## Infinite sets that admit fast exhaustive search (2007)

Venue: | In Proceedings of the 22nd Annual IEEE Symposium on Logic In Computer Science |

Citations: | 13 - 8 self |

### BibTeX

@INPROCEEDINGS{Escardó07infinitesets,

author = {Martín Escardó},

title = {Infinite sets that admit fast exhaustive search},

booktitle = {In Proceedings of the 22nd Annual IEEE Symposium on Logic In Computer Science},

year = {2007},

pages = {443--452},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Perhaps surprisingly, there are infinite sets that admit mechanical exhaustive search in finite time. We investigate three related questions: What kinds of infinite sets admit mechanical exhaustive search in finite time? How do we systematically build such sets? How fast can exhaustive search over infinite sets be performed? Keywords. Higher-type computability and complexity, Kleene–Kreisel functionals, PCF, Haskell, topology. 1.

### Citations

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Citation Context ... Cτ Cσ , where the product and exponential are calculated in the cartesian closed category of Hausdorff k-spaces. For a brief treatment of k-spaces, also known as compactly generated spaces, see e.g. =-=[14]-=-, and, for a more detailed one, see e.g. [9] or the references contained therein. A set is called clopen if it is both closed and open. Lemma 2.3. For every clopen U ⊆ Cσ there is a total predicate p ... |

479 | Domain theory
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Citation Context ... the products and exponentials are calculated in the cartesian closed category of continuous maps of Scott domains, where a Scott domain is an algebraic, bounded complete, and directed complete poset =-=[1]-=-. Total functionals. For each type σ, define, again by induction, a set Tσ ⊆ Dσ of total functionals and a relation ∼σ on Dσ as follows, where γ ranges over ground types: To = B, Tι = N, x ∼γ y ⇐⇒ x, ... |

405 |
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Citation Context ...definable, and, in order to do that, he first explicitly defined a search functional for the Cantor space. Here PCF is an applied simplytyped lambda-calculus with arithmetic and fixed-point recursion =-=[23, 19]-=-. Berger observed that, for partial functionals, PCF definability coincides with S1–S9 definability. Then Hyland informed the community that Gandy was aware of the S1–S9 definability of the Fan functi... |

275 |
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Citation Context ...ctive totality. Our results hold for the former but fail for the latter. This failure is to be expected: it is well known that, for the hereditarily effective notion, there is no total Fan functional =-=[3]-=-, and hence the set of total elements of the Cantor type cannot be exhaustible. Put another way, the above algorithms for the Fan functional are total in the Kleene–Kreisel sense, but not in the hered... |

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Citation Context ....1 with respect to the cited lemma on compactness of Sierpinskiexhaustible sets [6]. Examples 5.2. (1) There are non-compact, exhaustible sets. (So one needs to assume something such as totality.) By =-=[22, 20]-=-, any second-countable space, e.g. the real line R, can be embedded into the domain D = B ω . But R is a connected space, which is equivalent to saying that every continuous boolean-valued map defined... |

191 | A Natural Semantics for Lazy Evaluation
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Citation Context ...njectures hold, the algorithm developed in this section is faster, in practice, than untrained intuition would concede to be possible. 4.1 The model of computation We consider call-by-need evaluation =-=[15, 18, 30]-=-. For the language PCF, this is semantically equivalent to the call-by-name computational model. The essential operational difference of the call-by-name and the call-by-need computational models is t... |

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Citation Context ...us, and hence so is the composite i ◦ χU ◦ ρσ : Tσ → B, where i: B → B in the inclusion. Because T is dense in D (see e.g. [6]) and because Scott domains, and hence B, are densely injective (see e.g. =-=[12]-=-), by definition of injectivity this extends to a continuous function p: Dσ → B. Then p is total by construction, and the extension property amounts to the above set inclusions. A space is zero-dimens... |

49 |
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Citation Context ....1 with respect to the cited lemma on compactness of Sierpinskiexhaustible sets [6]. Examples 5.2. (1) There are non-compact, exhaustible sets. (So one needs to assume something such as totality.) By =-=[22, 20]-=-, any second-countable space, e.g. the real line R, can be embedded into the domain D = B ω . But R is a connected space, which is equivalent to saying that every continuous boolean-valued map defined... |

46 |
Programming in Haskell
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Citation Context ...sing experimental results and tentative theoretical explanations and conjectures (Section 4). These experiments are implemented in a fragment of the highertype functional programming language Haskell =-=[13]-=-, which is essentially the same as PCF (simply-typed lambda-calculus with arithmetic and fixed-point recursion) [29, 25]. Related work. Brouwer’s Fan functional gives the modulus of uniform continuity... |

45 | The call-by-need lambda calculus
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Citation Context ...njectures hold, the algorithm developed in this section is faster, in practice, than untrained intuition would concede to be possible. 4.1 The model of computation We consider call-by-need evaluation =-=[15, 18, 30]-=-. For the language PCF, this is semantically equivalent to the call-by-name computational model. The essential operational difference of the call-by-name and the call-by-need computational models is t... |

37 |
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Citation Context ... Then the set Tσ can be recovered from the relation ∼σ as x ∈ Tσ ⇐⇒ x ∼σ x, and the relation can be recovered from the set as x ∼σ y ⇐⇒ x ⊓ y ∈ Tσ ⇐⇒ x, y ∈ Tσ and x and y are bounded above (see e.g. =-=[5]-=- and [21]). In particular, ∼σ is an equivalence relation on Tσ. Computability. A partial functional is computable iff it is PCF-definable from parallel-or and parallel-exists [19]. This is a theorem, ... |

30 |
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Citation Context ...ly, and (3) the other algorithms diverge in this model (although, of course, they can be adapted to this model by the application of standard translation techniques of call-by-name into call-by-value =-=[24]-=-). These three issues regarding call-by-value computation should certainly be investigated in future research. 4.2 The experimental setup The main emphasis of Section 4 is asymptotic rather than actua... |

27 | Synthetic topology of data types and classical spaces
- Escardó
(Show Context)
Citation Context ...such that εi searches over Ki and all p defined on � i Ki, we have that Π(ε)(p)(n) ∈ Kn. Therefore Π(ε)(p) ∈ � i Ki.sWe don’t have a corresponding result for exhaustible sets (but see Theorem 6.1 and =-=[6, 8]-=-). The following consequence gives a uniform continuity principle, where α =n β means that αi = βi for all i < n, and α |n is defined by α |n(i) = αi for i < n and α |n(i) = ⊥ for i ≥ n. Corollary 4.6... |

27 | Comparing Cartesian closed categories of (core) compactly generated spaces. Topology and its Applications
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Citation Context ...e calculated in the cartesian closed category of Hausdorff k-spaces. For a brief treatment of k-spaces, also known as compactly generated spaces, see e.g. [14], and, for a more detailed one, see e.g. =-=[9]-=- or the references contained therein. A set is called clopen if it is both closed and open. Lemma 2.3. For every clopen U ⊆ Cσ there is a total predicate p ∈ (Dσ → B) such that ρ −1 σ (U) ⊆ p −1 (tt) ... |

24 | Lazy functional algorithms for exact real functionals, volume 1450
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Citation Context ...the result from left to right, but it has the disadvantage that evaluation of any entry forces evaluation of the previous entries, which is not the case for prod’’. We mention in passing that Simpson =-=[24]-=- applied Berger’s search operator to show that there is a sequential algorithm for Riemann integration when real numbers are implemented as infinite sequences of digits. Now, only the implementation p... |

22 |
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Citation Context ...False and True respectively, in less than 3s together in a 1.7GHz machine under the Glasgow Haskell compiler. This is in stark contrast with the algorithms we have been able to find in the literature =-=[4]-=-, which take at least 2680 computation steps, as they are exponential in the modulus of uniform continuity. We emphasize that the above algorithms, and all highertype algorithms developed in this pape... |

21 |
A type theoretical alternative to CUCH
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Citation Context ...definable, and, in order to do that, he first explicitly defined a search functional for the Cantor space. Here PCF is an applied simplytyped lambda-calculus with arithmetic and fixed-point recursion =-=[23, 19]-=-. Berger observed that, for partial functionals, PCF definability coincides with S1–S9 definability. Then Hyland informed the community that Gandy was aware of the S1–S9 definability of the Fan functi... |

16 | A.: “Comparing Functional Paradigms for Exact Real-number Computation
- Bauer, Escardó, et al.
- 2002
(Show Context)
Citation Context ...uction, and the extension property amounts to the above set inclusions. 3 A space is zero-dimensional iff it has a base of clopen sets. It is an open problem whether the spaces Cσ are zerodimensional =-=[2, 17]-=-. If they are, the following lemma becomes superfluous. The zero-dimensional reflection ZC of a space C is obtained by taking the same set of points and the clopen sets as a base. Lemma 2.4. ZCσ and C... |

14 | M.: “A Convenient Category of Domains
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Citation Context ...egory of compactly generated spaces. Additionally, the inclusion of Scott domains (the Scott-topology functor) into the category of compactly generated spaces preserves the cartesian-closed structure =-=[9, 2]-=-. Hence total and partial higher-type functionals coexist in the same cartesian closed category. One can envisage a higher-type system that simultaneously incorporates total and partial objects, and c... |

14 |
Computable and recursively countable functions of higher type
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Citation Context ...or instance, if D = N and each Ki is a finite subset of N defined uniformly in i, which is the case that has been considered in higher-type computability theory regarding the fan functional (see e.g. =-=[11]-=-). Here we have generalized this to arbitrary higher types D = Dσ. Remark 2.3.8. Berger’s algorithm [5] ε Berger : (B ω → B) → B for searching over the total elements of Bω , mentioned in the introduc... |

13 | Computability over the partial continuous functionals
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Citation Context ... total functionals. An interesting fact, which we don’t use, is that every total functional definable in PCF with parallel extensions is equivalent to one definable in PCF without parallel extensions =-=[16]-=-. Although the notion of totality plays a crucial role in the present work, most functionals we consider are not total by nature. But they produce total outputs for certain total inputs. For example, ... |

11 | Notions of computability at higher types. I
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Citation Context ... used to derive the algorithms of Section 6). 6. Characterization of searchability. 7. Experiments and tentative explanations. 8. Concluding remarks. 2. Higher-type computability As discussed in e.g. =-=[18, 13, 12]-=-, there are many equivalent approaches to higher-type computation. Kleene defined the total functionals directly, but it has been found more convenient to work with the larger collection of partial fu... |

10 | Operational Domain Theory and topology of a sequential language
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(Show Context)
Citation Context ...such that εi searches over Ki and all p defined on � i Ki, we have that Π(ε)(p)(n) ∈ Kn. Therefore Π(ε)(p) ∈ � i Ki.sWe don’t have a corresponding result for exhaustible sets (but see Theorem 6.1 and =-=[6, 8]-=-). The following consequence gives a uniform continuity principle, where α =n β means that αi = βi for all i < n, and α |n is defined by α |n(i) = αi for i < n and α |n(i) = ⊥ for i ≥ n. Corollary 4.6... |

8 | Full abstraction, totality and PCF
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(Show Context)
Citation Context ...e set Tσ can be recovered from the relation ∼σ as x ∈ Tσ ⇐⇒ x ∼σ x, and the relation can be recovered from the set as x ∼σ y ⇐⇒ x ⊓ y ∈ Tσ ⇐⇒ x, y ∈ Tσ and x and y are bounded above (see e.g. [5] and =-=[21]-=-). In particular, ∼σ is an equivalence relation on Tσ. Computability. A partial functional is computable iff it is PCF-definable from parallel-or and parallel-exists [19]. This is a theorem, but we ta... |

7 |
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Citation Context ... a computable retract of Cτ→ι for some τ. Topological aspects of the Kleene–Kreisel functionals. A proof of the following inductive topological characterization, attributed to Hyland, can be found in =-=[15]-=-. Lemma 2.2. Endow Tσ with the relative Scott topology and Cσ with the quotient topology of the surjection ρσ. 1. Cγ has the discrete topology for γ ground, 2. Cσ×τ = Cσ × Cτ and 3. Cσ→τ = Cτ Cσ , whe... |

5 | Comparing hierarchies of total functionals
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Citation Context ...uction, and the extension property amounts to the above set inclusions. 3 A space is zero-dimensional iff it has a base of clopen sets. It is an open problem whether the spaces Cσ are zerodimensional =-=[2, 17]-=-. If they are, the following lemma becomes superfluous. The zero-dimensional reflection ZC of a space C is obtained by taking the same set of points and the clopen sets as a base. Lemma 2.4. ZCσ and C... |

5 |
Computing with functionals—computability theory or computer science
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(Show Context)
Citation Context ...back to the 1950’s, as discussed in the related-work paragraph below. Our primary contribution is a comprehensive investigation of such sets from the point of view of higher-type computability theory =-=[18]-=-. We develop tools for systematically building them and a characterization: they are closed under intersections with decidable sets, under the formation of computable images and of finite and countabl... |

4 | Compact sets of functions and function rings - Gale - 1950 |

3 |
Call-by-need in token-passing nets
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- 2006
(Show Context)
Citation Context ...njectures hold, the algorithm developed in this section is faster, in practice, than untrained intuition would concede to be possible. 4.1 The model of computation We consider call-by-need evaluation =-=[15, 18, 30]-=-. For the language PCF, this is semantically equivalent to the call-by-name computational model. The essential operational difference of the call-by-name and the call-by-need computational models is t... |

2 |
On the ubiquity of certain type structures
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(Show Context)
Citation Context ... used to derive the algorithms of Section 6). 6. Characterization of searchability. 7. Experiments and tentative explanations. 8. Concluding remarks. 2. Higher-type computability As discussed in e.g. =-=[18, 13, 12]-=-, there are many equivalent approaches to higher-type computation. Kleene defined the total functionals directly, but it has been found more convenient to work with the larger collection of partial fu... |

1 | program for exhaustive search over infinite sets. http://www.cs.bham.ac.uk/ ∼ mhe/papers/ exhaustive.hs - Haskell - 2006 |

1 | of Oxford science publications - Topology - 1992 |