The mathematics and metamathematics of weak analysis Fernando - PowerPoint PPT Presentation

The mathematics and metamathematics of weak analysis Fernando Ferreira Universidade de Lisboa Measuring the Complexity of Computational Content: Weihrauch Reducibility and Reverse Analysis Schloss Dagstuhl September 20-25, 2015

Second-order arithmetic ◮ Z 2 : Hilbert-Bernays ◮ The big five : Π 1 1 -CA 0 ATR 0 ACA 0 WKL 0 RCA 0 ◮ RCA ⋆ 0 ◮ Weak analysis : BTPSA TCA 2 BTFA

Weak Analysis “To find a mathematically significant subsystem of analysis whose class of provably recursive functions consist only of the computationally feasible ones.” Wilfried Sieg (1988) ◮ BTFA: Base theory for feasible analysis Real numbers, continuous functions, intermediate-value theorem. With (versions of) weak K¨ onig’s lemma: Heine-Borel theorem, uniform continuity theorem. ◮ TCA 2 : Theory of counting arithmetic (analysis) Riemann integration and the fundamental theorem of calculus. ◮ BTPSA: Base theory for polyspace analysis

Basic set-up (fourteen open axioms) x ε = x x × ε = ε x ( y 0 ) = ( xy ) 0 x × y 0 = ( x × y ) x x ( y 1 ) = ( xy ) 1 x × y 1 = ( x × y ) x x 0 = y 0 → x = y x 1 = y 1 → x = y x ⊆ ε ↔ x = ε x ⊆ y 0 ↔ x ⊆ y ∨ x = y 0 x ⊆ y 1 ↔ x ⊆ y ∨ x = y 1 x 0 � = y 1 x 0 � = ε x 1 � = ε We abbreviate x ≤ y for 1 × x ⊆ 1 × y . We write x ≡ y for x ≤ y ∧ y ≤ x .

Basic set-up (induction on notation) We abbreviate x ⊆ ∗ y for ∃ w ( wx ⊆ y ) . A subword quantification is a quantification of the form ∀ x ⊆ ∗ t ( . . . ) or ∃ x ⊆ ∗ t ( . . . ) . Definition A Σ b 1 -formula is a formula of the form ∃ x ≤ t φ ( x ) , where φ is a subword quantification (sw.q.) formula. Note Σ b 1 -formulas define the NP-sets. Definition The theory Σ b 1 -NIA is the theory constituted by the basic fourteen axioms and the following form of induction on notation: φ ( ε ) ∧ ∀ x ( φ ( x ) → φ ( x 0 ) ∧ φ ( x 1 )) → ∀ x φ ( x ) , where φ ∈ Σ b 1 .

The polytime functions ◮ Initial functions C 0 ( x ) = x 0 and C 1 ( x ) = x 1 Projections Q ( x , y ) = 1 ↔ x ⊆ y ; Q ( x , y ) = 0 ∨ Q ( x , y ) = 1 ◮ Derived functions By generalized composition By bounded recursion on notation: f (¯ x , ǫ ) = g (¯ x ) f (¯ x , y 0 ) = h 0 (¯ x , y , f (¯ x , y )) | t (¯ x , y ) f (¯ x , y 1 ) = h 1 (¯ x , y , f (¯ x , y )) | t (¯ x , y ) , where t is a term of the language and q | t is the truncation of q at the length of t . Note We can introduce, via an extension by definitions, the polytime functions in the theory Σ b 1 - NIA . Actually, we can see the latter theory as the extension of a quantifier-free calculus PTCA .

Buss’ witness theorem Theorem (Buss) If Σ b 1 - NIA ⊢ ∀ x ∃ y θ ( x , y ) , where θ ∈ Σ b 1 , then there is a polytime description f such that PTCA ⊢ θ ( x , f ( x )) .

Polytime arithmetic in Σ b 1 -NIA ◮ N 1 : tally numbers (elements u such that u = 1 × u ). Model of I ∆ 0 , but one can also make definitions by bounded recursion on the tally part. ◮ N 2 : dyadic natural numbers of the form 1 w or ε , where w is a binary string ( w ∈ W ). Polytime arithmetic. ◮ D : dyadic rational numbers. Have the form �± , x , y � , where x (resp. y ) is ε or a binary string starting with 1 (resp. ending with 1). Dense ordered ring without extremes. ◮ Given n ∈ N 1 , 2 n is � + , 1 00 . . . 0 , ǫ � ; 2 − n is � + , ǫ, 00 . . . 0 1 � . � �� � � �� � n zeros n − 1 zeros ◮ D is not a field but it is always closed by divisions by tally powers of 2.

Buss’ theorem on bounded collection Definition A bounded formula is a formula obtained from the atomic formulas using propositional connectives and bounded quantifications, i.e., quantifications of the form ∃ x ≤ t φ ( x ) or ∀ x ≤ t φ ( x ) . These formulas define the predicates in the polytime hierarchy. Definition The bounded collection scheme B Σ 1 is constituted by the formulas: ∀ x ≤ a ∃ y ρ ( x , y ) → ∃ b ∀ x ≤ a ∃ y ≤ b ρ ( x , y ) , where ρ is a bounded formula. ◮ A Σ 1 -formula is a formula of the form ∃ x ρ ( x ) , where ρ is a bounded formula. These formulas define the recursively enumerable sets. Π 1 - formulas are defined dually. ◮ A Π 2 -formula is a formula of the form ∀ x ∃ y ρ ( x , y ) , where ρ is a bounded formula. Theorem (Buss) Σ b 1 - NIA + B Σ 1 is Π 2 -conservative over Σ b 1 - NIA .

The second-order theory BTFA Definition BTFA is the second-order theory whose axioms are Σ b 1 -NIA + B Σ 1 (allowing second-order parameters) plus the following recursive comprehension scheme: ∀ x ( ∃ y φ ( x , y ) ↔ ∀ z ϕ ( x , z )) → ∃ X ∀ x ( x ∈ X ↔ ∃ y φ ( x , y )) where φ is a ∃ Σ b 1 -formula and ϕ is a ∀ Π b 1 -formula, possibly with first and second-order parameters, and X does not occur in φ or ϕ . Theorem (FF) The theory BTFA is first-order conservative over Σ b 1 - NIA + B Σ 1 .

Simple consequences of recursive comprehension A function f : X �→ Y is given by a set of ordered pairs. We can state f ( x ) ∈ Z in two ways: x ∈ X ∧ ∃ y ( � x , y � ∈ f ∧ y ∈ Z ) x ∈ X ∧ ∀ y ( � x , y � ∈ f → y ∈ Z ) Thus, { x ∈ X : f ( x ) ∈ Z } exists in BTFA. Similar thing for the composition of two functions. Proposition The theory BTFA proves the ∃ Σ b 1 -path comprehension scheme, i.e., Path ( φ x ) → ∃ X ∀ x ( φ ( x ) ↔ x ∈ X ) , where φ is ∃ Σ b 1 -formula. Proof. φ ( x ) is equivalent to ∀ y ( y ≡ x ∧ y � = x → ¬ φ ( x )) .

Real numbers in BTFA Definition We say that a function α : N 1 �→ D is a real number if | α ( n ) − α ( m ) | ≤ 2 − n for all n ≤ m . Two real numbers α and β are said to be equal , and we write α = β , if ∀ n ∈ N 1 | α ( n ) − β ( n ) | ≤ 2 − n + 1 . The real number system is an ordered field. The relations α = β , α ≤ β , α + β ≤ γ , . . . are ∀ Π b 1 -formulas, while α � = β , α < β , . . . are ∃ Σ b 1 -formulas. A dyadic real number is a triple of the form �± , x , X � where x ∈ N 2 and X is an infinite path.

Continuous partial functions Definition Within BTFA, a (code for a) continuous partial function from R into R is a set of quintuples Φ ⊆ W × D × N 1 × D × N 1 such that: 1. if � x , n � Φ � y , k � and � x , n � Φ � y ′ , k ′ � , then | y − y ′ | ≤ 2 − k + 2 − k ′ ; 2. if � x , n � Φ � y , k � and � x ′ , n ′ � < � x , n � , then � x ′ , n ′ � Φ � y , k � ; 3. if � x , n � Φ � y , k � and � y , k � < � y ′ , k ′ � , then � x , n � Φ � y ′ , k ′ � ; where � x , n � Φ � y , k � stands for ∃ Σ b 1 -relation ∃ w � w , x , n , y , k � ∈ Φ , and where � x ′ , n ′ � < � x , n � means that | x − x ′ | + 2 − n ′ < 2 − n . Definition Let Φ be a continuous partial real function of a real variable. We say that a real number α is in the domain of Φ if � | α − x | < 2 − n ∧ � x , n � Φ � y , k � � ∀ k ∈ N 1 ∃ n ∈ N 1 ∃ x , y ∈ D .

Continuous functions (continued) Definition Let Φ be a continuous partial real function and let α be a real number in the domain of Φ . We say that a real number β is the value of α under the function Φ , and write Φ( α ) = β , if ∀ x , y ∈ D ∀ n , k ∈ N 1 [ � x , n � Φ � y , k � ∧ | α − x | < 2 − n → | β − y | ≤ 2 − k ] . Note Φ( α ) = β is a ∀ Π b 1 notion. Etc. Theorem ( BTFA) Let Φ be a continuous partial real function and let α in the domain of Φ . Then there is a dyadic real number β such that Φ( α ) = β . Moreover, this real number is unique. Corollary Every real number can be put in dyadic form.

Intermediate value theorem Theorem ( BTFA) If Φ is a continuous function which is total in the closed interval [ 0 , 1 ] and if Φ( 0 ) < 0 < Φ( 1 ) , then there is a real number α ∈ [ 0 , 1 ] such that Φ( α ) = 0 . ◮ The real system constitutes a real closed ordered field. ◮ Can define polynomials of tally degree as functions F : { i ∈ N 1 : i ≤ d } × N 1 → D such that, for every i ≤ d , the function γ i defined by γ i ( n ) = F ( i , n ) is a real number. ◮ Given P ( X ) = γ d X d + · · · + γ 1 X + γ 0 can define it as a continuous function. ◮ Generalize to series. Can introduce some transcendental functions. This has not been worked out.

Digression on interpretability in Robinson’s Q ◮ The theories I ∆ 0 + Ω n are interpretable in Robinson’s Q. ◮ Ω n + 1 means that the logarithmic part satisfies Ω n . The RSUV isomorphism characterizes the theory of the logarithmic part of a model (and vice-versa). ◮ Hence, lots of interpretability in Q. Basically, it includes any computations that take a (fixed) iterated exponential number of steps. The “fixed” is for the number of iterations. ◮ Note that I ∆ 0 + exp is not interpretable in Q. Theorem (FF) The theory BTFA is interpretable in Robinson’s Q . Corollary (FF / H. Friedman) Tarski’s theory of real closed ordered fields is interpretable in Q .