Jozef Gruska

Systolic trellis automatat

In the second part of this paper we discuss mainly various programming techniques for systolic automata. These techniques are used to derive theoretical results e.g. the closure of (homogeneous) systolic automaton under Boolean operations, or that every linear context free language can be accepted by a homogeneous systolic automaton and that their languages have time complexity 0(n2)

Some classifications of context-free languages

The basic definitions and notations of the theory of context-free grammars and languages (briefly grammars and languages) used in this paper are as in Ginsburg (1966). The classification of languages L according to the minimal number of variables in grammars for L was studied in Gruska (1967). In this paper some other classifications of grammars and languages are investigated. They are chosen in such a way as to characterize some aspects of our intuitive notion about complexity (of the description) of grammars and languages and their intrinsic structure. The classifications of languages are indicated by those of grammars. The intrinsic structure of a grammar G is characterized by the number and by the depth of the grammatical levels of G. A grammatical level Go of a grammar G is a maximal set of productions of G the left-side symbols of which are mutually dependent. The basic concepts of grammatical levels and classifications of grammars and languages are given in Sections 2 and 3. Only such classifications K are considered here, wherein for every grammar G (language L) K(G) (K(L)) is an integer. In this paper only nonnegative integers will be considered. A classification K is said to be connected in an alphabet Z if for every integer n there is a language L c Z* such that K(L) = n. Sections 4 to 6 provide the proofs that the classifications according to the number of variables, the number of productions, the number of grammatical levels, the number of non-elementary grammatical levels (that is, the grammatical levels with at least two variables) and the maximal depth of grammatical levels (that is, according to the maximal number of variables in grammatical levels) are connected in any alphabet with

Systolic Trellis automata: stability, decidability and complexity

Systolic trellis automata are models of hexagonally connected and triangularly shaped systolic arrays. This paper studies the problems of stability, decidability, and complexity for them. The original definition of systolic trellis automata requires that an input string is fed to a specific row of processors. Here it is shown that given a homogeneous trellis automaton we can construct an equivalent one (stable or superstable) which allows to feed the input string to any sufficiently long row of processors. Moreover, some closure and decidability results for trellis automata are established and the computational complexity of languages accepted by trellis automata is investigated.

Systolic automata for VLSI on balanced trees

SummarySystolic tree automata with a binary (or, more generally, balanced) underlying tree are investigated. The main emphasis is on input conditions, decidability, and characterization of acceptable languages.

On a family of L languages resulting from systolic tree automata

Abstract Recent work on systolic tree automata has given rise to a rather natural subfamily of EOL languages, referred to as systolic EOL languages in this paper. Systolic EOL languages possess some remarkable properties. While their family contains (because of its closure under Boolean operations) intuitively quite complicated languages, it still has decidable equivalence problem. Especially interesting is the fact that similar decision problems for slightly more general families lead to the celebrated open problems concerning Z -rational power series.

A few remarks on the index of context-free grammars and languages

A hierarchy of context-free grammars and languages with respect to the index of context-free grammars is established and the undecidability of the basic problems is proven.

One-Way Finite Automata with Quantum and Classical States

In this paper, we introduce and explore a new model of quantum finite automata (QFA). Namely, one-way finite automata with quantum and classical states (1QCFA), a one way version of two-way finite automata with quantum and classical states (2QCFA) introduced by Ambainis and Watrous in 2002 [3]. First, we prove that coin-tossing one-way probabilistic finite automata (coin-tossing 1PFA) [23] and one-way quantum finite automata with control language (1QFACL) [6] as well as several other models of QFA, can be simulated by 1QCFA. Afterwards, we explore several closure properties for the family of languages accepted by 1QCFA. Finally, the state complexity of 1QCFA is explored and the main succinctness result is derived. Namely, for any prime m and any e1 > 0, there exists a language L m that cannot be recognized by any measure-many one-way quantum finite automata (MM-1QFA) [12] with bounded error \(\frac{7}{9}+\epsilon_1\), and any 1PFA recognizing it has at last m states, but L m can be recognized by a 1QCFA for any error bound e > 0 with O(logm) quantum states and 12 classical states.

Synthesis, structure and power of systolic computations

Abstract A variety of problems related to systolic architectures, systems, models and computations are discussed. The emphases are on theoretical problems of a broader interest. Main motivations and interesting/important applications are also presented. The first part is devoted to problems related to synthesis, transformations and simulations of systolic systems and architectures. In the second part, the power and structure of tree and linear array computations are studied in detail. The goal is to survey main research directions, problems, methods and techniques in not too formal a way.

On the state complexity of semi-quantum finite automata

Some of the most interesting and important results concerning quantum finite automata are those showing that they can recognize certain languages with (much) less resources than corresponding classical finite automata \cite{Amb98,Amb09,AmYa11,Ber05,Fre09,Mer00,Mer01,Mer02,Yak10,ZhgQiu112,Zhg12}. This paper shows three results of such a type that are stronger in some sense than other ones because (a) they deal with models of quantum automata with very little quantumness (so-called semi-quantum one- and two-way automata with one qubit memory only); (b) differences, even comparing with probabilistic classical automata, are bigger than expected; (c) a trade-off between the number of classical and quantum basis states needed is demonstrated in one case and (d) languages (or the promise problem) used to show main results are very simple and often explored ones in automata theory or in communication complexity, with seemingly little structure that could be utilized.

Descriptional complexity (of languages) a short survey

The paper attempts (i) to present descriptional complexity as an identifiable part of the theory of complexity incorporating many diverse areas of research, (ii) to formulate basic problems and to survey some results (especially those concerning languages) in descriptional complexity, (iii) to discuss relation between descriptional and computational complexity.