An Inefficient Representation of the Empty Word

We show that Post's system of tag with alphabet $\{0,1\}$, deletion number 3, productions $0\rightarrow 00$ and $1\rightarrow 1101$, and initial string $111\cdots 1$ (330 times) converges to the empty word after a very large number of rewriting steps

A Fuzzy Approach to Erroneous Inputs in Context-Free Language Recognition

Using fuzzy context-free grammars one can easily describe a finite number of ways to derive incorrect strings together with their degree of correctness. However, in general there is an infinite number of ways to perform a certain task wrongly. In this paper we introduce a generalization of fuzzy context-free grammars, the so-called fuzzy context-free $K$-grammars, to model the situation of making a finite choice out of an infinity of possible grammatical errors during each context-free derivation step. Under minor assumptions on the parameter $K$ this model happens to be a very general framework to describe correctly as well as erroneously derived sentences by a single generating mechanism. Our first result characterizes the generating capacity of these fuzzy context-free $K$-grammars. As consequences we obtain: (i) bounds on modeling grammatical errors within the framework of fuzzy context-free grammars, and (ii) the fact that the family of languages generated by fuzzy context-free $K$-grammars shares closure properties very similar to those of the family of ordinary context-free languages. The second part of the paper is devoted to a few algorithms to recognize fuzzy context-free languages: viz. a variant of a functional version of Cocke-Younger- Kasami's algorithm and some recursive descent algorithms. These algorithms turn out to be robust in some very elementary sense and they can easily be extended to corresponding parsing algorithms

Generating All Permutations by Context-Free Grammars in Greibach Normal Form

We consider context-free grammars $G_n$ in Greibach normal form and, particularly, in Greibach $m$-form ($m=1,2$) which generates the finite language $L_n$ of all $n!$ strings that are permutations of $n$ different symbols ($n\geq 1$). These grammars are investigated with respect to their descriptional complexity, i.e., we determine the number of nonterminal symbols and the number of production rules of $G_n$ as functions of $n$. As in the case of Chomsky normal form these descriptional complexity measures grow faster than any polynomial function

Complete Symmetry in D2L Systems and Cellular Automata

We introduce completely symmetric D2L systems and cellular automata by means of an additional restriction on the corresponding symmetric devices. Then we show that completely symmetric D2L systems and cellular automata are still able to simulate Turing machine computations. As corollaries we obtain new characterizations of the recursively enumerable languages and of some space-bounded complexity classes

Fibonacci-like Differential Equations with a Polynomial Non-Homogeneous Part

We investigate non-homogeneous linear differential equations of the form $x''(t) + x'(t) - x(t) = p(t)$ where $p(t)$ is either a polynomial or a factorial polynomial in $t$. We express the solution of these differential equations in terms of the coefficients of $p(t)$, in the initial conditions, and in the solution of the corresponding homogeneous differential equation $y''(t) + y'(t) - y(t) = 0$ with $y(0) = y'(0) = 1$

A Characterization of ET0L and EDT0L Languages

There exists a PT0L language $L_0$ such that the following holds. A language $L$ is an ET0L language if and only if there exists a mapping $T$ induced by an a-NGSM (nondeterministic generalized sequential machine with accepting states) such that $L = T(L_0)$. There exists an infinite collection of EPDT0L languages $D_{mn}\subseteq\Sigma_{mn}^\star$ ($n\geq m\geq 1$) such that the family EDT0L is characterized in the following way. A language $L$ is an EDT0L language if and only if there exists $n\geq m\geq 1$, a homomorphism $h$ and a regular language $R \subseteq \Sigma_{mn}^\star$ such that $L = h(D_{mn} \cap R)$.\u

Towards Robustness in Parsing - Fuzzifying Context-Free Language Recognition

We discuss the concept of robustness with respect to parsing a context-free language. Our approach is based on the notions of fuzzy language, (generalized) fuzzy context-free grammar and parser / recognizer for fuzzy languages. As concrete examples we consider a robust version of Cocke-Younger-Kasami's algorithm and a robust kind of recursive descent recognizer

Permuting operations on strings: Their permutations and their primes

We study some length-preserving operations on strings that permute the symbol positions in strings. These operations include some well-known examples (reversal, circular or cyclic shift, shuffle, twist, operations induced by the Josephus problem) and some new ones based on Archimedes spiral. Such a permuting operation $X$ gives rise to a family $\{p(X,n)\}_{n\geq2}$ of similar permutations. We investigate the structure and the order of the cyclic group generated by such a permutation $p(X,n)$. We call an integer $n$ $X$-prime if $p(X,n)$ consists of a single cycle of length $n$ ($n\geq2$). Then we show some properties of these $X$-primes, particularly, how $X$-primes are related to $X^\prime$-primes as well as to ordinary prime numbers

Controlled Fuzzy Parallel Rewriting

We study a Lindenmayer-like parallel rewriting system to model the growth of filaments (arrays of cells) in which developmental errors may occur. In essence this model is the fuzzy analogue of the derivation-controlled iteration grammar. Under minor assumptions on the family of control languages and on the family of fuzzy languages in the underlying iteration grammar, we show (i) regular control does not provide additional generating power to the model, (ii) the number of fuzzy substitutions in the underlying iteration grammar can be reduced to two, and (iii) the resulting family of fuzzy languages possesses strong closure properties, viz. it is a full hyper-AFFL, i.e., a hyper-algebraically closed full Abstract Family of Fuzzy Languages