Linear Bounded Automata LBAs 1 Linear Bounded Automata

Linear Bounded Automata LBAs 1

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

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

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

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

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

A Universal Turing Machine 7

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

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

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

Description of Universal Turing Machine Tape Contents of State of 11

Alphabet Encoding Symbols: Encoding: 12

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

Transition Encoding Transition: Encoding: separator 14

Machine Encoding Transitions: Encoding: separator 15

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

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 17

Countable Sets 18

Infinite sets are either: • Countable • Uncountable 19

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

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

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

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

Better Approach 24

25

26

27

28

29

Rational Numbers: Correspondence: Positive Integers: 30

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

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

strings Enumeration Machine for output Finite time: 33

Enumeration Machine Configuration Time 0 Time 34

Time 35

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

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

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

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

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

Theorem: The set of all Turing Machines is countable 41

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

Enumeration Procedure: Repeat 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 43

Uncountable Sets 44

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

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

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

Elements of the powerset have the form: 48

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

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

Powerset element Encoding 51

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

New element: (Diagonal complement) 53

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

We have contradiction! Therefore the powerset is uncountable 55