29 Lecture
CS402
Midterm & Final Term Short Notes
Decidability
Decidability is a property of a problem or language, which refers to the ability to determine whether a given input belongs to the problem or language in a finite amount of time. A problem or language is said to be decidable if there exists an a
Important Mcq's
Midterm & Finalterm Prepration
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- Which of the following is an example of a decidable problem?
A) The halting problem
B) The traveling salesman problem
C) The sorting problem
D) The knapsack problem
Answer: C) The sorting problem
The complement of a decidable language is always:
A) Decidable
B) Undecidable
C) Finite
D) Regular
Answer: A) Decidable
Which of the following is an example of an undecidable problem?
A) Checking whether a given number is prime
B) Solving a system of linear equations
C) Computing the square root of a number
D) The halting problem
Answer: D) The halting problem
The Rice Theorem is used to:
A) Prove the undecidability of problems
B) Prove the decidability of problems
C) Classify problems according to their complexity
D) None of the above
Answer: A) Prove the undecidability of problems
Which of the following is an example of a language that is not decidable, but semi-decidable?
A) The set of even numbers
B) The set of prime numbers
C) The set of palindromes
D) The set of all Turing machines that halt on the empty input
Answer: D) The set of all Turing machines that halt on the empty input
The problem of deciding whether a given context-free grammar generates an infinite language is:
A) Decidable
B) Undecidable
C) Semi-decidable
D) Regular
Answer: B) Undecidable
Which of the following statements is true about decidable problems?
A) They are always polynomial-time solvable
B) They are always exponential-time solvable
C) They can be solved in any amount of time
D) None of the above
Answer: A) They are always polynomial-time solvable
The set of all regular languages is:
A) Decidable
B) Undecidable
C) Semi-decidable
D) None of the above
Answer: A) Decidable
Which of the following is an example of a problem that is not even semi-decidable?
A) The halting problem
B) The set of all Turing machines that halt on the empty input
C) The set of all context-free grammars
D) The set of all regular expressions
Answer: C) The set of all context-free grammars
Which of the following statements is true about semi-decidable problems?
A) They are always decidable
B) They are always undecidable
C) They can be solved in any amount of time
D) They can be solved in a finite amount of time, but may not always terminate
Answer: D) They can be solved in a finite amount of time, but may not always terminate
Subjective Short Notes
Midterm & Finalterm Prepration
Past papers included
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- Define decidability.
Answer: Decidability is a property of a problem or language, which refers to the ability to determine whether a given input belongs to the problem or language in a finite amount of time.
What is the difference between a decidable language and an undecidable language?
Answer: A decidable language is one for which there exists an algorithm that can determine whether a given input belongs to the language in a finite amount of time. An undecidable language, on the other hand, is one for which no such algorithm exists.
What is the halting problem?
Answer: The halting problem is the problem of determining, for a given input and program, whether the program will eventually halt or run forever.
What is the Rice Theorem used for?
Answer: The Rice Theorem is used to prove the undecidability of problems.
What is the difference between a decidable problem and a problem that is solvable in polynomial time?
Answer: A decidable problem is one for which there exists an algorithm that can determine whether a given input belongs to the problem in a finite amount of time. A problem that is solvable in polynomial time, on the other hand, is one for which there exists an algorithm that can solve the problem in a number of steps that is proportional to the input size raised to some fixed power.
What is the Church-Turing Thesis?
Answer: The Church-Turing Thesis states that any problem that can be solved by an algorithm can be solved by a Turing machine, and vice versa.
What is the difference between a language that is decidable and a language that is semi-decidable?
Answer: A language that is decidable is one for which there exists an algorithm that can determine whether a given input belongs to the language in a finite amount of time. A language that is semi-decidable, on the other hand, is one for which there exists an algorithm that can accept any input that belongs to the language, but may not halt on inputs that do not belong to the language.
Can a language be both decidable and semi-decidable?
Answer: Yes, a language can be both decidable and semi-decidable. An example of such a language is the set of even numbers.
What is the difference between decidability and computability?
Answer: Decidability refers to the ability to determine whether a given input belongs to a problem or language in a finite amount of time, while computability refers to the ability to solve a problem using an algorithm.
Is there a relationship between decidability and complexity?
Answer: Yes, there is a relationship between decidability and complexity. Decidable problems are typically those that are solvable in polynomial time, while undecidable problems are typically those that are at least as hard as the halting problem, which is known to be unsolvable in any amount of time.
Decidability is a fundamental concept in computer science that refers to the ability to determine whether a given input belongs to a particular problem or language in a finite amount of time. Decidability is closely related to the concept of computability, which refers to the ability to solve a problem using an algorithm.
A problem or language is said to be decidable if there exists an algorithm that can determine whether a given input belongs to the problem or language in a finite amount of time. Conversely, a problem or language is said to be undecidable if no such algorithm exists. One famous example of an undecidable problem is the halting problem, which asks whether a given program will eventually halt or run forever.
The Church-Turing thesis states that any problem that can be solved by an algorithm can be solved by a Turing machine, and vice versa. This thesis provides a framework for understanding the limits of what can and cannot be computed. In particular, the halting problem is known to be unsolvable by any algorithm, which implies that there are some problems that are inherently beyond the reach of computation.
Decidability has important implications for the study of algorithms and complexity theory. Decidable problems are typically those that can be solved in polynomial time, while undecidable problems are typically those that are at least as hard as the halting problem. Complexity theory studies the resources required to solve computational problems, and is concerned with classifying problems into complexity classes based on their difficulty.
In addition to decidability, there is also the related concept of semi-decidability, which refers to the ability to accept any input that belongs to a language, but may not halt on inputs that do not belong to the language. Semi-decidability is a weaker property than decidability, and is related to the concept of Turing recognizability.
Overall, decidability is a fundamental concept in computer science that underlies many of the theoretical foundations of the field. It provides a framework for understanding the limits of computation, and is an important tool for analyzing the complexity of algorithms and problems.