The Magic Number Problem for Subregular Language Families

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-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata accepting subregular languages like, for example, combinational languages, star-free, prefix-, suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free languages, showing that there are only trivial magic numbers, when they exist. For finite languages we obtain some partial results showing that certain numbers are non-magic.Comment: In Proceedings DCFS 2010, arXiv:1008.127

Inverse star, borders, and palstars

peer reviewedA language L is closed if L = L*. We consider an operation on closed languages, L-*, that is an inverse to Kleene closure. It is known that if L is closed and regular, then L-* is also regular. We show that the analogous result fails to hold for the context-free languages. Along the way we find a new relationship between the unbordered words and the prime palstars of Knuth, Morris, and Pratt. We use this relationship to enumerate the prime palstars, and we prove that neither the language of all unbordered words nor the language of all prime palstars is context-free

Free submonoids and minimal ω-generators of Rω

Let A be an alphabet and let R be a language in A+. An (¿-generator of -R" is a language G such that G" = R". The language Stab(-R") = {u G A* : ttiZ" Ç R"} is a submonoid of A*. We give results concerning the wgenerators for the case when Stab(Ru ) is a free submonoid which are not available in the general case. In particular, we prove that every ((»-generator of 22" contains at least one minimal w-generator of R". Furthermore these minimal w-generators are codes. We also characterize the w-languagea having only finite languages as minimal u-generators. Finally, we characterize the w- languages »-generated by finite prefix codes

The star height of reset-free events and strictly locally testable events

An algorithm is presented for determining the star height of reset-free events and strictly locally testable events

Algorithms for determining relative star height and star height

AbstractLet C = {R1, …, Rm} be a finite class of regular languages over a finite alphabet Σ. Let Δ = {b1, …, bm} be an alphabet, and δ be the substitution from Δ∗ into Σ∗ such that δ(bi) = Ri for all i (1 ≤ i ≤ m). Let R be a regular language over Σ which can be defined from C by a finite number of applications of the operators union, concatenation, and star. Then there exist regular languages over Δ which can be transformed onto R by δ. The relative star height of R w.r.t. C is the minimum star height of regular languages over Δ which can be transformed onto R by δ. This paper proves the existence of an algorithm for determining relative star height. This result obviously implies the existence of an algorithm for determining the star height of any regular language

Components and acyclicity of graphs. An exercise in combining precision with concision

Central to algorithmic graph theory are the concepts of acyclicity and strongly connected components of a graph, and the related search algorithms. This article is about combining mathematical precision and concision in the presentation of these concepts. Concise formulations are given for, for example, the reflexive-transitive reduction of an acyclic graph, reachability properties of acyclic graphs and their relation to the fundamental concept of “definiteness”, and the decomposition of paths in a graph via the identification of its strongly connected components and a pathwise homomorphic acyclic subgraph. The relevant properties are established by precise algebraic calculation. The combination of concision and precision is achieved by the use of point-free relation algebra capturing the algebraic properties of paths in graphs, as opposed to the use of pointwise reasoning about paths between nodes in graphs

Closure algorithms and the star-height problem of regular languages

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