Turing Machines: Understanding Recursively Enumerable and Decidable Languages, Slides of Theory of Computation

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BCSE304L TOC

Turing Machine

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Turing Machines are…

• (^) Very powerful (abstract) machines that could simulate any modern

day computer (although very, very slowly!)

• (^) Why design such a machine?
  • (^) If a problem cannot be “solved” even using a TM, then it implies that the

problem is undecidable

• (^) Computability vs. Decidability For every input, answer YES or NO

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A Turing Machine (TM)

• (^) M = (Q, ∑, , , q 0

,B,F)

…^ B^ B^ B^ X 1 X 2 X 3 …^ Xi …^ Xn B^ B …

Finite control Infinite tape with tape symbols B: blank symbol (special symbol reserved to indicate data boundary) Input & output tape symbols Tape head This is like the CPU & program counter Tape is the memory

Turing Machines

• (^) A Turing Machine (TM) has finite-state control (like PDA), and an

infinite read-write tape.

• (^) The tape serves as both input and unbounded storage device.
• (^) The tape is divided into cells, and each cell holds one symbol from the

tape alphabet.

• (^) There is a special blank symbol B. At any instant, all but finitely many

cells hold B.

• (^) Tape head sees only one cell at any instant. The contents of this cell

and the current state determine the next move of the TM.

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Way to check for Membership

• (^) Is a string w accepted by a TM?
• (^) Initial condition:
  • (^) The (whole) input string w is present in TM, preceded and followed by infinite

blank symbols

• (^) Final acceptance:
  • (^) Accept w if TM enters final state and halts
  • (^) If TM halts and not final state, then reject

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Example: L = {

n

n

| n≥1}

• (^) Strategy: w = 000111 … B B^0 00 1 1 1 B^ B … … B B X^00 1 1 1 B^ B … … B B X 0 0 Y 1 1 B B … … B B X^ X^0 Y^1 1 B^ B … B B X X 0 Y Y 1 B B … … B B X X X Y Y 1 B B … B B X X X Y Y Y B B … Accept B B X X X Y Y Y B B …

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TM for {

n

n

| n≥1}

Next Tape Symbol Curr. State

0 1 X Y B

q 0 (q 1 ,X,R) - - (q 3 ,Y,R) - q 1 (q 1 ,0,R) (q 2 ,Y,L) - (q 1 ,Y,R) - q 2 (q 2 ,0,L) - (q 0 ,X,R) (q 2 ,Y,L) - q 3 - - - (q 3 ,Y,R) (q 4 ,B,R) *q 4 - -- - - - Table representation of the state diagram *state diagram representation preferred

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TM for {

n

2n

| n≥1}

q 0 q 1

0  X,R

0  0,R

q 3

1  Y,L

Y  Y,L

0  0,L

X / X,R

q 4

Y  Y,R

Y  Y,R

q 5

B  B,R

Y  Y,R

q 2

1  Y,R

• (^) Draw a Turing machine for L = {a^i b^j c^k | i>j>k; k ≥ 1}

• (^) Draw Turing machines for L={w#w, w is from {0,1}*}}

A language is called Turing-recognizable or

recursively enumerable (r.e.) if some TM

recognizes it.

A language is called decidable or recursive

if some TM decides it.

recursive

languages

r.e.

languages

recursive

languages

r.e.

languages

Decidable Vs Recursive Enumerable

Whether the following languages are decidable?

1. {a

n

b

2n

| n≥1}

2. {ww | n≥1}

3. {a

n

b

2n

a

n

| n≥1}

4. {a

n

b

m

a

n

| n≥1}

5. {(ab)

n

c

n

(ad)

n

| n≥1}