9.1.1. A General Model of Computation¶

So far we have seen a few simple machine types, such as DFA, NFA, PDA, RegEx. And we have seen corresponding grammars: Regular grammars, context-free grammars, etc. There are some differences in the machines. DFAs are deterministic. NFAs add non-determinism, which simply means that there can be multiple choices out of a state for the same input character. The PDA adds the concept of a stack for storage.

These machines all have some similarities. They all take an input string. They all march across the string from left to right, one character at each step. They stop when they reach the end of the string and then make a simple decision: If the machine is in a final state, then it accepts the string, and otherwise it rejects the string. In other words, they are acceptors for some language. All that they can do is determine if a string is a member of the language, and the “more powerful” machines can accept or reject string from more “complicated” languages.

But there is a lot more that we typically expect when we talk about “computation”. Like, well, the ability to “compute” rather than simply accept. What does “compute” actually mean? For example, we would like to be able to take as input the arithmetic expression “$2 + 3$” and compute the value 5. This means that we need to be able to output that computed value 5. Or put another way, we can “compute” a value by taking the input string “$2 + 3$” and replacing it with the output string “5”.

In its most general form, we can think of everything that a computer does as taking some string as input, and then providing some string as output. Of course, modern peripheral devices like keyboards, mice, and computer monitors give us rich ways to express input strings (perhaps as button presses), and rich ways to interpret output strings (say, as pictures). But it’s not a huge stretch of the imagination to consider computation as converting an input string to an output string.

This concept of converting is far more powerful than simple accepting. A machine that takes an input and provides an output is called a transducer.

In the next section, we will introduce a simple machine, the Turing Machine, that is a transducer. It is only slightly more complicated than the machines that we have seen so far, and only slightly different in its operation. But these differences are significant. Ultimately, we will see that a Turing Machine can do any computation that even the most sophisticated modern computer can do.

9.1.2. Turing Machines¶

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<<<>>>

We now introduce the Turing Machine. A Turing Machine is a transducer, meaning that it can modify the input tape to generate an output. It is only slightly more complicated than the machines that we have seen so far, and only slightly different in its operation. But these differences are significant. Ultimately, we will see that a Turing Machine can do any computation that even the most sophisticated modern computer can do.

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1. #
2. a
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6. #
7. #
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1. #
2. a
3. a
4. a
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a,λ;R

λ,λ;S

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q1

q0

q1

q2

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λ,λ;S

a,a;R

b,b;R

c,c;R

λ,λ;S

c,c;R

c,c;R

λ,λ;S

b,b;R

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7. #

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9.1.3. Interpreting Turing Machines¶

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<<<>>>

A configuration for a Turing machine provides enough information to understand its current state of execution on a given string. We use the following notation:$(q, \underline{a}aba)$

This means that the TM is in state $q$, the tape contains $aaba$, and the read/write head position is on the underlined $a$. Recall that we assume at the start of processing input for any TM, the read/write head position is on the leftmost non-blank character.

Don't forget that the tape is infinite in both directions. So to the left of the leftmost 'a' in this configuration are an infinite number of blank squares, and to the right of the rightmost a is also an infinite number of blank squares.

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q1

a,λ;R

λ,λ;S

q0

q1

a,a;R

b,b;S

q0

q1

q2

I,I;R

#,I;R

#,#;S

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5. #

q0

q1

q2

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9.1.4. Turing-Decidable vs. Turing-Acceptable Languages¶

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<<<>>>

Recall that a Turing machine can accept or reject a string. We also noted that a Turing machine can go into an infinite loop. We will now examine that concept more closely.

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9.1.5. Making More Complicated Machines part 1¶

Obviously, Turing Machines can take an input and modify it. We will see examples of how this leads to powerful computational capability, even if it does not seem yet like they are so powerful. To get a quick idea of their power, consider the following relatively simple machine to accept $L(a^nb^nc^n)$. This is significant, because this language is in fact not context free. Which means that this simple Turing Machine is doing something that no DFA, NFA, or PDA can do!

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<<<>>>

We will see how the machine processes input string 'aabbcc'.

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a,A;R

B,B;R
a,a;R

b,B;R

C,C;R
b,b;R

c,C;L

a,a;L
B,B;L
C,C;L
b,b;L

A,A;R

B,B;R

B,B;R
C,C;R

#,#;L

C,c;L
B,b;L
A,a;L

#,#;R

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<<<>>>

While Turing machines might be able to do powerful things, when operating at the individual state level it can get rather difficult and tedious to program them. In fact, it might feel in some ways like writing machine code or assembly language.

The secret to success in modern software development is to build up more powerful tools, especially by packaging behavior together and manipulating the packages through specific interfaces.

We can hope to build up similar capability with Turing Machines.

$>M_1$

$M_2$

$M_3$

$a$

$b$

$>R$

$\overline{\#}$

$>L$

$\overline{\#}$

$\#$

$M$

$>L_{\#}$

$M$

$R$

$\# R^{2}_{\#} \sigma L^{2}_{\#} \sigma$

$R_{\#}$

$\overline{\#}$

$\#$

$>L_{\#}$

$R$

$L \sigma R$

$L\#$

$\overline{\#}$

$\#$

$>L_{\#}$

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<<<>>>

In this slideshow, we will see how a copy machine processes input string 'ab'.
The following machine is a copy machine that can transform #w# into #w#w#:
(Note: w represents the input string ('ab' in this case).)

$R$

$\#$

$R^{2}_{\#}$

$\sigma$

$L^{2}_{\#}$

$\sigma$

$R_{\#}$

$\overline{\#}$

$\#$

$>L_{\#}$

1. #
2. a
3. b
4. #
5. #
6. #
7. #

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