Turing Machines Variants

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Turing Machines Variants

• Zeph Grunschlag

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Announcement

• Midterms not graded yet
• Will get them back Tuesday

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Agenda

• Turing Machine Variants
• Non-deterministic TMs
• Multi-Tape

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Input-Output Turing Machines

• Input/output (or IO or transducing) Turing
  Machines, differ from TM recognizers in that they
  have a neutral halt state qhalt instead of the
  accept and reject halt states. The TM is then
  viewed as a string-function which takes initial
  tape contents u to whatever the non blank portion
  of the tape is when reaching qhalt . If v is the
  tape content upon halting, the notation fM (u)
  v is used.
• If M crashes during the computation, or enters
  into an infinite loop, M is said to be undefined
  on u.

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Input-Output Turing Machines

• When fM crashes or goes into an infinite loop
  for some contents, fM is a partial function
• If M always halts properly for any possible
  input, its function f is total (i.e. always
  defined).

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TM Notations

• There are three ways that Sipser uses to describe
  TM algorithms.
• High level pseudocode which explains how
  algorithm works without the technical snafoos of
  TM notation
• Implementation level describe how the TM
  operates on its tape. No need to mention states
  explicitly.
• Low-level description. One of
• Set of complete goto style instructions
• State diagram
• Formal description spell out the 7-tuple

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High-Level TMExample

• Let's for example describe a Turing Machine M
  which multiplies numbers by 2 in unary
• M "On input w 1n
• For each character c in w
• copy c onto the next available b blank space"

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Implementation-Level TMExample

• The idea is to carry out the high level
  description by copying each character to the end.
  We also need to keep track of which characters
  have already been copied (or were copies
  themselves) by distinguishing these characters.
  One way is to use a different character, say X.
• EG Lets see how 11111 is transformed.

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Implementation-Level TMExample

• So round by round, tape transformed as follows
• 11111
• X1111X
• XX111XX
• XXX11XXX
• XXXX1XXXX
• XXXXXXXXXX
• 1111111111

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Implementation-Level TMExample

• Implementation level describes what algorithm
  actually looks like on the Turing machine in a
  way that can be easily turned into a
  state-diagram
• Some useful subroutines
• fast forward
• move to the right while the given condition holds
  
• rewind
• move to the left while the given condition holds
• May need to add extra functionality
• Add if need to tell when end of tape is

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Implementation-Level TMExample

• M "On input w 1n
• HALT if no input
• Write in left-most position
• Sweep right and write X in first blank
• Sweep left through X-streak and 1-streak
• Go right
• If read X, go right and goto 9. Else,
  replace 1 by X, move right.
• If read X finished original w goto 8
  Else, goto 3
• Sweep to the right until reach blank, replace by
  X
• Sweep left replacing everything non-blank by 1
• HALT

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Implementation-Level TMExample

• At the low level the Turing Machine is
  completely described, usually using a state
  diagram

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?R
1?L
X?L
1?,R
?X,L
1?L
0
2
3
4
X?R
1X?R
1?R
1?X,R
X?1,L
5
6
?R
X?1,L
X?R
?1,L
?1,L
X?R
8
9
halt
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Non-Deterministic TMs

• A non-Deterministic Turing Machine N allows more
  than one possible action per given state-tape
  symbol pair.
• A string w is accepted by N if after being put on
  the tape and letting N run, N eventually enters
  qacc on some computation branch. 
• If, on the other hand, given any branch, N
  eventually enters qrej or crashes or enters an
  infinite loop on, w is not accepted.
• Symbolically as before
• L(N) x ? S ? accepting config. y, q0 x ?
  y
• (No change needed as ? need not be function)

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Non-Deterministic TMsRecognizers vs. Deciders

• N is always called a non-deterministic recognizer
  and is said to recognize L(N) furthermore, if in
  addition for all inputs and all computation
  branches, N always halts, then N is called a
  non-deterministic decider and is said to decide
  L(N).

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Non-Deterministic TMExample

• Consider the non-deterministic method
• void nonDeterministicCrossOut(char c)
• while()
• if (read blank) go left
• else
• if (read c)
• cross out, go right, return
• OR go right // even when reading c
• OR go left // even when reading c

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Non-Deterministic TMExample

• Using randomCross() put together a
  non-deterministic program
• 1. while(some character not crossed out)
• nonDeterministicCrossOut(0)
• nonDeterministicCrossOut(1)
• nonDeterministicCrossOut(2)
• 2. ACCEPT
• Q What language does this non-deterministic
  program recognize ?

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Non-Deterministic TMExample

• A x ? 0,1,2 x has the same no.
• of 0s as 1s as 2s
• Q Suppose q is the state of the TM while running
  inside nonDeterministicCrossOut(1) and q is
  the state of the TM inside nonDeterministicCrossOu
  t(2).
• Suppose that current configuration is
• u 0XX1Xq12X2
• For which v do we have u ? v ?

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Non-Deterministic TMExample

• A 0XX1Xq12X2 ?
• 0XX1qX12X2 0XX1X1q2X2 0XX1XXq 2X2
• These define 3 branches of computation tree
• Q Is this a non-deterministic TM decider?

0XX1Xq12X2
0XX1XXq 2X2
0XX1X1q2X2
0XX1qX12X2
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Non-Deterministic TMExample

• A No. This is a TM recognizer, but not a
  decider. nonDeterministicCrossOut() often enters
  an infinite branch of computation since can
  see-saw from right to left to right, etc. ad
  infinitum without ever crossing out anything.
  I.e., computation tree is infinite!
• Note If you draw out state-diagrams, you will
  see that the NTM is more compact, than TM version
  so there are some advantages to non-determinism!
  Later, will encounter examples of efficient
  nondeterministic programs for practically
  important problems, with no known efficient
  counterpart The P vs. NP Problem.

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NTMsKonigs Infinity Lemma

• For Problem 3.3 in Sipser the following fact is
  important
• If a NTM is a decider then given any input, there
  is a number h such that all computation branches
  involve at most h basic steps. I.e., computation
  tree has height h. Follows from
• Konigs Infinity Lemma An infinite tree with
  finite branching at each node must contain an
  infinitely long path from the root.
• Or really, the contrapositive is used A tree
  with no infinite paths, and with finite branching
  must itself be finite.

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Konigs Infinity LemmaProof Idea

• Idea is to smell-out where the infinite part of
  the tree is and go in that direction

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Konigs Infinity LemmaProof Idea

• Idea is to smell-out where the infinite part of
  the tree is and go in that direction

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Konigs Infinity LemmaProof Idea

• Idea is to smell-out where the infinite part of
  the tree is and go in that direction

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Konigs Infinity LemmaProof Idea

• Idea is to smell-out where the infinite part of
  the tree is and go in that direction

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Konigs Infinity LemmaProof Idea

• Idea is to smell-out where the infinite part of
  the tree is and go in that direction

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Konigs Infinity LemmaProof

• Proof. Given an infinite tree with finite
  branching construct an infinite path inductively
• Vertex v0 Take the root.
• Edge vn ? vn1 Suppose v0?v1vn-1?vn has been
  constructed and that the subtree from vn is
  infinite. Then one of vns finite no. of
  children, call it vn1, must have an infinite
  subtree, so add the edge vn ? vn1.

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Multi-tape TMs

• Often its useful to have several tapes when
  carrying out a computations. For example,
  consider a two tape I/O TM for adding numbers (we
  show only how it acts on a typical input)

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Multi Tape TMAddition Example
Input string
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
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Multi Tape TMAddition Example
HALT!
Output string
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Multitape TMsFormal Notation

• NOTE Sipsers multitape machines cannot pause
  on one of the tapes as above example. This isnt
  a problem since pausing 1-tape machines can
  simulate pausing k-tape machines, and non-pausing
  1-tape machines can simulate 1-tape pausing
  machines by adding dummy R-L moves for each
  pause.
• Formally, the d-function of a k-tape machine

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Multitape TMsConventions

• Input always put on the first tape
• If I/O machine, output also on first tape
• Can consider machines as string-vector
  generators. E.g., a 4 tape machine could be
  considered as outputting in (S)4