Dijkstra and Hoare monads in monadic computation

Jacobs, Bart

doi:10.1016/j.tcs.2015.03.020

Cited by 18 publications

(19 citation statements)

References 25 publications

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“…They can be described as a computations in the opposite category (CL ) op of complete lattices and meet-preserving functions. This is precisely like in the above 'logical' categories of the form (−) op , see also [43,42]. In general the predicates Pred(1), i.e.…”

Section: Predicates and Testsmentioning

confidence: 57%

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

Jacobs¹

2015

Logical Methods in Computer Science

Self Cite

140

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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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Section: Predicates and Testsmentioning

confidence: 57%

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

Jacobs¹

2015

Logical Methods in Computer Science

Self Cite

140

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“…The realization that predicate transformers form monads [Ahman et al 2017;Jacobs 2014Jacobs , 2015Swamy et al 2013Swamy et al , 2016 is the starting point to provide a uniform notion of specifications. Generalizing over prior work, we show that this is true not only for weakest precondition transformers, but also for strongest postconditions, and pairs of pre-and postconditions (see §4.1).…”

Section: Specification Monadsmentioning

confidence: 99%

“…As we shall see this construction is generic and leads to a categorical equivalence between Dijkstra monads and effect observations. In this section, we introduce more formally the notion of Dijkstra monad using dependent type theory, seen as the internal language of a comprehension category [Jacobs 1993], and then build a category DMon of Dijkstra monads. In order to compare this notion of Dijkstra monads to effect observations, we also introduce a category of monadic relations MonRel and show that there is an adjunction ∫ ⊣ pre : MonRel −→ DMon.…”

Section: Dijkstra Monads From Effect Observationsmentioning

confidence: 99%

“…We work in an extensional type theory with dependent products (x : A) → B, strong sums (x : A) × B, an identity type x = A y for x, y : A (where the type A is usually left implicit), a type of (proof-irrelevant) propositions P, and quotients of equivalence relations. This syntax is the internal language of a comprehension category [Jacobs 1993] with enough structure and we will write Type for any such category. This interpretation of type theory allows us to call any object Γ ∈ Type a type.…”

Section: Dijkstra Monads From Effect Observationsmentioning

confidence: 99%

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Dijkstra monads for all

Maillard

Ahman

Atkey

et al. 2019

Proc. ACM Program. Lang.

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This paper proposes a general semantic framework for verifying programs with arbitrary monadic side-effects using Dijkstra monads, which we define as monad-like structures indexed by a specification monad. We prove that any monad morphism between a computational monad and a specification monad gives rise to a Dijkstra monad, which provides great flexibility for obtaining Dijkstra monads tailored to the verification task at hand. We moreover show that a large variety of specification monads can be obtained by applying monad transformers to various base specification monads, including predicate transformers and Hoare-style preand postconditions. For defining correct monad transformers, we propose a language inspired by Moggi's monadic metalanguage that is parameterized by a dependent type theory. We also develop a notion of algebraic operations for Dijkstra monads, and start to investigate two ways of also accommodating effect handlers. We implement our framework in both Coq and F ⋆ , and illustrate that it supports a wide variety of verification styles for effects such as exceptions, nondeterminism, state, input-output, and general recursion.

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“…Dijkstra algorithm is widely used in finding the shortest path from a starting point to other vertices [12][13][14] . The advantage of the algorithm lies in finding out the vertex closest to the starting point, automatically taking it as the new starting point (knee point), and updating the original path information simultaneously.…”

Section: Establishment Of the Minimum Distance Matrixmentioning

confidence: 99%

A Mobile Robot Path Planning Based on Multiple Destination-points

Wang¹,

Dai²,

Li³

2017

dtcse

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Abstract. Path planning is an important aspect of mobile robot. This paper mainly studies the global path planning with multiple sub-path targets. Firstly, this paper build two initial distance matrix, then construct a minimum distance matrix by Dijkstra algorithm. Finally, the paper determine the destination nodes which have all minimum path. The path planning method can be used to some service robots, including: family service robots, medical robots, industrial robots.

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Dijkstra and Hoare monads in monadic computation

Cited by 18 publications

References 25 publications

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

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

Dijkstra monads for all

A Mobile Robot Path Planning Based on Multiple Destination-points

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