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Turing Machines

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Turing Machines

Chapter 3.1

- Turing Machines(TMs)
- Alan Turing
- Church-Turing Thesis

- Definitions
- Computation
- Configuration
- Recognizable vs. Decidable

- Examples
- Simulator

- Alan Turing

Alan Turing was one of the founding fathers of CS.

- His computer model the Turing Machine(1936)was the inspiration for the electronic computer that came two decades later
- Was instrumental in cracking the Nazi Enigma cryptosystem in WWII
- Invented the “Turing Test” used in AI
- The Turing Award. Pre-eminent award in Theoretical CS (called the “Nobel Prize” of CS)

- Thesis- Every effectively calculable function is a computable function
- Everything that is computable is computable by a Turing machine
- The thesis remains a hypothesis
- Despite the fact that it cannot be formally proven the Church–Turing thesis now has near-universal acceptance.

- Most powerful machine so far…
- Similar to a finite automata
- Uses infinite tape as memory
- Can both read from and write to the tape
- Read/write head can move left/right
- Accept/reject take affect immediately

- Similar to a finite automata
- Cannot solve all problems

- Sample Rules:
- If read 1, write 0, go right, repeat.
- If read 0, write 1, HALT!
- If read “”, write 1, HALT!

- Using these rules on a tape containing the reverse binary representation of 47 we obtain:

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

So the successor’s output on 111101 was 000011 which is the reverse binary representation of 48.

Similarly, the successor of 127 should be 128:

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

If read 1, write 0, go right, repeat.

If read 0, write 1, HALT!

If read “”, write 1, HALT!

- Draw the machine
- Description
- If read 1, write 0, go right, repeat.
- If read 0, write 1, HALT!
- If read “”, write 1, HALT!

- Description

- A string w is accepted by M if after being put on the tape then letting M run, M eventually enters the accept state. Therefore, w is an element of L(M) - the language accepted by M.
- We can formalize this notion as follows:

Suppose TM’s configuration at time t is given by uapxvwhere pis the current state, ua is what’s to the left of the head, x is what’s being read, and vis what’s to the right of the head.

If d(p,x) = (q,y,R) then write:

uapxv uayqv

With resulting configuration uaypv at time t+1. If, d(p,x) = (q,y,L) instead, then write:

uapxv uqayv

There are also two special cases:

- head is forging new ground –pad with the blank symbol
- head is stuck at left end –by def. head stays put
NOTE: “” is read as “yields”

Three possibilities occur on a given input w :

- The TM M eventually enters qacc and therefore halts and accepts. (w L(M) )
- The TM M eventually enters qrej orcrashes somewhere. M rejectsw . (w L(M) )
- Neither occurs! I.e., M never halts its computation and is caught up in an infinite loop, never reaching qacc or qrej. In this case w is neither accepted nor rejected. However, any string not explicitly accepted is considered to be outside the accepted language. (w L(M) )

- Recognizable- The TM recognizes the language but doesn’t necessarily reach an accept or reject state (could loop FOREVER )
- Decidable- is recognizable and guaranteed to reach an accept or reject state (without the possibility for an infinite loop )

Bit-shifting example

1) (25 pts) Give the full, formal description of a Turing Machine that accepts the following language:

L = { w#wR | where w consists of only 0s and 1s and has length at least 1. }

The input alphabet will be {0, 1, #}.

The tape alphabet will include 0, 1, #, B, and any other symbols you choose to include in it.

Give a simple intuitive description of what each state in the machine represents. And draw the diagram for this machine.

- Accepts 4 or 5 tuples
- (Start,input,NewState,output,Direction)
- (Start,input,NewState,Direction OR output)

- One tape
- http://ironphoenix.org/tril/tm/