Proof Mining for Nonexpansive Semigroups PhDs in Logic IX, Bochum 2017 Angeliki Koutsoukou-Argyraki Research Group Logic, Department of Mathematics TU Darmstadt, Germany
Origin of proof interpretations Hilbert’s 2nd problem (1900): Is mathematics consistent ?
Origin of proof interpretations Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨ odel (1931) : Impossible to prove the consistency of a theory T within T .
Origin of proof interpretations Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨ odel (1931) : Impossible to prove the consistency of a theory T within T . Let theories T 1 , T 2 with languages L ( T 1 ), L ( T 2 ) . T 2 is consistent relative to T 1 if it can be proved that if T 1 is consistent then T 2 is consistent.
Origin of proof interpretations Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨ odel (1931) : Impossible to prove the consistency of a theory T within T . Let theories T 1 , T 2 with languages L ( T 1 ), L ( T 2 ) . T 2 is consistent relative to T 1 if it can be proved that if T 1 is consistent then T 2 is consistent. Proof Interpretations originally developed for relative consistency proofs.
Origin of proof interpretations Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨ odel (1931) : Impossible to prove the consistency of a theory T within T . Let theories T 1 , T 2 with languages L ( T 1 ), L ( T 2 ) . T 2 is consistent relative to T 1 if it can be proved that if T 1 is consistent then T 2 is consistent. Proof Interpretations originally developed for relative consistency proofs. G¨ odel’s motivation: obtain a relative consistency proof for HA (and hence for PA).
Origin of proof interpretations Hilbert’s 2nd problem (1900): Is mathematics consistent ? G¨ odel (1931) : Impossible to prove the consistency of a theory T within T . Let theories T 1 , T 2 with languages L ( T 1 ), L ( T 2 ) . T 2 is consistent relative to T 1 if it can be proved that if T 1 is consistent then T 2 is consistent. Proof Interpretations originally developed for relative consistency proofs. G¨ odel’s motivation: obtain a relative consistency proof for HA (and hence for PA). G¨ odel’s functional ”Dialectica” Interpretation (1958): consistency of PA reduced to a quantifier-free calculus of primitive recursive functionals of finite type.
Proof Mining Shift of focus : G. Kreisel (1950’s): Unwinding of proofs ”What more do we know if we have proved a theorem by restricted means than if we merely know that it is true ?”
Proof Mining Shift of focus : G. Kreisel (1950’s): Unwinding of proofs ”What more do we know if we have proved a theorem by restricted means than if we merely know that it is true ?” Possible to obtain new quantitative/ qualitative information by logical analysis of proofs of statements of certain logical form.
Proof Mining Shift of focus : G. Kreisel (1950’s): Unwinding of proofs ”What more do we know if we have proved a theorem by restricted means than if we merely know that it is true ?” Possible to obtain new quantitative/ qualitative information by logical analysis of proofs of statements of certain logical form. Extraction of constructive information from non-constructive proofs.
Proof Mining T 1 transformed into T 2 by transforming every theorem φ ∈ L ( T 1 ) into φ I ∈ L ( T 2 ) via the proof interpretation I so that T 1 ⊢ φ ⇒ T 2 ⊢ φ I holds.
Proof Mining T 1 transformed into T 2 by transforming every theorem φ ∈ L ( T 1 ) into φ I ∈ L ( T 2 ) via the proof interpretation I so that T 1 ⊢ φ ⇒ T 2 ⊢ φ I holds. Then a given proof p of φ in T 1 is transformed into a proof p I of φ I in T 2 by a simple recursion over φ in T 1 .
Proof Mining T 1 transformed into T 2 by transforming every theorem φ ∈ L ( T 1 ) into φ I ∈ L ( T 2 ) via the proof interpretation I so that T 1 ⊢ φ ⇒ T 2 ⊢ φ I holds. Then a given proof p of φ in T 1 is transformed into a proof p I of φ I in T 2 by a simple recursion over φ in T 1 . This gives new quantitative information.
G¨ odel ’s functional ”Dialectica” interpretation (with negative translation) To every formula A in L (WE-HA ω ) we assign A S ≡ ∀ x ∃ y A S ( x , y ) where A S is quantifier-free.
G¨ odel ’s functional ”Dialectica” interpretation (with negative translation) To every formula A in L (WE-HA ω ) we assign A S ≡ ∀ x ∃ y A S ( x , y ) where A S is quantifier-free. By classical logic and
G¨ odel ’s functional ”Dialectica” interpretation (with negative translation) To every formula A in L (WE-HA ω ) we assign A S ≡ ∀ x ∃ y A S ( x , y ) where A S is quantifier-free. By classical logic and QF-AC : ∀ x ∃ y F 0 ( x , y ) → ∃ B ∀ x F 0 ( x , B ( x ))
G¨ odel ’s functional ”Dialectica” interpretation (with negative translation) To every formula A in L (WE-HA ω ) we assign A S ≡ ∀ x ∃ y A S ( x , y ) where A S is quantifier-free. By classical logic and QF-AC : ∀ x ∃ y F 0 ( x , y ) → ∃ B ∀ x F 0 ( x , B ( x ))
G¨ odel ’s functional ”Dialectica” interpretation (with negative translation) To every formula A in L (WE-HA ω ) we assign A S ≡ ∀ x ∃ y A S ( x , y ) where A S is quantifier-free. By classical logic and QF-AC : ∀ x ∃ y F 0 ( x , y ) → ∃ B ∀ x F 0 ( x , B ( x )) A S ↔ A .
G¨ odel ’s functional ”Dialectica” interpretation (with negative translation) To every formula A in L (WE-HA ω ) we assign A S ≡ ∀ x ∃ y A S ( x , y ) where A S is quantifier-free. By classical logic and QF-AC : ∀ x ∃ y F 0 ( x , y ) → ∃ B ∀ x F 0 ( x , B ( x )) A S ↔ A . Idea :
G¨ odel ’s functional ”Dialectica” interpretation (with negative translation) To every formula A in L (WE-HA ω ) we assign A S ≡ ∀ x ∃ y A S ( x , y ) where A S is quantifier-free. By classical logic and QF-AC : ∀ x ∃ y F 0 ( x , y ) → ∃ B ∀ x F 0 ( x , B ( x )) A S ↔ A . Idea : S extracts from a given proof p : p ⊢ ∀ x ∃ y A ( x , y ) an explicit effective functional that realizes A S :
G¨ odel ’s functional ”Dialectica” interpretation (with negative translation) To every formula A in L (WE-HA ω ) we assign A S ≡ ∀ x ∃ y A S ( x , y ) where A S is quantifier-free. By classical logic and QF-AC : ∀ x ∃ y F 0 ( x , y ) → ∃ B ∀ x F 0 ( x , B ( x )) A S ↔ A . Idea : S extracts from a given proof p : p ⊢ ∀ x ∃ y A ( x , y ) an explicit effective functional that realizes A S : ∀ x A S ( x , Φ( x )) .
Proof Mining Monotone functional interpretation (and negative translation) extracts a Φ ∗ such that:
Proof Mining Monotone functional interpretation (and negative translation) extracts a Φ ∗ such that: ∃ Y (Φ ∗ � Y ∧ ∀ x A S ( x , Y ( x ))) .
Proof Mining Monotone functional interpretation (and negative translation) extracts a Φ ∗ such that: ∃ Y (Φ ∗ � Y ∧ ∀ x A S ( x , Y ( x ))) . Majorizability
Proof Mining Monotone functional interpretation (and negative translation) extracts a Φ ∗ such that: ∃ Y (Φ ∗ � Y ∧ ∀ x A S ( x , Y ( x ))) . Majorizability x ∗ � N x : ≡ x ∗ ≥ x , x ∗ � ρ → τ x : ≡ ∀ y ∗ , y ( y ∗ � ρ y → x ∗ ( y ∗ ) � τ x ( y )) .
Proof Mining General logical metatheorems by Kohlenbach e t al use Dialectica and its variations (within specific formal frameworks).
Proof Mining General logical metatheorems by Kohlenbach e t al use Dialectica and its variations (within specific formal frameworks). Passage to the resulting interpretation survived by mathematical statements of the logical form ∀ x ∃ y A ∃ ( x , y ).
Proof Mining General logical metatheorems by Kohlenbach e t al use Dialectica and its variations (within specific formal frameworks). Passage to the resulting interpretation survived by mathematical statements of the logical form ∀ x ∃ y A ∃ ( x , y ). Metatheorems guarantee the extraction of explicit, computable bound on y from the proof.
Proof Mining General logical metatheorems by Kohlenbach e t al use Dialectica and its variations (within specific formal frameworks). Passage to the resulting interpretation survived by mathematical statements of the logical form ∀ x ∃ y A ∃ ( x , y ). Metatheorems guarantee the extraction of explicit, computable bound on y from the proof. Bounds are highly uniform : depend only on bounding information on the input data (majorants).
Proof Mining General logical metatheorems by Kohlenbach e t al use Dialectica and its variations (within specific formal frameworks). Passage to the resulting interpretation survived by mathematical statements of the logical form ∀ x ∃ y A ∃ ( x , y ). Metatheorems guarantee the extraction of explicit, computable bound on y from the proof. Bounds are highly uniform : depend only on bounding information on the input data (majorants). We will see examples of metatheorems adapted for specific mathematical situations.
Proof Mining By the logical metatheorems we cannot know a priori:
Proof Mining By the logical metatheorems we cannot know a priori: difficulty of extraction
Proof Mining By the logical metatheorems we cannot know a priori: difficulty of extraction complexity (but possible estimation by looking at proof)
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