Designing Finite Automata

San Skulrattanakulchai

Feb 15, 2016

Two Design Problems

Given an FA, describe its language precisely.
Given a formal definition of a regular language, design an FA that recognizes it.

Solving Problem 1 is an art. No general principles are known.

To solve Problem 2, work with state diagrams and follow these steps:

Get all the machine states. Each state memorizes one unique machine condition.

Pinpoint the accept states.

Draw the edges.

Exercises

Assume Σ = { a, b }. Design machines to recoqnize strings that

starts with a

ends with bb

starts with a and ends with b

starts with a or ends with a

if it starts with a, then it doesn't end with a

Exercises

does not start or end with a but a occurs somewhere

contains ab

contains ab and ba

contains ab or ba

contains neither aa nor bb

Exercises

contains (at least) an a

contains exactly one a

contains (at least) two a's

contains exactly two a's

negate each of the above

Remarks

We can always get the complement of a language of a DFA simply by exchanging the role of the accepting and the nonaccepting states. Does this technique work for NFAs?
The Product Construction can be used to obtain the language of binary set operations (like union, intersection, set difference, symmetric difference, etc) of two regular languages.