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

◮ Z2: Hilbert-Bernays ◮ The big five:

Π1

1-CA0

ATR0 ACA0 WKL0 RCA0

◮ RCA⋆ ◮ Weak analysis:

BTPSA TCA2 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¨

• nig’s lemma: Heine-Borel theorem,

uniform continuity theorem.

◮ TCA2: 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)

xε = x x × ε = ε x(y0) = (xy)0 x × y0 = (x × y)x x(y1) = (xy)1 x × y1 = (x × y)x x0 = y0 → x = y x1 = y1 → x = y x ⊆ ε ↔ x = ε x ⊆ y0 ↔ x ⊆ y ∨ x = y0 x ⊆ y1 ↔ x ⊆ y ∨ x = y1 x0 = y1 x0 = ε x1 = 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 Σb

1-formula is a formula of the form ∃x ≤ t φ(x), where φ is a

subword quantification (sw.q.) formula. Note Σb

1-formulas define the NP-sets.

Definition The theory Σb

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(x0) ∧ φ(x1)) → ∀xφ(x), where φ ∈ Σb

1.

The polytime functions

◮ Initial functions

C0(x) = x0 and C1(x) = x1 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, y0) = h0(¯ x, y, f(¯ x, y))|t(¯

x,y)

f(¯ x, y1) = h1(¯ 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 Σb

1-NIA. Actually, we can see the latter theory

as the extension of a quantifier-free calculus PTCA.

Buss’ witness theorem

Theorem (Buss) If Σb

1-NIA⊢ ∀x∃yθ(x, y), where θ ∈ Σb 1, then there is a polytime

description f such that PTCA ⊢ θ(x, f(x)).

Polytime arithmetic in Σb

1-NIA

◮ N1: 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.

◮ N2: dyadic natural numbers of the form 1w 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 ∈ N1, 2n is +, 1 00 . . . 0 n zeros

, ǫ; 2−n is +, ǫ, 00 . . . 0

n−1 zeros

1.

◮ D is not a field but it is always closed by divisions by tally powers

• f 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 BΣ1 is constituted by the formulas: ∀x ≤ a ∃yρ(x, y) → ∃b ∀x ≤ a ∃y ≤ b ρ(x, y), where ρ is a bounded formula.

◮ A Σ1-formula is a formula of the form ∃xρ(x), where ρ is a

bounded formula. These formulas define the recursively enumerable sets. Π1- formulas are defined dually.

◮ A Π2-formula is a formula of the form ∀x∃yρ(x, y),

where ρ is a bounded formula. Theorem (Buss) Σb

1-NIA + BΣ1 is Π2-conservative over Σb 1-NIA.

The second-order theory BTFA

Definition BTFA is the second-order theory whose axioms are Σb

1-NIA + BΣ1

(allowing second-order parameters) plus the following recursive comprehension scheme: ∀x (∃yφ(x, y) ↔ ∀zϕ(x, z)) → ∃X∀x (x ∈ X ↔ ∃yφ(x, y)) where φ is a ∃Σb

1-formula and ϕ is a ∀Πb 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 Σb

1-NIA + BΣ1.

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 ∃Σb

1-path comprehension scheme, i.e.,

Path(φx) → ∃X∀x(φ(x) ↔ x ∈ X), where φ is ∃Σb

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 : N1 → 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 ∈ N1|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 ∀Πb

1-formulas, while α = β, α < β, . . . are

∃Σb

1-formulas.

A dyadic real number is a triple of the form ±, x, X where x ∈ N2 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 × N1 × D × N1 such that:

• 1. if x, nΦy, k and x, nΦy′, k′, then |y − y′| ≤ 2−k + 2−k′;
• 2. if x, nΦy, k and x′, n′ < x, n, then x′, n′Φy, k;
• 3. if x, nΦy, k and y, k < y′, k′, then x, nΦy′, k′;

where x, nΦy, k stands for ∃Σb

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 ∀k ∈ N1∃n ∈ N1∃x, y ∈ D

• |α − x| < 2−n ∧ x, nΦy, k
• .

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 ∈ N1 [x, nΦy, k ∧ |α − x| < 2−n → |β − y| ≤ 2−k]. Note Φ(α) = β is a ∀Πb

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 ∈ N1 : i ≤ d} × N1 → D such that, for every i ≤ d, the function γi defined by γi(n) = F(i, n) is a real number.

◮ Given P(X) = γdX d + · · · + γ1X + γ0 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 + Ωn are interpretable in Robinson’s Q. ◮ Ωn+1 means that the logarithmic part satisfies Ωn. 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.

Weak K¨

• nig’s lemma

Given a formula φ(x), Tree(φx) abbreviates: ∀x∀y (φ(x) ∧ y ⊆ x → φ(y)) ∧ ∀b∃x ≡ b φ(x). Path(X) abbreviates: Tree((x ∈ X)x) ∧ ∀x∀y (x ∈ X ∧ y ∈ X → x ⊆ y ∨ y ⊆ x). Definition Weak K¨

• nig’s lemma for trees defined by bounded formulas, denoted

by Σ0-WKL, is the following scheme: Tree(φx) → ∃X (Path(X) ∧ ∀x (x ∈ X → φ(x))), where φ is a bounded formula and X is a new second-order variable.

Harrington’s conservation result adapted

Theorem (FF) The theory BTFA +Σ0-WKL is first-order conservative over BTFA. Proof idea. By Harrington forcing. However, the forcing conditions are the infinite trees defined by bounded formulas. BΣ1 is used to show the density

• f the class of trees with a stem of a given (nonstandard) height.

The Heine-Borel theorem

Definition (BTFA) A (code for an) open set U is a set U ⊆ W × D × N1. We say that a real number α is an element of U, and write α ∈ U, if ∃z ∈ D∃n ∈ N1(|α − z| < 2−n ∧ ∃ww, z, n ∈ U). Suppose that U is an open set and that [0, 1] ⊆ U. The Heine-Borel theorem states the existence of k ∈ N1 such that, for all α ∈ [0, 1], ∃z ∈ D, n ∈ N1, w ∈ W (z, n, w ≤ k ∧ |α − z| < 2−n ∧ w, z, n ∈ U). Theorem (BTFA) The Heine/Borel theorem for [0, 1] is equivalent to Πb

1-WKL.

The uniform continuity theorem

Definition Let Φ : [0, 1] → R be a (total) continuous function. We say that Φ is uniformly continuous if ∀k ∈ N1∃m ∈ N1∀α, β ∈ [0, 1](|α − β| ≤ 2−m → |Φ(α) − Φ(β)| < 2−k). Theorem (BTFA) The principle that every (total) real valued continuous function defined

• n [0, 1] is uniformly continuous implies WKL and is implied by

Πb

1-WKL.

The attainement of maximum

The Σb

1-IA induction principle is

φ(ǫ) ∧ ∀x(φ(x) → φ(S(x))) → ∀xφ(x), for φ ∈ Σb

1.

Note Over BTFA, the Σb

1-IA induction principle is equivalent to saying that

every non-empty bounded set X of W has a lexicographic maximum (minimum). Theorem Over BTFA + Σ0-WKL, the following are equivalent: (a) Every (total) real valued continuous function defined on [0, 1] has a maximum. (b) Every (total) real valued continuous function defined on [0, 1] has a supremum. (c) Σb

1-IA.

Counting and polyspace computability

The classe of polyspace computable functions is obtained by adding to the scheme generating the polytime computable functions the scheme of bounded recursion: f(¯ x, ǫ) = g(¯ x) f(¯ x, S(y)) = h(¯ x, y, f(y))|t(¯

x,y)

where S is the successor function in the lexicographic order. The classe of counting (hierarchy of counting functions) is obtained by adding instead the (weaker) scheme of counting: c(¯ x, ε) =

• if f(¯

x, ǫ) = 1 ε

• therwise

c(¯ x, S(y)) =

• S(c(¯

x, y)) if f(¯ x, S(y)) = 1 c(¯ x, y)

• therwise

Note c(¯ x, y) = #{w ≤l y : f(¯ x, w) = 1}.

Second-order bounded variables

X t, Y q, Z r: second-order bounded variables. They have a characteristic axiom: ∀X t∀y(y ∈ X t → y ≤ t) where y does not occur in the term t.

◮ The Σb,1 0 -formulas constitute the smallest class of formulas

containing the atomic formulas closed under bounded first-order

• quantifications. They define the (relativized) polytime hierarchy.

◮ A Σb,1 1 -formula is a formula of the form ∃X tφ(X t), where φ is a

Σb,1

0 -formula. Πb,1 1 -formulas are defined dually. ◮ The second-order bounded formulas constitute the smallest

class of formulas containing the atomic formulas and closed under first and second-order bounded quantifications.

Common axioms

◮ Basic fourteen axioms and characteristic axioms. ◮ Bounded comprehension for Σb,1 0 -formulas φ:

∀b∃X b∀x ≤ b(x ∈ X b ↔ φ(x)).

◮ Induction on notation for Σb,1 0 -formulas. Ordinary induction for

these formulas follows.

◮ Substitution scheme for Σb,1 0 -formulas:

∀x ≤ b∃X z φ(x, X z) → ∃Z q∀x ≤ b ˆ φ(x, Z q), where q is a concretely presented term and ˆ φ is obtained from φ by replacing s ∈ X z by x, s ∈ Z q.

Two second-order bounded theories

Definition The theory Σb,1

1 -NIA is the theory which adds to the commom axioms

induction on notation for Σb,1

1 -formulas.

Theorem (Buss) If Σb,1

1 -NIA ⊢ ∀x∃y φ(x, y), where φ is a Σb,1 1 -formula, then there is a

polyspace description f such that ∀xφ(x, f(x)). Definition The theory TCA (theory of counting arithmetic) is the theory which adds to the common axioms the axiom ∀z∀X z∃Z qCount(Z q, X z), where q is concrete and Count is a Σb,1

0 -formula which expresses that

Z q is the graph of the function x ❀ #{w ≤l z : w ∈ X z}. Theorem (Johannsen & Pollett) If TCA ⊢ ∀x∃y φ(x, y), where φ is a Σb,1

1 -formula, then there is a

description of a counting function f such that ∀xφ(x, f(x)).

Second-order bounded theories (continued)

Theorem To either Σb,1

1 -NIA or TCA, we can add the scheme of collection for

bounded second-order formulas and get a conservative extension with respect to sentences of the form ∀x∃yφ(x, y), where φ is a bounded second-order formula. Lemma The theory TCA proves bounded comprehension for ∆b,1

1 -formulas:

∀x ≤ b (φ(x) ↔ ϕ(x)) → ∃X b∀x (x ∈ X b ↔ φ(x)) where φ is a Σb,1

1 -formula and ϕ is a Πb,1 1 -formula.

The second-order theories

The second-order theories are framed in the language of second-order arithmetic. Second-order bounded variables are canonically interpreted in this language. Definition The theory BTPSA is the theory Σb,1

1 -NIA together with the scheme of

collection for bounded second-order formulas and the following recursive comprehension scheme: ∀x (∃yφ(x, y) ↔ ∀zϕ(x, z)) → ∃X∀x (x ∈ X ↔ ∃yφ(x, y)) where φ is a ∃Σb,1

1 -formula and ϕ is a ∀Πb,1 1 -formula.

The theory TCA2 is as above, but starting with TCA. Theorem (G. Ferreira & FF) The theory BTPSA (resp. TCA2) is conservative over the theory Σb,1

1 -NIA (resp. TCA) with the scheme of collection for bounded

second-order formulas.

Integration and counting

Given X ⊆ N2 a non-empty subset, let ΦX be the continuous function ✲ ✻ y x 1 2 3 4 5 ... 2 ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ where a “spike” follows x exactly when x ∈ X.

Integration and counting (continued)

The counting function f up to b is given simultaneously by: f = {x, n : x, n ∈ N2, x ≤ b + 1, x

0 ΦX(t)dt =R nR}

f = {x, n : x, n ∈ N2, x ≤ b +1, n− 1

2 <R

x

0 ΦX(t)dt <R nR + 1 2}

Get f by (the) recursive comprehension (available in BTFA). How does one prove that the two above definitions coincide? Can we show that, for each x ≤ b, x

0 ΦX(t)dt is equal to a (dyadic) natural

• number. By induction? Prima facie, we do have have this kind of

induction!

• 1. The unbounded quantifiers can be dealt by judicious uses of

bounded collection.

• 2. Σb

1-IA induction is available because we can prove that every

non-empty set has a minimum. Use the integration and the intermediate value theorem!

Counting and integration

◮ If we can count, then we can add: x

• w=0

f(w) = #{u : ∃w ≤2 x∃y <2 f(w) (u = w, y)}. Definition Let Φ be a continuous total function on [0,1]. A modulus of uniform continuity (m.u.c) is a strictly increasing function h : N1 → N1 such that ∀n ∈ N1∀α, β ∈ [0, 1](|α − β| ≤ 2−h(n) → |Φ(α) − Φ(β)| < 2−n). Note Over TCA2 + Σ0-WKL, if Φ is a continuous total function on [0,1] then Φ has a m.u.c.

Counting and integration (continued)

Definition (TCA2) Take Φ a continuous total function on [0,1] with a m.u.c. h. The integral of Φ between 0 and 1 is defined by 1 Φ(t)dt :=R lim

n Sn.

where, for all n ∈ N1, Sn = 2h(n)−1

w=0 1 2h(n) Φ( w 2h(n) , n). Here Φ(r, n) is a

suitable approximation of Φ(r). Note The above definition readily extends to integration for intervals with dyadic rational points as limits. Let d : D → D be: d(x) =      if x < 0 x if 0 ≤ x ≤ 1 1 if 1 < x

The fundamental theorem of calculus

Given Φ a continuous total function on [0,1] with m.u.c. h, define x, nΨy, k as follows: x, y ∈ D ∧ n, k ∈ N1 ∧

• d(x)

Φ(t)dt

• < 1

2k − 1 2n−m−1 , where m ∈ N1 is such that ∀α ∈ [0, 1] |Φ(α)| ≤ 2m.

◮ The above Ψ gives, within TCA2, the definition of the continuous

real function α ❀ α

0 Φ(t)dt. ◮ It is easy to prove that the derivative of Ψ at α is Φ(α).

On continuous functions

◮ Takeshi Yamazaki defined continuity via uniform approximations

• f piecewise linear functions. Uniform continuity is built in.

◮ What about defining continuity via uniform approximations of

polynomials? Do we get a nice theory of integration in BTFA?

◮ Weierstrass’ approximation theorem: every (uniformly)

continuous function on [0,1] is uniformly approximated by polynomials.

◮ Conjecture. Over BTFA (or close enough), Weierstrass’

approximation theorem is equivalent to the totality of exponentiation.

The FAN0 principle

Definition The FAN0 principle is the schema ∀X∃xφ(x, X) → ∃b∀X∃x ≤ b φ(x, X), where φ a second-order bounded formula (possibly with parameters) in which b does not occur. The contrapositive of FAN0 is known as strict Π1

1-reflection.

Theorem (Fernandes) The theory BTPSA+FAN0 (resp. TCA2+FAN0) is conservative over BTPSA (resp. TCA2) with respect to formulas without second-order unbounded quantifications. Proof. A forcing argument ` a la Harrington, where the forcing conditions are infinite trees (defined by second-order bounded formulas) of bounded sets X b (understood as encoding the “binary sequence”

• f its characteristic function).

References: bounded arithmetic

• S. Buss, Bounded Arithmetic. Bibliopolis, Naples, 1986.
• S. Buss, “A conservation result concerning bounded theories and

the collection axiom.” Proc. Amer. Math. Soc. 100, 709-716 (1987). F . Ferreira, “A note on a result of Buss concerning bounded theories and the collection scheme.” Portugaliae Mathematica 52, 331-336 (1995). F . Ferreira, “Polynomial time computable arithmetic.” In: Logic and Computation, W. Sieg (editor), Contemporary Mathematics 106, American Mathematical Society 1990, 137-156.

References: metamathematics of BTFA

F . Ferreira: Polynomial Time Computable Arithmetic and Conservative Extensions. Ph.D. thesis, Pennsylvania State University, 1988. F . Ferreira, “A feasible theory for analysis.” The Journal of Symbolic Logic 59, 1001-1011 (1994).

• A. Fernandes, F

. Ferreira & G. Ferreira, “Techniques in weak analysis for conservation results.” In: New Studies in Weak Arithmetics, P . C´ egielski, Ch. Cornaros and C. Dimitracopoulos (eds.), CSLI Publications (Stanford) and Presses Universitaires (Paris 12) 2013, 115-147. F . Ferreira & G. Ferreira, “Interpretability in Robinson’s Q.” The Bulletin of Symbolic Logic 19, 289-317 (2013).

References: weak analysis in BTFA

• A. Fernandes & F

. Ferreira, “Groundwork for weak analysis.” The Journal of Symbolic Logic 67, 557-578 (2002).

• A. Fernandes & F

. Ferreira, “Basic applications of weak K¨

• nig’s

lemma in feasible analysis.” In: Reverse Mathematics 2001, S. Simpson (editor), Association for Symbolic Logic / A K Peters 2005, 175-188.

• T. Yamazaki, “Reverse mathematics and weak systems of 0-1 for

feasible analysis.” In: Reverse Mathematics 2001, S. Simpson (editor), Association for Symbolic Logic / A K Peters 2005, 394-401.

• A. Fernandes, “The Baire category theorem over a feasible base

theory.” In: Reverse Mathematics 2001, S. Simpson (editor), Association for Symbolic Logic / A K Peters 2005, 164-174.

References: more metamathematics

• J. Johannsen & C. Pollett, “On proofs about threshold circuits and

counting hierarchies.” In: Thirteenth Annual IEEE Symposium on Logic in Computer Science, IEEE Press 1998,, 444-452.

• R. Bianconi, G. Ferreira & E. Silva, “Bounded theories for

polyspace computability.” Portugaliae Mathematica 70, 295-318 (2013).

• A. Fernandes, “Strict Π1

1-reflection in bounded arithmetic.”

Archive for Mathematical Logic 49, 17-34 (2009).

References: more weak analysis

F . Ferreira & Gilda Ferreira, “Counting as integration in feasible analysis.” Mathematical Logic Quarterly 52, 315-320 (2006). F . Ferreira & G. Ferreira, “The Riemann integral in weak systems

• f analysis.” Journal of Universal Computer Science 14,

908-937 (2008).

References: functional interpretations

• S. Cook & A. Urquhart, “Functional interpretations of feasibly

constructive arithmetic.” Annals of Pure and Applied Logic 63, 103-200 (1993). F . Ferreira & P . Oliva, “Bounded functional interpretation and feasible analysis.” Annals of Pure and Applied Logic 145, 115-129 (2007).

Thank you