Nominal Abstraction

Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such descriptions: the interpretation of atomic judgments through recursive definitions and an encoding of binding constructs via generic judgments. However, logics encompassing these two features do not currently allow for the definition of relations that embody dynamic aspects related to binding, a capability needed in many reasoning tasks. We propose a new relation between terms called nominal abstraction as a means for overcoming this deficiency. We incorporate nominal abstraction into a rich logic also including definitions, generic quantification, induction, and co-induction that we then prove to be consistent. We present examples to show that this logic can provide elegant treatments of binding contexts that appear in many proofs, such as those establishing properties of typing calculi and of arbitrarily cascading substitutions that play a role in reducibility arguments.Comment: To appear in the Journal of Information and Computatio

An extensional Kleene realizability semantics for the Minimalist Foundation

We build a Kleene realizability semantics for the two-level Minimalist Foundation MF, ideated by Maietti and Sambin in 2005 and completed by Maietti in 2009. Thanks to this semantics we prove that both levels of MF are consistent with the (Extended) formal Church Thesis CT. MF consists of two levels, an intensional one, called mTT and an extensional one, called emTT, based on versions of Martin-L\"of's type theory. Thanks to the link between the two levels, it is enough to build a semantics for the intensional level to get one also for the extensional level. Hence here we just build a realizability semantics for the intensional level mTT. Such a semantics is a modification of the realizability semantics in Beeson 1985 for extensional first order Martin-L\"of's type theory with one universe. So it is formalised in Feferman's classical arithmetic theory of inductive definitions. It is called extensional Kleene realizability semantics since it validates extensional equality of type-theoretic functions extFun, as in Beeson 1985. The main modification we perform on Beeson's semantics is to interpret propositions, which are defined primitively in MF, in a proof-irrelevant way. As a consequence, we gain the validity of CT. Recalling that extFun+ CT+ AC are inconsistent over arithmetics with finite types, we conclude that our semantics does not validate the full Axiom of Choice AC. On the contrary, Beeson's semantics does validate AC, being this a theorem of Martin-L\"of's theory, but it does not validate CT. The semantics we present here appears to be the best Kleene realizability semantics for the extensional level emTT of MF. Indeed Beeson's semantics is not an option for emTT since the full AC added to it entails the excluded middle