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What is “computable”?

• 1 + 1 = 2 is computable.

• 123456789987654321 ∼ (108)109 is computable. • π = 4(1− 13 + 15 − 17 + 19 − 111 + · · · )

= 3.141592653589793238462643383279502884197169 . . . and √

n7 + 1

are computable.

What is “computable”?

• 1 + 1 = 2 is computable.

• 123456789987654321 ∼ (108)109 is computable. • π = 4(1− 13 + 15 − 17 + 19 − 111 + · · · )

= 3.141592653589793238462643383279502884197169 . . . and √

n7 + 1

are computable.

What is “computable”?

• 1 + 1 = 2 is computable.

• 123456789987654321 ∼ (108)109 is computable.

• π = 4(1− 13 + 15 − 17 + 19 − 111 + · · · ) = 3.141592653589793238462643383279502884197169 . . .

and √ n7 + 1

are computable.

What is “computable”?

• 1 + 1 = 2 is computable.

• 123456789987654321 ∼ (108)109 is computable. • π = 4(1− 13 + 15 − 17 + 19 − 111 + · · · )

= 3.141592653589793238462643383279502884197169 . . . and √

n7 + 1

are computable.

What is “computable”?

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 ax2 + bx + c = 0:

x = −b±

√ b2 − 4ac

2a

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 ax2 + bx + c = 0:

x = −b±

√ b2 − 4ac

2a

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 ax2 + bx + c = 0:

x = −b±

√ b2 − 4ac

2a

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 ax2 + bx + c = 0:

x = −b±

√ b2 − 4ac

2a

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 ax2 + bx + c = 0:

x = −b±

√ b2 − 4ac

2a

Algorithm for bisecting an angle

Euclid (circ. 300 BC)

Elements: Book I, Proposition 9 Algorithm for bisecting an angle with ruler (straightedge) and compass

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Algorithm: A long history

《九章算术》

Nine Chapters on the Mathematical Art (∼ 800 BC–100 AD)

In Chapter 9: Solving a quadratic equation

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

• An algorithm can be simple or complex, short or very long.

• What is solvable (by an algorithm) is determined by the prescribed rules.

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.

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

• 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 2n.

• 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

• 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 2n.

• 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

• 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 2n.

• 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

• 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 2n.

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.

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

• 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?

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?

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

• 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

Outputs “1” if ax2 + bx + c = 0 has a real number solution

Outputs “0” otherwise.

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

Outputs “1” if ax2 + bx + c = 0 has a real number solution

Outputs “0” otherwise.

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

Outputs “1” if ax2 + bx + c = 0 has a real number solution

Outputs “0” otherwise.

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

Outputs “1” if ax2 + bx + c = 0 has a real number solution

Outputs “0” otherwise.

Basic facts