On Variants of Modified Bar Recursion Paulo Oliva Queen Mary, - - PowerPoint PPT Presentation

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On Variants of Modified Bar Recursion Paulo Oliva Queen Mary, - - PowerPoint PPT Presentation

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On Variants of Modified Bar Recursion On Variants of Modified Bar Recursion Paulo Oliva Queen Mary, University of London, UK (pbo@dcs.qmul.ac.uk) (Joint work with Mart n Escard o) Domains IX, Brighton 22 September 2008 On Variants of

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On Variants of Modified Bar Recursion

On Variants of Modified Bar Recursion

Paulo Oliva

Queen Mary, University of London, UK (pbo@dcs.qmul.ac.uk)

(Joint work with Mart´ ın Escard´

  • )

Domains IX, Brighton 22 September 2008

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SLIDE 2

On Variants of Modified Bar Recursion

Outline

1

Background

2

Three Realizability Bar Recursions BBC bar recursion Berger’s bar recursion Escardo’s bar recursion

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SLIDE 3

On Variants of Modified Bar Recursion Background

Outline

1

Background

2

Three Realizability Bar Recursions BBC bar recursion Berger’s bar recursion Escardo’s bar recursion

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SLIDE 4

On Variants of Modified Bar Recursion Background

Interpretions of Arithmetic and Analysis

PAω

N-trans

HAω

Dialec.

  • A-trans+mr

T

T + SBR PAω + AC0,ρ

N-trans

HAω + ACN

0,ρ Dialec.

  • A-trans+mr

T + MBR

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SLIDE 5

On Variants of Modified Bar Recursion Background

Primitive recursion vs Bar recursion

R(n)

τ

=

  • G

if n = 0 Hn(R(n − 1))

  • therwise

SBR(sρ∗)

τ

=

  • Gs

if Y (ˆ s) < |s| Hs(λxρ.SBR(s ∗ x))

  • therwise.
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SLIDE 6

On Variants of Modified Bar Recursion Background

Primitive recursion vs Bar recursion

R(n)

τ

=

  • G

if n = 0 Hn(R(n − 1))

  • therwise

SBR(sρ∗)

τ

=

  • Gs

if Y (ˆ s) < |s| Hs(λxρ.SBR(s ∗ x))

  • therwise.
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SLIDE 7

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

Outline

1

Background

2

Three Realizability Bar Recursions BBC bar recursion Berger’s bar recursion Escardo’s bar recursion

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SLIDE 8

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

∀n∃xA(n, x) → ∃f∀nA(n, fn)

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On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

∀n∃xA(n, x) → ∃f∀nA(n, fn) ∀n¬¬∃xAN(n, x) → ¬¬∃f∀nAN(n, fn)

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SLIDE 10

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

∀n∃xA(n, x) → ∃f∀nA(n, fn) ∀n¬¬∃xAN(n, x) → ¬¬∃f∀nAN(n, fn) ∀n((∃xAB(n, x) → B) → B) ∧ (∃f∀nAB(n, fn) → B) → B

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SLIDE 11

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

∀n∃xA(n, x) → ∃f∀nA(n, fn) ∀n¬¬∃xAN(n, x) → ¬¬∃f∀nAN(n, fn) ∀n((∃xAB(n, x) → B) → B) ∧ (∃f∀nAB(n, fn) → B) → B Given realizers for ∀n((∃xAB(n, x) → B) → B) ∃f∀nAB(n, fn) → B produce realizer for B.

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SLIDE 12

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

∀n∃xA(n, x) → ∃f∀nA(n, fn) ∀n¬¬∃xAN(n, x) → ¬¬∃f∀nAN(n, fn) ∀n((∃xAB(n, x) → B) → B) ∧ (∃f∀nAB(n, fn) → B) → B Given realizers for ∀n((∃xAB(n, x) → B) → B) ∀n∃xAB(n, x) → B produce realizer for B.

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SLIDE 13

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

∀n∃xA(n, x) → ∃f∀nA(n, fn) ∀n¬¬∃xAN(n, x) → ¬¬∃f∀nAN(n, fn) ∀n((∃xAB(n, x) → B) → B) ∧ (∃f∀nAB(n, fn) → B) → B Given realizers for ∀n((∃xAB(n, x) → B) → ∃xAB(n, x)) ∀n∃xAB(n, x) → B produce realizer for B.

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SLIDE 14

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

∀n∃xA(n, x) → ∃f∀nA(n, fn) ∀n¬¬∃xAN(n, x) → ¬¬∃f∀nAN(n, fn) ∀n((∃xAB(n, x) → B) → B) ∧ (∃f∀nAB(n, fn) → B) → B Given realizers for ∀n((A(n) → B) → A(n)) ∀nA(n) → B produce realizer for B.

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SLIDE 15

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

Given Hn : (A(n) → B) → A(n) Y : ∀nA(n) → B Produce a realiser for B (or ∀nA(n)).

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SLIDE 16

On Variants of Modified Bar Recursion Three Realizability Bar Recursions

The Challenge

Given Hn : (A(n) → B) → A(n) Y : ∀nA(n) → B Produce a realiser for B (or ∀nA(n)). Sketch of solution: (assume s(N×ρ)∗ : ∀n∈s A(n)) Ψ(s)(n)

ρ

=

  • s(n)

if n ∈ s Hn(λxρ.Y (Ψ(s ∗ n, x)))

  • therwise.
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SLIDE 17

On Variants of Modified Bar Recursion Three Realizability Bar Recursions BBC bar recursion

Berardi, Bezem, Coquand (BBC) functional

Ψ(s) = s @ λn.Hn(λx.Y (Ψ(s ∗ n, x))) Efficient Not easy to prove total Not easy to prove it is a realiser

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On Variants of Modified Bar Recursion Three Realizability Bar Recursions BBC bar recursion

Berger’s observation

Enough: Given Hn : (A(k) → B) → A(n) Y : ∀nA(n) → B Produce a realiser for ∀nA(n).

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SLIDE 19

On Variants of Modified Bar Recursion Three Realizability Bar Recursions BBC bar recursion

Berger’s observation

Enough: Given Hn : (A(k) → B) → A(n) Y : ∀nA(n) → B Produce a realiser for ∀nA(n). Sketch of solution: (assume sρ∗ : ∀n<|s| A(n)) Ψ(s)(n)

ρ

=

  • sn

if n < |s| Hn(λxρ.Y (Ψ(s ∗ x)))

  • therwise.
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SLIDE 20

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Berger’s bar recursion

Berger’s (MBR) functional

Ψ(s) = s @ λn.Hn(λx.Y (Ψ(s ∗ |s|, x))) Not very efficient Easy to prove total (by bar induction) Easy to prove it is a realiser (by bar induction)

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SLIDE 21

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Berger’s bar recursion

Question

Can we solve the original general problem efficiently with an easy proof of correctness?

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SLIDE 22

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion

Escardo’s trick

Given Hn : (A(n) → B) → A(n) Y : ∀nA(n) → B Produce a realiser for ∀nA(n). Sketch of solution: (assume sρ∗ : ∀n<|s| A(n)) Ψ(s)(n)

ρ

=

  • sn

if n < |s| Hn(λxρ.Y (Ψ(s ∗ n, x)))

  • therwise.
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SLIDE 23

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion

Escardo’s trick

Given Hn : (A(n) → B) → A(n) Y : ∀nA(n) → B Produce a realiser for ∀nA(n). Sketch of solution: (assume sρ∗ : ∀n<|s| A(n)) Ψ(s)(n)

ρ

=

  • sn

if n < |s| Hn(λxρ.Y (Ψ(s ∗ . . . ∗ n, x)))

  • therwise.
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SLIDE 24

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion

Escardo’s trick

Given Hn : (A(n) → B) → A(n) Y : ∀nA(n) → B Produce a realiser for ∀nA(n). Sketch of solution: (assume sρ∗ : ∀n<|s| A(n)) Ψ(s)(n)

ρ

=

  • sn

if n < |s| Hn(λxρ.Y (Ψ(s ∗ . . . ∗ n, x)))

  • therwise.

where . . . ≡ Ψ(s)[|s|, n − 1].

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SLIDE 25

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion

Escardo’s (CBR) bar recursion

Ψ(s) = s @ λn.Hn(λx.Y (Ψ(Ψ(s)(n) ∗ n, x))) Efficient Easy to prove total (by course-of-value bar induction) Easy to prove it is a realiser (by course-of-value bar induction)

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SLIDE 26

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion

Main results

SBR MBR CBR BBC

Known New Open

?

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SLIDE 27

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion

References

Provably recursive functionals of analysis Spector, Proc. Sym. in Pure Maths, 5:1–27, 1962 On the computational content of the axiom of choice Berardi, Bezem and Coquand, JSL, 63(2):600–622, 1998 Modified bar recursion and classical dependent choice Berger and Oliva, LNL, 20:89–107, 2005 Modified bar recursion Berger and Oliva, MSCS, 16:163–183, 2006 On variants on modified bar recursion Escardo and Oliva, in preparation