NFA and regex. the Boolean algebra of languages. regular expressions. Informatics 1 School of Informatics, University of Edinburgh

Transcription

1 NFA and regex cl the Boolean algebra of languages regular expressions Informatics

2 The intersection of two regular languages is regular L = even numbers L = odd numbers L = mod L = mod Informatics

3 The intersection of two regular languages is regular L = mod 3 L = mod 3 L = mod 3 Informatics 3

4 Two examples + Even binary numbers Input sequence is accepted if it ends with a zero. + Input sequence is accepted if it ends with a one. Odd binary numbers Informatics 4

5 Three examples Which binary numbers are accepted? + mod 3 Informatics 5

6 By three or not by three? divisible by three not divisible by three Informatics 6

7 The complement of a regular language is regular If A Σ* is recognised by M then A = Σ* \ A is recognised by M where M and M are identical except that the accepting states of M are the nonaccepting states of M and vice-versa Informatics 7

8 The intersection of two regular languages is regular divisible by 6 divisible by and divisible by 3 Informatics 8

9 The intersection of two regular languages is regular Run both machines in parallel? Build one machine that simulates two machines running in parallel! Keep track of the state of each machine. Informatics 9

10 The intersection of two regular languages is regular Informatics

11 intersection of languages run the two machines in parallel when a string is in both languages, both are in an accepting state

12 The intersection of two regular languages is regular = mod 3 mod 3 3 Informatics

13 The intersection of two regular languages is regular = 3 mod 3 mod mod Informatics 3

14 The intersection of two regular languages is regular = 3 mod 3 mod mod Informatics 4

15 The regular languages A Σ* form a Boolean Algebra Since they are closed under intersection and complement. Informatics 5

16 Non Determinism In a non-deterministic machine (NFA), each state may have any number of transitions with the same input symbol, leaving to different successor states. In a non-deterministic machine (NFA), each state may lead to any number of transitions with the same input symbol, leaving to different successor states., We have a transition relation Informatics 6

17 Non Determinism In a non-deterministic machine (NFA), each state may have any number of transitions with the same input symbol, leaving to different successor states.,,,,,,,, Informatics 7

18 Non Determinism We can simulate a non-deterministic machine using a deterministic machine by keeping track of the set of states the NFA could possibly be in.,,,,,,,, Informatics 8

19 Internal Transitions We sometimes add an internal transition ε to a nondeterministic machine (NFA)This is a state change that consumes no input. ε ε Informatics 9

20 Internal Transitions We sometimes add internal transitions labelled ε to a non-deterministic machine (NFA). This is a state change that consumes no input. It introduces non-determinism in the observed behaviour of the machine. ε ε ε* ε*, Informatics

21 NFA any number of start states and accepting states R S

22 sequence RS R S ε ε

23 alternation R S R S 3

24 iteration R* ε R ε ε 4