Decidability Sections 5 4 and 8 3 A

Decidability [Sections 5. 4 and 8. 3] A decision problem is a problem with a YES/NO answer. We have seen decision problems for - regular languages: - context free languages:

Decidability [Section 10. 5] 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 ?

Decidability Are all decision problems decidable ? (I. e. , is every language recursive ? ) [Section 10. 5]

The Halting Problem [Sections 11. 1 and 11. 2] 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.

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 [Sections 11. 2 -3] Input: a Turing machine T Answer: YES if T halts on the empty input, otherwise NO.

Variants of the Halting Problem [Sections 11. 2 -3] 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.

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 T 1 and T 2 Answer: YES if L(T 1) = L(T 2), otherwise NO. [Sections 11. 2 -3]

Rice’s Theorem [Section 11. 4] 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.

Post’s Correspondence Problem [Section 11. 5] 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: 1 11 10 0

Post’s Correspondence Problem Some more examples: 1 111 10 10 0 Another example: 10 101 011 11 101 011 [Section 11. 5]

Post’s Correspondence Problem [Section 11. 5] Thm: The Post’s Correspondence Problem is undecidable. Is PCP recursively enumerable?

Undecidable Problems for CFL’s [Section 11. 6] 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 a 1 | 2 A a 2 | … | k A ak | 1 a 1 | 2 a 2 | … | k ak B 1 B a 1 | 2 B a 2 | … | k B ak | 1 a 1 | 2 a 2 | … | k ak

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 G 1 and G 2, is L(G 1) = L(G 2) ? 4. Given two CFG’s G 1 and G 2, is L(G 1) Å L(G 2) = ; ? [Section 11. 6]