Closure under ?. 1. Page 2. Proposition 4. Regular Languages are closed under intersection i.e.
multiplication? subtraction? division? Closure of Regular Languages. Theorem 4.1 If L1 and L2 are regular languages then. L1. ? L2. L1. ?L2. L1L2. ¯. L1.
11 août 2000 FALSE. Let L1 = ?? and let L2 be any nonregular language over ?. (3) If L1L2 is regular and L1 is finite then ...
I'll fix them immediately. Thm. 4.1: If L1 and L2 are regular languages then so are L1 ?L2
If L? is regular and L2 is regular L1 must be regular. (c) If L is regular
Closure under ?. 1. Page 2. Proposition 4. Regular Languages are closed under intersection i.e.
multiplication? subtraction? division? Closure of Regular Languages. Theorem 4.1 If L1 and L2 are regular languages then. L1 ? L2. L1 ?L2. L1L2. ¯. L1.
Theorem 2.3.1 If L1 and L2 are regular languages then. L1. L2. L1L2. L*. 1. L1. L1. L2 are regular languages. Proof sketch. Union M1 = K1; ; 1; s1;
If L1 and L2 are regular languages so are L1 ? L2
L1 L2 = L1 ? L2 and regular languages are closed under intersection and complemen- tation. (B) If L ? ?? and L is finite
Thm 4 4: If L1 and L2 are regular languages then L1=L2 is regu-lar: The family of regular languages is closed under right quotient with a regular language Proof: 1 Assume that L1 and L2 are regular and let DFA M = (Q;?;–;q0;F) accept L1 2 We construct DFA Md = (Q;?;q0;Fc) as follows (a) For each qi 2 Q determine if there is a y 2
L1? L2is regular since regular languages are closed under complement ? L1? L2 is regular Proof via finite automata construction Let M? M?? and M be the formal definition of the finite automata that recognizes L1 L2 and L1? L2respectively M?=(Q? ? ?? q0? F?) M??=(Q???????? q0?? F??)
L1 and L2 are regular languages)9reg expr r1 and r2 s t L1 =L(r1)andL2=L(r2) r1 + r2 is r e denoting L1 [L2)closed under union r1r2 is r e denoting L1L2)closed under concatenation r 1 is r e denoting L 1)closed under star-closure 3
L1 and L2 are regular languages )9reg expr r andr s t L1 =L(r1) and L2=L(r2) r+r 2 is r e denoting L1[L2 )closed under union r1r2 is r e denoting L1L2 )closed under concatenation is r e denoting L 1 )closed under star-closurecomplementation:L1 is reg lang )9DFA M s t L1 = L(M)Construct M' s t
By de nition of regular languages we can say: If L1 and L2 are two regular languages then L1[L2 L1L2 L1 are regular 2 How about L1L2 L1 L2 ? also regular 3 Regular languages are closed under union concata-nation Kleene star set intersection set di erence etc 4 Given two FAs M1 and M2 can we construct new FAs to accept the the
If L and M are regular languages then so is L – M = strings in L but not M Proof: Let A and B be DFA’s whose languages are L and M respectively Construct C the product automaton of A and B Make the final states of C be the pairs where A-state is final but B-state is not