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1 Linear Bounded Automata LBAs

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2 Linear Bounded Automata are like Turing Machines with a restriction: The working space of the tape is the space of the input string

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3 Left-end marker Input string Right-end marker Working space of tape All computation is done between end markers

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4 We define LBA’s to be NonDeterministic Open Problem: NonDeterministic LBA’s have same power with Deterministic LBA’s ?

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5 Example languages accepted by LBAs: LBA’s have more power than NPDA’s

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6 Later in class we will prove: LBA’s have less power than Turing Machines

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7 A Universal Turing Machine

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8 Turing Machines are “hardwired” They execute only one program Limitation of Turing Machines: Real Computers are reprogrammable

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9 Solution: Universal Turing Machine is a reprogrammable machine simulates any other Turing Machine

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10 Universal Turing machine simulates any Turing machine Input of Universal Machine: Description of transitions of Initial tape contents of

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11 Universal Turing Machine Description of Tape Contents of State of

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12 Alphabet Encoding Symbols: Encoding:

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13 State Encoding States: Encoding: Head Move Encoding Move: Encoding:

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14 Transition Encoding Transition: Encoding: separator

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15 Machine Encoding Transitions: Encoding: separator

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16 Input of Universal Turing Machine: encoding of the simulated machine

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17 A Turing Machine is described with a string of 0’s and 1’s The set of Turing machines form a language: each string of the language is the encoding of a Turing Machine

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18 Countable Sets

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19 Infinite sets are either: Countable Uncountable

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20 Countable set: There is a one to one correspondence between elements of the set and positive integers

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21 Example: Even integers: The set of even integers is countable Positive integers: Correspondence: corresponds to

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22 Example:The set of rational numbers is countable Rational numbers:

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23 Naive Approach Rational numbers: Positive integers: Correspondence: Doesn’t work: we will never count numbers with nominator 2

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24 Better Approach

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30 Rational Numbers: Correspondence: Positive Integers:

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31 We proved: the set of rational numbers is countable by giving an enumeration procedure

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32 Definition An enumeration procedure for is a Turing Machine that generates any string of in finite number of steps Let be a set of strings

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33 Enumeration Machine for Finite time: strings output

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34 Enumeration Machine Configuration Time 0 Time

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35 Time

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36 A set is countable if there is an enumeration procedure for it

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37 Example: The set of all strings is countable We will describe the enumeration procedure

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38 Naive procedure: Produce the strings in lexicographic order: Doesn’t work: strings starting with will never be produced

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39 Better procedure: Produce all strings of length 1 Produce all strings of length 2.......... Proper Order

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40 Produce strings: Length 2 Length 3 Length 1 Proper Order

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41 Theorem: The set of all Turing Machines is countable

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42 Theorem: The set of all Turing Machines is countable Proof: Find an enumeration procedure for the set of Turing Machine strings Any Turing Machine is encoded with a string of 0’s and 1’s

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43 1. Generate the next string of 0’s and 1’s in proper order 2. Check if the string defines a Turing Machine if YES: print string on output if NO: ignore string Enumeration Procedure: Repeat

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44 Uncountable Sets

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45 A set is uncountable if it is not countable Definition:

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46 Theorem: Let be an infinite countable set. The powerset of is uncountable

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47 Proof: Since is countable, we can write Element of

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48 Elements of the powerset have the form:

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49 We encode each element of the power set with a string of 0’s and 1’s Powerset element Encoding

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50 Let’s assume for contradiction that the powerset is countable. We can enumerate the elements of the powerset

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51 Powerset element Encoding

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52 Take the powerset element whose bits are the complements if the diagonal

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53 New element: (Diagonal complement)

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54 The new element must be some This is impossible: The i-th bit must be the complement of itself

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55 We have contradiction! Therefore the powerset is uncountable

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