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On Computational Completeness of Semi-Conditional Matrix Grammars

Henning Fernau, Lakshmanan Kuppusamy, Indhumathi Raman

TL;DR

It is shown that certain semi-conditional matrix grammar families defined by restricting resources can generate all of the recursively enumerable languages.

Abstract

Matrix grammars are one of the first approaches ever proposed in regulated rewriting, prescribing that rules have to be applied in a certain order. Originally, they have been introduced by Ábrahám on linguistic grounds. In traditional regulated rewriting, the most interesting case shows up when all rules are context-free. Typical descriptional complexity measures incorporate the number of nonterminals or the matrix length, i.e., the number of rules per matrix. When viewing matrices as program fragments, it becomes natural to consider additional applicability conditions for such matrices. Here, we focus on attaching a permitting and a forbidden string to every matrix in a matrix grammar. The matrix is applicable to a sentential form~$w$ only if the permitting string is a subword in~$w$ and the forbidden string is not a subword in~$w$. We call such a grammar, where the application of a matrix is conditioned as described, a semi-conditional matrix grammar. We consider $(1)$ the maximal lengths of permitting and forbidden strings, $(2)$ the number of nonterminals, $(3)$ the number of conditional matrices, $(4)$ the maximal length of any matrix and $(5)$ the number of conditional matrices with nonempty permitting and forbidden strings, as the resources (descriptional complexity measures) of a semi-conditional matrix grammar. In this paper, we show that certain semi-conditional matrix grammar families defined by restricting resources can generate all of the recursively enumerable languages.

On Computational Completeness of Semi-Conditional Matrix Grammars

TL;DR

It is shown that certain semi-conditional matrix grammar families defined by restricting resources can generate all of the recursively enumerable languages.

Abstract

Matrix grammars are one of the first approaches ever proposed in regulated rewriting, prescribing that rules have to be applied in a certain order. Originally, they have been introduced by Ábrahám on linguistic grounds. In traditional regulated rewriting, the most interesting case shows up when all rules are context-free. Typical descriptional complexity measures incorporate the number of nonterminals or the matrix length, i.e., the number of rules per matrix. When viewing matrices as program fragments, it becomes natural to consider additional applicability conditions for such matrices. Here, we focus on attaching a permitting and a forbidden string to every matrix in a matrix grammar. The matrix is applicable to a sentential form~ only if the permitting string is a subword in~ and the forbidden string is not a subword in~. We call such a grammar, where the application of a matrix is conditioned as described, a semi-conditional matrix grammar. We consider the maximal lengths of permitting and forbidden strings, the number of nonterminals, the number of conditional matrices, the maximal length of any matrix and the number of conditional matrices with nonempty permitting and forbidden strings, as the resources (descriptional complexity measures) of a semi-conditional matrix grammar. In this paper, we show that certain semi-conditional matrix grammar families defined by restricting resources can generate all of the recursively enumerable languages.

Paper Structure

This paper contains 8 sections, 14 theorems, 23 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Our contribution.
  3. Definitions and Notation
  4. Normal Forms for Type-0 Grammars
  5. Main Results for SSCM Grammars
  6. Non-simple Semi-Conditional Matrix Grammars
  7. Computational Complete SCM Grammars with only Three Nonterminals
  8. Conclusions

Key Result

proposition 1

For each $\mathrm{RE}$ language $L$, $L\subseteq T^*$, there is a type-0 grammar of the form $G= (V, T, P\cup \{AB\to \lambda, CD\to \lambda\}, S)$ with $L(G)=L$, where $P$ contains only context-free rules and $N\coloneqq V\setminus T=\{S, A, B, C,D\}$. More specifically, the context-free rules are

Theorems & Definitions (26)

  • definition 1
  • proposition 1
  • proposition 2
  • proposition 3
  • proposition 4
  • proposition 5
  • theorem 1
  • proof
  • theorem 2
  • proof
  • ...and 16 more