Dalia Krieger

Every real number greater than 1 is a critical exponent

We prove that for every real number @a>1 there exists an infinite word w over a finite alphabet such that @a is the critical exponent of w.

Finding the Growth Rate of a Regular of Context-Free Language in Polynomial Time

We give an O(n+ t) time algorithm to determine whether an NFA with nstates and ttransitions accepts a language of polynomial or exponential growth. Given a NFA accepting a language of polynomial growth, we can also determine the order of polynomial growth in O(n+ t) time. We also give polynomial time algorithms to solve these problems for context-free grammars.

On stabilizers of infinite words

The stabilizer of an infinite word w over a finite alphabet @S is the monoid of morphisms over @S that fix w. In this paper we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements.

On critical exponents in fixed points of binary k -uniform morphisms

Let w be an infinite fixed point of a binary k-uniform morphism f, and let E(w) be the critical exponent of w. We give necessary and sufficient conditions for E(w) to be bounded, and an explicit formula to compute it when it is. In particular, we show that E(w) is always rational. We also sketch an extension of our method to non-uniform morphisms over general alphabets. MSC: 68R15.

FINDING THE GROWTH RATE OF A REGULAR OR CONTEXT-FREE LANGUAGE IN POLYNOMIAL TIME

We give an O(n + t) time algorithm to determine whether an NFA with n states and t transitions accepts a language of polynomial or exponential growth. Given an NFA accepting a language of polynomial growth, we can also determine the order of polynomial growth in O(n+t) time. We also give polynomial time algorithms to solve these problems for context-free grammars.

On Critical exponents in fixed points of k-uniform binary morphisms

Let w be an infinite fixed point of a binary k -uniform morphism f , and let E w be the critical exponent of w . We give necessary and sufficient conditions for E w to be bounded, and an explicit formula to compute it when it is. In particular, we show that E w is always rational. We also sketch an extension of our method to non-uniform morphisms over general alphabets.

Some remarks about stabilizers

We continue our study of stabilizers of infinite words over finite alphabets, begun in [D. Krieger, On stabilizers of infinite words, Theoret. Comput. Sci. 400 (2008), 169-181]. Let w be an aperiodic infinite word over a finite alphabet, and let Stab(w) be its stabilizer. We show that Stab(w) can be partitioned into the monoid of morphisms that stabilize w by finite fixed points and the ideal of morphisms that stabilize w by iteration. We also settle a conjecture given in the paper mentioned above, by showing that in some cases Stab(w) is infinitely generated. If the aforementioned ideal is nonempty, then it contains either polynomially growing morphisms or exponentially growing morphisms, but not both. Moreover, in the polynomial case, the degree of the polynomial is fixed. We also show how to compute the polynomial degree from the dependency graph of a polynomially growing morphism.

The critical exponent of the Arshon words

Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n ≥ 2, the critical exponent of the Arshon word of order n is given by (3n − 2)/(2n − 2), and this exponent is attained at position 1.