Chapter 9: Fundamental Physics

Section 12: Evolution of Networks

Graph grammars

The notion of generalizing substitutions for strings to the case of networks has been discussed in computer science since the 1960s—and a fair amount of formal work has been done on so-called graph grammars for specifying formal languages whose elements are networks. Even a good analog of regular languages has, however, not yet been found. But applications to constructing or verifying practical network-based system description schemes are quite often discussed. In mathematics, rather little is usually done with network substitutions, though the proof of the Four-Color Theorem in 1976 was for example based on showing that 300 or so possible replacement rules—if applied in an appropriate sequence—can transform any graph to have one of 1936 smaller subgraphs that require the same number of colors. (32 rules and 633 subgraphs are now known to be sufficient.)

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From Stephen Wolfram: A New Kind of Science [citation]