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CS 461 – Nov. 7

CS 461 – Nov. 7. Decidability concepts Countable = can number the elements  Uncountable = numbering scheme impossible  A TM undecidable Language classes Next { languages } uncountable, but { TMs } countable There are more languages than TMs! … Be on the lookout for ∞ rep’n.

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CS 461 – Nov. 7

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  1. CS 461 – Nov. 7 • Decidability concepts • Countable = can number the elements  • Uncountable = numbering scheme impossible  • ATM undecidable • Language classes • Next • { languages } uncountable, but { TMs } countable There are more languages than TMs! … • Be on the lookout for ∞ rep’n.

  2. Universal TM Let’s design “U” – the Universal TM: • Input consists of <M> and w: • <M> is the encoding of some TM • w is any (binary) string. • Assume: U is a decider (i.e. ATM is decidable.) U <M>,w M yes yes w no* no

  3. ATM solution • Start with U, the Universal Turing Machine • Suppose U decides ATM. Let’s build new TM D. • D takes in a Turing machine, and returns opposite of U. D <M> U no yes <M>,<M> no yes If M accepts its own string rep’n, D rejects <M>. If M doesn’t accept <M>, D accepts <M>. What does D do with <D> as input?

  4. For example Contradiction  The TM D can’t exist  So U is not a decider.

  5. In other words • Let U = universal TM. • Its input is a TM description <M> and a word <w>. • Determines if M accepts w. • Assume U halts for all inputs. (is a decider) • Create 2nd TM called D. • Its input is a TM description <M>. • Gives <M> to U as the TM to run as well as the input. • D returns the opposite of what U returns. • What happens when the input to D is <D>? • According to U, if D accepts <D>, U accepts, so D must reject! • According to U, if D rejects <D>, U rejects, so D must accept! • Both cases give a contradiction. • Thus, U is not a decider. ATM is undecidable.

  6. Language classes Working from the inside out: • Finite set • Regular • CFL (deterministic) • CFL (non-deterministic) • Decidable • Turing-recognizable • Outer space! • Yes – it’s possible for a language not to be recognized by any TM whatsoever • Note: all languages are countable (or finite).

  7. Language beyond TM • The set of all TM’s is countable. • Finite representation • The set of all languages is uncountable. • Infinite representation • Not enough TM’s to go around  There must be a language unrecognized by any TM. • Let’s find one!

  8. Other properties • 2 kinds of TMs  2 kinds of languages. • Turing-recognizable (a.k.a. recursively enumerable) • Example: ATM • Decidable (a.k.a. recursive) • Example: 0* • If L is decidable, then L’ is decidable. • If L and L’ are both Turing-recognizable,then L is decidable. (since either L or L’ must accept) • Therefore, the complement of ATM is not even Turing recognizable.

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