General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

State Machine:
1
1,R 1 1,R 1 1,L 1 1,L
1 0 0 0
1 0,R 2 0,R 3 1,L 4 0,L
5
0
1,R
c○: Michael Kohlhase 329
The computation of the turing machine is driven by the transition funciton: It starts in the initial
state, reads the character on the tape, and determines the next action, the character to write, and
the next state via the transition function.
Example Computation
T starts out in s1, replaces the first
1 with a 0, then
uses s2 to move to the right, skipping
over 1’s and the first 0 encountered.
s3 then skips over the next sequence
of 1’s (initially there are none) and
replaces the first 0 it finds with a 1.
s4 moves back left, skipping over 1’s
until it finds a 0 and switches to s5.
Step State Tape Step State Tape
1 s1 1 1 9 s2 10 0 1
2 s2 0 1 10 s3 100 1
3 s2 01 0 11 s3 1001 0
4 s3 010 0 12 s4 100 1 1
5 s4 01 0 1 13 s4 10 0 11
6 s5 0 1 01 14 s5 1 0 011
7 s5 0 101 15 s1 11 0 11
8 s1 1 1 01 — halt —
s5 then moves to the left, skipping over 1’s until it finds the 0 that was originally written by
s1.
It replaces that 0 with a 1, moves one position to the right and enters s1 again for another
round of the loop.
This continues until s1 finds a 0 (this is the 0 right in the middle between the two strings of
1’s) at which time the machine halts
c○: Michael Kohlhase 330
We have seen that a Turing machine can perform computational tasks that we could do in other
programming languages as well. The computation in the example above could equally be expressed
in a while loop (while the input string is non-empty) in SW, and with some imagination we could
even conceive of a way of automatically building action tables for arbitrary while loops using the
ideas above.
What can Turing Machines compute?
Empirically: anything any other program can also compute
Memory is not a problem (tape is infinite)
Efficiency is not a problem (purely theoretical question)
Data representation is not a problem (we can use binary, or whatever symbols we like)
All attempts to characterize computation have turned out to be equivalent
primitive recursive functions ([Gödel, Kleene])
lambda calculus ([Church])
186