## Adequacy for algebraic effects (2001)

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Venue: | In 4th FoSSaCS |

Citations: | 33 - 17 self |

### BibTeX

@INPROCEEDINGS{Plotkin01adequacyfor,

author = {Gordon Plotkin and Matija Pretnar},

title = {Adequacy for algebraic effects},

booktitle = {In 4th FoSSaCS},

year = {2001},

pages = {1--24}

}

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### Abstract

We present a logic for algebraic effects, based on the algebraic representation of computational effects by operations and equations. We begin with the a-calculus, a minimal calculus which separates values, effects, and computations and thereby canonises the order of evaluation. This is extended to obtain the logic, which is a classical firstorder multi-sorted logic with higher-order value and computation types, as in Levy’s call-by-push-value, a principle of induction over computations, a free algebra principle, and predicate fixed points. This logic embraces Moggi’s computational λ-calculus, and also, via definable modalities, Hennessy-Milner logic, and evaluation logic, though Hoare logic presents difficulties. 1

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Probabilistic nondeterminism
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142 | Towards a Mathematical Operational Semantics - Turi, Plotkin - 1997 |

129 |
A probabilistic powerdomain of evaluations
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- 1989
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Citation Context ...r (Smyth) and lower (Hoare) powerdomains. Probabilistic Nondeterminism We take T to be V, where V(P ) is the dcpo of all evaluations on P . This functor and its strong monad structure is discussed in =-=[14]-=-, and see, e.g., [15]; we just note that ηP (x) is the singleton evaluation. The semantics of probabilistic choice + is again an affine combination, this time of evaluations ν +X ν ′ = 1/2ν + 1/2ν ′ V... |

94 | Full abstraction for a simple parallel programming language
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- 1979
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Citation Context ...tions; it is in thatssense that they are purely formal. One might instead have introduced equations corresponding to the intended effects, e.g., semilattice equations for finite nondeterminism, as in =-=[12]-=- and then worked with operational semantics up to provable equality. The small step semantics proceeds by means of two kinds of transitions between closed terms, an unlabelled one and a labelled one M... |

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Citation Context ...ific effects, such as Hennessy-Milner logic [7] for concurrency, and more abstract approaches originating from the representation of computational effects with monads, such as Pitts’ evaluation logic =-=[17, 15, 16]-=-. (We define an embrace to be a translation, which preserves provable judgements. If the translation also reflects provable judgements, we call it a strong embrace.) Section 2 introduces the a-calculu... |

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- 2002
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Citation Context ...n two elements of any given type. The equations for the operations, for example saying that or is a semi-lattice operation, generate a free algebra monad, which is exactly the monad proposed by Moggi =-=[20]-=- to model the corresponding effect. This article proposes a logic for algebraic effects [22], and aims to show that it provides a rich framework, which embraces both approaches that have developed aro... |

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Citation Context ...ment already failed [23] to give a natural operational semantics for state. Such a treatment could fit well with Pitts’ ‘ad hoc’ approach to state [17]. 5 Recursion We sketch a version of Scott’s LCF =-=[30, 6]-=-, adapted to algebraic computational effects, but make no claim of definitiveness. The logic is an extension of our logic for algebraic effects over Set, based instead on the category ω-Cpo of ω-cpos ... |

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Citation Context ...Hoare) powerdomains.sProbabilistic Nondeterminism We take T to be V, where V(P ) is the dcpo of all evaluations on P ; this functor and its strong monad structure is discussed in [14], and see, e.g., =-=[15]; we just-=- note that ηP (x) is the singleton evaluation. The semantics of probabilistic choice + is again a linear combination, this time of evaluations ν +X ν ′ = 1/2ν + 1/2ν ′ V(P ) is a dcppo which ... |

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Citation Context ...types (actually at all types), it follows that no |M | (n) is total and hence that M �↓n. Essentially this result is in [29] and a stronger one for both call-by-name and call-by-value is in [10]; =-=see [3, 11, 9]-=- for work on call-by-name and nondeterminism. One can presumably also prove an adequacy theorem at other types, but the statement is more complex due to need for closure operators in the set-theoretic... |

36 |
Algebraic Semantics
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- 1981
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Citation Context ...tial as both ordinary and probabilistic powerdomains are considered.sOur treatment of operational semantics might well be seen as rather formal (cf. the formal results for recursion schemes in, e.g., =-=[8]-=-) and does not immediately specialise to the usual accounts for the examples at hand. In a way this has to be so: it is hard to imagine a theory which yields the natural operational semantics for any ... |

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Citation Context ... the idea that computations for an element of (say) a set A are modelled by elements of T A, where T is the monad. Plotkin and Power then proposed representing the effects by operations and equations =-=[19, 21, 23]-=- to get a uniform theory of effects that accounted for their source: we call such effects algebraic. All of the effects mentioned above are algebraic, with the notable exception of continuations [3], ... |

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Citation Context ...theorem for FPC with probabilistic choice was already proved in [13] (FPC can be viewed as an extension of our PCF with recursive types); for work on call-by-name and probabilistic nondeterminism see =-=[6]-=-. 7 Conclusions It is interesting to work out other examples. Printing provides one example, where one has a unary operation print a for each symbol a of an alphabet A, and for Set one can take the (n... |

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Citation Context ... the idea that computations for an element of (say) a set A are modelled by elements of T A, where T is the monad. Plotkin and Power then proposed representing the effects by operations and equations =-=[19, 21, 23]-=- to get a uniform theory of effects that accounted for their source: we call such effects algebraic. All of the effects mentioned above are algebraic, with the notable exception of continuations [3], ... |

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Citation Context ...istributive law λ : Dω(F + (X)) → F + (Dω(X))which makes this a monad; this way to combine the two forms of nondeterminism is used in a domain-theoretic context in [4]—modulo actions—and ment=-=ioned in [19]-=- where an interesting idea of restricting to affine sets of evaluations is advocated. In so far as we are successful with such examples, the question of how to treat other monads and their operations ... |

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Citation Context ...s τ are interpreted by models [τ ] of the theory L. The value types are interpreted by [Uτ ] = U [τ ] and in the obvious way in other cases, while computation types are interpreted by [F σ ] = F [σ ] =-=[1]-=- = 1 [τ 1 × τ 2 ] = [τ 1 ] × [τ 2 ] [σ → τ ] = [τ ] [σ ] , where the model structure is defined component-wise for M1 × M2 and point-wise for M A . The context x1 : σ1, . . . , xn : σn is interpreted ... |

19 | Call-by-push-value: A subsuming paradigm
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Citation Context ...ge variations, such as an extension with recursive types or call-by-name; for the latter it would be preferable to use a framework incorporating both parameter-calling mechanisms, such as Lévy’s CB=-=PV [18]-=-. More intriguingly, one would wish to reconcile this work with the co-algebraic treatment of operational semantics in [31] with its use of behaviour functors and co-monads contrasting with our use of... |

19 | A semantics for evaluation logic
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- 1995
(Show Context)
Citation Context ...ific effects, such as Hennessy-Milner logic [7] for concurrency, and more abstract approaches originating from the representation of computational effects with monads, such as Pitts’ evaluation logic =-=[17, 15, 16]-=-. (We define an embrace to be a translation, which preserves provable judgements. If the translation also reflects provable judgements, we call it a strong embrace.) Section 2 introduces the a-calculu... |

18 |
A mathematical semantics for a nondeterministic typed lambda-calculus
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- 1980
(Show Context)
Citation Context ...types (actually at all types), it follows that no |M | (n) is total and hence that M �↓n. Essentially this result is in [29] and a stronger one for both call-by-name and call-by-value is in [10]; =-=see [3, 11, 9]-=- for work on call-by-name and nondeterminism. One can presumably also prove an adequacy theorem at other types, but the statement is more complex due to need for closure operators in the set-theoretic... |

16 |
The Semantics of Call-by-value and Call-by-name in a Non-Deterministic Environment
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- 1980
(Show Context)
Citation Context ...at ground types (actually at all types), it follows that no |M | (n) is total and hence that M �↓n. Essentially this result is in [29] and a stronger one for both call-by-name and call-by-value is=-= in [10]-=-; see [3, 11, 9] for work on call-by-name and nondeterminism. One can presumably also prove an adequacy theorem at other types, but the statement is more complex due to need for closure operators in t... |

15 |
A representable approach to finite nondeterminism
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- 1997
(Show Context)
Citation Context ...ms of a suitable class of categorical structures appropriately extending the usual monadic view of the computational λ-calculus; this could as well have been based on closed Freyd categories [28] and=-= [2]-=- is a treatment of nondeterminism along such lines. Section 4 considers two examples: nondeterminism and probabilistic nondeterminism. We consider the full language with recursion in Section 5. Small ... |

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(Show Context)
Citation Context ...braic effects, but much clearly remains to be done. To mention one example, we expect to get an embrace of global evaluation logic [5], while we have not yet investigated the embrace of dynamic logic =-=[29]-=-. The question of how to account for computation deconstructors, such as exception handlers [1, 13, 21] also remains open, hence so does the question of what their logic may be. Beyond Set and ω-Cpo, ... |

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Citation Context ...all base types occurring in arities by countable sets. Then the effect theory gives rise to a countable discrete Lawvere ω-cpo theory L and an adjunction F ⊣ U : ModL(ω-Cpo) → ω-Cpo in a standard way =-=[11]-=-. 6 Future work We have presented some evidence of the expressiveness and strength of our logic of algebraic effects, but much clearly remains to be done. To mention one example, we expect to get an e... |

9 | Modularity and Dyads - Power, Robinson - 1999 |

9 | Computational adequacy in an elementary topos
- Simpson
- 1999
(Show Context)
Citation Context ...l general adequacy theorem can be proved. So we consider a call-by-value PCF with algebraic effects, an extension of the computational λ-calculus with operations, arithmetic and recursion (see, e.g.,=-= [32, 30]-=- for versions of call-byvalue PCF). We begin by treating the sublanguage without recursion. Section 2 presents both small step and big step operational semantics and also an associated evaluation func... |

9 | Combining Algebraic Effects with Continuations, Theoretical Computer Science 375
- Hyland, Levy, et al.
- 2007
(Show Context)
Citation Context ...hat accounted for their source: we call such effects algebraic. All of the effects mentioned above are algebraic, with the notable exception of continuations [3], which have to be treated differently =-=[9]-=-. ∗ Supported by EPSRC grant GR/586371/01 and a Royal SocietyWolfson Award Fellowship. † Supported by EPSRC grant GR/586371/01. In the algebraic approach, the arguments of an operation represent possi... |

8 |
A Syntactic Approach to Modularity
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- 1993
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Citation Context ...ther monads and their operations is the more pressing; exceptions, state and continuations all come immediately to mind. Possibly relevant here is the translational approach to defining operations in =-=[5],-=- but adapted to λc rather than the metalanguage; the idea would be to recover operational semantics via the translations. Ultimately, we would hope to incorporate the treatment ofsoperational semanti... |

8 |
Call-by-value and non-determinism
- Sieber
- 1993
(Show Context)
Citation Context ... x =⊥. Then ⊥ is in every [[|M | (n) ]](∗). Since no [[V ]](∗) is {⊥} at ground types (actually at all types), it follows that no |M | (n) is total and hence that M �↓n. Essentially this=-= result is in [29]-=- and a stronger one for both call-by-name and call-by-value is in [10]; see [3, 11, 9] for work on call-by-name and nondeterminism. One can presumably also prove an adequacy theorem at other types, bu... |

8 | Call-by-push-value: Decomposing call-by-value and call-byname
- Levy
(Show Context)
Citation Context ...s, thereby canonising the order of evaluation. In Section 3 it is extended to a classical first-order multi-sorted logic with higher-order value and computation types, as in Levy’s call-by-push-value =-=[12]-=-, a principle of induction over computations, a free algebra principle, and predicate fixed points. Next, in Section 4, we show that Moggi’s computational λ-calculus, and, via definable modalities, He... |

8 |
From Comodels to Coalgebras: State and Arrays
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- 2004
(Show Context)
Citation Context ...let xn be !ln in let z be t in let y1 be !l1 in . . . let yn be !ln in return〈⃗x, ⃗y〉](ϕ(⃗x) ⇒ ϕ ′ (⃗x, ⃗y)) . However, this does not seem natural to us. The answer may lie in a coalgebraic treatment =-=[27, 24]-=- of state, as an algebraic treatment already failed [23] to give a natural operational semantics for state. Such a treatment could fit well with Pitts’ ‘ad hoc’ approach to state [17]. 5 Recursion We ... |

7 | Monads and adjunctions for global exceptions
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- 2006
(Show Context)
Citation Context ...embrace of global evaluation logic [5], while we have not yet investigated the embrace of dynamic logic [29]. The question of how to account for computation deconstructors, such as exception handlers =-=[1, 13, 21]-=- also remains open, hence so does the question of what their logic may be. Beyond Set and ω-Cpo, and without yet looking for a logic over a general category, one could still ask for logics over catego... |

5 |
Some varieties of equational logic
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- 2006
(Show Context)
Citation Context ...were to describe a memory holding an infinite set of data by routinely generalising the operations to infinitary ones, we would be left with an infinitary syntax [19]. We take an alternative approach =-=[18]-=-, and allow each argument of an operation to be dependent on values of base types, for example lookupl((d : dat).updatel ′,d(ξ)) is an effect term for a computation that copies the datum d from l to l... |

5 | Tensors of Comodels and Models for Operational Semantics
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- 2008
(Show Context)
Citation Context ...let xn be !ln in let z be t in let y1 be !l1 in . . . let yn be !ln in return〈⃗x, ⃗y〉](ϕ(⃗x) ⇒ ϕ ′ (⃗x, ⃗y)) . However, this does not seem natural to us. The answer may lie in a coalgebraic treatment =-=[27, 24]-=- of state, as an algebraic treatment already failed [23] to give a natural operational semantics for state. Such a treatment could fit well with Pitts’ ‘ad hoc’ approach to state [17]. 5 Recursion We ... |

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3 |
Algebraic automata and context free sets
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- 1967
(Show Context)
Citation Context ...erminate. Theorem 2. Adequacy Suppose that M : σ. Then [[M ] = [[|M |]] Proof. This is an immediate consequence of Proposition 2 and Lemma 2 This result is very much in the spirit of Mezei and Wright =-=[19]-=-, and Theorem 4.26 of [8] is a similar result for recursive program schemes. Such results say that the denotational semantics of a program is that of the result of a preliminary symbolic computation. ... |

3 | Logic for Computational Effects: work in progress
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- 2003
(Show Context)
Citation Context ...or is a semi-lattice operation, generate a free algebra monad, which is exactly the monad proposed by Moggi [20] to model the corresponding effect. This article proposes a logic for algebraic effects =-=[22]-=-, and aims to show that it provides a rich framework, which embraces both approaches that have developed around specific effects, such as Hennessy-Milner logic [7] for concurrency, and more abstract a... |

2 |
Nondeterminism and Fully Abstract Models, in Informatique Théorique et
- Astesiano, Costa
- 1980
(Show Context)
Citation Context ...types (actually at all types), it follows that no |M | (n) is total and hence that M �↓n. Essentially this result is in [29] and a stronger one for both call-by-name and call-by-value is in [10]; =-=see [3, 11, 9]-=- for work on call-by-name and nondeterminism. One can presumably also prove an adequacy theorem at other types, but the statement is more complex due to need for closure operators in the set-theoretic... |