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Decidability
A decision problem is a problem with a YES/NO answer.
We have seen decision problems for
- regular languages:
- context free languages:
[Sections 5.4 and 8.3]
Decidability
A real-life decision problem (important e.g. for cryptography):
“Given is a number p, is p a prime ?”
How to view this problem as a language ?
Which class of languages describes decidable problems ?
[Section 10.5]
Decidability
Are all decision problems decidable ?
(I.e., is every language recursive ?)
[Section 10.5]
The Halting Problem
Input: a program P and an input x
Answer: YES if P halts on x, otherwise (i.e. P goes into an infinite loop on x), NO.
Note: the problem is typically phrased for Turing machines, i.e. P is a TM.
[Sections 11.1 and 11.2]
Variants of the Halting Problem
Input: a Turing machine T and an input x
Answer: YES if x 2 L(T), otherwise NO.
[Sections 11.2-3]
Variants of the Halting Problem
Input: a Turing machine T
Answer: YES if T halts on the empty input, otherwise NO.
[Sections 11.2-3]
Variants of the Halting Problem
Input: a Turing machine T, an input x, and a state q
Answer: YES if the computation of T on input x reaches the state q, otherwise NO.
[Sections 11.2-3]
Variants of the Halting Problem
Input: a Turing machine T
Answer: YES if L(T) = ;, otherwise NO.
[Sections 11.2-3]
Variants of the Halting Problem
Input: two Turing machines T1 and T2
Answer: YES if L(T1) = L(T2), otherwise NO.
[Sections 11.2-3]
Rice’s Theorem
Rice’s Thm:
Let R be a property on languages. If R holds for some but not all recursive enumerable languages, then the following problem is undecidable:
Input: a Turing machine T
Answer: YES if L(T) has the property R, otherwise NO.
[Section 11.4]
Post’s Correspondence Problem
Post’s Correspondence Problem (PCP) is a string matching problem:
Input: a collection of k types of dominoes, each domino contains two strings (i is at the top of the i-th domino, i is at the bottom; i=1,…,k).
Output: YES, if it’s possible to find a nonempty sequence of dominoes so that the string at the top matches the string on the bottom. (The domino types in the sequence can repeat.)
An easy example:
[Section 11.5]
1
11
10
0
Post’s Correspondence Problem
Some more examples:
Another example:
[Section 11.5]
1
111
10111
10
10
0
10
101
011
11
101
011
Post’s Correspondence Problem
Thm: The Post’s Correspondence Problem is undecidable.
Is PCP recursively enumerable?
[Section 11.5]
Undecidable Problems for CFL’s
Ambiguity of context-free grammars
Input: a CFG G
Output: YES if G is ambiguous, otherwise NO.
Consider an instance of a PCP: (1,1), (2,2), …, (k,k)
Create the following CFG:
S A | B
A 1 A a1 | 2 A a2 | … | k A ak | 1 a1 | 2 a2 | … | k ak
B 1 B a1 | 2 B a2 | … | k B ak | 1 a1 | 2 a2 | … | k ak
[Section 11.6]
Undecidable Problems for CFL’s
1. Given a CFG G, is G ambiguous ?
2. Given a CFG G, is L(G) = * ?
3. Given two CFG’s G1 and G2, is L(G1) = L(G2) ?
4. Given two CFG’s G1 and G2, is L(G1) Å L(G2) = ; ?
[Section 11.6]
