Juhani Karhumäki

Combinatorics of words

This is a survey on combinatorics of words to appear as a chapter in Handbook of Formal Languages. The topics covered in details are: defect effect, equations as properties of words, periodicity, finiteness conditions, avoidabilty and subword complexity.

Computer Science – Theory and Applications

Classical error-correcting codes deal with the problem of data transmission over a noisy channel. There are efficient error-correcting codes that work even when the noise is adversarial. In the interactive setting, the goal is to protect an entire conversation between two (or more) parties from adversarial errors. The area of interactive error correcting codes has experienced a substantial amount of activity in the last few years. In this talk we will introduce the problem of interactive errorcorrection and discuss some of the recent results. Finding All Solutions of Equations in Free Groups and Monoids with Involution Volker Diekert, Artur Jeż , and Wojciech Plandowski 1 Institut für Formale Methoden der Informatik, University of Stuttgart, Germany 2 Institute of Computer Science, University of Wroclaw, Poland 3 Max Planck Institute für Informatik, Saarbrcken, Germany 4 Institute of Informatics, University of Warsaw, Poland Abstract. The aim of this paper is to present a PSPACE algorithm The aim of this paper is to present a PSPACE algorithm which yields a finite graph of exponential size and which describes the set of all solutions of equations in free groups and monoids with involution in the presence of rational constraints. This became possible due to the recently invented recompression technique of the second author. He successfully applied the recompression technique for pure word equations without involution or rational constraints. In particular, his method could not be used as a black box for free groups (even without rational constraints). Actually, the presence of an involution (inverse elements) and rational constraints complicates the situation and some additional analysis is necessary. Still, the recompression technique is powerful enough to simplify proofs for many existing results in the literature. In particular, it simplifies proofs that solving word equations is in PSPACE (Plandowski 1999) and the corresponding result for equations in free groups with rational constraints (Diekert, Hagenah and Gutiérrez 2001). As a byproduct we obtain a direct proof that it is decidable in PSPACE whether or not the solution set is finite. * Supported by Humboldt Research Fellowship for Postdoctoral Researchers. 1 A full version of the present paper with detailed proofs can be found on arXiv. Algorithmic Meta Theorems for Sparse Graph Classes

The equivalence problem of multitape finite automata

Abstract Using a result of B.H. Neumann we extend Eilenbergs Equality Theorem to a general result which implies that the multiplicity equivalence problem of two (nondeterministic) multitape finite automata is decidable. As a corollary we solve a long standing open problem in automata theory, namely, the equivalence problem for multitape deterministic finite automata. The main theorem states that there is a finite test set for the multiplicity equivalence of finite automata over conservative monoids embeddable in a fully ordered group.

Finite Automata Computing Real Functions

A new application of finite automata as computers of real functions is introduced. It is shown that even automata with a restricted structure compute all polynomials, many fractal-like and other functions. Among the results shown, the authors give necessary and sufficient conditions for continuity, show that continuity and equivalence are decidable properties, and show how to compute integrals of functions in the automata representation.

On cube-free ω-words generated by binary morphisms

Abstract Let h be a morphism satisfying h(a) = ax for a letter a and a nonempty word x. Then h defines an infinite word (an ω-word) when applied iteratively starting from a. Such ω-words are considered in a binary case. It is shown that only biprefixes can generate cube-free ω-words, i.e. words which do not contain a word υ3, with υ ≠ λ, as a subword. The same does not hold true for fourth power-free ω-words, the counterexample being the ω-word defined by the Fibonaccimorphism: h(a) = ba, h(b) = a. As the main result it is proved that it is decidable whether a given morphism of the above form generates a cube-free ω-word. Moreover, it is shown that no more than 10 steps of iterations are needed to solve the problem.

Communication Complexity Method for Measuring Nondeterminism in Finite Automata

While deterministic finite automata seem to be well understood, surprisingly many important problems concerning nondeterministic finite automata (nfas) remain open. One such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata. In this paper the concept of communication complexity is applied in order to achieve progress in this problem area. The main results are as follows:(1) Deterministic communication complexity provides lower bounds on the size of nfas with bounded unambiguity. Applying this fact, the proofs of several results about nfas with limited ambiguity can be simplified and presented in a uniform way. (2) There is a family of languages KONk2 with an exponential size gap between nfas with polynomial leaf number/ambiguity and nfas with ambiguity k. This partially provides an answer to the open problem posed by B. Ravikumar and O. Ibarra (1989, SIAM J. Comput. 18, 1263-1282) and H. Leung (1998, SIAM J. Comput. 27, 1073-1082).

Some Decision Problems Concerning Semilinearity and Commutation

Let M be a class of automata (in a precise sense to be defined) and Mc the class obtained by augmenting each automaton in M with finitely many reversal-bounded counters. We show that if the languages defined by M are effectively semilinear, then so are the languages defined by Mc, and, hence, their emptiness problem is decidable. We give examples of how this result can be used to show the decidability of certain problems concerning the equivalence of morphisms on languages. We also prove a surprising undecidability result for commutation of languages: given a fixed two-element code K, it is undecidable whether a given context-free language L commutes with K, i.e., LK=KL.

Polynomial versus exponential growth in repetition-free binary words

It is known that the number of overlap-free binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7/3. More precisely, there are only polynomially many binary words of length n that avoid 7/3-powers, but there are exponentially many binary words of length n that avoid 7/3+-powers. This answers an open question of Kobayashi from 1986.

The equivalence of finite valued transducers (on HDT0L languages) is decidable

Abstract We show a generalization of the Ehrenfeucht Conjecture: for every language there exists a (finite) test set with respect to normalized k-valued finite transducers with bounded number of states. Further, we show that, for each HDT0L language, such a test can be effectively found. As a corollary we solve an open problem by Gurari and Ibarra: the equivalence problem for finite valued finite transducers is decidable. This is the first time the equivalence problem is shown to be decidable for a larger class of multivalued transducers.

The expressibility of languages and relations by word equations

Classically, several properties and relations of words, such as “being a power of the same word” can be expressed by using word equations. This paper is devoted to a general study of the expressive power of word equations. As main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations. In particular, “the primitiveness” and “the equal length” are such properties, as well as being “any word over a proper subalphabet”.