Section: New Results

Meta-theoretic results on type isomorphisms in the presence of sums

Participant : Danko Ilik.

Type isomorphisms are a pervasive notion of Theoretical Computer Science. In functional programming, two data types being isomorphic means that we can coerce data and programs back-and-forth between two specifications without loss of information. In Constructive Mathematics, two sets are of the same cardinality exactly when they are isomorphic as types. In the proof theory of intuitionistic logic, two formulas are strongly equivalent precisely when they are isomorphic as types.

However, the theory of simple types made from functions, products, and sums, is well understood only when we do not treat functions and sums at the same time. Fiore, Di Cosmo, and Balat [50] , presented a “negative” results: the theory of those type isomorphisms is not finitely axiomatizable. To establish the result, they used the work around the Tarski High School Algebra Problem from Mathematical Logic.

We showed that the picture is not so dark by presenting a positive result: the theory is recursively axiomatizable and decidable. The proofs exploit further the deep theory around Tarski's Problem. This work was presented at the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) in Vienna, Austria [23] .