## Introduction to Turing categories

Citations: | 4 - 1 self |

### BibTeX

@MISC{Cockett_introductionto,

author = {J. R. B. Cockett and Pieter J. W. Hofstra},

title = {Introduction to Turing categories},

year = {}

}

### OpenURL

### Abstract

### Citations

426 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...in a cartesian closed category is an object A together with embedding-retraction pairs 1 ⊳ A, A × A ⊳ A, A A ⊳ A. It is well-known that such an object is a model of untyped lambda calculus (see, e.g. =-=[21]-=-). A category where the search for non-trivial reflexive objects often leads to success is that of (directed-)complete partial orders (CPOs) and Scott-continuous maps. Given a reflexive object A in th... |

80 |
The effective topos
- Hyland
- 1982
(Show Context)
Citation Context ...ical setting in which certain aspects of recursion theory are reflected in a particularly nice way. This led to the discovery of the Recursive Topos by Mulry [32] and of the Effective Topos by Hyland =-=[19]-=-, as well as to the study of various categories of domains and effective domains (see [1] for an overview and references). The second line is a more low-level approach: one tries to find minimal categ... |

60 |
Realizability Toposes and Language Semantics
- Longley
- 1994
(Show Context)
Citation Context ...t U ⊆ ω codes a continuous function Pω −→ Pω. Among the continuous functions, there are those which are effective, i.e. whose modulus of continuity is, after coding, an r.e. subset. It turns out (see =-=[24]-=-), that the r.e. subsets of Pω form a sub-PCA of the Graph model. Finally, one may consider an extension T of combinatory logic, or of the lambda calculus; this always gives rise to a pair of syntacti... |

55 |
Categories of partial maps
- Robinson, Rosolini
- 1988
(Show Context)
Citation Context ...n categories, as well as techniques for constructing such categories. Also, better understanding of how partiality can be treated in categorical settings, due to the work by Rosolini, and Robinson in =-=[39, 37]-=- allowed for some improvements (mainly, [40]). Recently, a few other people have taken up this line of research: for example, in [45], the possibility of formulating notions of relative computability ... |

45 |
Continuity and Effectiveness in Topoi
- Rosolini
- 1986
(Show Context)
Citation Context ...n categories, as well as techniques for constructing such categories. Also, better understanding of how partiality can be treated in categorical settings, due to the work by Rosolini, and Robinson in =-=[39, 37]-=- allowed for some improvements (mainly, [40]). Recently, a few other people have taken up this line of research: for example, in [45], the possibility of formulating notions of relative computability ... |

43 |
Classical Recursion Theory, volume 125
- Odifreddi
- 1989
(Show Context)
Citation Context ...-known result that every model of the untyped lambda calculus may be realized as a reflexive 1A more elaborate and detailed account of various approaches as well as further references can be found in =-=[33]-=-. 2In fact, this was not formulated quite in this manner: it was shown that there was an object in the category such that the global sections of that object form a combinatory algebra in Sets. See als... |

30 | The Partial Lambda-Calculus - Moggi - 1988 |

24 | Local realizability toposes and a modal logic for computability
- Awodey, Birkedal, et al.
(Show Context)
Citation Context ...use of simplicity of the exposition and because the general version can easily be deduced when the necessity would ever arise. Relative realizability via pairs of PCAs was introduced by Awodey et.al. =-=[3]-=- and further studied by Birkedal and van Oosten (see [6]). In [17], it was recognized that it was the extra generality of having a relative PCA (there called a filter of designated truth-values) which... |

15 |
Dominical categories: recursion theory without elements
- Paola, Heller
- 1987
(Show Context)
Citation Context ...in Sets. See also the remarks in Section 4.4. 3sobject in a cartesian closed category. Much different in flavor is the well-known 1987 paper “Dominical categories: recursion theory without elements”, =-=[35]-=-, in which Roberto Di Paola and Alex Heller introduced a class of categories called recursion categories and showed how in these categories the basic aspects of recursion theory could be developed. Th... |

14 | F.: A functorial semantics for multi-algebras and partial algebras, with applications to syntax
- Corradini, Gadducci
- 2002
(Show Context)
Citation Context ...e of a partial equational theory; such theories are based on the term logic for cartesian restriction categories. Logics of this kind have been studied in various guises and levels of generality (see =-=[29, 34, 10, 42]-=-, just to name a few). One may consider models of PCL in any cartesian restriction category, and a model will be a Partial Combinatory Algebra (PCA), to be discussed in the next section. The classifyi... |

14 |
Generalized Banach-Mazur functionals in the topos of recursive sets
- Mulry
- 1982
(Show Context)
Citation Context ...in the first, the aim is to find a categorical setting in which certain aspects of recursion theory are reflected in a particularly nice way. This led to the discovery of the Recursive Topos by Mulry =-=[32]-=- and of the Effective Topos by Hyland [19], as well as to the study of various categories of domains and effective domains (see [1] for an overview and references). The second line is a more low-level... |

13 |
Algebraically generalized recursive function theory
- Strong
- 1968
(Show Context)
Citation Context ... computation on various kinds of abstract structures (e.g. [14, 13, 41]) 1 . In connection to the present work, we mention in particular Uniformly Reflexive Structures, developed by Wagner and Strong =-=[48, 46]-=-, and Moschovakis’ work on abstract computability [30] and (pre)computation theories [31]. Essentially, both approaches (which differ in presentation but mostly agree in content) single out a class of... |

12 | Notes on partial combinatory algebras
- Bethke
- 1988
(Show Context)
Citation Context ...mulation. This includes the homomorphisms of PCAs used in the model-theoretic context. Thus, any inclusion of PCAs (preserving a chosen basis) is a simulation, as is any quotient map (in the sense of =-=[4]-=-). The following Lemma gives an (atypical) example of how a morphism of PCAs can arise. It will be of use in Section 5.2. Lemma 5.3. Let F : C −→ D be a cartesian restriction functor and A = (A, •) a ... |

12 |
N.: Abstract first order computability I
- Moschovakis
- 1969
(Show Context)
Citation Context ... [14, 13, 41]) 1 . In connection to the present work, we mention in particular Uniformly Reflexive Structures, developed by Wagner and Strong [48, 46], and Moschovakis’ work on abstract computability =-=[30]-=- and (pre)computation theories [31]. Essentially, both approaches (which differ in presentation but mostly agree in content) single out a class of well-behaved combinatory structures and prove the ele... |

11 |
Restriction categories II: Partial map classification, Theoret
- Cockett, Lack
(Show Context)
Citation Context ... we introduce the general categorical setting in which we will be working, and fix some terminology and notation. All of the material discussed here can be found in more complete and detailed form in =-=[8, 9]-=-. 6sR.1 ff = f R.2 fg = gf whenever dom(f) = dom(g) R.3 gf = gf whenever dom(f) = dom(g) R.4 gf = fgf whenever cod(f) = dom(g) Table 1: Axioms for a restriction combinator 2.1 Categories of partial ma... |

10 |
Partiality, cartesian closedness and toposes
- Curien, Obtulowitz
- 1989
(Show Context)
Citation Context ...viewed as the study of cartesian closed (total) Turing categories. Also, we note that, more generally, one could work in a partial cartesian closed Turing category (for more on this notion, see, e.g. =-=[11]-=-, or [43]), and look for reflexive objects in there; those correspond to models of the untyped partial lambda calculus (see [29]). Hence in general cartesian closed Turing categories will correspond t... |

10 |
An existence theorem for recursive categories
- Heller
- 1990
(Show Context)
Citation Context ...their work to other categorytheoretic approaches to computability - out of 42 references in [35], only one is about category theory: MacLane’s CWM. Later publications by Heller, Di Paola and Montagna =-=[16, 36]-=-, and also by Lengyel [22], presented new examples of recursion categories, as well as techniques for constructing such categories. Also, better understanding of how partiality can be treated in categ... |

8 | Relative and modified relative realizability
- Birkedal, Oosten
- 2002
(Show Context)
Citation Context ...ral version can easily be deduced when the necessity would ever arise. Relative realizability via pairs of PCAs was introduced by Awodey et.al. [3] and further studied by Birkedal and van Oosten (see =-=[6]-=-). In [17], it was recognized that it was the extra generality of having a relative PCA (there called a filter of designated truth-values) which allows for describing realizability toposes purely in t... |

8 |
Cartesian closed categories of enumerations for effective typestructures
- Longo, Moggi
(Show Context)
Citation Context ...h is of interest to us; it started with the pioneering work by Eilenberg and Elgot [12], who first investigated recursiveness from the point of view of elementary category theory. Longo and Moggi, in =-=[27, 26]-=-, made significant contributions to this programme. While their main motivation seems to have been the development of categorical settings for the study of computability at higher types (as opposed to... |

8 | Partial Horn logic and cartesian categories
- Palmgren, Vickers
(Show Context)
Citation Context ...e of a partial equational theory; such theories are based on the term logic for cartesian restriction categories. Logics of this kind have been studied in various guises and levels of generality (see =-=[29, 34, 10, 42]-=-, just to name a few). One may consider models of PCL in any cartesian restriction category, and a model will be a Partial Combinatory Algebra (PCA), to be discussed in the next section. The classifyi... |

8 |
Uniformly reflexive structures: on the nature of Gödelizations and relative computability
- Wagner
- 1969
(Show Context)
Citation Context ... computation on various kinds of abstract structures (e.g. [14, 13, 41]) 1 . In connection to the present work, we mention in particular Uniformly Reflexive Structures, developed by Wagner and Strong =-=[48, 46]-=-, and Moschovakis’ work on abstract computability [30] and (pre)computation theories [31]. Essentially, both approaches (which differ in presentation but mostly agree in content) single out a class of... |

5 |
More existence theorems for recursion categories, Annals of Pure and Applied Logic 125
- Lengyel
- 2004
(Show Context)
Citation Context ...eoretic approaches to computability - out of 42 references in [35], only one is about category theory: MacLane’s CWM. Later publications by Heller, Di Paola and Montagna [16, 36], and also by Lengyel =-=[22]-=-, presented new examples of recursion categories, as well as techniques for constructing such categories. Also, better understanding of how partiality can be treated in categorical settings, due to th... |

5 | Classifying categories for partial equational logic
- Schroder
- 2003
(Show Context)
Citation Context ...e of a partial equational theory; such theories are based on the term logic for cartesian restriction categories. Logics of this kind have been studied in various guises and levels of generality (see =-=[29, 34, 10, 42]-=-, just to name a few). One may consider models of PCL in any cartesian restriction category, and a model will be a Partial Combinatory Algebra (PCA), to be discussed in the next section. The classifyi... |

4 |
Categorical simulations
- Cockett, Hofstra
(Show Context)
Citation Context ... h is then called a realizer for f. One is tempted to call such structure a Kleene category, since they are more suited for purposes of realizability than for computability (this will be discussed in =-=[7]-=-). Many things which are true for Turing categories are already true for Kleene categories. For example, in a Kleene category each object is a retract of the Kleene object, and hence one can normalize... |

4 |
On axiomatizing recursion theory
- Fenstad
- 1974
(Show Context)
Citation Context ...n). Or, if one takes closure properties of the recursive functions such as enumeration and parametrization as fundamental, then one is led to computation on various kinds of abstract structures (e.g. =-=[14, 13, 41]-=-) 1 . In connection to the present work, we mention in particular Uniformly Reflexive Structures, developed by Wagner and Strong [48, 46], and Moschovakis’ work on abstract computability [30] and (pre... |

4 |
editor. Handbook of Computability Theory, volume 140
- Griffor
- 1999
(Show Context)
Citation Context ...tural numbers which one wants to abstract. For 2sexample, one may replace the natural numbers by ordinals and generalize recursion theory to recursion on ordinals or admissible sets (see Chapter 3 in =-=[15]-=- for an overview, or [44] for an introduction). Or, if one takes closure properties of the recursive functions such as enumeration and parametrization as fundamental, then one is led to computation on... |

4 | Oosten Ordered partial combinatory algebras
- Hofstra, van
- 2006
(Show Context)
Citation Context ...t commutes. wφ×φ A 2 �� B 2 It is easily verified that relative PCAs in a fixed category C and simulations form a category, denoted PCA = PCA(C). (For the absolute case, proofs can be found in, e.g., =-=[24, 18]-=-; the relative case is hardly different and can be found in [17].) The category PCA is preorder-enriched. For two parallel simulations φ, ψ : (A, V) −→ (B, W), we define φ ⊢ ψ if and only if there exi... |

4 | On the ubiquity of certain total type structures
- Longley
- 2007
(Show Context)
Citation Context ...lt), one actually needs a relativised notion of computability based on pairs of PCAs; this seems to be more than a coincidence, since a similar construction has appeared in realizability, and also in =-=[25]-=-, in the context of computability at higher types. The correspondence between Turing categories and PCAs provides insight in both directions. On the one hand, examples of PCAs give rise to examples of... |

4 |
Axioms for computation theories
- Moschovakis
- 1969
(Show Context)
Citation Context ... the present work, we mention in particular Uniformly Reflexive Structures, developed by Wagner and Strong [48, 46], and Moschovakis’ work on abstract computability [30] and (pre)computation theories =-=[31]-=-. Essentially, both approaches (which differ in presentation but mostly agree in content) single out a class of well-behaved combinatory structures and prove the elementary recursion theoretic results... |

3 |
de Vrijer, Completing partial combinatory algebras with unique headnormal forms
- Bethke, Klop, et al.
- 1996
(Show Context)
Citation Context ...xample of a PCA is Kleene’s model K1; its underlying set is N, and application is Kleene-application as defined in the previous section. Combinators k and s are found using the Parameter Theorem (see =-=[5]-=-) for a precise account). Of course, Comp(K1) is the classical recursion category (first example in the previous section). Oracles. One may generalize this example by considering computability relativ... |

3 |
Some properties of the syntactic p-recursion categories generated by consistent, recursively enumerable extensions of Peano arithmetic
- Paola, Montagna
- 1991
(Show Context)
Citation Context ...their work to other categorytheoretic approaches to computability - out of 42 references in [35], only one is about category theory: MacLane’s CWM. Later publications by Heller, Di Paola and Montagna =-=[16, 36]-=-, and also by Lengyel [22], presented new examples of recursion categories, as well as techniques for constructing such categories. Also, better understanding of how partiality can be treated in categ... |

3 | The logic of the partial λ-calculus with equality
- Schröder
- 2004
(Show Context)
Citation Context ... the study of cartesian closed (total) Turing categories. Also, we note that, more generally, one could work in a partial cartesian closed Turing category (for more on this notion, see, e.g. [11], or =-=[43]-=-), and look for reflexive objects in there; those correspond to models of the untyped partial lambda calculus (see [29]). Hence in general cartesian closed Turing categories will correspond to partial... |

2 |
Fundamentals of Generalized Recursion Theory, volume 105 of Studies in Logic. North-Holland
- Fitting
- 1981
(Show Context)
Citation Context ...n). Or, if one takes closure properties of the recursive functions such as enumeration and parametrization as fundamental, then one is led to computation on various kinds of abstract structures (e.g. =-=[14, 13, 41]-=-) 1 . In connection to the present work, we mention in particular Uniformly Reflexive Structures, developed by Wagner and Strong [48, 46], and Moschovakis’ work on abstract computability [30] and (pre... |

2 |
All realizability is relative
- Hofstra
- 2006
(Show Context)
Citation Context ...on can easily be deduced when the necessity would ever arise. Relative realizability via pairs of PCAs was introduced by Awodey et.al. [3] and further studied by Birkedal and van Oosten (see [6]). In =-=[17]-=-, it was recognized that it was the extra generality of having a relative PCA (there called a filter of designated truth-values) which allows for describing realizability toposes purely in terms of th... |

2 |
Some reasons for generalizing recursion theory
- Kreisel
- 1971
(Show Context)
Citation Context ...s. In particular, various researchers started investigating to which extent one could replace the natural numbers by other structures and thus generalize recursion theory to wider settings (see, e.g. =-=[20]-=- for a lively discussion of why one would do such a thing). Such a generalization can take different forms: one typically starts by isolating certain structural aspects of recursive functions and the ... |

2 |
G��del numberings, principal morphisms, combinatory algebras" Prelim. version of this paper
- Longo, Moggi
- 1984
(Show Context)
Citation Context ...h is of interest to us; it started with the pioneering work by Eilenberg and Elgot [12], who first investigated recursiveness from the point of view of elementary category theory. Longo and Moggi, in =-=[27, 26]-=-, made significant contributions to this programme. While their main motivation seems to have been the development of categorical settings for the study of computability at higher types (as opposed to... |

2 |
A category theoretic characterization of functional completeness
- Longo, Moggi
- 1990
(Show Context)
Citation Context ...rization. Moreover, they studied these concepts in the context of models of the lambda calculus (being mainly interested in type structures, they concentrated on the cartesian closed case). Later, in =-=[28]-=-, this was generalized to arbitrary combinatory algebras. One of their ideas was to show that every combinatory algebra gave rise to a category with certain properties (which embody the notions of Göd... |

2 |
Representation theorems for special p-categories
- Rosolini
- 1988
(Show Context)
Citation Context ...ting such categories. Also, better understanding of how partiality can be treated in categorical settings, due to the work by Rosolini, and Robinson in [39, 37] allowed for some improvements (mainly, =-=[40]-=-). Recently, a few other people have taken up this line of research: for example, in [45], the possibility of formulating notions of relative computability in these settings is investigated, while [23... |

2 |
A relativization mechanism in recursion categories
- Stefani
- 1993
(Show Context)
Citation Context ...gorical settings, due to the work by Rosolini, and Robinson in [39, 37] allowed for some improvements (mainly, [40]). Recently, a few other people have taken up this line of research: for example, in =-=[45]-=-, the possibility of formulating notions of relative computability in these settings is investigated, while [23] adds to the study of recursion categories with specific additional structure (which is ... |

2 | A general form of relative recursion
- Oosten
- 2006
(Show Context)
Citation Context ...d G(q) are different qua lambda models they are isomorphic qua PCAs under simulation. See [24] for details. 6. Equivalences via lax simulations are weaker than via strict simulations; for example, in =-=[47]-=- an example of an equivalence between a total and a non-total PCA was given. This is impossible via strict simulations. 5.2 Equivalences We now have a notion of equivalence of (relative) PCAs; intuiti... |

1 |
Handbook for Logic in Computer Science, volume 3, chapter Domain theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...nice way. This led to the discovery of the Recursive Topos by Mulry [32] and of the Effective Topos by Hyland [19], as well as to the study of various categories of domains and effective domains (see =-=[1]-=- for an overview and references). The second line is a more low-level approach: one tries to find minimal categorical settings which embody certain elementary recursion theoretic phenomena. It is this... |

1 |
Asperti and Agata Ciabattoni. Effective applicative structures
- Andrea
- 1995
(Show Context)
Citation Context ... proved in this setting. Effective applicative structures. When, in the definition of a BRFT, one omits the requirements that the decision functions be present, one arrives at what Asperti and Longo (=-=[2]-=-) call an effective applicative structure. It is easily seen that this notion coincides with that of a combinatory complete applicative structure which is non-trivial (because we insist on the underly... |

1 | Locally connected recursion categories
- Lengyel
- 2006
(Show Context)
Citation Context ...40]). Recently, a few other people have taken up this line of research: for example, in [45], the possibility of formulating notions of relative computability in these settings is investigated, while =-=[23]-=- adds to the study of recursion categories with specific additional structure (which is needed for the formulation of certain recursiontheoretic concepts). Despite the evident close relationship betwe... |

1 | An abstract look at realizability
- Robinson, Rosolini
(Show Context)
Citation Context ...stingly, these can be described in terms of morphisms of generalized categories of assemblies (which, in turn, are constructed using a generalization of the F-construction by Robinson and Rosolini in =-=[38]-=-). It can then be shown that the 2-category of relative PCAs in a given base category is equivalent to the 2-category of Turing categories over the base. Of course, one of the central points of Turing... |

1 |
Reflexive structures
- Sanchis
- 1988
(Show Context)
Citation Context ...n). Or, if one takes closure properties of the recursive functions such as enumeration and parametrization as fundamental, then one is led to computation on various kinds of abstract structures (e.g. =-=[14, 13, 41]-=-) 1 . In connection to the present work, we mention in particular Uniformly Reflexive Structures, developed by Wagner and Strong [48, 46], and Moschovakis’ work on abstract computability [30] and (pre... |

1 |
course on admissible recursion theory
- Short
(Show Context)
Citation Context ...ants to abstract. For 2sexample, one may replace the natural numbers by ordinals and generalize recursion theory to recursion on ordinals or admissible sets (see Chapter 3 in [15] for an overview, or =-=[44]-=- for an introduction). Or, if one takes closure properties of the recursive functions such as enumeration and parametrization as fundamental, then one is led to computation on various kinds of abstrac... |