Threshold nets and cell-assemblies

Motivated by the cell-assemblies theory of the brain, we propose a new formal model of threshold nets (TN). TN are patterned after Petri nets, with a very different firing rule, which removes all tokens upon firing of a transition. The generative power of threshold nets, with and without inhibition, is compared with traditional families of languages. Excitatory TN languages are included by the noncounting regular languages and form an infinite hierarchy for increasing values of threshold. Inhibitory nets are included by the context-sensitive languages. Two new net operators, motivated by the phenomena of growth, learning and brain damage are introduced and compared with Boolean operators

Languages Generated by Iterated Idempotencies.

The rewrite relation with parameters m and n and with the possible length limit = k or :::; k we denote by w~, =kW~· or ::;kw~ respectively. The idempotency languages generated from a starting word w by the respective operations are wDAlso other special cases of idempotency languages besides duplication have come up in different contexts. The investigations of Ito et al. about insertion and deletion, Le., operations that are also observed in DNA molecules, have established that w5 and w~ both preserve regularity.Our investigations about idempotency relations and languages start out from the case of a uniform length bound. For these relations =kW~ the conditions for confluence are characterized completely. Also the question of regularity is -k n answered for aH the languages w- D 1 are more complicated and belong to the class of context-free languages.For a generallength bound, i.e."for the relations :"::kW~, confluence does not hold so frequently. This complicatedness of the relations results also in more complicated languages, which are often non-regular, as for example the languages WWithout any length bound, idempotency relations have a very complicated structure. Over alphabets of one or two letters we still characterize the conditions for confluence. Over three or more letters, in contrast, only a few cases are solved. We determine the combinations of parameters that result in the regularity of wDIn a second chapter sorne more involved questions are solved for the special case of duplication. First we shed sorne light on the reasons why it is so difficult to determine the context-freeness ofduplication languages. We show that they fulfiH aH pumping properties and that they are very dense. Therefore aH the standard tools to prove non-context-freness do not apply here.The concept of root in Formal Language ·Theory is frequently used to describe the reduction of a word to another one, which is in sorne sense elementary.For example, there are primitive roots, periodicity roots, etc. Elementary in connection with duplication are square-free words, Le., words that do not contain any repetition. Thus we define the duplication root of w to consist of aH the square-free words, from which w can be reached via the relation w~.Besides sorne general observations we prove the decidability of the question, whether the duplication root of a language is finite.Then we devise acode, which is robust under duplication of its code words.This would keep the result of a computation from being destroyed by dupli cations in the code words. We determine the exact conditions, under which infinite such codes exist: over an alphabet of two letters they exist for a length bound of 2, over three letters already for a length bound of 1.Also we apply duplication to entire languages rather than to single words; then it is interesting to determine, whether regular and context-free languages are closed under this operation. We show that the regular languages are closed under uniformly bounded duplication, while they are not closed under duplication with a generallength bound. The context-free languages are closed under both operations.The thesis concludes with a list of open problems related with the thesis' topics

On some modifications and applications of the post correspondence problem

The Post Correspondence Problem was introduced by Emil Post in 1946. The problem considers pairs of lists of sequences of symbols, or words, where each word has its place on the list determined by its index. The Post Correspondence Problem asks does there exist a sequence of indices so that, when we write the words in the order of the sequence as single words from both lists, the two resulting words are equal. Post proved the problem to be undecidable, that is, no algorithm deciding it can exist. A variety of restrictions and modifications have been introduced to the original formulation of the problem, that have then been shown to be either decidable or undecidable. Both the original Post Correspondence Problem and its modifications have been widely used in proving other decision problems undecidable. In this thesis we consider some modifications of the Post Correspondence Problem as well as some applications of it in undecidability proofs. We consider a modification for sequences of indices that are infinite to two directions. We also consider a modification to the original Post Correspondence Problem where instead of the words being equal for a sequence of indices, we take two sequences that are conjugates of each other. Two words are conjugates if we can write one word by taking the other and moving some part of that word from the end to the beginning. Both modifications are shown to be undecidable. We also use the Post Correspondence Problem and its modification for injective morphisms in proving two problems from formal language theory to be undecidable; the first problem is on special shuffling of words and the second problem on fixed points of rational functions

Decomposition and Descriptional Complexity of Shuffle on Words and Finite Languages

We investigate various questions related to the shuffle operation on words and finite languages. First we investigate a special variant of the shuffle decomposition problem for regular languages, namely, when the given regular language is the shuffle of finite languages. The shuffle decomposition into finite languages is, in general not unique. Thatis,therearelanguagesL^,L2,L3,L4withLiluL2= £3luT4but{L\,L2}^ {I/3, L4}. However, if all four languages are singletons (with at least two combined letters), it follows by a result of Berstel and Boasson [6], that the solution is unique; that is {L\,L2} = {L3,L4}. We extend this result to show that if L\ and L2 are arbitrary finite sets and Lz and Z-4 are singletons (with at least two letters in each), the solution is unique. This is as strong as it can be, since we provide examples showing that the solution can be non-unique already when (1) both L\ and L2 are singleton sets over different unary alphabets; or (2) L\ contains two words and L2 is singleton. We furthermore investigate the size of shuffle automata for words. It was shown by Campeanu, K. Salomaa and Yu in [11] that the minimal shuffle automaton of two regular languages requires 2mn states in the worst case (where the minimal automata of the two component languages had m and n states, respectively). It was also recently shown that there exist words u and v such that the minimal shuffle iii DFA for u and v requires an exponential number of states. We study the size of shuffle DFAs for restricted cases of words, namely when the words u and v are both periods of a common underlying word. We show that, when the underlying word obeys certain conditions, then the size of the minimal shuffle DFA for u and v is at most quadratic. Moreover we provide an efficient algorithm, which decides for a given DFA A and two words u and v, whether u lu u C L(A)

Clearing Restarting Automata

Restartovací automaty byly navrženy jako model pro redukční analýzu, která představuje lingvisticky motivovanou metodu pro kontrolu korektnosti věty. Cílem práce je studovat omezenější modely restartovacích automatů, které smí vymazat podřetězec nebo jej nahradit speciálním pomocným symbolem, jenom na základě omezeného lokálního kontextu tohoto podřetězce. Tyto restartovací automaty se nazývají clearing restarting automata. V práci jsou taktéž zkoumány uzávěrové vlastnosti těchto automatů, jejich vztah k Chomskeho hierarchii a možnosti učení těchto automatů na základě pozitivních a negativních příkladů.Restarting automata were introduced as a model for analysis by reduction which is a linguistically motivated method for checking correctness of a sentence. The goal of the thesis is to study more restricted models of restarting automata which based on a limited context can either delete a substring of the current content of its tape or replace a substring by a special symbol, which cannot be overwritten anymore, but it can be deleted later. Such restarting automata are called clearing restarting automata. The thesis investigates the properties of clearing restarting automata, their relation to Chomsky hierarchy and possibilities for machine learning of such automata from positive and negative samples.Department of Software and Computer Science EducationKatedra softwaru a výuky informatikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Regular expressions for muller context-free languages

Muller context-free languages (MCFLs) are languages of countable words, that is, labeled countable linear orders, generated by Muller context-free grammars. Equivalently, they are the frontier languages of (nondeterministic Muller-)regular languages of infinite trees. In this article we survey the known results regarding MCFLs, and show that a language is an MCFL if and only if it can be generated by a so-called µη-regular expression