## Some Varieties of Equational Logic (Extended Abstract), Algebra (2006)

Venue: | Meaning, and Computation, Essays Dedicated to Joseph A. Goguen on the Occasion of His 65th Birthday |

Citations: | 2 - 1 self |

### BibTeX

@INPROCEEDINGS{Plotkin06somevarieties,

author = {Gordon Plotkin},

title = {Some Varieties of Equational Logic (Extended Abstract), Algebra},

booktitle = {Meaning, and Computation, Essays Dedicated to Joseph A. Goguen on the Occasion of His 65th Birthday},

year = {2006},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

been a major theme of Joseph Goguen’s research, perhaps even the major theme. One strand of this work concerns algebraic datatypes. Recently there has been some interest in what one may call algebraic computation types. As we will show, these are also given by equational theories, if one only understands the notion of equational logic in somewhat broader senses than usual. One moral of our work is that, suitably considered, equational logic is not tied to the usual first-order syntax of terms and equations. Standard equational logic has proved a useful tool in several branches of computer science, see, for example, the RTA conference series [9] and textbooks, such as [1]. Perhaps the possibilities for richer varieties of equational logic discussed here will lead to further applications. We begin with an explanation of computation types. Starting around 1989, Eugenio Moggi introduced the idea of monadic notions of computation [11, 12]

### Citations

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Citation Context ...order syntax of terms and equations. Standard equational logic has proved a useful tool in several branches of computer science, see, for example, the RTA conference series [9] and textbooks, such as =-=[1]-=-. Perhaps the possibilities for richer varieties of equational logic discussed here will lead to further applications. We begin with an explanation of computation types. Starting around 1989, Eugenio ... |

766 | Notions of computation and monads
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Citation Context ...al logic discussed here will lead to further applications. We begin with an explanation of computation types. Starting around 1989, Eugenio Moggi introduced the idea of monadic notions of computation =-=[11, 12]-=- with the idea that, for appropriately chosen monads T on, e.g., Set, the category of sets, one thinks of T (X) as the type of computations of an element of X. For example, for side-effects one takes ... |

457 | Computational Lambda-calculus and monads
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Citation Context ...al logic discussed here will lead to further applications. We begin with an explanation of computation types. Starting around 1989, Eugenio Moggi introduced the idea of monadic notions of computation =-=[11, 12]-=- with the idea that, for appropriately chosen monads T on, e.g., Set, the category of sets, one thinks of T (X) as the type of computations of an element of X. For example, for side-effects one takes ... |

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Citation Context ...ion of) the convex powerdo⋆ This work has been done with the support of EPSRC grant GR/S86372/01 and a Royal Society-Wolfson research merit award.smain monad, originally defined only on a subcategory =-=[5]-=-. There are also lower, or Hoare, and upper, or Smyth, powerdomain monads; these are obtained by adding an additional axiom, viz: for the lower powerdomain, and: x ≤ x ∪ y x ≥ x ∪ y for the upper one.... |

34 | Algebraic operations and generic effects
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Citation Context ...otions of computation, where the operations and equations are primary and determine the monads. The computational importance of the operations is that it is they that give rise to the effects at hand =-=[16]-=-. Applications include the operational semantics of effects [14], their modular combination [7] and, prospectively, a general logic of effects [17]; see [18] for a survey. The examples demonstrate tha... |

33 | Adequacy for Algebraic Effects
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Citation Context ...imary and determine the monads. The computational importance of the operations is that it is they that give rise to the effects at hand [16]. Applications include the operational semantics of effects =-=[14]-=-, their modular combination [7] and, prospectively, a general logic of effects [17]; see [18] for a survey. The examples demonstrate that the algebraic theory of computation would benefit from a wider... |

31 |
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Citation Context ... equations and there is an evident notion of inequational, or ordered, equational logic over operations of finite arity, which has a straightforward completeness theorem using posets rather than sets =-=[3]-=-. The resulting ordered equational theories correspond to ordered Lawvere theories, in the sense of [23, 3]. These are not the same as the Pos-enriched Lawvere theories of [19], as the latter allow al... |

31 | Computational Effects and Operations: an Overview
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Citation Context ...they that give rise to the effects at hand [16]. Applications include the operational semantics of effects [14], their modular combination [7] and, prospectively, a general logic of effects [17]; see =-=[18]-=- for a survey. The examples demonstrate that the algebraic theory of computation would benefit from a wider means of expression than is provided by standard equational theories: one also needs to cons... |

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Citation Context ...erms. Perhaps one can restrict to considering only least fixed-points and work with a combination of the above ideas and the µ-calculus, for which, and associated logical and categorical results, see =-=[4, 8, 21]-=-. Whatever the difficulties are with finding the right logic, it is at least the case that the combination of parameterization, binding constructions and inequations, interpreted over ω-Cpo, is enough... |

25 |
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Citation Context ...erms. Perhaps one can restrict to considering only least fixed-points and work with a combination of the above ideas and the µ-calculus, for which, and associated logical and categorical results, see =-=[4, 8, 21]-=-. Whatever the difficulties are with finding the right logic, it is at least the case that the combination of parameterization, binding constructions and inequations, interpreted over ω-Cpo, is enough... |

16 | Enriched Lawvere theories
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Citation Context ... finite posets as arities of operations, not just the discrete ones. However they are the same as the Pos-enriched Lawvere theories of [10], equivalently the Pos-enriched discrete Lawvere theories of =-=[20]-=-. There is a natural generalisation to countable inequational logic, and the inequational theories of this logic correspond to the discrete countable Pos-theories (the countable case is the main one c... |

13 | The Logic of Recursive Equations
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Citation Context ...erms. Perhaps one can restrict to considering only least fixed-points and work with a combination of the above ideas and the µ-calculus, for which, and associated logical and categorical results, see =-=[4, 8, 21]-=-. Whatever the difficulties are with finding the right logic, it is at least the case that the combination of parameterization, binding constructions and inequations, interpreted over ω-Cpo, is enough... |

6 |
Enriched Lawvere theories. Theory and
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Citation Context ... monad TN(X) is the collection of all non-empty finite subsets of X. In general, the equational theories with operations of finite arity induce exactly those monads which have finite rank, see, e.g., =-=[19]-=-. In denotational semantics one typically employs a category of ordered structures, such as ω-Cpo, the category of ω-cpos, which are partial orders with lubs of increasing ω-chains, and with morphisms... |

5 | A domain-theoretic Banach-Alaoglu theorem
- Plotkin
(Show Context)
Citation Context ...nism [22]. These are the ω-Cpo-semimodules over the semiring R +, which latter is the ω-cpo of the nonnegative reals extended with a point at infinity, and endowed with the natural semiring structure =-=[13]-=-. Collections of such inequations induce the discrete finitary or countable ω-Cpo-theories, according to the arities of the operation symbols allowed. However there is a question as to what is the app... |

3 |
Semantic Domains for Combining Probability
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(Show Context)
Citation Context ...tinuous over the parameter ω-cpos as well as the algebra ω-cpo. A natural example is provided by d-cones, which arise when considering powerdomains for mixed ordinary and probabilistic nondeterminism =-=[22]-=-. These are the ω-Cpo-semimodules over the semiring R +, which latter is the ω-cpo of the nonnegative reals extended with a point at infinity, and endowed with the natural semiring structure [13]. Col... |

2 |
Logic for computational effects, in
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(Show Context)
Citation Context ...hat it is they that give rise to the effects at hand [16]. Applications include the operational semantics of effects [14], their modular combination [7] and, prospectively, a general logic of effects =-=[17]-=-; see [18] for a survey. The examples demonstrate that the algebraic theory of computation would benefit from a wider means of expression than is provided by standard equational theories: one also nee... |

1 |
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(Show Context)
Citation Context ... can argue that there the types should not be treated as algebraic since the natural operations are not even of the right type to be algebraic operations, and, further, the monad does not have a rank =-=[6]-=-. A more interesting case is that of local state, as opposed to the above global state, where one can declare new locations. This was treated using a monad over a presheaf category in [15]. The monad ... |

1 |
Twenty Years
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(Show Context)
Citation Context ...ot tied to the usual first-order syntax of terms and equations. Standard equational logic has proved a useful tool in several branches of computer science, see, for example, the RTA conference series =-=[9]-=- and textbooks, such as [1]. Perhaps the possibilities for richer varieties of equational logic discussed here will lead to further applications. We begin with an explanation of computation types. Sta... |

1 |
Varieties of Chain-Complete Algebras
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(Show Context)
Citation Context ...he Pos-enriched Lawvere theories of [19], as the latter allow all finite posets as arities of operations, not just the discrete ones. However they are the same as the Pos-enriched Lawvere theories of =-=[10]-=-, equivalently the Pos-enriched discrete Lawvere theories of [20]. There is a natural generalisation to countable inequational logic, and the inequational theories of this logic correspond to the disc... |