Relational Proof Interpretations Paulo Oliva Queen Mary University - PowerPoint PPT Presentation

Relational Proof Interpretations Paulo Oliva Queen Mary University of London Logic Colloquium Udine, 23 July 2018 Thanks to collaborators: Martín Escardó, Thomas Powell, Gilda Ferreira, Jaime Gaspar, Dan Hernest

proof interpretations classical (Dialectica, realizability,…) proof (extended) Brouwer, proofs-as-programs Bishop, Griffin, Krivine, Herbelin, … Bridges,… constructive proof computer programs proofs-as-programs

Plan relational approach • Part 1: Sets vs Relations 
 is more general ( intuitionistic logic ) • Part 2: Unification 
 interpretations (only) differ in treatment of ! A ( linear logic ) • Part 3: Games and Applications 
 ( classical logic ) higher-order games explain higher-order programs

Part 1: Sets vs Relations ( realizability vs dialectica )

n r ( s = t ) ≡ ( n = 0) ∧ ( s = t ) n r A ∧ B ≡ n 0 r A ∧ n 1 r B n r A ∨ B ≡ ( n 0 = 0 ∧ n 1 r A ) ∨ ( n 0 ≠ 1 ∧ n 1 r B ) n r A → B ≡ ∀ a ( a r A → { n }( a ) ↓ ∧ { n }( a ) r B ) n r ∃ z A ( z ) ≡ n 1 r A ( n 0 ) n r ∀ z A ( z ) ≡ ∀ x ({ n }( x ) ↓ ∧ { n }( x ) r A ( x )) Theorem (Kleene-Nelson). If HA ⊢ A then HA ⊢ n r A , for some numeral n

Realizability A { n : n r A } ! { { set of realizers of A sentence

Dialectica, vol. 12, 1958 x ∧ | B | w | A ∧ B | y , w x , v ≡ v | A | y x ) ∨ ( b ≠ 1 ∧ | B | w v ) | A ∨ B | y , w x , v , b ≡ ( b =0 ∧ | A | y | A → B | x , w ≡ → | B | w f , g x f ( x ) | A | g ( x , w ) | ∀ z A ( z )| y , s f ≡ f ( s ) | A ( s )| y | ∃ z A ( z )| y ≡ x , s x | A ( s )| y Theorem (Gödel). t for some term t ∈ T If HA ⊢ A then T ⊢ ∀ y | A | y

Dialectica Interpretation x } ! A { ( x , y ) : | A | y { { sentence relation between arguments and counter-arguments

Example α is eventually bounded α is bounded A ≡ ∃ n ∀ i ≥ n ( α ( i ) ≤ n ) → ∃ k ∀ j ( α ( j ) ≤ k ) f , g ≡ ( g ( n , j ) ≥ n → α ( g ( n , j )) ≤ n ) → α ( j ) ≤ f ( n ) | A | n , j = max{ n ,max { α ( i )| i < n }} f ( n ) = g ( n , j ) j

so… which one is better, sets or relations? relational approach is more general Realizability can also be presented in a ‘relational’ style

Kleene realizability n r A ∧ B ≡ n 0 r A ∧ n 1 r B n r A ∨ B ≡ ( n 0 = 0 ∧ n 1 r A ) ∨ ( n 0 ≠ 1 ∧ n 1 r B ) n r A → B ≡ ∀ a ( a r A → { n }( a ) ↓ ∧ { n }( a ) r B ) n r ∃ z A ( z ) ≡ n 1 r A ( n 0 ) n r ∀ z A ( z ) ≡ ∀ x ({ n }( x ) ↓ ∧ { n }( x ) r A ( x )) n r A iff Relational presentation ∀ a | A | a n n 0 ∧ | B | a 1 n 1 | A ∧ B | a ≡ n | A | a 0 n 1 ) ∨ ( n 0 ≠ 1 ∧ | B | a n 1 ) | A ∨ B | a n ≡ ( n 0 = 0 ∧ | A | a a 0 → ({ n }( a 0 ) ↓ ∧ | B | a 1 { n }( a 0 ) ) | A → B | a n ≡ ∀ b | A | b n 1 | ∃ z A ( z )| a ≡ n | A ( n 0 )| a { n }( a 0 ) | ∀ z A ( z )| a n ≡ { n }( a 0 ) ↓ ∧ | A ( a 0 )| a 1

Kreisel modified realizability x , v mr A ∧ B ≡ x mr A ∧ v mr B x , v , b mr A ∨ B ≡ ( b =0 ∧ x mr A ) ∨ ( b ≠ 0 ∧ v mr B ) f mr A → B ≡ ∀ x ( x mr A → f ( x ) mr B ) x , s mr ∃ z A ( z ) ≡ x mr A ( s ) f mr ∀ z A ( z ) ≡ ∀ x ( f ( x ) mr A ( x )) Relational presentation x ∧ | B | w | A ∧ B | y , w ≡ x , v v | A | y x mr A x ) ∨ ( b ≠ 1 ∧ | B | w v ) | A ∨ B | y , w ≡ ( b =0 ∧ | A | y x , v , b iff x → | B | w | A → B | x , w ≡ ∀ y | A | y f f ( x ) ∀ y | A | y x | ∀ z A ( z )| y , s ≡ f f ( s ) | A ( s )| y | ∃ z A ( z )| y x , s ≡ x | A ( s )| y

Gödel Dialectica interpretation x ∧ | B | w | A ∧ B | y , w x , v ≡ v | A | y x ) ∨ ( b ≠ 1 ∧ | B | w v ) | A ∨ B | y , w ≡ ( b =0 ∧ | A | y x , v , b A D ( x , y ) iff | A | y x | A → B | x , w f , g ≡ x → | B | w f ( x ) | A | g ( x , w ) | ∀ z A ( z )| y , s f ≡ f ( s ) | A ( s )| y | ∃ z A ( z )| y ≡ x , s x | A ( s )| y Kreisel modified realizability x ∧ | B | w | A ∧ B | y , w ≡ x , v v | A | y x ) ∨ ( b ≠ 1 ∧ | B | w v ) | A ∨ B | y , w x , v , b ≡ ( b =0 ∧ | A | y x mr A iff ∀ y | A | y x x → | B | w | A → B | x , w f ≡ ∀ y | A | y f ( x ) | ∀ z A ( z )| y , s ≡ f f ( s ) | A ( s )| y | ∃ z A ( z )| y ≡ x , s x | A ( s )| y P. Oliva, Unifying functional interpretations , NDJFL, 47 (2), 2006

Part 2: Linear Logic ( it’s all about the bang! )

Linear Logic A refinement of classical and intuitionistic logic A → B ! A ! B A & B A ∧ B A ⊗ B

call-by-name translation call-by-value translation A ° ⊗ B ° A * & B * ( A ∧ B ) * ≡ ( A ∧ B ) ° ≡ ! A * ⊕ ! B * A ° ⊕ B ° ( A ∨ B ) * ≡ ( A ∨ B ) ° ≡ ! A * ! B * !( A ° ! B ° ) ( A → B ) * ≡ ( A → B ) ° ≡ ( ∀ z A ) * ≡ ∀ z A * ( ∀ z A ) ° ! ∀ z A ° ≡ ( ∃ z A ) ° ∃ z A ° ( ∃ z A ) * ≡ ∃ z ! A * ≡ LL ⊢ A ° IL ⊢ A LL ⊢ A *

realizability ω ⊢ t mr A IL IL ⊢ A ( ⋅ ) * ( ⋅ ) * ω ⊢ ( t mr A ) * LL LL ⊢ A * ?

Interpretation of Linear Logic x ⊗ | B | w | A ⊗ B | y , w x , v ≡ v | A | y x ) ⊕ ( b ≠ 1 &| B | w v ) | A ⊕ B | y , w x , v , b ≡ ( b =0 &| A | y x ) ⊕ ( b ≠ 1 &| B | w v ) ≡ x , v | A & B | y , w , b ( b =0 &| A | y ≡ f , g x f ( x ) | A ! B | x , w | A | g ( x , w ) ! | B | w based on earlier work of | ∀ z A ( z )| y , s f ≡ f ( s ) | A ( s )| y de Paiva and Shirahata | ∃ z A ( z )| y x , s ≡ x | A ( s )| y P. Oliva, Modified realizability interpretation of classical linear logic , LICS 2007 G. Ferreira and P. Oliva, Functional interpretations of intuitionistic linear logic , Logical Methods in Computer Science, 7(1), 2011

modified realizability x mr A A ( ⋅ ) * ( ⋅ ) * ( x mr A ) * ⇔ ∀ y | A * | y x A * |! A | x ≡ ! ∀ y | A | y x interpretations (only) differ in treatment of ! A

!A Trans. Interpretation |! A | x ≡ ! ∀ y | A | y Kreisel modified x ( ⋅ ) * or ( ⋅ ) ° realizability x ≡ ! ∀ y ∈ a | A | y Diller-Nahm x |! A | a ( ⋅ ) * or ( ⋅ ) ° interpretation x ≡ !| A | a Gödel’s Dialectica x |! A | a ( ⋅ ) * or ( ⋅ ) ° interpretation |! A | x ≡ ! ∀ y | A | y x ⊗ ! A modified realizability ( ⋅ ) ° with truth |! A | x ≡ ! ∀ y | A | y x ⊗ ! A q-variant of ( ⋅ ) * modified realizability x ≡ ! ∀ y ∈ a | A | y x ⊗ ! A Diller-Nahm ( ⋅ ) ° |! A | a with truth J. Gaspar and P. Oliva, Proof interpretations with truth , MLQ, 56(6):591-610, 2010

Part 3: Applications ( classical logic and games )

How about classical logic, arithmetic and analysis? law of excluded middle double negation elimination A ∨ ¬ A ¬¬ A → A pre-linearity Peirce’s law ( A → B ) ∨ ( B → A ) (( A → B ) → B ) → A Markov principle Drinker’s paradox ¬ ∀ nD ( n ) → ∃ n ¬ D ( n ) ∃ x ( D ( x ) → ∀ yD ( y )) finite choice ∀ n < k ∃ i A ( n , i ) → ∃ s ∀ n < k A ( n , s n )

Gödel-Gentzen translation Kuroda translation ≡ ¬¬ P ≡ ( P ) G ( P ) K P A G ∧ B G A K ∧ B K ( A ∧ B ) G ≡ ( A ∧ B ) K ≡ ¬¬ ( A G ∨ B G ) A K ∨ B K ( A ∨ B ) G ≡ ( A ∨ B ) K ≡ A G → B G A K → B K ( A → B ) G ≡ ( A → B ) K ≡ ( ∀ z A ) G ≡ ∀ z A G ( ∀ z A ) K ≡ ∀ z ¬¬ A K ( ∃ z A ) G ≡ ¬¬ ∃ z A G ( ∃ z A ) K ≡ ∃ z A K IL ⊢ A G CL ⊢ A IL ⊢ ¬¬ A K G. Ferreira and P. Oliva, On the relation between various negative translations , Logic, Construction, Computation, vol 3, 227-258, 2012

classical CL ⊢ A proof negative translation IL ⊢ A G + proof interpretation ∃ t ∈ T T ⊢ | A G | y t computer programs

classical ∃ x X ∀ y R A ( x , y ) proof negative translation ¬¬ ∃ x X ∀ y R A ( x , y ) + proof ? interpretation program φ ( X → R ) → X ∀ p X → R A ( φ ( p ), p ( φ ( p ))) computer programs

higher-order games explain player higher-order programs program φ ( X → R ) → X move outcome ∃ x X ∀ y R A ( x , y ) x is a good move game continuation given outcome y ∀ p X → R A ( φ ( p ), p ( φ ( p ))) optimal optimal move outcome M. Escardó and P. Oliva, Sequential games and optimal strategies , Proc. of the Royal Society A, 467:1519-1545, 2011

Logical form Specifies ∃ x X ∀ y R A ( x , y ) player ∀ n ∃ x X ∀ y R A n ( x , y ) sequence of players finite choice finite game (bounded collection) unbounded game countable choice M. Escardó and P. Oliva, Selection functions, bar recursion, and backward induction , MSCS, 20 (2), pp .127-168, 2010 P. Oliva and T. Powell, A game-theoretic computational interpretation 
 of proofs in classical analysis , Gentzen's Centenary, 501-531, 2015

Summary • Realizability also has a “relational” presentation • Relational presentation allows for interpretation of LL and unification (including truth variants) • Classical proofs dealt with by combining interpretation with a negative translation • Classical proof (and higher-order programs) can be “explained” in terms of higher-order games