Introduction to the Theory of Computation John Paxton Montana State University Summer 2003

Humor Bush, Einstein and Picasso at the Pearly Gates Einstein dies and goes to heaven. At the Pearly Gates, Saint Peter tells him, "You look like Einstein, but you have NO idea the lengths that some people will go to sneak into Heaven. Can you prove who you really are?" Einstein ponders for a few seconds and asks, "Could I have a blackboard and some chalk?" Saint Peter snaps his fingers and a blackboard and chalk instantly appear. Einstein proceeds to describe with arcane mathematics and symbols his theory of relativity. Saint Peter is suitably impressed. "You really ARE Einstein!" he says. "Welcome to heaven!"

Humor The next to arrive is Picasso. Once again, Saint Peter asks for credentials. Picasso asks, "Mind if I use that blackboard and chalk?" Saint Peter says, "Go ahead." Picasso erases Einstein's equations and sketches a truly stunning mural with just a few strokes of chalk. Saint Peter claps. "Surely you are the great artist you claim to be!" he says. "Come on in!"

Humor Then Saint Peter looks up and sees George W. Bush. Saint Peter scratches his head and says, "Einstein and Picasso both managed to prove their identity. How can you prove yours?" George W. looks bewildered and says, "Who are Einstein and Picasso?" Saint Peter sighs and says, "Come on in, George."

Turing Machine A Turing maching is a 7-tuple (Q, , Γ, , q 0, q accept, q reject ) where 1.Q is the set of states 2.  is the input alphabet not containing the special blank symbol 3. Γ is the tape alphabet that contains both the blank symbol and   : Q x Γ -> Q x Γ x {L, R}

Turing Decidable A language is Turing decidable (or simply decidable) if some Turing machine decides it. A Turing machine can decide a language if it halts on all inputs.

Equivalent Turing Machine Variants There may be multiple tapes. A Turing machine may be nondeterministic.

Church-Turing Thesis Our intuitive notion of an algorithm is equivalent to the algorithms that can be performed on a Turing Machine. In other words, a Turing Machine is equivalent to the most powerful model of computation!

The Halting Problem Does a particular Turing Machine accept the input w? This problem is undecidable and can be proven using a diagonalization proof.

Classes of Languages regular context-free decidable all problems