1. THE CONVENTIONS 2 simple rules: Rule # 1: Rule # 2: RR “move to the right until you find  “ Note: first check. Then move (think of a “while”) “Never forget Rule # 1” (Mr Miogi, The Karate Kid) RR “move to the right until you find a symbol other than  “

2. Input and Output of Turing Machines When constructing Turing machines we have to establish conventions: For example, the Turing machine (  = {, a, b, ,  }): Does the following: when starting with a configuration of the form (s,  w), where w  (  - { , , })* then the final configuration is (h,  ww). …

3. Acceptance in Previous Machines Finite automata ConfigurationMachineAcceptance if (q,w) q: final state, w = e Pushdown automata (q,w,  ) q: final state, w =e  = e Pushdown automata accepting by empty stack (q,w,  ) w =e,  = e These machines are used to compute if a word belongs to a language

4. Input and Output of Turing Machines (2) Idea: Input: aa  b … a (  - { , })* state: s Output: ba  a … b (  - { , })* state: h 00

5. Functions (Again) Definition. A function, f, from A to B is a relation such that for every a  A there is one and only one b  A with f(a) = b This definition says that 2 things should happen: First, that “no one (in A) is left behind” A a d B Second, that each element in A has a unique representative in B A a d B Example: transitions in Turing machines

6. Functions (Again – Part 2) From the two restrictions, the first can be easily overcome: A Dom(f) = {a  A : there is a b  B with f(a) = b} One can restrict f to be from Dom(f) to B. In this situation one speaks of a partial function Example: f from Real to Real defined as f(x) = 1/x. Dom(f) = Real - {0} (Thus f is a partial function)

7. Input and Output of Turing Machines (3) Definition. Let: M = (S, , , s, H) be a Turing machine  0 =  - { , } Suppose that M starts with (s,  w) with w   0 * If the machine ends in a configuration (h,  u) with u   0 * and h  H, then: u is referred to as the output of M and write M(w) = u We can view M as a function f (from  0 * into  0 *), with f(w) = u. We say that M computes f. If M does not end for a w’   0 *, then we write M(w’) 

8. Turing-Computable Functions Definition. A function f:  0 *   0 * is said to be Turing computable if there exists a Turing machine M that computes it (i.e., for each w  dom(f), f(w)=M(w)) and dom(f) =  0 *. If dom(f) is an strict subset of  0 * (i.e., dom(f)   0 * and dom(f)   0 *), we say that f is partial Turing computable

9. Example of a Turing-Computable Function f(w) = wa where a is a fixed character in  0 and w   0 *

10. Example of a Turing-Computable Function (2) f(w) = ww where and w   0 * Let M be the machine that terminates in the configuration (h,  ww) when starting with (s,  w) (see Slide # 2) M …

11. Computing (Deciding) Languages Definition. Let L be a language in  0 * we define the characteristic function,  L, as follows:  L (w) = 1, if w  L  L (w) = 0, otherwise Definition. A language L is Turing-computable or decidable if its characteristic function,  L, is Turing- computable

12. Regular Languages are Turing- Computable (1) Steps to obtain a Turing machine accepting a regular language: Let L be a regular language on alphabet {a,b} Let A be a deterministic automata accepting L It is easy to simulate A with a Turing machine ML: ((p,a),q) ((p,a),(q,  )) For every favorable state f in A, add the transition: ((f,),(h,1))

13. Regular Languages are Turing- Computable (2) Use the machine ML as follows: ML 1 M1 1 M0 This Turing machine recognizes the language L Every regular language is Turing-computable (decidable)

14. Homework 4.9, 4.10, See slide # 10