## Notions of computability at higher types I (2005)

Venue: | In Logic Colloquium 2000 |

Citations: | 12 - 5 self |

### BibTeX

@INPROCEEDINGS{Longley05notionsof,

author = {John Longley},

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

booktitle = {In Logic Colloquium 2000},

year = {2005},

pages = {32--142},

publisher = {}

}

### OpenURL

### Abstract

We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.

### Citations

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Citation Context ...theory (which typically considers weaker systems). Both of these are significant areas of research in theoretical computer science. For detailed information on modern type theory, a good reference is =-=[Bar92]-=-; for higher type complexity, we refer the reader to the recent survey articles of Irwin, Kapron and Royer [IKR01a, IKR01b]. 2.2 Kleene computability: S1--S9 2.2.1 Kleene's work The first serious atte... |

294 |
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Citation Context ...es 1 and 2 1.1 Type 1 computability The main ideas concerning computability for type 1 functions of course date back to the development of basic recursion theory in the 1930s by Godel [God31], Church =-=[Chu36]-=-, Turing [Tur36, Tur37], Kleene [Kle36a, Kle36b] and Post [Pos36], who accumulated several characterizations of the class of (partial) recursive functions as mentioned in the introduction. The first e... |

278 | Foundations of Constructive Mathematics - Beeson - 1985 |

205 | Full abstraction for PCF
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(Show Context)
Citation Context ...ality in terms of dialogue games was attained around 1993, when three closely related models of PCF were obtained (simultaneously and more or less independently) by Abramsky, Jagadeesan and Malacaria =-=[AJM00]-=-, Hyland and Ong [HO00], and Nickau [Nic94]. The formal details can appear rather complicated, but the essential ideas can be easily grasped via an example. Usings#-calculus notation and some other sy... |

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Citation Context ...reditarily recursive operations (see Section 2.4). 16 that given by the extensional closed term model of System T itself. An argument due to Kreisel (appearing as one of the final set of exercises in =-=[Bar84]-=-) shows that the interpretation of System T in S induces a strictly smaller equivalence relation --- in computer science terms, this interpretation is not fully abstract. In [Loa97], Loader considers ... |

123 |
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Citation Context ...ut innocence correspond to notions of computation involving memory or state; a full abstraction result for Idealized Algol (with respect to the non-innocent, well-bracketed game model) is obtained in =-=[AHM98]-=-. Very recently, the present author has found some alternative constructions of the non-innocent, non-well-bracketed model, including a programming language characterization of this notion of computab... |

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(Show Context)
Citation Context ...his intensional character is also an important aspect of much of the work in computer science on the semantics of higher type computation: for instance, Berry and Curien's sequential algorithms model =-=[BC82], or the g-=-ame models of PCF [AJM00, HO00, Nic94]. Kleene's unimonotone semantics can be considered as one in which computable objects act only on computable objects, if "computable" here is understood... |

71 |
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(Show Context)
Citation Context ...al semantics of PCF and so were felt not to qualify as an independent mathematical characterization of sequentiality. Several other models for PCF were constructed: Berry's stable and bistable models =-=[Ber78b]-=-; the Berry-Curien sequential algorithms model [BC82, Cur93] based on Kahn and Plotkin's notion of concrete data structure [KP93]; and the BucciarelliEhrhardsstrongly stable model [BE91b, Ehr93]. (For... |

59 | Hypercoherence: A strongly stable model of linear logic
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Citation Context ... e#ective as could be desired, so that (for instance) equality of finite elements is decidable. Ehrhard later gave a simplified presentation of this type structure in the framework of hypercoherences =-=[Ehr93]-=-; a slightly more abstract analysis of the relevant structure was given in [BE93]. Another interesting characterization, given by Colson and Ehrhard in [CE94], showed that R in some sense arises natur... |

56 |
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(Show Context)
Citation Context ...ive notion of "sequential" computability, more generous than the PCF one. This class is in some sense the e#ective counterpart of the strongly stable functionals introduced by Bucciarelli an=-=d Ehrhard [BE91b]-=-, and later studied by Longley [Lon98] as the sequentially realizable (SR) functionals. The basic idea can be given by a simple example. Let M be the set of monotone computable functions N# # N# . and... |

54 | Game semantics
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(Show Context)
Citation Context ...heorem 5.2. Since then, the ideas of game semantics as developed by Abramsky et al have been successful in providing a semantic account of the expressivity of a number 63 of programming languages. In =-=[AM99]-=-, for instance, it is shown that by imposing or not imposing the well-bracketing and innocence conditions on strategies (see Section 3.3.3) one obtains a square of four categories of games correspondi... |

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(Show Context)
Citation Context ... proposed languages turn out to have the same expressivity as PCF+catch is already encouraging, but more significant is the following semantic characterization due to Cartwright, Curien and Felleisen =-=[CCF94]-=-: Theorem 5.2 The fully abstract term model for PCF+catch is isomorphic to the type structure over N# arising from the category of e#ective sequential algorithms (that is, the e#ective analogue of the... |

50 | An inverse of the evaluation functional for typed -calculus - Berger, Schwichtenberg - 1991 |

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37 |
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(Show Context)
Citation Context ...s or information systems [Ber93, SHLG94, Sch96, Nor99]. A particular focus of recent work has been the search for abstract formulations of the concept of totality for elements of such domains. Berger =-=[Ber93]-=- has shown the significance of the dual notions of densitysand codensity (= totality) for subsets of domains, and given generalized versions of the density theorem and Kreisel-Lacombe-Shoenfield theor... |

35 | Computable Analysis - Aberth - 1980 |

33 | Equilogical spaces
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(Show Context)
Citation Context ...is not surprising in view of the close connections between Scott domains and algebraic lattices. The categories PER(P#),PER(P# re ) also play a central role in recent work of Scott and his colleagues =-=[BBS98]-=-, who adopt the perspective that PERs on P# are essentially equivalent to countably based T 0 spaces equipped with an equivalence relation. Another interesting family of PCAs are the term models for u... |

32 |
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(Show Context)
Citation Context ...ction shows that many choices confront us if we wish to formulate a definition of higher type computability. For example: 1 1 Many of these points are also made previously in a survey article by Cook =-=[Coo90]-=-, whose point of view is close to ours in many respects. 1 . Domain of definition. Do we want to consider partial or total computable functions? Do we want them to act on partial functions of the next... |

25 | Local realizability toposes and a modal logic for computability, presented at
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(Show Context)
Citation Context ...egory of the modified realizability topos on A. In addition, several possible definitions of RC correspond to type structures in relative realizability models of the kind considered by Birkedal et al =-=[ABS00]-=-. (All the results alluded to above will be covered in more detail in Part II.) The general programme of trying to identify the total type structures arising from various PCAs was explicitly articulat... |

23 | Sequentiality in an extensional framework - Bucciarelli, Ehrhard - 1994 |

23 | Projecting sequential algorithms on strongly stable functions
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(Show Context)
Citation Context ... # # . From the above results it is still not obvious whether any of the non-PCFdefinable elements of R at higher types are sequentially computable in any reasonable sense. However, Ehrhard showed in =-=[Ehr96]-=- that the notion of strong stability admits a computational interpretation: every element of R can be in some sense computed by a Berry-Curien sequential algorithm. This line of investigation was cont... |

22 |
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(Show Context)
Citation Context ...finable in this way. More surprising is the fact that the fan functional # mentioned in Section 2.3 is PCF-definable, by means of a clever higher type recursion. This fact appeared in Berger's thesis =-=[Ber90]-=-; it was also known (independently) by Gandy. The following was conjectured by Cook [Coo90] and Berger [Ber93], and proved by Normann [Nor00a]. Theorem 3.18 The PCF-definable elements of C are exactly... |

15 | Collapsing partial combinatory algebras
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(Show Context)
Citation Context ... VI], where the cases of P# and P# re (giving rise to C and HRC respectively) were considered. Similar results for other graph models (such as the Plotkin-ScottEngeler models) were obtained by Bethke =-=[Bet88]-=-. As one might expect, it would seem that all natural PCAs based solely on a notion of continuous function application give rise to the type structure C, and their recursive analogues give rise to HRC... |

12 | Continuous functionals of dependent types and equilogical spaces
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(Show Context)
Citation Context ...with extending the construction of C and the associated results on domains, density and totality to transfinite types Martin-Lof-style dependent types with universes [Ber97, KN97]. Bauer and Birkedal =-=[BBar]-=- have shown how much of this material fits smoothly into the framework of Scott's equilogical spaces (see Section 4). Other recent results of Normann [Nor00a] and Plotkin [Plo97] involving C will be m... |

12 |
Sequential models of PCF: some contributions to the domain-theoretic approach to full abstraction
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(Show Context)
Citation Context ...d some important ideas, although most of these have been more clearly expressed and more fully explored in the subsequent computer science literature. A useful paper by Bucciarelli [Buc93a] (see also =-=[Buc93b]-=-) makes explicit the relationships between Kleene's work and subsequent developments in computer science, and in particular explores the connections between Kleene's oracles and Berry-Curien sequentia... |

10 |
Ehrhard T., “A theory of sequentiality
- Bucciarelli
- 1993
(Show Context)
Citation Context ...is decidable. Ehrhard later gave a simplified presentation of this type structure in the framework of hypercoherences [Ehr93]; a slightly more abstract analysis of the relevant structure was given in =-=[BE93]-=-. Another interesting characterization, given by Colson and Ehrhard in [CE94], showed that R in some sense arises naturally from the class of (infinitary) first order sequential functions N N # # N# .... |

9 | Continuous Functionals of Dependent and Transitive Types. Habilitationsschrift, Ludwig-Maximilians-Universitat Munchen - Berger - 1997 |

8 | Degrees of parallelism in the continuous type hierarchy
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- 1997
(Show Context)
Citation Context ...tinct degrees and some relationships between them: for instance, the parallel-or and exists operations represent incomparable degrees. A few results in a similar spirit appear in [Tra75]. Bucciarelli =-=[Buc95]-=- undertook a somewhat more systematic study of degrees of parallelism for first order functions; this line of investigation was pursued further by Lichtenthaler [Lic96]. Degrees of parallelism can be ... |

7 | Extensional embedding of a strongly stable model of PCF - Bucciarelli, Ehrhard - 1991 |

7 |
Sur les fonctions calculables
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(Show Context)
Citation Context ...duced the concept of a computable type 2 function. 1.2 Banach-Mazur functionals A very early definition of a class of type 2 functionals involving a notion of computability is due to Banach and Mazur =-=[BM37]-=- (see also [Maz63]): Definition 1.1 A total function F : (N # N) rec is Banach-Mazur if, for every total recursive function h : NN # N, the function #x.F (#y.h(x, y)) : N # N is total recursive. Notic... |

6 |
A relative PCF-definability result for strongly stable functions and some corollaries
- Ehrhard
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(Show Context)
Citation Context ... notion of strong stability admits a computational interpretation: every element of R can be in some sense computed by a Berry-Curien sequential algorithm. This line of investigation was continued in =-=[Ehr97]-=-, where Ehrhard showed that R is in fact the extensional collapse of the sequential algorithms model. A somewhat simpler characterization in the same vein was discovered independently by van Oosten [v... |

5 |
Another approach to sequentiality: Kleene’s unimonotone functions
- Bucciarelli
- 1993
(Show Context)
Citation Context ...his area contributed some important ideas, although most of these have been more clearly expressed and more fully explored in the subsequent computer science literature. A useful paper by Bucciarelli =-=[Buc93a]-=- (see also [Buc93b]) makes explicit the relationships between Kleene's work and subsequent developments in computer science, and in particular explores the connections between Kleene's oracles and Ber... |

4 | Equivalence of bar recursors in the theory of functionals of finite type - Bezem - 1988 |

4 | LCF should be lifted
- Bloom, Riecke
- 1989
(Show Context)
Citation Context ...a call-by-value evaluation strategy, which can be naturally interpreted in CBV type structures (the idea essentially appears in [Plo83, Chapter 3]). There is also a third possibility: the lazy PCF of =-=[BR89]-=-. The relationships between these languages were investigated by Sieber [Sie90], Riecke [Rie93] and Longley [Lon95, Chapter 6]. The picture that has emerged is that these languages are su#ciently inte... |

4 | On strong stability and higher-order sequentiality
- Ehrhard, Colson
- 1994
(Show Context)
Citation Context ...ture in the framework of hypercoherences [Ehr93]; a slightly more abstract analysis of the relevant structure was given in [BE93]. Another interesting characterization, given by Colson and Ehrhard in =-=[CE94]-=-, showed that R in some sense arises naturally from the class of (infinitary) first order sequential functions N N # # N# . (For functions of this type, there is evidently only one reasonable notion o... |

3 |
Isomorphisms between HEO and HRO
- Bezem
- 1985
(Show Context)
Citation Context ...ed by Gandy around 1965, that not all the HEOs are sequentially continuous in the sense of Definition 2.11. His ingenious construction of a counterexample at type 3 appears in [GH77, 8]. Later, Bezem =-=[Bez85a]-=- gave another characterization of HEO as the extensional collapse of HRO: Theorem 2.17 Define partial equivalence relations # # on each HRO # as follows:sx # 0 y i# x = y; and f # ### g i# fx # # gy w... |

2 |
Some combinatory algebras for sequential computation
- Abramsky, Longley
- 2000
(Show Context)
Citation Context ... well-bracketed strategies. The proof that it does so is non-trivial and to some extent furnishes the same kind of evidence for the status of Q as would be provided by the LongleyPhoa conjecture; see =-=[AL00]-=-. Meanwhile, van Oosten and Longley constructed the PCA B described in Section 3.4; the type structure described there is exactly the type structure over the obvious object N# in PER(B). It was this c... |

2 | Minimisation vs. Recursion on the Partial Continuous Functionals
- Berger
- 2000
(Show Context)
Citation Context ...of S1--S9: for instance, one can take Godel's System T extended with the minimization operatorsof ordinary recursion theory. This and related systems have indeed been considered recently by Berger in =-=[Ber00]-=-, where some preliminary expressivity results are obtained. We feel that the notion of computability given by this language is philosophically a very natural one and deserves further study --- certain... |

2 | Compact and majorizable functionals of finite type - Bezem - 1989 |

2 |
Computable functionals of arbitrary finite type
- Davis
- 1957
(Show Context)
Citation Context ... a new least element #. 3.1 Partial monotone functionals As far as we are aware, the earliest formulation of a class of hereditarily partial computable functionals at all finite types is due to Davis =-=[Dav59]-=-. Davis (together with Putnam) realized that some restriction on set-theoretic partial functionals was needed to obtain a good theory, and proposed (essentially) a call-by-name structure A of consiste... |

2 | The computation of primitive recursive terms of finite type, and primitive recursive realization - Dragalin - 1968 |

1 | Recursion in the superjump - Aczel, Hinman - 1974 |

1 |
Computability and Continuity in Finite Types
- Bergstra
- 1976
(Show Context)
Citation Context ...-sections for continuous type 2 objects. Early results in this vein were obtained by Grilliot [Gri71], who pointed out that such a 1-section is never closed under the ordinary jump operator. Bergstra =-=[Ber76]-=- showed that there are objects F # C 2 whose 1-section is not the 1-section of any f # C 1 (this improves on the result of Hinman mentioned above). Results relating to ordinal hierarchies for 1-sectio... |

1 |
The continuous functionals and E
- Bergstra
- 1978
(Show Context)
Citation Context ...ive schemata in the spirit of generalized recursion theory.) It is fairly easy to specify an infinite basis for RC consisting of partial recursively continuous functions, one for each type level (see =-=[Ber78a, 1]-=-). Normann [Nor79b, Nor81a] 28 gave a similar infinite basis consisting of total recursively continuous functionals (i.e. elements of RC), but showed that no finite subset of this su#ced; hence no fin... |

1 | The "real" ordinal of the 1-section of a continuous functional (abstract - Bergstra, Wainer - 1977 |

1 |
Hierarchies of predicates of finite types
- Clarke
- 1964
(Show Context)
Citation Context ...for 2-sections of 3 # and other type 3 objects. In [Kle63] Kleene outlined the construction of a hyperanalytic hierarchy for type 2 objects; this hierarchy was studied in detail by his student Clarke =-=[Cla64]-=-, who conjectured that it did not exhaust the type 2 objects computable in 3 #. This conjecture was confirmed by Moschovakis [Mos67], who however constructed an alternative hierarchy that does exhaust... |

1 | Computable numerations of morphisms - Ershov - 1971 |

1 | La theorie des enumerations - Ershov - 1970 |