LING/C SC/PSYC 438/538. Lecture 17 Sandiway Fong

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1 LING/C SC/PSYC 438/538 Lecture 7 Sandiway Fong

2 Today's Topic Review of ungraded homework Closure properties of FSA Practice! Homework out on Thursday

3 From last time Ungraded Homework 6 apply the set-of-states construction technique to the two machines on the ε- transition slide (repeated elow) self-check your answer: verify in each case that the machine produced is deterministic and accurately simulates its ε-transition counterpart a 2 3 a 2 3 ε ε

4 Ungraded Homework 6 Review Converting a NDFSA into a DFSA a 2 3 ε Note: this machine with an ε-transition is non-deterministic Note: this machine is deterministic {,3} {2} a {3} [Powerpoint animation]

5 Ungraded Homework 6 Review Converting a NDFSA into a DFSA a 2 3 Note: this machine with an ε-transition is non-deterministic ε Note: this machine is deterministic {,2} {2} a {3} [Powerpoint animation]

6 Formal (constructive) set-theoretic definition of a regular language Correspondence etween REs and Regular Languages concatenation (juxtaposition) union ( also [ ]) Kleene closure (*) Note: x + = xx*) Note: ackreferences are memory devices and thus are too powerful e.g. L = {ww} and prime numer testing (see earlier lectures)

7 Closure properties: i.e. do we still have a regular language afterward applying the operation? Properties not necessarily preserved higher up: e.g. context-free grammars as we ll see later

8 Textook gives one direction only case y case: a) Empty string ) Empty set c) Any character from the alphaet

9 Concatenation: Link final state of FSA to initial state of FSA 2 using an empty transition Note: empty transition ε can e deleted using the set of states construction

10 Kleene closure: repetition operator: zero or more times use empty transitions for loopack and ypass

11 Union: aka disjunction Non-deterministically run oth FSAs at the same time, accept if either one accepts

12 Other closure properties: Let s consider uilding the FSA machinery for each of these guys in turn

13 Other closure properties: Final state?

14 Other closure properties: Final state?

15 Other closure properties: Example: Σ* = {a,} need arcs for each character in Σ

16 Other closure properties: reverse arrows and swap initial/final

17 Regular Expressions from FSA Recall textook Exercise: find a RE for Examples (* denotes string not in the language): *a *a a λ (empty string) *aa aa

18 Regular Expressions from FSA Draw a FSA and convert it to a RE: 2 a 3 4 ε [Powerpoint Animation] * ( a+ )+ = +(a+)* ε

19 Regular Expressions from FSA Example Perl implementation: $s = "a a a aa aa"; while ($s =~ /\(+(a+)*)\/g) { print "<$ match!\n"; } Note: this doesn t include the empty string case Output: perl test.perl <a match! < match! <aa match! Note: recall /../g gloal flag for multiple matches

20 Converting FSA to REs Example: Give a RE for the FSA: State y-pass method:. Delete one state at a time 2. Calculate the possile paths passing through the deleted state 3. Add the regex calculated at each stage as an arc e.g. eliminate state 3 then 2

21 Converting FSA to REs eliminate state eliminate state 2 0( + 0 )* + 0( + 0 )* + Answer: (0( + 0 )* + )* [Powerpoint animation]

Lecture 3: Nondeterministic Finite Automata

Lecture 3: Nondeterministic Finite Automata September 5, 206 CS 00 Theory of Computation As a recap of last lecture, recall that a deterministic finite automaton (DFA) consists of (Q, Σ, δ, q 0, F ) where