What computability means?

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem.

What does the Unsolvability of the halting issue mean?

unsolvable algorithmic problem is the halting problem, which states that no program can be written that can predict whether or not any other program halts after a finite number of steps. The unsolvability of the halting problem has immediate practical bearing on software development.

Is the halting problem decidable?

The halting problem is theoretically decidable for linear bounded automata (LBAs) or deterministic machines with finite memory. A machine with finite memory has a finite number of configurations, and thus any deterministic program on it must eventually either halt or repeat a previous configuration: …

What is computability and Decidability?

Computability is a characteristic concept where we try to find out if we are able to compute every input of a particular problem. Decidability is a generalized concept where we try to find out if there is the Turing machine that accepts and halts for every input of the problem defined on the domain.

What are the characteristics of computability?

The basic characteristic of a computable function is that there must be a finite procedure (an algorithm) telling how to compute the function. The models of computation listed above give different interpretations of what a procedure is and how it is used, but these interpretations share many properties.

What is the difference between complexity and computability?

Computability is about what can be computed. Complexity is about how efficiently can it be computed.

What do you mean by decidable and undecidable problems?

The problems for which we can’t construct an algorithm that can answer the problem correctly in finite time are termed as Undecidable Problems. These problems may be partially decidable but they will never be decidable.

What did Alan Turing prove?

Turing showed, by means of his universal machine, that mathematics was also undecidable. To demonstrate this, Turing came up with the concept of “computable numbers,” which are numbers defined by some definite rule, and thus calculable on the universal machine.

Which of the following is decidable?

Which of the following are decidable? Explanation: (A) Intersection of two regular languages is regular and checking if a regular language is infinite is decidable.

What is TOC Decidability?

In terms of finite automata (FA), decidable refers to the problem of testing whether a deterministic finite automata (DFA) accepts an input string. A decidable language corresponds to algorithmically solvable decision problems.

Is a computable function Decidable?

Suppose N is decidable and M ≤m N via total computable reducing function f(x). Then N(f(x)) is total computable, since both N and f are total computable. Therefore, since M(x) = N(f(x)), it follows that M is also total computable, and hence decidable.

What do you mean by computability in automata?

Finite Automata. Computability theory, discussed in Part 1, is the theory of computation obtained when limitations of space and time are deliberately ignored. In automata theory, which we study in this chapter, computation is studied in a context in which bounds on space and time are entirely relevant.

Is the halting problem unsolvable?

This is the epochal paper where Turing defines Turing machines, formulates the halting problem, and shows that it (as well as the Entscheidungsproblem) is unsolvable. Sipser, Michael (2006). “Section 4.2: The Halting Problem”.

What is halting probability in Computer Science?

Gregory Chaitin has defined a halting probability, represented by the symbol Ω, a type of real number that informally is said to represent the probability that a randomly produced program halts. These numbers have the same Turing degree as the halting problem.

Is it possible to solve the halting problem with a program?

There are many programs that, for some inputs, return a correct answer to the halting problem, while for other inputs they do not return an answer at all. However the problem “given program p, is it a partial halting solver” (in the sense described) is at least as hard as the halting problem.

Can any arbitrary computable function be the halting function?

Therefore any arbitrary computable function f cannot be the halting function h . A typical method of proving a problem to be undecidable is to reduce it to the halting problem. For example, there cannot be a general algorithm that decides whether a given statement about natural numbers is true or false.