scholarly journals GENERATING ALL CIRCULAR SHIFTS BY CONTEXT-FREE GRAMMARS IN GREIBACH NORMAL FORM

2007 ◽  
Vol 18 (06) ◽  
pp. 1139-1149 ◽  
Author(s):  
PETER R. J. ASVELD
Keyword(s):  
Normal Form ◽  
Complexity Measures ◽  
Descriptional Complexity ◽  
Low Degree ◽  
Cyclic Shifts ◽  
Low Degree Polynomials ◽  
Context Free ◽  
Greibach Normal Form ◽  
Context Free Grammars

For each alphabet Σn = {a1,a2,…,an}, linearly ordered by a1 < a2 < ⋯ < an, let Cn be the language of circular or cyclic shifts over Σn, i.e., Cn = {a1a2 ⋯ an-1an, a2a3 ⋯ ana1,…,ana1 ⋯ an-2an-1}. We study a few families of context-free grammars Gn (n ≥1) in Greibach normal form such that Gn generates Cn. The members of these grammar families are investigated with respect to the following descriptional complexity measures: the number of nonterminals ν(n), the number of rules π(n) and the number of leftmost derivations δ(n) of Gn. As in the case of Chomsky normal form, these ν, π and δ are functions bounded by low-degree polynomials. However, the question whether there exists a family of grammars that is minimal w. r. t. all these measures remains open.

A Survey of Normal form Covers for Regular Grammars

1981 ◽  
Vol 4 (4) ◽  
pp. 761-776
Author(s):  
Anton Nijholt
Keyword(s):  
Normal Form ◽  
Special Form ◽  
Normal Forms ◽  
Context Free Grammar ◽  
Left Regular ◽  
Regular Grammar ◽  
Left And Right ◽  
Context Free ◽  
Greibach Normal Form

An overview is given of cover results for normal forms of regular grammars. Due to the special form of regular grammars and due to the results which are obtained it is sufficient to consider covering grammars in Greibach normal form. Among other things it is proved that any left-regular grammar can be left covered with a context-free grammar in Greibach normal form. All the cover results concerning the left- and right-regular grammars are listed, with respect to several types of covers, in a cover table.

Improved Descriptional Complexity Results for Simple Semi-Conditional Grammars

2021 ◽  
Vol 181 (2-3) ◽  
pp. 189-211
Author(s):  
Henning Fernau ◽  
Lakshmanan Kuppusamy ◽  
Rufus O. Oladele ◽  
Indhumathi Raman
Keyword(s):  
Normal Form ◽  
Maximum Length ◽  
Complexity Measures ◽  
Recursively Enumerable Language ◽  
Computational Completeness ◽  
Descriptional Complexity ◽  
Recursively Enumerable ◽  
Rewriting System ◽  
Complexity Results ◽  
Enumerable Language

A simple semi-conditional (SSC) grammar is a form of regulated rewriting system where the derivations are controlled either by a permitting string alone or by a forbidden string alone and this condition is specified in the rule. The maximum length i (j, resp.) of the permitting (forbidden, resp.) strings serves as a measure of descriptional complexity known as the degree of such grammars. In addition to the degree, the numbers of nonterminals and of conditional rules are also counted into the descriptional complexity measures of these grammars. We improve on some previously obtained results on the computational completeness of SSC grammars by minimizing the number of nonterminals and / or the number of conditional rules for a given degree (i, j). More specifically we prove, using a refined analysis of a normal form for type-0 grammars due to Geffert, that every recursively enumerable language is generated by an SSC grammar of (i) degree (2, 1) with eight conditional rules and nine nonterminals, (ii) degree (3, 1) with seven conditional rules and seven nonterminals (iii) degree (4, 1) with six conditional rules and seven nonterminals and (iv) degree (4, 1) with eight conditional rules and six nonterminals.

ON THE DESCRIPTIONAL COMPLEXITY OF METALINEAR CD GRAMMAR SYSTEMS

2005 ◽  
Vol 16 (05) ◽  
pp. 1011-1025
Author(s):  
BETTINA SUNCKEL
Keyword(s):  
Recursive Function ◽  
Trade Off ◽  
Descriptional Complexity ◽  
Grammar System ◽  
Trade Offs ◽  
Grammar Systems ◽  
Context Free ◽  
Context Free Grammars

Metalinear CD grammar systems are context-free CD grammar systems where each component consists of metalinear productions. The maximal number of nonterminals in all starting productions is referred to as the width of a CD grammar system. It is shown that between the class of CD grammar systems of width m + 1 and of width m there are savings concerning the size of the grammar system not bounded by any recursive function. This is called a non-recursive trade-off. Furthermore, it is proven that there are non-recursive trade-offs between the class of metalinear CD grammar systems of width m and the class of (2m - 1)-linear context-free grammars. In addition, some decidability results are presented.

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