ling 6932 topics in computational linguistics

1 LING 6932 Spring 2007

LING 6932 Topics in Computational Linguistics

Hana FilipLecture 2: Regular Expressions, Finite State Automata


Regular expressions

formulas for specifying text stringsHow can we search for any of these strings?

woodchuckwoodchucksWoodchuckWoodchucks

Figure from Dorr/Monz slides


Regular Expressions

Basic patterns of regular expressionsPerl-based syntax (slightly different from other notations for regular expressions as used in UNIX, for example)/Woodchuck/ matches any string containing the substring Woodchuck, if your search application returns entire lines, for example‘/’ notation used by Perl, NOT part of the RE

Google: Woodchuck Draft CiderProducers of Woodchuck Draft Cider in Spingfield, VT.www.woodchuck.com/ - 17k - Cached - Similar pages

Slide from Dorr/Monz


Regular Expressions

Regular expressions are CASE SENSITIVEThe pattern /woodchuck/ will not match the string WoodchuckDisjunction /[wW]oodchuck/



Regular Expressions

Ranges [A-Z]



Regular Expressions

Negation /[^a]/ ^: caret

‘match any single character except a’



Regular Expressions

Operators ? , * and +? (0 or 1)

/woodchucks?/ woodchuck or woodchucks/colou?r/ color or colour

* (0 or more)/oo*h!/ oh! or ooh! or ooooh!

+ (1 or more)

• /o+h!/ oh! or ooh! or ooooh!

related to the immediately preceding character or regular expression

*+

Stephen Cole Kleene Wild card ./beg.n/ begin or began or begun

any character between beg and n (except a carriage return)



Regular Expressions

Anchors ^ and $ start of line

/^[A-Z]/ “Ramallah, Palestine”

/^[^A-Z]/ “¿verdad?” “really?”

end of line

/\.$/ “It is over.”

/.$/ ?

Boundaries \b and \B/\bon\b/ “on my way” “Monday” (boundary)

/\Bon\b/ “automaton” (non-boundary)



Disjunction, Grouping, Precedence

Disjunction |

/yours|mine/ “it is either yours or mine”

/gupp(y|ies)/ “guppy” or “guppies”

Column 1 Column 2 Column 3 …How do we express this?/Column[0-9]*/ ‘space’ /(Column[0-9]*)*/ NOT a RE character

matches the word Column, followed by one number, followed by zero or more spaces, the whole pattern repeated any number of times (zero or more times)



Disjunction, Grouping, Precedence

Operator Precedence HierarchyParenthesis ()Counters * + ? Sequences and anchors the ^my end$Disjunction |

REs are greedy!They always match the largest string they can



Example

Find me all instances of the word “the” in a text.

/the/Misses capitalized examples

/[tT]he/Returns “other” or “theology”

/\b[tT]he\b/ matches “the” or “The”

/[â-zA-Z][tT]he[â-zA-Z]/

/(^|[â-zA-Z])[tT]he[â-zA-Z]/Matches “the_” or “the25”


Errors

The process we just went through was based on two fixing kinds of errors

Not matching things that we should have matched (The)

– False negatives

Matching strings that we should not have matched (there, then, other)

– False positives


Errors cont.

We’ll be telling the same story for many tasksReducing the error rate for an application often involves two antagonistic efforts:

Increasing accuracy (minimizing false positives)Increasing coverage (minimizing false negatives).


More complex RE example

Regular expressions for prices/$[0-9]+/

Doesn’t deal with fractions of dollars

/$[0-9]+\.[0-9][0-9]/Doesn’t allow $199, not at a word boundary

/\b$[0-9]+(\.[0-9]0-9])?\b)/


Advanced operators

Regular expression operators for counting

RE Match

{n} exactly n occurrences of the previous character or expression

{n,m} from n to m occurrences of the previous character or expression

{n, } at least n occurrences of the previous character or expression

/a\.{24}z/ a followed by 24 dots followed by z


Advanced operators

To refer to characters that are special themselves precede them with a backslash

RE Match Example Strings Matched

\* an asterisk “*” “K*A*P*L*A*N”

\. a period “.” “Dr.Livingston, I presume.”

\? A question mark “?” “Would you light my candle?”

\n a newline

\t tab


Advanced operators



Substitutions and Memory

Substitution operator s/regexp1/regexp2/ (UNIX, Perl)

s/colour/color/s/colour/color/g

Substitute as many times as possible!

Case insensitive matching

s/colour/color/i



Substitutions and Memory

Substitutions“the Xer they were, the Xer they will be”constrain the two X’s to be the same string

/the (.*)er they were, the $1er they will be/

/the (.*)er they (.*), the $1er they $2/

Using numbered memories or registers: $1, $2, etc. used to refer back to matches

An extended feature of regular expressions



Eliza [Weizenbaum, 1966]

User: Men are all alikeELIZA: IN WHAT WAYUser: They’re always bugging us about something or otherELIZA: CAN YOU THINK OF A SPECIFIC EXAMPLE?User: Well, my boyfriend made me come hereELIZA: YOUR BOYFRIEND MADE YOU COME HEREUser: He says I’m depressed much of the timeELIZA: I AM SORRY TO HEAR THAT YOU ARE DEPRESSED


Eliza-style regular expressions

s/.* YOU ARE (depressed|sad) .*/I AM SORRY TO HEAR YOU ARE \1/s/.* YOU ARE (depressed|sad) .*/WHY DO YOU THINK YOU ARE \1/s/.* all .*/IN WHAT WAY/s/.* always .*/CAN YOU THINK OF A SPECIFIC EXAMPLE/

Step 1: replace first person with second person references

s/\bI(’m | am)\b /YOU ARE/g

s/\bmy\b /YOUR/g

S/\bmine\b /YOURS/g

Step 2: use substitutions that look for relevant patterns in the input and create an appropriate output (reply)

Step 3: use scores to rank possible transformations



Summary on REs so far

Regular expressions are perhaps the single most useful tool for text manipulation

Dumb but ubiquitous

Eliza: you can do a lot with simple regular-expression substitutions


Three Views

Three equivalent formal ways to look at what we’re up to (thanks to Martin Kay)

Regular Expressions

Regular LanguagesFinite State Automata


Finite State Automata

Terminology: Finite State Automata, Finite State Machines, FSA, Finite AutomataRegular expressions are one way of specifying the structure of finite-state automata.FSAs and their close relatives are at the core of most algorithms for speech and language processing.


Finite-state Automata (Machines)

/^baa+!$/

q0 q1 q2 q3 q4

b a a !

a

state transitionfinalstate

baa! baaa! baaaa! baaaaa! ...



Sheep FSA

We can say the following things about this machineIt has 5 statesAt least b, a, and ! are in its alphabetq0 is the start stateq4 is the final (= accept) stateIt has 5 transitions


More Formally: Defining an FSA

You can specify an FSA by enumerating the following things.

a finite set of states: Q

a finite alphabet of symbols: the start state: q0

The set of accepting/final states: F such that FQ

A transition function (q,i) that maps Qx to Q

Given a state qQ and an input symbol i, (q,i) returns a new state q’Q.


Yet Another View

State-transition table


Recognition

Recognition is the process of determining if a string should be accepted by a machineOr… it’s the process of determining if a string is in the language we’re defining with the machineOr… it’s the process of determining if a regular expression matches a string


Recognition

Traditionally, (Turing’s idea, 1936) this process is depicted with a tape.

http://www.cs.princeton.edu/introcs/75turing/


Recognition - Execution

Start in the start stateExamine the current input in the active cellConsult the table: a finite table of instructions (a state transition diagram) that specifies exactly what action the machine takes at each stepGo to a new state and update the tape pointer.Until you run out of tape.


Input Tape

b a a a

q0 q1 q2 q3 q3 q4

!

0 1 2 3 4

b a a !a

ACCEPT



Input Tape

a b a ! b

q0

0 1 2 3 4

b a a !a

REJECT



Adding a failing state

q0 q1 q2 q3 q4

b a a !

a

qFa

!

b

! b ! b

b

a

!



Tracing D-Recognize


Key Points

Deterministic means that at each point in processing there is always one unique thing to do (no choices).D-recognize is a simple table-driven interpreterThe algorithm is universal for all unambiguous languages.To change the machine, you change the table.


Key Points

Deterministic Pattern Example: Consider a set of traffic lights; the sequence of lights is red - red/amber - green - amber - red. The sequence can be pictured as a state machine, where the different states of the traffic lights follow each other.

Each state is dependent solely on the previous state, so if the lights are green, an amber light will always follow - that is, the system is deterministic. Deterministic systems are relatively easy to understand and analyse, once the transitions are fully known.


Key Points

Crudely therefore… matching strings with regular expressions (a la Perl) is a matter of

translating the expression into a machine (table) and passing the table to an interpreter


Recognition as Search

You can view this algorithm as state-space search.States are pairings of tape positions and state numbers.Operators are compiled into the tableGoal state is a pairing with the end of tape position and a final accept state


Generative Formalisms

A formal Language is a model m which can both generate and recognize all and only the strings of a formal language; each string is composed of symbols from a finite set of symbols (alphabet)

L(m) ‘a formal language L characterized by the model m’Finite-state automata define formal languages (without having to enumerate all the strings in the language)The term Generative is based on the view that you can run the machine as a generator to get strings from the language.


Generative Formalisms

FSAs can be viewed from two perspectives:Acceptors that can tell you if a string is in the language (recognition)Generators to produce all and only the strings in the language (production)


Summary

Regular expressions are just a compact textual representation of FSAsRecognition is the process of determining if a string/input is in the language defined by some machine.

Recognition is straightforward with deterministic machines.

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