The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

The Query Complexity of Real Functionals Hugo Férée, Walid Gomaa, Mathieu Hoyrup

School on Computation and Computability in Dynamics ICTP Trieste, July 29th, 2014 Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

1 2 3 4

Introduction

Complexity of Norms Query Complexity Summary

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

1 2 3 4

Introduction

Complexity of Norms Query Complexity Summary

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Computing with Real Numbers (qn )n∈N

x if ∀n, |x − qn | ≤ 2−n

Computation model: Oracle Turing Machines Finite-time computation

−→ finite number of queries −→ finite knowledge of the input

Bounding computation time

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

(1)

⇐⇒ bounding computational power ⇐⇒ bounding number of queries The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Computing with Real Numbers Cont’d f ∈ C[0, 1] ⇐⇒ ∃µ, fQ :

modulus of continuity : µ : N → N

|x − y | ≤ 2−µ(n) =⇒ |f (x) − f (y )| ≤ 2−n

approximation function: fQ : Q × N → Q

|fQ (q, n) − f (q)| ≤ 2−n

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Computing with Real Numbers Cont’d Theorem

f : [0, 1] → R is computable w.r.t. an oracle ⇐⇒ f is continuous. ⇓ Complexity version ⇓

Theorem f : [0, 1] → R is polynomial time computable w.r.t. to an oracle ⇐⇒ its modulus of continuity is bounded by a polynomial. Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Second-Order Complexity [Kawamura& Cook] Regular functions

Σ is a finite alphabet

Oracle machine M realizes a functional: (Σ∗ → Σ∗ ) → (Σ∗ → Σ∗ ) φ 7→ ψ φ M (u): φ : Σ ∗ → Σ∗ φ is regular: |u| ≤ |v | ⇒ |φ(u)| ≤ |φ(v )| (consistency condition for complexity) u ∈ Σ∗

Mφ (u) = ψ(u)

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Second-Order Complexity Regular functions cont’d

The size of φ:

|φ| : N → N

Pairing of regular φ, ψ:

|φ|(n) = |φ(0n )|

hφ, ψi (0u) = φ(u)10|ψ(u)|

hφ, ψi (1u) = ψ(u)10|φ(u)|

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Second-Order Complexity Second-order polynomial

every c ∈ N is a second-order polynomial

every first-order variable n is a second-order polynomial

if P, Q are second-order polynomials, then so are: P + Q, PQ, and X (P), where X is a second-order variable P(X ) = X (x) + X (X (x) · X (x)) + 4 X (x) = x 3

P(X )(n) = n3 + n18 − n2 + 4

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Representation of f ∈ C[0, 1]

Modulus µ : N → N ⇒ regular µ¯ : Σ∗ → Σ∗ : µ¯(u) = 0µ(|u|)

Approximation function fQ : Q × N → Q ⇒ regular ¯fQ : Σ∗ × Σ∗ → Σ∗ : ¯fQ (u, v ) = a0n 1w

( n + dw fQ (du , |v |) = −n − 1 + dw

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

a=0 a=1

The Query Complexity of Real Functionals

(2)

Outline Introduction Complexity of Norms Query Complexity Summary

Representation of f ∈ C[0, 1] Cont’d D E ¯f = µ¯, ¯fQ

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Second-Order Complexity Running in polynomial time

Mφ (u) runs in poly-time if:

There exists a second-order polynomial P(n, X ) such that for any regular function φ : Σ∗ → Σ∗ and u ∈ Σ∗ , Mφ (u) halts in at most P(|u|, |φ|) steps.

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

1 2 3 4

Introduction

Complexity of Norms Query Complexity Summary

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Polynomial-Time Computable Norms

F is poly-time computable: F is not complete and

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Dependence of a Norm on a Point A norm F and a point α ∈ [0, 1]: dF ,α (n) =

sup{l : ∃f ∈ Lip1 , Supp(f ) ⊆ N (α, 1/l) and F (f ) > 2−n } α

|{z}

1/l

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

f

The Query Complexity of Real Functionals

Example

Outline Introduction Complexity of Norms Query Complexity Summary

F = ||.||∞

F = ||.||1

=⇒

=⇒

dF ,α (n) ∼ 2n

dF ,α (n) ∼ 2 2 n

Let Q = {q0 , q1 , . . .} be a dense subset of [0, 1]: Then

F (f ) =

dF ,qi (n) ≥ 2n−i , dF ,α (n) ≤

X |f (qi )|

(n + 1)2 , ε

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

i

2i

n≥i

for almost all α

The Query Complexity of Real Functionals

(3) (4)

(5)

Relevant Points

Outline Introduction Complexity of Norms Query Complexity Summary

Norm of f depends on the values of f at points of high dependence (α ∈ [0, 1] : dF ,α is large) Definition

α is relevant if ∃c > 0, ∀n,

dF ,α (n) ≥ c · 2 2 n

Rk = {α : ∀n, dF ,α (n) ≥ 2n−k }, All relevant points:

R=

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

relevant up to constant k

[

Rk

k

The Query Complexity of Real Functionals

Relevant Points

Outline Introduction Complexity of Norms Query Complexity Summary

Examples

Example

For ||.||∞ and ||.||1 , R = [0, 1]. Example P F (f ) = i relevant.

|f (qi )| . 2i

The {qi } is a dense subset of [0, 1]. The qi ’s are

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Relevant Points Cont’d

A norm should look everywhere to separate different functions: Theorem R is dense.

F (f ) up to some precision depends only on Rµf (k ) , so the points of Rµf (k ) are relevant to evaluate the norm of a function up to precision k : Theorem

f = 0 on R2µf (k ) =⇒ F (f ) ≤ c.2−k . µf : modulus of f

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

R is Dense (Proof)

F (h0 ) ≥ 2−c Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

R is Dense (Proof)

F (h1 ) ≥ 2−c−2 Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

R is Dense (Proof)

F (h2 ) ≥ 2−c−4

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

R is Dense (Proof)

F (hn ) ≥ 2−c−2n

dF ,α (2n + c) ≥ 2n+p =⇒ α ∈ R Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

1 2 3 4

Introduction

Complexity of Norms Query Complexity Summary

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Complexity of Norms

Complexity Restriction =⇒

Bounding computation time (power) and bounding the number and size of queries

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Query Complexity

Definition Qn : oracle calls of F on x 7→ 0 with precision 2−n . Definition F has a polynomial query complexity if it is computable by a relativized OTM with |Qn | ≤ P(n). Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Query Complexity Cont’d

Theorem If F has polynomial query complexity, then 1 2

Almost every point has a polynomial dependency (dF ,α ∈ P for almost all α). The set of relevant points has Lebesgue measure 0.

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Query Complexity Cont’d Proposition

F has polynomial query complexity n =⇒ ∃α, dF2,α (n) is bounded by a polynomial.

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

Query Complexity Cont’d Open question

Can generalize to any F : C[0, 1] → R?

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

One Oracle Access Case

Theorem The following are equivalent:

F is computable by a polynomial time machine doing only one oracle query ∀f , F (f ) = φ(f (α)) where: α ∈ Poly (R) φ ∈ Poly (R → R) φ is uniformly continuous

Open question

Generalization to any finite number of queries? Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Outline Introduction Complexity of Norms Query Complexity Summary

1 2 3 4

Introduction

Complexity of Norms Query Complexity Summary

Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals

Summary

Outline Introduction Complexity of Norms Query Complexity Summary

Set-up: Oracle Turing machine + Second-order complexity [Kawamura& Cook]

Introduced notions of dependence function dF ,α and relevant point Norm depends on values of high dependence (relevant) such points are dense Polynomial query complexity ⇒ almost all points have polynomial dependency Poly-time norms depend on few number of points, but depends significantly on them Hugo Férée, Walid Gomaa, Mathieu Hoyrup

The Query Complexity of Real Functionals