Computational completeness of simple semi-conditional insertion–deletion systems of degree (2,1)

Abstract

Insertion–deletion (or ins–del for short) systems are simple models of bio-inspired computing. They are well studied in formal language theory, especially regarding their computational completeness. This concerns the question if all recursively enumerable languages can be generated. This ultimately addresses the question if one can build general-purpose computers rooted in this formalism. The descriptional complexity of an ins–del system is typically measured by its size, a 6-dimensional tuple of non-negative integers $(e,e',e'';d,d',d'')$ where e is the maximum length of the insertion string, $e'$ (and $e''$) is the maximum length of the left (and right) context used for insertion; the last three parameters $d,d',d''$ are similarly understood for deletion rules. Computational completeness for ins–del systems can even be achieved with rule size (1, 1, 1; 1, 1, 1) but with no rule size strictly smaller than this. This fact has motivated to study ins–del systems in combination with regulation mechanisms. In this context, the six-tuple explained above is called the ID size of a system. Several regulations like graph-control, matrix and semi-conditional have been imposed on ins–del systems. Typically, the computational completeness results are obtained as trade-offs, reducing the ID size, say, to (1, 1, 0; 1, 1, 0) at the expense of increasing other measures of descriptional complexity. In this paper, we study simple semi-conditional ins–del systems, where an ins–del rule can be applied only in the presence or absence of substrings of the derivation string. This brings along two further natural parameters to measure descriptional complexity, namely, the maximum permitting string length p and the maximum forbidden string length f, usually summarized as the degree $d=(p,f)$. We show that simple semi-conditional ins–del systems of degree (2, 1) and with ID sizes $(1+e,e',e'';1+d,d',d'')$ are computationally complete for any $e,e',e'',d,d',d''\in \{0,1\}$, with $e+e'+e''=1$ and $d+d'+d''=1$. The obtained results complement and improve on the existing results known from the literature. To prove our results, we also show a new normal form for type-0 grammars that appears to be interesting in its own right.