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.
• 123456789987654321 ∼ (108)109 is computable.

What is “computable”?

• 1 + 1 = 2 is computable.
• 123456789987654321 ∼ (108)109 is computable.
• π = 4(1 − 1

3 + 1 5 − 1 7 + 1 9 − 1 11 + · · · )

= 3.141592653589793238462643383279502884197169 . . . and

• n7 + 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 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

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

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 2n.

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

• 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

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

• therwise.

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

anxn + an−1xn−1 + · · · + a1x + a0 = 0 has a solution in the complex numbers C.

Computable vs noncomputable: Examples

• Fundamental Theorem of Algebra: Any polynomial

anxn + an−1xn−1 + · · · + a1x + a0 = 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) = z2 + c, c ∈ C:

Julia set Jc for c = 0.300283+0.48857i f(z) = z2 + c; f (2)(z) = f(f(z)) = (z2 + c)2 + c; f (n+1)(z) = (f (n)(z))2 + c Jc = boundary of {z : f (n)(z) → ∞}.

Computable vs noncomputable: Examples

• Complex dynamical systems f(z) = z2 + c, c ∈ C:

Julia set Jc for c = 0.300283+0.48857i f(z) = z2 + c; f (2)(z) = f(f(z)) = (z2 + c)2 + c; f (n+1)(z) = (f (n)(z))2 + c Jc = boundary of {z : f (n)(z) → ∞}.

• (Braverman and Yampolsky 2006) There exist computable c’s

for which there is no TM to approximate Jc.

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
• utputs its theorems.

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
• utputs its theorems.

Left: Gódel and Einstein (Princeton, Auguest 1950)

Kurt Gödel (1906–1978) If arithmetic is consistent, then “consistency of arithmetic” is not an output of T.

Two Hilbert problems

Two Hilbert problems

• Problem 10. Produce an algorithm to decide whether a given

polynomial P(x1, . . . , xn) = 0 with integer coefficients has a solution in integers.

Two Hilbert problems

• Problem 10. Produce an algorithm to decide whether a given

polynomial P(x1, . . . , xn) = 0 with integer coefficients has a solution in integers.

• Examples:

Elliptic curve y2 = x3 + ax + b Fermat’s equation xn + yn = zn, n > 2

Two Hilbert problems

• Problem 10. Produce an algorithm to decide whether a given

polynomial P(x1, . . . , xn) = 0 with integer coefficients has a solution in integers.

• Examples:

Elliptic curve y2 = x3 + ax + b Fermat’s equation xn + yn = zn, n > 2

• Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia

Robinson (1961 to 1969): There is no TM for this.

Foundational issue

• Since TMs are not able to answer all questions, can we

strengthen the notion of TM to compute all mathematical truths?

Foundational issue

• Since TMs are not able to answer all questions, can we

strengthen the notion of TM to compute all mathematical truths?

• For example, write TMn as the nth TM. Consider
• A = {n : TMn has no output}

Foundational issue

• Since TMs are not able to answer all questions, can we

strengthen the notion of TM to compute all mathematical truths?

• For example, write TMn as the nth TM. Consider
• A = {n : TMn has no output}
• B = {n : TMn has infinitely many outrputs}

Foundational issue

• Since TMs are not able to answer all questions, can we

strengthen the notion of TM to compute all mathematical truths?

• For example, write TMn as the nth TM. Consider
• A = {n : TMn has no output}
• B = {n : TMn has infinitely many outrputs}
• C = {n : n = 2m0 · 3m1 & TMm0 and TMm1 eventually

produce the same outputs} · · ·

Foundational issue

• Since TMs are not able to answer all questions, can we

strengthen the notion of TM to compute all mathematical truths?

• For example, write TMn as the nth TM. Consider
• A = {n : TMn has no output}
• B = {n : TMn has infinitely many outrputs}
• C = {n : n = 2m0 · 3m1 & TMm0 and TMm1 eventually

produce the same outputs} · · ·

• There is No TM to compute A, B or C. In fact, C > B > A in

terms of “relative difficulty” to compute.

Foundational issue

• We can create a “larger” model of computation M∗ = all the

sets of TMs that our language can describe.

Foundational issue

• We can create a “larger” model of computation M∗ = all the

sets of TMs that our language can describe.

• But it can be proved that M∗ will still have limited

computational power.

Foundational issue

• We can create a “larger” model of computation M∗ = all the

sets of TMs that our language can describe.

• But it can be proved that M∗ will still have limited

computational power.

• So this model captures only a small fraction of the universe of

mathematical truths.

In the universe of sets, M∗ is a tiny collection.

Foundational issue

Foundational issue

• For example, there are uncountably many sets of natural

numbers but M∗ computes only countably many of them.

Foundational issue

• For example, there are uncountably many sets of natural

numbers but M∗ computes only countably many of them.

• This is a natural limitation of M∗ and hence of our ability to

compute.

Foundational issue

• For example, there are uncountably many sets of natural

numbers but M∗ computes only countably many of them.

• This is a natural limitation of M∗ and hence of our ability to

compute.

• To overcome this barrier, we need to go to “higher infinity”.

Foundational issue

• For example, there are uncountably many sets of natural

numbers but M∗ computes only countably many of them.

• This is a natural limitation of M∗ and hence of our ability to

compute.

• To overcome this barrier, we need to go to “higher infinity”.
• In the universe of higher infinity, there is a more powerful model
• f computation. But again there are sets of natural numbers it

cannot compute.

Back to the physical world...

• But what about Turing’s vision of AI by the year 2000?

Back to the physical world...

• But what about Turing’s vision of AI by the year 2000?

That’s another story.