## Computational interpretations of analysis via products of selection functions (2010)

Venue: | CIE 2010, INVITED TALK ON SPECIAL SESSION “PROOF THEORY AND COMPUTATION |

Citations: | 9 - 8 self |

### BibTeX

@MISC{Escardó10computationalinterpretations,

author = {Martín Escardó and Paulo Oliva},

title = {Computational interpretations of analysis via products of selection functions},

year = {2010}

}

### OpenURL

### Abstract

We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions.

### Citations

145 | Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes - Gödel - 1958 |

124 |
Metamathematical Investigations of Intuitionistic Arithmetic and Analysis
- Troelstra
- 1973
(Show Context)
Citation Context ...a a computational interpretation of the theory HA ω + AC N B + DNS. The fragment HA ω + AC N B, excluding the double negation shift, has a very straightforward (modified) realizability interpretation =-=[13]-=-, as well as a dialectica interpretation [1, 9]. The remaining challenge is to give a computational interpretation for DNS. A computational interpretation of DNS was first given by Spector [12], via t... |

74 |
Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics
- Spector
- 1962
(Show Context)
Citation Context ...tation [13], as well as a dialectica interpretation [1, 9]. The remaining challenge is to give a computational interpretation for DNS. A computational interpretation of DNS was first given by Spector =-=[12]-=-, via the dialectica interpretation. Spector devised a form of recursion on well-founded trees, nowadays known as bar recursion, and showed that the dialectica interpretation of DNS can be witnesses b... |

69 | Gödel’s functional (“Dialectica”) interpretation
- Avigad, Feferman
- 1998
(Show Context)
Citation Context ...y HA ω + AC N B + DNS. The fragment HA ω + AC N B, excluding the double negation shift, has a very straightforward (modified) realizability interpretation [13], as well as a dialectica interpretation =-=[1, 9]-=-. The remaining challenge is to give a computational interpretation for DNS. A computational interpretation of DNS was first given by Spector [12], via the dialectica interpretation. Spector devised a... |

54 | Program extraction from classical proofs
- Berger, Schwichtenberg
- 1994
(Show Context)
Citation Context ... We now prove the two assumptions of the bar induction: (i) ∀α∈S ∃k∀t≽[α](k) ∀nP (t, n), where t ≽ s means that t is an extension of the finite sequence 3 This is known as the (refined) A-translation =-=[5]-=-, and is useful to analyse proofs of Π 0 2 theorems in analysis.8 Martín Escardó and Paulo Oliva s. Given α we pick k to be the point of continuity of q on α. The result follows simply unfolding the ... |

53 |
Applied Proof Theory: Proof Interpretations and their Use in Mathematics
- Kohlenbach
- 2008
(Show Context)
Citation Context ...inary product of selection functions is precisely what is needed to solve Spector’s equations (which arise from the dialectica interpretation of full classical analysis). Theorem 1 (cf. lemma 11.5 of =-=[11]-=-). Let q : Π ∞ i=0 Xi → R and ω : Π ∞ i=0 Xi → N and ε: Π ∞ i=0 JXi be given. Define α := EPS 〈 〉(ω)(ε)(q) pn(x) := EPQ [α](n)∗x(ω)(ε)(q [α](n)∗x), identifying εs with ε |s|. Then, for all n ≤ ω(α) we... |

34 | On the computational content of the axiom of choice
- Berardi, Bezem, et al.
- 1998
(Show Context)
Citation Context ...s known as bar recursion, and showed that the dialectica interpretation of DNS can be witnesses by such recursion. A computational interpretation of DNS via realizability only came recently, first in =-=[2]-=-, via a non-standard form of realizability, and then in [3, 4], via Kreisel’s modified realizability. The realizability interpretation of DNS makes use of a new form of bar recursion, termed modified ... |

26 | Modified bar recursion and classical dependent choice
- Berger, Oliva
- 2005
(Show Context)
Citation Context ...terpretation of DNS can be witnesses by such recursion. A computational interpretation of DNS via realizability only came recently, first in [2], via a non-standard form of realizability, and then in =-=[3, 4]-=-, via Kreisel’s modified realizability. The realizability interpretation of DNS makes use of a new form of bar recursion, termed modified bar recursion.2 Martín Escardó and Paulo Oliva In this articl... |

13 | Infinite sets that admit fast exhaustive search
- Escardó
- 2007
(Show Context)
Citation Context ...∞ i=|s| Xi → R. IPS can be equivalently defined as IPSs(ε)(q)(n) X |s|+n = εs∗ts,n(λx X |s|+n.qts,n∗x(IPSs∗ts,n∗x(ε)(qts,n∗x))) where ts,n := [IPSs(ε)(q)](n). The functional IPS generalises Escardó’s =-=[6]-=- construction that selection functions for a sequence of spaces can be combined into a selection function for the product space. Proposition 1. IPS (with R = B and εs dependent only on |s|) is primiti... |

9 | Higher order reverse mathematics
- Kohlenbach
(Show Context)
Citation Context ... of selection functions. 1 Introduction Full classical analysis can be formalised using the language of finite types in Peano arithmetic PA ω extended with the axiom schema of full comprehension (cf. =-=[10]-=-) CA : ∃f N→B ∀n N (f(n) ↔ A(n)). As ∀n N (A(n) ∨ ¬A(n)) is equivalent to ∀n N ∃b B (b ↔ A(n)), full comprehension, in the presence of classical logic, follows from countable choice over the booleans ... |

8 | Selection functions, bar recursion, and backward induction
- Escardó, Oliva
(Show Context)
Citation Context ...h forms of bar recursion used to interpret classical analysis, via modified realizability and the dialectica interpretation, correspond to two closely related infinite products of selection functions =-=[8]-=-. Notation. We use X, Y, Z for variables ranging over types. Although in PA ω one does not have dependent types, we will develop the rest of the paper working with types such as Πi∈NXi rather than its... |

7 | Modified bar recursion
- Berger, Oliva
- 2006
(Show Context)
Citation Context ...terpretation of DNS can be witnesses by such recursion. A computational interpretation of DNS via realizability only came recently, first in [2], via a non-standard form of realizability, and then in =-=[3, 4]-=-, via Kreisel’s modified realizability. The realizability interpretation of DNS makes use of a new form of bar recursion, termed modified bar recursion.2 Martín Escardó and Paulo Oliva In this articl... |

1 | The peirce translation
- Escardó, Oliva
- 2010
(Show Context)
Citation Context ... by assumption (a) we get (s ∗ ts,n ∗ x ∗ IPSs∗ts,n∗x(ε)(qts,n∗x)) mr ∀nA(n). The proof is then concluded by the assumption that q mr ∀nA(n) → R. ✷ Remark 4. We analyse the J-shift in more details in =-=[7]-=-, where a proof translation based on the construction JX is also defined. 4.1 Relation to modified bar recursion We now show that IPS and modified bar recursion are in fact primitive recursively inter... |