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

Complexity Theory and the Operational Structure of Algebraic Programming Systems

An algebraic programming system is a language built from a fixed algebraic data abstraction and a selection of deterministic, and non-deterministic, assignment and control constructs. First, we give a detailed analysis of the operational structure of an algebraic data type, one which is designed to classify programming systems in terms of the complexity of their implementations. Secondly, we test our operational description by comparing the computations in deterministic and non-deterministic programming systems under certain space and time restrictions

Permuting Operations on Strings and the Distribution of Their Prime Numbers

Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation $X$ gives rise to a family $\{X_n\}_{n\geq2}$ of similar permutations. We call an integer $n$ {\em $X$-prime} if $X_n$ consists of a single cycle of length $n$ ($n\geq2$). For some instances of $X$ -- such as shuffle, twist, operations based on the Archimedes'spiral and on the Josephus problem -- we investigate the distribution of $X$-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures on the density of certain sets of prime numbers