This document discusses recursive and recursively enumerable (RE) languages. It defines recursive languages as those for which there exists a Turing machine that halts and accepts for strings in the language, and halts and rejects for strings not in the language. RE languages may include those where the Turing machine accepts strings in the language but does not always halt for strings not in the language. It proves that the complement of a recursive language is also recursive, but the complement of an RE language may not be recursive or RE. It introduces the universal language Lu, which is RE but not recursive, making the problem it defines undecidable.

1. 1
Undecidability that is RE
Deepak.M
1VE21CA010
CSE-AI
SRI VENKATESHWARA
COLLEGE
OF ENGINEERING

2. 2
9.2 An Undecidable Problem
That Is RE
 Concepts to be taught:
 RE languages are accepted (recognized) by TM’s.
 RE languages may be grouped into two classes:
 Class 1 (recursive language) --- each language L
in this class has a TM (thought as an algorithm)
which not only accepts strings of L, but also tells
us what strings are not in L by halting.
 Class 2 (RE but not recursive) --- each language L
in this class has a TM (not thought as an algorithm)
which accepts strings of L, but may not halt when
a given input string is not in L.

3. 3
9.2 An Undecidable Problem
That Is RE
 9.2.1 Recursive Languages
 Formally, a language L is recursive if L = L(M)
for some TM M such that:
(1) If wL, then M accepts (and therefore
halts).
(2) If wL, then M eventually halts, although it
never enters an accepting state (i.e., “reject”) .
 A TM of this type corresponds to the formal
notion of algorithm.

4. 4
9.2 An Undecidable Problem
That Is RE
9.2.1 Recursive Languages
 A given language L, regarded as a problem,
is called decidable if L is a recursive
language; and undecidable if not.
 The existence or nonexistence of an
algorithm to solve a problem (i.e., the
problem is decidable or undecidable) is more
important than the existence or nonexistence
of a TM to solve the problem.

5. 5
9.2 An Undecidable Problem
That Is RE
9.2.1 Recursive Languages
 A given language L, regarded as a
problem, is called decidable if L is a
recursive language; and undecidable if not.
Decidable prob.  recursive lang. L
 there is a TM M halts
with L as input

6. 6
9.2 An Undecidable Problem
That Is RE
 9.2.1 Recursive Languages
 Relationships among three classes of
languages (only Ld is studied so far)
Recursive
RE but not
Recursive Not RE
Ld
LuLd Lne
Figure 9.2
Le
Lu

7. 7
9.2 An Undecidable Problem
That Is RE
 9.2.1 Recursive Languages
 Lu, the universal language, will soon
be defined and proved not to be
recursive, though it is an RE
language.
 9.2.2 Complements of Recursive and
RE Languages
 A powerful tool of proving is language
complementation.
 We will show that the recursive language
is closed under complementation.

8. 8
9.2 An Undecidable Problem That
Is RE
 9.2.2 Complements of Recursive & RE Langs.
 Theorem 9.3: If a language L is recursive, so isL.
Proof. Prove by constructing a halting TM to acceptL.
Let TM M accepts L, which always halts. Construct a new
TMM for M as follows (illustrated by Fig. 9.3).
M
w
Accept
Reject
Accept
Reject
Figure 9.3
M

9. 9
9.2 An Undecidable Problem
That Is RE
 9.2.2 Complements of Recursive & RE langs.
Proof. (cont’d).
1. The accepting states of M are made
nonaccepting with no transitions, i.e., in these
states M will halt without accepting.
2.M has a new accepting state r & no transition
from r.
3. For each combination of a nonaccepting state of
M & a tape symbol of M such that M has no
transition (i.e., M halts without accepting), add a
transition to the accepting state r.
 Then if M halts, so isM, andM accepts strings
not accepted by M.

10. 10
9.2 An Undecidable Problem
That Is RE
 9.2.2 Complements of Recursive
& RE Languages
 Theorem 9.4
If a language L and its complement are RE,
then L is recursive, and so isL (by
Theorem 9.3).
Proof. See the text book. (Easy by Fig. 9.4 in
the next page.)

11. 11
9.2 An Undecidable Problem
That Is RE
 9.2.2 Complements of Recursive & RE
Languages
 Theorem 9.4
M1
w
Accept Accept
Reject
M2
Accept
Fig. 9.4.
Accept  halt

12. 12
9.2 An Undecidable Problem
That Is RE
 9.2.2 Complements of Recursive & RE langs.
 Of the 9 possibilities of placing L andL in Fig. 9.2,
only the following 4 are valid by Theorems 9.3 and
9.4:
 Both L andL are recursive (both in the inner ring
of Fig. 9.2)
 Neither L norL is RE. (both in the outer ring)
 L is RE but not recursive, andL is not in RE (one
in the middle ring, and the other in the outer ring)
 L is RE but not recursive, and L is not in RE (a
swap of above)

13. 13
9.2 An Undecidable Problem
That Is RE
 9.2.2 Complements of Recursive & RE langs.
 Of the 9 possibilities of placing L andL in Fig. 9.2,
only the following 4 are valid by Theorems 9.3 and
9.4:
Recursive
RE but not
Recursive Not RE
Recursive
RE but not
Recursive Not RE
Recursive
RE but not
Recursive Not RE
 
 


Case 1 Case 2 Cases 3 & 4

14. 14
9.2 An Undecidable Problem
That Is RE
 9.2.2 Complements of Recursive & RE langs.
 Example 9.5
Ld is not RE as shown before.
So,Ld cannot be recursive (not in the four
cases above). (Otherwise, Ld is recursive by Thm
9.4 and so is RE too by definition, contradiction!)
It can be shown thatLd is RE (omitted), just
like the way we show the universal language
Lu to be RE (shown next).
Note thatLd is the set of strings wi such that
the corresponding Mi accepts wi.

15. 15
9.2 An Undecidable Problem
That Is RE
 9.2.2 Complements of Recursive & RE langs.
 Example 9.5
 It can be shown thatLd is RE (omitted) but not recursive.
Recursive
RE but not
Recursive Not RE
Ld
LuLd Lne
Figure 9.2
Le
Lu

16. 16
9.2 An Undecidable Problem
That Is RE
 9.2.2 Complements of Recursive & RE langs.
 In conclusion,Ld is RE but not recursive and so
undecidable
 That is, although there is a TM forLd, the problem
defined byLd is undecidable (there is no
algorithm for it).
 The universal language Lu has the same property,
as proved next.

17. 17
9.2 An Undecidable Problem
That Is RE
 9.2.3 The Universal Language
 Definition:
The universal language Lu is the set of binary strings,
each of which encodes, in the notation of 9.1.2, a pair
(M, w), where M is a TM with the binary alphabet,
and w is a string in (0 + 1)*, such that w is in L(M).
Compare:Ld is the set of strings wi such that the
corresponding Mi accepts wi (note the numbering i
here)

18. 18
9.2 An Undecidable Problem
That Is RE
 9.2.4 Undecidability of the Universal
Language (cont’d)
Recursive
RE but not
Recursive Not RE
Ld
LuLd Lne
Figure 9.2
Le
Lu

19. THANK YOU
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