It is interesting to note that the very first papers related to isomorphism of types were written before the notion itself appeared. Their subjects were the study of equality of terms defined on numbers, the isomorphism of objects in certain categories and the invertibility of $\lambda$-terms, but not the isomorphism of types ‘as such’. One may cite the so-called ‘Tarsky High School Algebra Problem’: whether all identities between terms built from $+, x, \uparrow$, variables and constants are derivable from basic ‘high school equalities’, like $(xy)^z=x^zy^z$. The earliest publications related to this problem date from the 1940s, cf. Birkhoff (1940) – an extensive bibliography may be found in Burris and Yeats (2002).

We were all taught in high school that two objects $A$ and $B$ are isomorphic iff there exist two functions $f$ and $g$ such that

Type isomorphisms are pairs of functions $f:A\rightarrow B$ and $g:B\rightarrow A$ that are mutually inverse with respect to convertibility. They can be used in the context of type checking to perform type checking modulo type isomorphisms, by adopting the rule $$\frac{a:A}{a:B}$$ whenever $f:A\rightarrow B$ and $g:B\rightarrow A$ have been declared as type isomorphisms. Type isomorphisms may be viewed as a special instance of coercions that provide embeddings of one type into another. Indeed, type systems for coercive subtyping feature a rule $$\frac{a:A}{a:B}$$ whenever there exists a coercion $f:A\rightarrow B$. By declaring $f$ and $g$ as coercions for every type isomorphism $f:A\rightarrow B$ and $g:B\rightarrow A$, one can simulate type-checking modulo type isomorphisms. However, the proposed encoding relies on the possibility of declaring coercions $f:A\rightarrow B$ and $g:B\rightarrow A$ simultaneously. Such coercions, which we call back-and-forth coercions, are only allowed provided $f$ and $g$ are mutually inverse. In principle, type isomorphisms, viewed as back-and-forth coercions, could be used in the context of proof assistants based on dependent type theory in order to relate equivalent representations of mathematical notions, for example, the polar and cartesian representations of complex numbers. However, the coercions that map one representation into another are not mutually inverse because of the intensional nature of dependent type theories. Consequently, the standard concept of type isomorphisms has limited applicability in the context of proof assistants.

In order to circumvent this problem, we develop a computational interpretation of implicit coercions that allows for the definition of back-and-forth coercions without requiring them to be mutually inverse. We illustrate the usefulness of our approach in a number of formal developments that require us to navigate between different representations of mathematical objects or structures. We also discuss important meta-theoretical properties of our interpretation.

We study isomorphisms of inductive types (that is, recursive types satisfying a condition of strict positivity) in an extensional simply typed $\lambda$-calculus with product and unit types. We first show that the calculus enjoys strong normalisation and confluence. Then we extend it with new conversion rules ensuring that all inductive representations of the product and unit types are isomorphic, and such that the extended reduction remains convergent. Finally, we define the notion of a faithful copy of an inductive type and a corresponding conversion relation that also preserves the good properties of the calculus.

The first-order isomorphism problem is to decide whether two non-recursive types using product- and function-type constructors are isomorphic under the axioms of commutative and associative products, and currying and distributivity of functions over products. We show that this problem can be solved in $O(n \log^2 n)$ time and $O(n)$ space, where $n$ is the input size. This result improves upon the $O(n^2 \log n)$ time and $O(n^2)$ space bounds of the best previous algorithm. We also describe an $O(n)$ time algorithm for the linear isomorphism problem, which does not include the distributive axiom, thereby improving upon the $O(n \log n)$ time of the best previous algorithm for this problem.

We develop a matching algorithm for an equational theory with multiplication, exponentiation and a unit element. The algorithm is proved consistent, complete and minimal using techniques based on initial algebras.

The study of isomorphisms of types has, in the main, been carried out in an intuitionistic setting. We extend some of this work to classical logic for both call-by-name and call-by-value computations by means of polarised linear logic and game semantics. This leads to equational characterisations of these isomorphisms for all the propositional connectives.