Créer une présentation
Télécharger la présentation

Télécharger la présentation
## THE CONVENTIONS

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**THE CONVENTIONS**R R “move to the right until you find “ “move to the right until you find a symbol other than “ 2 simple rules: • Rule # 1: • Rule # 2: Note: first check. Then move (think of a “while”) “Never forget Rule # 1” (Mr Miogi, The Karate Kid)**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).**Acceptance in Previous Machines**Machine Configuration Acceptance if q: final state, w = e (q,w) Finite automata q: final state, w =e = e Pushdown automata (q,w,) Pushdown automata accepting by empty stack w =e, = e (q,w,) These machines are used to compute if a word belongs to a language**Input and Output of Turing Machines (2)**… a a b a state: s ( - {, })* … b a a b state: h ( - {, })* 0 Idea: Input: Output:**Functions (Again)**This definition says that 2 things should happen: Second, that each element in A has a unique representative in B First, that “no one (in A) is left behind” B A B A a a d d 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 Example: transitions in Turing machines**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)**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’)**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**Example of a Turing-Computable Function**f(w) = wa where a is a fixed character in 0 and w 0***Example of a Turing-Computable Function (2)**Let M be the machine that terminates in the configuration (h, ww) when starting with (s, w) (see Slide # 2) M f(w) = ww where and w 0* …**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**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))**Regular Languages are Turing-Computable (2)**1 M1 1 M0 Use the machine ML as follows: ML This Turing machine recognizes the language L Every regular language is Turing-computable (decidable)**Homework**• 4.9, 4.10, See slide # 10