Many will agree that identity sets are the most intriguing concept of intensional Martin-Löf type theory. For instance, it may appear surprising that their axiomatisation as an inductive family allows one to deduce the usual properties of equality, notably the replacement rule (Leibniz’s principle) which gives P(a′) from P(a) and a proof that a equals a′. This holds for arbitrary families of sets P, not only those corresponding to a predicate. This is not in conflict with decidability of type checking since if a equals a′ and p : P(a) then one does not in general have p : P(a′), but only subst(s, p) : P(a′) where s is the proof that a equals a′ and subst is defined from the eliminator for identity sets. It is a natural question to ask whether these translation functions subst(s, _) actually depend upon the nature of the proof s or, more generally, the question whether any two elements of an identity set are equal. We will call UIP(A) (t/niqueness of Identity Proofs) the following property. If a1, a2 are objects of type A then for any two proofs p and q of the proposition “a1 equals a2” we can prove that p and q are equal. More generally, UIP will stand for UIP(A) for all types A. Note that in traditional logical formalism a principle like UIP cannot even be expressed sensibly as proofs cannot be referred to by terms of the object language and thus are not within the scope of prepositional equality. The question of whether UIP is valid in intensional Martin-Löf type theory was open for a while, though it was commonly believed that UIP is underivable as any attempt for constructing a proof has failed (Coquand 1992; Streicher 1993; Altenkirch 1992). On the other hand, the intuition that a type is determined by its canonical objects might be seen as evidence for the validity of UIP as the identity sets have at most one canonical element corresponding to an instance of reflexivity.

No abstract

One of the goals of this paper is to demonstrate that denotational semantics is useful for operational issues like implementation of functional languages by abstract machines. This is exemplified in a tutorial way by studying the case of extensional untyped call-by-name λ-calculus with Felleisen's control operator [Cscr ]. We derive the transition rules for an abstract machine from a continuation semantics which appears as a generalization of the ¬¬-translation known from logic. The resulting abstract machine appears as an extension of Krivine's machine implementing head reduction. Though the result, namely Krivine's machine, is well known our method of deriving it from continuation semantics is new and applicable to other languages (as e.g. call-by-value variants). Further new results are that Scott's D∞-models are all instances of continuation models. Moreover, we extend our continuation semantics to Parigot's λμ-calculus from which we derive an extension of Krivine's machine for λμ-calculus. The relation between continuation semantics and the abstract machines is made precise by proving computational adequacy results employing an elegant method introduced by Pitts.

No abstract

Consistency with the formal Church's thesis, for short CT, and the axiom of choice, for short AC, was one of the requirements asked to be satisfied by the intensional level of a two-level foundation for constructive mathematics as proposed by the second author and G. Sambin in 2005.Here we show that this is the case for the intensional level of the twolevel Minimalist Foundation, for short MF, completed in 2009 by the second author. The intensional level of MF consists of an intensional type theory à la Martin-Löf, called mTT.The consistency of mTT with CT and AC is obtained by showing the consistency with the formal Church's thesis of a fragment of intensional Martin-Löf's type theory, called MLtt 1 , where mTT can be easily interpreted. Then to show the consistency of MLtt 1 with CT we interpret it within Feferman's predicative theory of non-iterative fixpoints ID 1 by extending the well known