Universal Models of Computation

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Universal Models of Computation

• Zeph Grunschlag

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Agenda

• Universal Models of Computation
• Recursive vs. Recursively Enumerable
• TM ? NTM ? k-tape TM ? k-track TM ? TM,
  therefore, all universal
• TM ? Queue-machine
• Exercise on HW8. Queue-machine ? TM

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Equivalence of TM Variants

• FAs universal any system for generating strings
  with strictly finite memory can be simulated by
  an FA.
• PDAs not as universal. CFGs split up into
  following hierarchy
• Deterministic CFL ? Unambiguous CFL ? CFL
• and much finer splittings are possible.
• TMs give a very natural class. Every reasonable
  TM variant gives rise to same language class.
  (Church-Turing Thesis)

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Recursive and Recursively Enumerable

• DEF A language is recursively enumerable if it
  is recognized by some Turing Machine. A language
  is recursive if it is decided by some Turing
  Machine.
• The terminology recursively enumerable comes
  from another TM model a TM which never halts
  (when describing an infinite language) and writes
  one string of the language at a time on an
  auxiliary printer.

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Recursive EnumeratorExample

• For example, at some point, the print-out of a
  recursive enumerator for pal might look like
• (empty string)
• a
• b
• aa
• bb
• aba
• bab
• aaa
• bbb

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Recursive EnumeratorExample

• Recursive-enumerators are equivalent to TM
  recognizers because of the following intuition
• Enumerator ? TM Given any input, you could
  simulate the enumerator. If the input ever
  appears on the list, accept.
• TM ? Enumerator Start with an enumerator for S
  and compose with the TM. Dovetail (i.e.
  multithread) trying to accept each element on the
  list. If any string accepted, write on
  enumerators output list.
• See Sipser p. 140141 for further details.

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Equivalence of k-tape vs. 1-tape

• The argument that k-tape TMs are universal, is
  fairly convincing. After all, the computers we
  use daily split their memory into several
  locations RAM, Hard-Disc, Floppy, CD-ROM. It
  is plausible that each memory unit could be
  simulated by a separate tape. The following
  theorem implies that the computer could still
  function fully (though probably with a loss of
  efficiency) if all the memory was merged onto a
  single tape

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Equivalence of k-tape vs. 1-tape

• THM Every k-tape TM can be simulated by a
  1-tape TM, and vice versa.
• Proof. Its clear that every 1-tape TM can be
  simulated by a k-tape TM. Simply ignore tapes
  no. 2 through k.
• On the other hand, to show that any k-tape TM can
  be simulated by a 1-tape TM, well show two
  parts
• Any k-tape TM can be simulated by a 2k-track TM.
• Any k-track TM can be simulated by a 1-tape TM.

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k-track TMs

• A k-track machine contains a tape whose cells are
  broken into k vertical sub-cells. As opposed to
  k-tape machines, there is only one reading head1,
  but it may read all k tracks and base its actions
  on these values.
• EG. A 2-tape machine
• vs. a 2-track machine

1 0 1 1 0 1 1
1 0 1
1 0 1 1 0 1 1
1 0 1
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k-track TMs

• A k-track machine contains a tape whose cells are
  broken into k vertical sub-cells. As opposed to
  k-tape machines, there is only one reading head1,
  but it may read all k tracks and base its actions
  on these values.
• EG. A 2-tape machine
• vs. a 2-track machine

1 0 1 1 0 1 1
1 0 1
1 0 1 1 0 1 1
1 0 1
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2k-tracks Simulating k-tapes

• We can simulate any k-tape TM by a k-track TM as
  follows
• Each tape of the k-tape TM is represented by one
  track.
• Each track representing a tape, is underneath a
  track which consists of all blanks, except for a
  single X which tells us which cells of the k-tape
  TM are active

X
1 0 1 1 0 1 1
X
1 0 1
1 0 1 1 0 1 1
1 0 1
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2k-tracks Simulating k-tapes

• For each move of k-tape TM, the 2k-track TM
  sweeps to right and then to left.
• During right-sweep, 2k-track TM notes the
  current state of the k-tape machine as well as
  the symbols read by each head, keeping this
  information in its finite control. Head back
  when passed all Xs.
• During the left-sweep, every X encountered
  causes corresponding track active cell to change
  in accordance to the memorized configuration of
  the k-tape machine.
• EG

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2k-tracks Simulating k-tapes
Corresponding simulation sequence Notable
1 0 1 1 0 1 1
1 0 1
X
1 0 1 1 0 1 1
X
1 0 1
k-tape TM transition
1 0 1 1 0 3 1
1 0 1 2
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2k-tracks Simulating k-tapes
Corresponding simulation sequence Notable
1 0 1 1 0 1 1
1 0 1
X
1 0 1 1 0 1 1
X
1 0 1
k-tape TM transition
1 0 1 1 0 3 1
1 0 1 2
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2k-tracks Simulating k-tapes
Corresponding simulation sequence Notable
1 0 1 1 0 1 1
1 0 1
X
1 0 1 1 0 1 1
X
1 0 1
k-tape TM transition
1 0 1 1 0 3 1
1 0 1 2
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2k-tracks Simulating k-tapes
Corresponding simulation sequence Notable
1 0 1 1 0 1 1
1 0 1
X
1 0 1 1 0 1 1
X
1 0 1
k-tape TM transition
1 0 1 1 0 3 1
1 0 1 2
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2k-tracks Simulating k-tapes
Corresponding simulation sequence Notable
1 0 1 1 0 1 1
1 0 1
X
1 0 1 1 0 1 1
X
1 0 1
k-tape TM transition
1 0 1 1 0 3 1
1 0 1 2
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2k-tracks Simulating k-tapes
Corresponding simulation sequence Notable
2nd head reads