1. Theory of Computation - Lecture 3 Regular Languages What is a computer? Complicated, we need idealized computer for managing mathematical theories... Hence: Computational Models. Comp. Models vary, depending on the features we want them to focus on. Simplest Comp. Model is Finite State Machine, or Finite Automaton Dr. Maamoun Ahmed

2. Finite Automaton Good models for computers with extremely limited memories. With such limited memories, what the computers can do? Give examples of such computers! …..

3. Example – Automatic Door Door Front pad Rear pad

4. FSM/FA of Auto-door CLOSEDOPEN FRONT NEITHER REAR BOTH NIETHER FRONT REAR BOTH

5. Cont. How many bits required for auto-door implementation? More examples: Elevator, what are the states and inputs? Dishwasher, Washing machine, Thermostats, digital watches and calculators, etc.... Finite Automata was in mind when all of above were designed!

6. Closer look at Finite Automata Abstracting, application-independent: Example: A finite automaton called M1 with 3 states: q1q2q3 1 0 0,1 0 1 State Diagram State Diagram

7. Closer Look – Cont. Start State Accept State Transitions - When 1101 is entered from the input to Start State (one by one, L2R), the output will be either Accept/Reject or for now, Yes/No respectively. Will 1101 output Accept or Reject in our example? What about 1? 01?11?101001001? What about 100? 0100? 1000000?

8. What strings will M1 accept? What will it reject? Can you describe the language consisting of all strings that M1 accepts? …..Go on!

9. Formal definition of a FA Why do we need a formal definition? 1- It is precise: Uncertain whether some FA allowed to have 0 accept states or not? Formal definition helps you decide. 2- It provides Notations, and they help you think and express your thoughts clearly. Formal definition says that FA is a list of five objects: Set of states, input alphabet, rules for moving, start state, and accept states What do we call a list of 5 elements ?

10. Formal definition of FA – cont. We define a FA to be a 5-tuple consisting of these five parts. Transition function ( δ) defines the rules of moving. Ex.: 2 states x, y. and an arrow from x to y with a label=1, then moving from x to y is ruled by reading 1 as an input, which can be written as : δ(x,1)=y

11. Formal definition of FA A finite automaton is a 5-tuple (Q, ∑, δ, q 0, F) where: Q is a finite set called the states ∑ is a finite set called the alphabet δ:Q X ∑ → Q is the transition function q 0 ∈ Q is the start state, and F ⊆ Q is the set of accept states(sometimes called Final states).

12. Example Go back to slide no.6 (M1 FA), we can describe M1 formally by writing M1 = (Q, ∑, δ, q1, F), where: 1. Q = {q1, q2, q3} 2. ∑ = {0,1} 3. δ is described as 0 1 q1 q2 q3 q2 q3 q2

13. Example – cont. 4. q1 is the start state, and 5. F = {q2} Note: A is a set of all strings that machine M accepts, we say: A is the language of machine M and write L(M)=A. We say: M recognizes A. A machine may accept several strings, but it always recognizes only one language. If the machine accepts no strings, it still recognizes one language, the empty language ϕ

14. Example – cont. So, in our example let: A = {w | w contains at least one 1 and an even number of 0s follows the last 1}. Then L(M1) = A, or equivalently, M1 recognizes A. Examples of FA (1.7 – 1.15) pages 37-40