Teori Bahasa dan Automata Lecture 1: Course Overview and Introduction By: Nur Uddin, Ph.D

Lecturer Name: Nur Uddin, PhD. Email: nur.uddin@upj.ac.id Education: PhD in Eng. Cybernetics, NTNU, Norway (April 2016) M. Eng in Mechanical Eng., GSNU, South Korea (Feb 2009) B. Eng in Aeronautics Eng., ITB, Indonesia (Mar 2002)

Theory of computation Theory of computation comprises the fundamental mathematical properties of computer hardware, software, and applications. Theory of computation is to answer: What are the fundamental capabilities and limitations of computers? Theory of computation consists of three areas: Automata Computability Complexity

Complexity Computer problems; Easy problems e.g.: sorting problem Hard problems e.g.: scheduling problem What makes some problems computationally hard and others easy? This is the central question of complexity theory.

Complexity (Cont’d) Several options to confront a computationally hard problem Find the root of the difficulty. Finding solutions that only approximate the perfect solution. Some problems are hard only in the worst case situation. Consider alternative types of computation. E.g.: randomized computation can speed up certain tasks.

Complexity Theory Complexity theory is an elegant scheme for classifying problems according to their computational difficulty. Complexity theory provides a method for giving evidence that certain problems are computationally hard, even if we are unable to prove it. In most fields, an easy computational problem is preferable to a hard one but not in cryptography. Cryptography is unusual because it requires computational problems to be hard, rather than easy.

Computability Theory During the first half of the twentieth century, mathematicians such as Kurt Godel, Alan Turing, and Alonzo Church discovered that certain basic problems cannot be solved by computers. One example of this phenomenon is the problem of determining whether a mathematical statement is true or false. The theories of computability and complexity are closely related. In complexity theory, the objective is to classify problems as easy ones and hard ones; whereas in computability theory, the problems is classified into solvable and unsolvable.

AUTOMATA THEORY Automata theory deals with the definitions and properties of mathematical models of computation. One model, called the finite automaton, is used in text processing, compilers, and hardware design. Another model, called the context-free grammar, is used in programming languages and artificial intelligence. Automata theory is an excellent place to begin the study of the theory of computation. The theories of computability and complexity require a precise definition of a computer. Automata theory provides formal definitions of computation.

Mathematical Notations and Terminology

Set Set Membership and non-membership ( ). Subset Infinite set Empty set

Set Operations Union ( ) Intersection ( ) Complement ( ) Venn diagram

Sequence and tuple A sequence of objects is a list of these objects in some order. E.g.: (7, 21, 57) is not the same as (57, 7, 21). Similarly, repetition does matter in a sequence, but it doesn’t matter in a set. Thus (7, 7, 21, 57) is different from (7, 21, 57) and (57, 7, 21)., whereas the set {7, 21, 57} is identical to the set {7, 7, 21, 57}.

Tuples Sequences may be finite or infinite. Finite sequences often are called tuples. A sequence with k elements is a k-tuple. Thus (7, 21, 57) is a 3-tuple.

Power set Sets and sequences may appear as elements of other sets and sequences. For example: The power set of A is the set of all subsets of A. If A is the set {0, 1}, the power set of A is the set { ∅, {0}, {1}, {0, 1} }. The set of all ordered pairs whose elements are 0s and 1s is { (0, 0), (0, 1), (1, 0), (1, 1) }.

Cartesian Product If A and B are two sets, the Cartesian product or cross product of A and B, written A × B, is the set of all ordered pairs wherein the first element is a member of A and the second element is a member of B. Example:

Function and Relation A function is an object that sets up an input–output relationship. A function also is called a mapping, and f(a) = b is said as f maps a to b. The set of possible inputs to the function is called its domain. The outputs of a function come from a set called its range. The notation for saying that f is a function with domain D and range R is:

Function: example 1

Function: example 2