Chapter 12 Theory of Computation

1. Chapter 12Theory of Computation Introduction to CS 1st Semester, 2014 Sanghyun Park

2. Outline • Functions and Their Computation • Turing Machines • Universal Programming Languages • A Noncomputable Function • Complexity of Problems

3. Functions and Their Computation • A function is a ______________ between a collection of possible input values and a collection of output values • The process of determining the particular output value that a function assigns to a given input is called _________ the function • Our question is whether we can always find a systemfor computing functions, regardless of their _________ • The answer is ___

4. Computable vs. Noncomputable Functions • There are functions that are so _______ that there is no well-defined, step-by-step process for determining their outputs based on their input values; these functions are said to be _____________ • The functions whose output values can be determined algorithmically from their input values are said to be ___________ • The study of computable and noncomputable functionsis an important undertaking in computer science

5. Turing Machines • In an effort to understand the ________ of machines, the Turing machine was proposed by Alan M. Turing in 1936 • A Turing machine consists of a ______ unit that can read and write symbols on a tape • The tape extends indefinitely at both ends and is divided into ____, each of which can contain a symbol • At any time during a Turning machine’s computation,the machine must be in one of a finite number of ______

6. Turing Machine’s Computation • A Turing machine’s computation consists of a sequence of steps that are executed by the control unit • Each step consists of • _________ the symbol in the current tape cell • _______ a symbol in that cell • Possibly _______ the read/write head one cell to the left or right • ________ states • The exact action to be performed is determined by a program that tells the control unit what to do based on the machine’s _____ and the ______ of the current tape ____

7. A Turing Machine forIncrementing a Value

8. * 1 0 1 * Machine State = START CurrentPosition * 1 0 1 * Machine State = ADD CurrentPosition * 1 0 0 * Machine State = CARRY CurrentPosition A Turing Machine Example (1/2)

9. * 1 1 0 * Machine State = RETURN CurrentPosition * 1 1 0 * Machine State = RETURN CurrentPosition * 1 1 0 * Machine State = HALT CurrentPosition A Turing Machine Example (2/2)

10. The Church-Turing Thesis (1/2) • The Turing machine in the preceding example can be used to compute the __________ function • A function that can be computed in this manner by a Turing machine is said to be ______ computable • Turing’s conjecture was that the Turing-computable functions were the same as the __________ functions;this conjecture is referred to as the Church-Turing thesis

11. The Church-Turing Thesis (2/2) • Today this thesis is widely ________; that is,the computable functions and the Turing-computable functions are considered ____ and the same • If a computational system can compute all the Turing-computable functions, it is considered to be as ________ as any computational system can be

12. Universal Programming Language • Most features in today’s high-level languages merely enhance ___________ rather than contribute to the fundamental _____ of the language • Our approach here is to describe a simple imperative programming language powerful enough to express programs for computing all the computable functions • A programming language with this power is called a _________ programming language • The language we present is quite simple; we call it Bare Bones in that it isolates a _______ set of requirements of a universal programming language

13. assignment loop structure The Bare Bones Languages • All variables are considered to be of type “___ pattern of arbitrary length” • Variable names must begin with a letter, which can be followed by any combination of letters and digits • Contains three assignment statements and one loop structure • _____name; • _____name; • _____name; • whilename not 0 do;..end;

14. Programming in Bare Bones A Bare Bones programfor “copy Today to Tomorrow” A Bare Bones programfor computing X * Y

15. The Universality of Bare Bones • Researchers have shown that the Bare Bones language can be used to express algorithms for computing ___ the Turing-computable functions • That is, any computable function can be computed by a program written in Bare Bones • Thus Bare Bones is a ________ programming language; if an algorithm exists for solving a problem, then that problem can be solved by some Bare Bones program • Bare Bones could ___________ serve as a general-purpose programming language

16. Halting Problem • The ______ problem is the problem of trying to predict in advance whether a program will _________ if started under certain conditions • Consider the simple Bare Bones programIf the initial value of X is __, then the program will halt;otherwise, the loop will be executed forever • It is easy in the above example to predict a program’s behavior; however, this task may be more complicatedor even impossible in some cases while X not 0 do; incr X; end;

17. Self-Reference • Whether a program ultimately halts can depend on the ______ values of its variables • We assign a program’s variables an initial value representing the _______ itself; that is, we assign the _________ version of a program as the value of its variables

18. Self-Termination • A Bare Bones program is self-terminating if its execution terminates when started with _____ as its input • The ______ problem is now precisely described as the problem of determining whether Bare Bones programs are or are not self-terminating • There is ___ algorithm for answering this question in general; thus, the solution to the halting problem lies _______ the capabilities of computers

19. Unsolvability of the Halting Problem (1/3)

20. Unsolvability of the Halting Problem (2/3)

21. Unsolvability of the Halting Problem (3/3) Therefore, the halting problem is __________

22. Complexity of Problems (1/2) • We are interested in the question of whether a solvable problem has a ________ solution • The complexity of a problem is determined by the properties of the algorithms that solve that problem • More precisely, the complexity of the _______ algorithm for solving a problem is considered to be the complexity of the problem itself • We measure an algorithm’s complexity in terms of the time required for its execution, which is proportional to the number of ______ that must be performed

23. Complexity of Problems (2/2) • The complexity of a problem is Θ(f(n)) if there is an algorithm with complexity of Θ(f(n)) for solving the problem and no other algorithm has a _____ complexity • Finding the best solution to a problem and knowing that it is the best is often a difficult problem itself;in such situations, big O notation is used • The complexity of a problem is O(f(n)) if it has a solution whose complexity is Θ(f(n)) but it could possibly have a ______ solution

24. Polynomial vs. Nonpolynomial Problems • “g(n) is bounded by f(n)” means that the graph of f(n) will be _______ the graph of g(n) for “large” values of n • A problem is a polynomial problem if the problem is in O(f(n)), where the expression f(n) is either a polynomial itself or bounded by a polynomial • The collection of all polynomial problems is denoted by P • Problems that are outside the class P are characterized as having extremely _____ execution times • Identifying the problems that belong to P is of major importance in computer science because it tells whether problems have _________ solutions

25. NP Problems • A problem that can be solved in polynomial time by a _______________ algorithm is called a nondeterministic polynomial problem, or an NP problem • It is customary to denote the class of NP problems by NP • All the problems in P are also in NP;However, whether all the NP problems are also in P isan _____ question