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

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´ o) Domains IX, Brighton 22 September 2008

On Variants of Modified Bar Recursion Outline Background 1 Three Realizability Bar Recursions 2 BBC bar recursion Berger’s bar recursion Escardo’s bar recursion

On Variants of Modified Bar Recursion Background Outline Background 1 Three Realizability Bar Recursions 2 BBC bar recursion Berger’s bar recursion Escardo’s bar recursion

� � On Variants of Modified Bar Recursion Background Interpretions of Arithmetic and Analysis Dialec. N - trans � HA ω PA ω � T A - trans + mr T + SBR Dialec. PA ω + AC 0 ,ρ � HA ω + AC N N - trans 0 ,ρ � T + MBR A - trans + mr

On Variants of Modified Bar Recursion Background Primitive recursion vs Bar recursion � if n = 0 G τ R ( n ) = H n ( R ( n − 1)) otherwise � G s if Y (ˆ s ) < | s | τ SBR ( s ρ ∗ ) = H s ( λx ρ . SBR ( s ∗ x )) otherwise .

On Variants of Modified Bar Recursion Background Primitive recursion vs Bar recursion � if n = 0 G τ R ( n ) = H n ( R ( n − 1)) otherwise � G s if Y (ˆ s ) < | s | τ SBR ( s ρ ∗ ) = H s ( λx ρ . SBR ( s ∗ x )) otherwise .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Outline Background 1 Three Realizability Bar Recursions 2 BBC bar recursion Berger’s bar recursion Escardo’s bar recursion

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge ∀ n ∃ xA ( n, x ) → ∃ f ∀ nA ( n, fn )

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge ∀ n ∃ xA ( n, x ) → ∃ f ∀ nA ( n, fn ) ∀ n ¬¬∃ xA N ( n, x ) → ¬¬∃ f ∀ nA N ( n, fn )

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge ∀ n ∃ xA ( n, x ) → ∃ f ∀ nA ( n, fn ) ∀ n ¬¬∃ xA N ( n, x ) → ¬¬∃ f ∀ nA N ( n, fn ) ∀ n (( ∃ xA B ( n, x ) → B ) → B ) ∧ ( ∃ f ∀ nA B ( n, fn ) → B ) → B

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge ∀ n ∃ xA ( n, x ) → ∃ f ∀ nA ( n, fn ) ∀ n ¬¬∃ xA N ( n, x ) → ¬¬∃ f ∀ nA N ( n, fn ) ∀ n (( ∃ xA B ( n, x ) → B ) → B ) ∧ ( ∃ f ∀ nA B ( n, fn ) → B ) → B Given realizers for ∀ n (( ∃ xA B ( n, x ) → B ) → B ) ∃ f ∀ nA B ( n, fn ) → B produce realizer for B .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge ∀ n ∃ xA ( n, x ) → ∃ f ∀ nA ( n, fn ) ∀ n ¬¬∃ xA N ( n, x ) → ¬¬∃ f ∀ nA N ( n, fn ) ∀ n (( ∃ xA B ( n, x ) → B ) → B ) ∧ ( ∃ f ∀ nA B ( n, fn ) → B ) → B Given realizers for ∀ n (( ∃ xA B ( n, x ) → B ) → B ) ∀ n ∃ xA B ( n, x ) → B produce realizer for B .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge ∀ n ∃ xA ( n, x ) → ∃ f ∀ nA ( n, fn ) ∀ n ¬¬∃ xA N ( n, x ) → ¬¬∃ f ∀ nA N ( n, fn ) ∀ n (( ∃ xA B ( n, x ) → B ) → B ) ∧ ( ∃ f ∀ nA B ( n, fn ) → B ) → B Given realizers for ∀ n (( ∃ xA B ( n, x ) → B ) → ∃ xA B ( n, x )) ∀ n ∃ xA B ( n, x ) → B produce realizer for B .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge ∀ n ∃ xA ( n, x ) → ∃ f ∀ nA ( n, fn ) ∀ n ¬¬∃ xA N ( n, x ) → ¬¬∃ f ∀ nA N ( n, fn ) ∀ n (( ∃ xA B ( n, x ) → B ) → B ) ∧ ( ∃ f ∀ nA B ( n, fn ) → B ) → B Given realizers for ∀ n (( A ( n ) → B ) → A ( n )) ∀ nA ( n ) → B produce realizer for B .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge Given : ( A ( n ) → B ) → A ( n ) H n Y : ∀ nA ( n ) → B Produce a realiser for B (or ∀ nA ( n ) ).

On Variants of Modified Bar Recursion Three Realizability Bar Recursions The Challenge Given : ( A ( n ) → B ) → A ( n ) H n Y : ∀ nA ( n ) → B Produce a realiser for B (or ∀ nA ( n ) ). Sketch of solution : (assume s ( N × ρ ) ∗ : ∀ n ∈ s A ( n ) ) � s ( n ) if n ∈ s ρ Ψ( s )( n ) = H n ( λx ρ .Y (Ψ( s ∗ � n, x � ))) otherwise .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions BBC bar recursion Berardi, Bezem, Coquand (BBC) functional Ψ( s ) = s @ λn.H n ( λx.Y (Ψ( s ∗ � n, x � ))) Efficient Not easy to prove total Not easy to prove it is a realiser

On Variants of Modified Bar Recursion Three Realizability Bar Recursions BBC bar recursion Berger’s observation Enough : Given H n : ( A ( k ) → B ) → A ( n ) Y : ∀ nA ( n ) → B Produce a realiser for ∀ nA ( n ) .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions BBC bar recursion Berger’s observation Enough : Given H n : ( A ( k ) → B ) → A ( n ) Y : ∀ nA ( n ) → B Produce a realiser for ∀ nA ( n ) . Sketch of solution : (assume s ρ ∗ : ∀ n< | s | A ( n ) ) � s n if n < | s | ρ Ψ( s )( n ) = H n ( λx ρ .Y (Ψ( s ∗ x ))) otherwise .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Berger’s bar recursion Berger’s (MBR) functional Ψ( s ) = s @ λn.H n ( 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 )

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?

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion Escardo’s trick Given : ( A ( n ) → B ) → A ( n ) H n Y : ∀ nA ( n ) → B Produce a realiser for ∀ nA ( n ) . Sketch of solution : (assume s ρ ∗ : ∀ n< | s | A ( n ) ) � if n < | s | s n ρ Ψ( s )( n ) = H n ( λx ρ .Y (Ψ( s ∗ � n, x � ))) otherwise .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion Escardo’s trick Given : ( A ( n ) → B ) → A ( n ) H n Y : ∀ nA ( n ) → B Produce a realiser for ∀ nA ( n ) . Sketch of solution : (assume s ρ ∗ : ∀ n< | s | A ( n ) ) � if n < | s | s n ρ Ψ( s )( n ) = H n ( λx ρ .Y (Ψ( s ∗ . . . ∗ � n, x � ))) otherwise .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion Escardo’s trick Given : ( A ( n ) → B ) → A ( n ) H n Y : ∀ nA ( n ) → B Produce a realiser for ∀ nA ( n ) . Sketch of solution : (assume s ρ ∗ : ∀ n< | s | A ( n ) ) � if n < | s | s n ρ Ψ( s )( n ) = H n ( λx ρ .Y (Ψ( s ∗ . . . ∗ � n, x � ))) otherwise . where . . . ≡ Ψ( s )[ | s | , n − 1] .

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion Escardo’s (CBR) bar recursion Ψ( s ) = s @ λn.H n ( 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 )

On Variants of Modified Bar Recursion Three Realizability Bar Recursions Escardo’s bar recursion Main results BBC ? MBR CBR Known New SBR Open

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