Logic of Fusion

Abstract

We pursue an extension of the Curry-Howard isomorphism of propositions and types by a correspondence of cut elimination and program fusion. In particular, we explore the repercussions of this extension in generic and transformational programming. It provides a logical interpretation of build fusion, or deforestation, in terms of the inductive and the coinductive datatypes. Viewed categorically, this interpretation leads to the novel structure of paranatural transformations. This is a modified version of functorial polymorphism, that played a prominent role in the work of Andre Scedrov.

D. Pavlovic—Supported by NSF and AFOSR.

Appendix: Proof of Proposition 3

Towards isomorphism (12), we define the maps

and show that they are inverse to each other.

Given $f: A \longrightarrow M_F$, the X-th component of $f'$ will be

where $k : (FX\Rightarrow X) \longrightarrow (M_F\Rightarrow X)$ maps the algebra structures $x:FX\rightarrow X$ to the catamorphisms . Formally, k is obtained by transposing the catamorphism for the F-algebra $\kappa$ on $(FX\Rightarrow X)\Rightarrow X$, obtained by transposing the composite

where arrow (i) is derived from the diagonal on $FX\Rightarrow X$, (ii) from the strength, while (iii) and (iv) are just evaluations.

Towards the definition of $\mathtt{build}$, for a paranatural $\varphi \ :\ A\times (FX\Rightarrow X) \longrightarrow X$ take

Composing the above two definitions, one gets the commutative square

Since $k\cdot \ulcorner \mu \urcorner = \ulcorner \mathrm{id}_M \urcorner$, the path around the square reduces to f, and yields $\mathtt{build}(f') = f$.

The converse $\mathtt{build}(\varphi )' = \varphi$ is the point-free version of lemma 1. It amounts to proving that the paranaturality of $\varphi$ implies (indeed, it is equivalent) to the commutativity of

where $\widetilde{\varphi }X$ is the transpose of $\varphi X$. Showing this is an exercise in cartesian closed structure. On the other hand, the path around the square is easily seen to be $\mathtt{build}(\varphi )'_X$.

To establish isomorphism (13), we internalize 15 similarly like we did 14 above. The natural correspondences

are defined

and

for $g:N_F\longrightarrow B$ and $\psi : X\times (X\Rightarrow FX) \longrightarrow B$. The arrow $\ell : (X\Rightarrow FX) \longrightarrow (X\Rightarrow FX)$ maps the coalgebra structures $x:X\rightarrow FX$ to the anamorphisms $[\negmedspace (x )\negmedspace ]:X \rightarrow N_F$. $\Box$