1

Introduction to Computability Theory

Lecture14: RecapProf. Amos Israeli

1. Regular languages – Finite automata.

2. Context free languages – Stack automata.

3. Decidable languages – Turing machines.

4. Undecidability.

5. Reductions.

Subjects

2

CFL-s Ex: 0| nba nn

RL-s Ex: 0| nan

The Language Hierarchy

3

Decidable Ex: 0| ncba nnn

Turing recognizableEx: TMA

Non Turing recognizableEx: TMA

1. Defined DFA-s and their languages.

2. Defined NFA-s and their languages.

3. Defined RE-s and their languages.

4. Showed all three are equivalent.

5. Proved the Pumping lemma and demonstrated its use to prove irregularity.

Regular Languages

4

Consider

a. Show that L is regular.

b. Present an RE for L.

Training Problem 1

5

*,,,0||,0|:| bayxyxyxL

a. Consider

Proof

6

*,,,0||,0|:| bayxyxyxL

a,b a,b-

a,ba,b

- - -

a,b,-

b. Consider

Proof

7

*,,,0||,0|:| bayxyxyxL

R ba *ba ba *ba

1. Defined CFG-s and their languages.

2. Defined Stack automata and their languages.

3. Showed that the two classes are equivalent.

4. Proved the Pumping lemma for CFL-s and demonstrated its use to prove languages to be non CFL.

Context Free Languages

8

Let

Show that L is context free.

Proof:

Training Problem 2

9

*,,|,|2||,0|:| bayxxyxyxL

baX |

XXRXR |XXRXS

1. Defined Turing machines, decidable languages and Turing recognizable languages.

2. Defined multi-tape TM-s and non deterministic TM-s, and showed their equivalence to ordinary TM-s.

3. Introduced the Church-Turing hypothesis.

Decidable Languages

10

Consider

a. Show that L is regular by presenting a DFA.

b. Show that L is CF by presenting a PDA.

c. Show that L is decidable by presenting a TM.

Training Problem 3

11

*1,0:00 xxL

Consider

a. Show that L is regular by presenting a DFA.

Training Problem 3

12

*1,0:00 xxL

00

0,1

Consider

b. Show that L is CF by presenting a PDA.

Training Problem 3

13

*1,0:00 xxL

0

0,1

0

Consider

c. Show that L is decidable by presenting a TM.

Training Problem 3

14

*1,0:00 xxL0,1

R,00 acceptq

rejectqR,11

R,00

R,11

1. Defined Cardinality of sets.

2. Showed that the cardinality of the rational numbers is equal to .

3. Used Diagonalization to show that the cardinality of infinite binary sequences is not equal to .

Undecidability

15

0ALEPH

0ALEPH

4. Showed that the cardinality of Turing recognizable languages is equal to .

5. Showed that the cardinality of languages is larger than .

6. Concluded the existence of a non Turing recognizable language.

Undecidability (cont.)

16

0ALEPH

0ALEPH

7. Defined and showed that it is undecidable.

Undecidability (cont.)

17

TMA

1. Defined reductions.

2. Used reductions to prove that , ,

, , and many other problems are undecidable.

3. Defined mapping reductions.

Reductions

18

TMHALT TME

TMEQ TMREGULAR

Consider the following problem:

Show that is undecidable.

Training Problem 4

19

TMSUBSET

NLMLNMNMSUBSETTM and s,-TM are ,,

We show a reduction from to .

Assume TM R is a decider for , let

S=“On input where N is a TM

1. Let M be the TM rejecting all its inputs.

2. if R accepts (meaning ) - accept, otherwise reject.”

Proof

20

MN ,

TMSUBSET

N

TME

TMSUBSET

MLNL

We conclude that S never loops and it accepts iff . In other words: S is a decider for . Since is undecidable, we conclude that is also undecidable.

QED

Other practice problems: Prove by reduction from and from .

Proof

21

TMSUBSET

TME

NL

TME

TMEQ TMA

Prove or disprove:

a. If L is Turing recognizable then L is undecidable.

Disprove: A Language L is Turing recognizable if there exists a TM, M, s.t. . If M, halts on every input then L is decidable. In other words: Every decidable language is also Turing recognizable.

Training Problem 5

22

LML

Prove or disprove:

b. If a Turing machine moves its head only to the right then it must halt.

Disprove:Present a state diagram of a TM that goes to the right forever.

Training Problem 5

23

Prove or disprove:

c. If a language A, is undecidable then its complement is also undecidable.

Training Problem 5

24

A

Prove: Assume towards a contradiction that is decidable and let M be a TM deciding it.

Consider TM M’ which is identical to M except that the accepting and rejecting states of M’ are switched. Clearly M’ accepts (rejects resp.) if and only if M rejects (accepts resp.), hence, M’ decides A, a contradiction. QED

Training Problem 5

25

A

An ordinary Turing machine may either change its current cell or leave it unchanged. A changer is a TM that always changes its current cell. Show that every Turing recognizable language is recognizable by a changer TM.

Training Problem 6

26

Let L be a Turing recognizable language and let M be a TM recognizing L, namely .

Let be M’s alphabet. Define a TM M’ whose alphabet is , where contains all the “barred” elements of . How should M’s transition function be changed in order to keep its functionality?

Proof

27

LML

Let Consider the following problem:

Show that is Turing recognizable.

Training Problem 7

28

TMOR

NLMLwNMwNMORTM and s,-TM are ,,,

Consider the following TM:

S=“On input where M,N are TM-s

1. Repeat 1.1 Run a single step of M on input w.

1.2 Run a single step of N on input w.

1.3 if either M or N accept - accept,

if both reject - reject.”

Proof

29

wNM ,,

We can conclude:

QED

Proof

30

NLwMLwNLMLw wS accepts