## Semantics for Algebraic Operations (2001)

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Venue: | Proc. MFPS 17, Electronic Notes in Thoeret. Comp. Sci |

Citations: | 10 - 3 self |

### BibTeX

@INPROCEEDINGS{Plotkin01semanticsfor,

author = {Gordon Plotkin and John Power},

title = {Semantics for Algebraic Operations},

booktitle = {Proc. MFPS 17, Electronic Notes in Thoeret. Comp. Sci},

year = {2001},

pages = {2001}

}

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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 #x : (Tx) v -# (Tx) w provides semantics for algebraic operations, which may be used to extend the usual monadic semantics of the computational #-calculus uniformly. 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. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with e#ects. 1

### Citations

733 | Notions of Computation and Monads
- Moggi
- 1991
(Show Context)
Citation Context ...ral definitions of algebraic operation. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with e#ects. 1 Introduction Eugenio Moggi, in =-=[8, 10]-=-, introduced the idea of giving a unified category theoretic semantics for computational e#ects such as nondeterminism, probabilistic nondeterminism, side-e#ects, input/output, and exceptions, by mode... |

440 | Computational lambda-calculus and monads
- Moggi
- 1989
(Show Context)
Citation Context ...ral definitions of algebraic operation. We outline examples and non-examples and we show that our definition also enriches one for call-by-name languages with e#ects. 1 Introduction Eugenio Moggi, in =-=[8, 10]-=-, introduced the idea of giving a unified category theoretic semantics for computational e#ects such as nondeterminism, probabilistic nondeterminism, side-e#ects, input/output, and exceptions, by mode... |

377 | Basic concepts of enriched category theory, volume 64
- Kelly
- 1982
(Show Context)
Citation Context ...shall build later. 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 =-=[6]-=- on enriched category theory. Implicitly using a 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 #... |

212 | A Powerdomain Construction
- Plotkin
- 1976
(Show Context)
Citation Context ... )# n i=1 # (T z) n y# Tx y# # x # f + # T z # z # commutes. For some examples of algebraic operations where C = V = Set, let T be the nonempty finite power-set monad with the binary choice operation =-=[12, 1]-=-; alternatively, let T be the monad for probabilistic nondeterminism with a probabilistic choice operation [4, 5]; or take T to be the monad for printing with printing operations [13]. Observe the non... |

165 |
An Abstract View of Programming Languages
- Moggi
- 1989
(Show Context)
Citation Context ...s for suitable V [16]: the result, in a more general setting than usual, characterises algebraic operations as generic e#ects. Moggi previously gave a semantic formulation of a notion of operation in =-=[9]-=-, with an analysis based on his computational metalanguage, but he only required naturality of the operations in C, and we know of no way to provide operational semantics in such generality. Our vario... |

136 |
Probabilistic Non-determinism
- Jones
- 1990
(Show Context)
Citation Context ... = Set, let T be the nonempty finite power-set monad with the binary choice operation [12, 1]; alternatively, let T be the monad for probabilistic nondeterminism with a probabilistic choice operation =-=[4, 5]-=-; or take T to be the monad for printing with printing operations [13]. Observe the non-commutativity in the latter example. One can, of course, generalise from Set to categories such as that of # - C... |

134 | G.: Towards a mathematical operational semantics
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ...uch as those for handling exceptions should be modelled. That might involve going beyond monads, as Moggi has suggested to us: one possibility involves the use of distributive laws along the lines of =-=[18]-=-; another involves dyads [17]; possibly, some combination may be appropriate. The paper is organised as follows. In Section 2, we recall and enrich the definition of algebraic operation given in [14] ... |

125 |
A probabilistic powerdomain of evaluations
- Jones, Plotkin
- 1989
(Show Context)
Citation Context ... = Set, let T be the nonempty finite power-set monad with the binary choice operation [12, 1]; alternatively, let T be the monad for probabilistic nondeterminism with a probabilistic choice operation =-=[4, 5]-=-; or take T to be the monad for printing with printing operations [13]. Observe the non-commutativity in the latter example. One can, of course, generalise from Set to categories such as that of # - C... |

34 |
Kan Extensions in Enriched Category Theory
- Dubuc
- 1970
(Show Context)
Citation Context ... as #-Cpo, one can also consider more exotic arities such as that given by Sierpinski space. For an enriched version of Lawvere's idea without the finitariness but with the restriction to C = V , see =-=[3]-=-. Once again, by use of parametrisation, we can avoid the closedness assumption on C here, yielding the stronger statement Theorem 3. Functoriality of - # - : C op T C T -# C in its first variable ind... |

30 | Adequacy for algebraic effects - Plotkin, Power - 2001 |

17 | Enriched lawvere theories,” Theory and Applications of Categories - Power - 1999 |

15 |
A representable approach to finite nondeterminism
- Anderson, Power
- 1997
(Show Context)
Citation Context ...sli V -category C T . We could equally formulate this in terms of an # This work has been done with the support of EPSRC grant GR/M56333. enriched version of closed F reyd-categories in the spirit of =-=[1]-=-. A preliminary version appears as [15]. The leading class of examples has T being generated by the operations subject to accompanying equations. Examples of such operations are nondeterministic choic... |

15 | Call-by-push-value: A subsuming paradigm
- Levy
- 1999
(Show Context)
Citation Context ...me languages with e#ects, so this formulation gives us an indication of how to generalise our analysis to call-by-name computation or perhaps to some combination of call-by-value and call-by-name, cf =-=[7]-=-. If C is V -closed and has equalisers, generalising Lawvere, the results of the previous section can equally be formulated as equivalences between algebraic operations and operations # (A,a) : U(A, a... |

15 | Enriched Lawvere Theories
- Power
- 2000
(Show Context)
Citation Context ...ts, which we believe provide strong evidence for the above choice of axioms. Our most interesting result is essentially about the relationship between V -monads and Lawvere V -theories for suitable V =-=[16]-=-: the result, in a more general setting than usual, characterises algebraic operations as generic e#ects. Moggi previously gave a semantic formulation of a notion of operation in [9], with an analysis... |

9 |
Modularity and Dyads
- Power, Robinson
- 1999
(Show Context)
Citation Context ...eptions should be modelled. That might involve going beyond monads, as Moggi has suggested to us: one possibility involves the use of distributive laws along the lines of [18]; another involves dyads =-=[17]-=-; possibly, some combination may be appropriate. The paper is organised as follows. In Section 2, we recall and enrich the definition of algebraic operation given in [14] and we exhibit some simple re... |

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

4 |
Adequacy for Algebraic E#ects
- Plotkin, Power
- 2001
(Show Context)
Citation Context ...ns for side-e#ects, read and write operations for interactive input/output, and operations for raising exceptions. A non-example is given by an operation for handling exceptions. In a companion paper =-=[14]-=-, we gave an unenriched version of the above definition together with a syntactic counterpart in terms of the computational #- calculus, and proved adequacy results. But such results leave some scope ... |