A Tangled Hierarchy of Graph-Constructing Graphs

The traditional construction paradigm of machine and tape is reformulated in a functionally homogeneous space of directed graph structures. Hierarchy-based roles, normally appointed to actors in a construction process, are dissolved and replaced by symmetric, level-less engagement. The separation between static (information carrying) and active (information processing) structures, imposed by mandate of the rules or physics in earlier theoretical models, results instead purely from graph topology. While encompassing traditional machine-tape paradigms as a special case, the formalism is shown to incorporate a wider class of construction relations. Exploiting its flexibility, a representation of a Turing machine is demonstrated, establishing computation universality. The concept of a “Tangled Construction Hierarchy” is introduced. Introduction The formalistic study of machine construction has its roots in the self-reproducing automata theory of mathematician John von Neumann (von Neumann, 1966). Inspired by the extreme “complication” of real living organisms, von Neumann sought to realize the emergence of complexityincreasing evolution in a functionally homogeneous medium governed only by local rules. With Stan Ulam he invented a formulation for this purpose, now known as cellular automata (CA) and widely adopted for the modeling of selfreplication (Sipper, 1998), in which the outputs and inputs of a construction process are fundamentally made of the same “stuff”. Embedded in this architecture von Neumann instantiated his automata theory in the form of a complicated universal construction machine capable of self-reproduction. By introducing static self-description (tape) and division of labour (translation/transcription), he thus demonstrated a loophole in the construction paradox stating that “a machine tool is more complicated than the elements that can be made with it” (von Neumann, 1966, p.79). Decades earlier, Turing (Turing, 1936) had initiated the study of computing machines on the basis of a similar machine-tape paradigm; von Neumann imported it to the realm of constructing machines. Hierarchical separation of machine and tape has since played an influential role in shaping the design of artificial construction and self-replication models (Mange and Sipper, 1998; McMullin, 2000), yet notable alternatives have been proposed. Hofstadter (Hofstadter, 1979), drawing inspiration from “the molecular logic of the living state” (Lehninger, 1976), blurred this separation in a typographical system he called “Typogenetics”. The players in this system are “strands”: strings of characters acting both as data to be manipulated and as an active “typographical enzyme” to be applied to other strands. Hofstadter called this mixing of levels a “Tangled Hierarchy” (Fig. 1), contrasting it with the case, typified by formal systems, in which there is a clear distinction between rules and the strings they apply to. Along similar lines, Laing (Laing, 1976) devised a system in which a pair of tapes undergo local sliding and state changes leading to self-reproduction via selfinspection. By means of transfer primitives, active and passive roles are arbitrarily exchangeable in this process, exemplifying a uniquely mixed-level style of execution. All of these models assign roles in a construction process employing the intuitive concept of levels. Describing his system, Hofstadter asserts that: “The two-way street which links ‘upper’ and ‘lower’ levels of Typogenetics shows that, in fact, neither strand nor enzyme can be thought of as being on a higher level than the other.” (Hofstadter, 1979, p.513) The word “level” is used here in the sense of containing the same information, interpreted as passive data in one case (strand), and as active process in the other (enzyme). von Neumann’s machine contains a related type of “levels”. As do strand and enzyme, the tape and machine it codes for, interpreted according to a set of transition rules1, contain the same information: one in a passive form, the other in an active one. By analogy, they are thus — according to Hofstadter’s use of the word — on the same “level”. 1Strictly speaking an embedded universal construction machine is also required in the initial configuration to carry out translation.