Remarks on Isomorphisms in Typed Lambda Calculi with Empty and Sum Types

Marcelo Fiore and Roberto Di Cosmo and Vincent Balat

To appear at Symposium on Logic in Computer Science (LICS'2002), Copenhagen, Denmark, July 22nd - 25th, 2002


Tarski asked whether the arithmetic identities taught in high school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question for the language of arithmetic expressions using a constant for the number one and the operations of product and exponentiation is affirmative, and the complete equational theory also characterizes isomorphism in the typed lambda calculus, where the constant for one and the operations of product and exponentiation respectively correspond to the unit type and the product and arrow type constructors. This paper studies isomorphisms in typed lambda calculi with empty and sum types from this viewpoint. We close an open problem by establishing that the theory of type isomorphisms in the presence of product, arrow, and sum types (with or without the unit and/or the empty type) is not finitely axiomatizable. Further, we observe that for type theories with arrow, empty and sum types the correspondence between isomorphism and arithmetic equality generally breaks down, but that it still holds in the particular case of type isomorphism with the empty type and equality with zero.

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