From Machine and Tape to Structure and Function: Formulation of a Reflexively Computing System

Abstract

The relationship between structure and function is explored via a system of labeled directed graph structures upon which a single elementary read/write rule is applied locally. Boundaries between static (information-carrying) and active (information-processing) objects, imposed by mandate of the rules or physics in earlier models, emerge instead as a result of a structure-function dynamic that is reflexive: objects may operate directly on their own structure. A representation of an arbitrary Turing machine is reproduced in terms of structural constraints by means of a simple mapping from tape squares and machine states to a uniform medium of nodes and links, establishing computation universality. Exploiting flexibility of the formulation, examples of other unconventional "self-computing" structures are demonstrated. A straightforward representation of a kinematic machine system based on the model devised by Laing is also reproduced in detail. Implications of the findings are discussed in terms of their relation to other formal models of computation and construction. It is argued that reflexivity of the structure-function relationship is a critical informational dynamic in biochemical systems, overlooked in previous models but well captured by the proposed formulation.

A graph-based reflexive artificial chemistry

“…It has been shown elsewhere that, via incorporation of a simple overwrite mechanism into the rules of the system, the form of Fig. 3(b) is sufficient to simulate a Turing machine (Salzberg, 2006). This equivalence will not be elaborated upon as it is our goal here to formulate a constructive chemistry and not a computational formalism.…”

“…The term "engagement" has been coined in earlier work to describe an interaction between entities in which informational dynamics are inherently codependent (Salzberg, 2006;Salzberg et al, 2004). This co-dependency arises from the fact that exclusively "active" or "passive" objects, upon the basis of which systems such as the FSM and Turing machine are described, are replaced by dynamics on a space of elementary structures with no a priori functional role.…”

“…It is not, however, abandoned: individual components may themselves play an active or passive role, but only at a locally referential scale. In what follows, we review the steps, described in Salzberg (2006), that lead from FSM to engagement system.…”

“…The dotted circle is the output of the last step, the solid circle the output of the upcoming step. With a simple overwrite mechanism described in Salzberg (2006), the system shown above simulates a Turing machine.…”

On the reachability of a version of graph-rewriting system

“…There have been studies that deal with dynamic structures based on discrete symbol dynamics. Examples are an L-system [3], extensions of cellular automata for varying structures, e.g., [1], and studies using graphs [6,5] based on graphrewriting systems [4].…”