## An operational domain-theoretic treatment of recursive types (2006)

Venue: | in: Twenty-Second Mathematical Foundations of Programming Semantics |

Citations: | 2 - 2 self |

### BibTeX

@INPROCEEDINGS{Ho06anoperational,

author = {Weng Kin Ho},

title = {An operational domain-theoretic treatment of recursive types},

booktitle = {in: Twenty-Second Mathematical Foundations of Programming Semantics},

year = {2006}

}

### OpenURL

### Abstract

We develop a domain theory for treating recursive types with respect to contextual equivalence. The principal approach taken here deviates from classical domain theory in that we do not produce the recursive types via the usual inverse limits constructions- we have it for free by working directly with the operational semantics. By extending type expressions to endofunctors on a ‘syntactic ’ category, we establish algebraic compactness. To do this, we rely on an operational version of the minimal invariance property. In addition, we apply techniques developed herein to reason about FPC programs. Key words: Operational domain theory, recursive types, FPC, realisable functor, algebraic compactness, generic approximation lemma, denotational semantics 1

### Citations

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Citation Context ... category C § is denoted by ˘ C and its diagonal subcategory A is denoted by C D . 7 This is a generalisation of R. Bird’s approximation lemma [3], which in turn generalises the well-known take lemma =-=[4]-=-. 13 ✷sHo Here we need to introduce an auxiliary type - the vertical natural numbers ω, defined by ω = µX.X⊥. For each x : ω, its successor is defined by succ(x) = fold ◦ up(x). We define 0 := ⊥, n :=... |

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Citation Context ... we assume certain pre-requisites from the reader which include the operational semantics of the call-by-name version of FPC ([17]), theory of recursive domain equations ([2,26]) and category theory (=-=[18]-=-). 2 Preliminaries We choose to work with a call-by-name version of FPC 3 (Fixed Point Calculus). The most salient part of the operational semantics of FPC has to do with the following typing and eval... |

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Citation Context ... rationally SFP if it has an SFP structure. 6 In [1], the category C § is denoted by ˘ C and its diagonal subcategory A is denoted by C D . 7 This is a generalisation of R. Bird’s approximation lemma =-=[3]-=-, which in turn generalises the well-known take lemma [4]. 13 ✷sHo Here we need to introduce an auxiliary type - the vertical natural numbers ω, defined by ω = µX.X⊥. For each x : ω, its successor is ... |

124 |
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Citation Context ...version of Freyd’s algebraic compactness and the underlying domain structure of the contextual order. Our study also makes use of A.M. Pitts’ operational tools (which can be traced back to works like =-=[12,14,15,19]-=-) developed in [22] which are recorded in Section 2. We work with the diagonal category A built from the category of closed FPC types C. Details of this are discussed in Section 3 in conjunction with ... |

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Citation Context ...version of Freyd’s algebraic compactness and the underlying domain structure of the contextual order. Our study also makes use of A.M. Pitts’ operational tools (which can be traced back to works like =-=[12,14,15,19]-=-) developed in [22] which are recorded in Section 2. We work with the diagonal category A built from the category of closed FPC types C. Details of this are discussed in Section 3 in conjunction with ... |

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Citation Context ...◦ [unfold S(�σ) ] holds. By classical domain theory, the least endomorphism [e] on [S(�σ)] must be id[[S(�σ)]]. But Lemma 3.4 below asserts that e(x) = x for all x : S(�σ). Thus by the Context Lemma (=-=[20]-=-), h = idS(�σ). ✷ Lemma 3.4 Let e : τ → τ be a closed term such that [e] = id[[τ]] in the Scott-model. Then for all x : τ, e(x) = x. Proof. For all x : τ, it holds that [e(x)] = [e]([x]) = id[[τ]]([x]... |

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Citation Context ...ossibilities. (i) By adapting the methods in [5], construct a purely operational proof of the minimal invariance property (Lemma 3.3) in our setting. (ii) Investigate the implications of Pitts’ work (=-=[21]-=-) on relational properties of domains to our operational domain theory (c.f. similar to that of [5] but with emphasis on the functorial status of FPC type expressions). (iii) Develop an operational do... |

99 | Equality in lazy computation systems
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Citation Context ...version of Freyd’s algebraic compactness and the underlying domain structure of the contextual order. Our study also makes use of A.M. Pitts’ operational tools (which can be traced back to works like =-=[12,14,15,19]-=-) developed in [22] which are recorded in Section 2. We work with the diagonal category A built from the category of closed FPC types C. Details of this are discussed in Section 3 in conjunction with ... |

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Citation Context ...ng its morphism part. Functoriality of FPC type expressions can be established in a manner similar to that of basic type expressions (c.f. Proposition 3.1). ✷ 4 Algebraic compactness operationally In =-=[8]-=-, P.J. Freyd introduced the notion of algebraic compactness to capture the bifree nature of the canonical solution to the domain equation X = F X in that “every endofunctor (on cpo-enriched categories... |

75 | Operationally-based Theories of Program Equivalence
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Citation Context ... compactness and the underlying domain structure of the contextual order. Our study also makes use of A.M. Pitts’ operational tools (which can be traced back to works like [12,14,15,19]) developed in =-=[22]-=- which are recorded in Section 2. We work with the diagonal category A built from the category of closed FPC types C. Details of this are discussed in Section 3 in conjunction with the extension of fo... |

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Citation Context ...those of C ∗ but whose morphisms are strict morphism pairs. Despite this restriction, there is no need to contract on our class of realisable functors, owing to the following lemma. Lemma 4.1 (Freyd, =-=[7]-=-) If T : (C ∗ ) n+1 → C ∗ is a locally monotone functor, then it preserves strict morphism-pairs. In particular, so does every functor that realises a type expression. Proof. Essentially the same as L... |

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Citation Context ...X is covariant and Y is contravariant in X → Y ). (ii) A particular type variable may be covariant and contravariant (e.g. X in X → X). The usual solution to this problem, following the work of Freyd =-=[9]-=- is to work with the category C op × C. However, as discussed in Section 4, this is not possible for FPC-definable functors. We work instead with a full subcategory C ∗ of this. Define C ∗ , the diago... |

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Citation Context ...tes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcssHo many authors have already exported domain-theory into the study of the operational order. Amongst these are I.A. Mason, et al, =-=[19]-=-, as well as L. Birkedal and R. Harper [5] who, in particular, gave a relational interpretation to recursive types in an operational setting. This paper complements with [5] in that we focus on the co... |

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Citation Context ...lds that [e(x)] = [e]([x]) = id[[τ]]([x]) = [x]. The desired result then follows from the computational adequacy of the Scottmodel which in turn can be proven by adapting the proof of Theorem 7.14 of =-=[10]-=- to our call-by-name setting. ✷ Remark 3.5 Although a semantic trick has been employed in the above proof, the result is purely operational. It is interesting to note that there 6 ✷sHo is indeed a pur... |

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Citation Context ... [22] and are thus omitted in this paper. 3 The categorical setting This section gives an account of the categorical framework within which our theory is organised. Our approach, largely adapted from =-=[1]-=-, turns out to be the appropriate option among others. We carefully explain this in two stages: (i) understand the basic type expressions (i.e., without the function types) as functors, and then (ii) ... |

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Citation Context ...mation lemma” [16]. In Section 6, running examples taken from [22] demonstrate the versatility of this proof method as compared to others such as bisimulation techniques ([12,22]) and program fusion (=-=[11]-=-). Throughout the discussion we assume certain pre-requisites from the reader which include the operational semantics of the call-by-name version of FPC ([17]), theory of recursive domain equations ([... |

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Citation Context ...heory (c.f. similar to that of [5] but with emphasis on the functorial status of FPC type expressions). (iii) Develop an operational domain-theory that caters for non-deterministic languages, such as =-=[13]-=-. 19sAcknowledgement Ho My gratefulness goes to my supervisor, M.H. Escardó, whose deep insight and gentle advice I have greatly benefited from. I must also acknowledge T. Streicher for a series of di... |

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and full abstraction for FPC
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Citation Context ... techniques ([12,22]) and program fusion ([11]). Throughout the discussion we assume certain pre-requisites from the reader which include the operational semantics of the call-by-name version of FPC (=-=[17]-=-), theory of recursive domain equations ([2,26]) and category theory ([18]). 2 Preliminaries We choose to work with a call-by-name version of FPC 3 (Fixed Point Calculus). The most salient part of the... |

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Recursive Types in Kleisli Categories, Unpublished note
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Citation Context ...d by a direct proof by induction on types. That would deem to fail because the least fixed point of a type expression is not in any way built from those of its constituents. Remark 3.6 Theorem 5.2 of =-=[25]-=- seems to suggest that the Plotkin’s uniformity can be used to prove that the morphism fold : σ[µX.σ/X] → µX.σ is a special invariant (which is equivalent to our operational minimal invariance propert... |

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