A fibration semantics for extended term graph rewriting
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7.1 INTRODUCTION In this chapter, we reexamine the problem of providing a categorical semantics for the core of the general term graph rewriting language DACTL. Partial success in this area has been obtained by describing graph rewrites as certain kinds of pushout. See Ken87, HP88, HKP88, Ken91]. Nevertheless, none of these constructions successfully describe the whole of the operational models of BvEG + 87] where term graph rewriting was introduced, or of its generalization in the language DACTL itself GKSS88, GHK + 88, GKS91, Ken90]. The main stumbling blocks for all of these attempts have been examples such as the I combinator root:IIa]) a when applied to a circular instance of itself x:IIx]. None of the hitherto proposed categorical formulations of TGR adequately capture the DACTL version of the rewrite (which is, reasonably enough, a null action), nor do they give a convincing story of their own (generally speaking the result of the rewrite is undeened). The aim of this chapter is to describe how these deeciencies may be overcome by using a diierent approach to the categorical semantics of rewriting. Instead of pushouts, we use a Grothendieck oppbration. Now Grothendieck oppbrations have strong universal properties, too strong to be applicable to all DACTL rewrites. Accordingly, a less universal construction describes the full operational core of DACTL rewriting. It turns out that the circular I example sits in between these two extremes. In outline, the rest of the chapter is as follows. Section 7.2 describes the free rewriting core of the original DACTL model. Section 7.3 describes the categorical construc
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