Today we will learn…

• Recap exercises
• TM configuration and configuration history
• TM acceptance
• Variants of Turing Machines
  • Multi-tape
  • Nondeterministic

Section 3.1, 3.2, and 3.3

Supplementary material

• Professor Harry Porter's video
• Professor Dan Gusfield's video
• Turing Machines, Stanford Encyclopedia of Philosophy

Exercise 1

Convert the following grammar into a PDA

Exercise 2

Show that if $L_1 \cup L_2$ is not context-free, then either $L_1$ is not context-free or $L_2$ is not context free.

Proof.

Exercise 3

We know that $L_2 = \{ w \mid w = a^n b^n c^n \lor |w| \text{ is even} \}$ is not context free.

Show that $L_3 = \{ a^n b^n c^n \mid n \ge 0 \}$ is not context-free without using the Pumping Lemma for CF or the Theorem of non-CF from Lecture 11.

Proof.

Turing Machines

Definition 3.3

Configuration

A configuration is a snapshot of a computation. That is, it contains all information necessary to resume (or replay) a computation from any point in time.

A configuration consists of

• the tape
• the head of the tape
• the current state

Example 2

Example 1 configuration

Configuration history

Acceptance

A Turing machine

• accepts a string if there is a configuration history that reaches the accept state.
• rejects a string if there is a configuration history that reaches the reject state.
• rejects a string if it never reaches an accept or reject states
  This means that for any configuration of any length, there is no accept nor a reject state.

The acceptance algorithm

• halts when the machine is in an accept or reject state
  This is different than NFAs/PDAs which can enter and leave the accept state.

Exercise

Give a Turing Machine that recognizes words of an even length.

NFA

TM that recognizes words of an even length

(online)

Exercise

What language does this TM recognize? (online)

Alan Turing and the Turing Machine

• No computers at the time (1936)
• Alan Turing was researching into the foundations of mathematics
• Original intent: capture all possible processes which can be carried out in computing a number$^\dagger$
  
• What about non-numerical problems?
• How do Turing machines capture all general and effective procedures which determine whether something is the case or not.

Section 3.3

The Church-Turing thesis

• Any algorithm can be represented by an equivalent Turing machine

• A problem is computable if, and only if, there exists a Turing Machine that recognizes it.

• Turing Machines are equivalent to $\lambda$-terms

The Universal Turing Machine

A Turing Machine that is capable of simulating any other Turing Machine

• Let $U$ be a TM.
• Given some TM $M$ and some input $w$, we can encoded as an input string, which we represent as $\langle M,w \rangle$

Note that the Universal TM is a regular TM. This computability model is expressive enough to simulate itself.

Alan Turing's impact on modern computers

• Modern computers: von Neumann’s EDVAC design
• Fundamental idea of the EDVAC design: stored-programs
  Manipulation of programs as data
• Universal Turing Machines pioneer the idea of stored programs

TM are used to reflect on the limits and potentials of general-purpose computers by engineers, mathematicians and logicians (Module 3)

The TM tape only grows to the right

• An important thing to note is that TMs have a tape that grows only to the right
• In turingmachine.io, the tape actually grows both ways

Multi-tape Turing Machine

• A variation of the Turing Machine with multiple tapes
• The control may issue each head to move: forward, backward, or skip

Turing Machines $\iff$ Multitape Turing Machines

• ($\Leftarrow$) Multitape Turing Machines trivially recognize the same language as Turing Machine
  (let the number of tapes be 1)

• ($\Rightarrow$) How can a single tape encode multitape?

Simulating a multitape

Tape encoding

• Concatenate the three tapes together
• Delimit each tape with a character that is not in the alphabet $\sharp$
• "Tag" the character to encode each tape head (virtual heads), eg $\cdot\text{a}$
• The tape head always sits in the beginning of the tape

Simulating a multitape

Operation

• To move the $i$-th head, read the tape from the beginning until you read $\sharp$ a total of $i$ times and then seek until you find the marked character

• If the virtual head $i$ hits the end of the tape $\sharp$, then shift the rest of the tape to the right and insert a blank character ␣

Nondeterministic Turing Machines (NTM)

A machine can follow more than one transitions for the same input:

Consequence

Deterministic can only have one outgoing edge per character read, a nondeterministic machine can have multiple edges

Configurations in a TM

In a deterministic TM, a configuration history is linear

abc q1 aac -> abcx q2 ac -> abcxa q2 c -> abcx q2 ay

Configurations in a NTM

In a nondeterministic TM, a configuration history is a tree!

Nondeterministic Turing Machines

• Accept: when any branch reaches $q_{start}$

• Reject: when all branches reach $q_{reject}$

• To find a single acceptance state we need to search the computation tree

TM $\iff$ NTM

• Given an NTM, say $N$ we show how to construct a TM, say $D$
• If $N$ accepts on any branch, then $D$ halts and accepts
• If $N$ rejects on every branch, then $D$ halts and rejects

Intuition

Simulate all branches of the computation; search for any node with an accept state.

Question: If we are searching a search tree, and there may exist infinite branches (due to loops), how should we search the tree: DFS or BFS?

Addressing configuration history

• We can use a sequence of numbers to uniquely identify each node of the configuration history

Using a TM to simulate a NTM

Use 3 tapes

1. Initial input: One tape for the input
2. Simulation tape: Where we will be executing an address
3. Address tape: An ever growing number that uniquely identifies where we are in the tree

How many choices at each step?

1. Copy tape 1 to tape 2
2. Simulate TM with address from the address tape; if it reaches an accepted state, then ACCEPT, otherwise continue
3. Increment address (next BFS-wise) and go to 1