Karel Culik

On real-time cellular automata and trellis automata

SummaryIt is shown that f(n)-time one-way cellular automata are equivalent to f(n)-time trellis automata, the real-time one-way cellular automata languages are closed under reversal, the 2n-time one-way cellular automata are equivalent to real-time cellular automata and the latter are strictly more powerful than the real-time one-way cellular automata.

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.

Image compression using weighted finite automata

Abstract We introduce Weighted Finite Automata (WFA) as a tool to define real functions, in particular, greyness functions of grey-tone images. Mathematical properties and the definition power of WFA have been studied by Culik and Karhumaki. Their generative power is incomparable with Barnsleys Iterative Function Systems. Here, we given an automatic encoding algorithm that converts an arbitrary grey-tone-image (a digitized photograph) into a WFA that can regenerate it (with or without information loss). The WFA seems to be the first image definition tool with such a relatively simple encoding algorithm.

Computation theoretic aspects of cellular automata

Abstract Cellular automata may be viewed as computational models of complex systems. They also can be seen as continuous functions on a particular family of compact metric spaces. This paper surveys results largely motivated by an attempt to answer questions posed by the latter approach by means of techniques from the former.

On the limit sets of cellular automata

The limit sets of cellular automata, defined by Wolfram, play an important role in applications of cellular automata to complex systems. A number of results on limit sets are proved, considering both finite and infinite configurations of cellular automata. The main concern of this paper is with testing membership and (essential) emptiness of limit sets for linear and two-dimensional cellular automata.

A note on some tree similarity measures

Given two structures of the same type, one standard question is: how close are these two structures to each other? One example is the string-to-string correction problem (see [3,7,9] for example), a second is syntax-error repairing in parsers (see [ 1,5]), and a third is the similarity of two dendrograms [2,6,8] It is this third example we are concerned with in the present note. We consider labelled and unlabelled trees and search trees of the same size n. We show that two trees of the same type are O(n) and O(n log n) distance apart, for unlabelled and labelled trees respectively. The basis for the distance measure is the interchange or rotation tree transformation.

State Complexity of Basic Operations on Finite Languages

The state complexity of basic operations on regular languages has been studied in [9,10,11]. Here we focus on finite languages. We show that the catenation of two finite languages accepted by an m- state and an n-state DFA, respectively, with m > n is accepted by a DFA of (m - n + 3)2n-2 - 1 states in the two-letter alphabet case, and this bound is shown to be reachable. We also show that the tight upperbounds for the number of states of a DFA that accepts the star of an n-state finite language is 2n-3 + 2n-4 in the two-letter alphabet case. The same bound for reversal is 3 ? 2p-1 - 1 when n is even and 2p - 1 when n is odd. Results for alphabets of an arbitrary size are also obtained. These upper-bounds for finite languages are strictly lower than the corresponding ones for general regular languages.

The Decidability of the Equivalence Problem for DOL-Systems*

FRISK Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI The language and sequence equivalence problem for DOL-systems is shown to be decidable. In an algebraic formulation the sequence equivalence problem for DOL-systems can be stated as follows: Given homomorphisms h, and h, on a free monoid z1* and a word D from Z*, is %rn(r,) = &“(o) for all n > O?

An aperiodic set of 13 Wang tiles

Abstract A new aperiodic tile set containing only 13 tiles over 5 colors is presented. Its construction is based on a recent technique developed by Kari. The tilings simulate the behavior of sequential machines that multiply real numbers in balanced representations by real constants.

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.