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Decidability

[Sections 5.4 and 8.3]. Decidability. A decision problem is a problem with a YES/NO answer. We have seen decision problems for regular languages: context free languages:. [Section 10.5]. Decidability. A real-life decision problem (important e.g. for cryptography):

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Decidability

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  1. [Sections 5.4 and 8.3] Decidability • A decision problem is a problem with a YES/NO answer. • We have seen decision problems for • regular languages: • context free languages:

  2. [Section 10.5] 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 ?

  3. [Section 10.5] Decidability Are all decision problems decidable ? (I.e., is every language recursive ?)

  4. [Section 10.5] Decidability Are all decision problems decidable ? NO. A diagonalization argument:

  5. [Sections 11.1 and 11.2] 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.

  6. [Sections 11.2-3] Variants of the Halting Problem Input: a Turing machine T and an input x Answer: YES if x 2 L(T), otherwise NO.

  7. [Sections 11.2-3] Variants of the Halting Problem Input: a Turing machine T and an input x Answer: YES if x 2 L(T), otherwise NO.

  8. [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.

  9. [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.

  10. [Sections 11.2-3] Variants of the Halting Problem Input: a Turing machine T Answer: YES if L(T) = ;, otherwise NO.

  11. [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.

  12. [Section 11.4] 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 T has the property R, otherwise NO.

  13. [Section 11.5] 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: 1 10 11 0

  14. [Section 11.5] Post’s Correspondence Problem Some more examples: Another example: 1 10111 10 111 10 0 10 011 101 101 11 011

  15. [Section 11.5] Post’s Correspondence Problem Thm: The Post’s Correspondence Problem is undecidable. Is PCP recursively enumerable?

  16. [Section 11.6] 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

  17. [Section 11.6] Undecidable Problems for CFL’s • Given a CFG G, is G ambiguous ? • Given a CFG G, is L(G) = * ? • Given two CFG’s G1 and G2, is L(G1) = L(G2) ? • Given two CFG’s G1 and G2, is L(G1) Å L(G2) = ; ?

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