Václav Koubek

Mathematical Foundations of Computer Science 2004

We introduce and analyse a simple model of genome evolution. It is based on two fundamental evolutionary events: gene loss and gene duplication. We are mainly interested in asymptotic distributions of gene families in a genome. This is motovated by previous work which consisted in fitting the available genomic data into, what is called paralog distributions. Two approaches are presented in this paper: continuous and discrete time models. A comparison of them is presented too – the asymptotic distribution for the continuous time model can be seen as a limit of the discrete time distributions, when probabilities of gene loss and gene duplication tend to zero. We view this paper as an intermediate step towards mathematically settling the problem of characterizing the shape of paralog distribution in bacterial genomes.

On the greatest fixed point of a set functor

Abstract The greatest fixed point of a set functor is proved to be (a) a metric completion and (b) a CPO-completion of finite iterations. For each (possibly infinitary) signature Σ the terminal Σ-coalgebra is thus proved to be the coalgebra of all Σ-labelled trees; this is the completion of the set of all such trees of finite depth. A set functor is presented which has a fixed point but does not have a greatest fixed point. A sufficient condition for the existence of a greatest fixed point is proved: the existence of two fixed points of successor cardinalities.

A reduct-and-closure algorithm for graphs

The transitive closure of (V,E) is the graph Clos(V,E) = (V,C) where (x,y) £C iff there is a path p:x-~y in (V,E) with ILpil>0. The transitive reduct Red(V,E) of an acyclic graph (V,E) is the least graph (V,R) with Clos(V,R) = Clos(V,E) . The number o f e l emen t s o f a f i n i t e s e t X w i l l be denoted t h r o u ghout by IX1 . If Alg is an algorithm processing graphs, Time (Alg(V,E)) denotes the number of steps taken by Alg to process (V,E) . We write Time(Alg(V,E)) ~ 0(f(V,E)) , for a function f:K -~ N from some class of graphs to the non-negative integers N , if there exists ceN such that the inequality Time(Alg(V,E)) ~ c.f(V,E) holds for all (V,E)~ K . We then also say that Alg needs O(f(V,E)) time to process a member (V,E) of K , or that Alg has the time complexity O(f) on K . The main objective of the present note is to describe an algorithm Recl computing simultaneously both Red(V,E) and Clos(V,E) , for (V,E) acyclic, with the time complexity Time(Recl(V,E)) ~ O(IV I .~R~+ +lE l ) comparing f a v o r a b l y wi th the a l g o r i t h m s d e s c r i b e d i n [1~ , [2J , [ 5S , [6]. We also describe some useful modifications of this basic algorithm.

Least Fixed Point of a Functor

We exhibit a con- struction of the LFP, generalizing the Knaster-Tarski formula lub(F”(O)},,, : lubs are substituted by well-ordered colimits and n is allowed to be an arbitrary ordinal. Related LFP constructions, always restricted to 1z E w, have been considered by various authors [6, 11, 13, 141. The advantage of the present approach is its effectiveness: Whenever a functor

CATEGORICAL CONSTRUCTIONS OF FREE ALGEBRAS, COLIMITS, AND COMPLETIONS OF PARTIAL ALGEBRAS

1.1. Given a base category K and a functor F: K -, K, we shall consider the category A(F) of F-algebras and the category PA(F) of partial F-algebras (for definitions, see 1.2). The category A(F) has been studied in a lot of papers [2-7, 13, 171 see also [ll, 121 in connection with the categorical universal algebra and automata and control .theory. We shall deal with the following problems: (1) Existence and construction of free algebras in A(F). (2) Existence and construction of colimits in A(F). (3) Existence and construction of left adjoints to functors A(F) -j A(G) induced by transformations G += F. (4) Completions of partial algebras and other properties of the category PA(F). (5) Cocompleteness of the category of algebras for a triple (F, q, p) in K. The present paper is based on the thesis [15] of the second author which generalizing categorical constructions [l, 4, 7, 131 and classical constructions of universal algebra attempts to form a general theory of constructions of free algebras, colimits etc. Some results of [ 151 are improved and the approach is applied to partial algebras and to algebras for a triple. As a technical tool, we shall embed A(F) into the category A*(F) [15] of algebraized chains which possess convenient properties (Sections 2-4). The category PA(F) will be investigated by means of the embedding into the category GPA(F) of generalized partial algebras. Notice that categories A*(F) and GPA(F) form “completions” of A(F): they have free algebras and are cocomplete. The usefulness

Cartesian closed initial completions

Abstract We construct cartesian closed extensions of concrete categories with special (topological) properties. As a consequence we find a necessary and sufficient condition for a concrete category to have finitely productive, cartesian closed initial completion. Finally, we exhibit a topological category, not satisfying this condition; this gives a negative answer to the problem of Herrlich and Nel whether each topological category has a cartesian closed topological bull [6]. These results have been announced in [1].

On the minimum order of graphs with given semigroup

Abstract Denote by M(n) the smallest positive integer such that for every n-element monoid M there is a graph G with at most M(n) vertices such that End(G) is isomorphic to M. It is proved that 2 (1 + o(1))n log 2 n ≤M(n)≤n · 2n + O(n) . Moreover, for almost all n-element monoids M there is a graph G with at most 12 · n · log2n + n vertices such that End(G) is isomorphic to M.

Parallel algorithms for connected components in a graph

The aim of this paper is to give new implementations of parallel algorithms for a construction of connected components of a graph As a model of parallel computation we use parallel random-access machine (PRAM). We assume that all processors share the same memory, eaeh proeessor can perform any arithmetic, Boolean or logical operations in one time unit and all instructions executed in parallel are identical (Single Instruction Stream Multiple Data Stream SIMD) Using of the o@mmon memory leads to different conflicts. There exist three models with respect to a solution of these conflicts: The exclusive read exclusive write (~REW) model at most one processor can read from or write to a given memory location at a time. It is the weakest model. The concurrent-read exclusive-write (CREW) model an arbitrary number of processors can read from a given memory location but at most one processor can write to it at a time. It is the model most frequently used. The concurrent-read concurrent-write (CRCW) model an arbitrary number of processors can read from or write to a given memory location at a time (in the latter case one processor succeeds but we do not know in advance which see [5] and ~ 9] ) It is the most powerful model~

Fast algorithms constructing minimal subalgebras, congruences, and ideals in a finite algebra

Abstract Fast algorithms are presented which find the minimal nontrivial congruences, subalgebras, and ideals of a finite algebra, given by tables of its operations. Our method involves finding objects which are minimal among all nontrivial objects, i.e., objects distinct from the least one. The running times of the presented algorithms are O( n r +1 ), in case of congreunces, and O( n r ), in case of subalgebras and ideals, where n is the number of elements of the algebra and r is the maximal arity of its operations. For the minimal nontrivial congruences in groups and rings, better algorithms are presented working in O( n 2 ) time. The algorithm for minimal nontrivial congruences is used for a test whether a given algebra is simple, or subdirectly irreducible. This algorithm is both a generalization and an improvement of that for sequential automata, published by the present authors (1981).

RANK PROBLEMS FOR COMPOSITE TRANSFORMATIONS

Let (X, F) be a pair consisting of a finite set X and a set F of transformations of X, and, let and F(l) denote, respectively, the semigroup generated by F and the part of consisting of the transformations determined by a generator sequence of length no more than a given integer l. We show the following: • The problem whether or not, for a given pair (X, F) and a given integer r, there is an idempotent transformation of rank r in is PSPACE-complete. • For each fixed r≥1, it is decidable in a polynomial time, for a given pair (X, F), whether or not contains an idempotent transformation of rank r, and, if yes then a generator sequence of polynomial length composing to an idempotent transformation of rank r can be obtained in a polynomial time. • For each fixed r≥1, the problem whether or not, for a given (X, F) and l, there is an idempotent transformation of rank r in F(l) is NP-complete. • For each fixed r≥2, to decide, for a given (X, F), whether or not contains a transformation of rank r is NP-hard.