Questions and Answers
Which model of computation is most accurate for describing a general-purpose computer?
What type of problems does finite automata primarily address?
What do decision problems represent in the context of models of computation?
Which of the following is an example of a decision problem?
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If set A is a subset of set B, which of the following must be true?
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In set notation, which symbol denotes that an element is not a member of a set?
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What is an infinite set?
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What is the main purpose of studying the limits of a model of computation?
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To pass the course, students must achieve which minimum weighted average in their midterm and final exam?
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What is the maximum weight of the final exam in the overall course grade?
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What happens if an assignment is submitted 3 days late?
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How many students are allowed to work together on an assignment?
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What is the weight of quizzes in the overall grading system?
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What is the consequence of providing your work to another student?
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If a student misses the midterm exam without a legitimate reason, what grade will they receive?
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What is the total percentage weight allocated to class participation?
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What condition must be met for a graph to be considered a tree?
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Which of the following correctly defines a directed path?
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If Σ1 = {0, 1}, which of the following is a valid string over Σ1?
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What is the length of the empty string ε?
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Which of the following languages is a subset of Σ* for the alphabet {a, b}?
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Which statement is true regarding string concatenation?
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For two languages L1 and L2 over the alphabet Σ, which operations also result in a language over Σ?
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What does the notation $n_σ(x)$ represent?
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What is the result of concatenating the languages {a, aa} and {ε, b, ab}?
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What does the notation L* denote for a language L?
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Which operation denotes the conjunction in Boolean logic?
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What is an example of a tautology?
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Which of the following represents a contradiction?
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In the expression 'P if and only if Q', what does 'if' signify?
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What operation is designated by the symbol ⊕ in Boolean logic?
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If a language L is defined as {ab, bab}* ∪ {b}{ba}{ab}, what can this language produce?
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What does the statement 'P if Q' imply?
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Which of the following is a method to prove that two sets A and B are equal?
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What is demonstrated by the proof of the sum of the degrees of all nodes in a graph G being even?
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How is a proof by contradiction structured?
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In the proof regarding integers, what outcome is derived when assuming n^3 + 5 is odd?
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What are the steps involved in proof by induction?
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What does closure property for integers indicate?
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Which statement is accurate regarding the inductive proof's base case?
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What happens if a student submits an assignment 2 days late?
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What is the total weight of the assignments in the overall course grade?
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Under what circumstances can a midterm exam be shifted to account for a missed score?
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How must quizzes be taken according to course policies?
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What is the penalty for providing your work to another student?
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What characterizes a computationally hard problem?
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What does automata theory primarily focus on?
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Which of the following statements is true regarding the classification of problems?
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What is the significance of understanding the difficult aspects of computer problems?
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In the context of problem solving in computing, which factor determines how resources are utilized?
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What is a characteristic feature of decision problems in the context of models of computation?
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Which model of computation is primarily used in programming languages?
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In set theory, which statement is true about a subset?
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What type of output does a computation problem typically require?
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Which of the following is a limitation of finite automata?
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What is the role of sets in mathematical notation?
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What does the symbol ∈ signify in set theory?
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Which condition must be true for a problem to be classified as a function problem?
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When referring to models of computation, what is a characteristic of Turing machines?
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What is an example of a decision problem based on set theory?
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Students must achieve a score of 50% or more in the weighted average of midterm and final exam to pass the course.
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Assignments can be submitted up to 5 days late with a penalty of 10% for each day.
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Group assignments can have a maximum of 3 students working together.
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Class participation contributes a bonus of 5% to a student's total grade.
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If a student misses the midterm exam for any reason, they will receive a score of 0.
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Finite automata are primarily used in hardware design.
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Model of computation involves classification and computation problems but excludes the proof of limits.
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A set is a collection of different elements represented as a whole.
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In set notation, the symbol ∈ indicates that an element is not a member of a set.
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An infinite set contains a finite number of elements.
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A directed graph is strongly connected if there is a directed path connecting every two nodes.
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An alphabet must be an infinite set consisting of symbols.
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The length of the string '01001' over the alphabet Σ1 = {0, 1} is 5.
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The string ordering for all strings over the alphabet {0,1} begins with (0, 1, ε).
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A language is a finite subset of strings over an alphabet.
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If x = ab and y = bab, then the concatenation of x and y is yx = ababab.
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The string formed by reversing 'abc' is 'abc'.
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The empty string ε is a member of every language defined over an alphabet.
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For two languages L1 and L2 over the alphabet Σ, the intersection L1 ∩ L2 is also a language over Σ.
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The notation $n_σ(x)$ represents the number of occurrences of every symbol in the string x.
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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.
Recommended Textbooks
- 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.
Recommended Textbooks
- 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.
