Formal Language Theory Quiz
48 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Which of the following best describes the relationship between a character and a symbol?

  • A symbol is a specific type of character.
  • Characters and symbols represent different concepts with no connection.
  • A character is a specific type of symbol. (correct)
  • Characters and symbols are interchangeable terms.

What is the correct order of the Chomsky hierarchy classes from smallest to largest?

  • Type 1, Type 2, Type 3, Type 0
  • Type 0, Type 1, Type 2, Type 3
  • Type 2, Type 3, Type 0, Type 1
  • Type 3, Type 2, Type 1, Type 0 (correct)

Which type of automata is associated with Context-Free Languages?

  • Pushdown Automata (correct)
  • Turing Machine
  • Linear Bounded Automata
  • Finite Automata

If Σ = {a, b}, what is the result of Σ^2?

<p>{aa, ab, ba, bb} (D)</p> Signup and view all the answers

What is the defining characteristic of a language in the context of formal language theory?

<p>It is a subset of Σ*. (B)</p> Signup and view all the answers

What is the relationship between a token and a string?

<p>A token is a specific instance of a string. (C)</p> Signup and view all the answers

Which of the following is NOT a type of language in the Chomsky hierarchy?

<p>Procedural Language (D)</p> Signup and view all the answers

What does Σ* represent?

<p>The set of all possible strings over an alphabet. (B)</p> Signup and view all the answers

If every state in a DFA is non-final, what is the language recognized by that DFA?

<p>The empty set, denoted as ∅ or {} (B)</p> Signup and view all the answers

Given a language L, which of the following is true about L and its complement L?

<p>L ∪ L = Σ* and L ∩ L = ∅ (B)</p> Signup and view all the answers

What is the minimum number of states required in a DFA to recognize a language where all strings have length equal to n or less than or equal to n?

<p>n + 2 states (B)</p> Signup and view all the answers

What is the minimum number of states required in a minimum DFA, that accepts the language consisting of strings over alphabet {a, b}, where the number of 'a's is divisible by m and the number of 'b's is divisible by n?

<p>m * n states (A)</p> Signup and view all the answers

Consider a language where the nth symbol of every string (from the beginning) over the alphabet {a, b} is 'a'. What is the number of states needed in the minimal DFA?

<p>n + 2 states (A)</p> Signup and view all the answers

Which statement is true concerning the relationship between DFAs and NFAs for a given regular language?

<p>For every regular language, there exists only one unique minimum DFA. (A)</p> Signup and view all the answers

If a language requires that the nth symbol from the start of any string over alphabet {a,b} is 'a', how many states are needed in the minimum NFA?

<p>n + 1 states (B)</p> Signup and view all the answers

How does the minimum number of states needed for a DFA differ for strings of length |w| = n and |w| ≥ n?

<p>The first requires n+2 and the second requires n+1 states. (D)</p> Signup and view all the answers

According to the given rules, what is the result of the operation a + ∅?

<p>a (C)</p> Signup and view all the answers

What is the result of the Kleene closure operation applied to the empty set, ∅*?

<p>∈ (C)</p> Signup and view all the answers

Which of the following is a correct representation of a positive closure, 'R⁺'?

<p>R¹ + R² + R³ + ... (C)</p> Signup and view all the answers

Which of these operations result in a?

<p>a ∙ ∈ (A)</p> Signup and view all the answers

According to provided properties, is concatenation commutative?

<p>No (A)</p> Signup and view all the answers

Given the language L = {aⁿbᵐ | n, m ≥ 0}, which of the following is the correct regular expression?

<p>a<em>b</em> (C)</p> Signup and view all the answers

What is the equivalent regular set for ∈ + ∈?

<p>{∈} (A)</p> Signup and view all the answers

Based on the provided properties, which expression acts as an annihilator for concatenation?

<p>∅ (B)</p> Signup and view all the answers

What is the key difference between the transition functions of a Nondeterministic Finite Automaton (NFA) and a Deterministic Finite Automaton (DFA)?

<p>NFA uses a transition function $Q \times \Sigma \rightarrow 2^Q$, while DFA uses $Q \times \Sigma \rightarrow Q$. (D)</p> Signup and view all the answers

Consider a language where all strings have a length of exactly $n$ over the alphabet {a, b}. How many states are required in a minimal DFA and NFA?

<p>DFA: $n+2$, NFA: $n+1$ (A)</p> Signup and view all the answers

For a language where the $n^{th}$ symbol from the end of any string is ‘a’, what is the number of states in the minimum DFA and NFA, respectively?

<p>DFA: $2n$, NFA: $n+1$ (C)</p> Signup and view all the answers

How does the number of states in a DFA relate to the number of states in an equivalent NFA, and what process do we apply to achieve this?

<p>A NFA with n states results in a DFA with at most $2^n$ states, using the subset construction method. (C)</p> Signup and view all the answers

Which of the following is true regarding paths for valid and invalid strings in an NFA?

<p>Valid strings have one path, and invalid strings have zero or more paths. (A)</p> Signup and view all the answers

Which description accurately reflects the relationship between Moore and Mealy machines?

<p>In Moore machines, output is associated with states, while in Mealy machines, output is associated with transitions. (A)</p> Signup and view all the answers

Given the language {w | w ∈ {a, b}*, |w| ≥ n}, how many states are required for a minimal NFA and DFA respectively?

<p>NFA: n+1, DFA: n+2 (C)</p> Signup and view all the answers

Which statement is correct about the interrelation of different Finite Automata (FA)?

<p>Every DFA is an NFA, but not every NFA is a DFA. (C)</p> Signup and view all the answers

Which of the following languages is considered a regular language?

<p>L = {a*} over Σ = {a} (C)</p> Signup and view all the answers

Which language is NOT a regular language?

<p>L = { anbn | n &gt;= 0} (D)</p> Signup and view all the answers

Which of these languages is described as a non-regular language?

<p>L = {a^2 | n &gt; 0} over Σ = {a} (C)</p> Signup and view all the answers

The language L = {a^n! | n >= 100} is classified as what type of language?

<p>Non-regular language (C)</p> Signup and view all the answers

Which of the following languages is a regular language?

<p>L = {a2n |n &gt;= 0} (C)</p> Signup and view all the answers

Which language is classified as a DCFL (Deterministic Context-Free Language)?

<p>L = { ambnc* | c &lt;= n } (B)</p> Signup and view all the answers

Which language is NOT a Context-Free Language (CFL)?

<p>All of the above are CFLs (D)</p> Signup and view all the answers

Given the language L = {aPrime}*, how would it be categorized?

<p>Regular language (D)</p> Signup and view all the answers

Which of the following is NOT equivalent to a standard Turing Machine (TM)?

<p>Finite Automata with one stack (C)</p> Signup and view all the answers

If a Turing Machine always halts, and uses a linearly bounded tape, what type of language does it accept?

<p>Context-Sensitive Language (D)</p> Signup and view all the answers

What is the key difference between a Deterministic Turing Machine (DTM) and a Non-Deterministic Turing Machine (NTM) based on their transition functions?

<p>DTM transitions are always to a single state whereas NTM can be to multiple states. (D)</p> Signup and view all the answers

What is another name for Unrestricted Grammar (UG)?

<p>Phase structure grammar (A)</p> Signup and view all the answers

Which of the following is true about Left Linear Grammar (LLG)?

<p>It has the form V → V T* | T* (B)</p> Signup and view all the answers

Which of the following describes a Turing Machine that always halts?

<p>It is a Halting Turing Machine (HTM). (A)</p> Signup and view all the answers

A Turing Machine (TM) with what specific restriction will only accept regular languages?

<p>The TM tape is read-only and the head is unidirectional (D)</p> Signup and view all the answers

What type of language is accepted by a Linear Bounded Automaton (LBA)?

<p>Context-Sensitive languages (B)</p> Signup and view all the answers

Flashcards

String

A sequence of characters from a given alphabet. Imagine it as a chain of letters or symbols.

Alphabet

A set of symbols used in a language, like letters, numbers, or punctuation marks. Think of them as the building blocks for words and sentences.

Universal Language (Σ*)

A collection of all possible strings formed from a specific alphabet.

Language

A set of strings that follows a specific set of rules or grammar. It's a subset of the universal language.

Signup and view all the flashcards

Chomsky Hierarchy

Formal systems for describing languages that classify them based on their complexity. It's like a hierarchy of languages.

Signup and view all the flashcards

Regular Language

A type of language in the Chomsky Hierarchy that is characterized by simple patterns and can be recognized by a finite state machine.

Signup and view all the flashcards

Context-Free Language

A type of language in the Chomsky Hierarchy that is characterized by more complex grammar and can be recognized by a pushdown automaton. The language must be defined by its grammar.

Signup and view all the flashcards

Context-Sensitive Language

A type of language in the Chomsky Hierarchy that is characterized by even more complex grammar and can be recognized by a linear bounded automaton.

Signup and view all the flashcards

Complement of a language

The set of strings that are not in a particular language.

Signup and view all the flashcards

Intersection of two languages

The set of strings that are in both languages.

Signup and view all the flashcards

Union of two languages

The set of strings that are in at least one of the languages.

Signup and view all the flashcards

Finite Automaton

A finite state machine with a finite number of states and transitions.

Signup and view all the flashcards

Deterministic Finite Automaton (DFA)

A type of finite automaton with a single start state and no transitions on empty input (epsilon).

Signup and view all the flashcards

Non-Deterministic Finite Automaton (NFA)

A type of finite automaton that allows multiple transitions on the same input symbol and transitions on empty input (epsilon).

Signup and view all the flashcards

Minimum DFA

The minimum number of states needed to recognize a language.

Signup and view all the flashcards

Equivalence classes

The number of states in the minimum DFA equals the number of equivalence classes in the language.

Signup and view all the flashcards

DFA States for 'a' at the end

The number of states in a minimum Deterministic Finite Automaton (DFA) that recognizes a language where the nth symbol from the end of a string must be 'a'.

Signup and view all the flashcards

NFA States for 'a' at the end

The number of states in a minimum Nondeterministic Finite Automaton (NFA) that recognizes a language where the nth symbol from the end of a string must be 'a'.

Signup and view all the flashcards

Concatenation

The operation that combines two regular expressions. It concatenates the strings represented by the two expressions. It's represented by the '.' symbol.

Signup and view all the flashcards

Kleene Star (R*)

An operation that repeats a regular expression any number of times, including zero. It's represented by the '*' symbol.

Signup and view all the flashcards

Positive Closure (R+)

An operation that repeats a regular expression at least once. It's represented by the '+' symbol.

Signup and view all the flashcards

Halting Turing Machine (HTM)

A Turing Machine (TM) that always halts for any input. It is a more restricted type compared to a general TM.

Signup and view all the flashcards

Linear Bounded Automaton (LBA)

A Turing Machine (TM) with a tape whose length is limited by a linear function of the input size. It ensures efficiency and predictability.

Signup and view all the flashcards

Recursive Language

The class of languages recognized by a TM that halts for all inputs. These languages are also called decidable languages.

Signup and view all the flashcards

Recursively Enumerable (RE) Language

The class of languages recognized by a TM. These languages are also called partially decidable or semi-decidable languages.

Signup and view all the flashcards

Non-Deterministic Turing Machine (NTM)

A type of TM that can choose among multiple possible transitions based on the current state and input symbol. It's like a machine with multiple choices.

Signup and view all the flashcards

Deterministic Turing Machine (DTM)

A type of TM that has a single possible transition for each state and input symbol. Imagine it as a predictable machine that follows a single path.

Signup and view all the flashcards

Transition Function ()

The set of instructions that a TM follows, specifying the transition based on the current state and the symbol read from the tape.

Signup and view all the flashcards

Unrestricted Grammar (UG)

The type of grammar used by a TM. It allows for unrestricted rewrites of the input string. This gives it a lot of power for analyzing and generating languages.

Signup and view all the flashcards

L = {anbm | m = n, m > 10}

The language consists of all strings with an equal number of 'a's and 'b's, where the number of 'a's is greater than 10. For example, 'aaaaabbbbb' is in the language, but 'aabb' is not.

Signup and view all the flashcards

L = {ambn | gcd (m, n) = 1}

The language contains all strings where the greatest common divisor (GCD) of the number of 'a's and 'b's is 1. For example, 'aaabbb' is in the language (GCD of 3 and 3 is 1), but 'aaaaabbbb' is not (GCD of 5 and 4 is 1).

Signup and view all the flashcards

L = {ambn | LCM(m, n) = 1}

The language contains all strings where the least common multiple (LCM) of the number of 'a's and 'b's is 1. The LCM of 1 and any number is 1. This occurs only when both the number of 'a's and 'b's are 1, leading to a regular language with the pattern 'ab'.

Signup and view all the flashcards

L = {anbn | n >= 0}

The language consists of all strings with an equal number of 'a's and 'b's. For example, 'ab', 'aabb', 'aaabbb' are all in the language.

Signup and view all the flashcards

L = {anb2n | n >= 0}

The language contains strings where the number of 'b's is twice the number of 'a's. For example, 'aaabbbb' is in the language, but 'ab' and 'aaabb' are not.

Signup and view all the flashcards

L = {ambn | m=even, n =odd} OR L = {a2nb2n | n = even}

The language includes all strings where the number of 'a's is even and the number of 'b's is odd, or the language can be expressed as strings containing an even number of 'a's followed by an even number of 'b's. For example, 'aabb', 'aaaaabbbbb' are in the language, but 'ab', 'aaab' are not.

Signup and view all the flashcards

L = {a*} over Σ = {a}

The language consists of any string made up of only 'a's, including the empty string. Example: '', 'a', 'aa', 'aaa' are all valid strings.

Signup and view all the flashcards

L = {a2n | n >= 0} over Σ = {a}

The language includes all strings where the number of 'a's is a multiple of 2. For example, 'aa', 'aaaa', 'aaaaaa' etc., are all in the language.

Signup and view all the flashcards

Study Notes

Theory of Computation

  • This subject matter involves the study of computation, languages, and automata.
  • It deals with the limitations and capabilities of computers.

Index

  • Basics of Theory of Computation: Sections 7.1-7.5
  • Finite Automata: Sections 7.6-7.21
  • Push Down Automata: Sections 7.22-7.29
  • Turning Machine: Sections 7.30-7.39

Basics of Theory of Computation

  • Symbol: A unique, indivisible unit.
  • Alphabet: A finite set of symbols. Examples include the English alphabet (a-z), binary alphabet (0, 1), decimal alphabet (0-9).
  • String: A sequence of symbols defined over an alphabet. Strings over alphabets can be empty (null), one character (length 1), or more complex (variable length). Examples are English words, binary sequences, or decimal numbers.
  • Operations on Strings:
    • Length: The number of symbols in a string. For a string 'w', the length is |w|.
    • Reversal: The reverse order of symbols. If 'w' = 'abb', then 'wR' = 'bba'.
    • Concatenation: Joining two strings together. If 'w1' = 'a' and 'w2' = 'ba', then 'w1w2' = 'aba'.
    • Prefixes: A substring that starts at the beginning of a string (including the empty string).
    • Suffixes: A substring that ends at the end of a string (including the empty string).
    • Substrings: Any contiguous sequence of symbols within a string.

Finite Automata

  • Finite Automata (FA): Describes or represents a regular language. Two types exist:
    • Acceptor: Accepts or rejects given input strings.
    • Transducer: Produces output strings for given input strings.
  • Finite Automata (FA) components:
    • Set of states (Q)
    • Alphabet (Σ)
    • Transition function
    • Initial state (q0)
    • Set of final states (F)

Push Down Automata

  • Push-down Automata is a type of Finite Automata (FA) that accepts Context Free Languages.

Turing Machine

  • Turing Machine is a type of Finite Automata that accepts recursively enumerable languages.
  • Turing Machines have components:
    • States (Q)
    • Input alphabet (Σ)
    • Tape alphabet (Γ)
    • Transition function
    • Start state (q0)
    • Final state(s)
  • There are two types of Turing Machines:
    • Deterministic Turing Machines (DTM)
    • Non-Deterministic Turing Machines (NDTM)

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Quiz Team

Related Documents

Theory of Computation PDF

Description

Test your understanding of formal language theory with this quiz that covers concepts from the Chomsky hierarchy to automata theory. Challenge yourself with questions about symbols, tokens, and various types of languages. Perfect for students of computer science and linguistics!

More Like This

Use Quizgecko on...
Browser
Open
Browser
Browser