3 research outputs found
Proof Nets, Coends and the Yoneda Isomorphism
Proof nets provide permutation-independent representations of proofs and are
used to investigate coherence problems for monoidal categories. We investigate
a coherence problem concerning Second Order Multiplicative Linear Logic (MLL2),
that is, the one of characterizing the equivalence over proofs generated by the
interpretation of quantifiers by means of ends and coends.
We provide a compact representation of proof nets for a fragment of MLL2
related to the Yoneda isomorphism. By adapting the "rewiring approach" used in
coherence results for star-autonomous categories, we define an equivalence
relation over proof nets called "re-witnessing". We prove that this relation
characterizes, in this fragment, the equivalence generated by coends.Comment: In Proceedings Linearity-TLLA 2018, arXiv:1904.0615
The Yoneda Reduction of Polymorphic Types (Extended Version)
In this paper we explore a family of type isomorphisms in System F whose
validity corresponds, semantically, to some form of the Yoneda isomorphism from
category theory. These isomorphisms hold under theories of equivalence stronger
than beta-eta-equivalence, like those induced by parametricity and
dinaturality. We show that the Yoneda type isomorphisms yield a rewriting over
types, that we call Yoneda reduction, which can be used to eliminate
quantifiers from a polymorphic type, replacing them with a combination of
monomorphic type constructors. We establish some sufficient conditions under
which quantifiers can be fully eliminated from a polymorphic type, and we show
some application of these conditions to count the inhabitants of a type and to
compute program equivalence in some fragments of System F
Intersection Type Distributors
Building on previous works, we present a general method to define proof
relevant intersection type semantics for pure lambda calculus. We argue that
the bicategory of distributors is an appropriate categorical framework for this
kind of semantics. We first introduce a class of 2-monads whose algebras are
monoidal categories modelling resource management, following Marsden-Zwardt's
approach. We show how these monadic constructions determine Kleisli
bicategories over the bicategory of distributors and we give a sufficient
condition for cartesian closedness. We define a family of non-extentional
models for pure lambda calculus. We then prove that the interpretation of
lambda terms induced by these models can be concretely described via
intersection type systems. The intersection constructor corresponds to the
particular tensor product given by the considered free monadic construction. We
conclude by describing two particular examples of these distributor-induced
intersection type systems, proving that they characterise head-normalization.Comment: Submitte