------------------------------------------------ COMP SCI 731 - Lecture 4 - Tuesday, Nov 7, 2000 ------------------------------------------------ Today: o The TM model o P The TM model ------------ To make things precise we will need a computational model. Most popular model is a TM. Popular because of it's simplicity. Draw a picture. Definition: A TM is a tuple M = (Q, Sigma, Gamma, delta) * Q is a finite set , "states". It contains three special states: q_0, q_Y, q_N. These are all different from each other. * Sigma is the "input alphabet". It is a finite non-empty set of characters. * Gamma is the "tape alphabet". It is a superset of Sigma. It contains one special character, b, called the blank symbol. This symbol is not an element of Sigma. * delta: Q-{q_Y,q_n} x Gamma -> Q x Gamma x {-1,+1} is the "transition function". It tells the machine where to move. Finite Control | | \|/ read/write head ------------------------------------------- | | | | | | | | | | Tape ------------------------------------------- -4 -3 -2 -1 0 1 2 3 4 Describe the operation of a TM. Describe the conventions for how the input is presented to the TM. Example: * Make a TM to decide if its input is even. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Problem 2. Specify a TM that, when started on a blank tape, visits every single tape cell. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ P - P = { L: L is a language and there exists a TM M s.t. * x in L ==> M(x) says YES * x not in L ==> M(x) says NO * For some polynomial poly, for all x, #steps_M(x) <= poly(|x|) Discuss the robustness of P to changes in the model of computation. Discuss the sense in which P has been a good approximation, for practice, to what is computable by "reasonable" algorithms.