Abstract Syntax: Substitution and Binders

• Authors: John Power
• Venue: Electronic Notes in Theoretical Computer Science (ENTCS)
• Year: 2007

We summarise Fiore et al's paper on variable substitution and binding, then axiomatise it. Generalising their use of the category F of finite sets to model untyped cartesian contexts, we let S be an arbitrary pseudo-monad on Cat and consider (S1)^o^p: this generality includes linear contexts, affine contexts, and contexts for the Logic of Bunched Implications. Given a pseudo-distributive law of S over the (partial) pseudo-monad T"c"o"c-=[(-)^o^p,Set] for free cocompletions, one can define a canonical substitution monoidal structure on the category [(S1)^o^p,Set], generalising Fiore et al's substitution monoidal structure for cartesian contexts: this provides a natural substitution structure for the above examples. We give a concrete description of this substitution monoidal structure in full generality. We then give an axiomatic definition of a binding signature, then state and prove an initial algebra semantics theorem for binding signatures in full generality, once again extending the definitions and theorem of Fiore et al. A delicate extension of the research includes the category Pb(Inj^o^p,Set) studied by Gabbay and Pitts in their quite different analysis of binders, which we compare and contrast with that of Fiore et al.