Diagonalization

Automata

Author

Prof. Calvin

Sketch

  • Diagonalization
    • (It’s non-trivial)

Goal

\[ \neg\text{Decidable}(A_{TM} := \{\langle M,w \rangle \space | \space w \in M:\text{TM}\}) \]

  • We need some way to say this.
  • We term this notion undecidability
  • We introduce a technique diagonalization first.
    • It is the only known technique to show undecidability.

The Size of Infinity

Set Theory

  • Diagonalization emerged from set and number theory.
    • My first diagonalization, and perhaps yours, was over \(\mathbb{R}\)
  • First deployed by Georg Cantor in 1891
    • Showing \(|\mathbb{N}| \lt |\mathbb{R}|\)
  • Famously showed Gödel’s incompleteness
  • Famously (as here) showed Turing’s Entscheidungsproblem

Functions

  • One way to show infinite sets have the same or different sizes is using functions.
  • We claim that a set of infinite size is of the same infinite size as some other set if there exists some function that is:
    • one-to-one, and
    • onto.

Notation

  • We introduce notations \(f : A \rightarrow B\)
    • This is logicially equivalent to \(f : A \times B\), but
    • Captures a different meaning.

Injectivity

  • We term a one-to-one function to be injective:
  • Said function:

maps distinct elements of its domain to distinct elements of its codomain

  • \(\forall x_1, x_2 : x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)

Graphically

finite_automata cluster_X A cluster_Y B 1 1 w w 1->w 2 2 x x 2->x 3 3 z z 3->z y y

Injectivity

  • Every element of the function’s codomain
    • Also called “set of destination”
    • \(\{B | f: A \rightarrow B\}\)
  • Is the image
    • The image of value \(x \in A\) is the single value \(f(x) \in B\)
  • Of at most one element of the domain
    • The set of inputs,
    • \(\{A | f: A \rightarrow B\}\)

Examples

  • \(f(x) = x^2 : \mathbb{N} \rightarrow \mathbb{N}\)
  • \(f(x) = x^3 : \mathbb{R} \rightarrow \mathbb{R}\)
  • Python ord which takes a finite set of printable characters to \(\mathbb{N}\)
[ord(a) for a in 'abcdefghijklmnopqrstuvwxyz!@#$%^&*()'][::5]
[97, 102, 107, 112, 117, 122, 37, 41]
  • Social security number is a injection from people with them to numbers.

Nonexamples

  • \(f(x) = x^2 : \mathbb{R} \rightarrow \mathbb{R}\)
[x*x for x in range(-1,2)]
[1, 0, 1]
  • \(f(x) = x^2 - x : \mathbb{N} \rightarrow \mathbb{Z}\)
[x*x-x for x in range(3)]
[0, 0, 2]
  • Emails-to-humans, e.g. I have:

Surjectivity

  • We term a function to be surjective if:

for every element \(y\) of the function’s codomain, there exists at least one element \(x\) in the function’s domain such that \(f(x) = y\)

  • \(\forall y \in B : \exists x \in A : y = f(x)\)

Graphically

finite_automata cluster_X A cluster_Y B 1 1 w w 1->w 2 2 x x 2->x 3 3 y y 3->y 4 4 4->y

Properties of Surjectivity

  • Every element of the function’s codomain
    • Also called ‘set of destination’
    • \(\{B | f: A \rightarrow B\}\)
  • Is the image of at least one element of the domain
    • The set of inputs,
    • \(\{A | f: A \rightarrow B\}\)

Properties of Surjectivity

  • Can also say
    • The codomain
      • \(\{B | f: A \rightarrow B\}\)
    • Is the image
      • Different meaning!
      • Image of a set, not an element.
      • “The image of \(A\) under \(f\)” is the set \(\{f(x) \in B | x \in A\}\)

Examples

  • \(f(x) = |x| : \mathbb{Z} \rightarrow \mathbb{N}\)
  • \(f(x) = \ln(x) : \mathbb{R} \rightarrow \mathbb{R}\)
  • Python chr which takes a finite subset of infinite numerical values but maps to every printing character.
[chr(a) for a in range(0,10000,1000)]
['\x00', 'Ϩ', 'ߐ', 'ஸ', 'ྠ', 'ᎈ', 'ᝰ', '᭘', 'ὀ', '⌨']
  • Names are surjections - every name a person, not unique.

Nonexamples

  • \(f(x) = x^2 : \mathbb{R} \rightarrow \mathbb{R}\)
    • No value such that \(f(x) = -1 \in \mathbb{R}\)
  • \(f(x) = 2x : \mathbb{N} \rightarrow \mathbb{N}\)
    • No value such that \(f(x) = 3\)
  • English words to Latin character spellings
    • There exists no English word xxxyzx

Bijective

  • A function is bijective if it is injective and surjective.
    • \(\forall x_1, x_2 : x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)
    • \(\forall y \in B : \exists x \in A : y = f(x)\)
  • Or perhaps, the bijective functions are the intersection of the injective and surjective functions.

Graphically

finite_automata cluster_X A cluster_Y B 1 1 w w 1->w 2 2 x x 2->x 3 3 y y 3->y 4 4 z z 4->z

Bijective

finite_automata cluster_X A cluster_Y B 1 1 w w 1->w 2 2 x x 2->x 3 3 y y 3->y 4 4 z z 4->z

Injective

finite_automata cluster_X A cluster_Y B 1 1 w w 1->w 2 2 x x 2->x 3 3 z z 3->z y y

Surjective

finite_automata cluster_Y B cluster_X A 1 1 w w 1->w 2 2 x x 2->x 3 3 y y 3->y 4 4 4->y

Neither

finite_automata cluster_X A cluster_Y B 1 1 w w 1->w 2 2 x x 2->x 3 3 z z 3->z 4 4 4->z y y

Terms

  • Bijective functions are also called bijections
  • Also called “one-to-one correspondence”
    • I avoid this term as an injective function is a one-to-one function
    • I just use the unambigious terms when at all possible.

Examples

  • \(f(x) = x : \mathbb{R} \rightarrow \mathbb{R}\)
  • \(f(x) = x^3 : \mathbb{R} \rightarrow \mathbb{R}\)
  • This for \(\mathbb{N} \rightarrow \mathbb{Z}\)
from itertools import count
f = lambda x: x // 2 * (x % 2 * 2 - 1)
Z = (f(x) for x in count(1))
[next(Z) for _ in range(10)]
[0, -1, 1, -2, 2, -3, 3, -4, 4, -5]
  • Wait a minute!

Size of infinity

from itertools import count
f = lambda x: x // 2 * (x % 2 * 2 - 1)
Z = (f(x) for x in count(1))
[next(Z) for _ in range(10)]
[0, -1, 1, -2, 2, -3, 3, -4, 4, -5]
  • \(\forall n \in \mathbb{N}, n \in \mathbb{Z}\)
    • \(\mathbb{N} \subseteq \mathbb{Z}\)
  • \(\exists x \in \mathbb{Z} : x \notin \mathbb{N}\)
    • \(\mathbb{N} \subset \mathbb{Z}\)
  • \(\exists \text{ bijection } f: \mathbb{N} \rightarrow \mathbb{Z}\)
  • \(|\mathbb{N}| = |\mathbb{Z}|\)

Countability

  • \(|\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}|\)
  • We term these countably infinite.
    • See Pythonic count
  • \(\mathbb{Q}\) is non-trivial, I’m not aware of a close form.
    • We’ll write it out real quick.

\(\mathbb{Q}\)

  • Restrict to \(\mathbb{Q^+} = \{ \dfrac{m}{n} | m,n \in \mathbb{N}\}\)
  • Set the fractions out within a table.
1 2 3 4 \(\ldots\)
1 \(\dfrac{1}{1}\) \(\dfrac{1}{2}\) \(\dfrac{1}{3}\) \(\dfrac{1}{4}\)
2 \(\dfrac{2}{1}\) \(\dfrac{2}{2}\) \(\dfrac{2}{3}\) \(\dfrac{2}{4}\)
3 \(\dfrac{3}{1}\) \(\dfrac{3}{2}\) \(\dfrac{3}{3}\) \(\dfrac{3}{4}\)
4 \(\dfrac{4}{1}\) \(\dfrac{4}{2}\) \(\dfrac{4}{3}\) \(\dfrac{4}{4}\)

\(\mathbb{Q}\)

  • Populate a set.
  • Careful with duplicates!
  • Careful with placing infinite many values before another value!
1 2 3 4 \(\ldots\)
1 \(\dfrac{1}{1}\) \(\dfrac{1}{2}\) \(\dfrac{1}{3}\) \(\dfrac{1}{4}\)
2 \(\dfrac{2}{1}\) \(\dfrac{2}{2}\) \(\dfrac{2}{3}\) \(\dfrac{2}{4}\)
3 \(\dfrac{3}{1}\) \(\dfrac{3}{2}\) \(\dfrac{3}{3}\) \(\dfrac{3}{4}\)
4 \(\dfrac{4}{1}\) \(\dfrac{4}{2}\) \(\dfrac{4}{3}\) \(\dfrac{4}{4}\)

\(\mathbb{Q}\)

pair = lambda i,j: (i+j-2)*(i+j-1)//2+j
for i in range(1,5):
  print([pair(i,j) for j in range(1,5)])
[1, 3, 6, 10]
[2, 5, 9, 14]
[4, 8, 13, 19]
[7, 12, 18, 25]
  • Basically just load the fractions into a stack
  • Check they aren’t in the ordering yet
  • Add them if they aren’t.
1 2 3 4
1 \(\dfrac{1}{1}\) \(\dfrac{1}{2}\) \(\dfrac{1}{3}\) \(\dfrac{1}{4}\)
2 \(\dfrac{2}{1}\) \(\dfrac{2}{3}\)
3 \(\dfrac{3}{1}\) \(\dfrac{3}{2}\) \(\dfrac{3}{4}\)
4 \(\dfrac{4}{1}\) \(\dfrac{4}{3}\)

Countability Lemma

\[ |\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}| \]

Proof.

1 2 3 4
1 \(\dfrac{1}{1}\) \(\dfrac{1}{2}\) \(\dfrac{1}{3}\) \(\dfrac{1}{4}\)
2 \(\dfrac{2}{1}\) \(\dfrac{2}{3}\)
3 \(\dfrac{3}{1}\) \(\dfrac{3}{2}\) \(\dfrac{3}{4}\)
4 \(\dfrac{4}{1}\) \(\dfrac{4}{3}\)

Diagonalization Lemma

\[ |\mathbb{R}| \gt |\mathbb{N}| \]

  • We will assume \(\mathbb{R}\) is countable and find a contradiction.
  • We note that e.g., \(3.14159 \in \mathbb{R} \cap \mathbb{Q}\)
  • We note that e.g., \(3.14159\ldots = \pi \in \mathbb{R} \setminus \mathbb{Q}\)

A bijection

  • Let’s take all the real numbers in \(\mathbb{R}\)
    • We note that are the numbers expressible with potentially infinite strings.
    • Decimal or binary is fine.

A listing

\(n\) \(f(n): \mathbb{N} \rightarrow \mathbb{R}\)
1
2
3
4
  • If there is some bijection to \(\mathbb{N}\), we can make such a listing!
  • You are welcome to try!

A listing

\(n\) \(f(n): \mathbb{N} \rightarrow \mathbb{R}\)
1 \(\pi = 3.14159\ldots\)
2 \(e = 2.71828\ldots\)
3 \(0 = 0.00000\ldots\)
4 \(x = 1.41421\ldots\)
5 \(y = 0.14286\ldots\)
6 \(z = 0.20788\ldots\)
7 \(v = 1.23456\ldots\)
  • Missing:
    • 4…
    • 4.8…
    • 4.81…
    • 4.815…
    • 4.8159…
    • 4.81599…
    • 4.815998…

Diagonalization Lemma

\[ |\mathbb{R}| \gt |\mathbb{N}| \]

\(Proof.\)

  • 4.815998…
  • That is, take an proposed bijection, and differ in the \(i\)’th digit from the \(i\)’th value.
  • In fact, \(|\mathbb{R} = |2^\mathbb{N}|\)

Coming Attractions

  • That is all for now!
    • But… we treated \(\mathbb{R}\) (strings) as bigger than \(\mathbb{N}\) (lists) from a language based argument!
    • We have been storing computers as strings!
    • We have been writing computers that study strings!

\[ \neg\text{Decidable}(A_{TM} := \{\langle M,w \rangle \space | \space w \in M:\text{TM}\}) \]

Thank you!

  • Next lecture, on undecidability, posted Monday to YouTube!
  • Homework 1, on automata, the Monday after
    • Assuming my flight etc etc