Regular Expressions andFinite State Automata

Gul Agha2104 SChttp://www.cs.uiuc.edu/class/fa05/cs421/current/

Based in part on slides by Mattox Beckman, as updated by Vikram Adve and Elsa Gunter

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Language Syntax

Syntax is the description of which strings of symbols are meaningful expressions in a languageIt takes more than syntax to understand a language; need meaning (semantics) tooSyntax is the entry point

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Elements of Syntax

Character set – previously always ASCII, now often 64 character setsKeywords – usually reservedSpecial constants – cannot be assigned toIdentifiers – can be assigned toOperator symbolsDelimiters (parenthesis, braces, brackets,)Blanks (aka white space)

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Elements of Syntax

ExpressionsType expressionsDeclarations (in functional languages)Statements (in imperative languages)Subprograms

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Elements of Syntax

ModulesInterfacesClasses (for object-oriented languages)

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Formal Language Descriptions

Regular expressions, regular grammars, finite state automataContext-free grammars, BNF grammars, syntax diagrams

Whole family more of grammars and automata – covered in automata theory

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Grammars

Grammars are formal descriptions of which strings over a given character set are in a particular languageLanguage designers write grammarLanguage implementers use grammar to know what programs to acceptLanguage users use grammar to know how to write legitimate programs

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Regular Expressions

Start with a given character set –a, b, c…

Each character is a regular expressionThe regular expression b represents the set { b }

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Regular Expressions

If x and y are regular expressions, then xy is a regular expression

xy represents the set of all strings made concatenating a string in x with a string in y

If x and y are regular expressions, then x | yis a regular expression

x | y represents the set of strings in x or in y (or both)

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Regular Expressions

If x is a regular expression, then so is ( x )( x ) represents the same thing as x

If x is a regular expression, then so is x*x* represents strings made by concatenating zero or more strings from x

εrepresents the empty set

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Example Regular Expressions

(0|1)*1The set of all strings of 0’s and 1’s ending in 1, {1, 01, 11,…}

a*b(a*)The set of all strings of a’s and b’s with exactly one b

((01) |(10))*You tell me

Regular expressions (equivalently, regular grammars) important for lexing, breaking strings into recognized words

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Example: Lexing

Regular expressions good for describing lexemes (words) in a programming language

Identifier = (a | b | … | z | A | B | … | Z) (a | b |… | z | A | B | … | Z | 0 | 1 | … | 9)*Digit = (0 | 1 | … | 9)Number = (1 | … | 9)(0 | … | 9) | ~ (1 | … | 9)(0 | … | 9) Keywords: if = if, while = while,…

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Implementing Regular Expressions

Regular expressions can generate strings in a languageNot so good for recognizing when a string is in languageRegular expressions: which option to choose, how many repetitions to makeAnswer: finite state automata

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Finite State Automata

A finite state automata over an alphabet is:a directed graphedges are labeled with elements of alphabet,

or empty stringsome nodes (or states), marked as finalone node marked as start state

Syntax of FSA

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Example FSA

0 11

0Start State

Final State

Final State

1

0

10

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Deterministic FSA’s

If FSA has for every state exactly one edge for each letter in alphabet then FSA is deterministic

No edge labeled with ε

In general FSA in non-deterministic.NFSA also allows edges labeled by ε

Deterministic FSA special kind of non-deterministic FSA

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DFSA Language Recognition

Think of a DFSA as a board game; DFSA is boardYou have string as a deck of cards; one letter on each cardStart by placing a disc on the start state

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DFSA Language Recognition

Move the disc from one state to next along the edge labeled the same as top card in deck; discard top card

When you run out of cards:if you are in final state, you win; string is in languageif you are not in a final state, you lose; string is not in language

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DFSA Language Recognition -Summary

Given a string (over the alphabet)Start at start stateMove over edge labeled with first letter to new stateRemove first letter from stringRepeat until string goneIf end in final state then string in language

Semantics of FSA

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Example DFSA

Regular expression: (0 | 1)* 1Deterministic FSA

0 1

1

0

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Example DFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0 1

1

0

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Example DFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0 1

1

0

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Example DFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0 1

1

0

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Example DFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0 1

1

0

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Example DFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0 1

1

0

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Example DFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0 1

1

0

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Example DFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0 1

1

0

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NFSA generalize DFSA in two ways:Include edges labeled by ε

Allows process to non-deterministically change state

Each state can have zero, one or more edges labeled by each letter

Given a letter, non-deterministically choose an edge to use

Non-deterministic FSA’s

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NFSA Language Recognition

Play the same game as with DFSAFree move: move across an edge with empty string label without discarding cardWhen you run out of letters, if you are in final state, you win; string is in languageYou can take one or more moves back and try againIf have tried all possible paths without success, then you lose; string not in language

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FSA Language RecognitionMove the disc from one state to next if edge between labeled the same as top card in deck; discard top card Free move: move across an edge with empty string label without discarding cardWhen you run out of cards, if you are in final state, you win; string is in languageYou can take a move back and try anotherIf you have cards left and you have tried all possible edges without success, then you lose; string not in language

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Example NFSA

Regular expression: (0 | 1)* 1Non-deterministic FSA

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1Guess

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1Backtrack

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1Guess again

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1Guess

0

1

1

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Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1Backtrack

Example NFSA

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1Guess again

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1

0

1

1

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Example NFSA

Regular expression: (0 | 1)* 1Accepts string 0 1 1 0 1Guess (Hurray!!)

0

1

1

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Rule Based Execution

SearchWhen stuck backtrack to last point with choices remainingExecuting the NFSA in last example was example of rule based executionFSA’s are rule-based programs; transitions between states (labeled edges) are rules; set of all FSA’s is programming language