Imre Simon

Recognizable Sets with Multiplicities in the Tropical Semiring

The last ten years saw the emergence of some results about recognizable subsets of a free monoid with multiplicities in the Min-Plus semiring. An interesting aspect of this theoretical body is that its discovery was motivated throughout by applications such as the finite power property, Eggans classical star height problem and the measure of the nondeterministic complexity of finite automata. We review here these results, their applications and point out some open problems.

On semigroups of matrices over the tropical semiring

The tropical semiring M consists of the set of natural numbers extended with infinity, equipped with the operations of taking minimums (as semiring addition) and addition (as semiring multiplication). We use factorization forests to prove finiteness results related to semigroups of matrices over M. Our method is used to recover results of Hashiguchi, Leung and the author in a unified combinatorial framework

Factorization forests of finite height

Abstract We introduce factorization forests. It is shown that the vertex set of a dense factorization forest of finite height satisfies a bounded gap Ramsey-type property of words. The main result is that every morphism from a free semigroup to a finite semigroup S admits a Ramseyan factorization forest of height at most 9|S|. Techniques for constructed factorization forests are developed.

On finite semigroups of matrices

Abstract Finite semigroups of n by n matrices over the naturals are characterized both by algebraic and combinatorial methods. Next we show that the cardinality of a finite semigroup S of n by n matrices over a field is bounded by a function depending only on n , the number of generators of S and the maximum cardinality of its subgroups. As a consequence, given n and k , there exist, up to isomorphism, only a finite number of finite semigroups of n by n matrices over the rationals, generated by at most k elements. Among other applications to Automaton Theory, we show that it is decidable whether the behavior of a given N – Σ automaton is bounded.

String Matching Algorithms and Automata

In this paper we study the structure of finite automata recognizing sets of the form A*p, for some word p, and use the results obtained to improve the Knuth-Morris-Pratt string searching algorithm. We also determine the average number of nontrivial edges of the above automata.

Compression and Entropy

The connection between text compression and the measure of entropy of a source seems to be well known but poorly documented. We try to partially remedy this situation by showing that the topological entropy is a lower bound for the compression ratio of any compressor. We show that for factorial sources the 1978 version of the Ziv-Lempel compression algorithm achieves this lower bound.

The Product of Rational Languages

The very basic operation of the product of rational languages is the source of some of the most fertilizing problems in the Theory of Finite Automata. Indeed, attempts to solve McNaughtons star-free problem, Eggans star-height problem and Brzozowskis dot-depth problem, all three related to the product, already led to many deep and ever expanding connections between the Theory of Finite Automata and other parts of Mathematics, such as Combinatorics, Algebra, Topology, Logic and even Universal Algebra. We review some of the most significant results of the area, obtained during the last 35 years, and try to show their contribution to our understanding of the product.

Sequence Comparision: Some Theory and Some Practice

A brief survey of the theory and practice of sequence comparison is made focusing on diff, the UNIX1 file difference utility.

A note on the triangle conjecture

Abstract We prove some particular cases of the following conjecture of Perrin and Schutzenberger, known as “the triangle conjecture.” Let A = {a, b} be a two-letter alphabet, d a positive integer and let Bd = {aibaj| 0 ⩽ i + j ⩽ d}. If X ⊂ Bd is a code, then |X| ⩽ d + 1.

Properties of factorization forests

It has been proved that every morphism f: A+ → S, with S a finite semigroup, admits a Ramseyan factorization forest of height at most 9|S|. In this paper we show that, up to a constant factor, this result is best possible. More precisely, we show that if S is a finite rectangular band and f(A) = S then every Ramseyan factorization forest admitted by f has height at least |S|.