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LTL types FRP: Lineartime temporal logic propositions as types, proofs as functional reactive programs
 In Proc. ACM Workshop Programming Languages meets Program Verification
, 2012
"... Functional Reactive Programming (FRP) is a form of reactive programming whose model is pure functions over signals. FRP is often expressed in terms of arrows with loops, which is the type class for a Freyd category (that is a premonoidal category with a cartesian centre) equipped with a premonoid ..."
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Cited by 14 (3 self)
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Functional Reactive Programming (FRP) is a form of reactive programming whose model is pure functions over signals. FRP is often expressed in terms of arrows with loops, which is the type class for a Freyd category (that is a premonoidal category with a cartesian centre) equipped with a premonoidal trace. This type system suffices to define the dataflow structure of a reactive program, but does not express its temporal properties. In this paper, we show that Lineartime Temporal Logic (LTL) is a natural extension of the type system for FRP, which constrains the temporal behaviour of reactive programs. We show that a constructive LTL can be defined in a dependently typed functional language, and that reactive programs form proofs of constructive LTL properties. In particular, implication in LTL gives rise to stateless functions on streams, and the “constrains ” modality gives rise to causal functions. We show that reactive programs form a partially traced monoidal category, and hence can be given as a form of arrows with loops, where the type system enforces that only decoupled functions can be looped.
A general framework for sound and complete FloydHoare logics
 ACM Transactions on Computational Logic (TOCL
, 2009
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Partially traced categories
, 1107
"... This paper deals with questions relating to Haghverdiand Scott’s notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced ..."
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This paper deals with questions relating to Haghverdiand Scott’s notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every partially traced
Hoare Logic in the Abstract
"... Abstract. We present an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. We first identify a particular class of functors – which we call ‘verification functors ’ – between traced symmetric monoidal categories and subcategories ..."
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Abstract. We present an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. We first identify a particular class of functors – which we call ‘verification functors ’ – between traced symmetric monoidal categories and subcategories of Preord (the category of preordered sets and monotone mappings). We then give an abstract definition of Hoare triples, parametrised by a verification functor, and prove a single soundness and completeness theorem for such triples. In the particular case of the traced symmetric monoidal category of while programs we get back Hoare’s original rules. We discuss how our framework handles extensions of the Hoare logic for while programs, e.g. the extension with pointer manipulations via separation logic. Finally, we give an example of how our theory can be used in the development of new Hoare logics: we present a new sound and complete set of Hoarelogiclike rules for the verification of linear dynamical systems, modelled via stream circuits. 1
Partially traced categories
"... This paper deals with questions relating to Haghverdi and Scott’s notion of partially traced categories. The main result is a representationtheorem for such categories: we provethat everypartiallytraced categorycan be faithfully embedded in a totally traced category. Also conversely, every symmetric ..."
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This paper deals with questions relating to Haghverdi and Scott’s notion of partially traced categories. The main result is a representationtheorem for such categories: we provethat everypartiallytraced categorycan be faithfully embedded in a totally traced category. Also conversely, every symmetric monoidal subcategory ofatotallytracedcategoryispartiallytraced, sothis characterizesthe partiallytracedcategoriescompletely. The main technique we use is based on Freyd’s paracategories, along with a partial version of Joyal, Street, and Verity’s Intconstruction.
On Geometry of Interaction for Polarized Linear Logic
, 2014
"... We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment (without structural rules) of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multipoints to various categorical models of GoI. Multi ..."
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We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment (without structural rules) of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multipoints to various categorical models of GoI. Multipoints are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing, as well as playing a fundamental role in allowing us to construct concrete polarized GoI models. Our approach to polarized GoI involves two independent studies, based on different categorical perspectives of GoI. (i) Inspired by the work of Abramsky, Haghverdi, and Scott, a polarized GoI situation is dened which considers multipoints added to a traced monoidal category with an appropriate re
exive object U. Categorical versions of Girard's Execution formula (taking into account the multipoints) are dened, as well as the GoI
Quantum Turing automata
"... A denotational semantics of quantum Turing machines having a quantum control is defined in the ..."
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A denotational semantics of quantum Turing machines having a quantum control is defined in the