Theory of Computation: Question Set – 23
Can a Turing machine compute the factorial of a number?
Yes, a Turing machine can compute the factorial of a number by iterating through a loop that multiplies the current value by the previous value until it reaches the desired input.
What is the Church-Turing thesis?
The Church-Turing thesis is the hypothesis that any function that can be computed by an algorithmic process can also be computed by a Turing machine. This thesis has been widely accepted in the computer science community and is considered a fundamental principle of the theory of computation.
Can a Turing machine compute a non-computable function?
No, a Turing machine cannot compute a non-computable function since by definition such a function cannot be computed by any algorithmic process, including a Turing machine.
What is the difference between a deterministic Turing machine and a non-deterministic Turing machine?
A deterministic Turing machine has only one possible transition for each symbol read from the tape and each state it can be in, while a non-deterministic Turing machine may have multiple possible transitions for the same inputs. Non-deterministic Turing machines can be more powerful than deterministic ones, but they are harder to simulate and verify.
Can a Turing machine simulate a computer program?
Yes, a Turing machine can simulate a computer program by using its tape to represent the computer’s memory and its state to keep track of the program’s execution. This simulation allows the Turing machine to perform the same computations as the computer program, although it may be slower and less efficient.
How can you prove that a problem is undecidable using a Turing machine?
To prove that a problem is undecidable using a Turing machine, you need to show that there is no algorithmic process that can solve the problem on all possible inputs. This can be done by constructing a Turing machine that simulates any possible algorithm for the problem and then showing that there are inputs for which the machine never halts, or that the machine halts with the wrong answer.
Can a Turing machine recognize a context-sensitive language?
Yes, a Turing machine can recognize a context-sensitive language by using its tape to keep track of the context of the input string and its state to apply the appropriate productions of a context-sensitive grammar. This is achieved by using a multi-tape Turing machine or a single-tape Turing machine with a larger alphabet.
Can a Turing machine recognize an infinite language?
Yes, a Turing machine can recognize an infinite language if it has a finite description or if there is an algorithmic process that generates its strings. However, if the language is not recursively enumerable, there is no Turing machine that can recognize it.