Context-Sensitive Grammars - Chapter 4
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Questions and Answers

What is the form for the string where the conversion is performed?

  • ai C k B j
  • ai C j B k
  • ai B j C k (correct)
  • ai B k C j

    • All context-sensitive languages are decidable languages.

    True

    What is the initial transition rule of the context-sensitive grammar for converting CB to BC?

    CB → CD

    A linear bounded automaton is a type of __________ where the tape head is restricted only to the input.

    <p>Turing Machine</p> Signup and view all the answers

    Match the automaton definitions with their descriptions:

    <p>Linear Bounded Automaton = Tape head restricted to input Context-Free Language = Language generated by context-free grammar Context-Sensitive Language = Language where production rules need context Decidable Language = Language for which there exists an algorithm to determine membership</p> Signup and view all the answers

    Which of the following grammars includes both ε (epsilon) transitions?

    <p>C → CC</p> Signup and view all the answers

    The theorem states that a language L is context-sensitive if there exists a linear bounded automaton M such that L(M) = L.

    <p>True</p> Signup and view all the answers

    What conversion is performed on B's when simulating swaps?

    <p>Convert B's to b's</p> Signup and view all the answers

    What does the Universal Turing Machine (TM) do when given input < M, w >?

    <p>Simulates the TM M on input w.</p> Signup and view all the answers

    ATM is a decidable language.

    <p>False</p> Signup and view all the answers

    What contradiction arises from the construction of TM D?

    <p>TM D accepts if D doesn't accept &lt; D &gt; and rejects if D accepts &lt; D &gt;.</p> Signup and view all the answers

    A language is decidable if it is both Turing-recognizable and __________.

    <p>co-Turing-recognizable</p> Signup and view all the answers

    Match the following concepts with their descriptions:

    <p>H = Decider for the ATM problem D = Constructed TM which contradicts itself M1 = Recognizer for ATM M2 = Another recognizer for ATM</p> Signup and view all the answers

    Which statement about the Turing machines M1 and M2 is true?

    <p>M1 is a recognizer for ATM, but M2 contradicts it.</p> Signup and view all the answers

    ATM is Turing-recognizable.

    <p>True</p> Signup and view all the answers

    What role did the Universal TM play in computer architecture?

    <p>It inspired the von Neumann computer architecture.</p> Signup and view all the answers

    What does the Turing Machine D do when given input < D >?

    <p>Both A and B</p> Signup and view all the answers

    The Turing Machine H only accepts if M does not accept w.

    <p>False</p> Signup and view all the answers

    What key concept does the Barber Paradox illustrate?

    <p>A self-referential paradox</p> Signup and view all the answers

    The Turing Machine H is used as a __________ in the construction of D.

    <p>decider</p> Signup and view all the answers

    What significant issue did Bertrand Russell identify that affected Frege's logic?

    <p>The contradiction in class membership</p> Signup and view all the answers

    Match the following terms with their descriptions:

    <p>ATM = The set of all Turing machines that accept a given input H = The decider for ATM D = The machine that contradicts H's decision Barber Paradox = A paradox about self-reference in shaving behavior</p> Signup and view all the answers

    What happens if the barber shaves himself according to the paradox?

    <p>The barber contradicts his own rules</p> Signup and view all the answers

    Frege continued to pursue the logical foundations of arithmetic after discovering the contradiction.

    <p>False</p> Signup and view all the answers

    If the barber does not shave himself, he must be shaved by himself.

    <p>True</p> Signup and view all the answers

    What is the cardinality of the set {1, 3, 5, 7}?

    <p>4</p> Signup and view all the answers

    The theorem states that __________ is undecidable.

    <p>ATM</p> Signup and view all the answers

    A function f : A → B is called __ if it never maps two different elements to the same place.

    <p>one-to-one</p> Signup and view all the answers

    Match the following terms with their definitions:

    <p>Cardinality = Number of elements in a set One-to-one function = Never maps two different elements to the same place Onto function = Hits every element in the codomain Correspondence = A function that is both one-to-one and onto</p> Signup and view all the answers

    What is an example of irrational numbers that challenged the Pythagorean belief?

    <p>√2 and π</p> Signup and view all the answers

    Two sets A and B are considered the same size if there is a one-to-one, onto function from A to B.

    <p>True</p> Signup and view all the answers

    Who did Bertrand Russell write to regarding the paradox he discovered?

    <p>Frege</p> Signup and view all the answers

    Which of the following statements is true regarding the set of all languages?

    <p>The set of all languages is uncountable.</p> Signup and view all the answers

    Every language has a unique characteristic sequence, which is a finite binary string.

    <p>False</p> Signup and view all the answers

    What is the notation used to represent the set of all infinite binary sequences?

    <p>B</p> Signup and view all the answers

    The proof shows that by using a bijection with N, we can construct a sequence that differs from every sequence on the right-hand side by flipping the ______ bit.

    <p>i-th</p> Signup and view all the answers

    Which of the following sequences is included in the set Σ∗?

    <p>10</p> Signup and view all the answers

    The size of the set of all languages is equal to the size of the set of finite binary sequences.

    <p>False</p> Signup and view all the answers

    What is the general result shown by the theorem regarding the set of all languages?

    <p>Uncountable</p> Signup and view all the answers

    What is the main purpose of the decider M as described?

    <p>To simulate two different recognizers in parallel</p> Signup and view all the answers

    Theorem ATM states that ATM is T-recognizable.

    <p>False</p> Signup and view all the answers

    What can be used to construct a new sequence that differs from each sequence in a countable bijection?

    <p>Flipping the i-th bit in the i-th sequence</p> Signup and view all the answers

    The set of all languages is __________.

    <p>uncountable</p> Signup and view all the answers

    Which of the following statements about the set of infinite binary sequences is true?

    <p>It is uncountable.</p> Signup and view all the answers

    Any language has a unique characteristic sequence.

    <p>True</p> Signup and view all the answers

    What contradiction arises from assuming that ATM is T-recognizable?

    <p>It would imply that we can decide ATM, which is impossible.</p> Signup and view all the answers

    Study Notes

    Administrivia

    • Midterm results available today
    • Midterm feedback due today
    • Second midterm exam date vote today
    • Reading assignment: Chapter 4
    • Previous lecture topics: Turing Machines, Decidable problems
    • Today's lecture topics: Context-Sensitive Grammars, LBAs, decidable, recognizable, undecidable languages

    Context-Sensitive Grammars

    • Context-free grammar rules: a → β, where a is a single variable and β is a string of variables and terminals.
    • Example context-free grammar rules: S → E | F, E → 0E0 | 1E1 | ε, F → 0F0 | 1F1 | 0 | 1
    • Context-sensitive grammar (CSG) rules: substitutions of variables must be surrounded by context on either or both sides.
    • Context-sensitive grammar definition: a 4-tuple (V, Σ, R, S), where:
      • V is a finite set of variables.
      • Σ is a finite set of terminals, Σ∩V = 0
      • R is a finite set of rules of the form αΑβ → αγβ where A ∈ V, α, β ∈ (V∪Σ)*, and γ ∈ (V∪Σ)+
      • S ∈ V is the start variable
      • S ∈ ε is permitted

    CSG Example

    • Language L = {aibjck | 1 ≤ i ≤ j ≤ k}
    • Method to construct a CSG for this language: start by constructing a CFG for the language {ai(b c)jck-j | 1 ≤ i ≤ j ≤ k} and then swap b's and c's using CSG rules.
    • Example: CFG for {ai(bc)jck-j | 1 ≤ i ≤ j ≤ k}: S→ TXC, T → aTBC | abc, X → BCX | ε, C → CC | ε
    • Solution: Use variables B and C instead of terminals b and c; use CSG rules to simulate a swap.

    Linear Bounded Automata (LBA)

    • Designed to handle context-sensitive languages.
    • Tape head restricted to the input string.
    • If the head tries to move off the input string, the head stays in place.

    Decidability

    • ADFA = {< B, w > | B is a DFA that accepts input string w} is decidable
    • ANFA = {< B, w > | B is a NFA that accepts input string w} is decidable
    • EDFA = {< A > | A is a DFA and L(A) = ∅} is decidable
    • EQDFA = {< A, B > | A and B are DFAs and L(A) = L(B)} is decidable
    • ACFG = {< G, w > | G is a CFG that generates string w} is decidable

    CFLs and Decidable Languages

    • Every context-free language is decidable

    ATM and Undecidability

    • ATM = {< M, w > | M is a TM that accepts input w} is undecidable.

    The Universal TM

    • UTM = "On input < M, w >, Simulate M on input w."
    • UTM is recognizable but not a decider

    Paradox

    • Barber Paradox: A barber shaves all those, and those only, who do not shave themselves.
    • Contradiction: If the barber shaves himself, he doesn't; if he doesn't shave himself, he does. No solution exists.
    • Russell's Paradox: The set of all sets that are not members of themselves. Leads to inconsistency in set theory.

    Comparing Infinities

    • The size of a finite set is the number of elements.
    • E.g., {1, 3, 5, 7} has size 4 and {a, b, c} has size 3.
    • A function f : A → B is 1-to-1 (injective) if different elements in A are mapped to different elements in B.
    • A function f : A → B is onto (surjective) if every element in B is mapped to by at least one element in A.
    • Sets A and B are the same size if there's a one-to-one, onto function (bijection) between them.
    • Example: The set of odd natural numbers is the same size as the set of all natural numbers.
    • There exists a bijection from N to O. Examples of bijections f:N->O includes f(n) = 2n - 1

    Other Topics

    • Infinite binary sequences are uncountable
    • The set of all languages is uncountable
    • The set of all Turing Machines is countable
    • There exists a language that is not Turing-recognizable

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    07-Lecture PDF - Context-Sensitive Grammars

    Description

    This quiz covers context-sensitive grammars as discussed in Chapter 4 of the reading assignment. Key concepts include the definition of context-sensitive grammar, rules, and examples, along with a review of related topics such as context-free grammars. Test your understanding of these important language structures.

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