Which of the below listed statements are true? (select all that apply) Select one or more: O The Pumping lemma for regular languages is used to prove non-regularity of languages. O Every context-free language (CFL) is at the same time a regular language (RL). O The language oi13, where i,j > o is regular. O A grammar is ambiguous, if for every accepting word, there exists exactly one possible derivation.

Which of the below listed statements are true? (select all that apply) Select one or more: O The Pumping lemma for regular languages is used to prove non-regularity of languages. O Every context-free language (CFL) is at the same time a regular language (RL). O The language oi13, where i,j > o is regular. O A grammar is ambiguous, if for every accepting word, there exists exactly one possible derivation.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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1. Prove that the language L(G) is not regular where G is the following context-free grammar: G = ({S,A,B,C}, {a,b}, {S->bB|C, A>b|C, B->Sb, C->aAa }, S). Note: You must first determine L(G).
1 1.12 3 4. 5. . ... Theorem: Let G= (V,T,P.S) be a rghtlimear grammar. Then there exists a language L(G) which is accepted by a FA. i. , the language generated from the regular grammar is regular language. 1/1 lilil
(1) Fill the blanks to complete the following sentences that describe the relations between regular expression, regular language, and regular gramma. You can also use your own statements to describe the relations. there is an NFA machine to (i) A language L is regular (ii) A language L is regular (ii1)A language L is regular it. there is a regular expression r to it. there is a regular grammar G to it.
Question 3 a) Consider the grammar E → E + E | E ∗ E | (E) | id. I. Change ambiguous grammar to non-ambiguous? II. Give a context-free grammar for generating the set of palindromes over the alphabet a,b. III. Give a context-free grammar for generating all strings over a,b such that | a |= 2∗ | b
Context-free grammars are sometimes used to model natural languages. In this problem you will model a fragment of the English language using context-free grammars. Consider the following English sentences: The girl is pretty. The girl that the boy likes is pretty. The girl that the boy that the clerk pushed likes is pretty. The girl that the boy that the clerk that the girl knows pushed likes is pretty. This is a special type of sentence built from a subject (The girl), a relative pronoun (that) followed by another sentence, a verb (is) and an adjective (pretty). Give a context-free grammar G that models this special type of sentence. Your terminals should be words or sequences of words like pretty or the girl. Is the language of G regular? If so, write a regular expression for it. If not, prove using the pumping lemma for regular languages. Can you give an example of a sentence that is in G but does not make sense in common English?
(a) Exhibit a derivation of the string aaaa using the following phrase structured grammar: S → YZY Z → BZC| E ВС Сав Ва — ав аС — Са BY → Y ҮС — Y Y → E (b) Find a match for the following instance of the Post Correspondence Problem (PCP): {=} {三目圖} b a bb
1. Which of the following regular expressions best describes L(G), where G is the left-linear grammar { S → Ab | Ba , A → a| Aa , B → b| Bb }? A. aa*b + bb*a B. ab* + ba* C. (a+b)(a* + b*) 2. Given an NFA M with n states and an arbitrary string x of length p, determining whether or not x is in L(M) can be done in A. O(n+p) В. О(2а + р) C. O(np) D. O(p") steps. 3. What is the simplest regular expression equivalent to (a*b*c* + c*b*a*)* ? A. (a + b+ c)* B. (abc + cba)* C. It is already in its simplest form. 4. What does the egrep pattern "b?[ab]{1,2}" translate into? A. (b + ɛ)(a + b+ aa + ab + ba + bb) B. (bal + bb1 + ba2 + bb2) C. (al + a2 + b1 + b2 + ba1 + ba2 + bb1 + bb2)
2. Are the following languages context-free or non-context-free? • If a language is context-free, give a context-free grammar that generates it. • If a language is not context-free, prove this using the pumping lemma. A = { a*b'c*d² |k > i > 0, l>0 } B = {a²i+1fk+1c3l (2k+1q* | i > 1, k > 1, l > 1 } (a) (b)
When is a context free grammar ambiguous? Show that the following grammar is ambiguous: S> AS | A A> Ab | aAb | ab. Construct an equivalent unambiguous grammar.
1. Use the algorithm described in class that takes a CFG and converts it into a CNF (Chomsky Normal Form) grammar that generates the same language as the original grammar, except possibly for the string A, to answer the following question. As part of your answer, you should say which variables are nullable, and you should give the appropriate grammar after each step of the algorithm. There are four steps. Give a CFG (context-free grammar) in CNF (Chomsky Normal Form) that generates the same language, except possibly for the string A, as the following grammar: SAB AB →bAa A → aaB bb
Formal language Let’s define languages: L1 = {wan | w is an arbitrary string containing a’s and b’s, of length n}, Σ = {a, b} L2 = {w | 01∗0(11 + 00)∗} , Σ = {0, 1} For every language, define a context free grammar (CFG) for it.
NAT Lang processing: Please explain in detail Consider a CFG representing a phrase structure grammar for a natural language such as English, including the following rule (the period is part of the rule): S -> NP VP . In this rule, S, NP, and VP are non-terminal symbols, but we will assume that the period is a terminal symbol specifically representing a period (as opposed to some other end-of-sentence marker). So, we can interpret this rule as stating that a sentence can have the form of a noun phrase followed by a verb phrase followed by a period. (To implement a parser for grammars with rules like this, the tokenizer would have to treat periods as separate tokens.) Assume that you need to convert this grammar to Chomsky normal form (CNF), perhaps with the CKY parsing algorithm in mind. Using the procedure discussed in class, show the rules that might result from the conversion.