THEORY OF COMPUTATION Komate AMPHAWAN 1

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DESCRIPTION-1 Studies concepts of:  grammars,  automata,  languages,  computability and complexity 3

DESCRIPTION-2 the relationship between automata and various classes of languages Turing machine and equivalent models of computation the Chomsky hierarchy context-free grammar, push-down automata, Pumping lemmas and variants, closure properties and decision properties; parsing algorithms. 4

Computability What is “Computation”? What is Computable?  Problem of determining whether a mathematical statement is true or false. 5

Evaluation and Grading Midterm exam40% Final exam40% Quiz10% Class attendance10% 6

Text-1 “Introduction to Languages and the Theory of Computation”—John C. Martin,4 ed,

Text-2 “Introduction to the Theory of Computation”—Michael Sipser, 2 ed,

Main topic We shall study different types of theoretical machines that are mathematical models for actual physical processes. 9

Machine Model-1 10

Machine Model-2 11

Machine Model-3 12

Prerequisites Set Theory Relations and Functions „ Relations and Functions Elementary Graph Theory Boolean Logic Proof Mechanisms 13

REVIEW… ON BASIC MATHEMATICS 14

SET 15

Sets-1 A set is a collection of "things," called the elements or members of the set. It is essential to have a criterion for determining, for any given thing, whether it is or is not a member of the given set. This criterion is called the membership criterion of the set. 16

Sets-2 There are two common ways of indicating the members of a set:  List all the elements, e.g. {a, e, i, o, u}  Provide some sort of an algorithm or rule, such as a grammar 17

Sets-3 Notation:  To indicate that x is a member of set S, we write x S  We denote the empty set (the set with no members) as { } or   If every element of set A is also an element of set B, we say that A is a subset of B, and write A  B  If every element of set A is also an element of set B, but B also has some elements not contained in A, we say that A is a proper subset of B, and write A  B 18

Operations on Sets The union of sets A and B, written A  B, is a set that contains everything that is in A, or in B, or in both. The intersection of sets A and B, written A  B, is a set that contains exactly those elements that are in both A and B. The set difference of set A and set B, written A - B, is a set that contains everything that is in A but not in B. 19

Additional terminology The cardinality of a set A, written |A|, is the number of elements in a set A. The powerset of a set Q, written 2 Q, is the set of all subsets of Q. The notation suggests the fact that a set containing n elements has a powerset containing 2 n elements. Two sets are disjoint if they have no elements in common, that is, if A  B = . 20

RELATIONS AND FUNCTIONS 21

Relations and Functions-1 A relation on sets S and T is a set of ordered pairs (s, t), where  s  S (s is a member of S),  t  T,  S and T need not be different,  The set of all first elements (s) is the domain of the relation, and  The set of all second elements is the range of the relation. 22

Relations and Functions-2 A relation is a function iff every element of S occurs once and only once as a first element of the relation. A relation is a partial function iff every element of S occurs at most once as an element of the relation. 23

GRAPHS 24

Graph-1 A graph consists of two sets  A set V of vertices (or nodes), and  A set E of edges (or arcs). An edge consists of a pair of vertices in V. If the edges are ordered, the graph is a digraph (a contraction of "directed graph"). 25

Graph-2 A walk is a sequence of edges, where the finish vertex of each edge is the start vertex of the next edge. Example: (a, e), (e, i), (i, o), (o, u). A path is a walk with no repeated edges. A simple path is a path with no repeated vertices. 26

TREES 27

Trees-1 A tree is a kind of digraph:  It has one distinguished vertex called the root;  There is exactly one path from the root to each vertex; and  The level of a vertex is the length of the path to it from the root. 28

Trees-2 Terminology: if there is an edge from A to B, then A is the parent of B, and B is the child of A. A leaf is a node with no children. The height of a tree is the largest level number of any vertex. 29

PROOF TECHNIQUES 30

Importance Because this is a formal subject, the textbook is full of proofs Proofs are encapsulated understanding You may be asked to learn a very few important proofs 31

Proof by induction Prove something about P1 (the basis) Prove that if it is true for Pn, then it is true for Pn+1 (the inductive assumption) Conclude that it is true for all P 32

Proof by contradiction also called reduction ad absurdum Assume some fact P is false Show that this leads to a contradiction Conclude P must be true 33