LING/C SC/PSYC 438/538 Lecture 11 Sandiway Fong. Administrivia Homework 3 graded.

LING/C SC/PSYC 438/538

Lecture 11Sandiway Fong

Administrivia

• Homework 3 graded

Last Time

1. Introduced Regular Languages– can be generated by regular expressions – or Finite State Automata (FSA)– or regular grammars --- not yet introduced

2. Deterministic and non-deterministic FSA3. DFSA can be easily encoded in Perl:

– hash table for the transition function– foreach loop over a string (character by character)– conditional to check for end state

4. NDFSA can be converted into DFSA– example of the set of states construction– Practice: ungraded homework exercise

Ungraded Homework Exercise

• do not submit, do the following exercise to check your understanding

– apply the set-of-states construction technique to the two machines on the ε-transition slide (repeated below)

– self-check your answer: • verify in each case that the machine produced is deterministic and

accurately simulates its ε-transition counterpart

a

ε

b>

a

ε

b>

Ungraded Homework Exercise Review

• Converting a NDFSA into a DFSA

1

a

ε

2 3

b>

{1,3} {2}a b {3}>

Note: this machine with an ε-transitionis non-deterministic

Note: this machine isdeterministic

Ungraded Homework Exercise Review

• Converting a NDFSA into a DFSA

1

a

ε

2 3

b>

{1,2} {2}a b {3}

b

Note: this machine with an ε-transitionis non-deterministic

Note: this machine isdeterministic >

Last TimeRegular Languages• Three formalisms

– All formally equivalent (no difference in expressive power)– i.e. if you can encode it using a RE, you can do it using a FSA or regular grammar, and so

on …

Regular Grammars

FSA Regular Expressions

Regular Languages

talk more about formalequivalence later today…

Perl regular

expressions

stuff out here

Perl Regular Expressions• Perl regex can include backreferences to groupings (i.e. \1, etc.)

– backreferences give Perl regexs expressive power beyond regular languages:

• the set of prime numbers is not a regular languageLprime = {2, 3, 5, 7, 11, 13, 17, 19, 23,.. }

can be proved using the Pumping Lemma for regular languages

(later)

can have regular Perl

code inside a regex

Backreferences and FSA• Deep question:

– why are backreferences impossible in FSA?

s x

y

aa

b

b

>

Example:Suppose you wanted a machine that accepted /(a+b+)\1/

One idea: link two copies of the machine together

x2

y2

a

a

b

b

y

Doesn’t work!Why?

• Perl implementation:– how to modify it get the backreference effect?

Regular Languages and FSA• Formal (constructive) set-theoretic definition of a regular language

• Correspondence between REs and Regular Languages• concatenation (juxtaposition)• union (| also [ ])• Kleene closure (*) = (x+ = xx*)

• Note:• backreferences are memory devices and thus are too powerful• e.g. L = {ww} and prime number testing (earlier slides)

Regular Languages and FSA

• Other closure properties:

• Not true higher up: e.g. context-free grammars as we’ll see later

Equivalence: FSA and RegexsTextbook gives one direction only• Case by case:

a) Empty stringb) Empty setc) Any character from the alphabet

Equivalence: FSA and Regexs

• Concatenation:

– Link final state of FSA1 to initial state of FSA2 using an empty transition

Note: empty transition can be eliminated using the set of states construction(see earlier slides in this lecture)

Equivalence: FSA and Regexs

• Kleene closure:

– repetition operator: zero or more times– use empty transitions for loopback and bypass

Equivalence: FSA and Regexs• Union: aka disjunction

– Non-deterministically run both FSAs at the same time, accept if either one accepts

Regular Languages and FSA

• Other closure properties:

Let’s consider building the FSA machinery for each of these guys in turn…

Regular Languages and FSA

• Other closure properties:

Regular Languages and FSA

• Other closure properties:

Regular Languages and FSA

• Other closure properties:

Regular Languages and FSA

• Other closure properties:

Regular Expressions from FSA

Textbook Exercise: find a RE for

Examples (* denotes string not in the language):*ab *bababλ (empty string)bb*babababab

Regular Expressions from FSA

• Draw a FSA and convert it to a RE:

1 2 3 4>

b

ab b

b

ε

b* ab+( )+

[PowerpointAnimation]

= b+(ab+)*| ε

b

Regular Expressions from FSA

• Perl implementation:

$s = "ab ba bab bb baba babab";

while ($s =~ /\b(b+(ab+)*)\b/g) { print "<$1> match!\n";}

• Output:

perl test.perl<bab> match!<bb> match!<babab> match!

Note:doesn’t include the empty stringcase

Note: /../g global flagfor multiple matches