Question

In: Computer Science

Given an alphabet Σ, what are the languages L over Σ for which L ∗ is...

Given an alphabet Σ, what are the languages L over Σ for which L ∗ is finite? How many such languages exist?

Solutions

Expert Solution

If Σ is an alphabet, and L ⊆ Σ∗, then L is a (formal) language over Σ.
Language: A (possibly infinite) set of strings all of which are chosen from some
Σ∗
A language over Σ need not include strings with all symbols of Σ
Thus, a language over Σ is also a language over any alphabet that is a superset
of Σ
Examples:

  • Programming language C
  • Legal programs are a subset of the possible strings that can be formed from the alphabet of the language (a subset of ASCII characters)
  • English or French
  • The language of all strings consisting of
    n 0s followed by n 1s (n≥0): {ǫ, 01, 0011, 000111, . . .}
  • The set of strings of 0s and 1s with an equal number of each:{ǫ, 01, 10, 0011, 0101, 1001, . . .}
  • Σ∗ is a language for any alphabet Σ
  • ∅, the empty language, is a language over any alphabet
  • {ǫ}, the language consisting of only the empty string, is also a language over any alphabet
  • {w | w consists of an equal number of 0 and 1}

Related Solutions

3. Are the following languages A and B over the alphabet Σ = {a, b, c,...
3. Are the following languages A and B over the alphabet Σ = {a, b, c, d} regular or nonregular? • For a language that is regular, give a regular expression that defines it. • For a nonregular language, using the pumping lemma prove that it is not regular. (a) A = {d 2j+1c k+1 | j ≥ k ≥ 0} · {c r+2b 2s+3 | r ≥ 0 and s ≥ 0} (b) B = {a 2j+2b k+3c j+1...
Let L be the set of all languages over alphabet {0}. Show that L is uncountable,...
Let L be the set of all languages over alphabet {0}. Show that L is uncountable, using a proof by diagonalization.
1. Prove the language L is not regular, over the alphabet Σ = {a, b}. L...
1. Prove the language L is not regular, over the alphabet Σ = {a, b}. L = { aib2i : i > 0} 2) Prove the language M is not regular, over the alphabet Σ = {a, b}. M = { wwR : w is an element of Σ* i.e. w is any string, and wR means the string w written in reverse}. In other words, language M is even-length palindromes.
4. Construct regular expressions for the following languages over the alphabet {a, b}: a. Strings that...
4. Construct regular expressions for the following languages over the alphabet {a, b}: a. Strings that do not begin with an “a”. b. Strings that contain both aa and bb as substrings. 5. Let ? = {? ?? ?? ? | ? > ? + ?}. a. List all strings of length 7. (use power notation: i.e. aabbbbaaaaaaaa is a2b 4a 8 ) b. Use the Pumping Lemma for Regular Languages to prove that L is not regular.
2. Let S be the set of strings over the alphabet Σ = {a, b, c}...
2. Let S be the set of strings over the alphabet Σ = {a, b, c} defined recursively by (1) λ ∈ S and a ∈ S; and (2) if x ∈ S, then bxc ∈ S. Recall that λ denotes the empty string which has no letters and has length 0. List all strings of S which are length at most seven. 3. Prove the following theorem by induction: For every integer n ≥ 1, 1 · 2 +...
Automata Question. Over the alphabet Σ = {a, b}: 1) Give a DFA, M1, that accepts...
Automata Question. Over the alphabet Σ = {a, b}: 1) Give a DFA, M1, that accepts a Language L1 = { w | w has exactly 2 a’s } 2) Give a DFA, M2, that accepts a Language L2 = { w | w has at least 2 b’s } 3) Give acceptor for L1 intersection L2 4) Give acceptor for L1 - L2
Consider the language L1 over alphabet Σ = { 0, 1 } where the production rules...
Consider the language L1 over alphabet Σ = { 0, 1 } where the production rules for L1 are as follows: S → TT S → U T → 0T T → T0 T→ 1 U → 0U00 U → 1 Q → λ P → QU Transform this grammar into Chomsky Normal Form, consistent with the CNF specification in the Quick Reference, and using only Variables { S, T, U, V, W, X }. Implement that CNF grammar in...
Use Context-Free Pumping Lemma to prove that that following languages over the alphabet {'x', 'y', 'z'}...
Use Context-Free Pumping Lemma to prove that that following languages over the alphabet {'x', 'y', 'z'} are NOT context-free (a) {xjy2jzj : j > 0} (b) { xmynzk : m, n, k ≥ 0 and k = min(m,n) }
Describe the languages specified by the following regular expressions: 1. \\(_+)/ 2. (\(740\)...-...)|(...-...) (The alphabet is...
Describe the languages specified by the following regular expressions: 1. \\(_+)/ 2. (\(740\)...-...)|(...-...) (The alphabet is {1,2,3,4,5,6,7,8,9,0})
Theory of computation: Consider the alphanumeric alphabet Σ = {a, b, . . . , z,...
Theory of computation: Consider the alphanumeric alphabet Σ = {a, b, . . . , z, A, B, . . . , Z, 0, 1, . . . , 9} and let L be the language of all regular expressions over Σ: L = {w ∈ Σ ∪ {∅,(,), ∪, ·, * }* | w is a syntactically legal regular expression over Σ} Give an unambiguous context-free grammar that generates L. The grammar should use the following precedence levels, from...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT