Inhabitation for Non-idempotent Intersection Types

The inhabitation problem for intersection types in the lambda-calculus is known to be undecidable. We study the problem in the case of non-idempotent intersection, considering several type assignment systems, which characterize the solvable or the strongly normalizing lambda-terms. We prove the decidability of the inhabitation problem for all the systems considered, by providing sound and complete inhabitation algorithms for them

A Coq Library of Undecidable Problems

International audienceWe propose a talk on our library of mechanised reductions to establish undecidability results in Coq. The library is a collaborative effort, growing constantly and we are seeking more outside contributors willing to work on undecidability results in Coq

Hilbert's Tenth Problem in Coq (Extended Version)

We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem -- in our case by a Minsky machine -- is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer. Additionally, we prove the reverse direction and show that every Diophantine relation is recognisable by $\mu$-recursive functions and give a certified compiler from $\mu$-recursive functions to Minsky machines.Comment: submitted to LMC

Enumerating proofs of positive formulae

We provide a semi-grammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This method is complete in the sense that each normal proof-term of the formula is produced by some scheme generated by the grammar. As a corollary, we get a similar description of the set of normal proofs of positive formulae for a large class of theories including simple type theory and System F

Typability and type inference in atomic polymorphism

It is well-known that typability, type inhabitation and type inference are undecidable in the Girard-Reynolds polymorphic system F. It has recently been proven that type inhabitation remains undecidable even in the predicative fragment of system F in which all universal instantiations have an atomic witness (system Fat). In this paper we analyze typability and type inference in Curry style variants of system Fat and show that typability is decidable and that there is an algorithm for type inference which is capable of dealing with non-redundancy constraints.The second author acknowledges the support of FCT — Fundação para a Ciência e a Tecnologia under the projects UIDB/04561/2020, UIDB/00408/2020 and UIDP/00408/2020, and she is also grateful to CMAFcIO — Centro de Matemática, Aplicações Fundamentais e Investigação Operacional and to LASIGE — Computer Science and Engineering Research Centre (Universidade de Lisboa).info:eu-repo/semantics/publishedVersio

Typability and type inference in atomic polymorphism

It is well-known that typability, type inhabitation and type inference are undecidable in the Girard-Reynolds polymorphic system F. It has recently been proven that type inhabitation remains undecidable even in the predicative fragment of system F in which all universal instantiations have an atomic witness (system Fat). In this paper we analyze typability and type inference in Curry style variants of system Fat and show that typability is decidable and that there is an algorithm for type inference which is capable of dealing with non-redundancy constraints.The second author acknowledges the support of FCT — Fundação para a Ciência e a Tecnologia under the projects UIDB/04561/2020, UIDB/00408/2020 and UIDP/00408/2020, and she is also grateful to CMAFcIO — Centro de Matemática, Aplicações Fundamentais e Investigação Operacional and to LASIGE — Computer Science and Engineering Research Centre (Universidade de Lisboa).info:eu-repo/semantics/publishedVersio

Ticket Entailment is decidable

Submitted on 06/09/2010 to Math. Struct. in Comp. Science. Accepted for publication on 12/19/2011. Last revision on 03/06/2012We prove the decidability of Ticket Entailment. Raised by Anderson and Belnap within the framework of relevance logic, this question is equivalent to the question of the decidability of type inhabitation in simply-typed combinatory logic with the partial basis BB'IW. We solve the equivalent problem of type inhabitation for the restriction of simply-typed lambda-calculus to hereditarily right-maximal terms