There is a Turing machine program with the property that for any function f : N → N on the natural numbers, including non-computable functions, there is a model of arithmetic or set theory inside of which the function computed by agrees exactly with on all standard finite input. …
Are all problems computable?
Hilbert believed that all mathematical problems were solvable, but in the 1930’s Gödel
Are all functions computable?
Every such function is computable
. It is not known whether there are arbitrarily long runs of fives in the decimal expansion of π, so we don’t know which of those functions is f. Nevertheless, we know that the function f must be computable.)
Are all continuous functions computable?
A famous result in intuitionistic mathematics is that
all real-valued total functions are continuous
. Since the requirements for a function to be admitted intuitionistically is that it must define a procedure or algorithm, all functions are computable.
How do you prove that a function is computable?
The function f such that f(n) = 1 if there is a sequence of at least n consecutive fives in the decimal expansion of π, and
f(n) = 0 otherwise
, is computable.
What things are not computable?
A non-computable is a problem for which there is no algorithm that can be used to solve it. Most famous example of a non-computablity (or undecidability
What makes a problem Undecidable?
In computability theory, an undecidable problem is a
type of computational problem
What makes a function computable?
Computable functions are the formalized analogue of the intuitive notion of algorithms, in the sense that a function is computable
if there exists an algorithm that can do the job of the function, i.e. given an input of the function domain it can return the corresponding output
. …
What is effectively computable function?
[kəm¦pyüd·ə·bəl ′fəŋk·shən] (mathematics)
A function whose value can be calculated by some Turing machine in a finite number of steps
. Also known as effectively computable function.
What is Uncomputable?
Uncomputable:
function that cannot be computed by any Turing machine
. ● Extra important now to differentiate functions vs. programs/Turing machines. Theorem 9.5: Uncomputable Functions. There exists a function that is not computable by any Turing machine.
Are undecidable problems solvable?
The corresponding informal problem is that of deciding whether a given number is in the set. A decision problem A is called decidable or
effectively solvable if A is a recursive set and undecidable otherwise
.
What is an undecidable problem example?
Examples – These are few important Undecidable Problems:
Whether a CFG generates all the strings or not
? As a CFG generates infinite strings, we can’t ever reach up to the last string and hence it is Undecidable. … Since we cannot determine all the strings of any CFG, we can predict that two CFG are equal or not.
Which problems are decidable?
Definition:
A decision problem that can be solved by an algorithm that halts on all inputs in a finite number of steps
. The associated language is called a decidable language. Also known as totally decidable problem
How do you prove halting problem is undecidable?
Proof by contradiction
: Assume we have a procedure HALTS that takes as input a program P and input data D and answers yes if P halts on input D and no otherwise.
How do you fix halting problems?
Halting problem is perhaps the most well-known problem that has been proven to be undecidable; that is,
there is no program that can solve the halting problem for
general enough computer programs.
What is the difference between decidable and undecidable problems?
A decision problem is decidable if there exists a decision algorithm for it. Otherwise it is undecidable. To show that a decision problem is decidable it is
sufficient to give an algorithm for it
.
