127 research outputs found
Relational Parametricity and Control
We study the equational theory of Parigot's second-order
λμ-calculus in connection with a call-by-name continuation-passing
style (CPS) translation into a fragment of the second-order λ-calculus.
It is observed that the relational parametricity on the target calculus induces
a natural notion of equivalence on the λμ-terms. On the other hand,
the unconstrained relational parametricity on the λμ-calculus turns
out to be inconsistent with this CPS semantics. Following these facts, we
propose to formulate the relational parametricity on the λμ-calculus
in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc
The Internal Operads of Combinatory Algebras
We argue that operads provide a general framework for dealing with
polynomials and combinatory completeness of combinatory algebras, including the
classical -algebras, linear -algebras, planar
-algebras as well as the braided -algebras. We show that every extensional combinatory algebra gives rise to
a canonical closed operad, which we shall call the internal operad of the
combinatory algebra. The internal operad construction gives a left adjoint to
the forgetful functor from closed operads to extensional combinatory algebras.
As a by-product, we derive extensionality axioms for the classes of combinatory
algebras mentioned above
Models of sharing graphs: a categorical semantics of let and letrec
To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is first-order acyclic sharing graphs represented by let-syntax, and others are extensions with higher-order constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and trace
Traced Monads and Hopf Monads
A traced monad is a monad on a traced symmetric monoidal category that lifts
the traced symmetric monoidal structure to its Eilenberg-Moore category. A
long-standing question has been trying to provide a characterization of traced
monads without explicitly mentioning the Eilenberg-Moore category. On the other
hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are
invertible. For compact closed categories, symmetric Hopf monads are precisely
the kind of monads that lift the compact closed structure to their
Eilenberg-Moore categories. Since compact closed categories and traced
symmetric monoidal categories are closely related, it is a natural question to
ask what is the relationship between Hopf monads and traced monads. In this
paper, we introduce trace-coherent Hopf monads on traced monoidal categories,
which can be characterised without mentioning the Eilenberg-Moore category. The
main theorem of this paper is that a symmetric Hopf monad is a traced monad if
and only if it is a trace-coherent Hopf monad. We provide many examples of
trace-coherent Hopf monads, such as those induced by cocommutative Hopf
algebras or any symmetric Hopf monad on a compact closed category. We also
explain how for traced Cartesian monoidal categories, trace-coherent Hopf
monads can be expressed using the Conway operator, while for traced coCartesian
monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We
also provide separating examples of traced monads that are not Hopf monads, as
well as symmetric Hopf monads that are not trace-coherent.Comment: Restructured the paper based on anonymous reviewer's suggestion
Finite Dimensional Vector Spaces Are Complete for Traced Symmetric Monoidal Categories
Abstract. We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists astrongtracedfunctorintoFinVectk which distinguishes them. Therefore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVectk.
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