## Notions of computability at higher types II (2001)

Venue: | In preparation |

Citations: | 2 - 2 self |

### BibTeX

@INPROCEEDINGS{Longley01notionsof,

author = {John Longley},

title = {Notions of computability at higher types II},

booktitle = {In preparation},

year = {2001},

pages = {pages}

}

### OpenURL

### Abstract

ntroduce some simple general theory to allow us to talk about notions of higher-type computable functional. The following definitions (with minor variations) appear frequently in the literature. Definition 1.1 (Weak partial type structures) A weak partial type structure, or weak PTS A [over a set X], consists of the following data: . for each type #, a set A # of elements of type # [equipped with a canonical bijection A 0 # = X], . for each #, # , a partial application function ## : A ### A # # A # . We usually omit type subscripts from application operations, and often write x y simply as xy. By convention, w

### Citations

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Citation Context ... models (see e.g. [Mit96, Section 4.5]). Total type structures are called combinatory type structures in [Bar84, Appendix A], and weak total type structures are called typed applicative structures in =-=[Mit96]-=-. A weak PTS A is called a partial type frame (cf. [Mit96, Section 4.5]) if each A ### is a set of partial functions from A # to A # , and each ## is ordinary function application. It is easy to see t... |

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Citation Context ..., if each ## is a total function. (iii) A (weak) PTS A is extensional if for all #, # and all f, g # A ### we have (#x # A # .fx # gx) =# f = g. Extensional (total) type structures were introduced in =-=[Hen50]-=- and so are often called Henkin models (see e.g. [Mit96, Section 4.5]). Total type structures are called combinatory type structures in [Bar84, Appendix A], and weak total type structures are called t... |

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Citation Context ...second model K 2 [KV65]. . The e#ective partial continuous functionals (corresponding to PCF ++ ) arise from the e#ective analogues of each of the above PCAs, as well as from Kleene's first model K 1 =-=[Kle45]-=-. . The hereditarily sequential functionals arise from various PCAs recently constructed by Abramsky (see [Lon98a]). They also from PCAs obtained by solving various recursive domain equations in known... |

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Citation Context ...f certain impure #-calculi (see [MRS99]). Moreover, the Longley-Phoa Conjecture asserts that this type structure also arises from the pure term model # 0 /T for any semi-sensible #-theory T (see e.g. =-=[Lon95]-=-). . The sequentially realizable (SR) functionals arise from van Oosten's combinatory algebra B [Oos97], and from the combinatory algebra A constructed by Abramsky (see [Lon98a]). They also arise from... |

60 | Hereditarily sequential functionals
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Citation Context ...type structures are: . The partial continuous functionals: that is, the finite type structure arising from the familiar Scott domain model [Sco93]. . The hereditarily sequential functionals of Nickau =-=[Nic94]-=-: this coincides with the finite type structure arising from the fully abstract game models for PCF due to Abramsky, Hyland et al [AJM00, HO00]. . The strongly stable functionals of Bucciarelli and Eh... |

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Citation Context ... coincides with the finite type structure arising from the fully abstract game models for PCF due to Abramsky, Hyland et al [AJM00, HO00]. . The strongly stable functionals of Bucciarelli and Ehrhard =-=[BE91]-=-: these coincide with the sequentially realizable functionals of Longley [Lon98b]. Intuitively, the type structure of hereditarily sequential functionals is smaller than the other two (more precisely,... |

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Citation Context ...t a weak PTS is extensional i# it is isomorphic to a partial type frame. The significance of the conditions involving k, s in Definition 1.2 is shown by the following result, essentially due to Curry =-=[Cur30]-=-. Given a weak PTS A, let us write E(A) for the set of well-typed formal expressions over A freely generated from constants (corresponding to elements of A) and typed variables via formal application.... |

41 |
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Citation Context ... . All of these PCAs will reappear later in the paper. 2.1.1.6 Realizability over T # Ershov; Berger? 2.1.1.7 Kleene's definition via associates Finally we give Kleene's original definition of C from =-=[Kle59]-=-. Here the intensional objects involved are known as associates; modulo trivialities, these are essentially realizers drawn from a PCA known as Kleene's second model or K 2 (introduced in [KV65]). Sin... |

36 | The sequentially realizable functionals
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Citation Context ...models for PCF due to Abramsky, Hyland et al [AJM00, HO00]. . The strongly stable functionals of Bucciarelli and Ehrhard [BE91]: these coincide with the sequentially realizable functionals of Longley =-=[Lon98b]-=-. Intuitively, the type structure of hereditarily sequential functionals is smaller than the other two (more precisely, it is a subquotient of each of the others): partial continuous sequentially real... |

26 | Prelogical relations
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Citation Context ...losed under composition, and this is one reason for favouring our definition of morphism over the concept of logical relation. In fact, our morphisms correspond to pre-logical relations as studied in =-=[HS99]-=-, where the reader may find further details arguments for the advantages of pre-logical over logical relations. We also have the following simple facts: Proposition 1.18 Suppose A is an extensional PT... |

23 |
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(Show Context)
Citation Context ...use on the way towards type structures based on e#ectivity. For a more detailed study of this type structure and its characterizations, with proofs of most of the main results, we refer the reader to =-=[Hyl79]-=-. Another more recent survey is [Nor99]. 2.1.1 Characterizations of the continuous functionals There are very many ways to characterize the type structure C of continuous functionals, providing ample ... |

22 |
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Citation Context ...tion in Section 2.2 below. 2.1.1.4 Kreisel's definition via neighbourhoods We now give Kreisel's construction of C via a notion of continuity based on neighbourhoods. In Kreisel's original definition =-=[Kre59], "ne-=-ighbourhoods" were first constructed rather syntactically as formal expressions; and the points themselves were obtained essentially as limits of shrinking sequences of neighbourhoods. We give he... |

17 |
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Citation Context ...t. In C: non-computability of Fan functional (Tait). Normann: no finite basis. Conclusion: a bit odd. 3 Partial functionals Not written yet. Meanwhile, here are some relevant paragraphs borrowed from =-=[Lon99]: At present, it see-=-ms that there are essentially three di#erent finite type structures that occur widely in nature, each of which comes in both a "full continuous" and an "e#ective" flavour. All six ... |

10 |
The Foundations of Intuitionistic Mathematics
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Citation Context ...from [Kle59]. Here the intensional objects involved are known as associates; modulo trivialities, these are essentially realizers drawn from a PCA known as Kleene's second model or K 2 (introduced in =-=[KV65]-=-). Since the latter observation is not really explicit in the literature, we here approach the definition of C via K 2 . Elements of K 2 are total functions N # N. As a first step, we can regard a ele... |

7 | Recursion on the countable functionals, volume 811 of Lec. Not - Normann - 1980 |

4 | LCF should be lifted
- Bloom, Riecke
- 1989
(Show Context)
Citation Context ...puter scientists under these names. For instance, the language PCF has been considered in three flavours: the original call-by-name version [Plo77], a call-by-value version [SF90], and a lazy version =-=[BR89]-=-. Term models for these languages naturally give rise to call-by-name, call-by-value and lazy type structures respectively. The general form of call-by-name lambda calculi was more or less indicated i... |

4 | Full abstraction and universality via realisability
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- 1999
(Show Context)
Citation Context ...ed by Abramsky (see [Lon98a]). They also from PCAs obtained by solving various recursive domain equations in known fully abstract models of PCF, such as categories of games or sequential domains (see =-=[MRS99]-=-). . The e#ective hereditarily sequential functionals (i.e. the PCF-definable functionals) arise from the e#ective analogues of any of these, and from the term models of certain impure #-calculi (see ... |

3 |
Recursion theory on the countable functionals
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(Show Context)
Citation Context ... of K with respect to equality on N. In fact, it is easy to show that K is pre-extensional, so that C is a quotient of K. That this definition of C is equivalent to the others given here is proved in =-=[Hyl75]. 2.1.1.5 -=-Realizability over P# We now give three characterizations of C as the type structure realized by certain "continuous" partial combinatory algebras as indicated in Example 1.16: Scott's P#, P... |

3 |
Realizability models for sequential computation, In preparation; an incomplete draft is available from the authorâ€™s home
- Longley
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(Show Context)
Citation Context ...e#ective analogues of each of the above PCAs, as well as from Kleene's first model K 1 [Kle45]. . The hereditarily sequential functionals arise from various PCAs recently constructed by Abramsky (see =-=[Lon98a]-=-). They also from PCAs obtained by solving various recursive domain equations in known fully abstract models of PCF, such as categories of games or sequential domains (see [MRS99]). . The e#ective her... |

3 | Notions of computability at higher types III
- Longley
- 2001
(Show Context)
Citation Context ...r types II John Longley April 6, 2001 In Part I of this series of papers [Lon01a] we gave a historical survey of the study of notions of higher-type computability. In the present paper and its sequel =-=[Lon01b]-=-, we undertake a more systematic exposition of notions of higher-type computability, showing how many of the existing ideas and results can be fitted into a coherent framework. In Part II we will rest... |