Combining Turing Machines

1
Combining Turing Machines

• Reading 9.2-10.4

2
Combining TMs

• We can string together Turing Machine
  computations in order to compute more complicated
  functions.
• f(x) xy if x gt y 0 if x lt y
• Answer Use Comparer calculation to end up in qx
  or qy.
• qx begins the adder computation
• qy erases all other ones and puts a zero at the
  end.

3
If - else

• We now can do some if-else constructions
• If a then qj else qk
• Read an a, then go to qj without writing or
  moving the head (that is, move right, then left)
• If the symbol was not an a, go to qk without
  changing anything.

4
Subroutines

• Call a subroutine in a Turing Machine.
• Want to stop computation A, go to a separate part
  of the tape, and do computation B. Then return
  to computation A in the same place on the tape.
• How it works
• Before calling B, A copies inputs into a shared
  tape space (indicated by a separator)
• A transitions into the initial state of B
• B uses its tape space to compute
• B writes relevant output to the shared space
• Bs final state now points back into a state in A.

5
Example x times y

• Initially, the tape should have leading zeroes,
  ones denoting x, a zero, then ones denoting y.
• At the end, it should have ones denoting xy at
  the beginning of the input.
• 1. Repeat until there are no more 1s in x
• Find a 1 in x and replace it with another symbol,
  a
• Replace the leftmost 0 by 0y.
• 2. Replace all as with 1s.

6
Turings Thesis

• Can Turing Machines do anything computers can do?
• Hmm - depends on the definition of computer!
• Turings thesis(mid-30s)
• Any computation carried out by mechanical means
  can be carried out by a Turing Machine.

7
Turings thesis true or false?

• Cant prove it, because we cant foresee what
  mechanical means people might come up with in
  the future.
• Could disprove it with one counterexample...

8
Evidence Supporting Turings thesis

• Anything that can be done on an existing digital
  computer can be done on a Turing Machine
• We can prove this - just write a TM for each
  machine instruction.
• No one has been able to suggest a problem
  solvable by an algorithm but not a TM
• Many alternative models of digital computers have
  been proposed - but none are more powerful than
  TMs

9
Algorithm definition

• An algorithm for a function is a Turing Machine
  that halts with the correct answer on the tape
  for any input in the domain.

10
Other types of automata

• Given another type of automata, how do you know
  which is more powerful?
• Simulation - show how anything done with one type
  of automata can be done with the other
• Example I can simulate with a PDA any FSA.
  Therefore PDAs as powerful (maybe more) than FSAs.

11
Example

• Any FSA can be written as a PDA
• So PDAs are at least as powerful.

12
Variations on TMs

• Add a stay-option
• Tape is only unbounded in one direction
• Separate read-only tape for input (off-line TM)
• Multiple storage tapes
• Multi-dimensional tape (extends infinitely in 2
  dimensions)
• Non-deterministic TMs (not presently possible
  with todays computers)
• All these are equivalent automata to Turing
  Machines!

13
Non-deterministic TMs

• An interesting class of TMs...
• Equivalent in power to TMs (so everything
  computable by a ND-TM is computable by a TM)
• BUT, it takes more steps on a TM than a ND-TM.
  We think it takes exponentially more steps...
• This is where the classes P and NP come from...
• P set of all problems computable by a TM in
  polynomial time
• NP set of all problems computable by a ND-TM in
  polynomial time
• P NP? Unknown...

14
Detour into 331...

• NP-Complete set of all problems in NP that are
  polynomial-time reducible to ALL problems in NP.
• Meaning, if you can solve 1 NPC problem in
  polynomial time, you can solve ALL NP problems in
  polynomial time, and P NP
• Cooks Theorem SAT is NP-Complete
• How did this work? He showed that every ND-TM
  has an equivalent SAT expression!!
• From that it is easy to reduce other NP-problems
  to SAT to prove that they too are NPC
• Hamiltonian paths (Traveling salesman)
• Cliques

15
Universal Turing Machines

• Are computers more powerful because they are
  programmable?
• Universal Turing Machine takes its own
  specification as input.
• Are Universal Turing Machines more powerful?

16
Universal TMs

• Universal TMs have been constructed from
  multi-tape TMs. One tape gives the input, one
  gives the internal storage, one tape is the
  program telling the machine what to do next.
• Since multi-tape TMs are equivalent to TMs,
  Universal TMs are equivalent to TMs!