Computing Functions with Turing Machines

Transcription

1 CS 30 - Lecture 20 Combining Turing Machines and Turing s Thesis Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata Euivalence of NFA and DFA Regular Expressions Regular Grammars Properties of Regular Languages Languages that are not regular and the pumping lemma Context Free Languages Context Free Grammars Derivations: leftmost, rightmost and derivation trees Parsing and ambiguity Simplifications and Normal Forms Nondeterministic Pushdown Automata Pushdown Automata and Context Free Grammars Deterministic Pushdown Automata Pumping Lemma for context free grammars Properties of Context Free Grammars Turing Machines Definition and Accepting Languages Today: Computing Functions, Combining Machines, and Turing s Thesis Standard Turing Machine The machine we described is the standard: Deterministic Infinite tape in both directions Computing Functions with Turing Machines Tape is the input/output file

2 A function f (w) has: A function may have many parameters: Domain: D Result Region: S Example: Addition function w D f (w) f ( w) S f ( x, y) = x + y Integer Domain Decimal: 5 Definition: A function f is computable if there is a Turing Machine M such that: Binary: 0 Unary: We prefer unary representation: easier to manipulate with Turing machines Initial configuration w 0 initial state For all w D Final configuration f (w) f final state Domain 2

3 In other words: A function is computable if there is a Turing Machine M such that: Initial Configuration f 0 w f f ( w) For all w D Final Configuration Domain The function Turing Machine: Example f x, y) = x + y Input string: ( is computable x0y Output string: xy0 x, y are integers unary unary x y x y Start 0 Start 0 0 initial state 0 initial state x + y The 0 is the delimiter that separates the two numbers Finish 0 f final state 3

4 The 0 helps when we use the result for other operations Turing machine for function f ( x, y) = x + y Finish x + y 0 f final state 0 0, R, L 0, L 2 4, L, R Execution Example: x = y = (2) (2) Time 0 x y 0 0 Time 0 0 0, L Final Result x + y 0 4 0, R 0, L 2 0, L 4, R 4

5 Time 0 Time , L, L 0, R 0, L 2 0, L 4, R 0, R 0, L 2 0, L 4, R Time 3 Time 4, L, L 0, R 0, L 2 0, L 4, R 0, R 0, L 2 0, L 4, R 5

6 Time 5 Time 6 2, L, L 0, R 0, L 2 0, L 4, R 0, R 0, L 2 0, L 4, R Time 7 0 Time 8 0, L, L 0, R 0, L 2 0, L 4, R 0, R 0, L 2 0, L 4, R 6

7 Time 9 0 Time 0 0, L, L 0, R 0, L 2 0, L 4, R 0, R 0, L 2 0, L 4, R Time 0 Time 2 0 4, L, L 0, R 0, L 2 0, L 4, R 0, R 0, L 2 0, L HALT & accept 4, R 7

8 The function Another Example f ( x) = 2x is computable x is integer Start 0 x initial state Turing Machine: Input string: x unary 2x Output string: xx unary Finish f final state Turing Machine Pseudocode for f ( x) = 2x Turing Machine for f ( x) = 2x Replace every with $ Repeat: Find rightmost $, replace it with $, R, L Go to right end, insert Until no more $ remain, L 0 $, R 2, R, L 8

9 Start 0 $, R Example, L Finish Another Example if x > y f ( x, y) 0 if x y The function = is computable, L 0 $, R 2, R, L Turing Machine for Input: x0y f ( x, y) = Output: or 0 if x > y 0 if x y Turing Machine Pseudocode: Repeat Match a from x with a from Until all of x or y is matched x If a from is not matched erase tape, write else erase tape, write 0 y ( x > y) ( x y) 9

10 Combining Turing Machines input Block Diagram Turing Machine output Example: f ( x, y) = x + y 0 if if x > y x y x, y Comparer x, y x > y Adder x + y Turing s Thesis x y Eraser 0 0

11 Computer Science Law: Turing s thesis: Any computation carried out by mechanical means can be performed by a Turing Machine A computation is mechanical if and only if it can be performed by a Turing Machine (930) There is no known model of computation more powerful than Turing Machines Definition of Algorithm: An algorithm for function f (w) is a Turing Machine which computes f (w) Algorithms are Turing Machines When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm

12 Read What s Next Linz Chapter,2., 2.2, 2.3, (skip 2.4), 3, 4, 5, 6., 6.2, (skip 6.3), 7., 7.2, 7.3, (skip 7.4), 8, 9, 0., 0.2, and 0.3 JFLAP Chapter, 2., (skip 2.2), 3, 4, 5, 6, 7, (skip 8), 9 Next Lecture Topics From 0., 0.2 and 0.3 Turing Machine Variations Quiz 3 in Recitation on Wednesday /2 Covers Linz 7., 7.2, 7.3, (skip 7.4), 8, and JFLAP 5,6,7 Closed book, but you may bring one sheet of 8.5 x inch paper with any notes you like. Quiz will take the full hour Homework Homework Due Today New Homework Available Friday Morning New Homework Due Next Thursday 2