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1 Turing Machines Variants Zeph Grunschlag

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2 Announcement Midterms not graded yet Will get them back Tuesday

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3 Agenda Turing Machine Variants Non-deterministic TM’s Multi-Tape

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4 Input-Output Turing Machines Input/output (or IO or transducing) Turing Machines, differ from TM recognizers in that they have a neutral halt state q halt 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 q halt. If v is the tape content upon halting, the notation f M (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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5 Input-Output Turing Machines When f M crashes or goes into an infinite loop for some contents, f M 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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6 TM Notations There are three ways that Sipser uses to describe TM algorithms. 1) High level –pseudocode which explains how algorithm works without the technical snafoos of TM notation 2) Implementation level –describe how the TM operates on its tape. No need to mention states explicitly. 3) Low-level description. One of: Set of complete “goto” style instructions State diagram Formal description: spell out the 7-tuple

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7 High-Level TM Example Let's for example describe a Turing Machine M which multiplies numbers by 2 in unary: M = "On input w = 1 n For each character c in w copy c onto the next available b blank space"

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8 Implementation-Level TM Example 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: Let’s see how is transformed.

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9 Implementation-Level TM Example So round by round, tape transformed as follows: X1111X XX111XX XXX11XXX XXXX1XXXX XXXXXXXXXX

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10 Implementation-Level TM Example 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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11 Implementation-Level TM Example M = "On input w = 1 n 1. HALT if no input 2. Write $ in left-most position 3. Sweep right and write X in first blank 4. Sweep left through X-streak and 1-streak 5. Go right 6. If read X, go right and goto 9. Else, replace 1 by X, move right. 7. If read X [[finished original w]] goto 8 Else, goto 3 8. Sweep to the right until reach blank, replace by X 9. Sweep left replacing everything non-blank by HALT

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12 Implementation-Level TM Example At the low level the Turing Machine is completely described, usually using a state diagram: 0 halt RR 2 1 $,R $|X R 1|X R 34 1L1L 1 X,R 1,L X,L XLXL 1L1L R1R XRXR XRXR 9 X 1,L $ 1,L $R$R 9 X 1,L

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13 Non-Deterministic TM’s 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 q acc on some computation branch. If, on the other hand, given any branch, N eventually enters q rej or crashes or enters an infinite loop on, w is not accepted. Symbolically as before: L(N) = { x * | accepting config. y, q 0 x * y } (No change needed as need not be function)

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14 Non-Deterministic TM’s Recognizers 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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15 Non-Deterministic TM Example 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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16 Non-Deterministic TM Example 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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17 Non-Deterministic TM Example A: {x {0,1,2}* | x has the same no. of 0’s as 1’s as 2’s } Q: Suppose q is the state of the TM while running inside nonDeterministicCrossOut(‘1’) and q’ is the state of the TM inside nonDeterministicCrossOut(‘2’). Suppose that current configuration is u = 0XX1Xq12X2 For which v do we have u v ?

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18 Non-Deterministic TM Example A: 0XX1Xq12X2 0XX1qX12X2 | 0XX1X1q2X2 | 0XX1XXq’ 2X2 These define 3 branches of computation tree: Q: Is this a non-deterministic TM decider? 0XX1XXq’ 2X2 0XX1Xq12X2 0XX1X1q2X20XX1qX12X2

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19 Non-Deterministic TM Example 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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20 NTM’s Konig’s 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: Konig’s 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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21 Konig’s Infinity Lemma Proof Idea Idea is to “smell-out” where the infinite part of the tree is and go in that direction:

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22 Konig’s Infinity Lemma Proof Idea Idea is to “smell-out” where the infinite part of the tree is and go in that direction:

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23 Konig’s Infinity Lemma Proof Idea Idea is to “smell-out” where the infinite part of the tree is and go in that direction:

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24 Konig’s Infinity Lemma Proof Idea Idea is to “smell-out” where the infinite part of the tree is and go in that direction:

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25 Konig’s Infinity Lemma Proof Idea Idea is to “smell-out” where the infinite part of the tree is and go in that direction:

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26 Konig’s Infinity Lemma Proof Proof. Given an infinite tree with finite branching construct an infinite path inductively: Vertex v 0 : Take the root. Edge v n v n+1 : Suppose v 0 v 1 …v n-1 v n has been constructed and that the subtree from v n is infinite. Then one of v n ’s finite no. of children, call it v n+1, must have an infinite subtree, so add the edge v n v n+1.

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27 Multi-tape TM’s Often it’s 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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28 Multi Tape TM Addition Example $101$1011 Input string

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29 Multi Tape TM Addition Example $101$1011

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30 Multi Tape TM Addition Example $101$1011

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31 Multi Tape TM Addition Example $101$1011

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32 Multi Tape TM Addition Example $101$1011

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33 Multi Tape TM Addition Example $101$1011 $

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34 Multi Tape TM Addition Example $101$1011 $1

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35 Multi Tape TM Addition Example $101$1011 $10

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36 Multi Tape TM Addition Example $101$1011 $101

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37 Multi Tape TM Addition Example $101$1011 $1011

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38 Multi Tape TM Addition Example $101$1011 $1011

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39 Multi Tape TM Addition Example $101$101 $1011

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40 Multi Tape TM Addition Example $101$10 $1011

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41 Multi Tape TM Addition Example $101$1 $1011

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42 Multi Tape TM Addition Example $101$ $1011

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43 Multi Tape TM Addition Example $101 $1011

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44 Multi Tape TM Addition Example $100 $1010

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45 Multi Tape TM Addition Example $100 $1000

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46 Multi Tape TM Addition Example $000 $1000

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47 Multi Tape TM Addition Example 0000 $0000

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48 Multi Tape TM Addition Example

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49 Multi Tape TM Addition Example

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50 Multi Tape TM Addition Example

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51 Multi Tape TM Addition Example

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52 Multi Tape TM Addition Example Output string HALT!

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53 Multitape TM’s Formal Notation NOTE: Sipser’s multitape machines cannot pause on one of the tapes as above example. This isn’t 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 -function of a k-tape machine:

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54 Multitape TM’s Conventions 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 ( *) 4

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