A Measure of Space for Computing over the Reals Paulin Jacob e de - PowerPoint PPT Presentation

A Measure of Space for Computing over the Reals Paulin Jacob´ e de Naurois LIPN - Universit´ e Paris XIII A Measure of Space for Computing over the Reals – p. 1/24

Plan of the Talk The BSS Model of Computation over the Reals Michaux’s Result - Computing in Constant Space Koiran’s Weak Model A Weak Measure of Space A Measure of Space for Computing over the Reals – p. 2/24

Motivation of Blum, Shub et Smale Provide a theoretical framework for studying calculability and complexity properties for natural problems and algorithms over real numbers, in particular, problems of numerical analysis, geometry, topology... The BSS Model of Computation over the Reals – p. 3/24

Motivation of Blum, Shub et Smale Provide a theoretical framework for studying calculability and complexity properties for natural problems and algorithms over real numbers, in particular, problems of numerical analysis, geometry, topology... Example: . . . . . . . . . . . . . . . . . . . . . . . . ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Newton’s Method for finding a zero of a function. The BSS Model of Computation over the Reals – p. 3/24

The Model A BSS Machine is essentially a Turing Machine over R , such that The BSS Model of Computation over the Reals – p. 4/24

The Model A BSS Machine is essentially a Turing Machine over R , such that the tape cells hold arbitrary numbers in R The BSS Model of Computation over the Reals – p. 4/24

The Model A BSS Machine is essentially a Turing Machine over R , such that the tape cells hold arbitrary numbers in R some computation nodes compute an arithmetical operation + , − , ∗ , / , with unbounded precision, at unit cost, The BSS Model of Computation over the Reals – p. 4/24

The Model A BSS Machine is essentially a Turing Machine over R , such that the tape cells hold arbitrary numbers in R some computation nodes compute an arithmetical operation + , − , ∗ , / , with unbounded precision, at unit cost, some constant nodes write a constant of R on the tape, The BSS Model of Computation over the Reals – p. 4/24

The Model A BSS Machine is essentially a Turing Machine over R , such that the tape cells hold arbitrary numbers in R some computation nodes compute an arithmetical operation + , − , ∗ , / , with unbounded precision, at unit cost, some constant nodes write a constant of R on the tape, some branch nodes branch on a test “ a ≤ b ”, at unit cost, The BSS Model of Computation over the Reals – p. 4/24

The Model A BSS Machine is essentially a Turing Machine over R , such that the tape cells hold arbitrary numbers in R some computation nodes compute an arithmetical operation + , − , ∗ , / , with unbounded precision, at unit cost, some constant nodes write a constant of R on the tape, some branch nodes branch on a test “ a ≤ b ”, at unit cost, some shift nodes move the scanning head on the tape, some copy nodes duplicate the content of some cells. The BSS Model of Computation over the Reals – p. 4/24

Input - Output Convention: One Input Tape, one Work Tape, one Output Tape. Inputs and Outputs are vectors in R ∗ = � R n , n ∈ N The BSS Model of Computation over the Reals – p. 5/24

Input - Output Convention: One Input Tape, one Work Tape, one Output Tape. Inputs and Outputs are vectors in R ∗ = � R n , n ∈ N Decision Problems - or Languages - are subsets of R ∗ . The BSS Model of Computation over the Reals – p. 5/24

Calculability There exist universal BSS machines. The BSS Model of Computation over the Reals – p. 6/24

Calculability There exist universal BSS machines. The Halting problem is undecidable. The BSS Model of Computation over the Reals – p. 6/24

Calculability There exist universal BSS machines. The Halting problem is undecidable. The Mandelbrot Set is undecidable. The BSS Model of Computation over the Reals – p. 6/24

Calculability There exist universal BSS machines. The Halting problem is undecidable. The Mandelbrot Set is undecidable. The set of points that converge under Newton’s algorithm is undecidable. The BSS Model of Computation over the Reals – p. 6/24

Sequential Time Complexity Unit measure of time: # of computation steps The BSS Model of Computation over the Reals – p. 7/24

Sequential Time Complexity Unit measure of time: # of computation steps P R : subsets of R ∗ decided in polynomial time The BSS Model of Computation over the Reals – p. 7/24

Sequential Time Complexity Unit measure of time: # of computation steps P R : subsets of R ∗ decided in polynomial time NP R : subsets of R ∗ decided in non-deterministic polynomial time (existential witnesses in R ∗ ) The BSS Model of Computation over the Reals – p. 7/24

Sequential Time Complexity Unit measure of time: # of computation steps P R : subsets of R ∗ decided in polynomial time NP R : subsets of R ∗ decided in non-deterministic polynomial time (existential witnesses in R ∗ ) coNP R : subsets of R ∗ decided in non-deterministic polynomial time (universal witnesses in R ∗ ) The BSS Model of Computation over the Reals – p. 7/24

Sequential Time Complexity Unit measure of time: # of computation steps P R : subsets of R ∗ decided in polynomial time NP R : subsets of R ∗ decided in non-deterministic polynomial time (existential witnesses in R ∗ ) coNP R : subsets of R ∗ decided in non-deterministic polynomial time (universal witnesses in R ∗ ) EXP R : subsets of R ∗ decided in exponential time. The BSS Model of Computation over the Reals – p. 7/24

Complexity (2) There exist natural NP R and coNP R -complete problems. Ex: 4FEAS R (existence of a zero for a real polynomial of degree 4) is NP R -complete (reductions in P R ). The BSS Model of Computation over the Reals – p. 8/24

Complexity (2) There exist natural NP R and coNP R -complete problems. Ex: 4FEAS R (existence of a zero for a real polynomial of degree 4) is NP R -complete (reductions in P R ). Inclusions: NP R P R EXP R coNP R Question: P R = NP R ? The BSS Model of Computation over the Reals – p. 8/24

Algebraic Circuits An algebraic circuit C computes F C : R n → R m . x y z − π ≥ 0 output if x − y ≥ 0 then z else π , n = 3 , m = 1 The BSS Model of Computation over the Reals – p. 9/24