# The Magic Number Problem for Subregular Language Families

@article{Holzer2012TheMN,
title={The Magic Number Problem for Subregular Language Families},
author={Markus Holzer and Sebastian Jakobi and Martin Kutrib},
journal={Int. J. Found. Comput. Sci.},
year={2012},
volume={23},
pages={115-131}
}
• Published 7 August 2010
• Mathematics
• Int. J. Found. Comput. Sci.
We investigate the magic number problem, that is, the question whether there exists a minimal n-state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has alpha states, for all n and alpha satisfying n less or equal to alpha less or equal to exp(2,n). A number alpha not satisfying this condition is called a magic number (for n). It was shown in [11] that no magic numbers exist for general regular languages, while in [5] trivial and non…
13 Citations

## Figures from this paper

### New metrics for finite automaton complexity and subregular language hierarchies

It is shown that the nearly acyclic NFAs are exactly the NFAs with finite depth path width, and an infinite hierarchy exists within the most powerful new family, generalized word-definite languages.

### State Complexity of Prefix Distance of Subregular Languages

• Computer Science
J. Autom. Lang. Comb.
• 2016
Upper bounds and matching lower bounds for the size of the minimal deterministic finite automaton (DFA) needed for the radius k prefix distance neighbourhood of an n state DFA that recognizes, respectively, a finite, a prefix-closed and aprefix-free language.

### COMPLEXITY OF PREFIX DISTANCE OF SUBREGULAR LANGUAGES

• Computer Science
• 2017
Upper bounds and matching lower bounds are given for the size of the minimal deterministic finite automaton (DFA) needed for the radius k prefix distance neighbourhood of an n state DFA that recognizes, respectively, a finite, a prefix-convex, aprefix-closed, a suffix-free, and a right ideal language.

### A Review on State Complexity of Individual Operations

• Computer Science
• 2011
This report reviews some of the results of state complexity of individual operations for regular and some subregular languages on the basis of subset construction and nondeterministic state complexity.

### Networks of evolutionary processors: the power of subregular filters

• Computer Science
Acta Informatica
• 2012
The use of filters from the class of ordered, non-counting, power-separating, circular, suffix-closed regular, union-free, definite, and combinational languages is as powerful as the use of arbitrary regular languages and yields networks that can generate all the recursively enumerable languages.

### Networks of Evolutionary Processors with Subregular Filters

• Computer Science
LATA
• 2011
The use of filters from the class of ordered, non-counting, power-separating, circular, suffix-closed regular, union-free, definite and combinational languages is as powerful as the use of arbitrary regular languages and yields networks that can generate all the recursively enumerable languages.

### The Complexity of Languages Resulting from the Concatenation Operation

• Computer Science
J. Autom. Lang. Comb.
• 2016
We prove that for all m, n, and $$\alpha$$ with $$1 \le \alpha \le f(m,n)$$, where f(m, n) is the state complexity of the concatenation operation, there exist a minimal m-state DFA A and a minimal

## References

SHOWING 1-10 OF 33 REFERENCES

### Deterministic blow-ups of minimal NFA's

The paper treats the question whether there always exists a minimal nondeterministic finite automaton of n states whose equivalent minimal deterministic finite automaton has α states for any integers

### Deterministic Blow-Ups of Minimal Nondeterministic Finite Automata over a Fixed Alphabet

• Computer Science
Developments in Language Theory
• 2007
It follows that in the case of a four-letter alphabet, there are no "magic numbers" i.e., the holes in the hierarchy, as for all integers n and α such that n ⩽ α ⦽ 2n.

### Magic Numbers and Ternary Alphabet

An n-state nondeterministic finite automaton with a three-letter input alphabet that requires exactly α deterministic states is defined.

### On the State Complexity of Complements, Stars, and Reversals of Regular Languages

The deterministic and nondeterministic state complexity of complements, stars, and reversals of regular languages are examined and an exponential number of values that are non-magic are obtained, which improves a similar result of Geffert.

### Nondeterministic Finite Automata-Recent Results on the Descriptional and Computational Complexity

• Computer Science
CIAA
• 2008
This paper discusses recent developments relevant to NFAs related problems like, for example, simulation of and by several types of finite automata, minimization and approximation, size estimation of minimal NFAs, and state complexity of language operations.

### On the Bounds for State-Set Size in the Proofs of Equivalence Between Deterministic, Nondeterministic, and Two-Way Finite Automata

• F. R. Moore
• Computer Science, Mathematics
IEEE Transactions on Computers
• 1971
The bounds on state-set size in the proofs of the equivalence between nondeterministic and deterministic finite automata and between two-way and one-way deterministic finite automata are considered.