Turing Machines string x, will be left on the tape when the machine - - PDF document

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Turing Machines

Computation and programming

Computation with Turing Machines

• Computing functions with TMs

– The result of the function applied to an input string x, will be left on the tape when the machine halts.

• Note: it doesn’t matter if the machine halts in a final

state.

– Functions with multiple arguments can be placed on the input tape with arguments separated by some character.

Computation with Turing Machines

• Turing’s original use of his machine was to

calculate integer valued functions.

– Integers were represented in unary as blocks of a single symbol. – Example

• 000000 would represent 6
• 0000 would represent 4

– In our text

• 0s are used as the unary symbol
• 1s are used to separate arguments

Computation with Turing Machines

• Computing functions with TMs

– Formally,

• Let M = (Q, Σ, Γ , δ, q0, B, F) be a TM and let f be a

partial function on Σ*. We say T computes f if for every x ∈ Σ* where f is defined:

– q0x a* pf(x)

• And for every other x, T fails to halt on input x.

Computation with Turing Machines

• Computing functions with TMs

– Formally (with multiple arguments)

• Let M = (Q, Σ, Γ , δ, q0, B, F) be a TM and let f be a

partial function on (Σ*) k. We say T computes f if for every (x1, x2. …, xk) where f is defined:

– q0x1 1x2… 1xk a* pf(x1, x2. …, xk)

• And for every other k-tuple, T fails to halt on input

x.

Computation with Turing Machines

• Example

– Proper subtraction:

• m – n = max (m-n, 0)
• Use 0 as unary symbol
• M will start with 0m10n
• M will halt with 0(m-n) on it’s tape

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Computation with Turing Machines

• Example

– Proper subtraction:

• Basic idea:

– Finds leftmost 0 and replaces with blank, – Finds first 1 – Finds first 0 after 1, replaces with a 1. – Repeat until » When searching for first 0 after the 1 it doesn’t find it » Cannot find leftmost 0.

Computation with Turing Machines

• Proper subtraction

Computation with Turing Machines

• Questions?

Combining Turing Machines

• Suppose an algorithm has a number of tasks

to perform

– Each task has its own TM

• Much like subroutines or functions

– They can be combined into a single TM

• T1T2 is the composite TM
• T1 → T2

Combining Turing Machines

• Composite TMs

– T1T2

• Start at start state of T1
• For any move that causes T1 to enter the halt state,

have T1 move to the start state of T2.

• So T2 will get executed immediately after T1 as long

as T1 will halt on a given input.

– Allows one to define subroutines.

Let’s try a non-context free language

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Combining Turing Machines

• Useful “subroutines”

– Copy TM

• Creates a copy of a “block” of 0s to the right of the input

– 0i1q10n10j a* 0i1q50n10j+n

• Basic idea

– Marks first 0 with an X – Move right to first blank and write a 0 – Move left until X is found – Repeat until all 0’s are converted to X – When finished, coverts Xs back to 0s.

Combining Turing Machines

• Copy TM
• See in action

Combining Turing Machines

• Useful “subroutines”

– Something to note

• The TM must be set up to be in the correct

configuration before delete is “called”

• I.e. the tape head must be at the character to be

deleted.

– Questions?

Computation with Turing Machines

• Characteristic function of a set:

– For any language L, the characteristic function is defined as:

• 1 if x ∈ L
• 0 otherwise

– If a TM computes the characteristic function of a language, it will always halt with either a 1 or 0 left on the tape

• This is stronger than saying that x is accepted by the TM since

if x ∉ L, for a TM that accepts L, there is no guarantee that the the machine will halt.

Turing Machine Variants

• Some variants of the TM

– Multitaped TMs

• Have multiple tapes
• With a multiple tape heads that is read/writing a

different position on each tape at any one given time.

Turing Machine Variants

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Turing Machine Variants

• Multitape TM

– Move of a multitape TM

• Will depend upon:

– The state of the machine – The symbol being read on all of the tape heads

• Result:

– Each of the tape heads will write a new symbol on the tape – Each of the tape heads will move left, right, or stationery – Tape heads can move independently.

Multitape TM

• Example

– {wwR | w is a string in {a,b}* } – Even length palandromes – 2-tapes

• Step 1: Copy contents of tape 1 to tape 2
• Step 2: position head of tape 1 at beginning of tape

and head of tape 2 at end

• Step 3: Compare strings character by character

Multitape TM Multitape TM

• Let’s go to the video tape

Turing Machine Variants

• Multitape TM

– Thankfully, the multiTape TM can be shown to be equivalent to the “vanilla” TM.

• You can simulate the workings of a multitape TM
• n a single tape TM.
• See Theorem 8.9 in text.

Turing Machine Variants

• Some other variants of the TM

– Non-deterministic TMs

• More than 1 move is possible from any given

configuration.

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Turing Machine Variants

• Some variants of the TM

– Semi-infinite TM

• What some books calls the “basic” TM.
• Input tape is bounded on the left

Turing Machine Variants

• Thankfully, all these variants can be shown

to be equivalent to the “basic” TM.

– Specifically, non-determinism does not add extra computing power to a TM

• Much like FAs
• But unlike PDAs

– Multitape TMs will be handy in future lectures.

• No better or worse than the basic TM.

Summary

• Turing Machines
• Computation on TMs
• Subroutines
• TM variants

– Non-deterministic – Multitape TM

• Questions

To come

• Next time:

– The classes of languages associated with TMs – Chomsky Heirarchy.