Basic Regular Expressions. Introduction. Introduction to Computability. Theory. Motivation. Lecture4: Regular Expressions

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1 Introducton to Computablty Theory Lecture: egular Expressons Prof Amos Israel Motvaton If one wants to descrbe a regular language, La, she can use the a DFA, Dor an NFA N, such L ( D = La that that Ths s not always very convenent Consder for example the regular expresson descrbng the language of bnary strngs contanng a sngle 3 Introducton egular languages are defned and descrbed by use of fnte automata In ths lecture, we ntroduce egular Expressons as an euvalent way, yet more eleganto descrbe regular languages Basc egular Expressons A egular Expresson(E n short s a strng of symbols that descrbes a regular Language Let Σ be an alphabet For each σ Σ he symbol σ s an E representng the set { σ } ε { ε } The symbol s an E representng the set (The set contanng the empty strng The symbol φ s an E representng the empty set

2 Inductve Constructon Let and be two regular expressons representng languages L and L, resp The strng ( s a regular expresson representng the set L L ( L The strng s a regular expresson representng the set L o L The strng ( s a regular expresson representng the set L 5 Inductve Constructon - emarks Ths nductve defnton also dctates the way we wll prove theorems: For any theorem T Stage :Prove Tcorrect for all base cases Stage : Assume Ts correct for and ( ( Prove correctness for,, and ( 7 Inductve Constructon - emarks Note that n the nductve part of the defnton larger E-s are defned by smaller ones Ths ensures that the defnton s not crcular 6 Some Useful Notaton Let be a regular expresson: The strng + represents, and t also holds that + { ε } = The strng k represents 3 The strng represents σ σ, The Language represented by s denoted by L ( k tmes Σ {, σ k } 8

3 Precedence ules The star ( operaton has the hghest precedence The concatenaton ( o operaton s second on the preference order The unon ( operaton s the least preferred Parentheses can be omtted usng these rules 9 Examples Σ Σ - all words ng and endng wth the same letter ( ε -all strngs of forms,,, and,,, φ = -Aset concatenated wth the empty set yelds the empty set φ =φ φ = { ε} - Examples Σ Σ ( w w contans str as a substrng Σ Σ str ( { w w contans a sngle } { w w has at least a sngle } { } + w every n w s followed by at least a sngle ( ΣΣ { w w s of even length } Euvalence Wth Fnte Automata egular expressons and fnte automata are euvalent n ther descrptve power Ths fact s expressed n the followng Theorem: Theorem A set s regular f and only f t can be descrbed by a regular expresson The proof s by two Lemmata,(Lemmas:

4 Lemma -> If a language Lcan be descrbed by regular expresson then L s regular,, 3 Inducton Bass For any σ Σ he expresson σ descrbes the set { σ } σ, recognzed by: The set represented by the expresson s recognzed by: ε 3 The set represented by the expresson φ s recognzed by: 5 Proofs Usng Inductve Defnton The proof follows the nductve defnton of E-s as follows: Stage : Prove correctness for all base cases Stage : Assume correctness for and, and show ts correctness for, and ( ( ( The Inducton Step Now, we assume that and represent two regular sets and clam that, o and represent the correspondng regular sets The proof for ths clam s straght forward usng the constructons gven n the proof for the closure of the three regular operatons 6

5 Examples Show that the followng regular expressons represent regular languages: ( ab a ( a b aba To be demonstrated on the Blackboard 7 Proof Stages The proof follows the followng stages: Defne Generalzed Nondetermnstc Fnte Automaton (GNFA n short Show how to convert any DFA to an euvalent GNFA 3 Show an algorthm to convert any GNFA to an euvalent GNFA wth states, Convert a -state GNFA to an euvalent E 9 Lemma <- If a language Ls regular then Lcan be descrbed by some regular expresson 8 Propertes of a Generalzed NFA A GNFA s a fnte automaton n whch each transton s labeled wth a regular expresson over the alphabet Σ A sngle ntal statewth all possble outgong transtons and no ncomng trans 3 A sngle fnal state wthout outgong trans, A sngle transton between every two states, ncludng self loops

6 Example of a Generalzed NFA ab aa a ab ba b φ ab b ( aa accept Example of a GNFA Computaton Consder abba or bb or abbbaaaaabbbbb ab aa a ab ba b φ ab b ( aa accept 3 A Computaton of a GNFA A computatonof a GNFA s smlar to a computaton of an NFA except: In each step, a GNFA consumes a block of symbolsthat matches the E on the transton used by the NFA Convertng a DFA (or NFA to a GNFA Converson s done by a very smple process: Add a new state wth an ε -transton from the new state to the old state Add a new acceptng state wth ε -transton from every old acceptng state to the new acceptng state

7 Convertng a DFA to a GNFA (Cont eplace any transton wth multple labels by a sngle transton labeled wth the unon of all labels 5 Add any mssng transton, ncludng self transtons; label the added transton by φ 5 Stage : Convert Dto a GNFA Add new states c a, b accept 7 Stage : Convert Dto a GNFA Start wth D c a, b 6 Stage : Convert Dto a GNFA Make the ntal state and accept the fnal state c a, ε ε b accept 8

8 Stage : Convert Dto a GNFA 3 eplace mult label transtons by ther unon c a ε ε b accept 9 ppng a state from a GNFA The fnal element needed for the proof s a procedure n whch for any GFN G, any state of G, not ncludng and accept, can be rpped off G, whle preservng L G ( Ths s demonstrated n the next slde by consderng a general state, denoted by rp, and an arbtrary par of states, and j, as demonstrated n the next slde: 3 Stage : Convert Dto a GNFA Add all mssng transtons and label them φ c a ε ε b accept 3 emovng a state from a GNFA Before ppng After ppng ( ( ( 3 j 3 rp 3 Note:Ths should be done for every par of outgong and ncomng outgong rp j

9 Ellaboraton ( ( 3 Consder the E, representng all strngs that enable transton from va rp to j What we want to do s to augment the egular expresson of transton (, namely, j, so These strngs can pass through ( Ths s, j done by settng t to rp ( ( ( Elaboraton Assume the followng stuaton: In order to rp, all pars rp of ncomng and outgong transtons should be consdered n the way showed on the prevous slde namely consder ( t ( 5( ( 5( 3 ( t t 3 t 3 5 one after the other After that can be rpped whle preservng L, ( G t rp rp t t 5 35 Ellaboraton Note: In order to acheve an euvalent GNFA n whch s dsconnected, rp rp ths procedure should be carred out separately, for every par of transtons of the form (, rp and, Then can be removed, as rp demonstrated on the next slde: ( rp j 3 3 In Partcular eplace wth ( 3 a 3 rp 36 j j

10 A (half? Formal Proof of Lemma<- The frst step s to formally defne a GNFA Each transton should be labeled wth an E Defne the transton functon as follows: δ : ( { } Q { } ( Σ Q accept E where E Σ denotes all regular expressons over Σ Note: The def of δ s dfferent then for NFA 37 GNFA A Formal Defnton A Generalzed Fnte Automaton s a 5-tupple Q, Σ, δ,, accept where: Q s a fnte set called the states Σ s a fnte set called the alphabet 3 δ : ( Q { accept } ( Q { } E Σ s the transton functon ( Q s the state, and 5 accept Q s the accept state 39 Changes n δ Defnton Note: The defnton of δ as: ( { } { } ( Σ δ : Q accept Q E s dfferent than the orgnal defntons (For DFA and NFA In ths defnton we rely on the fact that every states (except and accept are connected n both drectons 38 GNFA Defnng a Computaton A GNFA accepts a strng Σ f w = w w and there exsts a seuence of states accept, satsfyng: For each, k, w L (, where =δ (,, or n other words, s the expresson on the arrow from to + w w k

11 Procedure CONVET Procedure CONVET takes as nput a GNFA G wth k states If k = then these states must be and accept, and the algorthm returns δ (, accept If k > he algorthm converts Gto an euvalent G wth k states by use of the rppng procedure descrbed before ecap In ths lecture we: Motvated and defned regular expressons as a more concse and elegant method to represent regular Languages Proved that FA-s (Determnstc as well as Nondetermnstc and E-s s dentcal by: Defned GNFA s Showed how to convert a DFA to a GNFA 3 Showed an algorthm to converted a GNFA wth K states to an euvalent GNFA wth K states 3 Procedure CONVET ( G Convert k Q G ; If ( k = return δ (, accept ; 3 rp GetandomState ( Q G ; Q ' Q G rp ; 5 For any Q ' accept and any Q ' δ '(, j ( ( ( 3 for = δ (, rp = δ ( rp, rp 3 = δ ( r rp, = δ (, j return G ' = ( Q ', Σ, δ ',, accept ;

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