Introduction to Computer Science

Transcription

1 Introduction to Computer Science CSCI 109 China Tianhe-2 Andrew Goodney Fall 2017 Lecture 11: Abstract Machines and Theory Nov. 13th, 2017

2 Schedule 1

3 Abstract machines and theory uwhat is theoretical computer science? ufinite state machines uthe Turing machine u The Church-Turing thesis u The halting problem uthe analysis of algorithms uthe analysis of problems Reading: St. Amant Ch. 8 2

4 What is theoretical computer science? uthe study of what can be computed and how efficiently it can be computed ubased on abstract mathematical models uinsight into the behavior of a real computer any real computer 3

5 Finite state machines Up On Crying Fed Hungry Asleep Down Up More hungry Hungry Down Off Light switch Awake, not crying 1 day old infant 10 day old infant 4

6 Finite state machines Add M Add M Add M Add M M+1 M+2 M+3 M+4 Add F Add F Add F M=F Add F Add F Add F Add F Add F Add M F+1 F+2 F+3 F+4 Add M Add M Add M Add M Add F St. Amant pp

7 Finite state machine limitations uif a finite state machine with n states is given n+1 symbols, it will revisit one state at least once uno way to tell if revisited state is being visited for the 1 st time or 5 th time uthe pigeonhole principle uproblems an FSM cannot solve: v Given a string of As and Bs, tell if the number of As and Bs are the same v Tell if a string is a palindrome 6

8 The Turing machine u An infinite tape, with squares marked on it u Each square can be blank or hold a symbol u A machine moves over the tape u When positioned over a square, the machine can v v v v Read a symbol from the tape Erase a symbol from the tape Write a symbol onto the tape (erasing what was on the square) Do nothing u The machine can move either one square to right or left 7

9 Simple Turing Machine with 1 symbol 8

10 Alan Turing 9

11 Turing Machine Representation: Table This machine has 4 symbols: 0, 1, _ and * L means the head moves to the left R means the head moves to the right 10

12 Turing Machine Representation: Table 11

13 Turing Machine Representation: Table 12

14 Turing Machine Representation: Table write_symbol, head_move_direction, state_update 13

15 Turing Machine Representation: Table write_symbol, head_move_direction, state_update 14

16 Same Machine as a State Diagram read_symbol,write_symbol,head_position_move 15

17 Same Machine as a State Diagram read_symbol,write_symbol,head_position_move 16

18 17

19 Example execution u Assume the machine always begins in state 0 with the head positioned on the rightmost non-blank square u The machine stops when it is in state 4 (called an accept state) u The machine stops when it is in state 5 (called a reject state) u How does the execution look if the tape has an input of 10 written on it at the beginning? 18

20 Behavior of the machine on input 10 19

21 Behavior of the machine on input 10 20

22 Behavior of the machine on input

23 What does this Turing machine do? uchecks if a binary string has the same number of 1 s and 0 s and enters the accept state if it does, otherwise it enters the reject state. 22

24 Turing Machine Programming uwriting programs for a Turing machine is very tedious uhowever 23

25 The Church-Turing thesis Turing machines are capable of solving any effectively solvable algorithmic problem Turing machines are thus excellent models for what (all) real computers are capable of doing 24

26 Problems not effectively computable uare there problems that are not effectively computable? u Yes, infinite such problems ucould these be computable someday? uperhaps, but not on a Turing-equivalent computer 25

27 A halting question udeciding whether a program will halt (terminate) udoes the following program halt? Input number While number is not 0 Print number number = number 1 26

28 The halting problem udeciding whether a given program will halt (terminate) on an arbitrary input uin 1936, Turing proved that a general algorithm to the halting problem for all possible programs over all possible inputs cannot exist The halting problem is undecidable 27

29 Proof sketch for halting problem u Want to decide if program A halts on input a u Need to write a program, lets call it halter v Inputs: A, a v Output: Yes (if program A halts on input a) or No (if program A does not halt on input a) halter(a,a) if(some complex computation)is true Return Yes else Return No 28

30 The halting problem is undecidable uin 1936, Turing proved that a general algorithm to the halting problem for all possible programs over all possible inputs cannot exist uin other words, a general version of halter (one that works for all programs and their inputs) does not exist 29

31 Proof by contradiction u Assume that a general version of halter exists u Show that this assumption leads to a contradiction u Invent a new program called clever clever(program,input) result = halter(program,input) if result is No return Yes else loop forever 30

32 Proof by contradiction uwhat does clever do when given itself as input? clever(program,input) result=halter(program,input) if result is No else return Yes loop forever 31

33 Proof by contradiction u If halter says that clever halts, then clever loops forever (which means it doesn t halt) clever(program,input) result=halter(program,input) if result is No else return Yes u If halter says loop forever that clever does not halt then Conclusion: The program clever halts and halter cannot exist returns Yes 32

34 Russell s Paradox u In a hypothetical small town: v There is one male barber v The barber shaves all those and only those men in town who do not shave themselves v All men stay clean shaven either by shaving themselves or going to the barber u Who shaves the barber? u More generally: v Let R be the set of all sets that are not members of themselves v If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves 33

35 Analysis of problems u Study of algorithms illuminates the study of classes of problems u If a polynomial time algorithm exists to solve a problem then the problem is called tractable u If a problem cannot be solved by a polynomial time algorithm then it is called intractable u This divides problems into three groups: v Problems with known polynomial time algorithms v Problems that are proven to have no polynomial-time algorithm v Problems with no known polynomial time algorithm but not yet proven to be intractable 34

36 Tractable and Intractable u Tractable problems (P) v v v Sorting a list Searching an unordered list Finding a minimum spanning tree in a graph u Might be (in)tractable v Subset sum: given a set of numbers, is there a subset that adds up to a given number? v Travelling salesperson: n cities, n! routes, find the shortest route u Intractable v Listing all permutations (all possible orderings) of n numbers These problems have no known polynomial time solution However no one has been able to prove that such a solution does not exist 35

37 Tractability, Intractability, Undecidability u Properties of problems (NOT properties of algorithms ) u Tractable: problem can be solved by a polynomial time algorithm (or something more efficient) u Intractable: problem cannot be solved by a polynomial time algorithm (all solutions are proven to be more inefficient than polynomial time) u Unknown: not known if the problem is tractable or intractable (no known polynomial time solution, no proof that a polynomial time solution does not exist) u Undecidable: decision problem for which there is (provably) no algorithmic solution on a Turing machine v v Non-existence Not a matter of efficiency 36

38 Abstract machines and theory uwhat is theoretical computer science? ufinite state machines uthe Turing machine u The Church-Turing thesis u The halting problem uthe analysis of algorithms uthe analysis of problems Reading: St. Amant Ch. 8 37

39 Overview Problems Solution: Algorithms + Data Structures Pseudocode Low-level Low-level instructions Low-level instructions instructions Executions managed by Operating System Program Program Program Program Compile to CPU, Memory, Disk, I/O 38

40 Quiz #6 39