Chapter 12Theory of Computation

Introduction to CS

1st Semester, 2014 Sanghyun Park

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

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 ___

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

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 ______

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 ____

A Turing Machine forIncrementing a Value

A Turing Machine Example (1/2)

* 1 0 1 *

CurrentPosition

Machine State = START

* 1 0 1 *

CurrentPosition

Machine State = ADD

* 1 0 0 *

CurrentPosition

Machine State = CARRY

A Turing Machine Example (2/2)

* 1 1 0 *

CurrentPosition

Machine State = RETURN

* 1 1 0 *

CurrentPosition

Machine State = RETURN

* 1 1 0 *

CurrentPosition

Machine State = HALT

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

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

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

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; while name not 0 do;

.

.end;

assignment

loop structure

Programming in Bare Bones

A Bare Bones programfor computing X * Y

A Bare Bones programfor “copy Today to Tomorrow”

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

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 program

If 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;

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

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

Unsolvability of the Halting Problem (1/3)

Unsolvability of the Halting Problem (2/3)

Unsolvability of the Halting Problem (3/3)

Therefore, the halting problem is __________

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

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

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

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