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Regular Expressions and Automata

August 302012

Lecture #2

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Rule-based vs Statistical Approaches

• Rule-based = linguistic• Statistical – “learn” from a large corpus that has been

marked up with the phenomina you are studying• For what problems is rule-based better suited and

when is statistical better?– Identifying proper names– Distinguishing a biography from a dictionary entry– Answering Questions

• May depend on how much good training data is available

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Rule-based vs Statistical Approaches• Even when using statistical – many tasks easily done

with rules: tokenization, sentence breaking, morphology

• Some PARTS of a task may be done with rules: e.g., rules may be used to extract features, and then statistical learning methods might combine those features to perform a task.

• How much knowledge of language do our algorithms need to do useful NLP?

80/20 rule:– Claim: 80% of NLP can be done with simple methods (low

hanging fruit

– The last 20% can be more difficult to attain – when should we worry about that??

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Today

• Review some of the simple representations and ask ourselves how we might use them to do interesting and useful things– Regular Expressions– Finite State Automata

• How much can you get out of a simple tool?

• Should also think about the limits of these approaches:– How far can they take us?– When are they enough?– When do we need something more?

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Regular Expressions

• Can be viewed as a way to specify:– Search patterns over text string– Design of a particular kind of machine, called a Finite State

Automaton (FSA)

• These are really equivalent

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Uses of Regular Expressions in NLP

• As grep, perl: Simple but powerful tools for large corpus analysis and ‘shallow’ processing– What word is most likely to begin a sentence?– What word is most likely to begin a question?– In your own email, are you more or less polite than the

people you correspond with?

• With other unix tools, allow us to– Obtain word frequency and co-occurrence statistics– Build simple interactive applications (e.g., Eliza)

• Regular expressions define regular languages or sets

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Regular Expressions

• Simple but powerful tools for “shallow” processing of a document of “corpus”– What word begins a sentence?– What words begin a question– Identify all NPs

• These simple tools can enable us to– Build simple interactive applications (e.g., Eliza)– Recognize date, time, money… expressions– Recognize Named Entities (NE): people names, company names– Do morphological analysis

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Regular Expressions

• Regular Expression: Formula in algebraic notation for specifying a set of strings

• String: Any sequence of alphanumeric characters– Letters, numbers, spaces, tabs, punctuation marks

• Regular Expression Search– Pattern: specifying the set of strings we want to search for– Corpus: the texts we want to search through

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Simple Example

• Sequence of simple characters:

RE DESCRIPTION USES

/This/ Matches the string “This”

Finding the word “this” starting a sentence

/this/ Matches the string “this”

Finding the word “this” internal to a sentence.

/[Tt]his/ Matches either “This” or “this” -- disjunction

Finding the word “this” anywhere in a sentence.

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Some Examples

RE Description Uses

/./ Wild card; Any char except <cr>

A non-blank line?

/a/ Any ‘a’ Line with words?

/[ab]/ A choice

/[a-z]/ l.c. char (range) Common noun?

/[A-Z]/ u.c. char Proper noun?

/[^?.!]/ Neg of set Not S-final punc

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RE Description Uses?

/a*/ Zero or more a’s Optional doubled modifiers

/a+/ One or more a’s Non-optional...

/a?/ Zero or one a’s Optional...

/cat|dog/ ‘cat’ or ‘dog’ Words modifying pets

/^cat\.$/ A line that contains only cat. ^anchors beginning, $ anchors end of line.

??

/\bun\B/ Beginnings of longer strings

Words prefixed by ‘un’

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happier and happier, fuzzier and fuzzier/ (.+)ier and \1ier /

Morphological variants of ‘puppy’/pupp(y|ies)/

E.G.RE

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Optionality and Repetition

• /[Ww]oodchucks?/ matches woodchucks, Woodchucks, woodchuck, Woodchuck

• /colou?r/ matches color or colour• /he{3}/ matches heee• /(he){3}/ matches hehehe• /(he){3,} matches a sequence of at least 3 he’s

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Operator Precedence Hierarchy

1. Parentheses ()

2. Counters * + ? {}

3. Sequence of Anchors the ^my end$

4. Disjunction |

Examples

/moo+/

/try|ies/

/and|or/

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A Simple Exercise

• Write a regular expression to find all instances of the determiner “the”:

/the//[tT]he//\b[tT]he\b//(^|[^a-zA-Z][tT]he[^a-zA-Z]/

The recent attempt by the police to retain their current rates of pay has not gathered much favor with the southern factions.

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A Simple Exercise

• Write a regular expression to find all instances of the determiner “the”:

/the//[tT]he//\b[tT]he\b//(^|[^a-zA-Z][tT]he[^a-zA-Z]/

The recent attempt by the police to retain their current rates of pay has not gathered much favor with

southern factions.

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A Simple Exercise

• Write a regular expression to find all instances of the determiner “the”:

/the//[tT]he//\b[tT]he\b//(^|[^a-zA-Z][tT]he[^a-zA-Z]/

The recent attempt by the police to retain their current rates of pay has not gathered much favor with the southern factions.

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A Simple Exercise

• Write a regular expression to find all instances of the determiner “the”:

/the//[tT]he//\b[tT]he\b//(^|[^a-zA-Z][tT]he[^a-zA-Z]/

The recent attempt by the police to retain their current rates of pay has not gathered much favor with the southern factions.

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A Simple Exercise

• Write a regular expression to find all instances of the determiner “the”:

/the//[tT]he//\b[tT]he\b//(^|[^a-zA-Z][tT]he[^a-zA-Z]/

The recent attempt by the police to retain their current rates of pay has not gathered much favor with the southern factions.

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A Simple Exercise

• Write a regular expression to find all instances of the determiner “the”:

/the//[tT]he//\b[tT]he\b//(^|[^a-zA-Z])[tT]he[^a-zA-Z]/

The recent attempt by the police to retain their current rates of pay has not gathered much favor with the southern factions.

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The Two Kinds of Errors

• The process we just went through was based on fixing errors in the regular expression

– Errors where the instances were included (when they should not have been) – False positives

– Errors where some of the instances were missed (judged to not be instances when they should have been) – False negatives

• This is pretty much going to be the story of the rest of the course!

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Errors

• We’ll be telling the same story for many tasks, all semester. Reducing the error rate for an application often involves two antagonistic efforts: Increasing accuracy, or precision,

(minimizing false positives) Increasing coverage, or recall,

(minimizing false negatives).

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Substitutions (Transductions)

• Sed or ‘s’ operator in Perl– s/regexp1/pattern/– s/I am feeling (.++)/You are feeling \1?/– s/I gave (.+) to (.+)/Why would you give \2 \1?/

Next Several Slides from McKeown 2009

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ELIZA: Substitutions Using Memory

User: Men are all alike.

ELIZA: IN WHAT WAY

s/.* all .*/IN WHAT WAY/

They’re always bugging us about something or other.

ELIZA: CAN YOU THINK OF A SPECIFIC EXAMPLE

s/.* always .*/CAN YOU THINK OF A SPECIFIC EXAMPLE/

User: My boyfriend says I’m depressed.

ELIZA: I AM SORRY TO HEAR YOU ARE DEPRESSED

s/.* I’m (depressed|sad) .*/I AM SORRY TO HEAR YOU ARE \1/

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Using RE’s: Examples• Predictions from a news corpus:

– Which candidate for Governor of California is mentioned most often in the news? Is going to win?

– What stock should you buy?– Which White House advisers have the most power?

• Language use:– Which form of comparative is more frequent: ‘oftener’ or

‘more often’?– Which pronouns are conjoined most often?– How often do sentences end with infinitival ‘to’?– What words most often begin and end sentences?– What’s the most common word in your email? Is it different

from your neighbor?

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• Personality profiling: – Are you more or less polite than the people you correspond

with?– With labeled data, which words signal friendly messages vs.

unfriendly ones?

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Finite State Automata

• Regular Expressions (REs) can be viewed as a way to describe machines called Finite State Automata (FSA, also known as automata, finite automata).

• FSAs and their close variants are a theoretical foundation of much of the field of NLP.

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Finite State Automata

• FSAs recognize the regular languages represented by regular expressions– SheepTalk: /baa+!/

q0 q4q1 q2 q3

b a

a

a !

• Directed graph with labeled nodes and arc transitions

•Five states: q0 the start state, q4 the final state, 5 transitions

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Formally

• FSA is a 5-tuple consisting of– Q: set of states {q0,q1,q2,q3,q4} : an alphabet of symbols {a,b,!}– q0: a start state– F: a set of final states in Q {q4} (q,i): a transition function mapping Q x to Q

q0 q4q1 q2 q3

b a

a

a !

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State Transition Table for SheepTalk

StateInput

b a !

0 1 Ø Ø

1 Ø 2 Ø

2 Ø 3 Ø

3 Ø 3 4

4 Ø Ø Ø

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Recognition• Recognition (or acceptance) is the process of

determining whether or not a given input should be accepted by a given machine.– Whether the string is in the language defined by the machine

• In terms of REs, it’s the process of determining whether or not a given input matches a particular regular expression.– Whether the string is in the language defined by the expression

• Traditionally, recognition is viewed as processing an input written on a tape consisting of cells containing elements from the alphabet.

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Recognition

• Recognition (or acceptance) is the process of determining whether or not a given input should be accepted by a given machine.

• Or… it’s the process of determining if as string is in the language we’re defining with the machine

• In terms of REs, it’s the process of determining whether or not a given input matches a particular regular expression.

• Traditionally, recognition is viewed as processing an input written on a tape consisting of cells containing elements from the alphabet.

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• FSA recognizes (accepts) strings of a regular language– baa!– baaa!– baaa!– …

• Tape metaphor: a rejected input

b!aba

q0

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Recognition

• Simply a process of starting in the start state• Examining the current input• Consulting the table• Going to a new state and updating the tape pointer.• Until you run out of tape.

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D-Recognize

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b a a a a !

q0

StateInput

b a !

0 1 Ø Ø

1 Ø 2 Ø

2 Ø 3 Ø

3 Ø 3 4

4 Ø Ø Ø

q3 q3 q4q1 q2 q3

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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 interpreter• The algorithm is universal for all unambiguous

languages.– To change the machine, you change the table.

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Key Points

• Crudely therefore… matching strings with regular expressions (ala Perl) is a matter of – translating the expression into a machine (table) and – passing the table to an interpreter

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Recognition as Search

• You can view this algorithm as a degenerate kind of state-space search.

• States are pairings of tape positions and state numbers.

• Operators are compiled into the table• Goal state is a pairing with the end of tape position

and a final accept state• Its degenerate because?

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Formal Languages

• Formal Languages are sets of strings composed of symbols from a finite set of symbols.

• Finite-state automate define formal languages (without having to enumerate all the strings in the language)

• Given a machine m (such as a particular FSA) L(m) means the formal language characterized by m.

– L(Sheeptalk FSA) = {baa!, baaa!, baaaa!, …} (an infinite set)

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Generative Formalisms

• The term Generative is based on the view that you can run the machine as a generator to get strings from the language.

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

language

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Three Views

• Three equivalent formal ways to look at what we’re up to (not including tables – and we’ll find more…)

Regular Expressions

Regular LanguagesFinite State Automata

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Determinism

• Let’s take another look at what is going on with d-recognize.

• In particular, let’s look at what it means to be deterministic here and see if we can relax that notion.

• How would our recognition algorithm change?

• What would it mean for the accepted language?

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Determinism and Non-Determinism

• Deterministic: There is at most one transition that can be taken given a current state and input symbol.

• Non-deterministic: There is a choice of several transitions that can be taken given a current state and input symbol. (The machine doesn’t specify how to make the choice.)

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Non-Deterministic FSAs for SheepTalk

q0 q4q1 q2 q3

b aa

a !

q0 q4q1 q2 q3

b a a !

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FSAs as Grammars for Natural Language

q2 q4 q5q0 q3q1 q6

the rev mr

dr

hon

pat l. robinson

ms

mrs

Can you use a regexpr to capture this too?

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Equivalence

• Non-deterministic machines can be converted to deterministic ones with a fairly simple construction (essentially building “set states” that are reached by following all possible states in parallel)

• That means that they have the same power; non-deterministic machines are not more powerful than deterministic ones

• It also means that one way to do recognition with a non-deterministic machine is to turn it into a deterministic one.

• Problems: translating gives us a not very intuitive machine, and this machine has LOTS of states

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Non-Deterministic Recognition

• In a ND FSA there exists at least one path directed through the machine by a string that is in the language defined by the machine that leads to an accept condition..

• But not all paths directed through the machine by an accept string lead to an accept state. It is OK for some paths to lead to a reject condition.

• In a ND FSA no path directed through the machine by a string outside the language leads to an accept condition.

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Non-Deterministic Recognition• So success in a non-deterministic recognition occurs

when a path is found through the machine that ends in an accept.

• However, being driven to a reject condition by an input does not imply it should be rejected.

• Failure occurs only when none of the possible paths lead to an accept state.

• This means that the problem of non-deterministic recognition can be thought of as a standard search problem.

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The Problem of Choice• Choice in non-deterministic models comes up again

and again in NLP.

Several Standard Solutions• Backup (search, this chapter)

– Save input/state of machine at choice points– If wrong choice, use this saved state to back up and try

another choice

• Lookahead– Look ahead in the input to help make a choice

• Parallelism– Look at all choices in parallel

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Backup

• After a wrong choice leads to a dead-end (either no input left in a non-accept state, or no legal transitions), return to a previous choice point to pursue another unexplored choice.

• Thus, at each choice point, the search process needs to remember the (unexplored) choices.

• Standard State Space Search.• State = (FSA node or machine state, tape-position)

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Example

b a a a ! \

q0 q1 q2 q2 q3 q4

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ND-Recognize Code

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ExampleAgenda:

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Example

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Key Points

• States in the search space are pairings of tape positions and states in the machine.

• By keeping track of as yet unexplored states, a recognizer can systematically explore all the paths through the machine given an input.

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Infinite Search

• If you’re not careful such searches can go into an infinite loop.

• How?

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Why Bother?

• Non-determinism doesn’t get us more formal power and it causes headaches so why bother?– More natural solutions– Machines based on construction are too big

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Compositional Machines

• Formal languages are just sets of strings

• Therefore, we can talk about various set operations (intersection, union, concatenation)

• This turns out to be a useful exercise

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Union

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Concatenation

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Negation

• Construct a machine M2 to accept all strings not accepted by machine M1 and reject all the strings accepted by M1 Invert all the accept and not accept

states in M1

• Does that work for non-deterministic machines?

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Intersection

• Accept a string that is in both of two specified languages

• An indirect construction… A^B = ~(~A or ~B)

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Why Bother?

• ‘FSAs can be useful tools for recognizing – and generating – subsets of natural language – But they cannot represent all NL phenomena (Center

Embedding: The mouse the cat ... chased died.)

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Summing Up• Regular expressions and FSAs can represent

subsets of natural language as well as regular languages– Both representations may be impossible for humans to

understand for any real subset of a language– But they are very easy to use for smaller subsets– They are extremely useful for certain subproblems (e.g.,

morphological analysis).

• Next time: Read Ch 3• For fun:

– Think of ways you might characterize features of your email using only regular expressions