Throughout the history of Formal Languages, one of the research directions has always been to describe computational completeness using only a small amount of possibly scarce resources. We review some of these results in the form of an essay.

Parsimonious Computational Completeness

A matrix insertion-deletion system (or matrix ins-del system) is described by a set of insertion-deletion rules presented in matrix form, which demands all rules of a matrix to be applied in the given order. These systems were introduced to model very simplistic fragments of sequential programs based on insertion and deletion as elementary operations as can be found in biocomputing. We are investigating such systems with limited resources as formalized in descriptional complexity. A traditional descriptional complexity measure of such a matrix ins-del system is its size $$s=(k;n,i',i'';m,j',j'')$$
 , where the parameters from left to right represent the maximal matrix length, maximal insertion string length, maximal length of left contexts in insertion rules, maximal length of right contexts in insertion rules; the last three are deletion counterparts of the previous three parameters. We call the sum $$n+i'+i''+m+j'+j''$$
 the sum-norm of s. We show that matrix ins-del systems of sum-norm 4 and sizes (3; 1, 0, 0;  1, 2, 0), (3; 1, 0, 0;  1, 0, 2), (2; 1, 2, 0;  1, 0, 0), (2; 1, 0, 2;  1, 0, 0), and (2; 1, 1, 1;  1, 0, 0) describe the recursively enumerable languages. Moreover, matrix ins-del systems of sizes (3; 1, 1, 0;  1, 0, 0), (3; 1, 0, 1;  1, 0, 0), (2; 2, 1, 0;  1, 0, 0) and (2; 2, 0, 1;  1, 0, 0) can describe at least the regular closure of the linear languages. In fact, we show that if a matrix ins-del system of size s can describe the class of linear languages $$\mathrm {LIN}$$
 , then without any additional resources, matrix ins-del systems of size s also describe the regular closure of $$\mathrm {LIN}$$
 . Finally, we prove that matrix ins-del systems of sizes (2; 1, 1, 0;  1, 1, 0) and (2; 1, 0, 1;  1, 0, 1) can describe at least the regular languages.

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On the generative capacity of matrix insertion-deletion systems of small sum-norm

With good biological motivation, we add substitutions as a further type of operations to (in particular, context-free) insertion-deletion systems. This way, we obtain new characterizations of and normal forms for context-sensitive and recursively enumerable languages.

Insertion-deletion systems with substitutions I

We discuss substitutions as a further type of operations, added to matrix insertion-deletion systems. For such systems, we additionally discuss the effect of appearance checking. This way, we obtain new characterizations of the families of context-sensitive and the family of recursively enumerable languages. To reach computational completeness, not much context is needed for systems with appearance checking.

Adding Matrix Control: Insertion-Deletion Systems with Substitutions III

Insertion-deletion systems have been introduced as a formalism to model operations that find their counterparts in ideas of bio-computing, more specifically, when using DNA or RNA strings and biological mechanisms that work on these strings. So-called matrix control has been introduced to insertion-deletion systems in order to enable writing short program fragments. We discuss substitutions as a further type of operation, added to matrix insertion-deletion systems. For such systems, we additionally discuss the effect of appearance checking. This way, we obtain new characterizations of the family of context-sensitive and the family of recursively enumerable languages. Not much context is needed for systems with appearance checking to reach computational completeness. This also suggests that bio-computers may run rather traditionally written programs, as our simulations also show how Turing machines, like any other computational device, can be simulated by certain matrix insertion-deletion-substitution systems.

https://www.mdpi.com/1999-4893/14/5/131/pdf

Regulated rewriting is one of the classical areas in Formal Languages, as tree automata are a classical topic. Somewhat surprisingly, there have been no attempts so far to combine both areas. Here, we start this type of research, introducing regulated tree automata, proving in particular characterizations of the yields of such regulated automata.

Martin Vu

Papers

Adding Matrix Control: Insertion-Deletion Systems with Substitutions III

Adding Matrix Control: Insertion-Deletion Systems with Substitutions III

Regulated Tree Automata