Non determinism through type isomorphism

Several non-deterministic extensions to the λ -calculus have been proposed, e.g. [6, 7, 10–12, 24]. In these approaches, the parallel composition (sometimes called the must-convergent parallel composition) is such that if r and s are two λ -terms, the term r+ s (also written r ‖ s) represents the computation that runs either r or s non-deterministically. It is common to consider in these approaches the associativity and commutativity of the operator +. Indeed the interpretation “either r or s runs” shall not prioritise any of them, and so “either s or r runs” must be represented by the same term. Moreover, (r+ s)t can run either rt or st, which is the same expressed by rt+ st. Extra equivalences (or rewrite rules, depending on the presentation) are set up to account for such an interpretation, e.g. (r+ s)t ↔ rt+ st. This right distributivity can alternatively be seen as the one of function sum: (f+ g)(x) is defined pointwise as f(x)+ g(x). This is the approach of the algebraic lambda-calculi [3, 26], two independently introduced algebraic extensions which resulted strongly related afterwards [4,15]. In these algebraic calculi, a scalar pondering each ‘choice’ is considered in addition to the sum of terms. Because of these equivalences between terms, it is natural to think that a typed version must allow some equivalences at the type level. Definitely, if r and s are typed with types A and B respectively, it is natural to expect that whatever connective tie these types in order to type r+ s, it must be commutative and associative. An independent stream of research is the study of isomorphisms between types for several languages (see [13] for a reference). For example, we know that the propositions A∧B and B∧A are equiprovable: one is provable if and only if the other is, but they do not have the same proofs. If r is a proof of A and s is a proof of B, then 〈r,s〉 is a proof of A∧B while 〈s,r〉 is a proof of B∧A. Despite that both proofs can be derived from the same hypotheses, they are not the same. In this paper, we show how the non-determinism arises naturally in a classic context only by introducing some equivalences between types. These equivalences, nevertheless, will be chosen among valid, well-known isomorphisms. In order to consider these isomorphic types as equivalent, we need to design a proof system such that they have the same proofs, or conversely, in order to consider these terms to be equivalent, we need to make these isomorphic types to be equivalent. Formally, two types A and B are isomorphic if there are two conversion functions f of type A ⇒ B and g of type B ⇒ A, such that g( f (x)) = x for any x of type A and f (g(y)) = y for any y of type B. Hence, in this system the conversion functions f and g should become and identity function. In other words, we take the quotient of the set of propositions by the relation