Lecture 8 - Turing Machines

Question 1

What does a Turing Machine consist of?

Answer 1

• finite alphabet (includes blank symbols)
• unbounded tape of squares, each square can hold one single symbol
• tape is unbounded in both directions
• we can read and write to the squares
• a start state
• two halt states Sy and Sn - depending on output
• the transition function / program

Question 2

What does the transition relation in a Turing Machine do?

Answer 2

in state s , we can read a symbol on the tape , move to another state , overwrite the symbol and move left or right on the tape

Question 3

What happens if we don’t want to overwrite the symbol we read?

Answer 3

we simply write it back to the square before moving onto the next state (

Question 4

What happens if T halts in Sy?

Answer 4

The answer is yes

Question 5

What happens if T halts in Sn?

Answer 5

The answer is no.

Question 6

Main way of answering TM questions

Answer 6

pseudocode

Question 7

What does a turing machine actually compute?

Answer 7

accepts a language that actually computes a functions f(x)

Question 8

What does a function computed by turing return?

Answer 8

1 if the input is in the language or 0 otherwise

Question 9

What needs to be added if our TM if we are computing a function?

Answer 9

• we need a set of halting states
• a function f(x) = y
  where x is the input , y is the tape on output

Question 10

How can we output 1 or 0 from the TM?

Answer 10

instead of entering Sy we wrie 1 on the tape and enter a halt state

instead of entering Sn we write 0 on the tape and enter a halt state

Question 11

How should you go about designing a turing machine for functions?

Answer 11

Try a couple of examples, work out pattern

Question 12

When is a language Turing-recognizable?

Answer 12

If some TM recognises it with the given input x. The turing machine will halt in Sy.

Non-recognisable means halts in Sn or fails to halt.

Question 13

When is a language Turing-decidable?

Answer 13

When the tm decides it given an input string. The turing machine will halt in Sy - with a yes.

Non decidable means No is returned or fails to halt.

decidable is recognisable , but with a decision problem

Question 14

Is every decidable language recognisable?

Answer 14

Yes, but not every recognisable language is decidable.

Question 15

When is something Turing Computable?

Answer 15

There is a turing machine M such that for any input the machine M halts with the output f(x).

Question 16

How can turing machines be enhanced?

Answer 16

• adding more tapes than one
• having a 2d tape
• could operate non-deterministically

Question 17

Do enhacements increase the power of the turing machine?

Answer 17

NO
everything recognisable , decidable or computable with an enhanced turing machine, is also recognisable , decidable or computable with a basic turing machine

Question 18

When does non-determinism add power?

Answer 18

When dealing with a pushdown automata.