Scalars, Monads, and Categories

Coumans, Dion; Jacobs, Bart

doi:10.1093/acprof:oso/9780199646296.003.0007

Cited by 25 publications

(43 citation statements)

References 14 publications

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“…In [13] it is shown that a monad S is additive iff the coproducts (0, +) of its Kleisli category K (S) are biproducts iff the products (1, ×) of its category EM(S) of Eilenberg-Moore algebras are biproducts.…”

Section: Examplementioning

confidence: 99%

“…An effectus is a relatively simple category, with finite coproducts and a final object, satisfying some elementary properties: certain squares have to be pullbacks and certain parallel maps have to be jointly monic, see (25) and (13) below. These effectuses have been introduced in [30], and give rise to a rich theory that includes quantum computation, see the overview paper [10].…”

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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Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

From probability monads to commutative effectuses

Jacobs

2018

Journal of Logical and Algebraic Methods in Programming

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“…Given a simple Segala system c : X → P(A × DX), its EM-extension F EM (c) from Lemma 3, obtained via the EM-law ρ from (8), is the same as the coalgebra c # : DX → P(A × DX) described in Equation (7).…”

Section: Simple Segala Systems In Em-stylementioning

confidence: 99%

“…The extension natural tranformation follows from additivity of the multiset monad M C (like in (2), see [8]), in:…”

Section: Weighted Automatamentioning

confidence: 99%

Trace Semantics via Determinization

Jacobs

Silva

Sokolova

2012

Coalgebraic Methods in Computer Science

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Abstract. This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories, stemming from an initial algebra in the underlying category. This approach requires some non-trivial assumptions, like dcpo enrichment, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). More recently, it has been noticed that trace semantics can also arise by first performing a determinization construction. In this paper, we develop a systematic approach, in which the two approaches correspond to different orders of composing a functor and a monad, and accordingly, to different distributive laws. The relevant final coalgebra that gives rise to trace semantics does not live in a Kleisli category, but more generally, in a category of Eilenberg-Moore algebras. In order to exploit its finality, we identify an extension operation, that changes the state space of a coalgebra into a free algebra, which abstractly captures determinization of automata. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.

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“…-Fundamental to our approach is a (partial) semiring structure on the set T1, studied in [8,3,2]. On the one hand, its (partial) addition operation induces an order on T1 which is used to generalise the notion of predicate typically employed in the semantics of modal logics, by considering predicates valued in T1.…”

Section: Introductionmentioning

confidence: 99%

A Coalgebraic Approach to Linear-Time Logics

Cîrstea

2014

Lecture Notes in Computer Science

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Abstract. We extend recent work on defining linear-time behaviour for state-based systems with branching, and propose modal and fixpoint logics for specifying linear-time temporal properties of states in such systems. We model systems with branching as coalgebras whose type arises as the composition of a branching monad and a polynomial endofunctor on the category of sets, and employ a set of truth values induced canonically by the branching monad. This yields logics for reasoning about quantitative aspects of linear-time behaviour. Examples include reasoning about the probability of a linear-time behaviour being exhibited by a system with probabilistic branching, or about the minimal cost of a linear-time behaviour being exhibited by a system with weighted branching. In the case of non-deterministic branching, our logic supports reasoning about the possibility of exhibiting a given linear-time behaviour, and therefore resembles an existential version of the logic LTL.

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Scalars, Monads, and Categories

Cited by 25 publications

References 14 publications

From probability monads to commutative effectuses

From probability monads to commutative effectuses

Trace Semantics via Determinization

A Coalgebraic Approach to Linear-Time Logics

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