9.3. Regular Languages

Transcription

1 9.3. REGULAR LANGUAGES Regulr Lnguges Properties of Regulr Lnguges. Recll tht regulr lnguge is the lnguge ssocited to regulr grmmr, i.e., grmmr G = (N, T, P, σ) in which every production is of the form: where A, B N, T. A or A B or A λ, Regulr lnguges over n lphet T hve the following properties (recll tht λ = empty string, αβ = conctention of α nd β, α n = α conctented with itself n times ): 1., {λ}, nd {} re regulr lnguges for ll T. 2. If L 1 nd L 2 re regulr lnguges over T the following lnguges lso re regulr: L 1 L 2 = {α α L 1 or α L 2 } L 1 L 2 = {αβ α L 1, β L 2 } L 1 = {α 1... α n α k L 1, n N}, T L 1 = {α T α / L 1 }, L 1 L 2 = {α α L 1 nd α L 2 }. We justify the ove clims out L 1 L 2, L 1 L 2 nd L 1 s follows. We lredy know how to comine two grmmrs (see 9.2.4) L 1 nd L 2 to otin L 1 L 2, L 1 L 2 nd L 1, the only prolem is tht the rules given in section do no hve the form of regulr grmmr, so we need to modify them slightly (we use the sme nottion s in section 9.2.4): 1. Union Rule: Insted of dding σ σ 1 nd σ σ 2, dd ll productions of the form σ RHS, where RHS is the right hnd side of some production (σ 1 RHS) P 1 or (σ 2 RHS) P Product Rule: Insted of dding σ σ 1 σ 2, use σ 1 s strting symol nd replce ech production (A ) P 1 with A σ 2 nd (A λ) P 1 with A σ Closure Rule: Insted of dding σ σ 1 σ nd σ λ, use σ 1 s strting symol, dd σ 1 λ, nd replce ech production (A ) P 1 with A σ 1 nd (A λ) P 1 with A σ 1.

2 9.3. REGULAR LANGUAGES Regulr Expressions. Regulr lnguges cn e chrcterized s lnguges defined y regulr expressions. Given n lphet T, regulr expression over T is defined recursively s follows: 1., λ, nd re regulr expressions for ll T. 2. If R nd S re regulr expressions over T the following expressions re lso regulr: (R), R + S, R S, R. In order to use fewer prentheses we ssign those opertions the following hierrchy (from do first to do lst):,, +. We my omit the dot: α β = αβ. Next we define recursively the lnguge ssocited to given regulr expression: L( ) =, L(λ) = {λ}, L() = {} for ech T, L(R + S) = L(R) L(S), L(R S) = L(R)L(S) L(R ) = L(R) (lnguge product), (lnguge closure). So, for instnce, the expression represents ll strings of the form n m with n 0, m > 0, ( + c) is the set of strings consisting of ny numer of s followed y or c, ( + ) is the set of strings over {, } thn strt with nd end with, etc. Another wy of chrcterizing regulr lnguges is s sets of strings recognized y finite-stte utomt, s we will see next. But first we need generliztion of the concept of finite-stte utomton Nondeterministic Finite-Stte Automt. A nondeterministic finite-stte utomton is generliztion of finite-stte utomton so tht t ech stte there might e severl possile choices for the next stte insted of just one. Formlly nondeterministic finite-stte utomton consists of 1. A finite set of input symols I. 2. A finite set of sttes S. 3. A next-stte function f : S I P(S). 4. A suset A of S of ccepting sttes.

3 5. An initil stte σ S REGULAR LANGUAGES 141 We represent the utomton A = (I, S, f, A, σ). We sy tht nondeterministic finite-stte utomton ccepts given string of input symols if in its trnsition digrm there is pth from the strting stte to n ccepting stte with its edges leled y the symols of the given string. A pth (which we cn express s sequence of sttes) whose edges re leled with the symols of string is sid to represent the given string. Exmple: Consider the nondeterministic finite-stte utomton defined y the following trnsition digrm: strt σ C F This utomton ccepts precisely the strings of the form n m, n 0, m > 0. For instnce the string is represented y the pth (σ, σ, σ, C, C, F ). Since tht pth ends in n ccepting stte, the string is ccepted y the utomton. Next we will see tht there is precise reltion etween regulr grmmrs nd nondeterministic finite-stte utomt. Regulr grmmr ssocited to nondeterministic finite-stte utomton. Let A e non-deterministic finite-stte utomton given s trnsition digrm. Let σ e the initil stte. Let T e the set of inputs symols nd let N e the set of sttes. Let P e the set of productions S xs if there is n edge leled x from S to S nd S λ if S is n ccepting stte. Let G e the regulr grmmr G = (N, T, P, σ). Then the set of strings ccepted y A is precisely L(G). Exmple: For the nondeterministic utomton defined ove the corresponding grmmr will e:

4 9.3. REGULAR LANGUAGES 142 T = {, }, N = {σ, C, F }, with the productions σ σ, σ C, C C, C F, F λ. The string cn e produced like this: σ σ σ C C F. Nondeterministic finite-stte utomton ssocited to given regulr grmmr. Let G = (N, T, P, σ) e regulr grmmr. Let I = T S = N {F }, where F / N T f(s, x) = {S S xs P } {F S x P } A = {F } {S S λ P }. Then the nondeterministic finite-stte utomton A = (I, S, f, A, σ) ccepts precisely the strings in L(G) Reltionships Between Regulr Lnguges nd Automt. In the previous section we sw tht regulr lnguges coincide with the lnguges ccepted y nondeterministic finite-stte utomt. Here we will see tht the term nondeterministic cn e dropped, so tht regulr lnguges re precisely those ccepted y (deterministic) finite-stte utomt. The ide is to show tht given ny nondeterministic finite-stte utomt it is possile to construct n equivlent deterministic finite-stte utomt ccepting exctly the sme set of strings. The min result is the following: Let A = (I, S, f, A, σ) e nondeterministic finite-stte utomton. Then A is equivlent to the finite-stte utomton A = (I, S, f, A, σ ), where 1. S = P(S). 2. I = I. 3. σ = {σ}. 4. A = {X S X A }. 5. f (X, x) = f(s, x), f (, x) =. S X

5 9.3. REGULAR LANGUAGES 143 Exmple: Find (deterministic) finite-stte utomton A equivlent to the following nondeterministic finite-stte utomton A: strt σ C F Answer: The set of input symols is the sme s tht of the given utomton: I = I = {, }. The set of sttes is the set of susets of S = {σ, C, F }, i.e.: S = {, {σ}, {C}, {F }, {σ, C}, {σ, F }, {C, F }, {σ, C, F }}. The strting stte is {σ}. The ccepting sttes of A re the elements of S contining some ccepting stte of A: A = {{F }, {σ, F }, {C, F }, {σ, C, F }}. Then for ech element X of S we drw n edge leled x from X to f(s, x) (nd from to ): S X strt {σ} {C} {σ, C, F } {σ, F } {σ, C} {F } {C, F } We notice tht some sttes re unrechle from the strting stte. After removing the unrechle sttes we get the following simplified version of the finite-stte utomton: strt {σ} {C} {C, F }

6 9.3. REGULAR LANGUAGES 144 So, once proved tht every nondeterministic finite-stte utomton is equivlent to some deterministic finite-stte utomton, we otin the min result of this section: A lnguge L is regulr if nd only if there exists finite-stte utomton tht ccepts precisely the strings in L.