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## Language Recognition (12.4) Longin Jan Latecki Temple University

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**Language Recognition (12.4)**Longin Jan Latecki Temple University Based on slides by Costas Busch from the course http://www.cs.rpi.edu/courses/spring05/modcomp/and …**Three Equivalent Representations**Regular expressions Each can describe the others Regular languages Finite automata • Kleene’s Theorem: • For every regular expression, there is a deterministic finite-state automaton that defines the same language, and vice versa.**EXAMPLE 1**• Consider the language { ambn| m, n N}, which is represented by the regular expression a*b*. • A regular grammar for this language can be written as follows: • S | aS | B • B b | bB.**NFAs Regular grammarsThus, the language recognized by**FSA is a regular language • Every NFA can be converted into a corresponding regular grammar and vice versa. • Each symbol A of the grammar is associated with a non-terminal node of the NFA sA, in particular, start symbol • S is associated with the start state sS. • Every transition is associated with a grammar production: • T(sA,a) = sB A aB. • Every production B is associated with final state sB.**Equivalent FSA and regular grammar, Ex. 4, p. 772.**G=(V,T,S,P) V={S, A, B, 0, 1} with S=s0, A=s1, and B=s2, T={0,1}, and productions are S 0A | 1B | 1 | λ A 0A | 1B | 1 B 0A | 1B | 1 | λ**Kleene’sTheorem**Languages Generated by Regular Expressions LanguagesRecognizedby FSA**We will show:**Languages Generated by Regular Expressions LanguagesRecognizedby FSA Languages Generated by Regular Expressions LanguagesRecognizedby FSA**For any regular expression**the language is recognized by FSA (= is a regular language) Proof by induction on the size of Proof - Part 1 Languages Generated by Regular Expressions LanguagesRecognizedby FSA**NFAs**regular languages Induction Basis • Primitive Regular Expressions:**Inductive Hypothesis**• Assume • for regular expressions and • that • and are regular languages**Inductive Step**• We will prove: Are regular Languages**We need to show:**Regular languages are closed under: Union Concatenation Star By inductive hypothesis we know: and are regular languages This fact is illustrated in Fig. 2 on p. 821.**Therefore:**Are regular languages And trivially: is a regular language**For any regular language there is**a regular expression with Proof - Part 2 Languages Generated by Regular Expressions LanguagesRecognizedby FSA Proof by construction of regular expression**Since is regular take the**NFA that accepts it Single final state**From construct the equivalent**Generalized Transition Graph in which transition labels are regular expressions Example:**In General**• Removing states:**The final transition graph:**The resulting regular expression:**Three Equivalent Representations**Regular expressions Each can describe the others Regular languages Finite automata