Decidability

1 definition found

From The Free On-line Dictionary of Computing (27 SEP 03) [foldoc]:

  decidability
       
           A property of sets for which one can determine
          whether something is a member or not in a {finite} number of
          computational steps.
       


          Decidability is an important concept in {computability
          theory}.  A set (e.g. "all numbers with a 5 in them") is said
          to be "decidable" if I can write a program (usually for a
          {Turing Machine}) to determine whether a number is in the set
          and the program will always terminate with an answer YES or NO
          after a finite number of steps.
       
          Most sets you can describe easily are decidable, but there are
          infinitely many sets so most sets are undecidable, assuming
          any finite limit on the size (number of instructions or number
          of states) of our programs.  I.e. how ever big you allow your
          program to be there will always be sets which need a bigger
          program to decide membership.
       
          One example of an undecidable set comes from the {halting
          problem}.  It turns out that you can encode every program as a
          number: encode every symbol in the program as a number (001,
          002, ...) and then string all the symbol codes together.  Then
          you can create an undecidable set by defining it as the set of
          all numbers that represent a program that terminates in a
          finite number of steps.
       
          A set can also be "semi-decidable" - there is an {algorithm}
          that is guaranteed to return YES if the number is in the set,
          but if the number is not in the set, it may either return NO
          or run for ever.
       
          The {halting problem}'s set described above is semi-decidable.
          You decode the given number and run the resulting program.  If
          it terminates the answer is YES.  If it never terminates, then
          neither will the decision algorithm.
       
          (1995-01-13)