205 research outputs found
Enriched Lawvere Theories for Operational Semantics
Enriched Lawvere theories are a generalization of Lawvere theories that allow
us to describe the operational semantics of formal systems. For example, a
graph enriched Lawvere theory describes structures that have a graph of
operations of each arity, where the vertices are operations and the edges are
rewrites between operations. Enriched theories can be used to equip systems
with operational semantics, and maps between enriching categories can serve to
translate between different forms of operational and denotational semantics.
The Grothendieck construction lets us study all models of all enriched theories
in all contexts in a single category. We illustrate these ideas with the
SKI-combinator calculus, a variable-free version of the lambda calculus.Comment: In Proceedings ACT 2019, arXiv:2009.0633
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
Modalities, Cohesion, and Information Flow
It is informally understood that the purpose of modal type constructors in
programming calculi is to control the flow of information between types. In
order to lend rigorous support to this idea, we study the category of
classified sets, a variant of a denotational semantics for information flow
proposed by Abadi et al. We use classified sets to prove multiple
noninterference theorems for modalities of a monadic and comonadic flavour. The
common machinery behind our theorems stems from the the fact that classified
sets are a (weak) model of Lawvere's theory of axiomatic cohesion. In the
process, we show how cohesion can be used for reasoning about multi-modal
settings. This leads to the conclusion that cohesion is a particularly useful
setting for the study of both information flow, but also modalities in type
theory and programming languages at large
Tensor of Quantitative Equational Theories
We develop a theory for the commutative combination of quantitative effects, their tensor, given as a combination of quantitative equational theories that imposes mutual commutation of the operations from each theory. As such, it extends 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 show that under certain assumptions on the quantitative theories the free monad that arises from the tensor of two theories is the categorical tensor of the free monads on the theories. As an application, we provide the first algebraic axiomatizations of labelled Markov processes and Markov decision processes. Apart from the intrinsic interest in the axiomatizations, it is pleasing they are obtained compositionally by means of the sum and tensor of simpler quantitative equational theories
Diagrammatic Inference
Diagrammatic logics were introduced in 2002, with emphasis on the notions of
specifications and models. In this paper we improve the description of the
inference process, which is seen as a Yoneda functor on a bicategory of
fractions. A diagrammatic logic is defined from a morphism of limit sketches
(called a propagator) which gives rise to an adjunction, which in turn
determines a bicategory of fractions. The propagator, the adjunction and the
bicategory provide respectively the syntax, the models and the inference
process for the logic. Then diagrammatic logics and their morphisms are applied
to the semantics of side effects in computer languages.Comment: 16 page
String diagram rewrite theory II: Rewriting with symmetric monoidal structure
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras
Handling Fibred Algebraic Effects
International audienceWe study algebraic computational effects and their handlers in the dependently typed setting. We describecomputational effects using a generalisation of Plotkin and Pretnar’s effect theories, whose dependentlytyped operations allow us to capture precise notions of computation, e.g., state with location-dependent storetypes and dependently typed update monads. Our treatment of handlers is based on an observation that theirconventional term-level definition leads to unsound program equivalences being derivable in languages thatinclude a notion of homomorphism. We solve this problem by giving handlers a novel type-based treatmentvia a new computation type, the user-defined algebra type, which pairs a value type (the carrier) with a set ofvalue terms (the operations), capturing Plotkin and Pretnar’s insight that effect handlers denote algebras. Wethen show that the conventional presentation of handlers can be routinely derived, and demonstrate that thistype-based treatment of handlers provides a useful mechanism for reasoning about effectful computations.We also equip the resulting language with a sound denotational semantics based on families fibrations
- …