Foundations of Computing Fall 2024
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Questions and Answers

Which model of computation is most accurate for describing a general-purpose computer?

  • Context-free grammars
  • Decision trees
  • Turing machines (correct)
  • Finite automata

    • What type of problems does finite automata primarily address?

  • Machine learning models
  • Integer computations
  • Text processing and compilers (correct)
  • Sorting algorithms

    • What do decision problems represent in the context of models of computation?

  • Problems requiring a yes or no answer (correct)
  • Problems requiring numerical solutions
  • Problems involving complex algorithms
  • Problems with multiple outcomes

    • Which of the following is an example of a decision problem?

    <p>Is the number n prime?</p> Signup and view all the answers

    If set A is a subset of set B, which of the following must be true?

    <p>Every member of A is also a member of B</p> Signup and view all the answers

    In set notation, which symbol denotes that an element is not a member of a set?

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

    What is an infinite set?

    <p>A set containing infinitely many elements</p> Signup and view all the answers

    What is the main purpose of studying the limits of a model of computation?

    <p>To determine what problems can be solved using it</p> Signup and view all the answers

    To pass the course, students must achieve which minimum weighted average in their midterm and final exam?

    <p>50%</p> Signup and view all the answers

    What is the maximum weight of the final exam in the overall course grade?

    <p>35%</p> Signup and view all the answers

    What happens if an assignment is submitted 3 days late?

    <p>The assignment will not be accepted and receive a score of zero.</p> Signup and view all the answers

    How many students are allowed to work together on an assignment?

    <p>Up to 2 students.</p> Signup and view all the answers

    What is the weight of quizzes in the overall grading system?

    <p>12%</p> Signup and view all the answers

    What is the consequence of providing your work to another student?

    <p>You will be awarded a zero for the submitted work.</p> Signup and view all the answers

    If a student misses the midterm exam without a legitimate reason, what grade will they receive?

    <p>0%</p> Signup and view all the answers

    What is the total percentage weight allocated to class participation?

    <p>3%</p> Signup and view all the answers

    What condition must be met for a graph to be considered a tree?

    <p>It must be connected and have no simple cycles.</p> Signup and view all the answers

    Which of the following correctly defines a directed path?

    <p>A path in which all arrows point in the same direction.</p> Signup and view all the answers

    If Σ1 = {0, 1}, which of the following is a valid string over Σ1?

    <p>01001</p> Signup and view all the answers

    What is the length of the empty string ε?

    <p>0</p> Signup and view all the answers

    Which of the following languages is a subset of Σ* for the alphabet {a, b}?

    <p>{a, ab, aab}</p> Signup and view all the answers

    Which statement is true regarding string concatenation?

    <p>|xy| equals |x| plus |y|.</p> Signup and view all the answers

    For two languages L1 and L2 over the alphabet Σ, which operations also result in a language over Σ?

    <p>L1 ∪ L2, L1 ∩ L2, and L1 - L2</p> Signup and view all the answers

    What does the notation $n_σ(x)$ represent?

    <p>The number of occurrences of the symbol σ in the string x.</p> Signup and view all the answers

    What is the result of concatenating the languages {a, aa} and {ε, b, ab}?

    <p>{a, ab, aab, aa, aaab}</p> Signup and view all the answers

    What does the notation L* denote for a language L?

    <p>The language of strings that can be formed by concatenating zero or more strings in L</p> Signup and view all the answers

    Which operation denotes the conjunction in Boolean logic?

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

    What is an example of a tautology?

    <p>p ∨ ¬p</p> Signup and view all the answers

    Which of the following represents a contradiction?

    <p>p ∧ ¬p</p> Signup and view all the answers

    In the expression 'P if and only if Q', what does 'if' signify?

    <p>P only if Q, meaning P cannot be true without Q also being true</p> Signup and view all the answers

    What operation is designated by the symbol ⊕ in Boolean logic?

    <p>Exclusive OR operation</p> Signup and view all the answers

    If a language L is defined as {ab, bab}* ∪ {b}{ba}{ab}, what can this language produce?

    <p>All strings from two patterns involving 'a' and 'b'</p> Signup and view all the answers

    What does the statement 'P if Q' imply?

    <p>If Q is true, then P is true.</p> Signup and view all the answers

    Which of the following is a method to prove that two sets A and B are equal?

    <p>Prove that every member of A is in B and every member of B is in A.</p> Signup and view all the answers

    What is demonstrated by the proof of the sum of the degrees of all nodes in a graph G being even?

    <p>Each edge connects two distinct nodes.</p> Signup and view all the answers

    How is a proof by contradiction structured?

    <p>Assume the theorem is false and find a contradiction.</p> Signup and view all the answers

    In the proof regarding integers, what outcome is derived when assuming n^3 + 5 is odd?

    <p>A contradiction with an even integer is found.</p> Signup and view all the answers

    What are the steps involved in proof by induction?

    <p>Prove P(1) is true and assume P(i) to show P(i+1) is true.</p> Signup and view all the answers

    What does closure property for integers indicate?

    <p>The sum of any two integers is an integer.</p> Signup and view all the answers

    Which statement is accurate regarding the inductive proof's base case?

    <p>It serves as the starting point for induction.</p> Signup and view all the answers

    What happens if a student submits an assignment 2 days late?

    <p>A 15% deduction is applied for each day late.</p> Signup and view all the answers

    What is the total weight of the assignments in the overall course grade?

    <p>28%</p> Signup and view all the answers

    Under what circumstances can a midterm exam be shifted to account for a missed score?

    <p>If the student has a legitimate documented reason.</p> Signup and view all the answers

    How must quizzes be taken according to course policies?

    <p>Individually during the lecture time.</p> Signup and view all the answers

    What is the penalty for providing your work to another student?

    <p>Both students will receive a zero on the assignment.</p> Signup and view all the answers

    What characterizes a computationally hard problem?

    <p>It is primarily easy except in the worst-case scenarios.</p> Signup and view all the answers

    What does automata theory primarily focus on?

    <p>The definitions and properties of mathematical models of computation.</p> Signup and view all the answers

    Which of the following statements is true regarding the classification of problems?

    <p>Some problems cannot be solved by any computer, such as certain mathematical statements.</p> Signup and view all the answers

    What is the significance of understanding the difficult aspects of computer problems?

    <p>It allows for the possibility of settling for an imperfect but workable solution.</p> Signup and view all the answers

    In the context of problem solving in computing, which factor determines how resources are utilized?

    <p>The simplicity or complexity of the problem at hand.</p> Signup and view all the answers

    What is a characteristic feature of decision problems in the context of models of computation?

    <p>They are represented by sets of strings.</p> Signup and view all the answers

    Which model of computation is primarily used in programming languages?

    <p>Context-free grammars</p> Signup and view all the answers

    In set theory, which statement is true about a subset?

    <p>A subset can be an empty set.</p> Signup and view all the answers

    What type of output does a computation problem typically require?

    <p>A specific value for certain inputs.</p> Signup and view all the answers

    Which of the following is a limitation of finite automata?

    <p>Inability to handle context-free grammars.</p> Signup and view all the answers

    What is the role of sets in mathematical notation?

    <p>To represent grouped objects as units.</p> Signup and view all the answers

    What does the symbol ∈ signify in set theory?

    <p>Membership of an element in a set.</p> Signup and view all the answers

    Which condition must be true for a problem to be classified as a function problem?

    <p>It involves a specific output for each input.</p> Signup and view all the answers

    When referring to models of computation, what is a characteristic of Turing machines?

    <p>They can simulate any algorithmically computable function.</p> Signup and view all the answers

    What is an example of a decision problem based on set theory?

    <p>Determining if a number is prime.</p> Signup and view all the answers

    Students must achieve a score of 50% or more in the weighted average of midterm and final exam to pass the course.

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

    Assignments can be submitted up to 5 days late with a penalty of 10% for each day.

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

    Group assignments can have a maximum of 3 students working together.

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

    Class participation contributes a bonus of 5% to a student's total grade.

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

    If a student misses the midterm exam for any reason, they will receive a score of 0.

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

    Finite automata are primarily used in hardware design.

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

    Model of computation involves classification and computation problems but excludes the proof of limits.

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

    A set is a collection of different elements represented as a whole.

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

    In set notation, the symbol ∈ indicates that an element is not a member of a set.

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

    An infinite set contains a finite number of elements.

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

    A directed graph is strongly connected if there is a directed path connecting every two nodes.

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

    An alphabet must be an infinite set consisting of symbols.

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

    The length of the string '01001' over the alphabet Σ1 = {0, 1} is 5.

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

    The string ordering for all strings over the alphabet {0,1} begins with (0, 1, ε).

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

    A language is a finite subset of strings over an alphabet.

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

    If x = ab and y = bab, then the concatenation of x and y is yx = ababab.

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

    The string formed by reversing 'abc' is 'abc'.

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

    The empty string ε is a member of every language defined over an alphabet.

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

    For two languages L1 and L2 over the alphabet Σ, the intersection L1 ∩ L2 is also a language over Σ.

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

    The notation $n_σ(x)$ represents the number of occurrences of every symbol in the string x.

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

    Study Notes

    Course Overview

    • Instructor: Azam Asilian Bidgoli, from the Department of Computer Science.
    • Office hours are on Wednesdays from 16:00-17:00 by appointment in N2076J, Science Building.
    • Class meets on Mondays and Wednesdays from 17:30-18:50.
    • Michael Sipser - "Introduction to the Theory of Computation", 3rd edition.
    • John C. Martin - "Introduction to Languages and Theory of Computation", 4th edition.

    Course Requirements

    • Minimum 50% average required across midterm and final exam to pass.
    • Grading breakdown:
      • Assignments: 28% (7% each for four assignments)
      • Quizzes: 12%
      • Midterm Exam: 25%
      • Final Exam: 35%
      • Class Participation: 3% (bonus)

    Policies

    • Assignments accepted up to 2 days late with a 15% penalty per day. No credit after 2 days.
    • Quizzes taken individually during class time; schedules announced one week prior.
    • Group assignments allowed, limited to 2 students per group, with registration required.
    • Plagiarism will result in a zero grade; sharing work is academic misconduct.

    Exam Policies

    • Missing a midterm results in a score of 0 unless a legitimate reason is documented.

    Models of Computation

    • Finite Automata: Used in text processing, compilers, and hardware design.
    • Context-Free Grammars: Important for programming languages and AI.
    • Turing Machines: A more accurate representation of general-purpose computing.

    Problem Classification

    • Decision Problem: Asks if an input is of type A (e.g., Is n prime?).
    • Function Problem: Seeks specific output for given input (e.g., finding the minimum cost of a Hamiltonian tour).

    Mathematical Notions

    • Sets: Groups of objects with elements represented as units (e.g., S = {7, 21, 57}).
    • Membership: Denoted with symbols ∈ (member of) and ∉ (not a member).
    • Subset: A is a subset of B if all members of A are also members of B (A ⊆ B).
    • Graphs: Trees are connected graphs without cycles. Directed graphs have arrows, and strongly connected graphs have paths between all nodes.

    Strings and Languages

    • Alphabet: A nonempty finite set of symbols (e.g., Σ = {0, 1}).
    • String: A finite sequence of symbols from an alphabet.
    • Language: A set of strings over an alphabet, denoted as Σ*.
    • Concatenation: Combining strings (e.g., if x = ab and y = bab, then xy = abbab).

    Boolean Logic

    • Operations: Includes NOT (¬), AND (∧), OR (∨), XOR (⊕), equality, and implication (→).
    • Truth Tables: Display logical relationships between variables.
    • Tautology: Always true (e.g., p ∨ ¬p).
    • Contradiction: Always false (e.g., p ∧ ¬p).

    Proof Techniques

    • Finding Proofs: A statement "P if and only if Q" needs to establish two directions: P ⇒ Q and Q ⇒ P.
    • Proof by Contradiction: Assume a statement is false, leading to a contradiction.
    • Proof by Induction: Establish base case P(1) and show P(k) leads to P(k+1).

    Example Proofs

    • Degree Sum in Graphs: Every edge connected to two nodes ensures that the sum of degrees is even (2e).
    • Example of Proof by Contradiction: For integers n, if n³ + 5 is odd, then n must be even.

    Creating Languages

    • Operations on Languages: Languages can be combined using union (L1 ∪ L2), intersection (L1 ∩ L2), and concatenation (L1 L2).
    • Exponential Notation: ak = a concatenated k times; L* represents all combinations of strings from L.

    These notes cover the essentials of coursework, important definitions, and proof techniques relevant to Foundations of Computing.

    Course Overview

    • Course titled "Foundations of Computing" for Fall 2024.
    • Instructor: Azam Asilian Bidgoli; office located in N2076J, Science Building.
    • Office hours on Wednesdays from 16:00-17:00 by appointment.
    • Classes held on Mondays and Wednesdays from 17:30-18:50.

    Course Requirements

    • Passing requires a minimum of 50% in combined midterm and final exam scores, evaluated as (weighted midterm + weighted final) ≥ 30.
    • Grades distribution includes:
      • Four Assignments: 28% (7% each)
      • Quizzes: 12%
      • Midterm Exam: 25%
      • Final Exam: 35%
      • Class Participation: 3% (Bonus)

    Policies on Assignments and Exams

    • Assignments accepted late for up to 2 days, incurring a 15% deduction per late day.
    • Quizzes taken individually during lectures; schedules announced one week in advance.
    • Assignments permitted in groups of up to 2 students, requiring registration.
    • No credit after 2 days late on assignments; plagiarism results in a score of zero.

    Midterm Policies

    • Missed midterms graded as 0 unless a legitimate reason is provided.
    • Weight shift for the final may apply for confirmed exceptional circumstances.

    Key Questions in Computing

    • What problems can computers solve?
    • What resources are required for problem-solving?
    • Are certain problems inherently harder than others?

    Central Areas of Focus

    • Automata: Study of abstract machines and the problems they can solve.
    • Computability: Classification of problems as solvable or unsolvable.
    • Complexity: Understanding why some problems are computationally difficult, contrasting easy problems like sorting with harder ones like scheduling.

    Computability Theory

    • Distinction between solvable and unsolvable problems.
    • Example of an unsolvable problem: Determining the truth of all mathematical statements.

    Automata Theory

    • Focuses on definitions and properties of computation models.
    • Serves as an initial study area for computational foundations.

    Models of Computation

    • Finite Automata: Used in text processing and hardware design.
    • Context-Free Grammars: Applicable in programming languages and AI.
    • Turing Machines: Detailed model representing general-purpose computers.

    Mathematical Notions and Terminology

    • Sets: Groups of objects, members identified with symbols ∈ (member of) and ∉ (not member of).
    • Graphs: Definitions of connected graphs and directed graphs; strongly connected if all nodes are interlinked by directed paths.

    Strings and Languages

    • Defined as finite sequences of symbols from a given alphabet.
    • Examples of languages include distinct subsets like the empty language and specific patterns.

    Operations on Strings and Languages

    • Operations include concatenation, union, intersection, and set difference.
    • Language notation includes L* representing all concatenated strings formed from L.

    Boolean Logic

    • Constructs based on TRUE and FALSE values.
    • Operations include NOT (¬), AND (∧), OR (∨), XOR (⊕), and implication (→).
    • Truth tables analyze logical expressions.

    Finding Proofs

    • Involves constructing two-part statements in the form "P if and only if Q" (P ↔ Q).
    • Forward direction established through the implication "If P is true, then Q is true" (P ⇒ Q).

    Course Overview

    • Instructor: Azam Asilian Bidgoli, from the Department of Computer Science.
    • Office hours are on Wednesdays from 16:00-17:00 by appointment in N2076J, Science Building.
    • Class meets on Mondays and Wednesdays from 17:30-18:50.
    • Michael Sipser - "Introduction to the Theory of Computation", 3rd edition.
    • John C. Martin - "Introduction to Languages and Theory of Computation", 4th edition.

    Course Requirements

    • Minimum 50% average required across midterm and final exam to pass.
    • Grading breakdown:
      • Assignments: 28% (7% each for four assignments)
      • Quizzes: 12%
      • Midterm Exam: 25%
      • Final Exam: 35%
      • Class Participation: 3% (bonus)

    Policies

    • Assignments accepted up to 2 days late with a 15% penalty per day. No credit after 2 days.
    • Quizzes taken individually during class time; schedules announced one week prior.
    • Group assignments allowed, limited to 2 students per group, with registration required.
    • Plagiarism will result in a zero grade; sharing work is academic misconduct.

    Exam Policies

    • Missing a midterm results in a score of 0 unless a legitimate reason is documented.

    Models of Computation

    • Finite Automata: Used in text processing, compilers, and hardware design.
    • Context-Free Grammars: Important for programming languages and AI.
    • Turing Machines: A more accurate representation of general-purpose computing.

    Problem Classification

    • Decision Problem: Asks if an input is of type A (e.g., Is n prime?).
    • Function Problem: Seeks specific output for given input (e.g., finding the minimum cost of a Hamiltonian tour).

    Mathematical Notions

    • Sets: Groups of objects with elements represented as units (e.g., S = {7, 21, 57}).
    • Membership: Denoted with symbols ∈ (member of) and ∉ (not a member).
    • Subset: A is a subset of B if all members of A are also members of B (A ⊆ B).
    • Graphs: Trees are connected graphs without cycles. Directed graphs have arrows, and strongly connected graphs have paths between all nodes.

    Strings and Languages

    • Alphabet: A nonempty finite set of symbols (e.g., Σ = {0, 1}).
    • String: A finite sequence of symbols from an alphabet.
    • Language: A set of strings over an alphabet, denoted as Σ*.
    • Concatenation: Combining strings (e.g., if x = ab and y = bab, then xy = abbab).

    Boolean Logic

    • Operations: Includes NOT (¬), AND (∧), OR (∨), XOR (⊕), equality, and implication (→).
    • Truth Tables: Display logical relationships between variables.
    • Tautology: Always true (e.g., p ∨ ¬p).
    • Contradiction: Always false (e.g., p ∧ ¬p).

    Proof Techniques

    • Finding Proofs: A statement "P if and only if Q" needs to establish two directions: P ⇒ Q and Q ⇒ P.
    • Proof by Contradiction: Assume a statement is false, leading to a contradiction.
    • Proof by Induction: Establish base case P(1) and show P(k) leads to P(k+1).

    Example Proofs

    • Degree Sum in Graphs: Every edge connected to two nodes ensures that the sum of degrees is even (2e).
    • Example of Proof by Contradiction: For integers n, if n³ + 5 is odd, then n must be even.

    Creating Languages

    • Operations on Languages: Languages can be combined using union (L1 ∪ L2), intersection (L1 ∩ L2), and concatenation (L1 L2).
    • Exponential Notation: ak = a concatenated k times; L* represents all combinations of strings from L.

    These notes cover the essentials of coursework, important definitions, and proof techniques relevant to Foundations of Computing.

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    Description

    This quiz covers the key concepts and theories of computing as introduced in the Foundations of Computing course for Fall 2024. Based on the recommended textbooks by Michael Sipser and John C. Martin, the quiz will test your understanding of computation theories and languages. Prepare to engage with foundational concepts that are essential for any computer science student.

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