What is meant by a computable function?

What is meant by a computable function?

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.

How do you prove that a function is computable?

Now, consider g(p). As ϕp(x)↓ for all x≥1, g(p)=1 if and only if ϕp(p)↓ by the definition of ϕp, which is actually the function g. Hence, if g would be computable, the halting problem would be computable as well. Therefore, we reach a contradiction.

What are the functions of Turing machine?

A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model’s simplicity, it is capable of implementing any computer algorithm.

What is a computable program?

A function f: Σ* Σ* is computable if there is a. program P that when executed on an ideal computer (one with infinite memory), computes f. That is, for all strings x in Σ*, f(x) = P(x).

What is computable problem?

A mathematical problem is computable if it can be solved in principle by a computing device. Some common synonyms for “computable” are “solvable”, “decidable”, and “recursive”. Hilbert believed that all mathematical problems were solvable, but in the 1930’s Gödel, Turing, and Church showed that this is not the case.

Is every function computable?

I’d like to share a simple proof I’ve discovered recently of a surprising fact: there is a universal algorithm, capable of computing any given function!

What is a computable problem?

Are all Turing machines computable?

Assuming that the Turing machine notion fully captures computability (and so that Turing’s thesis is valid), it is implied that anything which can be “computed”, can also be computed by that one universal machine. Conversely, any problem that is not computable by the universal machine is considered to be uncomputable.

How is the transition function of a TM is defined as?

δ is a transition function which maps Q × T → Q × T × {L,R}. Depending on its present state and present tape alphabet (pointed by head pointer), it will move to new state, change the tape symbol (may or may not) and move head pointer to either left or right. q0 is the initial state. F is the set of final states.

Is Ackermann function computable?

The Ackermann function is the simplest example of a well-defined total function which is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive (Dötzel 1991).

Are all computable functions continuous?

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. This seems to suggest that all computable functions are continuous.

What are non computable functions?

Yet there are also problems and functions that are non-computable (or undecidable or uncomputable), meaning that there exists no algorithm that can compute an answer or output for all inputs in a finite number of simple steps.

What does effective computability mean?

is effectively computable if there is an effective procedure or algorithm that correctly calculates f. An effective procedure is one that meets the following specifications.

Why we use TM in TOC?

A single tape Turing machine has a single infinite tape, which is divided into cells. The tape symbols are present in these cells. A finite control is present, which controls the working of Turing machines based on the given input. The Finite control has a Read/write head, which points to a cell in tape.

What are the special features of TM?

There are various features of the Turing machine:

  • It has an external memory which remembers arbitrary long sequence of input.
  • It has unlimited memory capability.
  • The model has a facility by which the input at left or right on the tape can be read easily.
  • The machine can produce a certain output based on its input.

Is the Turing machines countably infinite set?

Since every Turing machine can be encoded as a finite string over a finite alphabet Q ∪ Σ ∪ Γ ∪ {R, L}, the set of all Turing machines is countable. On the other hand, every language over Σ is uniquely described as an infinite 0,1-sequence.