Tero Harju

Splicing semigroups of dominoes and DNA

Abstract We introduce semigroups of dominoes as a tool for working with sets of linked strings. In particular, we are interested in splicing semigroups of dominoes. In the special case of alphabetic (symbol-to-symbol-linked) dominoes the splicing semigroups are essentially equivalent to the splicing systems introduced by Head to study informational macromolecules, specifically to study the effect of sets of restriction enzymes and ligase that allow DNA molecules to be cleaved and reassociated to produce further molecules. Our main result is that in the case of alphabetic dominoes the splicing semigroup generated from an initial regular set is again regular. This implies positive solution of two open problems stated by Head, namely the regularity of splicing systems and the decidability of their membership problem.

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.

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.

The theory of 2-structures : a framework for decomposition and transformation of graphs

The theory of 2-structures provides a convenient framework for decomposition and transformation of mathematical systems where one or several different binary relationships hold between the objects of the system. In particular, it forms a useful framework for decomposition and transformation of graphs.The decomposition methods presented in this book correspond closely to the top-down design methods studied in computer science. The transformation methods considered here have a natural interpretation in the dynamic evolution of certain kinds of communication networks. From the mathematical point of view, the clan decomposition method presented here, also known as modular decomposition or substitution decomposition, is closely related to the decomposition by quotients in algebra. The transformation method presented here is based on labelled 2-structures over groups, the theory of which generalizes the well-studied theory of switching classes of graphs.This book is both a text and a monograph. As a monograph, the results concerning the decomposition and transformation of 2-structures are presented in a unified way. In addition, detailed notes on references are provided at the end of each chapter. These notes allow the reader to trace the origin of many notions and results, and to browse through the literature in order to extend the material presented in the book.To facilitate its use as a textbook, there are numerous examples and exercises which provide an opportunity for the reader to check his or her understanding of the discussed material. Furthermore, the text begins with preliminaries on partial orders, semigroups, groups and graphs to the extent needed for the book.

ON THE UNDECIDABILITY OF FREENESS OF MATRIX SEMIGROUPS

We slightly improve the result of Klarner, Birget and Satterfield, showing that the freeness of finitely presented multiplicative semigroups of 3×3 matrices over ℕ is undecidable even for triangular matrices. This is achieved by proving a new variant of Post correspondence problem. We also consider the freeness problem for 2×2 matrices. On the one hand, we show that it cannot be proved undecidable using the above methods which work in higher dimensions, and, on the other hand, we give some evidence that its decidability, if so, is also a challenging problem.

Characterizing the Micronuclear Gene Patterns in Ciliates

Abstract The process of gene assembly in ciliates is one of the most complex examples of DNA processing known in any organism, and it is fascinating from the computational point of view—it is a prime example of DNA computing in vivo. In this paper we continue to investigate the three molecular operations (ld, hi , and dlad ) that were postulated to carry out the gene assembly process in the intramolecular fashion. In particular, we focus on the understanding of the IES/ MDS patterns of micronuclear genes, which is one of the important goals of research on gene assembly in ciliates. We succeed in characterizing for each subset S of the three molecular operations those patterns that can be assembled using operations in S. These results enhance our understanding of the structure of micronuclear genes (and of the nature of molecular operations). They allow one to establish both similarity and complexity measures for micronuclear genes.

Theory of 2-Structures

From the combination of knowledge and actions, someone can improve their skill and ability. It will lead them to live and work much better. This is why, the students, workers, or even employers should have reading habit for books. Any book will give certain knowledge to take all benefits. This is what this theory of 2 structures tells you. It will add more knowledge of you to life and work better. Try it and prove it.

Undecidability Bounds for Integer Matrices using Claus Instances

There are several known undecidable problems for 3 × 3 integer matrices the proof of which use a reduction from the Post Correspondence Problem (PCP). We establish new lower bounds in the number of matrices for the mortality, zero in the left upper corner, vector reachability, matrix reachability, scalar reachability and freeness problems. Also, we give a short proof for a strengthened result due to Bell and Potapov stating that the membership problem is undecidable for finitely generated matrix semigroups R ⊆ ℤ4×4 whether or not kI4 ∈ R for any given |k| > 1. These bounds are obtained by using the Claus instances of the PCP.

Binary (generalized) post correspondence problem

We give a new proof for the decidability of the binary Post Correspondence Problem (PCP) originally proved in 1982 by Ehrenfeucht, Karhumki and Rozenberg. Our proof is complete and somewhat shorter than the original proof although we use the same basic. Copyright 2002 Elsevier Science B.V. All rights reserved.

Mortality in Matrix Semigroups

We present a new shorter and simplified proof for the undecidability of the mor- tality problem in matrix semigroups, originally proved by Paterson in 1970. We use the clever coding technique introduced by Paterson to achieve also a new res- ult, the undecidability of the vanishing (left) upper corner. Since our proof for the undecidability of the mortality problem uses only 8 matrices, a new bound for the dimension for the undecidability of the mortality in the two generator matrix semigroup is achieved.