Characterizations of Recursively Enumerable Languages by Means of Insertion Grammars (original) (raw)

Discrete Applied Mathematics, 2000

We investigate the context-free languages whose complements are also context-free. We call them strongly context-free languages. The family of strongly linear languages is similarly deÿned. After examining the closure properties of the family of strongly context-free languages, we prove that any slender context-free language is strongly linear. We then show that there are languages of a bounded complexity in terms of the number of non-terminals or productions necessary to generate them, whereas the complexity of their complements is arbitrarily large. ?

Context-free insertion–deletion systems

Theoretical Computer Science, 2005

We consider a class of insertion-deletion systems which have not been investigated so far, those without any context controlling the insertion-deletion operations. Rather unexpectedly, we found that context-free insertion-deletion systems characterize the recursively enumerable languages. Moreover, this assertion is valid for systems with only one axiom, and also using inserted and deleted strings of a small length. As direct consequences of the main result we found that set-conditional insertion-deletion systems with two axioms generate any recursively enumerable language (this solves an open problem), as well as that membrane systems with one membrane having contextfree insertion-deletion rules without conditional use of them generate all recursively enumerable languages (this improves an earlier result). Some open problems are also formulated.

On a classification of context-free languages

Kybernetika (Praha), 1967

The set E of strings is said to be definable (strongly definable) if there is a context-free grammar G such that E is the set of all terminal strings generated from the initial symbol (from all non terminal symbols) of G. The classification of definable and strongly definable sets in dependence on minimal number of nonterminal symbols needed for their generation is given.

Invariance: A Theoretical Approach for Coding Sets of Words Modulo Literal (Anti)Morphisms

Combinatorics on Words, 2017

Let A be a finite or countable alphabet and let θ be literal (anti)morphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under θ (θ-invariant for short). We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin θ-invariant codes, maximality and completeness are two equivalent notions. We prove that a similar property holds for some special families of θ-invariant codes such as prefix (bifix) codes, codes with a finite (two-way) deciphering delay, uniformly synchronous codes and circular codes. For a special class of involutive antimorphisms, we prove that any regular θ-invariant code may be embedded into a complete one.

Universal recursively enumerable sets of strings

Theoretical Computer Science, 2011

The main topic of the present work are universal machines for plain and prefix-free description complexity and their domains. It is characterised when an r.e. set W is the domain of a universal plain machine in terms of the description complexity of the spectrum function s W mapping each non-negative integer n to the number of all strings of length n in W ; furthermore, a characterisation of the same style is given for supersets of domains of universal plain machines. Similarly the prefix-free sets which are domains or supersets of domains of universal prefix-free machines are characterised. Furthermore, it is shown that the halting probability Ω V of an r.e. prefix-free set V containing the domain of a universal prefix-free machine is Martin-Löf random, while V may not be the domain of any universal prefix-free machine itself. Based on these investigations, the question whether every domain of a universal plain machine is the superset of the domain of some universal prefix-free machine is discussed. A negative answer to this question had been presented at CiE 2010 by Mikhail Andreev, Ilya Razenshteyn and Alexander Shen, while this paper was under review.

Separating the Classes of Recursively Enumerable Languages Based on Machine Size

International Journal of Foundations of Computer Science, 2015

In the late nineteen sixties it was observed that the r.e. languages form an infinite proper hierarchy [Formula: see text] based on the size of the Turing machines that accept them. We examine the fundamental position of the finite languages and their complements in the hierarchy. We show that for every finite language L one has that L, [Formula: see text] for some [Formula: see text] where m is the length of the longest word in L, c is the cardinality of L, and [Formula: see text]. If [Formula: see text], then [Formula: see text] for some [Formula: see text]. We also prove that for every n, there is a finite language Ln with [Formula: see text] such that [Formula: see text] but Ln, [Formula: see text] for some [Formula: see text]. Several further results are shown that how the hierarchy can be separated by increasing chains of finite languages. The proofs make use of several auxiliary results for Turing machines with advice.