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 (A)

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 (A), X → BCX (B), S → TXC (C)</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 (A)</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. (B)</p> Signup and view all the answers

ATM is a decidable language.

<p>False (B)</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. (D)</p> Signup and view all the answers

ATM is Turing-recognizable.

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

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

<p>False (B)</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 (C)</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 (A)</p> Signup and view all the answers

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

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

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

<p>True (A)</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 (B)</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 (A)</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. (B)</p> Signup and view all the answers

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

<p>False (B)</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 (B), 0000 (D)</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 (B)</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 (B)</p> Signup and view all the answers

Theorem ATM states that ATM is T-recognizable.

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

Any language has a unique characteristic sequence.

<p>True (A)</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

Flashcards

Linearly Bounded Automaton (LBA)

A Turing machine with a tape head that can only move within the bounds of the input string. It cannot move beyond either end of the input.

LBA and Context-Sensitive Languages

A language is context-sensitive if and only if there exists an LBA that accepts it. In other words, you can construct an LBA to recognize any context-sensitive language.

Relationship between Context-Free and Context-Sensitive Languages

Every context-free language is also a context-sensitive language. This means that all languages handled by context-free grammars are also recognized by LBAs.

Context-Sensitive Languages are Decidable

Every context-sensitive language is decidable. This means there exists an algorithm that can determine whether any given string belongs to the language or not.

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Paradox

A problem that arises when a statement or condition leads to a contradiction, regardless of whether it is true or false.

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Acceptance Turing Machine (ATM)

A machine that can accept a description of a Turing Machine (TM) and an input string and decide whether the TM accepts the input string.

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TM D (Contradiction Machine)

A Turing Machine (TM) designed specifically to contradict its own behavior, leading to a paradox and proving the undecidability of ATM.

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Undecidability

A problem is undecidable if there is no algorithm that can solve it for all possible inputs.

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Decider (H) for ATM

A Turing Machine that can determine whether another Turing Machine will accept a given input.

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The Barber Paradox

The barber has a rule: He shaves all those who do not shave themselves. This creates a paradox because if the barber shaves himself, he breaks the rule, but if he doesn't, he must shave himself according to the rule.

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Decidability

A problem is decidable if there exists an algorithm that can solve it for all possible inputs.

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Turing Machine (TM)

A system of rules and symbols that define the behavior of a computer program.

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Russell's Paradox

A set that is not a member of itself.

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One-to-One Function

A function that maps each element of set A to a unique element in set B, ensuring no two elements in A map to the same element in B.

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Onto Function

A function that maps every element in set B to at least one element in set A.

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Correspondence

A function that is both one-to-one and onto, establishing a perfect pairing between every element in set A and set B.

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Cardinality

A way to determine the size of a set by counting its elements.

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Set-Theoretic Paradox

A paradox that arises when considering sets that contain themselves.

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Comparing Infinities

The idea that different sets can have different sizes, even if they contain infinitely many elements.

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Self-Membership

A set that is a member of itself, leading to a contradiction.

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What is ATM language

ATM is a language that consists of pairs of Turing Machines (TM) and input strings, where the TM accepts the input string. The TM accepts the input string using a finite number of steps and comes to a specific accept state.

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Is ATM recognizable?

ATM is recognizable, meaning there exists a Turing Machine that can accept all pairs of TM and input strings where the TM accepts the string. This Turing Machine simulates the TM on the input string and accepts if the simulated TM reaches the accept state.

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Is ATM decidable?

ATM is undecidable, meaning there is no Turing Machine that can always decide whether a given pair of TM and input string is in ATM. This can be proven by contradiction. Assuming ATM is decidable, we can construct a Turing Machine that contradicts its own behavior.

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Decidability and Recognizability

A language L is decidable if and only if both L and its complement (¬L) are Turing recognizable. This means there exist Turing Machines that accept all strings in L and ¬L, respectively.

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Turing Recognizable

A language is Turing-recognizable if there exists a Turing Machine that can accept all strings in the language. This means the Turing Machine will halt and reach an accept state for any string in the language.

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Is ATM Turing recognizable?

ATM (Accepting Turing Machine) is not Turing recognizable, meaning there is no Turing Machine that can accept all pairs of TM and input strings where the TM accepts the string. This is because constructing a Turing Machine that can recognize ATM would lead to a contradiction.

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What is the Universal TM?

The Universal Turing Machine (UTM) is a Turing Machine that can simulate any other Turing Machine, including itself. This concept is foundational to the concept of the stored-program computer and the von Neumann architecture, which is the dominant computer architecture today.

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T-recognizable Language

A language is T-recognizable if there exists a Turing machine that accepts all strings in the language and halts on some strings not in the language.

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Decider Turing Machine

A decider Turing machine accepts all strings in the language and halts (either accepting or rejecting) on all strings.

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Decidable Language

If there is a Turing machine that can decide whether a string is in a language, the language is decidable.

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Set of Languages - Uncountable

The set of all languages is uncountable.

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ATM - Not T-recognizable

The language ATM (Accepting Turing Machine) is not T-recognizable.

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Infinite Binary Sequences - Uncountable

The set of all infinite binary sequences is uncountable.

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T-recognizable Language - Definition

A language is T-recognizable if there exists a Turing machine that accepts all strings in the language and may run forever on strings that are not in the language.

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Language - Not T-recognizable

A language is not T-recognizable if there is no Turing machine that can accept all strings in the language and halt on some strings not in the language.

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String

A sequence of characters, often used to represent text or data.

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Formal Language

A formal system that uses rules to generate a set of strings (words or sentences) that are considered valid in the language.

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Language

A collection of all possible strings that can be generated by the given formal language.

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String (in computer science context)

A finite sequence of symbols from a given alphabet.

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Language (in computer science context)

A string is a sequence of symbols drawn from an alphabet. An infinite set of such symbols is called a language.

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Σ*

The set of all possible strings (including the empty string) that can be formed from an alphabet.

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B

A set of all infinite binary sequences, where each sequence is an infinite string of 0s and 1s.

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Characteristic Sequence

A unique infinite binary string that represents a particular language by indicating the presence or absence of each string in the language (1 for present, 0 for absent)

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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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