From monoids to near-semirings

Rivas, Exequiel; Jaskelioff, Mauro; Schrijvers, Tom

doi:10.1145/2790449.2790514

Cited by 10 publications

(3 citation statements)

References 17 publications

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“…We can also verify that the morphisms m and u form a MonadPlus structure [29]. They obviously form a monoid, so it is left to verify two additional laws: left distributivity and left zero (or, in the language of Plotkin and Power [27], that m and u are algebraic).…”

Section: Free Eilenberg-moore Monoidsmentioning

confidence: 99%

“…The fact that ⌊m⌋ ⊗ is a morphism between monoids follows from the construction of Cayley representation of monoids in monoidal categories [28,29]. The retraction r : (MA ⇒ MA) → MA is defined as follows:…”

Section: A5 Theorem 16mentioning

confidence: 99%

“…The triple MA ⇒ MA, comp, ident is a monoid, since it is a special case of the general construction A ⇒ A, comp, ident ; see, for example, Rivas et al [29]. We need to verify that p has the Eilenberg-Moore property.…”

Section: A4 Theorem 15mentioning

confidence: 99%

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Eilenberg–Moore Monoids and Backtracking Monad Transformers

Piróg

2016

Electron. Proc. Theor. Comput. Sci.

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We develop an algebraic underpinning of backtracking monad transformers in the general setting of monoidal categories. As our main technical device, we introduce Eilenberg-Moore monoids, which combine monoids with algebras for strong monads. We show that Eilenberg-Moore monoids coincide with algebras for the list monad transformer ('done right') known from Haskell libraries. From this, we obtain a number of results, including the facts that the list monad transformer is indeed a monad, a transformer, and an instance of the MonadPlus class. Finally, we construct an Eilenberg-Moore monoid of endomorphisms, which, via the codensity monad construction, yields a continuation-based implementationà la Hinze.

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Section: Free Eilenberg-moore Monoidsmentioning

confidence: 99%

Section: A5 Theorem 16mentioning

confidence: 99%

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Eilenberg–Moore Monoids and Backtracking Monad Transformers

Piróg

2016

Electron. Proc. Theor. Comput. Sci.

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Equational Theories and Monads from Polynomial Cayley Representations

Piróg

Polesiuk

Sieczkowski

2019

Lecture Notes in Computer Science

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We generalise Cayley's theorem for monoids by providing an explicit formula for a (multi-sorted) equational theory represented by the type P X → X, where P is an arbitrary polynomial endofunctor with natural coefficients. From the computational perspective, examples of effects given by such theories include backtracking nondeterminism (obtained with the original Cayley representation X → X), finite mutable state (obtained with n → X, for a constant n), and their different combinations (via n × X → X or X n → X). Moreover, we show that monads induced by such theories are implementable using the type formers available in programming languages based on a polymorphic λ-calculus, both as compositions of algebraic datatypes and as continuation-like monads. We give a set-theoretic model of the latter in terms of Barr-dinatural transformations. We also introduce CayMon, a tool that takes a polynomial as an input and generates the corresponding equational theory together with the two implementations of the induced monad in Haskell.

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Effect Systems Revisited—Control-Flow Algebra and Semantics

Mycroft

Orchard

Petříček

2015

Lecture Notes in Computer Science

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Abstract. Effect systems were originally conceived as an inference-based program analysis to capture program behaviour-as a set of (representations of) effects. Two orthogonal developments have since happened. First, motivated by static analysis, effects were generalised to values in an algebra, to better model control flow (e.g. for may/must analyses and concurrency). Second, motivated by semantic questions, the syntactic notion of set-(or semilattice-) based effect system was linked to the semantic notion of monads and more recently to graded monads which give a more precise semantic account of effects. We give a lightweight tutorial explanation of the concepts involved in these two threads and then unify them via the notion of an effect-directed semantics for a control-flow algebra of effects. For the case of effectful programming with sequencing, alternation and parallelism-illustrated with music-we identify a form of graded joinads as the appropriate structure for unifying effect analysis and semantics.

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From monoids to near-semirings

Cited by 10 publications

References 17 publications

Eilenberg–Moore Monoids and Backtracking Monad Transformers

Eilenberg–Moore Monoids and Backtracking Monad Transformers

Equational Theories and Monads from Polynomial Cayley Representations

Effect Systems Revisited—Control-Flow Algebra and Semantics

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