ray12 Posted March 16, 2010 Share Posted March 16, 2010 prove the formula: (111*)* = (11 + 111)* thank you for helping Link to comment Share on other sites More sharing options...
vordhosbn Posted March 16, 2010 Share Posted March 16, 2010 [math](111^1)^1 \neq (11 + 111)^1[/math] Link to comment Share on other sites More sharing options...
ray12 Posted March 16, 2010 Author Share Posted March 16, 2010 this is kleene star it can not be replaced by power 1, the formula (111*)* = (11 + 111)* is correct but i need to prove it.. Link to comment Share on other sites More sharing options...
Aeternus Posted March 25, 2010 Share Posted March 25, 2010 I assume your alphabet is [math]\{1\}[/math]? In this case, you can just proceed by induction on the length [math]n[/math] of the string. Take a couple of base cases to make it simple. It makes things much simpler if you first of all reduce it to (111*)* = 111*, since clearly 111* has any string of length 2 or greater, which is the same as (111*)*. In the case [math]n = 0[/math], then due to the outer most Kleene Star in both cases, the empty word is in both languages. In case [math]n = 1[/math], then as we have no string in either expression of length less than 2, no amount or concatenation or choice will result in a string of length one, so "1" is not in either language. In case [math]n = 2[/math], then this is clearly in both languages from the definitions of Kleene star and +. In case [math]n = 3[/math], then this is clearly in both languages from the definitions of Kleene star and +. Assume for the induction hypothesis that we have the string [math]S[/math] of size [math]n[/math] in the language of both, and the string S' of size [math]n+1[/math] also in the language, and [math]n \ge 3[/math], then consider what happens if we have the string S'' of length [math]n+2[/math]. Clearly S'' is in the language 111* (as this matches any string of length 2 or greater). In case [math]n+2[/math] is odd, then we have that S of length [math]n[/math] is in the language (11 + 111)* and by the definition of the Kleene star S11 is also in the language, and the same argument holds if [math]n+2[/math] is even. Basically, from 11 and 111, you can make any odd or even sized string of 1s of length greater than 2 by simply adding 11 to either repeatedly (to make all the even or odd sizes. Link to comment Share on other sites More sharing options...
Recommended Posts
Create an account or sign in to comment
You need to be a member in order to leave a comment
Create an account
Sign up for a new account in our community. It's easy!
Register a new accountSign in
Already have an account? Sign in here.
Sign In Now