Automata and Formal Languages - CM0081 Algebraic Laws for Regular Expressions

Transcription

1 Automata and Formal Languages - CM0081 Algebraic Laws for Regular Expressions Andrés Sicard-Ramírez Universidad EAFIT Semester

2 Algebraic Laws for Regular Expressions Definition (Equivalence of regular expressions) Two regular expressions with variables are equivalent if whatever languages we substitute for the variables, the results of the two expressions are the same language. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 2/33

3 Algebraic Laws for Regular Expressions Definition (Equivalence of regular expressions) Two regular expressions with variables are equivalent if whatever languages we substitute for the variables, the results of the two expressions are the same language. Notation Let L, M and N be regular expression variables. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 3/33

4 Algebraic Laws for Regular Expressions Definition (Equivalence of regular expressions) Two regular expressions with variables are equivalent if whatever languages we substitute for the variables, the results of the two expressions are the same language. Notation Let L, M and N be regular expression variables. Sugar syntax L + def = LL, L? def = ε + L. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 4/33

5 Algebraic Laws for Regular Expressions Some laws for union (L + M) + N = L + (M + N) L + = + L = L L + M = M + L L + L = L (associativity) (identity) (commutativity) (idempotence) Remark: There is no inverse for union. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 5/33

6 Algebraic Laws for Regular Expressions Some laws for concatenation (LM)N = L(M N) Lε = εl = L LM M L L = L = (associativity) (identity) (non-commutativity) ( is the annihilator for concatenation) Remark: There is no inverse for concatenation. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 6/33

7 Algebraic Laws for Regular Expressions Some laws for union and concatenation L(M + N) = LM + LN (L + M)N = LN + LM (distributive) (distributive) Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 7/33

8 Algebraic Laws for Regular Expressions Some laws for closure (L ) = L = ε ε = ε (ε + L) = L L = L + + ε (idempotence) Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 8/33

9 Simplification of Regular Expressions Example 0 + (ε + 1)(ε + 1) 0 = 0 + (ε + 1)1 0 ((ε + L) = L ) = 0 + (ε )0 (distributive) = 0 + ( )0 (identity) = 0 + ( )0 (def. L + ) = (equivalence) = 1 0 (equivalence) Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 9/33

10 Let E and F be two regular expressions with the same set of variables {L 1,, L n }. To test if E = F : 1. Convert E and F to concrete regular expressions C and D, replacing each L i by a different symbol a i, for i = 1, 2,, n. 2. Test whether L(C) = L(D). If so, then E = F, and if not E F. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 10/33

11 Let E and F be two regular expressions with the same set of variables {L 1,, L n }. To test if E = F : 1. Convert E and F to concrete regular expressions C and D, replacing each L i by a different symbol a i, for i = 1, 2,, n. 2. Test whether L(C) = L(D). If so, then E = F, and if not E F. Observation We are proving by example! Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 11/33

12 Example Prove of disprove that L = L L. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 12/33

13 Example Prove of disprove that L = L L. 1. We replace the variable L by the concrete regular expression a. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 13/33

14 Example Prove of disprove that L = L L. 1. We replace the variable L by the concrete regular expression a. 2. a a a. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 14/33

15 Example Prove of disprove that L = L L. 1. We replace the variable L by the concrete regular expression a. 2. a a a. 3. Because L(a ) = L(a a ), we conclude that L = L L. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 15/33

16 Example Prove of disprove that L + ML = (L + M)L. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 16/33

17 Example Prove of disprove that L + ML = (L + M)L. 1. We replace the variables L and M by the concrete regular expressions a and b respectively. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 17/33

18 Example Prove of disprove that L + ML = (L + M)L. 1. We replace the variables L and M by the concrete regular expressions a and b respectively. 2. a + ba (a + b)a. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 18/33

19 Example Prove of disprove that L + ML = (L + M)L. 1. We replace the variables L and M by the concrete regular expressions a and b respectively. 2. a + ba (a + b)a. 3. aa L(a + ba) and aa L((a + b)a) L(a + ba) L((a + b)a) L + ML (L + M)L Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 19/33

20 Example (Hopcroft, Motwani and Ullman [2007], Exercise d) Prove of disprove that (L + M) M = (L M). Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 20/33

21 Example (Hopcroft, Motwani and Ullman [2007], Exercise d) Prove of disprove that (L + M) M = (L M). 1. We replace the variables L and M by the concrete regular expressions a and b respectively. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 21/33

22 Example (Hopcroft, Motwani and Ullman [2007], Exercise d) Prove of disprove that (L + M) M = (L M). 1. We replace the variables L and M by the concrete regular expressions a and b respectively. 2. (a + b) b (a b). Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 22/33

23 Example (Hopcroft, Motwani and Ullman [2007], Exercise d) Prove of disprove that (L + M) M = (L M). 1. We replace the variables L and M by the concrete regular expressions a and b respectively. 2. (a + b) b (a b). 3. Since ε (a + b) b and ε (a b) (L + M) M (L M) Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 23/33

24 Example (counter-example) Extensions of the previous test beyond regular expressions may fail. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 24/33

25 Example (counter-example) Extensions of the previous test beyond regular expressions may fail. 1. Add to the regular expression operators. 2. Test if L M N = L M. 3. From L = a, M = b and N = c, we should conclude that L M N = L M, that is, the property is true. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 25/33

26 Example (counter-example) Extensions of the previous test beyond regular expressions may fail. 1. Add to the regular expression operators. 2. Test if L M N = L M. 3. From L = a, M = b and N = c, we should conclude that L M N = L M, that is, the property is true. 4. The property is false. For example, if L = M = a and N = then L M N L M. 5. Therefore, the test is not valid! Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 26/33

27 Derivative of a Regular Expression Definition (Derivative of a language by a symbol) Let L Σ be a language and a Σ a symbol. We define a\l (derivative of L by a) by a\l = {x Σ ax L}. Brzozowski, J. A. (1964). Derivates of Regular Expressions. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 27/33

28 Derivative of a Regular Expression Definition (Derivative of a language by a symbol) Let L Σ be a language and a Σ a symbol. We define a\l (derivative of L by a) by Examples a\l = {x Σ ax L}. a\{abab, abba} = {bab, bba}, a\l(ab ) = L(b ), b\l(ab ) =. Brzozowski, J. A. (1964). Derivates of Regular Expressions. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 28/33

29 Derivative of a Regular Expression Definition (Derivative of a regular expression by a symbol) We define recursively a\e (derivative of the regular expression E by a Σ) by a\ =, a\ε =, a\a = ε, a\b = for a b, Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 29/33

30 Derivative of a Regular Expression Definition (Derivative of a regular expression by a symbol) We define recursively a\e (derivative of the regular expression E by a Σ) by a\ =, a\ε =, a\a = ε, a\b = for a b, a\(e + F ) = a\e + a\f, (a\e)f + a\f, a\(ef ) = { (a\e)f, a\(e ) = (a\e)e. if ε L(E), otherwise, Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 30/33

31 Derivative of a Regular Expression Definition (Derivative of a regular expression by a string) We define recursively w\e (derivative of the regular expression E by w Σ ) by ε\e = E, ax\e = a\(x\e). Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 31/33

32 Derivative of a Regular Expression Definition (Derivative of a regular expression by a string) We define recursively w\e (derivative of the regular expression E by w Σ ) by ε\e = E, ax\e = a\(x\e). Theorem (Brzozowski s Theorem 4.2) w L(E) ε L(w\E). Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 32/33

33 References Brzozowski, J. A. (1964). Derivates of Regular Expressions. Journal of the ACM 11.4, pp Hopcroft, J. E., Motwani, R. and Ullman, J. D. (2007). Introduction to Automata theory, Languages, and Computation. 3rd ed. Pearson Education. Automata and Formal Languages - CM0081. Algebraic Laws for Regular Expressions 33/33