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

Venue: | In Logic Colloquium 2000 |

Citations: | 11 - 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

686 |
Introduction to Metamathematics
- Kleene
- 1952
(Show Context)
Citation Context ...cally replaced by primitive recursions at type 2; thus, the Ackermann function, though not primitive recursive in the usual sense, could be defined by a type 2 primitive recursion. In his famous book =-=[Kle52]-=-, Kleene gave schemata for defining primitive, total and partial recursive functions uniformly in a finite list of functions of type (N # N) (47, 58, 63), and made explicit the possibility of regardin... |

518 | Lambda calculi with types
- Barendregt
- 1992
(Show Context)
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... |

385 |
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme.” Monatshefte für Mathematik und Physik
- Gödel
- 1931
(Show Context)
Citation Context ...utability at types 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 introduct... |

263 | Foundations of Constructive Mathematics - Beeson - 1985 |

263 |
An unsolvable problem of elementary number theory
- Church
(Show Context)
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... |

197 |
On full abstraction for PCF
- Hyland, Ong
(Show Context)
Citation Context ...ely decidable", so that Church's 1936 thesis (IM 62) will apply with the higher types included (as well as to partial functions, IM p. 332). (This has been called "Kleene's problem" by =-=Hyland and Ong [HO00]-=-.) Like Platek, Kleene sought to avoid the anomalies of the original S1--S9 theory by developing a theory of hereditarily partial objects. However, whereas Platek was interested primarily in questions... |

192 | Full abstraction for pcf
- Abramsky, Jagadeesan, et al.
- 2000
(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... |

154 |
The Lambda Calculus: its Syntax and Semantics. NorthHolland, revised edition
- Barendregt
- 1984
(Show Context)
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 ... |

116 |
A fully abstract game semantics for general references
- Abramsky, Honda, et al.
- 1998
(Show Context)
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... |

114 | Computability and Unsolvability - Davis - 1958 |

84 |
Sequential algorithms on concrete data structures, Theor
- Berry, Curien
- 1982
(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... |

82 |
On the interpretation of intuitionistic number theory
- Kleene
- 1945
(Show Context)
Citation Context ...Later, PER(K 1 ) was identified as an important subcategory of Hyland's e#ective topos [Hyl82], a categorical model for higher order logic based on Kleene's realizability interpretation of arithmetic =-=[Kle45]-=-. (More generally, PER(A) arises as a subcategory of the realizability topos RT(A); we will refer loosely to categories such as PER(A), RT(A) and their close relatives as realizability models.) Here, ... |

67 |
Stable models of typed lambda-calculi
- Berry
- 1978
(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... |

67 | General Recursive Functions of Natural Numbers’, Mathematische Annalen 112 - Kleene - 1936 |

63 |
Recursion-Theoretic Hierarchies
- Hinman
- 1978
(Show Context)
Citation Context ...ied in [Gan67b, Pla71, AH74]. Superjump operators of even higher types have been considered by Harrington [Har73, Har74]. Several of the above hierarchy results are covered in detail in Hinman's book =-=[Hin78]-=-. 2.3 The total continuous functionals 2.3.1 Early work: Kleene and Kreisel Whereas computations over the full set-theoretic type structure S seem to have connections with the metamathematics of class... |

61 | Full abstraction for functional languages with control - Laird - 1997 |

60 |
Realizability Toposes and Language Semantics
- Longley
- 1994
(Show Context)
Citation Context ... rise to an object N# which may be defined (as a PER) by N# (m, n) # m . 0 # n . 0. The morphisms N # N# then correspond precisely to the partial recursive functions N # N, as we might have hoped. In =-=[Lon95]-=-, Longley developed more explicitly the idea that di#erent PCAs embody di#erent notions of computability. Longley and Simpson [LS97] developed a version of synthetic domain theory that applied uniform... |

59 | Hypercoherences: a strongly stable model of linear logic
- Ehrhard
- 1993
(Show Context)
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... |

59 | Extension de la notion de fonction recursive aux fonctions d’une ou plusieurs variables reelles - Lacombe - 1955 |

57 |
Computable functionals
- Grzegorczyk
- 1955
(Show Context)
Citation Context ...t seem to have been noted in the literature of the time. In [Grz55b] Grzegorczyk showed that his definition was equivalent to an Herbrand-Godel style definition in terms of an equational calculus. In =-=[Grz55a] he invest-=-igated some properties of these functionals, notably a continuity property and the existence of certain computable "modulus of continuity" functionals. The relationship between computability... |

57 |
Recursive functionals and quantifiers of finite types
- Kleene
- 1959
(Show Context)
Citation Context ...s finite paths of arbitrary length (hence, by Konig's Lemma, K contains infinite paths. 13 . K contains no recursive infinite paths. The Kleene tree is so named after its appearance as Theorem LII of =-=[Kle59b]-=-, although examples of the same phenomenon also appear in [Lac55d] and [Zas62, ZC62]. We may now obtain a total e#ective operation # that does not extend to a total recursive function on N N , by defi... |

53 |
First steps in synthetic domain theory
- Hyland
- 1991
(Show Context)
Citation Context ...ivalent to e#ective Scott domains) embeds fully in a model of intuitionistic ZF set theory based on Kleene realizability (a model closely related to PER(K 1 )). Later work by Rosolini [Ros86], Hyland =-=[Hyl90]-=- and many other researchers sought to give an axiomatic account of synthetic domain theory: a set of conditions on a model of constructive set theory (usually a topos) which su#ce to ensure that it co... |

52 |
Sequentiality and Strong Stability
- Bucciarelli, Ehrhard
- 1991
(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... |

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

50 | Lambda-definability and recursiveness - Kleene - 1936 |

49 | Fully abstract semantics for observably sequential languages
- Cartwright, Curien, et al.
- 1994
(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... |

48 | Game semantics
- Abramksy, McCusker
- 1998
(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... |

46 | PCF extended with real numbers
- Escardó
- 1996
(Show Context)
Citation Context ...nd (see [Abe80, Bee85]), and classical computable analysis on the other (see [PER89, Wei00]). Interest in real number computability has also recently been reawakened within computer science (see e.g. =-=[Esc96]-=-), and exact real number computation appears as an attractive application area for higher 5 type programming. Higher types over the reals and the associated notions of computability have recently been... |

44 |
Theories of finite type related to mathematical practice
- Feferman
- 1977
(Show Context)
Citation Context ...o73], who focused on realizability and related interpretations. In a somewhat similar spirit is Feferman's use of computable objects of higher type to give interpretations for theories of finite type =-=[Fef77b]. Although-=- in this case the logical systems are classical, they are typically intended to reflect "semi-constructivist" viewpoints that su#ce to support most of mathematical practice. 0.1.2 Descriptiv... |

42 |
Countable functionals
- Kleene
- 1959
(Show Context)
Citation Context ... the total continuous functionals has been motivated by interest in more constructive interpretations. The type structure C of total continuous functionals was obtained around the same time by Kleene =-=[Kle59a]-=- (who called them the countable functionals) and Kreisel [Kre59] (who called them the continuous functionals). Both definitions were rather complicated and neither were obviously natural, but they bot... |

41 |
Hereditarily majorizable functionals of finite type
- Howard
- 1973
(Show Context)
Citation Context ...beyond System T itself and make use of some of the other type structures we shall discuss in this paper. Other mathematical results pertaining specifically to System T include the theorems of Howard (=-=[How73]-=-; see also [Gir87, annex 7.A]) that all T-definable functionals are hereditarily majorizable, and (hence) that there is no T-definable modulus of extensionality functional. A further mathematical anal... |

39 | Observable sequentiality and full abstraction - Cartwright, Curien, et al. - 1992 |

36 |
Total sets and objects in domain theory
- Berger
- 1993
(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... |

36 | Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals - Bezem - 1985 |

36 | The sequentially realizable functionals
- Longley
(Show Context)
Citation Context ...ty, more generous than the PCF one. This class is in some sense the e#ective counterpart of the strongly stable functionals introduced by Bucciarelli and 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 consider the function F : M # N# defi... |

35 | Concrete domains
- Kahn, Antipolis, et al.
- 1993
(Show Context)
Citation Context ... 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 a detailed survey of this material and further references, we recommend [Ong95]). This line of research gave much insight into t... |

33 | Computable Analysis - Aberth - 1980 |

32 | A semantic analysis of control - Laird - 1999 |

31 | Equilogical spaces
- Bauer, Birkedal, et al.
(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... |

31 |
Computational complexity of higher type functions
- Cook
- 1991
(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... |

29 |
Model C of partial continuous functionals
- Ershov
- 1977
(Show Context)
Citation Context ...nalogous result for e#ective type structures --- that the extensional collapse of P e# gives rise to HEO --- was proved in [Ers76b]. Ershov's main results on P and P e# are conveniently summarized in =-=[Ers77a]-=-. Some further discussion of the relationship between the partial and total type structures also appears in Sections 9 and 10 of [GH77]. A streamlined presentation of the main results on f 0 -spaces a... |

29 |
Proofs of strong normalization
- Gandy
- 1980
(Show Context)
Citation Context ...ndently that System T is strongly normalizing --- that is, for any System T term, all reduction paths terminate yielding the same normal form. (A remarkable alternative proof was later given by Gandy =-=[Gan80]-=-.) This result, whose proofs depend explicitly or implicitly on transfinite induction up to # 0 , shows that the extensional term model for System T (consisting of closed System T terms modulo extensi... |

28 |
Interprétation functionelle et élimination des coupures de l’arithmétique d’ordre supérieur
- Girard
- 1972
(Show Context)
Citation Context ...ce with HRC as defined via Kleene associates (see Definition 2.15). Another descendant of Kreisel's definition of HEO was Girard's category PER of partial equivalence relations on the natural numbers =-=[Gir72]-=-. We will give the definition of this category in Section 4; meanwhile, we simply remark that HEO can be naturally seen as the finite type structure over N in the cartesian closed category PER. 3 Part... |

25 | Finitary PCF is not decidable
- Loader
- 2001
(Show Context)
Citation Context ...ully abstract model satisfying the Jung-Stoughton criterion existed, it would follow that observational equivalence for finitary PCF was decidable. However, the latter statement was refuted by Loader =-=[Loa96]-=- using a delicate encoding of semi-Thue problems. Loader's result represents one of the most important advances in our understanding of PCF: it closes the door on a large class of attempts at the full... |

24 | Local realizability toposes and a modal logic for computability
- Awodey, Birkedal, et al.
(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 | Filter Spaces and Continuous Functions - Hyland - 1979 |

22 |
Totale Objekte und Mengen in der Bereichstheorie
- Berger
- 1990
(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... |

22 | Projecting sequential algorithms on strongly stable functions
- Ehrhard
- 1996
(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 |
Hierarchies of number-theoretic predicates
- Kleene
- 1955
(Show Context)
Citation Context ...onals computable from a given object. The starting point for this endeavour is the fact that the 1-section of 2 # consists of exactly the hyperarithmetic or # 1 1 functions, which (as shown by Kleene =-=[Kle55b]-=-) can be classified according to their logical complexity by means of an ordinal hierarchy of height # CK 1 , known as the hyperarithmetic hierarchy. Kleene in [Kle63] introduced the general idea of s... |

22 |
Interpretation of analysis by means of functionals of finite type
- Kreisel
- 1959
(Show Context)
Citation Context ...out need for an interpretation of System T in another type structure. (This idea was already implicit in a remark of Kreisel [Kre59, 3.4].) The Dialectica interpretation also inspired work by Kreisel =-=[Kre59]-=- and later Spector [Spe62] on interpretations for analysis (essentially second order arithmetic) in a similar spirit, but these go beyond System T itself and make use of some of the other type structu... |