## Tensors of Comodels and Models for Operational Semantics

Citations: | 4 - 1 self |

### BibTeX

@MISC{Plotkin_tensorsof,

author = {Gordon Plotkin and John Power},

title = {Tensors of Comodels and Models for Operational Semantics},

year = {}

}

### OpenURL

### Abstract

In seeking a unified study of computational effects, in particular in order to give a general operational semantics agreeing with the standard one for state, one must take account of the coalgebraic structure of state. Axiomatically, one needs a countable Lawvere theory L, a comodel C, typically the final one, and a model M, typically free; one then seeks a tensor C ⊗ M of the comodel with the model that allows operations to flow between the two. We describe such a tensor implicit in the abstract category theoretic literature, explain its significance for computational effects, and calculate it in leading classes of examples, primarily involving state.

### Citations

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(Show Context)
Citation Context ..., free cocompletion, tensor. 1 Introduction Over the past decade, in collaboration with a number of other researchers, and following Eugenio Moggi’s seminal monadic approach to notions of computation =-=[10]-=-, we have been developing an algebraic theory of computational effects. This theory emphasises the operations that give rise to the effects at hand, and the equations that hold between them: see [14] ... |

379 | Basic concepts of enriched category theory, volume 64
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(Show Context)
Citation Context ...the category Cocomp(E ′ , E) is the category of colimit preserving functors from E ′ to E and all natural transformations between them. A proof of this appears in the enriched setting in Kelly’s book =-=[6]-=-, in which it plays a central rôle. The inverse equivalence sends a functor H : D −→ E to its left Kan extension LanYH, which can be described in more elementary terms as follows: ∫dɛD (5) (LanY H)(F ... |

42 |
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(Show Context)
Citation Context ...t evidently applies to the category Set. Moreover, since we know from [3] that Mod(L ⊗ LS, Set) is equivalent to Mod(LS, Mod(L, Set)) then, as Mod(L, Set) satisfies the conditions of the theorem, see =-=[1]-=-, Mod(L ⊗ LS, Set)) is canonically equivalent to Mod(L, Set). So every model of Mod(L ⊗ LS, Set) has the form A S , with LS-structure given as before and with L-structure given pointwise, and maps of ... |

31 |
Structures defined by finite limits in the enriched context 1,” Cahiers de Topologie et Géométrie Différentielle
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(Show Context)
Citation Context ...: D −→ E, the functor sending an object X of E to E(H−, X) : Dop −→ Set is the right adjoint to LanY H. There are numerous refinements of Theorem 4.1. A refinement in the direction we need appears in =-=[7]-=- and tells us the following. Theorem 4.2 Let D be a small category with countable coproducts. Then for any cocomplete locally small category E, composition with the Yoneda embedding induces an equival... |

30 | Adequacy for algebraic effects
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(Show Context)
Citation Context ...em: see [14] for an overview. One goal of this project was to give an axiomatic account of the various methods of combining effects [3,4,5]. Another, indeed the focus of the first paper of the series =-=[11]-=-, was to develop a unified theory of structural operational semantics for effects; unfortunately, however, the axiomatics of [11] had the 1 Supported by EPSRC grant GR/586372/01 and a Royal Society-Wo... |

26 | Computational effects and operations: an overview
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(Show Context)
Citation Context ... [10], we have been developing an algebraic theory of computational effects. This theory emphasises the operations that give rise to the effects at hand, and the equations that hold between them: see =-=[14]-=- for an overview. One goal of this project was to give an axiomatic account of the various methods of combining effects [3,4,5]. Another, indeed the focus of the first paper of the series [11], was to... |

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7 |
From comodels to coalgebras: State and arrays
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(Show Context)
Citation Context ...t accounting for the example of state, or for any combination of effects including state. More recently, we have begun to appreciate the importance of coalgebra in understanding the dynamics of state =-=[16]-=- and in this paper we start to rectify the situation by developing the combination of algebra and coalgebra which we believe will be needed. In modelling state, one typically has assignment and derefe... |

2 |
et al. Combining algebraic effects with continuations
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(Show Context)
Citation Context ... rise to the effects at hand, and the equations that hold between them: see [14] for an overview. One goal of this project was to give an axiomatic account of the various methods of combining effects =-=[3,4,5]-=-. Another, indeed the focus of the first paper of the series [11], was to develop a unified theory of structural operational semantics for effects; unfortunately, however, the axiomatics of [11] had t... |

2 |
to appear) Algebraic Operations and Generic Effects. Applied Categorical Structures
- Plotkin, Power
(Show Context)
Citation Context ...sli category aTS(X) : TS(X) −→ TS(X) Loc×V 2 dTS(X) : TS(X) V −→ TS(X) LocPlotkin and Power and the maps g:= and g! modelling assignment and dereferencing appear as the corresponding generic effects =-=[13]-=-. Next, as we recall in Section 3, a comodel of LS, i.e., a countable coproduct preserving functor from L op S to Set, amounts to a set Y together with cooperations aY : (Loc × V ) × Y −→ Y dY : Loc ×... |

1 | et al, Two-dimensional monad theory, JPAA 59 - Blackwell - 1989 |

1 |
et al, Combining effects: sum and tensor
- Hyland
- 2006
(Show Context)
Citation Context ...esponds to the theory: Ld + (LS ⊗ LN) 16Plotkin and Power where Ld is the theory of a unary operation d with no equations and LN is the theory of a semilattice, to model nondeterministic choice; see =-=[4]-=- for more details. However, in this case the operational semantics itself becomes more complicated involving not a single evaluation to produce a value, together with a final state, but a series of th... |

1 |
et al, Coalgebraic Semantics for Timed Processes
- Kick
- 2006
(Show Context)
Citation Context ...y of left M-sets. For an arbitrary countable Lawvere theory L, the tensor product of L with LM generates the monad TL(M × −) on Set [4]. One use of this theory is for resources, e.g., timed processes =-=[4,8]-=-; there the monoid is typically the positive reals, or the natural numbers, with addition. Another is write-only memory, where for example, one takes the theory generated by an operation a : 1 → Loc ×... |

1 |
State Monads and their Algebras
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- 407
(Show Context)
Citation Context ...ely, however, the two calculations go through for ωCpo and so we can account for recursion, see [3,4], including the lifting monad among the effects not involving comodels. We also employ a result of =-=[9]-=- which may not enrich; however we only use it to add motivation to results that do enrich: see Theorem 5.1 and Corollary 5.2 and the discussions thereafter. 2 Models of Countable Lawvere Theories In t... |

1 |
Enriched Lawvere Theories, Theo. and Apps. of Categories
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(Show Context)
Citation Context ...e to express ourselves through the course of the paper, in terms of ordinary countable Lawvere theories, and models and comodels in Set. One would wish everything to generalise to Lawvere V -theories =-=[15]-=- for those V that are locally countably presentable as cartesian closed categories, such as ωCpo. For the most part it does, and we comment as appropriate. Example 6.3 and succeeding consist of two ca... |