Affine Monads and Side-Effect-Freeness

Jacobs, Bart

doi:10.1007/978-3-319-40370-0_5

Cited by 10 publications

(10 citation statements)

References 21 publications

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“…It then digs deeper into affineness and introduces a slightly stronger notion, called 'strong affiness', following [33]. We describe basic properties and examples.…”

Section: Affineness and Strong Affiness Of Monadsmentioning

confidence: 99%

“…Thanks are due to Kenta Cho, Robert Furber, Mathys Rennela, Bram Westerbaan, and Fabio Zanasi for helpful discussions on the topic of the paper, and to the anonymous referees of [33,34], and of this paper, for suggesting several improvements.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…It should lead to a characterisation of (categorical) models of probabilistic computation. This paper combines two earlier conference publications [33,34] into a single integrated account. It introduces in a step-by-step manner successive properties of monads that ensure that their Kleisli categories are commutative effectuses.…”

Section: Introductionmentioning

confidence: 99%

“…Affineness of a monad T means that it preserves the final object: T (1) ∼ = 1. The property 'strong affineness' comes from [33], where it is used to prove a bijective correspondence between predicates and side-effect-free instruments. The relation between predicates and associated actions (instruments / coalgebras) comes from quantum theory in general, and effectus theory in particular.…”

Section: Introductionmentioning

confidence: 99%

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From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

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Effectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These 'probabilistic' models are called commutative effectuses, and are the focus of attention here. The paper describes the main known 'probability' monads: the monad of discrete probability measures, the Giry monad, the expectation monad, the probabilistic power domain monad, the Radon monad, and the Kantorovich monad. It also introduces successive properties that a monad should satisfy so that its Kleisli category is a commutative effectus. The main properties are: partial additivity, strong affineness, and commutativity. It is shown that the resulting commutative effectus provides a categorical model of probability theory, including a logic using effect modules with parallel and sequential conjunction, predicate-and state-transformers, normalisation and conditioning of states.

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“…It then digs deeper into affineness and introduces a slightly stronger notion, called 'strong affiness', following [33]. We describe basic properties and examples.…”

Section: Affineness and Strong Affiness Of Monadsmentioning

confidence: 99%

Section: Acknowledgementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

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“…• the tensor unit is a final object 1, so that tensors ⊗ have projections π 1 : X ⊗Y → X ⊗ 1 ∼ = X; this makes the setting affine, see [22,23];…”

Section: Markov Categories With Colimitsmentioning

confidence: 99%

Multinomial and Hypergeometric Distributions in Markov Categories

Jacobs

2021

Electron. Proc. Theor. Comput. Sci.

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Markov categories, having tensors with copying and discarding, provide a setting for categorical probability. This paper uses finite colimits and what we call uniform states in such Markov categories to define a (fixed size) multiset functor, with basic operations for sums and zips of multisets, and a graded monad structure. Multisets can be used to represent both urns filled with coloured balls and also draws of multiple balls from such urns. The main contribution of this paper is the abstract definition of multinomial and hypergeometric distributions on multisets, as draws. It is shown that these operations interact appropriately with various operations on multisets.1. Probabilistic programming languages that incorporate updating (conditioning) and/or higher order features, see e.g. [10,11,12,32,34,13].2. The compositional approach to Bayesian networks [8,16] and to Bayesian reasoning [9,27,25].3. The use of diagrammatic methods in (quantum) foundations and probability, see [7] for an overview.4. Study of 'probability monads', e.g. in [30,23]. Axiomatisation of disintegration as key probabilistic technique, see e.g. [17,3,19,18], and also [5].6. Exploration of categorical structures, such as compact closed categories [1,33] or effectuses [21,4].These topics cover both ordinary (classical) probability as well as quantum probability.An issue that we are particularly interested in is the interplay between multisets (a.k.a. bags) and (probability) distributions (see e.g. [26]). Multisets play a fundamental role in probability theory, for instance as representations of urns with coloured balls, and also of draws from such urns. More generally, in learning, collections of data items, possibly occurring multiple times, are properly represented as multisets. Multinomial and hypergeometric distributions assign probabilities to draws from an urn,

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Shades of Iteration: From Elgot to Kleene

Goncharov

2023

Lecture Notes in Computer Science

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Affine Monads and Side-Effect-Freeness

Cited by 10 publications

References 21 publications

From probability monads to commutative effectuses

From probability monads to commutative effectuses

Multinomial and Hypergeometric Distributions in Markov Categories

Shades of Iteration: From Elgot to Kleene

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