Turing Machine

$$ TM = (S, T \subseteq S \times \Gamma \rightarrow S \times \Gamma \times \text{\em dir}, q0 \in S, q{Accept} \subseteq S) $$

$S$ is a finite set (the “in-the-head” processing states)
$\Gamma$ is a finite set (symbols that can be writte on the tape)
$\text{\em dir} = { \text{\bf Left}, \text{\bf Right}, \text{\bf Halt} }$ is the direction to move on the tape.

An execution of a Turing Machine, $TM = (S, T \subseteq S \times \Gamma \rightarrow S \times \Gamma \times \text{\em dir}, q0 \in S, q{Accept} \subseteq S)$, is a (possibly infinite) sequence of configurations, $(x_0, x_1, \ldots)$ where $x_i \in \text{\em Tsil} \times S \times \text{\em List}$ (elements of the lists are in the finite set of symbols, $\Gamma$), such that:

  - $x_0 = (\text{\bf null}, q_0, \text{\bf input})$
  - and all transitions follow the rules (need to be specified in detail).

Machines and Languages

A language is a (possibly infinite) set of finite strings.

$\Sigma$ = alphabet, a finite set of symbols
$L \subseteq \Sigma^{*}$

$$L_{XOR} = { x_0x_1\ldots x_n + y_0y_1\ldots y_n = z_0z_1\ldots z_n \$ | x_i \in {0, 1}, y_i \in {0, 1}, z_i = x_i \oplus y_i }$$

Turing Machine Computation

A Turing Machine, $M = (S, T \subseteq S \times \Gamma \rightarrow S \times \Gamma \times \text{\em dir}, q0 \in S, q{Accept} \subseteq S)$, accepts a string x, if there is an execution of M that starts in configuration $(\text{\bf null}, q_0, x)$, and terminates in a configuration, $(l, q_f, r)$, where $qf \in q{Accept}$.

A Turing Machine, $M = (S, T \subseteq S \times \Gamma \rightarrow S \times \Gamma \times \text{\em dir}, q0 \in S, q{Accept} \subseteq S)$, recognizes a language $L$, if for all strings $s \in L$, $M$ accepts $s$, and there is no string $t \notin L$ such that $M$ accepts $t$.

TuringMachine = namedtuple('TuringMachine', ['rules', 'q_0', 'q_accepting'])

def simulate(tm, starttape):
    tape = starttape
    headpos = 0
    currentstate = tm.q_0

    while True:
        readsym = tape[headpos]
        if (currentstate, readsym) in tm.rules:
            (nextstate, writesym, dir) = tm.rules[(currentstate, readsym)]
            tape[headpos] = writesym
            currentstate = nextstate
        else:
            return currentstate in tm.q_accepting # no rule, halt

        if dir == 'L':
            headpos = max(headpos - 1, 0)
        elif dir == 'R':
            headpos += 1
            if len(tape) <= headpos: tape.append('_') # blank
        else: # assert dir == 'Halt'
            return currentstate in tm.q_accepting
           
xorm = TuringMachine(
    rules = { ('S', '0'): ('R0', '-', 'R'), ('S', '1'): ('R1', '-', 'R'),
              ('S', '+'): ('C', '+', 'R'),
              ('R0', '0'): ('R0', '0', 'R'), ('R0', '1'): ('R0', '1', 'R'),
              ('R1', '0'): ('R1', '0', 'R'), ('R1', '1'): ('R1', '1', 'R'),
              ('R0', '+'): ('X0', '+', 'R'), ('R1', '+'): ('X1', '+', 'R'),
              ('X0', 'X'): ('X0', 'X', 'R'), ('X1', 'X'): ('X1', 'X', 'R'),
              ('X0', '0'): ('Y0', 'X', 'R'), ('X0', '1'): ('Y1', 'X', 'R'),
              ('X1', '0'): ('Y1', 'X', 'R'), ('X1', '1'): ('Y0', 'X', 'R'),
              ('Y0', '0'): ('Y0', '0', 'R'), ('Y0', '1'): ('Y0', '1', 'R'),
              ('Y1', '0'): ('Y1', '0', 'R'), ('Y1', '1'): ('Y1', '1', 'R'),
              ('Y0', '='): ('Z0', '=', 'R'), ('Y1', '='): ('Z1', '=', 'R'),
              ('Z0', 'X'): ('Z0', 'X', 'R'), ('Z1', 'X'): ('Z1', 'X', 'R'),
              ('Z0', '0'): ('B', 'X', 'L'),  ('Z1', '1'): ('B', 'X', 'L'),
              ('B', '0'): ('B', '0', 'L'),   ('B', '1'): ('B', '1', 'L'),
              ('B', '+'): ('B', '+', 'L'),   ('B', 'X'): ('B', 'X', 'L'),
              ('B', '='): ('B', '=', 'L'),   ('B', '-'): ('S', '-', 'R'),
              ('C', 'X'): ('C', 'X', 'R'),   ('C', '='): ('C', '=', 'R'), 
              ('C', '$'): ('Accept', '$', 'Halt') },
    q_0 = 'S', q_accepting = { 'Accept' })    

simulate(xorm, list("00+10=10$"))

Cantor’s Continuum “Hypothesis”

Aleph-naught: $\aleph_0 = |\mathbb{N}|$ is the smallest infinite cardinal number.

Recall: $\omega$ is the smallest infinite ordinal. The first ordinal after $0, 1, 2, \cdots$.

$$2^{\aleph_0} = |pow(\mathbb{N})| = | [0, 1] | = | \mathbb{R} | = | {0, 1}^{\omega} | > |\mathbb{N} $$

What does it mean to say it is proven to not be possible to settle the question of whether $\aleph_1 = 2^{\aleph_0}$ with the ZFC axioms?

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On Computable Numbers

The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. (Alan Turing, On Computable Numbers, with an Application to the Entscheidungsproblem, 1936.)

Is $\tau$ computable (by Turing’s definition)? ($\tau = \frac{\text{Circumference of circle}}{\text{radius of circle}}$)

What are the problems with using the State Machine to model computation: $$M = (S, G \subseteq S \times S, q_0 \in S)$$

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What was Turing attempting to model in defining what we know call a Turing Machine?

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$$ TM = (S, T \subseteq S \times \Gamma \rightarrow S \times \Gamma \times \text{\em dir}, q0 \in S, q{Accept} \subseteq S) $$

$S$ is a finite set (the “in-the-head” processing states)
$\Gamma$ is a finite set (symbols that can be writte on the tape)
$\text{\em dir} = { \text{\bf Left}, \text{\bf Right}, \text{\bf Halt} }$ is the direction to move on the tape.

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What is the cardinality of the set of all Turing Machines?

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Prove that there are numbers that cannot be output by any Turing Machine.

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Infinite Sets Recap

Definition. A set $C$ is countable if and only if there exists a surjective function from $\mathbb{N}$ to $C$. (That is, $\le 1$ arrow out from $\mathbb{N}$, $ge 1$ arrow in to $C$.)

Definition. A set $C$ is countably infinite if and only if there exists a bijection between $C$ and $\mathbb{N}$.

Cantor’s Theorem

For all sets, $S$, $| pow(S) | > | S |$.

What does this mean for $| \mathbb{N} |$?

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What is a real number?

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Show there is a bijection between $[0, 1)$ and $pow(\mathbb{N})$.

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Countable and Uncountable Sets

Definition. A set $S$ is countably infinite if and only if there exists a bijection between $S$ and $\mathbb{N}$.

Definition. A set $S$ is uncountable, if there exists no bijection between $S$ and $\mathbb{N}$.

The power set of $A$ ($\textrm{pow}(A)$)is the set of all subsets of $A$: $$ B \in \textrm{pow}(A) \iff B \subseteq A. $$

For all finite sets $S$, $|pow(S)| = 2^{|S|}$.

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For all sets $S$, $|pow(S)| > |S|$.

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Prove $pow(\mathbb{N})$ is uncountable.

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$\text{bitstrings} = \forall n \in \mathbb{N} . {0, 1}^n$.

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Ordinal and Cardinal Numbers

$\omega$ is the smallest infinite ordinal. The first ordinal after $0, 1, 2, \cdots$.

What is the difference between an ordinal and cardinal number?

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What should $2\omega$ mean?

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Is $\text{InfiniteBitStrings} = {0, 1}^\omega$ countable?

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Prove the number of real numbers in the interval $[0, 1]$ is uncountable.

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What set is bigger than $\mathbb{R}$?

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Aleph-naught: $\aleph_0 = |\mathbb{N}|$ is the smallest infinite cardinal number.

How is $\omega$ different from $\aleph_0$?

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$$2^{\aleph_0} = |pow(\mathbb{N})| = | [0, 1] | = | \mathbb{R} | = | \text{InfiniteBitStrings} | $$

What is necessary to conclude that it is not possible to settle the question of whether $\aleph_1 = 2^{\aleph_0}$ with the ZFC axioms?