9.01Additional Material

Structure of Models & Methodology

The class of models studied here represent a simplification and generalization of the trivalent graph models introduced in [1:9] and [87] (see also [148]).

The methodology of computational exploration used here has been developed particularly in [5][31][1]. Some exposition of the methodology has been given in [149].

The class of models studied here can be viewed as generalizing or being related to a great many kinds of abstract systems. One class is graph rewriting systems, also known as graph transformation systems or graph grammars (e.g. [150]). The models here are generalizations of both the double-pushout and single-pushout approaches. Note that the unlabeled graphs and hypergraphs studied here are different from the typical cases usually considered in graph rewriting systems and their applications.

Multiway systems as used here were explicitly introduced and studied in [1:p204] (see also [1:p938]). Versions of them have been invented many times, most often for strings, under names such as semi-Thue systems [151], string rewriting systems [152], term rewriting systems [65], production systems [153], associative calculi [154] and canonical systems [153][155].

Connections to Physics Theories

An outline of applying models of a type very similar to those considered here was given in [1:9]. Some additional exposition was given in [156][157][158]. The discussion here contains many new ideas and developments, explored in [159].

For a survey of ultimate models of physics, see [1:p1024]. The possibility of discreteness in space has been considered since antiquity [160][161][162][163]. Other approaches that have aspects potentially similar to what is discussed here include: causal dynamical triangulation [164][165][166], causal set theory [167][168][169], loop quantum gravity [170][171], pregeometry [172][173][174], quantum holography [175][176][177], quantum relativity [178], Regge calculus [179], spin networks [180][181][182][183][184], tensor networks [185], superrelativity [186], topochronology [187], topos theory [188], twistor theory [128]. Other discrete and computational approaches to fundamental physics include: [189][190][191][192][193][194][195][196].

The precise relationships among these approaches and references and the current work are not known. In some cases it is expected that conceptual motivations may be aligned; in others specific mathematical structures may have direct relevance. The latter may also be the case for such areas as conformal field theory [197], higher-order category theory [198], non-commutative geometry [199], string theory [200].