Turing Machines Context-Free Languages n b n R a ww Regular - PDF document

Review • Languages and Grammars – Alphabets, strings, languages • Regular Languages – Deterministic Finite and Nondeterministic Automata – Equivalence of NFA and DFA and Minimizing a DFA CS 301 - Lecture 19 – Regular Expressions – Regular Grammars Turing Machines – Properties of Regular Languages – Languages that are not regular and the pumping lemma • Context Free Languages – Context Free Grammars Fall 2008 – Derivations: leftmost, rightmost and derivation trees – Parsing and ambiguity – Simplifications and Normal Forms – Nondeterministic Pushdown Automata – Pushdown Automata and Context Free Grammars – Deterministic Pushdown Automata – Pumping Lemma for context free grammars – Properties of Context Free Grammars • Turing Machines The Language Hierarchy ww ? n n n ? a b c Turing Machines Context-Free Languages n b n R a ww Regular Languages a * a * b * 1

Languages accepted by Tape A Turing Machine Turing Machines ...... ...... ww n n n a b c Read-Write head Context-Free Languages Control Unit n b n R a ww Regular Languages a * a * b * ...... ...... The Tape No boundaries -- infinite length Read-Write head ...... ...... The head at each time step: Read-Write head 1. Reads a symbol 2. Writes a symbol The head moves Left or Right 3. Moves Left or Right 2

Example: Time 1 Time 0 ...... ...... a b k c ...... ...... a a c b Time 2 Time 1 ...... ...... f ...... ...... a c k a c b k a b 1. Reads 1. Reads f 2. Writes k 2. Writes 3. Moves Left 3. Moves Right The Input String Input string Input string Blank symbol Blank symbol ...... ...... ...... ...... a a c a a c b b ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ head head Head starts at the leftmost position Remark: the input string is never empty of the input string 3

Example: Time 1 States & Transitions Write ...... ...... Read a a c Move Left b ◊ ◊ ◊ ◊ ◊ q 1 a b , L → current state q q 1 2 Move Right a b , R → q q a b , R → 2 1 q q 1 2 Example: Time 1 Time 1 ...... ...... ...... ...... a a c a a c b b ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ q q 1 1 Time 2 Time 2 ...... ...... ...... ...... a c a c b b b b ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ q q 2 2 a b , R a b , L → → q q q q 1 2 1 2 4

Example: Time 1 Determinism ...... ...... a a c b Turing Machines are deterministic ◊ ◊ ◊ ◊ ◊ Not Allowed Allowed q 1 a b , R q a b , R q → → 2 Time 2 2 ...... ...... g a c b b ◊ ◊ ◊ ◊ q q 1 1 q q 3 a d , L 3 q b d , L → → 2 g , R ◊ → q q No lambda transitions allowed 1 2 Partial Transition Function Example: Halting ...... ...... a a c b ◊ ◊ ◊ ◊ ◊ The machine halts if there are q no possible transitions to follow 1 Allowed: a b , R q → 2 No transition q 1 for input symbol c q 3 b d , L → 5

Example: Final States q q Allowed 1 2 ...... ...... a b a c ◊ ◊ ◊ ◊ ◊ q 1 Not Allowed q q 1 2 a b , R q → 2 No possible transition • Final states have no outgoing transitions q 1 HALT!!! q b d , L 3 • In a final state the machine halts → Acceptance Turing Machine Example If machine halts A Turing machine that accepts the language: Accept Input in a final state aa * If machine halts a a , R → in a non-final state Reject Input or , L ◊ → ◊ q q If machine enters 0 1 an infinite loop 6

a a a a a a Time 0 Time 1 ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ q q 0 0 a a , R a a , R → → , L , L ◊ → ◊ ◊ → ◊ q q q q 0 1 0 1 Time 2 a a a Time 3 a a a ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ q q 0 0 a a , R a a , R → → , L , L ◊ → ◊ ◊ → ◊ q q q q 0 1 0 1 7

Rejection Example a a a Time 4 ◊ ◊ ◊ ◊ q a a Time 0 b 1 ◊ ◊ ◊ ◊ q 0 a a , R → Halt & Accept a a , R → , L ◊ → ◊ q q 0 1 , L ◊ → ◊ q q 0 1 Infinite Loop Example Time 1 a a b ◊ ◊ ◊ ◊ q 0 b b , L → No possible Transition a a , R → Halt & Reject a a , R → , L ◊ → ◊ q q 0 1 , L ◊ → ◊ q q 0 1 8

Time 0 a a Time 1 a a b b ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ q q 0 0 b b , L b b , L → → a a , R a a , R → → , L , L ◊ → ◊ ◊ → ◊ q q q q 0 1 0 1 Time 2 a a b Time 2 a a b ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ q 0 q 0 Time 3 a a b ◊ ◊ Infinite loop ◊ ◊ q 0 b b , L → a a a a , R Time 4 b → ◊ ◊ ◊ ◊ q 0 , L ◊ → ◊ q q 0 1 Time 5 a a b ◊ ◊ ◊ ◊ q 0 9

Turing Machine Example Because of the infinite loop: n b n Turing machine for the language { a } • The final state cannot be reached • The machine never halts y y , R y y , L → q → 4 y y , R → a a , R a a , L → → , L ◊ → ◊ • The input is not accepted y y , R a x , R b y , L → → → q q q q 3 0 2 1 x x , R → a a x a Time 0 b b Time 1 b b ◊ ◊ ◊ ◊ ◊ ◊ q q 0 1 y y , L y y , L y y , R y y , R → → q → q → 4 4 y y , R y y , R → a a , R a a , L → a a , R a a , L → → → → , L , L ◊ → ◊ ◊ → ◊ y y , R a x , R b y , L y y , R a x , R b y , L → → → → → → q q q q q q q q 3 0 1 2 3 0 1 2 x x , R x x , R → → 10

Time 2 x a Time 3 x a y b b b ◊ ◊ ◊ ◊ ◊ ◊ q q 1 2 y y , R y y , L y y , R y y , L → → q → q → 4 4 y y , R y y , R → a a , R a a , L → a a , R a a , L → → → → , L , L ◊ → ◊ ◊ → ◊ y y , R a x , R y y , R a x , R b y , L b y , L → → → → → → q q q q q q q q 3 0 2 3 0 2 1 1 x x , R x x , R → → x a y x a y Time 4 b Time 5 b ◊ ◊ ◊ ◊ ◊ ◊ q q 2 0 y y , L y y , L y y , R y y , R → → q → q → 4 4 y y , R y y , R → a a , R a a , L → a a , R a a , L → → → → , L , L ◊ → ◊ ◊ → ◊ y y , R a x , R b y , L y y , R a x , R b y , L → → → → → → q q q q q q q q 3 0 1 2 3 0 1 2 x x , R x x , R → → 11

Time 6 x x y Time 7 x x y b b ◊ ◊ ◊ ◊ ◊ ◊ q q 1 1 y y , R y y , L y y , R y y , L → → q → q → 4 4 y y , R y y , R → a a , R a a , L → a a , R a a , L → → → → , L , L ◊ → ◊ ◊ → ◊ y y , R a x , R y y , R a x , R b y , L b y , L → → → → → → q q q q q q q q 3 0 2 3 0 2 1 1 x x , R x x , R → → x x y y x x y y Time 8 Time 9 ◊ ◊ ◊ ◊ ◊ ◊ q q 2 2 y y , L y y , L y y , R y y , R → → q → q → 4 4 y y , R y y , R → a a , R a a , L → a a , R a a , L → → → → , L , L ◊ → ◊ ◊ → ◊ y y , R a x , R b y , L y y , R a x , R b y , L → → → → → → q q q q q q q q 3 0 1 2 3 0 1 2 x x , R x x , R → → 12

Time 10 x x y y Time 11 x x y y ◊ ◊ ◊ ◊ ◊ ◊ q q 0 3 y y , R y y , L y y , R y y , L → → q → q → 4 4 y y , R y y , R → a a , R a a , L → a a , R a a , L → → → → , L , L ◊ → ◊ ◊ → ◊ y y , R a x , R y y , R a x , R b y , L b y , L → → → → → → q q q q q q q q 3 0 2 3 0 2 1 1 x x , R x x , R → → x x y y x x y y Time 12 Time 13 ◊ ◊ ◊ ◊ ◊ ◊ q q 3 4 Halt & Accept y y , L y y , L y y , R y y , R → → q → q → 4 4 y y , R y y , R → a a , R a a , L → a a , R a a , L → → → → , L , L ◊ → ◊ ◊ → ◊ y y , R a x , R b y , L y y , R a x , R b y , L → → → → → → q q q q q q q q 3 0 1 2 3 0 1 2 x x , R x x , R → → 13

Observation: Formal Definitions If we modify the n b n machine for the language { a } for Turing Machines we can easily construct n n n { a b c } a machine for the language Transition Function Transition Function a b , R c d , L → → q q q q 1 2 1 2 ( q , a ) ( q , b , R ) ( q , c ) ( q , d , L ) δ = δ = 1 2 1 2 14