(definition) Definition: A language for which membership can be decided by an algorithm that halts on all inputs in a finite number of steps --- equivalently, can be recognized by a Turing machine that halts for all inputs. Also known as recursive language, totally decidable language.

What is decidable and undecidable language?

The Turing machine will halt every time and give an answer(accepted or rejected) for each and every string input. A language 'L' is decidable if it is a recursive language. ... If P2 was decidable, then P1 would also be decidable but that becomes a contradiction because P1 is known to be undecidable.

Is every language decidable?

1. (a) True, since every regular language is context-free, every context-free language is decidable, and every decidable language is Turing-recognizable.

What is the difference between a decidable language and a Turing recognizable language?

A language is said to be Decidable if there is a Machine that will accept strings in the language and reject strings not in the language. A Language is called Turing Recognizable if some Turing Machine recognizes it. ... It is a stronger condition than Turing-Recognizable.

What is meant by a Decidable problem?

(definition) 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, algorithmically solvable, recursively solvable.

Decidability and Undecidability

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How do you show a problem is decidable?

Prove that the language it recognizes is equal to the given language and that the algorithm halts on all inputs. To prove that a given language is Turing-recognizable: Construct an algorithm that accepts exactly those strings that are in the language. It must either reject or loop on any string not in the language.

When a problem is decidable?

A problem is said to be Decidable if we can always construct a corresponding algorithm that can answer the problem correctly. We can intuitively understand Decidable problems by considering a simple example. Suppose we are asked to compute all the prime numbers in the range of 1000 to 2000.

Can a language be unrecognizable but decidable?

If no such Turing machine exists, L is unrecognizable. A basic fact about these definitions is: A language L is decidable if and only both L and ¯L are recognizable. Your language ATM is one way of stating the halting problem (there are many equivalent ways).

Which of the following is decidable problem?

1) This is a variation of Turing Machine Halting problem and it is undecidable. 2)CFL are not closed under complement so it is undecidable. 3) Complement of Regular languages is also regular. ... 4) Recursvie language are closed under complement,so it is decidable.

Can a Turing machine decide a language?

A language is recursively enumerable (generated by Type-0 grammar) if it is accepted by a Turing machine. A TM decides a language if it accepts it and enters into a rejecting state for any input not in the language. A language is recursive if it is decided by a Turing machine.

How do you know if a language is decidable?

A language is decidable if and only if it and its complement are recognizable. Proof. If a language is decidable, then its complement is decidable (by closure under complementation). Either w ∈ L, or w ∈ L .

Are NP languages decidable?

All languages in P and NP are decidable. Any undecidable language, such as the halting problem, cannot be in NP. The nondeterministic time hierarchy theorem can be used to show that there are languages in NEXP that are not in NP.

Is recursive language decidable?

Recursive languages are also called decidable. ... This type of language was not defined in the Chomsky hierarchy of (Chomsky 1959). All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recursive.

What are semi-decidable languages?

To say that a language L is semi-decidable means that there is an algorithm that accepts precisely the strings in L; however, for a string x /∈ L, the algorithm will either reject or not halt. In order for this intuition to be valid, we have to accept the Church-Turing Thesis.

Are Turing machines decidable?

A language is called Decidable or Recursive if there is a Turing machine which accepts and halts on every input string w. Every decidable language is Turing-Acceptable.

What is difference between Decidability and Undecidability?

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.

What language does Turing machine Mcq use?

Explanation: A language L is recursively enumerable if there is a turing machine that accepts L, and recursive if there is a TM that recognizes L. (Sometimes these languages are alse called Turing-acceptable and Turing-decidable respectively). 3.

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 the intersection of CFL and regular language?

The intersection of a context-free language and a regular language is context- free (Theorem 3.5. 2). The idea of the proof is to simulate a push-down automaton and a finite state automaton in parallel and only accept if both machines accept.

Are decidable languages closed under union?

Theorem 6: The set of Turing-decidable languages is closed under union, intersection, and complement.

Why is ATM not decidable?

D rejects (D), but then H accepted (D,(D)) and hence D accepted (D), contradiction! So D cannot exist, so H cannot exist either (D was built from H). This means that ATM is undecidable.

What is a Turing unrecognizable language?

~A_TM is the canonical example of a Turing-unrecognizable language. This means there does not exist a Turing Machine which will accept the set of all machine-string pairs <M,w> such that M does NOT halt when run on w. The proof of this is very short: Lemma: A_TM is Turing-recognizable.

What is decidable problem example?

Examples. A classic example of a decidable decision problem is the set of prime numbers. It is possible to effectively decide whether a given natural number is prime by testing every possible nontrivial factor.

What does Undecidability mean in literature?

: not capable of being decided : not decidable …

What do you mean by Decidability in logic?

In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined.