Type 3 Grammars and Context-Free Languages Quiz

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Questions and Answers

What rule should be added to the CFG for each state qi of the DFA when an input a leads to state qj?

Ri ! aRj (correct)

Ri ! a

Ri ! Rj

Ri ! RjRj

In the context of Type 3 Grammars, which of the following is a characteristic of the strings in the language L1?

Strings can only contain 0s

Strings must contain at least three 1s (correct)

Strings must contain at least two 1s

Strings can only be of even length

Which grammar represents the language L2, where the length of w is odd and its middle symbol is a 0?

S2 ! 0|0S1|1S0|1S1

S2 ! 0|1S0|0S1|1S1

S2 ! 0|0S0|1S0|1S1 (correct)

S2 ! 0|1S1|1S0

What is a defining feature of Type 3 grammars concerning their unambiguity?

Exactly one way to generate a string Signup and view all the answers

Which grammar correctly describes strings with more a's than b's in the language L3?

S3 ! TaT | aTb | bTa | a | ✏ Signup and view all the answers

What defines a context-free language (CFL)?

A language generated by a context-free grammar. Signup and view all the answers

Which of the following is an example of a non-terminal in the given grammar?

hNOUNi Signup and view all the answers

What does the CYK algorithm determine?

Whether a string can be generated by a context-free grammar. Signup and view all the answers

Which statement about context-free grammars is true?

They are used to specify the syntax of programming languages. Signup and view all the answers

Which type of grammar is used for regular languages?

Type 3 grammars Signup and view all the answers

In the grammar, which symbol represents a verb phrase?

hVERBihPREP.PHRASEi Signup and view all the answers

What does it mean when a grammar is said to generate a language?

It produces valid strings of that language. Signup and view all the answers

Which of the following is NOT a component of the grammar provided?

hPRONOUNi Signup and view all the answers

What is the primary function of a pushdown automaton?

To accept context-free languages using a stack. Signup and view all the answers

Which of the following operations is not a regular operation for regular languages?

Reversal Signup and view all the answers

In the Pumping Lemma for Regular Languages, what must hold true about the segment 'y' when a string 's' is divided into three parts?

|y| must be greater than zero. Signup and view all the answers

How do you prove that a language is non-regular using the Pumping Lemma?

By demonstrating a contradiction regarding the required conditions. Signup and view all the answers

What characterizes a regular expression?

It can be formed from simple symbols and operations on them. Signup and view all the answers

According to the properties of regular languages, when is a string considered to be accepted by an automaton?

When the automaton reaches a final state after processing it. Signup and view all the answers

Which of the following denotes a non-regular language characteristic?

It cannot be pumped according to the Pumping Lemma. Signup and view all the answers

What does the symbol ε (epsilon) represent in regular expressions?

The empty string. Signup and view all the answers

Which of the following is a defining characteristic of context-free languages (CFLs)?

Defined using context-free grammars Signup and view all the answers

What type of grammar is recognized as Type 2 in Chomsky’s hierarchy?

Context-free grammar Signup and view all the answers

Which language is an example of a non-regular language?

{0n 1n | n > 0} Signup and view all the answers

What is the main distinction between deterministic pushdown automata (DPDAs) and non-deterministic pushdown automata (NPDAs)?

DPDAs and NPDAs are not equivalent in power Signup and view all the answers

What constitutes a context-free grammar?

A 4-tuple including a finite set of variables Signup and view all the answers

Which of the following statements is true regarding context-free grammars and their derivations?

Derivation can involve multiple steps via rules Signup and view all the answers

In the context-free grammar G = ({S}, {0, 1}, R, S), what is the language generated by the rule S → 0S1 | ε?

{0^n 1^n | n ≥ 0} Signup and view all the answers

What does the notation uAv → uwv represent in the context of a context-free grammar?

u produced by A is combined with strings w and v Signup and view all the answers

What does the special symbol $ signify when placed on the stack?

The stack is empty. Signup and view all the answers

In the language L = {ai bj ck | i, j, k ≥ 0 and i = j or i = k}, what are the requirements for the parameters i, j, and k?

i and j can be equal, or i and k can be equal. Signup and view all the answers

What is the defining structure of a context-free grammar?

A tuple containing variables, terminals, rules, and a start variable. Signup and view all the answers

What property does Chomsky Normal Form provide in relation to context-free grammars?

Facilitates polynomial time algorithms to decide string generation. Signup and view all the answers

In the notation a, b !c, what does '!' indicate in the context of grammar rules?

A production rule application. Signup and view all the answers

Given the language L = {ww R | w ∈ {0, 1}*}, what does w R represent?

A version of w with its characters reversed. Signup and view all the answers

What happens if a PDA is in an accept state but the input has not been fully processed?

The PDA rejects the input. Signup and view all the answers

When simplifying rules in context-free grammars, what is a potential benefit of using Greibach Normal Form?

Improved efficiency in parsing algorithms. Signup and view all the answers

What is a characteristic of a grammar in Greibach Normal Form?

Every rule must be of the form A → aA1A2...Ak. Signup and view all the answers

Which of the following correctly describes Chomsky Normal Form?

Rules must either be of the form A → BC or A → a. Signup and view all the answers

What is the purpose of converting a CFG to Chomsky Normal Form?

To simplify the parsing of the grammar. Signup and view all the answers

Which step is NOT part of the conversion process to CNF?

Transforming all variables into terminals. Signup and view all the answers

What restriction is applied to variables B and C in Chomsky Normal Form?

They cannot be the start variable. Signup and view all the answers

In the CNF conversion process, what is removed in Phase 2?

All ε-productions for variables. Signup and view all the answers

Which statement is true regarding CFGs and their representation in CNF?

All context-free languages can be generated by a CFG in CNF. Signup and view all the answers

What is the significance of introducing a new start variable during the CNF conversion?

It ensures that the new grammar remains in Chomsky Normal Form. Signup and view all the answers

Study Notes

Administrivia

Previous topics: regular expressions, properties of regular languages, pumping lemma
Current topics: context-free languages, pushdown automata, grammars
Reading assignment: Chapter 2
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Anonymous feedback requested

Recall

Automaton language: set of strings accepted by the automaton
Computation (in this context): determining whether a string belongs to a language (decision problem)
Finite Automata (FA) variants: deterministic, non-deterministic
Regular language: a language recognized by some FA
Regular operations: union, concatenation, Kleene star
Regular expressions: used to describe regular languages
Non-regular languages: languages that cannot be described by regular expressions, related to the pumping lemma

Regular Expressions

Definition R is a regular expression if:
- R is a for a character a in the alphabet Σ
- R is ε
- R is ∅
- R is (R₁ U R₂) where R₁ and R₂ are regular expressions
- R is (R₁ R₂) where R₁ and R₂ are regular expressions
- R is (R₁*) where R₁ is a regular expression

Pumping Lemma (PL) for Regular Languages

Theorem: If A is a regular language, there exists a number p (pumping length) such that any string s in A with length at least p can be divided into three pieces xyz satisfying these conditions:
- xy^iz ∈ A, for each i ≥ 0
- y is greater than 0
- xy is less than or equal to p

Proving a Language Non-Regular

Use PL to prove a language isn't regular
Given language L and p
Choose a string s of length greater than p in L
Break down the string into 3 parts xyz to match the PL criteria
Show a contradiction for every possible breakdown of the string will satisfy the 3 conditions

Examples of Non-Regular Languages

L = {0ⁿ1ⁿ | n > 0}
L = {ww | w ∈ {0,1}*}
L = {0ⁱ1ʲ | i > j}

Context-Free Languages

Characterized by context-free grammars (CFGs)
Also recognized by pushdown automata (PDAs)
PDAs have memory (a stack)
Example of CFL: L = {0ⁿ1ⁿ | n > 0}
FAs, DPDAs, and NPDAs aren't equally powerful
Related to type 2 grammars in Chomsky's hierarchy

Context-Free Grammars (CFGs)

Definition: A 4-tuple (V, Σ, R, S) where
- V: finite set of variables
- Σ: finite set of terminals
- R: finite set of rules of the form α → β
- S: start variable (element of V)
Example grammar: G = ({S}, {0, 1}, R, S)
R : set of rules: S → 0S1, S → ε
Language of G: {0ⁿ1ⁿ | n ≥ 0}

Context Free Grammars (Continued)

Derivation rules:
- If u, v, and w are strings of variables and terminals, and A → w is a rule, then uAv yields uwv.
- u ⇒ v if u = v or there exists a sequence u₁ ,u₂, ..., uₖ such that u₁ ⇒ u₂ ⇒ ... ⇒ uₖ = v.
Language of a CFG: {w ∈ Σ* | S ⇒ w} (all strings w that can be derived from the start variable S)

CFGs (Additional Considerations)

Parsing and Parse Trees: Compilers and interpreters use parsers to extract program meaning before compilation
CFGs can be used to build a parser

Context-Free Languages and Grammars

Examples of strings generated by the specific grammar rules

CFLs and CFGs

CFGs specify language syntax
Programming languages like Python and Java use CFGs for parsing
CYK algorithm (O(n³|G|)) uses dynamic programming to determine if a string can be generated by a CFG
Context-free language: a language that can be generated by a CFG

Grammars

Grammars provide another way to describe a language (unlike automata which recognize a language)
Regular languages can be described using finite automata, regular expressions, or type 3 grammars
Context-free languages can be described using context-free grammars (also type 2)
Certain languages can be recognized by either a finite automaton or a pushdown automaton, and can also be generated by grammar type 3 or 2 respectively

Type 3 Grammars - Example

Converting DFAs to equivalent CFGs
Create a variable for each DFA state Rᵢ
If (qᵢ, a) = qⱼ, in the DFA, then Rᵢ → aRⱼ in the CFG
If qᵢ is an acceptor state, then Rᵢ → ε

Designing Grammars

Examples of designing grammars for specific languages
L₁ = {w | w contains at least three 1s}:
- S₁ → R₁R₁R₁R
- R → 0R|1R|ε

Designing Grammars - Examples Continued

L₂ = {w | length of w is odd and middle symbol is 0}
L₃ = {w | w has more a's than b's}

Ambiguity

Type 3 grammars and languages are unambiguous (only 1 way)
Type 2 grammars (CFGs) and context-free languages can be ambiguous (more than 1 way)
Ambiguity can lead to multiple meanings of programs, undesirable in programming languages
A grammar is ambiguous if there exists a string with multiple parse trees

Ambiguity (Continued)

Leftmost derivation: every step replaces the leftmost remaining variable
Ambiguity definition: A string w is generated ambiguously in a CFG G if it has two or more distinct leftmost derivations
Ambiguous CFG: CFG that generates strings ambiguously

Pushdown Automata (PDAs)

Definition: A 6-tuple (Q, Σ, Γ, δ, q₀, F)
- Q: Finite set of states
- Σ: Input alphabet
- Γ: Stack alphabet
- δ: Transition function

Computation of a PDA

M accepts w if:
- Initial state: r₀ = q₀ and s₀ = ε
- Transition rule (rᵢ₊₁, b) ∈ δ (rᵢ, wᵢ₊₁, a), where sᵢ = a⋅t
- Final state: rₙ ∈ F Where (rᵢ₊₁, b) ∈ δ(rᵢ, wᵢ₊₁, a) for i ∈ [0, n − 1]
- a, b ∈ Γ* and t ∈ Γ*

PDA Technicalities

Empty stack test: adds a special symbol ($) to the stack
End of input: Acceptor mechanism only activates when the entire input string has been processed
Understanding PDA transition descriptions: meaning of characters (a, b, c) in transition descriptions (input symbols, stack top symbols, ε, etc.)

Let's Build a PDA for a Language

Example language L = {aⁱbʲcᵏ | i, j, k > 0 and i = j or i = k}
- Building a PDA to accept that language(Graphical example)
Another example L = {wwR | w ∈ {0,1}*}
- Building a PDA to accept that language (Graphical Example)

Context-Free Grammars and Normal Forms

Definition: A 4-tuple (V, Σ, R, S) where:
- V: finite set of variables
- Σ: finite set of terminals
- R: finite set of rules (α → β where α is a variable and β is a string of variables and terminals)
- S: start variable (element of V)
Example grammar

Normal Forms for Type 2 Grammars

Chomsky Normal Form (CNF): simplifies CFG rules, string size proportional to number of rules used
- Polynomial-time algorithm to determine if a string is derivable
Greibach Normal Form (GNF): makes proving a language can be recognized by a PDA in linear time easier

Chomsky Normal Form

Definition: CFG in CNF has rules where every rule is either of the form A → BC or A → a, where a is a terminal, and A, B, and C are variables (except that B and C may not be the start variable; S).
Compare/contrast with right linear grammars.

Any CFL has a CFG in CNF

Theorem: Every context-free language can be generated by a CFG in CNF
Proof outline:
- Start with a CFG for a context-free language
- Successively replace rules that don't fit CNF criteria with equivalent ones.
  - Add a new start variable
  - Eliminate epsilon rules
  - Eliminate unit rules
  - Put rest of rules into CNF form

CFG to CNF Example

Step-by-step transformations of a CFG to CNF. (graphical examples in the form of diagrams and rules)

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