2. 1.Introduction a)What is Automata b)Automaton and types of automata c)History, Turing Machine d) Why study Automata Theory e)Applications & Uses Dept. of Computer Science & IT, FUUAST Automata Theory 2 Automata Theory I  Software for designing and checking digital circuits.  Lexical analyzer of compilers.  Pattern Recognition. Finding words and patterns in large bodies of text, e.g. in web pages.  Verification of systems with finite number of states, e.g. communication protocols.

3. 3 2.Mathematics a)Fundamentals of Set Theory i.Sets and subsets ii.Union A  B = { x: x  A or x  B} iii.Intersection A  B = { x: x  A and x  B} iv.Difference A - B = { x: x  A and x  B} v.Complement A = {x: x  A} Automata Theory I Dept. of Computer Science & IT, FUUAST Automata Theory

4. 4 vi.Properties of Set Operations Idempotency  A  A = A  A  A = A Commutativity  A  B = B  A  A  B = B  A Associativity  (A  B)  C = A  (B  C)  (A  B)  C = A  (B  C) Fundamentals of Set Theory Continued ………… Automata Theory I Dept. of Computer Science & IT, FUUAST Automata Theory

5. 5 vii.DeMorgan’s Laws: A - (B  C) = (A – B)  (A – C) A - (B  C) = (A – B)  (A – C) (A  B) = A  B (A  B) = A  B Fundamentals of Set Theory Continued ………… Absorption  (A  B)  A = A  (A  B)  A = A Distributivity  (A  B)  C = (A  C)  (B  C)  (A  B)  C = (A  C)  (B  C) Automata Theory I Dept. of Computer Science & IT, FUUAST Automata Theory

6. 6 viii.Cardinality  A  = number of elements in set A ix.Disjoint Sets A  B =  x.Power Sets 2 A, set of all subsets of A = P(A) xiCartesian Product A  B = {(x, y) : x  A and y  B} xiiRelations S = {a,b,c,d,e} and T= {w,x,y,z}, R is a Relation on S and T = {(a,y),(c,w),(c,z),(d,y)} Fundamentals of Set Theory Continued ………… Automata Theory I Dept. of Computer Science & IT, FUUAST Automata Theory

7. 7 Automata Theory I xiii.Equivalence Relation A subset R of A  A if 1.(a, a)  R for all a  A (Reflexive) 2.If (a,b)  R then (b,a)  R (Symmetric) 3.If (a,b)  R and (b,c)  R then (a,c)  R (Transitive) xiv.Partition of a Set Partition P of S is a collection of nonempty subsets {A i } of S such that 1.Each a  S belongs to some A i, 2.If A i  A j then A i  A j =  Fundamentals of Set Theory Continued ………… Dept. of Computer Science & IT, FUUAST Automata Theory

8. 8 xv.Functions and kinds One-to-One (Injection) Onto(Surjection) One-to-One Onto(Bijection) Automata Theory I Fundamentals of Set Theory Continued ………… b) Induction c) Deduction d) Boolean Logic e) Graphs f) Trees Dept. of Computer Science & IT, FUUAST Automata Theory

9. 9 Automata Theory I Basic Automata Concepts 3.Basic Automata Concepts a) Alphabets (  ) and power of Alphabets ( b)Strings c)Languages ( L ) a) Alphabets (  ) and power of Alphabets (  P ) Strings c)Languages ( L ), L={w   * : w has some property} Kleene Star L * is set of all strings obtained from concatenating zero or more strings from L d) Grammar, G=(V,T,P,S). i.V is a set of Variables ii.T is a set of Terminal Symbols iii.P is Production Rule iv.S is a set of start Variables Dept. of Computer Science & IT, FUUAST Automata Theory

10. 10 Automata Theory I Dept. of Computer Science & IT, FUUAST Automata Theory The End