ANJUMAN COLLEGE OF ENGINEERING TECHNOLOGY Department of Computer

ANJUMAN COLLEGE OF ENGINEERING & TECHNOLOGY Department of Computer Science & Engineering Theoretical Foundations of Computer Sciences Prof. Imteyaz Shahzad

Syllabus: Ø Unit 1: Mathematical preliminaries –Sets, operations, relations, strings, closure of relation, countability and diagonalization, induction and proof methods- pigeon-hole principle , concept of language, formal grammars, Chomsky hierarchy. Ø Unit 2: Finite Automaton, regular languages, deterministic & non deterministic finite automata, ϵ-closures, minimization of automata, equivalence, Moore and Mealy machine. Ø Unit 3: Regular expression, identities, Regular grammar, right linear, left linear, Arden theorem, Pumping lemma for regular sets, closure & decision properties for regular sets, Context free languages, parse trees and ambiguity, reduction of CFGS, Normal forms for CFG.

ØUnit 4: Push down Automata (PDA), non-determinism, acceptance by two methods and their equivalence, conversion of PDA to CFG, CFG to PDAs, closure and decision properties of CFLs, pumping lemma for CFL. ØUnit 5: Turing machines, TM as acceptor, TM as transducers, Variations of TM, linear bounded automata, TM as computer of function. ØUnit 6: Recursively enumerable (r. e. ) set, recursive sets, Decidability and solvability, Post correspondence Problem (PCP), Introduction to recursive function theory, primitive recursive functions, Ackerman function.

COURSE OUTCOMES: Ø CO 1: Classify the concept of languages and automata. Ø CO 2: Explain the formal relationships among machines, languages and grammars. Ø CO 3: Construct Regular Grammar and normal forms for CFG. Ø CO 4: Design and develop finite automata for given regular language. Ø CO 5: Design Push Down Automata, Turing Machine for given languages Ø CO 6: Demonstrate use of computability, decidability, recursive function theory through problem solving

Turing machines

Turing machine consist of 7 TUPLE: M=(Q, Ʃ, ┌ , δ, qo, B, F) Q= Set of Finite State Ʃ= Set of Alphabet ┌ = Finite Set Tap Symbol δ = Transition Function Mapping From Q x┌ to Q x┌ x (L, R) q 0= Initial State B= Blank Symbol F= Output Mapping Function

n n L={a b │n≥ 0}

Transition Table a b B Q 0 Q 1, B, R --------- Q 4, B, S Q 1, a, R Q 1, b, R Q 2, B, L Q 2 --------- Q 3, B, L --------- Q 3, a, L Q 3, b, L Q 0, B, L Q 4 -----------------

Design a Turing Machine For 2’s Compliment of binary number.

Transition Table 0 1 B Q 0, 0, R Q 0, 1, R Q 1, B, L Q 1 Q 0, 1, L Q 2, 1, L ----------- Q 2, 1, L Q 2, 0, L Q 3, B, S Q 3 ----------------------

L={WCWr│W=(a, b)*and Wr is a reverse of W }

THANK YOU