1 Introduction to Computability Theory Lecture14: Recap Prof. Amos Israeli.
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Transcript of 1 Introduction to Computability Theory Lecture14: Recap Prof. 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
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
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
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 ,,