## Applications of the Kleene-Kreisel Density Theorem to Theoretical Computer Science (2006)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Normann06applicationsof,

author = {Dag Normann},

title = {Applications of the Kleene-Kreisel Density Theorem to Theoretical Computer Science},

year = {2006}

}

### OpenURL

### Abstract

The Kleene-Kreisel density theorem is one of the tools used to investigate the denotational semantics of programs involving higher types. We give a brief introduction to the classical density theorem, then show how this may be generalized to set theoretical models for algorithms accepting real numbers as inputs and finally survey some recent applications of this generalization. 1

### Citations

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Citation Context ...archy, we use domain theory, initiated by Scott [31] and by Ershov [9]. We will start with a brief introduction to domain theory. See Stoltenberg-Hansen, Lindström and Griffor [33], Abramsky and Jung =-=[1]-=- or Amadiou and Curien [2] for detailed introductions. 2.1 Scott-domains A complete partial ordering, a cpo for short, is a partial ordering (X, ⊑) such that each directed subset has a least upper bou... |

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235 |
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Citation Context ...ves that for each compact a of type σ there is a total Φ of type σ such that a is consistent with (Φ)m, where m ≥ n is as in Lemma 10. 2.5 Kleene-schemes The Kleene schemes S1 - S9 were introduced in =-=[12]-=-. They can be seen as nine clauses in a grand inductive definition defining the relation {e}(φ1, . . . , φk} � a where each φi will be a functional of pure type and a ∈ N. The interpretation of these ... |

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Citation Context ..., but only indications of proofs. For a general overview of the interplay between the CT-approaches and the TCS-approaches to computations in higher types in general, see Normann [20]. In 1959 Kleene =-=[13]-=- and Kreisel [14] introduced what is now known as the Kleene-Kreisel continuous functionals. The motivation behind the two papers were different, the two approaches were different, and in fact, the tw... |

30 |
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Citation Context ...sed P CF - term t of base type is interpreted as an element �= ⊥, then we may rewrite t to the constant for the interpretation using the conversion rules. The proof of 1. goes back to Platek’s thesis =-=[27]-=-, where it is proved in the discontinuous case. A proof can also be found in Moldestad [18]. 2. is also known since Platek [27], and is observed in Scott [31] and Plotkin [28]. It is relevant to us to... |

27 |
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Citation Context ... the density theorem. We will not discuss the details here. The Scott model is not fully abstract for P CF . This means that there are compacts in the Scott model that are not P CF -definable. Milner =-=[17]-=- showed that there exist one, and up to isomorphism, only one fully abstract model for P CF consisting of algebraic domains. Of course we may define the class of hereditarily total objects in Milner’s... |

23 | Lazy Functional Algorithms for Exact Real Functionals
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Citation Context ...ct Cf = {g ∈ N N | g ≤ f} in N N . Independently, Berger [4] showed that there is a representative Φ ∈ H(3) for the fan functional that is S1 - S9 - computable in the Scott - hierarchy sense. Simpson =-=[32]-=- used this to show that we may write a P CF -program for Riemann integration if the reals are represented by the elements of a given σ-compact subset of N N . We will not go into detail, but the point... |

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Citation Context ...tency (boundedness) is an equivalence relation on pairs from H(σ) b) Each compact a ∈ D(σ) has an extension to an element of H(σ). The density theorem is proved over and over again in the literature, =-=[13, 14, 9, 4]-=- and in surveys in general. In the proof, a) and b) are proved by simultaneous induction on the type. In order to prove b) one proves that for each compact a of type σ there is a total Φ of type σ suc... |

22 |
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Citation Context ...tions of proofs. For a general overview of the interplay between the CT-approaches and the TCS-approaches to computations in higher types in general, see Normann [20]. In 1959 Kleene [13] and Kreisel =-=[14]-=- introduced what is now known as the Kleene-Kreisel continuous functionals. The motivation behind the two papers were different, the two approaches were different, and in fact, the two concepts of cou... |

21 |
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Citation Context ...s model, we may use this representation, its higher type semantics and a P CF -like language to give a semantics to typed computations involving reals. This idea is discussed further in DiGianantonio =-=[6, 7, 8]-=- and Simpson [32]. 18sLet us define the hereditarily total functionals REPσ with the equivalence relation ∼σ for each type σ: Definition 21 We let R I (0) be as above, and define R I (σ) for all finit... |

16 | The continuous functionals of finite type over the reals
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Citation Context ... elements of type σ, the equivalence ≈ R σ on HR(σ) and the extensional collapse hierarchy {Ct R(σ)}σ type with the corresponding identification functions ρ R σ . Details may be found e.g. in Normann =-=[23]-=-. It is easy to see that each Ct R(σ) is organized into a topological vectorspace in a natural way. By induction on the type, we also see that each compact p in R(σ) will determine a closed, convex su... |

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Citation Context ...ice of representation of reals as data will have any influence on what is considered to be a continuous, total functional of higher type. The problem was first addressed by Bauer, Escardó and Simpson =-=[3]-=-, and they obtained some partial results. Normann [26] showed that the coincidence problem is equivalent to a topological problem about the KleeneKreisel functionals: Is the topology generated by all ... |

14 |
Computable and recursively countable functions of higher type
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Citation Context ...e, this has a dramatical effect on the computational power of S1 - S9. Tait [34] observed that the fan functional is not S1 - S9 - computable over the Kleene-Kreisel functionals. Later, Martin Hyland =-=[10, 11]-=- showed that the functional Γ, defined by Gandy, is not S1 - S9 - computable relative to the fan functional. Finally, Normann [19] showed that for any type k ≥ 3 and any continuous Φ of type k there i... |

13 | Computability over the partial continuous functionals
- Normann
(Show Context)
Citation Context ...eisel functionals that have computable associates also have hereditarily total representatives that are P CF -computable. It turned out that this is true in a rather strong sense: Theorem 12 (Normann =-=[21]-=-) Let σ be a finite type. There is a P CF -definable function EV AL of type (ι → ι) → σ such that whenever a ∈ H(σ) and f enumerates (the Gödel numbers of) the set of compacts in D(σ) bounded by a, th... |

10 |
An abstract data type for real numbers
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Citation Context ...s model, we may use this representation, its higher type semantics and a P CF -like language to give a semantics to typed computations involving reals. This idea is discussed further in DiGianantonio =-=[6, 7, 8]-=- and Simpson [32]. 18sLet us define the hereditarily total functionals REPσ with the equivalence relation ∼σ for each type σ: Definition 21 We let R I (0) be as above, and define R I (σ) for all finit... |

9 |
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Citation Context ...on H(σ) as follows: 1. ≈ι and ≈o are the identity relations on N and B respectively. 2. If σ = τ → δ and f and g are in H(σ), we let f ≈σ g ⇔ ∀x ∈ H(τ)∀y ∈ H(τ)(x ≈τ y ⇒ f(x) ≈ g(y)). Longo and Moggi =-=[16]-=- observed that we with ease may prove that x ≈σ y ⇔ x ⊓ y ∈ H(σ) when x and y are in H(σ). It then follows that ≈σ is an equivalence relation, something that had been seen as a consequence of the dens... |

8 | Real number computability and domain theory
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(Show Context)
Citation Context ...s model, we may use this representation, its higher type semantics and a P CF -like language to give a semantics to typed computations involving reals. This idea is discussed further in DiGianantonio =-=[6, 7, 8]-=- and Simpson [32]. 18sLet us define the hereditarily total functionals REPσ with the equivalence relation ∼σ for each type σ: Definition 21 We let R I (0) be as above, and define R I (σ) for all finit... |

8 |
Full abstraction, totality and
- Plotkin
- 1999
(Show Context)
Citation Context ...cts in Milner’s model as well, consistency will be an equivalence relation on these objects, so we may even form the extensional collapse of the hereditarily total elements of Milner’s model. Plotkin =-=[29]-=- showed as a corollary of Theorem 12 that this construction leads us to the Kleene-Kreisel continuous functionals. Thus Milner’s construction is an alternative characterization. Recently Longley [15] ... |

8 |
Admissible representations of limit spaces
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- 2000
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Citation Context ...aches coincide, i.e. if the typed structures defined via these approaches are the same. 19sDue to characterizations of the extensional hierarchy (Normann [23]) and the T T E-based hierarchy (Schröder =-=[30]-=-) as the one obtained from the reals in the category of limit spaces, these two approaches are known to be equivalent. Thus the real coincidence problem is if the domain theoretical approaches based o... |

6 |
Hierarchies of Total Functionals over the Reals, Theoretical Computer Science 316
- Normann
- 2004
(Show Context)
Citation Context ...or each type σ = τ → δ, each φ ∈ Ct(σ) and a ∈ Ct(τ) we have that πσ(φ)(πτ(a)) = πδ(φ). 20sThis theorem, and an analogue result for the hierarchy based on the intensional representation, is proved in =-=[24]-=- We will not give the proof here, but discuss where the density theorem is used. Ideally we would like to replace the (possible) use of a (nonexistent) continuous projection from CtR(σ) to Ct(σ) with ... |

6 |
A type-theoretical alternative to ISWIM, CUCH, OWHY. Theoretical Computer Science, 121:422–440
- Scott
- 1993
(Show Context)
Citation Context ...ain gap. In CS it is important to model partiality, so the main structure is a hierarchy of partial continuous functionals. In order to define this hierarchy, we use domain theory, initiated by Scott =-=[31]-=- and by Ershov [9]. We will start with a brief introduction to domain theory. See Stoltenberg-Hansen, Lindström and Griffor [33], Abramsky and Jung [1] or Amadiou and Curien [2] for detailed introduct... |

5 | Comparing hierarchies of total functionals
- Normann
(Show Context)
Citation Context ...influence on what is considered to be a continuous, total functional of higher type. The problem was first addressed by Bauer, Escardó and Simpson [3], and they obtained some partial results. Normann =-=[26]-=- showed that the coincidence problem is equivalent to a topological problem about the KleeneKreisel functionals: Is the topology generated by all continuous functions φ : Ct(n) → R zero dimensional fo... |

4 |
Maximal and everywhere defined functionals
- Ershov
- 1974
(Show Context)
Citation Context ...s important to model partiality, so the main structure is a hierarchy of partial continuous functionals. In order to define this hierarchy, we use domain theory, initiated by Scott [31] and by Ershov =-=[9]-=-. We will start with a brief introduction to domain theory. See Stoltenberg-Hansen, Lindström and Griffor [33], Abramsky and Jung [1] or Amadiou and Curien [2] for detailed introductions. 2.1 Scott-do... |

4 | On the ubiquity of certain total type structures
- Longley
- 2007
(Show Context)
Citation Context ... [29] showed as a corollary of Theorem 12 that this construction leads us to the Kleene-Kreisel continuous functionals. Thus Milner’s construction is an alternative characterization. Recently Longley =-=[15]-=- showed that under reasonable general assumptions constructions of an extensional hierarchy of total continuous functionals containing N N at type 1 tends to end up with the Kleene-Kreisel functionals... |

4 |
Continuity properties of partial recursive functionals of finite type
- Tait
(Show Context)
Citation Context ... . , φn) is total, while this is relaxed in the Scott hierarchy since non-total functionals are present there. As we will see, this has a dramatical effect on the computational power of S1 - S9. Tait =-=[34]-=- observed that the fan functional is not S1 - S9 - computable over the Kleene-Kreisel functionals. Later, Martin Hyland [10, 11] showed that the functional Γ, defined by Gandy, is not S1 - S9 - comput... |

3 |
Recursion on the countable functionals
- Hyland
- 1979
(Show Context)
Citation Context ...e, this has a dramatical effect on the computational power of S1 - S9. Tait [34] observed that the fan functional is not S1 - S9 - computable over the Kleene-Kreisel functionals. Later, Martin Hyland =-=[10, 11]-=- showed that the functional Γ, defined by Gandy, is not S1 - S9 - computable relative to the fan functional. Finally, Normann [19] showed that for any type k ≥ 3 and any continuous Φ of type k there i... |

2 |
The continuous functionals: computations, recursions and degrees, Annals of Pure and Applied Logic 21
- Normann
- 1981
(Show Context)
Citation Context ...putable over the Kleene-Kreisel functionals. Later, Martin Hyland [10, 11] showed that the functional Γ, defined by Gandy, is not S1 - S9 - computable relative to the fan functional. Finally, Normann =-=[19]-=- showed that for any type k ≥ 3 and any continuous Φ of type k there is a continuous Ψ of type k that is not S1 - S9 - computable relative to Φ and any continuous functional of lower type. However, if... |

2 | Computing with functionals - computability theory or computer science
- NORMANN
(Show Context)
Citation Context ...me formal definitions, but only indications of proofs. For a general overview of the interplay between the CT-approaches and the TCS-approaches to computations in higher types in general, see Normann =-=[20]-=-. In 1959 Kleene [13] and Kreisel [14] introduced what is now known as the Kleene-Kreisel continuous functionals. The motivation behind the two papers were different, the two approaches were different... |

1 |
Calculabilité sur les réels
- DeJaeger
- 2003
(Show Context)
Citation Context ...be a probability distribution on {1, . . . , n}. Then n� µ(i)ai ∈ V i=1 p σ . We are now going to discuss the density theorem, originally due to Normann [23] but with an alternative proof in DeJaeger =-=[5]-=-: Theorem 15 Let p be a compact in R(σ). Then there is an element a ∈ H R(σ) extending p. Since we have not given any detailed definitions, we will not give a detailed proof. We will however indicate ... |

1 |
The Cook-Berger Problem - A Guide to the Solution
- Normann
- 2000
(Show Context)
Citation Context ...H(σ) and f enumerates (the Gödel numbers of) the set of compacts in D(σ) bounded by a, then EV AL(f) ∈ H(σ) and EV AL(f) ⊑ a. It will lead too far to give the details of the construction, see [21] or =-=[22]-=-. In [22] we only prove the theorem for type 3. There are some obstacles that are not visible at this level. Here we will explain how the density theorem is used. For types ≤ 2 the theorem is quite ea... |

1 |
Definability and reducibility in higher types over the reals, to appear in the proceedings og Logic Colloquium ’03
- Normann
(Show Context)
Citation Context ...nt spaces over domains, old and contemporary literature is too vast for us to be specific about it. It is also well known that Polish spaces will be homeomorphic to Gδsubsets of [0, 1] ω . In Normann =-=[25]-=- we consider finite types where arbitrary Polish spaces may be used as base types, interpreted in the category of limit spaces. We then show that the hierarchy {Ct R(σ)}σ type is adequate for interpre... |