method for the Unambiguity of Sets Defined by Number Systems

- MathematicsJ. Univers. Comput. Sci.
- 1995

We show that it is decidable, given a number system N whether or not there is an unambiguous number system equivalent to N.

On Number Systems with Finite Degree of Ambiguity

- MathematicsInf. Comput.
- 1998

Abstract We show that it is decidable whether or not a given n -recognizable set is representable by a number system having finite degree of ambiguity. As a corollary we obtain an algorithm for…

LOGIC AND p-RECOGNIZABLE SETS OF INTEGERS

- Computer Science, Mathematics
- 1994

Cobham’s theorem is focused on which characterizes the sets recognizable in dierent bases p and on its generalization to N m due to Semenov.

On number systems with negative digits

- Mathematics, Computer Science
- 1989

It is shown that the set of nonnegative integers represented by a number system -lf = (n,rn1,. . . ,mo) is n-recognizable and the equivalence problem for number systems is decidable.

Syntactic Complexity of Ultimately Periodic Sets of Integers and Application to a Decision Procedure

- MathematicsFundam. Informaticae
- 2012

We compute the cardinality of the syntactic monoid of the language 0a repb(m$\mathbb{N}$) made of base b expansions of the multiples of the integer m. We also give lower bounds for the syntactic…

Enumeration and Decidable Properties of Automatic Sequences

- Mathematics, Computer ScienceDevelopments in Language Theory
- 2011

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many…

Ultimate Periodicity of b-Recognisable Sets: A Quasilinear Procedure

- MathematicsDevelopments in Language Theory
- 2013

It is decidable if a set of numbers, whose representation in a base b is a regular language, is ultimately periodic. This was established by Honkala in 1986.

Syntactic Complexity of Ultimately Periodic Sets of Integers

- MathematicsLATA
- 2011

The cardinality of the syntactic monoid of the language 0* repb(mN) made of base b expansions of the multiples of the integer m is computed and lower bounds are given for any (ultimately) periodic set of integers written in base b.

An efficient algorithm to decide periodicity of b-recognisable sets using LSDF convention

- Computer ScienceICALP
- 2017

It is shown that it can be verified in linear time if a minimal automaton meets this description of the minimal automata that accept periodic sets, and this yields a procedure to decide whether an automaton with n states accepts an ultimately periodic set of nonnegative integers.

A Decision Problem for Ultimately Periodic Sets in Non-standard Numeration Systems

- Mathematics, Computer ScienceMFCS
- 2008

A decision procedure is obtained under some hypothesis about the considered numeration system and an analogous decision result is obtained for a particular class of abstract numeration systems built on an infinite regular language.

## References

SHOWING 1-5 OF 5 REFERENCES

Automata, languages, and machines. A

- Computer SciencePure and applied mathematics
- 1974

and Number Systems, Theoret

- Comput. Sci., Vol. 22,
- 1983