Convexity, Duality and Effects

Jacobs, Bart

doi:10.1007/978-3-642-15240-5_1

Cited by 58 publications

(60 citation statements)

References 22 publications

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“…This is the category of Eilenberg-Moore algebras of the distribution monad D, see [26,28] for details. Such state-and-effect triangles are studied systematically in [32,31] as formalisation of the fundamental (dual adjoint) relationship between states and predicates/effects, and between state transformers and predicate transformers in programming semantics and logic [15].…”

Section: Emodmentioning

confidence: 99%

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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Section: Emodmentioning

confidence: 99%

From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

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“…Such convex sets can also be described in terms of 'weighted sums' x + r y, interpreted as rx + (1 − r)y, see e.g. [76,78,38]. Morphisms in EM(D M ) are affine maps; they preserve such convex sums i m i x i .…”

Section: Preliminaries On Effect Algebras Effect Modules and Convexmentioning

confidence: 99%

“…5] (based on [38]) a (dual) adjunction between convex sets over [0, 1] and effect modules over [0, 1] is described. This adjunction exists in fact for an arbitrary effect monoid M -instead of [0, 1] -by using M as dualising object.…”

Section: Preliminaries On Effect Algebras Effect Modules and Convexmentioning

confidence: 99%

New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Jacobs¹

2015

Logical Methods in Computer Science

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Abstract. Intuitionistic logic, in which the double negation law ¬¬P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold, especially in quantitative logics for probabilistic and quantum systems. The algebraic notions of effect algebra and effect module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X → 1 + 1 carry such effect module structure, and can be used as predicates. Maps of this form X → 1 + 1 are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C * -algebra or Hilbert space.In quantum foundations the duality between states and effects (predicates) plays an important role. This duality appears in the form of an adjunction in our categorical setting, where we use maps 1 → X as states. For such a state ω and a predicate p, the validity probability ω |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature.Measurement from quantum mechanics is formalised categorically in terms of 'instruments', using Lüders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C * -algebras, side-effects appear exclusively in the noncommutative (properly quantum) case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems, in which the above mentioned elements occur.

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“…These are sets X in which for each formal convex sum i r i |x i there is an actual convex sum i r i x i ∈ X. Morphisms in Conv preserve such convex sums, and are often called affine functions. A convex set can be defined alternatively as a barycentric algebra [29], see [12] for the connection. Similarly, we write Conv ≤1 = EM(D ≤1 ) for the category of subconvex sets, in which subconvex sums exist.…”

Section: Linear and (Sub)convex Computationmentioning

confidence: 99%

Dijkstra and Hoare monads in monadic computation

Jacobs

2015

Theoretical Computer Science

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The Dijkstra and Hoare monads have been introduced recently for capturing weakest precondition computations and computations with pre-and post-conditions, within the context of program verification, supported by a theorem prover. Here we give a more general description of such monads in a categorical setting. We first elaborate the recently developed view on program semantics in terms of a triangle of computations, state transformers, and predicate transformers. Instantiating this triangle for different computational monads T shows how to define the Dijkstra monad associated with T , via the logic involved.Subsequently we give abstract definitions of the Dijkstra and Hoare monad, parametrised by a computational monad. These definitions presuppose a suitable (categorical) predicate logic, defined on the Kleisli category of the underlying monad. When all this structure exists, we show that there are maps of monads (Hoare) ⇒ (State) ⇒ (Dijkstra), all parametrised by a monad T .

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Convexity, Duality and Effects

Cited by 58 publications

References 22 publications

From probability monads to commutative effectuses

From probability monads to commutative effectuses

New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Dijkstra and Hoare monads in monadic computation

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