What is computable? What is computable? 1 + 1 = 2 is computable. - PowerPoint PPT Presentation

What is “computable”?

What is “computable”? • 1 + 1 = 2 is computable.

What is “computable”? • 1 + 1 = 2 is computable. • 123456789 987654321 ∼ ( 10 8 ) 10 9 is computable.

What is “computable”? • 1 + 1 = 2 is computable. • 123456789 987654321 ∼ ( 10 8 ) 10 9 is computable. • π = 4 ( 1 − 1 3 + 1 5 − 1 7 + 1 9 − 1 11 + · · · ) = 3 . 141592653589793238462643383279502884197169 . . . and � n 7 + 1 are computable.

What is “computable”?

Algorithm : The heart of computation

Algorithm : The heart of computation • A “computable” operation is prescribed by an algorithm.

Algorithm : The heart of computation • A “computable” operation is prescribed by an algorithm. • An algorithm is a set of rules that can be executed step by step.

Algorithm : The heart of computation • A “computable” operation is prescribed by an algorithm. • An algorithm is a set of rules that can be executed step by step. • Algorithm is not just about numerical computation.

Algorithm : The heart of computation • A “computable” operation is prescribed by an algorithm. • An algorithm is a set of rules that can be executed step by step. • Algorithm is not just about numerical computation. • An algorithm for solving ax 2 + bx + c = 0: √ b 2 − 4 ac x = − b ± 2 a

Algorithm for bisecting an angle Euclid (circ. 300 BC) Elements : Book I, Proposition 9 Algorithm for bisecting an angle with ruler (straightedge) and compass

. . . . . . . . . . . . . . . . . Algorithm: A long history Nine Chapters on the Mathematical In Chapter 9: Solving a quadratic . . . . . . . . . . . . . . . . . . . . . . . equation 《九章算术》 Art ( ∼ 800 BC–100 AD)

Algorithm: A long history Diophantus (210–295 AD) “Father of algebra”: Solved quadratic equations in his book Arithmetica

Algorithm: A long history René Descartes Algebraic formula for solution of a quadratic equation first appeared in his La Geométrie (1637).

Algorithm: The heart of computation

Algorithm: The heart of computation • An algorithm can be simple or complex, short or very long.

Algorithm: The heart of computation • An algorithm can be simple or complex, short or very long. • What is solvable (by an algorithm) is determined by the prescribed rules.

Negative solution

Negative solution • Trisection of an angle is not solvable by ruler and compass.

Negative solution • Trisection of an angle is not solvable by ruler and compass. • (Pierre Wentzel (1837)) Every angle constructed using ruler and compass corresponds to a root of a minimal polynomial of some degree 2 n .

Negative solution • Trisection of an angle is not solvable by ruler and compass. • (Pierre Wentzel (1837)) Every angle constructed using ruler and compass corresponds to a root of a minimal polynomial of some degree 2 n . • Trisecting an angle is impossible in general since it corresponds to root of a cubic polynomial (e.g. trisecting 20 ◦ = π/ 9 not possible).

Negative solution • Solution of a polynomial of degree ≥ 5 by the method of radicals is not possible .

Negative solution • Solution of a polynomial of degree ≥ 5 by the method of radicals is not possible . Evariste Galois (1812–1832) • Created Galois Theory (published 1846)) that revolutionized algebra.

A key question: I. Existence

A key question: I. Existence • Given that a mathematical problem has a solution, how does one “compute” a solution ?

A key question: I. Existence • Given that a mathematical problem has a solution, how does one “compute” a solution ? • Example. If f is a continuous function and f ( a ) < 0 < f ( b ) , then f ( c ) = 0 for some c ∈ [ a , b ] . How to find c ?

Alan Turing (1912–1954) Formulated the concept of algorithm and computation on a Turing machine

Turing machine: Basic model of computation

Basic facts about TM

Basic facts about TM • A Turing machine (TM) is defined by a set of instructions.

Basic facts about TM • A Turing machine (TM) is defined by a set of instructions. • Not every input has an output, and different inputs may have the same output.

Basic facts about TM • A Turing machine (TM) is defined by a set of instructions. • Not every input has an output, and different inputs may have the same output. • Different TMs may perform the same task.

Basic facts about TM • A Turing machine (TM) is defined by a set of instructions. • Not every input has an output, and different inputs may have the same output. • Different TMs may perform the same task. • We can “code” a problem into a TM. Example: A TM that on input a , b , c if ax 2 + bx + c = 0 has a real number solution Outputs “1” Outputs “0” otherwise.

Church-Turing thesis

Church-Turing thesis Intuitively computable ⇐ ⇒ Executable by a Turing machine

Church-Turing thesis Intuitively computable ⇐ ⇒ Executable by a Turing machine • Every Turing machine is a computer program.

Church-Turing thesis Intuitively computable ⇐ ⇒ Executable by a Turing machine • Every Turing machine is a computer program. • “Intuitive” is subjective while “computer program” is precise.

Church-Turing thesis Intuitively computable ⇐ ⇒ Executable by a Turing machine • Every Turing machine is a computer program. • “Intuitive” is subjective while “computer program” is precise. • Equating the two is a leap of faith in our perception of truth.

Church-Turing thesis Intuitively computable ⇐ ⇒ Executable by a Turing machine • Every Turing machine is a computer program. • “Intuitive” is subjective while “computer program” is precise. • Equating the two is a leap of faith in our perception of truth. The collection of TMs is the basic “model of computation” (Von Neumann architecture).

The central concern of mathematics

The central concern of mathematics Decide if a mathematical statement is TRUE or FALSE.

The central concern of mathematics Decide if a mathematical statement is TRUE or FALSE. • Historically, mathematics took the algorithmic approach. Abstraction came much later.

The central concern of mathematics Decide if a mathematical statement is TRUE or FALSE. • Historically, mathematics took the algorithmic approach. Abstraction came much later. • Can algorithmic approach answer every mathematical question?

Computable vs noncomputable: Examples

Computable vs noncomputable: Examples • Fundamental Theorem of Algebra : Any polynomial a n x n + a n − 1 x n − 1 + · · · + a 1 x + a 0 = 0 has a solution in the complex numbers C .

Computable vs noncomputable: Examples • Fundamental Theorem of Algebra : Any polynomial a n x n + a n − 1 x n − 1 + · · · + a 1 x + a 0 = 0 has a solution in the complex numbers C . • (Tanaka and Yamazaki 2001) If the coefficients are computable, then there is a TM that computes a solution.

Computable and noncomputable: Examples

Computable and noncomputable: Examples • Brouwer’s Fixed Point Theorem : Every continuous function f from the unit circle into itself has a fixed point, i.e. an a such that f ( a ) = a .

Computable and noncomputable: Examples • Brouwer’s Fixed Point Theorem : Every continuous function f from the unit circle into itself has a fixed point, i.e. an a such that f ( a ) = a . • (Shioji and Tanaka 1990) There is a computable continuous function with no TM to compute a fixed point.

Computable vs noncomputable: Examples

Computable vs noncomputable: Examples • Complex dynamical systems f ( z ) = z 2 + c , c ∈ C : Julia set J c for c = 0 . 300283 + 0 . 48857 i f ( z ) = z 2 + c ; f ( 2 ) ( z ) = f ( f ( z )) = ( z 2 + c ) 2 + c ; f ( n + 1 ) ( z ) = ( f ( n ) ( z )) 2 + c J c = boundary of { z : f ( n ) ( z ) �→ ∞} .

Computable vs noncomputable: Examples • Complex dynamical systems f ( z ) = z 2 + c , c ∈ C : Julia set J c for c = 0 . 300283 + 0 . 48857 i f ( z ) = z 2 + c ; f ( 2 ) ( z ) = f ( f ( z )) = ( z 2 + c ) 2 + c ; f ( n + 1 ) ( z ) = ( f ( n ) ( z )) 2 + c J c = boundary of { z : f ( n ) ( z ) �→ ∞} . • (Braverman and Yampolsky 2006) There exist computable c ’s for which there is no TM to approximate J c .

David Hilbert (1862–1943) In 1900, Hilbert proposed 23 mathematical problems. The development of mathematics in the new century was greatly influenced by in- vestigations of these problems.

Two Hilbert problems

Two Hilbert problems • Problem 2. Prove that arithmetic is consistent, i.e. free of contradiction.

Two Hilbert problems • Problem 2. Prove that arithmetic is consistent, i.e. free of contradiction. • There is a TM T that inputs the (Peano) axioms of arithmetic and outputs its theorems.