We now consider again the sentence A ≡ ∀ x ∃ y ∀ z (P(x , y) ∨ ¬ P(x , z)) , In contrast to A , the Herbrand normal form A H of A � � A H ≡ ∃ y P(x , y) ∨ ¬ P(x , g(y)) allows one to construct a list of candidates (uniformly in x , g ) for ‘ ∃ y ’, namely ( c , g ( c )) for any constant c (also ( x , g ( x ))) � � � � A H , D : ≡ P(x , c) ∨ ¬ P(x , g(c)) ∨ P(x , g(c)) ∨ ¬ P(x , g(g(c))) Proof Mining: Proof Interpretations and Their Use in
We now consider again the sentence A ≡ ∀ x ∃ y ∀ z (P(x , y) ∨ ¬ P(x , z)) , In contrast to A , the Herbrand normal form A H of A � � A H ≡ ∃ y P(x , y) ∨ ¬ P(x , g(y)) allows one to construct a list of candidates (uniformly in x , g ) for ‘ ∃ y ’, namely ( c , g ( c )) for any constant c (also ( x , g ( x ))) � � � � A H , D : ≡ P(x , c) ∨ ¬ P(x , g(c)) ∨ P(x , g(c)) ∨ ¬ P(x , g(g(c))) � �� � ∈ TAUT is a tautology. Proof Mining: Proof Interpretations and Their Use in
J. Herbrand’s Theorem (‘Th´ eor` eme fondamental’, 1930) Theorem Let A ≡ ∃ x 1 ∀ y 1 ∃ x 2 ∀ y 2 A qf (x 1 , y 1 , x 2 , y 2 ) . Then: PL ⊢ A iff there are terms s 1 , . . . , s k , t 1 , . . . , t n (built up out of the constants and variables of A and the index functions used for the formation of A H ) such that k n � � � � A H , D : ≡ A qf s i , f(s i ) , t j , g(s i , t j ) i=1 j=1 is a tautology. A H , D is called a Herbrand Disjunction . Proof Mining: Proof Interpretations and Their Use in
J. Herbrand’s Theorem (‘Th´ eor` eme fondamental’, 1930) Theorem Let A ≡ ∃ x 1 ∀ y 1 ∃ x 2 ∀ y 2 A qf (x 1 , y 1 , x 2 , y 2 ) . Then: PL ⊢ A iff there are terms s 1 , . . . , s k , t 1 , . . . , t n (built up out of the constants and variables of A and the index functions used for the formation of A H ) such that k n � � � � A H , D : ≡ A qf s i , f(s i ) , t j , g(s i , t j ) i=1 j=1 is a tautology. A H , D is called a Herbrand Disjunction . Note that the length of this disjunction is fixed: k · n . Proof Mining: Proof Interpretations and Their Use in
J. Herbrand’s Theorem (‘Th´ eor` eme fondamental’, 1930) Theorem Let A ≡ ∃ x 1 ∀ y 1 ∃ x 2 ∀ y 2 A qf (x 1 , y 1 , x 2 , y 2 ) . Then: PL ⊢ A iff there are terms s 1 , . . . , s k , t 1 , . . . , t n (built up out of the constants and variables of A and the index functions used for the formation of A H ) such that k n � � � � A H , D : ≡ A qf s i , f(s i ) , t j , g(s i , t j ) i=1 j=1 is a tautology. A H , D is called a Herbrand Disjunction . Note that the length of this disjunction is fixed: k · n . The terms s i , t j can be extracted from a given PL-proof of A . Proof Mining: Proof Interpretations and Their Use in
Herbrand’s Theorem continued Replacing in A H , D all terms ‘ g ( s i , t j )’, ‘ f ( s i )’, by new variables (treating larger terms first) results in another tautological disjunction A Dis s.t. A can be inferred from A by a direct proof . Proof Mining: Proof Interpretations and Their Use in
An example (Ulrich Berger) Consider the open first-order theory T in the language of first-order logic with equality and a constant 0 and two unary function symbols S , f . The only non-logical axiom of T is ∀ x ( S ( x ) � = 0) . Proposition � T ⊢ ∃ x f(S(f(x))) � = x) . Proof Mining: Proof Interpretations and Their Use in
An example (Ulrich Berger) Consider the open first-order theory T in the language of first-order logic with equality and a constant 0 and two unary function symbols S , f . The only non-logical axiom of T is ∀ x ( S ( x ) � = 0) . Proposition � T ⊢ ∃ x f(S(f(x))) � = x) . Proof: Suppose that � � ∀ x f(S(f(x))) = x , then f is injective, but also (since S ( x ) � = 0) surjective on { x : x � = 0 } and hence non-injective. Contradiction! ✷ Proof Mining: Proof Interpretations and Their Use in
Analyzing the above proof yields the following Herbrand terms: 3 � PL ⊢ (S(s) � = 0) → (f(S(f(t j ))) � = t j ) , j=1 where t 1 := 0 , t 2 := f(0) , t 3 := S(f(f(0))) , s := f(f(0)) . ✷ Proof Mining: Proof Interpretations and Their Use in
Remark For sentences A ≡ ∀ x ∃ y ∀ z A qf (x , y , z) , A Dis can be written in the form A qf (x , t 1 , b 1 ) ∨ A qf (x , t 2 , b 2 ) ∨ . . . ∨ A qf (x , t k , b k ) , where the b i are new variables and t i does not contain any b j with i ≤ j (used by Luckhardt’s analysis of Roth’s theorem, see below). Proof Mining: Proof Interpretations and Their Use in
Remark For sentences A ≡ ∀ x ∃ y ∀ z A qf (x , y , z) , A Dis can be written in the form A qf (x , t 1 , b 1 ) ∨ A qf (x , t 2 , b 2 ) ∨ . . . ∨ A qf (x , t k , b k ) , where the b i are new variables and t i does not contain any b j with i ≤ j (used by Luckhardt’s analysis of Roth’s theorem, see below). Herbrand’s theorem immediately extends to first-order theories T whose non-logical axioms G 1 , . . . , G n are all purely universal. Proof Mining: Proof Interpretations and Their Use in
Theorem (Roth 1955) An algebraic irrational number α has only finitely many exceptionally good rational approximations, i.e. for ε > 0 there are only finitely many q ∈ I N such that Z : (p , q) = 1 ∧ | α − pq − 1 | < q − 2 − ε . R(q) : ≡ q > 1 ∧ ∃ !p ∈ Z Proof Mining: Proof Interpretations and Their Use in
Theorem (Roth 1955) An algebraic irrational number α has only finitely many exceptionally good rational approximations, i.e. for ε > 0 there are only finitely many q ∈ I N such that Z : (p , q) = 1 ∧ | α − pq − 1 | < q − 2 − ε . R(q) : ≡ q > 1 ∧ ∃ !p ∈ Z Theorem (Luckhardt 1985/89) The following upper bound on # { q : R ( q ) } holds: # { q : R(q) } < 7 3 ε − 1 log N α + 6 · 10 3 ε − 5 log 2 d · log(50 ε − 2 log d) , where N α < max(21 log 2h( α ) , 2 log(1 + | α | )) and h is the logarithmic absolute homogeneous height and d = deg ( α ) . Independently: Bombieri and van der Poorten 1988. Proof Mining: Proof Interpretations and Their Use in
Limitations Techniques work only for restricted formal contexts: mainly purely universal (‘algebraic’) axioms, restricted use of induction, no higher analytical principles. Proof Mining: Proof Interpretations and Their Use in
Limitations Techniques work only for restricted formal contexts: mainly purely universal (‘algebraic’) axioms, restricted use of induction, no higher analytical principles. Require that one can ‘guess’ the correct Herbrand terms: in general n+1 = 2 2 k procedure results in proofs of length 2 | P | n , where 2 k n ( n cut complexity). Proof Mining: Proof Interpretations and Their Use in
Towards generalizations of Herbrand’s theorem Allow functionals Φ(x , f) instead of just Herbrand terms: Let’s consider again the example � � A ≡ ∀ x ∃ y ∀ z T(x , x , y) ∨ ¬ T(x , x , z)) . Proof Mining: Proof Interpretations and Their Use in
Towards generalizations of Herbrand’s theorem Allow functionals Φ(x , f) instead of just Herbrand terms: Let’s consider again the example � � A ≡ ∀ x ∃ y ∀ z T(x , x , y) ∨ ¬ T(x , x , z)) . A H can be realized by a computable functional of type level 2 which is defined by cases: � c if ¬ T(x , x , g(c)) Φ(x , g) := g(c) otherwise . Proof Mining: Proof Interpretations and Their Use in
Towards generalizations of Herbrand’s theorem Allow functionals Φ(x , f) instead of just Herbrand terms: Let’s consider again the example � � A ≡ ∀ x ∃ y ∀ z T(x , x , y) ∨ ¬ T(x , x , z)) . A H can be realized by a computable functional of type level 2 which is defined by cases: � c if ¬ T(x , x , g(c)) Φ(x , g) := g(c) otherwise . From this definition it easily follows that � � ∀ x , g T(x , x , Φ(x , g)) ∨ ¬ T(x , x , g(Φ(x , g)) . Φ satisfies G. Kreisel’s no-counterexample interpretation! Proof Mining: Proof Interpretations and Their Use in
If A is not provable in PL but e.g. in PA more complicated functionals are needed (Kreisel 1951): Proof Mining: Proof Interpretations and Their Use in
If A is not provable in PL but e.g. in PA more complicated functionals are needed (Kreisel 1951): Let ( a n ) be a nonincreasing sequence in [0 , 1] . Then, clearly, ( a n ) is convergent and so a Cauchy sequence which we write as: N ∀ i , j ∈ [n; n + m] ( | a i − a j | ≤ 2 − k ) , (1) ∀ k ∈ I N ∃ n ∈ I N ∀ m ∈ I where [ n ; n + m ] := { n , n + 1 , . . . , n + m } . Proof Mining: Proof Interpretations and Their Use in
If A is not provable in PL but e.g. in PA more complicated functionals are needed (Kreisel 1951): Let ( a n ) be a nonincreasing sequence in [0 , 1] . Then, clearly, ( a n ) is convergent and so a Cauchy sequence which we write as: N ∀ i , j ∈ [n; n + m] ( | a i − a j | ≤ 2 − k ) , (1) ∀ k ∈ I N ∃ n ∈ I N ∀ m ∈ I where [ n ; n + m ] := { n , n + 1 , . . . , n + m } . Then the (partial) Herbrand normal form of this statement is N I N ∃ n ∈ I N ∀ i , j ∈ [n; n + g(n)] ( | a i − a j | ≤ 2 − k ) . (2) ∀ k ∈ I N ∀ g ∈ I Proof Mining: Proof Interpretations and Their Use in
By E. Specker 1949 there exist computable such sequences ( a n ) even in Q ∩ [0 , 1] without computable bound on ‘ ∃ n ’ in (1) . Proof Mining: Proof Interpretations and Their Use in
By E. Specker 1949 there exist computable such sequences ( a n ) even in Q ∩ [0 , 1] without computable bound on ‘ ∃ n ’ in (1) . By contrast, there is a simple (primitive recursive) bound Φ ∗ ( g , k ) on (2) (also referred to as ‘metastability’ by T.Tao): Proof Mining: Proof Interpretations and Their Use in
By E. Specker 1949 there exist computable such sequences ( a n ) even in Q ∩ [0 , 1] without computable bound on ‘ ∃ n ’ in (1) . By contrast, there is a simple (primitive recursive) bound Φ ∗ ( g , k ) on (2) (also referred to as ‘metastability’ by T.Tao): Proposition Let ( a n ) be any nonincreasing sequence in [0 , 1] then N I N ∃ n ≤ Φ ∗ (g , k) ∀ i , j ∈ [n; n+g(n)] ( | a i − a j | ≤ 2 − k ) , ∀ k ∈ I N ∀ g ∈ I where g (2 k − 1) (0) with ˜ Φ ∗ (g , k) := ˜ g(n) := n + g(n) . Moreover, there exists an i < 2 k such that n can be taken as ˜ g ( i ) (0) . Proof Mining: Proof Interpretations and Their Use in
Remark The previous result can be viewed as a polished form of a Herbrand disjunction of variable (in k ) length : 2 k − 1 � � g (i) (0)) | ≤ 2 − k � | a ˜ g (i) (0) − a ˜ . g(˜ i=0 Proof Mining: Proof Interpretations and Their Use in
Remark The previous result can be viewed as a polished form of a Herbrand disjunction of variable (in k ) length : 2 k − 1 � � g (i) (0)) | ≤ 2 − k � | a ˜ g (i) (0) − a ˜ . g(˜ i=0 Corollary (T. Tao’s finite convergence principle) ∀ k ∈ I N , g : I N → I N ∃ M ∈ I N ∀ 1 ≥ a 0 ≥ . . . ≥ a M ≥ 0 ∃ N ∈ I N � N + g(N) ≤ M ∧ ∀ n , m ∈ [N , N + g(N)]( | a n − a m | ≤ 2 − k �� . g (2 k ) (0) . One may take M := ˜ Proof Mining: Proof Interpretations and Their Use in
An Example from Ergodic Theory X Hilbert space , f : X → X linear and � f(x) � ≤ � x � for all x ∈ X . n � 1 f ( i ) ( x ) A n ( x ) := n + 1S n ( x ) , where S n ( x ) := ( n ≥ 0 ) i = 0 Proof Mining: Proof Interpretations and Their Use in
An Example from Ergodic Theory X Hilbert space , f : X → X linear and � f(x) � ≤ � x � for all x ∈ X . n � 1 f ( i ) ( x ) A n ( x ) := n + 1S n ( x ) , where S n ( x ) := ( n ≥ 0 ) i = 0 Theorem (von Neumann Mean Ergodic Theorem) For every x ∈ X , the sequence ( A n ( x )) n converges. Proof Mining: Proof Interpretations and Their Use in
An Example from Ergodic Theory X Hilbert space , f : X → X linear and � f(x) � ≤ � x � for all x ∈ X . n � 1 f ( i ) ( x ) A n ( x ) := n + 1S n ( x ) , where S n ( x ) := ( n ≥ 0 ) i = 0 Theorem (von Neumann Mean Ergodic Theorem) For every x ∈ X , the sequence ( A n ( x )) n converges. Avigad/Gerhardy/Towsner (TAMS 2010): in general no computable rate of convergence . But: Prim. rec. bound on metastable version! Proof Mining: Proof Interpretations and Their Use in
An Example from Ergodic Theory X Hilbert space , f : X → X linear and � f(x) � ≤ � x � for all x ∈ X . n � 1 f ( i ) ( x ) A n ( x ) := n + 1S n ( x ) , where S n ( x ) := ( n ≥ 0 ) i = 0 Theorem (von Neumann Mean Ergodic Theorem) For every x ∈ X , the sequence ( A n ( x )) n converges. Avigad/Gerhardy/Towsner (TAMS 2010): in general no computable rate of convergence . But: Prim. rec. bound on metastable version! Theorem (Garrett Birkhoff 1939) Mean Ergodic Theorem holds for uniformly convex Banach spaces. Proof Mining: Proof Interpretations and Their Use in
By logical metatheorems (see Lecture II tomorrow!): Theorem (K./Leu ¸ stean, Ergodic Theor. Dynam. Syst. 2009) X uniformly convex Banach space, η a modulus of uniform convexity and f : X → X as above, b > 0. Then for all x ∈ X with � x � ≤ b , all ε > 0 , all g : I N → I N : � � ∃ n ≤ Φ( ε, g , b , η ) ∀ i , j ∈ [n; n + g(n)] � A i (x) − A j (x) � < ε , Proof Mining: Proof Interpretations and Their Use in
By logical metatheorems (see Lecture II tomorrow!): Theorem (K./Leu ¸ stean, Ergodic Theor. Dynam. Syst. 2009) X uniformly convex Banach space, η a modulus of uniform convexity and f : X → X as above, b > 0. Then for all x ∈ X with � x � ≤ b , all ε > 0 , all g : I N → I N : � � ∃ n ≤ Φ( ε, g , b , η ) ∀ i , j ∈ [n; n + g(n)] � A i (x) − A j (x) � < ε , where Φ( ε, g , b , η ) := M · ˜ h (K) (0) , with � � � 16b � � ε � ε b M := , γ := 16 η , K := , ε 8b γ h , ˜ ˜ h : I N → I N , h(n) := 2(Mn + g(Mn)) , h(n) := max i ≤ n h(i) . Computable rate of convergence iff the norm of limit is computable ! Proof Mining: Proof Interpretations and Their Use in
Bounding the number of fluctuations We say that ( x n ) admits k ε -fluctuations if there are i 1 ≤ j 1 ≤ . . . i k ≤ j k s.t. � x j n − x i n � ≥ ε for n = 1 , . . . , k . Proof Mining: Proof Interpretations and Their Use in
Bounding the number of fluctuations We say that ( x n ) admits k ε -fluctuations if there are i 1 ≤ j 1 ≤ . . . i k ≤ j k s.t. � x j n − x i n � ≥ ε for n = 1 , . . . , k . As a corollary to our analysis of Birkhoff’s proof, Avigad and Rute showed Theorem (Avigad, Rute (ETDS 2015)) ( A n ( x )) admits at most 2 log(M) · b ε + b γ · 2 log(2M) · b ε + b γ many fluctuations. Proof Mining: Proof Interpretations and Their Use in
Bounding the number of fluctuations We say that ( x n ) admits k ε -fluctuations if there are i 1 ≤ j 1 ≤ . . . i k ≤ j k s.t. � x j n − x i n � ≥ ε for n = 1 , . . . , k . As a corollary to our analysis of Birkhoff’s proof, Avigad and Rute showed Theorem (Avigad, Rute (ETDS 2015)) ( A n ( x )) admits at most 2 log(M) · b ε + b γ · 2 log(2M) · b ε + b γ many fluctuations. Partly possible because Birkhoff’s proof only uses boundedly many (in the data) instances of the law-of-excluded-middle for ∃ -statements! Proof Mining: Proof Interpretations and Their Use in
Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Proof Mining: Proof Interpretations and Their Use in
Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Direct example: Infinitary Pigeonhole Principle (IPP): � � ∀ n ∈ I N ∀ f : I N → C n ∃ i ≤ n ∀ k ∈ I N ∃ m ≥ k f(m) = i . Proof Mining: Proof Interpretations and Their Use in
Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Direct example: Infinitary Pigeonhole Principle (IPP): � � ∀ n ∈ I N ∀ f : I N → C n ∃ i ≤ n ∀ k ∈ I N ∃ m ≥ k f(m) = i . IPP causes arbitrary primitive recursive complexity , but (IPP) H � � ∀ n ∈ I N ∀ f : I N → C n ∀ F : C n → I N ∃ i ≤ n ∃ m ≥ F(i) f(m) = i Proof Mining: Proof Interpretations and Their Use in
Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Direct example: Infinitary Pigeonhole Principle (IPP): � � ∀ n ∈ I N ∀ f : I N → C n ∃ i ≤ n ∀ k ∈ I N ∃ m ≥ k f(m) = i . IPP causes arbitrary primitive recursive complexity , but (IPP) H � � ∀ n ∈ I N ∀ f : I N → C n ∀ F : C n → I N ∃ i ≤ n ∃ m ≥ F(i) f(m) = i has trivial n.c.i.-solution for ‘ ∃ i ’, ‘ ∃ m ’: M(n , f , F) := max { F(i) : i ≤ n } and I(n , f , F) := f(M(n , f , F)) . Proof Mining: Proof Interpretations and Their Use in
Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Direct example: Infinitary Pigeonhole Principle (IPP): � � ∀ n ∈ I N ∀ f : I N → C n ∃ i ≤ n ∀ k ∈ I N ∃ m ≥ k f(m) = i . IPP causes arbitrary primitive recursive complexity , but (IPP) H � � ∀ n ∈ I N ∀ f : I N → C n ∀ F : C n → I N ∃ i ≤ n ∃ m ≥ F(i) f(m) = i has trivial n.c.i.-solution for ‘ ∃ i ’, ‘ ∃ m ’: M(n , f , F) := max { F(i) : i ≤ n } and I(n , f , F) := f(M(n , f , F)) . M , I do not reflect true complexity of IPP! Proof Mining: Proof Interpretations and Their Use in
Problems of the no-counterexample interpretation For principles F ∈ ∃∀∃ n.c.i. no longer ‘correct’. C n := { 0 , 1 , . . . , n } . Direct example: Infinitary Pigeonhole Principle (IPP): � � ∀ n ∈ I N ∀ f : I N → C n ∃ i ≤ n ∀ k ∈ I N ∃ m ≥ k f(m) = i . IPP causes arbitrary primitive recursive complexity , but (IPP) H � � ∀ n ∈ I N ∀ f : I N → C n ∀ F : C n → I N ∃ i ≤ n ∃ m ≥ F(i) f(m) = i has trivial n.c.i.-solution for ‘ ∃ i ’, ‘ ∃ m ’: M(n , f , F) := max { F(i) : i ≤ n } and I(n , f , F) := f(M(n , f , F)) . M , I do not reflect true complexity of IPP! Related problem: bad behavior w.r.t. modus ponens! Proof Mining: Proof Interpretations and Their Use in
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Proof Mining: Proof Interpretations and Their Use in
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Interpretation C I contains the additional information , Proof Mining: Proof Interpretations and Their Use in
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Interpretation C I contains the additional information , Construct by recursion on P a new proof P I of C I . Proof Mining: Proof Interpretations and Their Use in
A Modular Approach: Proof Interpretations Interpret the formulas A in P : A �→ A I , Interpretation C I contains the additional information , Construct by recursion on P a new proof P I of C I . Our approach is based on novel forms and extensions of: K. G¨ odel’s functional interpretation! Proof Mining: Proof Interpretations and Their Use in
G¨ odel’s functional interpretation in five minutes G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that Proof Mining: Proof Interpretations and Their Use in
G¨ odel’s functional interpretation in five minutes G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that A Sh ≡ ∀ x ∃ y A Sh (x , y) , where A Sh is quantifier-free , Proof Mining: Proof Interpretations and Their Use in
G¨ odel’s functional interpretation in five minutes G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that A Sh ≡ ∀ x ∃ y A Sh (x , y) , where A Sh is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A Sh ≡ A . Proof Mining: Proof Interpretations and Their Use in
G¨ odel’s functional interpretation in five minutes G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that A Sh ≡ ∀ x ∃ y A Sh (x , y) , where A Sh is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A Sh ≡ A . A ↔ A Sh by classical logic and quantifier-free choice in all types QF-AC : ∀ a ∃ b F qf (a , b) → ∃ B ∀ a F qf (a , B(a)) . Proof Mining: Proof Interpretations and Their Use in
G¨ odel’s functional interpretation in five minutes G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that A Sh ≡ ∀ x ∃ y A Sh (x , y) , where A Sh is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A Sh ≡ A . A ↔ A Sh by classical logic and quantifier-free choice in all types QF-AC : ∀ a ∃ b F qf (a , b) → ∃ B ∀ a F qf (a , B(a)) . x , y are tuples of functionals of finite type over the base types of the system at hand. Proof Mining: Proof Interpretations and Their Use in
A Sh ≡ ∀ u ∃ x A Sh (u , x) , B Sh ≡ ∀ v ∃ y B Sh (v , y) . Proof Mining: Proof Interpretations and Their Use in
A Sh ≡ ∀ u ∃ x A Sh (u , x) , B Sh ≡ ∀ v ∃ y B Sh (v , y) . (Sh1) P Sh ≡ P ≡ P Sh for atomic P Proof Mining: Proof Interpretations and Their Use in
A Sh ≡ ∀ u ∃ x A Sh (u , x) , B Sh ≡ ∀ v ∃ y B Sh (v , y) . (Sh1) P Sh ≡ P ≡ P Sh for atomic P (Sh2) ( ¬ A) Sh ≡ ∀ f ∃ u ¬ A Sh (u , f(u)) Proof Mining: Proof Interpretations and Their Use in
A Sh ≡ ∀ u ∃ x A Sh (u , x) , B Sh ≡ ∀ v ∃ y B Sh (v , y) . (Sh1) P Sh ≡ P ≡ P Sh for atomic P (Sh2) ( ¬ A) Sh ≡ ∀ f ∃ u ¬ A Sh (u , f(u)) � � (Sh3) (A ∨ B) Sh ≡ ∀ u , v ∃ x , y A Sh (u , x) ∨ B Sh (v , y) Proof Mining: Proof Interpretations and Their Use in
A Sh ≡ ∀ u ∃ x A Sh (u , x) , B Sh ≡ ∀ v ∃ y B Sh (v , y) . (Sh1) P Sh ≡ P ≡ P Sh for atomic P (Sh2) ( ¬ A) Sh ≡ ∀ f ∃ u ¬ A Sh (u , f(u)) � � (Sh3) (A ∨ B) Sh ≡ ∀ u , v ∃ x , y A Sh (u , x) ∨ B Sh (v , y) (Sh4) ( ∀ z A) Sh ≡ ∀ z , u ∃ x A Sh (z , u , x) Proof Mining: Proof Interpretations and Their Use in
A Sh ≡ ∀ u ∃ x A Sh (u , x) , B Sh ≡ ∀ v ∃ y B Sh (v , y) . (Sh1) P Sh ≡ P ≡ P Sh for atomic P (Sh2) ( ¬ A) Sh ≡ ∀ f ∃ u ¬ A Sh (u , f(u)) � � (Sh3) (A ∨ B) Sh ≡ ∀ u , v ∃ x , y A Sh (u , x) ∨ B Sh (v , y) (Sh4) ( ∀ z A) Sh ≡ ∀ z , u ∃ x A Sh (z , u , x) � � (Sh5) (A → B) Sh ≡ ∀ f , v ∃ u , y A Sh (u , f(u)) → B Sh (v , y) Proof Mining: Proof Interpretations and Their Use in
A Sh ≡ ∀ u ∃ x A Sh (u , x) , B Sh ≡ ∀ v ∃ y B Sh (v , y) . (Sh1) P Sh ≡ P ≡ P Sh for atomic P (Sh2) ( ¬ A) Sh ≡ ∀ f ∃ u ¬ A Sh (u , f(u)) � � (Sh3) (A ∨ B) Sh ≡ ∀ u , v ∃ x , y A Sh (u , x) ∨ B Sh (v , y) (Sh4) ( ∀ z A) Sh ≡ ∀ z , u ∃ x A Sh (z , u , x) � � (Sh5) (A → B) Sh ≡ ∀ f , v ∃ u , y A Sh (u , f(u)) → B Sh (v , y) (Sh6) ( ∃ zA) Sh ≡ ∀ U ∃ z , f A Sh (z , U(z , f) , f(U(z , f))) Proof Mining: Proof Interpretations and Their Use in
A Sh ≡ ∀ u ∃ x A Sh (u , x) , B Sh ≡ ∀ v ∃ y B Sh (v , y) . (Sh1) P Sh ≡ P ≡ P Sh for atomic P (Sh2) ( ¬ A) Sh ≡ ∀ f ∃ u ¬ A Sh (u , f(u)) � � (Sh3) (A ∨ B) Sh ≡ ∀ u , v ∃ x , y A Sh (u , x) ∨ B Sh (v , y) (Sh4) ( ∀ z A) Sh ≡ ∀ z , u ∃ x A Sh (z , u , x) � � (Sh5) (A → B) Sh ≡ ∀ f , v ∃ u , y A Sh (u , f(u)) → B Sh (v , y) (Sh6) ( ∃ zA) Sh ≡ ∀ U ∃ z , f A Sh (z , U(z , f) , f(U(z , f))) (Sh7) (A ∧ B) Sh ≡ ∀ n , u , v ∃ x , y (n=0 → A Sh (u , x)) ∧ (n � =0 → B Sh (v , y)) � � ↔ ∀ u , v ∃ x , y A Sh (u , x) ∧ B Sh (v , y) . Proof Mining: Proof Interpretations and Their Use in
Sh extracts from a given proof p p ⊢ ∀ x ∃ y A qf ( x , y ) an explicit effective functional Φ realizing A Sh , i.e. ∀ x A qf ( x , Φ ( x )) . Proof Mining: Proof Interpretations and Their Use in
3. Monotone functional interpretation (K.1996) Monotone Sh extracts Φ ∗ such that � � Φ ∗ � Y ∧ ∀ x A Sh ( x , Y ( x )) ∃ Y , Proof Mining: Proof Interpretations and Their Use in
3. Monotone functional interpretation (K.1996) Monotone Sh extracts Φ ∗ such that � � Φ ∗ � Y ∧ ∀ x A Sh ( x , Y ( x )) ∃ Y , where � is some suitable notion of being a ‘bound’ that applies to higher order function spaces (W.A. Howard) � N x : ≡ x ∗ ≥ x , x ∗ � I x ∗ � ρ → τ x : ≡ ∀ y ∗ , y(y ∗ � ρ y → x ∗ (y ∗ ) � τ x(y)) . Also relevant: bounded functional interpretation (F. Ferreira, P. Oliva) Proof Mining: Proof Interpretations and Their Use in
Tao on a finitary approach to analysis ‘it is common to make a distinction between “hard”, “quantitative”, or “finitary” analysis on the one hand, and “soft”, “qualitative”, or “infinitary” analysis on the other hand.’ ...‘It is fairly well known that the results obtained by hard and soft analysis resp. can be connected to each other by various “correspondence principles” or “compactness principles”. It is however my belief that the relationship between the two types of analysis is much deeper.’ ...’There are rigorous results from proof theory which can allow one to automatically convert certain types of qualitative arguments into quantitative ones...’ (T. Tao: Soft analysis, hard analysis, and the finite convergence principle, 2007) Proof Mining: Proof Interpretations and Their Use in
Literature 1) Kohlenbach, U., Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. xx+536pp., Springer Heidelberg-Berlin, 2008. 2) Kreisel, G., Macintyre, A., Constructive logic versus algebraization I. In: Troelstra, A.S., van Dalen, D. (eds.), Proc. L.E.J. Brouwer Centenary Symposium (Noordwijkerhout 1981), North-Holland (Amsterdam), pp. 217-260 (1982). 3) Luckhardt, H., Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken. J. Symbolic Logic 54 , pp. 234-263 (1989). 4) Tao, T., Soft analysis, hard analysis, and the finite convergence principle. In: ‘Structure and Randomness. AMS, 298pp., 2008’. 5) Special issue of ‘Dialectica’ on G¨ odel’s interpretation with contributions e.g. by Ferreira, Kohlenbach, Oliva, 2008. Proof Mining: Proof Interpretations and Their Use in
Lecture II Proof Mining: Proof Interpretations and Their Use in
Recall from Lecture I: G¨ odel’s functional interpretation G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that Proof Mining: Proof Interpretations and Their Use in
Recall from Lecture I: G¨ odel’s functional interpretation G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that A Sh ≡ ∀ x ∃ y A Sh (x , y) , where A Sh is quantifier-free , Proof Mining: Proof Interpretations and Their Use in
Recall from Lecture I: G¨ odel’s functional interpretation G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that A Sh ≡ ∀ x ∃ y A Sh (x , y) , where A Sh is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A Sh ≡ A . Proof Mining: Proof Interpretations and Their Use in
Recall from Lecture I: G¨ odel’s functional interpretation G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that A Sh ≡ ∀ x ∃ y A Sh (x , y) , where A Sh is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A Sh ≡ A . A ↔ A Sh by classical logic and quantifier-free choice QF-AC . Proof Mining: Proof Interpretations and Their Use in
Recall from Lecture I: G¨ odel’s functional interpretation G¨ odel’s functional interpretation D combined with Krivine’s negative translation N results in an interpretation Sh = D ◦ N (Streicher/K.07) A �→ A Sh (Shoenfield variant) such that A Sh ≡ ∀ x ∃ y A Sh (x , y) , where A Sh is quantifier-free , For A ≡ ∀ x ∃ y A qf (x , y) one has A Sh ≡ A . A ↔ A Sh by classical logic and quantifier-free choice QF-AC . Sh extracts from a given proof p p ⊢ ∀ x ∃ y A qf (x , y) an explicit effective functional Φ realizing A Sh , i.e. ∀ x A qf (x , Φ(x)) . Proof Mining: Proof Interpretations and Their Use in
Basic facts about functional interpretation Peano arithmetic in all finite types PA ω has a functional interpretation by primitive recursive functionals in higher types in the sense of Hilbert (1926), G¨ odel (1941,1958). Proof Mining: Proof Interpretations and Their Use in
Basic facts about functional interpretation Peano arithmetic in all finite types PA ω has a functional interpretation by primitive recursive functionals in higher types in the sense of Hilbert (1926), G¨ odel (1941,1958). Full classical analysis PA ω +dependent choice has functional interpretation by bar recursive functionals (Spector 1962). Proof Mining: Proof Interpretations and Their Use in
Basic facts about functional interpretation Peano arithmetic in all finite types PA ω has a functional interpretation by primitive recursive functionals in higher types in the sense of Hilbert (1926), G¨ odel (1941,1958). Full classical analysis PA ω +dependent choice has functional interpretation by bar recursive functionals (Spector 1962). PRA ω +weak K¨ onigs lemma has functional interpretation by ordinary primitive recursive functionals in the sense of Kleene (K.1992). Proof Mining: Proof Interpretations and Their Use in
Basic facts about functional interpretation Peano arithmetic in all finite types PA ω has a functional interpretation by primitive recursive functionals in higher types in the sense of Hilbert (1926), G¨ odel (1941,1958). Full classical analysis PA ω +dependent choice has functional interpretation by bar recursive functionals (Spector 1962). PRA ω +weak K¨ onigs lemma has functional interpretation by ordinary primitive recursive functionals in the sense of Kleene (K.1992). Systems of bounded arithmetic have functional interpretation by basic feasible functionals (Cook, Urquhart 1993). Proof Mining: Proof Interpretations and Their Use in
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