Grad Students Only

1
Grad Students Only

• Please see me after class about project
• Integration Paper
• 5 topics to me by next week, paper due before
  final
• Sample Topics (see web site for more samples)
• Programming Languages
• Construction of Hardware Automata
• Protocol Design
• A.I.
• Microprocessors
• 2 pages ½ page (12 pt. font, double-spaced)
• MS Word (submit via email)

2
8b "Busy Beaver" Problem Variations on Basic
Turing Machine 10.1,10.2,10.3

• BB(n) Function
• Equivalence of Classes of Automata
• Simulation of One Type of T.M. by Another
• Multiple Tracks on Tape
• Multi-Tape T.M.
• 2-Dimensional "Tape"
• Non-Deterministic T.M.

3
BB on Simulator

• Run BB(2), BB(3), BB(4), BB(5)
• http//cs.union.edu/csc140

4
BB(n) Function

• BB(1) 1
• BB(2) 4
• BB(3) 6

• BB(4) 13
• BB(5) ? 4,098
• BB(6) ? 95,524,079

Note There are 63,403,380,965,376 possible
5-state BB machines and 232,218,265,089,212,000
possible 6-state BB machines. We must run each
one for millions or billions or zillions of steps
to see how many 1s are on the tape if and when
it halts
5
BB(n) is not-computable

• BB(12) ? 640964096409640964 166 times!
• Proof is beyond the scope of this course

6
Equivalence of Classes of Automata

• Added features do not change power of class of
  machines if they can be simulated on
  single-track, single-head machine

7
Simulation of One Type of T.M. by Another

• If simulated machine halts, so does the
  simulation
• If simulated machine never halts, neither does
  the simulation

8
Multiple Tracks on Tape

• Simulate multiple tracks on a one-track T.M.
• The Trick
• Define new symbols consisting of ordered pairs
  (triples, etc.) of symbols from cells on
  multi-track machine

9
Multi-Tape T.M.

• Multi-Tracks for each tape. Extra tracks for
  keeping track of each read/write head
• Note that we have to scan the tracks to find the
  pointer to the appropriate tape
• Could use some sort of marker indicating leftmost
  and rightmost visited cell thus far
• Note also how SLOW this simulation will be...

10
Multiple Tapes allow "addressable" memory

• If we wanted to solve a problem such as, say, z
  x y
• Put x on one tape, y on second tape
• Answer on third tape

11
2-Dimensional "Tape" page 261-62

• Move Left, Right, Up, Down
• Simulate with 2-track machine which keeps track
  of tape symbols of simulated machine together
  with "address" (1,2) or (10,-3), say.
• Since address may be arbitrarily large, need to
  use delimiters to separate "cells" of simulated
  machine
• could use (1,2) beneath simulated symbol--book
  uses 12
• if at (1,2) and told to move R, need to find
  (1,3)
• if (1,3) cell doesnt exist already, create it...
• Note how LONG it takes to simulate simple moves

12
Deterministic Simulation of Non-Deterministic
T.M.

• List all possible current configurations on
  separate tapes (adding new "tapes" as you go)
• Simulate on 2-dimensional tape
• Each tape will represent current config. of 1
  choice
• Determine maximum number of choices for any one
  move, count in this number base (on a separate
  tape) where each digit indicates which choice of
  move at each stage in simulation
• All possible 1-move simulations first...
• Talk about SSSLLLLOOOOWWWWW

13
In-Class ExerciseExercise 10.2.5 page 262

• Precursor to "Universal" Turing Machine
• Work in groups of 2 or 3
• See example on next slide (peeking OK)
• First 2 groups done, put on board

14
Action of Simulator

• To simulate ?(q7,a)(q4,b,L)

15
TM Simulator (Darius Version)Ignoring symbol on
top track
qi/?,L
?(qi,) (qj,,L)
State for j0
0
?/q0,S
qi/?,L
?/qn,S
qi/?,L
qi/?,L
?/q1S
n
qi/?,L
1
State for j1
?/qn-1,S
?/q2,S
n-1
2

Same "j" states for moves Right. Problem More
than 6 states
16
TM Simulator (Nikhil Version)Ignoring Symbol on
Top Track
?(qi,) (qj,,L)
qi/qj,L
increment new location
?/q0,R
qn/qn1,R
qn1/qn,L
decrement old location
q0/?,L
2 More States for moves Right and we're done!
17
In-Class ExerciseSpecific Example

• Original TM QA,B, ?0,1 let 0 represent
  blank
• ?(A,0)(B,0,L) ?(A,1)(A,1,R)
• ?(B,0)(A,1,R) ?(B,1)(A,1,L)
• Simulator Q p0,pL,pR,pL',pR' ? (0,A),
  (0,B), (0,?), (1,A), (1,B), (1,?)
• ?(p0,(0,A))(pL,(0,B),L) ?(p0,(1,A))(pR,(1,A),R)
  
• ?(p0,(0,B))(pR,(1,A),R) ?(p0,(1,B))(pL,(1,A),L)
  
• ?(pL,(0,?))(pL',(0,A),R) ?(pL,(0,A),R)(pL',(0,B
  ),R)
• ?(pL,(1,?))(pL',(1,A),R) ?(pL,(1,A),R)(pL',(1,B
  ),R)
• ?(pL',(0,B))(pL,(0,A),L) ?(pL',(0,A),L)(p0,(0,
  ?),L)
• ?(pL',(1,B))(pL,(1,A),L) ?(pL',(1,A),L)(p0,(1,
  ?),L)
• Repeat previous 4 lines for pR and pR' (moving
  opposite directions)

18
In-Class ExerciseGeneral Answer (1 of 2)

• The simulator uses a two-track tape.
• The simulated tape on top, and the state of the
  simulated machine underneath.
• You only need one state (call it p0) to simulate
  ?(qi,b)(qj,e,R) for any change of states and
  symbols.
• However, to simulate the move ?(qi,b)(qj,e,R)
  the simulator cannot simply ?(p0,(b,qi))(p0,(e,qj
  ),R) since that would put the R/W head of the
  simulator over the correct cell but leave the
  simulated state under the wrong cell.

19
In-Class ExerciseGeneral Answer (2 of 2)

• Hence we need additional states to "move" the
  state of the simulated machine under the R/W
  head.
• Something like ?(p0,(b,qi))(pR,(e,qj),R), then
  pR would go back and pick up the simulated state
  before returning to state p0.
• Similarly, ?(p0,(b,qi))(pL,(e,qj),L) for
  ?(qi,b)(qj,e,L).
• To copy the state of the simulated machine,
  decrement the spot it is coming from, increment
  the spot it is going to--continue until the from
  spot is decremented down past first state to
  blank. (q8? q7? q6? q5?q4?q3?q2?q1?q0?blank)
• Note that the simulator must have all states of
  the simulated machine as part of its (lower-half)
  tape alphabet.