774 research outputs found
Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory
We describe categorical models of a circuit-based (quantum) functional
programming language. We show that enriched categories play a crucial role.
Following earlier work on QWire by Paykin et al., we consider both a simple
first-order linear language for circuits, and a more powerful host language,
such that the circuit language is embedded inside the host language. Our
categorical semantics for the host language is standard, and involves cartesian
closed categories and monads. We interpret the circuit language not in an
ordinary category, but in a category that is enriched in the host category. We
show that this structure is also related to linear/non-linear models. As an
extended example, we recall an earlier result that the category of W*-algebras
is dcpo-enriched, and we use this model to extend the circuit language with
some recursive types
Healthiness from Duality
Healthiness is a good old question in program logics that dates back to
Dijkstra. It asks for an intrinsic characterization of those predicate
transformers which arise as the (backward) interpretation of a certain class of
programs. There are several results known for healthiness conditions: for
deterministic programs, nondeterministic ones, probabilistic ones, etc.
Building upon our previous works on so-called state-and-effect triangles, we
contribute a unified categorical framework for investigating healthiness
conditions. We find the framework to be centered around a dual adjunction
induced by a dualizing object, together with our notion of relative
Eilenberg-Moore algebra playing fundamental roles too. The latter notion seems
interesting in its own right in the context of monads, Lawvere theories and
enriched categories.Comment: 13 pages, Extended version with appendices of a paper accepted to
LICS 201
Exploring the Boundaries of Monad Tensorability on Set
We study a composition operation on monads, equivalently presented as large
equational theories. Specifically, we discuss the existence of tensors, which
are combinations of theories that impose mutual commutation of the operations
from the component theories. As such, they extend the sum of two theories,
which is just their unrestrained combination. Tensors of theories arise in
several contexts; in particular, in the semantics of programming languages, the
monad transformer for global state is given by a tensor. We present two main
results: we show that the tensor of two monads need not in general exist by
presenting two counterexamples, one of them involving finite powerset (i.e. the
theory of join semilattices); this solves a somewhat long-standing open
problem, and contrasts with recent results that had ruled out previously
expected counterexamples. On the other hand, we show that tensors with bounded
powerset monads do exist from countable powerset upwards
Fibrational induction meets effects
This paper provides several induction rules that can be used to prove properties of effectful data types. Our results are semantic in nature and build upon Hermida and Jacobs’ fibrational formulation of induction for polynomial data types and its extension to all inductive data types by Ghani, Johann, and Fumex. An effectful data type μ(TF) is built from a functor F that describes data, and a monad T that computes effects. Our main contribution is to derive induction rules that are generic over all functors F and monads T such that μ(TF) exists. Along the way, we also derive a principle of definition by structural recursion for effectful data types that is similarly generic. Our induction rule is also generic over the kinds of properties to be proved: like the work on which we build, we work in a general fibrational setting and so can accommodate very general notions of properties, rather than just those of particular syntactic forms. We give examples exploiting the generality of our results, and show how our results specialize to those in the literature, particularly those of Filinski and Støvring
Generic Trace Semantics via Coinduction
Trace semantics has been defined for various kinds of state-based systems,
notably with different forms of branching such as non-determinism vs.
probability. In this paper we claim to identify one underlying mathematical
structure behind these "trace semantics," namely coinduction in a Kleisli
category. This claim is based on our technical result that, under a suitably
order-enriched setting, a final coalgebra in a Kleisli category is given by an
initial algebra in the category Sets. Formerly the theory of coalgebras has
been employed mostly in Sets where coinduction yields a finer process semantics
of bisimilarity. Therefore this paper extends the application field of
coalgebras, providing a new instance of the principle "process semantics via
coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page
Distributive Laws and Decidable Properties of SOS Specifications
Some formats of well-behaved operational specifications, correspond to
natural transformations of certain types (for example, GSOS and coGSOS laws).
These transformations have a common generalization: distributive laws of monads
over comonads. We prove that this elegant theoretical generalization has
limited practical benefits: it does not translate to any concrete rule format
that would be complete for specifications that contain both GSOS and coGSOS
rules. This is shown for the case of labeled transition systems and
deterministic stream systems.Comment: In Proceedings EXPRESS/SOS 2014, arXiv:1408.127
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