## A Combinatory Algebra for Sequential Functionals of Finite Type (1997)

Venue: | University of Utrecht |

Citations: | 21 - 2 self |

### BibTeX

@TECHREPORT{Oosten97acombinatory,

author = {Jaap Van Oosten},

title = {A Combinatory Algebra for Sequential Functionals of Finite Type},

institution = {University of Utrecht},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

It is shown that the type structure of finite-type functionals associated to a combinatory algebra of partial functions from IN to IN (in the same way as the type structure of the countable functionals is associated to the partial combinatory algebra of total functions from IN to IN), is isomorphic to the type structure generated by object N (the flat domain on the natural numbers) in Ehrhard's category of "dI-domains with coherence", or his "hypercoherences". AMS Subject Classification: Primary 03D65, 68Q55 Secondary 03B40, 03B70, 03D45, 06B35 Introduction PCF , "Godel's T with unlimited recursion", was defined in Plotkin's paper [16]. It is a simply typed -calculus with a type o for integers and constants for basic arithmetical operations, definition by cases and fixed point recursion. More importantly, there is a special reduction relation attached to it which ensures (by Plotkin's "Activity Lemma") that all PCF -definable higher-type functionals have a sequential, i.e. non-paral...

### Citations

397 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ...or his "hypercoherences". AMS Subject Classification: Primary 03D65, 68Q55 Secondary 03B40, 03B70, 03D45, 06B35 Introduction PCF , "Godel's T with unlimited recursion", was defined=-= in Plotkin's paper [16]-=-. It is a simply typed -calculus with a type o for integers and constants for basic arithmetical operations, definition by cases and fixed point recursion. More importantly, there is a special reducti... |

197 |
On full abstraction for PCF
- Hyland, Ong
(Show Context)
Citation Context ...an important overlap with higher-type recursion theory as it was recognized (I believe, initially by Robin Gandy, whose insights were transmitted by Martin Hyland and partially laid down in the paper =-=[9]) that Kle-=-ene's late attempts ([10, 11, 12, 13]) to formalize the notion of a recursive functional of higher type, had much in common with the "full abstraction problem" for PCF . As far as I am aware... |

80 |
The effective topos
- Hyland
- 1982
(Show Context)
Citation Context ... oe j oe typeg of section 2, and hence the corresponding part of Ehrhard's Hypercoherences, fits into the realizability topos generated by the combinatory algebra B (for realizability toposes consult =-=[8, 6]-=-). Let us call it Eff(B). An 10 important subcategory (the subcategory of ::-separated objects) of Eff(B) can be described as follows: Let B-Set be the category with objects pairs (X; EX ) where X is ... |

66 | Malacaria P., “Full Abstraction for PCF (extended abstract
- Abramsky, Jagadeesan
- 1994
(Show Context)
Citation Context ...ructure is intrinsic. This should be of interest to Synthetic Domain Theory ([7]). Thirdly it raises the question whether maybe more models of PCF (including the fully abstract game models of [9] and =-=[1]-=-)can be induced in this way by combinatory algebras. Finally I should admit to an oversight: one of the stumbling blocks for me in carrying out the analysis reported in this paper, was my initial fail... |

57 |
Recursive functionals and quantifiers of finite types
- Kleene
- 1959
(Show Context)
Citation Context ...ype recursion theory as it was recognized (I believe, initially by Robin Gandy, whose insights were transmitted by Martin Hyland and partially laid down in the paper [9]) that Kleene's late attempts (=-=[10, 11, 12, 13]) to forma-=-lize the notion of a recursive functional of higher type, had much in common with the "full abstraction problem" for PCF . As far as I am aware however, the exact relationship between Kleene... |

53 |
First steps in synthetic domain theory
- Hyland
- 1991
(Show Context)
Citation Context ...re. Secondly, it shows that Ehrhard's "strongly stable" model of PCF lives inside a realizability topos where its domain structure is intrinsic. This should be of interest to Synthetic Domai=-=n Theory ([7]-=-). Thirdly it raises the question whether maybe more models of PCF (including the fully abstract game models of [9] and [1])can be induced in this way by combinatory algebras. Finally I should admit t... |

52 |
Sequentiality and Strong Stability
- Bucciarelli, Ehrhard
- 1991
(Show Context)
Citation Context ...rom IN to IN (yielding a total combinatory algebra), gives a type structure of higher-type functionals which coincides with the relevant part of Ehrhard and Bucciarelli's "dI-domains with coheren=-=ce" ([3, 4, 2]-=-). Research supported by the Dutch National Research Foundation NWO 1 This could be interesting for a number of reasons. First, it provides another handle on Ehrhard's work, which is complicated and r... |

22 | Projecting sequential algorithms on strongly stable functions
- Ehrhard
- 1996
(Show Context)
Citation Context ...rom IN to IN (yielding a total combinatory algebra), gives a type structure of higher-type functionals which coincides with the relevant part of Ehrhard and Bucciarelli's "dI-domains with coheren=-=ce" ([3, 4, 2]-=-). Research supported by the Dutch National Research Foundation NWO 1 This could be interesting for a number of reasons. First, it provides another handle on Ehrhard's work, which is complicated and r... |

20 |
The System F of Variable Types
- Girard
- 1986
(Show Context)
Citation Context ...every such pair (B; v) defines a function F which is given as F ff for some (non-unique) ff. We call the pair (B; v) the trace of F . This is in harmony with usage in the literature of this term, cf. =-=[5]-=-. When is a base B of a continuous function the set of leaves of a sequential tree? Answer: Proposition 1.4 B is the set of leaves of a sequential tree if and only if for each nonempty finite subset B... |

1 |
A relative PCF-definability result, manuscript
- Ehrhard
- 1996
(Show Context)
Citation Context ...rom IN to IN (yielding a total combinatory algebra), gives a type structure of higher-type functionals which coincides with the relevant part of Ehrhard and Bucciarelli's "dI-domains with coheren=-=ce" ([3, 4, 2]-=-). Research supported by the Dutch National Research Foundation NWO 1 This could be interesting for a number of reasons. First, it provides another handle on Ehrhard's work, which is complicated and r... |

1 |
Recursion on the Countable Functionals
- Norman
- 1980
(Show Context)
Citation Context ...ionship between Kleene's work and the work on PCF still remains to be clarified. An important model of Kleene's axioms is provided by the so-called "continuous (or countable) functionals" (s=-=ee, e.g., [15]). They ar-=-ise, in a standard way, as the type structure coming from the partial combinatory algebra of "Kleene's function realizability" (introduced in [14]). This is a partial combinatory algebra str... |

1 |
A New Category?, manuscript
- Scott
- 1996
(Show Context)
Citation Context ...theories of domains saw the light, inducing the father of domain theory, Dana Scott, to lament that "there are too many proposed categories of domains and [: : : ] their study has become too arca=-=ne" ([17]-=-), a judgement with which it is hard to disagree. Although most interest in the semantics of PCF was shown by computer scientists, it became clear that there is an important overlap with higher-type r... |