21 research outputs found
A Linear Category of Polynomial Diagrams
We present a categorical model for intuitionistic linear logic where objects
are polynomial diagrams and morphisms are simulation diagrams. The
multiplicative structure (tensor product and its adjoint) can be defined in any
locally cartesian closed category, whereas the additive (product and coproduct)
and exponential Tensor-comonoid comonad) structures require additional
properties and are only developed in the category Set, where the objects and
morphisms have natural interpretations in terms of games, simulation and
strategies.Comment: 20 page
Indexed induction and coinduction, fibrationally.
This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a sound coinduction rule for any data type arising as the final coalgebra of a functor, thus relaxing Hermida and Jacobsâ restriction to polynomial data types. For this we introduce the notion of a quotient category with equality (QCE), which both abstracts the standard notion of a fibration of relations constructed from a given fibration, and plays a role in the theory of coinduction dual to that of a comprehension category with unit (CCU) in the theory of induction. Second, we show that indexed inductive and coinductive types also admit sound induction and coinduction rules. Indexed data types often arise as initial algebras and final coalgebras of functors on slice categories, so our key technical results give sufficent conditions under which we can construct, from a CCU (QCE) U : E -> B, a fibration with base B/I that models indexing by I and is also a CCU (QCE)
General Recursion and Formal Topology.
Comment: In Proceedings PAR 2010, arXiv:1012.455
Indexed Induction and Coinduction, Fibrationally
This paper extends the fibrational approach to induction and coinduction
pioneered by Hermida and Jacobs, and developed by the current authors, in two
key directions. First, we present a dual to the sound induction rule for
inductive types that we developed previously. That is, we present a sound
coinduction rule for any data type arising as the carrier of the final
coalgebra of a functor, thus relaxing Hermida and Jacobs' restriction to
polynomial functors. To achieve this we introduce the notion of a quotient
category with equality (QCE) that i) abstracts the standard notion of a
fibration of relations constructed from a given fibration; and ii) plays a role
in the theory of coinduction dual to that played by a comprehension category
with unit (CCU) in the theory of induction. Secondly, we show that inductive
and coinductive indexed types also admit sound induction and coinduction rules.
Indexed data types often arise as carriers of initial algebras and final
coalgebras of functors on slice categories, so we give sufficient conditions
under which we can construct, from a CCU (QCE) U:E \rightarrow B, a fibration
with base B/I that models indexing by I and is also a CCU (resp., QCE). We
finish the paper by considering the more general case of sound induction and
coinduction rules for indexed data types when the indexing is itself given by a
fibration
Interfaces as functors, programs as coalgebrasâA final coalgebra theorem in intensional type theory
AbstractIn [P. Hancock, A. Setzer, Interactive programs in dependent type theory, in: P. Clote, H. Schwichtenberg (Eds.), Proc. 14th Annu. Conf. of EACSL, CSLâ00, Fischbau, Germany, 21â26 August 2000, Vol. 1862, Springer, Berlin, 2000, pp. 317â331, URL ăciteseer.ist.psu.edu/article/hancock00interactive.htmlă; P. Hancock, A. Setzer, Interactive programs and weakly final coalgebras in dependent type theory, in: L. Crosilla, P. Schuster (Eds.), From Sets and Types to Topology and Analysis. Towards Practicable Foundations for Constructive Mathematics, Oxford Logic Guides, Clarendon Press, 2005, URL ăwww.cs.swan.ac.uk/âŒcsetzer/ă] Hancock and Setzer introduced rules to extend Martin-Löf's type theory in order to represent interactive programming. The rules essentially reflect the existence of weakly final coalgebras for a general form of polynomial functor. The standard rules of dependent type theory allow the definition of inductive types, which correspond to initial algebras. Coalgebraic types are not represented in a direct way. In this article we show the existence of final coalgebras in intensional type theory for these kind of functors, where we require uniqueness of identity proofs (UIP) for the set of states S and the set of commands C which determine the functor. We obtain the result by identifying programs which have essentially the same behaviour, viz. are bisimular. This proves the rules of Setzer and Hancock admissible in ordinary type theory, if we replace definitional equality by bisimulation. All proofs [M. Michelbrink, Verifications of final coalgebra theorem in: Interfaces as Functors, Programs as CoalgebrasâA Final Coalgebra Theorem in Intensional Type Theory, 2005, URL ăwww.cs.swan.ac.uk/âŒcsmichel/ă] are verified in the theorem prover agda [C. Coquand, Agda, Internet, URL ăwww.cs.chalmers.se/âŒcatarina/agda/ă; K. Peterson, A programming system for type theory, Technical Report, S-412 96, Chalmers University of Technology, Göteborg, 1982], which is based on intensional Martin-Löf type theory
Infinite Types, Infinite Data, Infinite Interaction
We describe a way to represent computable functions between coinductive types as particular transducers in type theory. This generalizes earlier work on functions between streams by P. Hancock to a much richer class of coinductive types. Those transducers can be defined in dependent type theory without any notion of equality but require inductive-recursive definitions. Most of the properties of these constructions only rely on a mild notion of equality (intensional equality) and can thus be formalized in the dependently typed language Agda
Interaction laws of monads and comonads
We introduce and study functor-functor and monad-comonad interaction laws as
mathematical objects to describe interaction of effectful computations with
behaviors of effect-performing machines. Monad-comonad interaction laws are
monoid objects of the monoidal category of functor-functor interaction laws. We
show that, for suitable generalizations of the concepts of dual and Sweedler
dual, the greatest functor resp. monad interacting with a given functor or
comonad is its dual while the greatest comonad interacting with a given monad
is its Sweedler dual. We relate monad-comonad interaction laws to stateful
runners. We show that functor-functor interaction laws are Chu spaces over the
category of endofunctors taken with the Day convolution monoidal structure.
Hasegawa's glueing endows the category of these Chu spaces with a monoidal
structure whose monoid objects are monad-comonad interaction laws