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**Regular Languages Regular Expressions Finite-State Automata**

Torbjörn Lager, Stockholm University Regular Languages Regular Expressions Finite-State Automata Välkommen. Fokus på verktyg? Nej! Inte så mycket morfologi

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**Languages Sets of strings Example: {“ac” “abc” “abbc” “abbbc” ...}**

FST - Torbjörn Lager, UU

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**Finite-State Automata**

Directed graphs consisting of states, and arcs labelled with symbols. For example: 2 1 a c b FST - Torbjörn Lager, UU

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**Regular expressions Example: [a b* c] Note:**

Atomic symbols (e.g. ‘a’) denote languages (e.g. {“a”}) Operations (here: concatenation and kleene-star) are operations over languages [a b* c] FST - Torbjörn Lager, UU

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**Regular expressions, regular languages and automata**

denote compile into Regular languages Finite-state automata generate accept FST - Torbjörn Lager, UU

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**Regular expressions, regular languages and automata**

[a b* c] denotes compiles into b {“ac” “abc” “abbc” ...} a c 1 2 generates accepts FST - Torbjörn Lager, UU

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**Regular expression operators**

Concatenation A B Union A | B Iteration (Kleene-star) A* Difference A - B Intersection A & B Grouping of expressions [A] FST - Torbjörn Lager, UU

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**Examples a b {“ab”} a [b|c] {“ab” “ac”} a* [b|c]* {“” “a” “ab” ... }**

[a|b] & [b|c] {“b”} [a|b] - b {“a”} [a|b] - [b|a] {} FST - Torbjörn Lager, UU

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**Special symbols ? The any symbol ?* The universal language**

[] The empty-string language 0 (or “”) The empty string (epsilon) FST - Torbjörn Lager, UU

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**Regular expression operators**

Optionality (A) Kleene-plus A+ Complement ~A Containment $A Restriction A => B _ C FST - Torbjörn Lager, UU

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**Examples a (b) c {“ac” “abc”} a b+ c {“abc” “abbc” “abbbc” ...}**

~[a b c] {“” ... “a” ... “ab” ... “abca” ..} $[a|b] {“a” ... “abba” ... “abcd” ...} b => a _ c {“” ... “a” ... “ccc” ... “abc” ..} FST - Torbjörn Lager, UU

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**Component technologies in FST**

Word lists and lexica Tokenisers Morphological analysers Part-of-speech taggers Parsers detta är sånt som vi ska göra under laborationer FST - Torbjörn Lager, UU

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**Applications of FST Named-entity recognition Information extraction**

Corpus linguistics Spelling- and grammar checking Speech processing applications FST - Torbjörn Lager, UU

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**The Xerox Finite-State Tool**

Compiles extended regular expressions into finite-state machines (automata and transducers) Allows the user to display, examine and modify the machines FST - Torbjörn Lager, UU

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**Non-deterministic FSAs**

At least one state has more than one transition leading from it labelled with the same symbol b a b 1 2 FST - Torbjörn Lager, UU

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**Determinization of FSAs**

Any non-deterministic FSA can be transformed into an equivalent deterministic FSA. Example: Determinize for efficiency! b b a b a b 1 2 1 2 FST - Torbjörn Lager, UU

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Minimization of FSAs Any (deterministic) FSA can be transformed into an equivalent FSA that has a minimal number of states. Minimize for space! Storleken på automaten växer i värsta fall linjärt med storleken på språket FST - Torbjörn Lager, UU

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**Representing word lists**

Think of a word list as a regular language Use the calculus of regular expressions to query and update the wordlist Determinize for speed! Minimize for space! FST - Torbjörn Lager, UU

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**Various equivalences (A) = A|[] A+ = A A* A+ = A* - [] A - B = A & ~B**

De två sista är De-morgans lagar! FST - Torbjörn Lager, UU

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**Various equivalences A - A = ~[?*] A | ~[?*] = A A [] = A**

FST - Torbjörn Lager, UU

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**Important theoretical results**

Kleene’s theorem (concerning FSAs) Closure properties of regular languages and regular relations Decidability FST - Torbjörn Lager, UU

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**FSAs and regular expressions**

Kleene’s theorem: Any language recognised by an FSA is denoted by a regular expression and any language denoted by a regular expression can be recognised by a FSA. FST - Torbjörn Lager, UU

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**Nondeterministic FSA with epsilon transitions**

Regex to FSA to regex Nondeterministic FSA Deterministic FSA Nondeterministic FSA with epsilon transitions Regular expressions Picture adapted from Hopcroft & Ullman 1979 FST - Torbjörn Lager, UU

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**From regular expressions to finite-state automata**

The only really necessary operators: Disjunction Concatenation Iteration Sidenote: Compare regular grammars: A --> x B A --> x (where A and B are nonterminals, and where x is a sequence of terminals) FST - Torbjörn Lager, UU

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**Closure properties of regular languages**

A set is said to be closed under an operation iff applying the operation to members of the set will never take us outside the set Example: if A and B are regular languages, then [A|B] is always regular. Therefore regular languages are closed under union. FST - Torbjörn Lager, UU

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**Decidability Given one automaton A: Given two automata A1 and A2:**

Is the string S a string in L(A) ? Does L(A) contain any strings at all ? Is L(A) equivalent to ?* ? Given two automata A1 and A2: Is L(A1) a subset of L(A2) ? Are L(A1) and L(A2) equivalent ? Do L(A1) and L(A2) overlap ? FST - Torbjörn Lager, UU

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FST - Torbjörn Lager, UU

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Language and Automata Theory

Language and Automata Theory

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