1. Turing Machine Finite state automaton –Limitation: finite amount of memory –Prevents recognizing languages that are not regular {0 n 1 n |n = 0,1,2,…} Pushdown automation –Unlimited memory : stack –Equivalent to context free grammers –Can recogninze A = {0 n 1 n |n = 0, 1, 2,….} –Can’t recognize context sensitive grammer (B = {a n b n c n |n = 0, 1, 2,….}) Linear bounded automata –More powerful than pushdown models. –Recognize context sensitive grammer. –Cannot recognize all the languages generated by phrase –structure grammers.

2. Turing Machines Defn: A Turing Machine T = (S,I,f,s 0 ) –s 0 = starting state –S = finite set of states –f = partial function (S x I X {R, L}) –I = An alphabet contating the blank symbol B Action of TM f (s, x) = (S’,x’,d) 1.Enters the state S’ 2.Writes the symbol x’ in the current cell, erasing x and 3.Moves right one cell if d=R or moves left one cell if d=L. –TM is usually specified with a set of five- tuples of the form (S,x,S’,x’,d) TM to recognize sets Defn: Let V be a subset of an alphabet T. A Turing Machine T=(S,I,f,s 0 ) recognizes a string x in V* if and only if T, starting in the initial positon when x is written on the tape, halts in the final state. T is said to recognize a subset A of V* if x is recognized by T if and only if x belongs to A. Non-regular set –{0 n 1 n |n = 0,1,2,…} –V = {0,1} and I = {0,I,M} –M: marker (auxiliary tape symbol)

3. Turing Machines Computing functions –TM as a computer that finds the values of a partial function. –Functions defined on integers? Different types of TM –Multiple TM Allow tape to be two-dimensional Read different cells simultaneously Nondeterministic TM: allow (state, tape sysmbol) pair as fist elements in more than one five-tuple of the TM Restrict: (can reduce the capabilites of TM) –tape to be infinite in only one dimension. –Tape alphabet to have only two symbols. The Church-Turing thesis: –Any probelm which can be solved with an effective algorithm, there is a TM that can solve this problem.