## Algebraic Operations and Generic Effects (2003)

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Venue: | Applied Categorical Structures |

Citations: | 33 - 7 self |

### BibTeX

@ARTICLE{Plotkin03algebraicoperations,

author = {Gordon Plotkin and John Power},

title = {Algebraic Operations and Generic Effects},

journal = {Applied Categorical Structures},

year = {2003},

volume = {11},

pages = {2003}

}

### Years of Citing Articles

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### Abstract

Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V-category C with cotensors and a strong V-monad T on C, we investigate axioms under which an ObC-indexed family of operations of the form α_x : (Tx)^ν → (Tx)^ω provides semantics for algebraic operations on the computational λ-calculus. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. In particular, we define the notion of generic effect and show that to give a generic effect is equivalent to giving an algebraic operation. We further show how the usual monadic semantics of the computational λ-calculus extends uniformly to incorporate generic effects. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with e#ects.

### Citations

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(Show Context)
Citation Context ...e V-natural transformations with components ax,y,z : (x⊗y)⊗z −→ x ⊗ (y ⊗ z), lx : I ⊗ x −→ x, rx : x ⊗ I −→ x, and cx,y : x ⊗ y −→ y ⊗ x, subject to the usual axioms for a symmetric monoidal category =-=[10,12]-=-. A monoidal V-category C is V-closed if for every object x of C, the V-functor − ⊗ x : C −→ C has a right V-adjoint. Note that if C is a monoidal V-category, the underlying ordinary category C0 of C ... |

732 | Notions of computation and monads
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- 1991
(Show Context)
Citation Context ...formly to incorporate generic effects. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with effects. 1 Introduction Eugenio Moggi, in =-=[13,15]-=-, introduced the idea of giving a unified category theoretic semantics for computational effects, modelling each of them in the Kleisli category of an appropriate strong monad T on a base category C w... |

607 |
The Definition of Standard ML (Revised
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- 1997
(Show Context)
Citation Context ...stematic distinction between computations and values. It represents a fragment of a call-by-value programming language; in particular, it was designed to model fragments of Milner et al’s language ML =-=[17]-=-. We describe it in Appendix A. However, the calculus does not contain operations, the constructs that actually create the effects. For example, for binary nondeterminism, one has a binary ⋆ This work... |

441 | Computational lambda-calculus and monads
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- 1989
(Show Context)
Citation Context ...formly to incorporate generic effects. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with effects. 1 Introduction Eugenio Moggi, in =-=[13,15]-=-, introduced the idea of giving a unified category theoretic semantics for computational effects, modelling each of them in the Kleisli category of an appropriate strong monad T on a base category C w... |

379 | Basic concepts of enriched category theory, volume 64
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- 1982
(Show Context)
Citation Context ...ads, and cotensors We assume that V is a complete and cocomplete symmetric monoidal closed category: those are the conditions on V required for the preponderance of results of Kelly’s definitive book =-=[10]-=- on enriched category theory. Implicitly usingsa larger universe, the category V-CAT of locally small V-categories has a symmetric monoidal structure, with A ∗ B having object set ObA × ObB, with: (A ... |

214 | A powerdomain construction
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(Show Context)
Citation Context ...take instead V = Set and C = ωCpo or Dcpo (dcpos are partial orders with lubs of all directed sets). Example 1. Nondeterminism The nonempty finite power-set monad T supports a binary choice operation =-=[18,1]-=-: ∨x : (Tx) 2 −→ Tx where ∨x(u, v) = u ∪ v. It is routine to verify the coherence condition. This generalises from Set to ωCpo, where the power-domain can be characterised as giving the free semilatti... |

165 |
An abstract view of programming languages
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- 1990
(Show Context)
Citation Context ...er. One should note that in such cases as interactive input/output and state, these operations may be infinitary (see below). Moggi’s computational metalanguage does contain operations, and his paper =-=[14]-=- includes semantics for them, but he only demanded naturality of the operations in C, and he did not develop a body of theory in support of that semantics. Here, by demanding the stronger coherence co... |

136 |
Probabilistic Non-determinism
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(Show Context)
Citation Context ...d just as for nondeterminism, the semantic operation +r models a corresponding probabilistic choice operation symbol +r. This also generalises from Set, here to the probabilistic power-domain on Dcpo =-=[8,9, 6]-=-. Example 3. Exceptions The monad Tx = x+E for exceptions supports E evident nullary operations, one for each e in E; these model the nullary operation symbols raisee for raising exceptions. For each ... |

125 |
A probabilistic powerdomain of evaluations
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(Show Context)
Citation Context ...d just as for nondeterminism, the semantic operation +r models a corresponding probabilistic choice operation symbol +r. This also generalises from Set, here to the probabilistic power-domain on Dcpo =-=[8,9, 6]-=-. Example 3. Exceptions The monad Tx = x+E for exceptions supports E evident nullary operations, one for each e in E; these model the nullary operation symbols raisee for raising exceptions. For each ... |

97 |
On closed categories of functors
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(Show Context)
Citation Context ...atural operation: blockx : [L, Tx] → Tx for declaring, initialising and then using a new location; here we are using a second closed structure on C, namely Day’s convolution closed monoidal structure =-=[3]-=-. Rather than give explicit definitions of these operations here, we take them as defined in terms of the corresponding generic effects, considered below. For partiality, we would instead use a catego... |

34 |
Kan Extensions in Enriched Category Theory
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- 1970
(Show Context)
Citation Context ...are not sets at all, but are presheaves (see Example 11), hence our generalisation from Set. For an enriched version of Lawvere’s idea without the finitariness but with the restriction to C = V , see =-=[4]-=-. The result as we have formulated it above is an elegant instance of the enriched Yoneda lemma [10]. But once again, using parametrisation but losing a little elegance, we can generalise it to avoid ... |

30 | Adequacy for algebraic effects
- Plotkin, Power
- 2001
(Show Context)
Citation Context ...on of algebraic operations, which we support by equivalence theorems to indicate definitiveness of the axioms, and which are further supported by our development of a unified operational semantics in =-=[20]-=-. In all cases we can go further, taking the monad T to be generated by the operations subject to accompanying equations; this idea is explored in [22,7]. Of the various operations, handle is of a dif... |

22 | Probabilistic domains
- Heckmann
- 1994
(Show Context)
Citation Context ...d just as for nondeterminism, the semantic operation +r models a corresponding probabilistic choice operation symbol +r. This also generalises from Set, here to the probabilistic power-domain on Dcpo =-=[8,9, 6]-=-. Example 3. Exceptions The monad Tx = x+E for exceptions supports E evident nullary operations, one for each e in E; these model the nullary operation symbols raisee for raising exceptions. For each ... |

19 | Combining computational effects: commutativity and sum
- Hyland, Plotkin, et al.
- 2002
(Show Context)
Citation Context ...evelopment of a unified operational semantics in [20]. In all cases we can go further, taking the monad T to be generated by the operations subject to accompanying equations; this idea is explored in =-=[22,7]-=-. Of the various operations, handle is of a different computational character and, although natural, it is not algebraic. Andrzej Filinski (personal communication) describes handle as a deconstructor,... |

15 |
A representable approach to finite nondeterminism
- Anderson, Power
- 1997
(Show Context)
Citation Context ...e Kleisli V-category CT. Enrichment allows us to employ complex arities, i.e., objects of V , as in the case of local state—see below. (Enrichment by, e.g., V = ωCpo allows us to handle recursion, cf =-=[1]-=-, but that is a rather different matter, not involving complex arities; here ωCpo is the category of small ω-cpos, i.e, the category of posets with sups of ω-chains.) Parametrisation allows us to mode... |

15 | Call-by-push-value: A subsuming paradigm
- Levy
- 1999
(Show Context)
Citation Context ...notion of algebraic operation in terms of the V-category of algebras T-Alg. The co-Kleisli category of the comonad on T-Alg induced by the monad T is used to model call-by-name languages with effects =-=[11]-=-, so this formulation gives us an indication of how to generalise our analysis to call-by-name computation or perhaps to some combination of callby-value and call-by-name, cf [11]. If C is V-closed an... |

15 | Enriched Lawvere Theories
- Power
- 2000
(Show Context)
Citation Context ...l λ-calculus, and we proved adequacy results: these required naturality in CT. Our most interesting result is essentially about the relationship between V-monads and Lawvere V-theories for suitable V =-=[23,7]-=-: the result, in a more general setting than usual, characterises algebraic operations via generic effects. The general idea of generic effect seems to be new, although particular examples are known a... |

7 | Semantics for Algebraic Operations (extended abstract - Plotkin, Power - 2001 |

5 | On the call-by-value CPS transform and its semantics
- Führmann, Thielecke
- 2004
(Show Context)
Citation Context ...xtend directly. Our impression of continuations is that they are more naturally seen as a logical construct, following the CurryHoward correspondence, than one amenable to our algebraic treatment, cf =-=[5]-=-. They should therefore be treated separately, but the question of their integration with the algebraic case remains. We should also like to extend and integrate this work with work addressing other a... |

2 | Models for the Computational Lambda Calculus - Power - 2001 |