The millennium question over the reals, the complex numbers and - - PowerPoint PPT Presentation

Technion, Spring semester 2013 238900-13

The millennium question over the reals, the complex numbers and other general structures.

Research seminar 238900-13 Johann A. Makowsky∗

∗ Faculty of Computer Science,

Technion - Israel Institute of Technology, Haifa, Israel janos@cs.technion.ac.il

Graph polynomial project: http://www.cs.technion.ac.il/∼janos/RESEARCH/gp-homepage.html

File:title 1

Technion, Spring semester 2013 238900-13-1

Lecture 1: Summary

• Introducing the topic.
• Register machines for arbitrary rings R.
• Basic observations concerning unit cost over R.
• Defining decidability DECR over R.
• Defining PR and NPR over R.
• Proving computability of NPR ⊂ DECR over the real and complex num-

bers using quantifier elimination (QE).

• What is QE?

File:lec-1 2

Technion, Spring semester 2013 238900-13-2

Lecture 2: Summary

We shall consider two cases where PR = NPR. 0n one case we add the FORTRAN-function: sin(x), in the oder case we disregard multiplication and the order.

• Adding sin(x) to R.

R = R, +, ×, <, sin x, 0, 1. We define Psin, NPsin and DECsin in the natural way.

• Disregarding multiplication and order and test only for equality:

R = R, +, 0, 1 We define Plin, NPlin and DEClin in the natural way. Theorem:(Klaus Meer) Psin = NPsin and Plin = NPlin.

Klaus Meer,

• A note on a P = NP for a restricted class of real machines,

Journal of Complexity 8 (1992), 451-453

• Real number models under various sets of operations,

Journal of Complexity 9 (1993), 366-372 File:lec-2 3

Technion, Spring semester 2013 238900-13-2

Lecture 2: Summary (contd)

We shall also discuss adding other FORTRAN functions: exp, log, sin

• Adding exp(x) to R.

R = R, +, ×, <, exp(x), 0, 1. We define Pexp and NPexp in the natural way.

• Adding two functions:

F1 = {exp(x), sin(x)} or F2 = {exp(x), log(x)} to R. R = R, +, ×, <, exp(x), log(x), sin(x)0, 1. We define PF1, PF2, NPF1 and NPF2 in the natural way. Theorem:(Mihai Prunescu) Pexp = NPexp and PF1 = NPF1. Note: PF2=NPF2 remains open.

Mihai Prunescu,

• P = NP for a the reals with Various Analytic Functions.

Journal of Complexity 17 (2001), 17-26 File:lec-2 4

Technion, Spring semester 2013 238900-13-2

Exercise: Computability of Z in R

We want to decide whether the sets Z ⊂ R and Q ⊂ R are computable.

• The decision problems for the sets Z and Q are computable in BSS over

the field R, but there is no bound on the length of the computation.

• Are the problems (R, Z) and (R, Q) in NPR?
• The problems (R, Z) and (R, Q) are in Psin. In fact they are computable

in constant time. We use the fact that sin(k · π) = 0 for k ∈ Z.

File:lec-2 5

Technion, Spring semester 2013 238900-13-2

The set AMeer

We look at the set AMeer = A = {t ∈ [0, 2π] : ∃k ∈ N (k · t 2π ∈ N)} . We note that

• t ∈ A iff

t 2π ∈ Q ∩ [0, 1], hence A is countable and dense in [0, 2π].

• The problem (R, [0, 2π]) is in Psin.

Here we use a constant a = 2π.

• We study the problem (R, AMeer), respectively ([0, 2π], AMeer).

File:lec-2 6

Technion, Spring semester 2013 238900-13-2

([0, 2π], AMeer) is in NPsin

• Input x ∈ R always has size 1.
• We show that using a single guess ([0, 2π], AMeer) can be solved

in constant time.

• 1. GUESS k ∈ R.
• 2. TEST k ≥ 0 and then sin(kπ) = 0.
• 3. Two yes show that k ∈ N.
• 4. TEST sin(k·x

2 ) = 0.

• 5. Yes shows that x ∈ A.

Q.E.D.

File:lec-2 7

Technion, Spring semester 2013 238900-13-2

Real-analytic functions

S.G. Krantz and H.R. Parks, A Primer for Real-Analytic Functions, Birkh¨ auser, 2002 (2nd edition)

A function f : R → R is real-analytic on an open set U ⊆ R if for all x ∈ U

• f has derivatives of all orders at x, and
• for every a ∈ U there is a neighborhood a ∈ V ⊂ U such that for all x ∈ V

f agrees with its Taylor series, i.e., f(x) =

• n

f(n)(a) n! (x − a)n Examples: polynomials, 1

x, sin, cos, exp, log are real analytic.

The function f(x) = x ∈ Q 1 x ∈ R − Q is not real-analytic.

File:lec-2 8

Technion, Spring semester 2013 238900-13-2

Properties of real-analytic functions

Proposition:(Classical)

• The set of real-analytic functions on U is closed under

scalar multiplication, pointwise addition multiplication and composition.

• The reciprocal of an analytic function that is nowhere zero is analytic.
• The inverse of an invertible analytic function whose derivative is nowhere

zero is analytic.

• Assume rn are distinct zeroes of f and lim rn = r is in a connected

component Dr the domain D of f. Then f(x) = 0 for all x ∈ Dr. Hence, if f is not constant on Dr, it has at most countable many zeroes in Dr.

File:lec-2 9

Technion, Spring semester 2013 238900-13-2

([0, 2π], AMeer) is not in DECsin hence not Psin.

There is no deterministic program using sin(x) which always terminates and decides ([0, 2π], AMeer).

• We proceed by contradiction.
• We do not allow division, and discuss later what effect divsion has.
• Assume we have a program which decides ([0, 2π], AMeer). It may have a

fixed number of constants, c1, . . . , cs ∈ R.

File:lec-2 10

Technion, Spring semester 2013 238900-13-2

Evaluating paths in the computation tree

Let γ be a path in the (unwound) computation tree of a program over R and sin.

• γ evaluates a term Tγ(x, c1, . . . , cs).
• Tγ(x, c1, . . . , cs) represents a a real fγ(x) function which is real-analytic.
• Since the program always terminates, there are at most countably many

paths γ.

• We can replace functions fγ(x) which are identically 0 by constant as-

signments.

• Let So there at most countably many values x ∈ R for which fγ(x) = 0.

File:lec-2 11

Technion, Spring semester 2013 238900-13-2

Heading for the contradiction

We now look at the set B = {t ∈ [0, 2π] : ∃γ (fγ(t) = 0)} We note:

• B is countable.
• Let t0 ∈ [0, 2π] − B. So for all γ we have fγ(t0) = 0.
• Since each path γ is finite there is some open set U(t0) ⊆ R such that
• ur program gives the saem answer for all inputs x ∈ U(t0), i.e.,

U(t0) ⊆ A or U(t0) ⊆ [0, 2π] − A.

• But this is impossible since A is countable and dense in [0, 2π].

Q.E.D.

File:lec-2 12

Technion, Spring semester 2013 238900-13-2

Meer’s Theorem

We have shown Theorem:(K. Meer 1992) Let F be any set of real-analytic functions such that (R, Z) ∈ NPF. Then PF = NPF. Problem: What limitations does the requirement (R, Z) ∈ NPF impose?

• For a set D ⊆ R the problem is in NPF iff it is existentially first order de-

finable using addition, multiplication, order, constants and unary function symbols for functions from F.

• For F = {exp(x)} we have (R, Z) ∈ NPF.

This is so, because every set D ⊆ R with (R, D) ∈ NPexp has only finitely many connected components.

• L. van den Dries and C. Miller, The field of reals with restricted analytic functions and

exponentiation, Israel Journal of Mathematics (1994). File:lec-2 13

Technion, Spring semester 2013 238900-13-2

The role of division

Our proof shows really: Theorem: (Meer-Prunescu) Let D ⊆ R and R − D be dense in [a, b] ⊂ R. Then ([a, b], D) ∈ DECF for any set F of real-analytic functions.

• This allows to include division, although division is not a total function.
• To obtain Then PF = NPF we still need to show that ([a, b], D) ∈ NPF.
• To show that Pexp = NPexp Prunescu uses a different approach.

File:lec-2 14

Technion, Spring semester 2013 238900-13-2

Semi-algebraic sets

Let D ⊂ Rn.

• D is semi-algebraic if it is the solution set of a quantifierfree first order formula over

the ordered field of R.

• A function f : Rn → R is semi-algebraic, if its graph is a semi-algebraic set.
• A function f is essentially non-semi-algebraic if for no open set U ⊆ Rn the function f|U

is semi-algebraic.

• Let F be a set of real-analytic functions. f is semi-analytic if it is the solution set of a

quantifierfree first order formula over the ordered field of R with function symbols for functions from F.

• f is sub-analytic if it is the solution set of a first order formula over the ordered field
• f R with function symbols for functions from F.
• A real-analytic function is tame on U if it has no analytical singularities on the boundaries
• f U.
• A total real-analytic function f is always tame.

File:lec-2 15

Technion, Spring semester 2013 238900-13-2

Prunescu’s Theorem

Theorem: (M. Prunescu 2001) Let F be a set of real-analytic tame functions containing at least one function which is essentially non semi-algebraic. Then PF = NPF. Comments:

• F0 = {exp} is tame.

F1 = {exp, log} is not tame. F2 = {exp, log |(1,∞)} is tame.

• AP runescu = {(x, y, z) ∈ R3 : y > 0 ∧ z = y · exp(x

y)}

• AP runescu ∈ NPFi for i = 0, 1, 2.
• AP runescu ∈ PF0.

AP runescu ∈ PF1 in constant time. AP runescu ∈ PF2 but AP runescu ∈ DECF2.

File:lec-2 16

Technion, Spring semester 2013 238900-13-2

Still to be done (as projects)

• Meer’s Theorem: Plin = NPlin.
• Note that: P<

lin = NP< lin is still open.

• However, it is known that NP<

lin ⊆ DEC< lin.

This is shown using quantifier elimination.

• Prunescu’s Theorem for tame functions.

File:lec-2 17