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.

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

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

For example … M1YesNo Yes M2No Yes M3YesNoYesNo M4YesNo … D YesNoYesUh-oh Contradiction  The TM D can’t exist  So U is not a decider.

In other words Let U = universal TM. –Its input is a TM description and a word. –Determines if M accepts w. –Assume U halts for all inputs. (is a decider) Create 2 nd TM called D. –Its input is a TM description. –Gives 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 ? –According to U, if D accepts, U accepts, so D must reject! –According to U, if D rejects, U rejects, so D must accept! –Both cases give a contradiction. –Thus, U is not a decider. A TM is undecidable.

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).

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!

Other properties 2 kinds of TMs  2 kinds of languages. –Turing-recognizable (a.k.a. recursively enumerable) Example: A TM –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 A TM is not even Turing recognizable.