tba
play

tba Arnold Beckmann Department of Computer Science University of - PowerPoint PPT Presentation

Nov 11, 2023 •336 likes •780 views

Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content tba Arnold Beckmann Department of Computer Science University of Wales Swansea UK 5 April 2008 Workshop in Honour of Wilfried Buchholz 60th Birthday Munich

  1. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content tba Arnold Beckmann Department of Computer Science University of Wales Swansea UK 5 April 2008 Workshop in Honour of Wilfried Buchholz’ 60th Birthday Munich Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  2. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Proof Notations for Bounded Arithmetic Arnold Beckmann (joint work with Klaus Aehlig) Department of Computer Science University of Wales Swansea UK 5 April 2008 Workshop in Honour of Wilfried Buchholz’ 60th Birthday Munich Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  3. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Outline of talk Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Arnold Beckmann tba

  4. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Language of Bounded Arithmetic (BA) Language of first order arithmetic similar to Peano Arithmetic Non-logical symbols: { 0 , 1 , + , · , ≤} + {| . | , # , . . . } | x | = length of binary representation of x 2 | x |·| y | produces polynomial growth rate x # y = Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  5. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Language of Bounded Arithmetic (BA) Language of first order arithmetic similar to Peano Arithmetic Non-logical symbols: { 0 , 1 , + , · , ≤} + {| . | , # , . . . } | x | = length of binary representation of x 2 | x |·| y | produces polynomial growth rate x # y = Bounded Formulas: ˆ Σ b 1 : ∃ x 1 ≤ s 1 ∀ y ≤ | t | ϕ ( x 1 , y ) ˆ Σ b 2 : ∃ x 1 ≤ s 1 ∀ x 2 ≤ s 2 ∃ y ≤ | t | ϕ ( x 1 , x 2 , y ) . . . with quantifier-free ϕ Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  6. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Bounded Arithmetic theories Induction: Φ-Ind : ϕ (0) ∧ ∀ x ( ϕ ( x ) → ϕ ( x + 1)) → ∀ x ϕ ( x ) ϕ (0) ∧ ∀ x ( ϕ ( x ) → ϕ ( x + 1)) → ∀ x ϕ ( | x | ) Φ-LInd : where ϕ ∈ Φ Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  7. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Bounded Arithmetic theories Induction: Φ-Ind : ϕ (0) ∧ ∀ x ( ϕ ( x ) → ϕ ( x + 1)) → ∀ x ϕ ( x ) ϕ (0) ∧ ∀ x ( ϕ ( x ) → ϕ ( x + 1)) → ∀ x ϕ ( | x | ) Φ-LInd : where ϕ ∈ Φ BASIC = a set of open formulas defining the non-logical symbols. Theories: Pick a set of formulas and an induction scheme, form the theory BASIC + all instances of induction for formulas from the set just picked. BASIC + ˆ S 1 Σ b Examples: = 1 -LInd 2 BASIC + ˆ S 2 Σ b = 2 -LInd 2 Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  8. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Definable functions f is ˆ there exists ϕ ∈ ˆ Σ b 1 -definable in S 1 Σ b iff 1 such that 2 ◮ f ( x ) = y ⇐ ⇒ N � ϕ ( x , y ) ◮ S 1 2 ⊢ ∀ x ∃ y ϕ ( x , y ) Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  9. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Definable functions f is ˆ there exists ϕ ∈ ˆ Σ b 1 -definable in S 1 Σ b iff 1 such that 2 ◮ f ( x ) = y ⇐ ⇒ N � ϕ ( x , y ) ◮ S 1 2 ⊢ ∀ x ∃ y ϕ ( x , y ) Theorem (Buss ’86) f is ˆ Σ b 1 -definable in S 1 f ∈ FP iff 2 Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  10. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Definable functions – the general case bounded polynomial time arithmetic hierarchy of f is ˆ Σ b i -definable in T theories functions iff there exists ϕ ∈ ˆ Σ b i such that S 3 FP Σ p ◮ f ( x ) = y ˆ Σ b 2 2 3 ⇐ ⇒ N � ϕ ( x , y ) S 2 FP Σ p ˆ Σ b 1 2 2 ◮ T ⊢ ∀ x ∃ y ϕ ( x , y ) S 1 ˆ Σ b FP 2 1 Arnold Beckmann tba

  11. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Definable search problems Theorem (Buss ’86) 2 = FP Σ p ˆ Σ b i -definable functions in S i i − 1 Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  12. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Definable search problems Theorem (Buss ’86) 2 = FP Σ p ˆ Σ b i -definable functions in S i i − 1 Theorem (Kraj´ ıˇ cek’93) 2 = FP Σ p ˆ Σ b i +1 -definable multi-functions in S i i [ wit , O (log n )] Arnold Beckmann

  13. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Definable search problems Theorem (Buss ’86) 2 = FP Σ p ˆ Σ b i -definable functions in S i i − 1 Theorem (Kraj´ ıˇ cek’93) 2 = FP Σ p ˆ Σ b i +1 -definable multi-functions in S i i [ wit , O (log n )] Theorem (Buss, Kraj´ ıˇ cek’94) 2 = projection of PLS Σ p ˆ Σ b i − 1 -definable multi-functions in S i i − 2 Arnold Beckmann

  14. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Independence results Main open problem for bounded arithmetic: Does the hierarchy of bounded arithmetic theories collapse? Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  15. Bounded Arithmetic Dynamic Ordinals Bounded Arithmetic Proof Notations Definable functions Computational Content Independence results Main open problem for bounded arithmetic: Does the hierarchy of bounded arithmetic theories collapse? Theorem (Kraj´ ıˇ cek, Pudl´ ak, Takeuti ’91, Kraj´ ıˇ cek ’93) If the levels of the polynomial time hierarchy of predicates (PH) are separated, then the levels of bounded arithmetic theories (BA) are separated as well. In particular, if Σ p i +2 � = Π p 2 � = S i +1 i +2 , then S i . 2 Theorem (Buss ’95, Zambella ’96) BA collapses iff PH collapses provable in BA Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  16. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Dynamic ordinals – a picture dynamic (log n ) O (1) ordinals propositional proof systems S 1 2 bounded polynomial ptime arithmetic time functions hierarchy Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  17. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Dynamic ordinals – a picture dynamic (log n ) O (1) ordinals propositional proof systems S 1 2 bounded polynomial ptime arithmetic time functions hierarchy Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  18. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Proposed future work at MFO’05: Adapt finitary notations for infinitary derivations to Bounded Arithmetic setting. Wilfried Buchholz. Notation systems for infinitary derivations. Archive for Mathematical Logic , 30:277–296, 1991. Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  19. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Proposed future work at MFO’05: Adapt finitary notations for infinitary derivations to Bounded Arithmetic setting. Wilfried Buchholz. Notation systems for infinitary derivations. Archive for Mathematical Logic , 30:277–296, 1991. Klaus Aehlig and Arnold Beckmann. On the computational complexity of cut-reduction. Accepted for publication at LICS 2008. Full version available as Technical Report CSR15-2007, Department of Computer Science, Swansea University, December 2007. http://arxiv.org/abs/0712.1499 . Arnold Beckmann (joint work with Klaus Aehlig) Proof Notations for Bounded Arithmetic

  20. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Finitary Proof System BA ⋆ if � ∆ ∈ BASIC ( Ax ∆ ) ∆ A 0 A 1 A k ( � k ( � A 0 ∧ A 1 ) ( k ∈ { 0 , 1 } ) A 0 ∨ A 1 ) A 0 ∧ A 1 A 0 ∨ A 1 A x ( y ) A x ( t ) ( � y ( � t ( ∀ x ) A ) ( ∃ x ) A ) ( ∀ x ) A ( ∃ x ) A ¬ F , F y (s y ) ( IND y , t F ) ¬ F y (0) , F y (2 | t | ) ¬ F , F y (s y ) ( IND y , n , i ( n , i ∈ N ) ) F ¬ F y ( n ) , F y ( n + 2 i ) C ¬ C ( Cut C ) ∅ Arnold Beckmann tba

  21. Bounded Arithmetic Dynamic Ordinals Proof Notations Computational Content Proof Notations for Bounded Arithmetic H BA : set of closed BA ⋆ -derivations For h ∈ H BA define, following translation into propositional logic tp( h ): denoted last inference h [ j ]: denoted j th subderivation | h | : size = number of inference symbols occurring in h o( h ): height of denoted derivation tree Using auxiliary induction inference symbols ( IND y , n , i ) we can F ensure | h [ i ] | ≤ | h | Arnold Beckmann tba

Recommend

More recommend