1. FORMAL LANGUAGES AND AUTOMATA THEORY
C K Nagpal
M.Tech. Ph.D.
Associate Professor
Computer Engg.
YMCA University of Science & Technology

2. Chapter 8 THE PITFALL OF ALGORITHMIC COMPUTING: UNDECIDABILITY
In these slides, we will cover the following topics:
• What a decision problem is
• Recursive and recursively enumerable languages
Coding the Turing Machine
Decision Problems Related to Turing machines
Reduction (P1 ≤ P2)
Rice’s Theorem
• Decision problem relating to context-free grammars
• Post correspondence problem
Constructing CFG Using PCP
PCP Unsolvability

3. Recursive & Recursively Enumerable Languages
Recursive Language L :- TM halts for every string w over ∑*. Stops to say yes if w belongs to L and stops to say no if w does not belong to L. (Turing decidable language)
Recursively Enumerable Language L : TM halts for every string w belonging to L and stopping is not sure for if w does not belong to L. ( Turing acceptable language)

4. Recursive & Recursively Enumerable Languages (Contd.)
Languages which are neither recursive nor RE
RE languages which are not recursive
Recursive languages
Universe of languages

5. Some Conclusions (Theorems 8.1-8.7)
If a language L is recursive then its complement L’ is also recursive.
If two languages L1 and L2 are recursive their union L1 U L2 is also recursive.
If two languages L1 and L2 are recursive their intersection L1 ∩ L2 is also recursive.
If a language L and its complement L’ are both recursively enumerable then L is recursive.
Let L1, L2, L3,……, Ln be the set of recursively enumerable languages over ∑ such that they form partition set over ∑*. Then each of language L1, L2, L3,……, Ln is recursive.
If two languages L1 and L2 are recursive enumerable then their union L1 U L2 is also recursive enumerable.
If two languages L1 and L2 are recursive enumerable then their intersection L1 ∩ L2 is also recursive enumerable.

6. Coding the Turing Machine
A Turing machine can be coded in the form binary string
Lv : Set of all binary strings which form the valid code for a TM
Le = {M | M is the code of Turing machine and L(M) = ϕ}
Lne = {M | M is the code of Turing machine and L(M) ≠ ϕ}
Ld = {M | M is a Turing machine and M belongs to L(M)}
Lnd = {M | M is a Turing machine and M does not belong to L(M)}
Conclusions ( Theorem )
The language Lv is recursive
The language Lnd is not recursively enumerable
The language Ld is not recursive

7. Decision Problems Related to Turing machines
Lu : Language of Universal Turing Machine
Lu = {x | x = (M, w) and w ϵ L(M)}
Theorem
The language Lu is recursively enumerable but not recursive
Given a Turing machine M and a string w, the problem ‘Does M accepts w?’ is undecidable

8. Reduction (P1 ≤ P2) The language Lne is not recursive
Process of converting the instances of a problem P1 to the corresponding instances of a problem P2 such that they have the same answer, thereby solving the problem P2 through the problem P1.
The symbol for reduction is ≤.
If the problem P1 reduces to P2, then it is denoted as P1 ≤ P2.
Theorems based upon reduction ( Theorem )
The halting problem of a Turing machine is undecidable
The language Lne is recursively enumerable
The language Lne is not recursive
The language Le is not recursively enumerable
( Rice’s theorem) Every non-trivial property of a recursively enumerable language is undecidable.

9. Rice’s Theorem (Implications)
Undecidable Problems
Given a Turing machine M, does L(M) contain any string?
2. Given a Turing machine M, does L(M) contain at least two strings?
3. Given a Turing machine M, is L(M) finite?
4. Given a Turing machine M, is L(M) = Σ* ?
5. Given a Turing machine M, is L(M) a context-free language?
6. Given a Turing machine M, is L(M) a regular language?

10. Decision Problems Related to CFG
Theorem
Given a string x and a CFG G = (V, Σ, P, S), the problem ‘does x ϵ L(G )?’ is decidable.
Given a CFG G = (V, Σ, P, S), the problem ‘Is L(G) = ϕ?’ is decidable.
Given a CFG G = (V, Σ, P, S), the problem ‘Is L(G) finite?’ is decidable.

11. Post Correspondence Problem (PCP)
The Post Correspondence Problem is unsolvable, there is no algorithm to check if a given instance of the PCP has a solution.

12. Constructing CFG Using PCP

13. PCP Unsolvability ( Implications)
Theorems
Given two CFGs Gx and Gy, the problem ‘Is L(Gx) ∩L(Gy) ≠ϕ?’ is undecidable.
Given a CFG G, the problem ‘Is G ambiguous?’ is undecidable.
Given two CFGs Gx and Gy, the problem ‘Is L(Gx) = L(Gy)?’ is undecidable.
Given two CFGs Gx and Gy, the problem ‘Is L(Gx) a subset of L(Gy)?’ is undecidable.