Context Free languages are accepted by pushdown automata but not by finite automata. Context free languages can be generated by context free grammar which has the form :. Their union says either of two conditions to be true.

So it is also context free language. L2 will also be context free. So it can be accepted by pushdown automata, hence context free. Their intersection says both conditions need to be true, but push down automata can compare only two. So it cannot be accepted by pushdown automata, hence not context free. Deterministic PDA has only one move from a given state and input symbol, i. It can be recognized by Deterministic PDA. So, it can only be implemented by NPDA.

Question : Consider the language L1,L2,L3 as given below. L1 is a regular language C. All the three languages are context free D. Turing machine can be used to recognize all the three languages.

L1 contains all strings with any no. So, it can be accepted by PDA. L2 contains strings with n no. It can also be accepted by PDA. So, option A is correct. Option B says that L1 is regular. Option C says L1, L2 and L3 are context free. L3 languages contains all strings with n no. So option C is not correct. Option D is correct as Turing machine can be used to recognize all the three languages.With correct knowledge and ample experience, this question becomes very easy to solve.

North node 0 degrees capricornDecidable language -A decision problem P is said to be decidable i. Undecidable language -— A decision problem P is said to be undecidable if the language L of all yes instances to P is not decidable or a language is undecidable if it is not decidable.

Raffaella niroAn undecidable language maybe a partially decidable language or something else but not decidable. If a language is not even partially decidablethen there exists no Turing machine for that language. Partially decidable or Semi-Decidable Language -— A decision problem P is said to be semi-decidable i. The Turing machine will halt every time and give an answer accepted or rejected for each and every string input. All decidable languages are recursive languages and vice-versa.

One way to solve decidability problems is by trying to reduce an already known undecidable problem to the given problem.

### Decidable and Undecidable Problems Table | TOC

By reducing a problem P1 to P2, we mean that we are trying to solve P1 by using the algorithm used to solve P2. If we can reduce an already known undecidable problem P1 to a given problem P2then we can surely say that P2 is also undecidable. If P2 was decidable, then P1 would also be decidable but that becomes a contradiction because P1 is known to be undecidable. Now lets try to reduce the Halting problem to the State Entry problem. A Turing machine only halts when a transition function?

Change every undefined function? Note that the state Q can only be reached when the Turing machine halts. Suppose we have have an algorithm for solving the State Entry problem which will halt every time and tell us whether state Q can be reached or not. By telling us that we can or cannot reach state Q every time, it is telling us that the Turing machine will or will not halt, every time. But we know that is not possible because the halting problem is undecidable.

That means that our assumption that there exists an algorithm which solves the State Entry problem and halts and gives us an answer every time, is false. Hence, the state entry problem is undecidable.

Both Turing machines will halt and give us an answer. Since this system of two Turing machines and a modified AND gate will always stop, this problem is a decidable problem. There are a lot of questions on this topic. There is no universal algorithm to solve them.

Most of the questions require unique and ingenious proofs. Here is where experience is needed. By solving a lot of these problems, one can become very quick in coming up with proofs for these problems on the spot.

**Decidable - Undecidable - Complexity Theory - NP Completeness - TOC - THEORY OF COMPUTATION - part-3**

So, keep practicing. For eg. This article has been contributed by Nitish Joshi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. Writing code in comment? Please use ide. Lets start with some definitions:- Decidable language -A decision problem P is said to be decidable i.

Improved By : VaibhavRai3. Load Comments.The following problems are undecidable problems:. CS Engineering. Home Computer Sc.

48 vs 72 hour fast redditStudy Notes on Undecidability. If non-accepting, it may or may not halt i. Either decidable or partially decidable Decidable Problem If there is a Turing machine that decides the problem, called as Decidable problem.

A decision problem that can be solved by an algorithm that halts on all inputs in a finite number of steps.

## Decidable and Undecidable problems in Theory of Computation

A problem is decidable, if there is an algorithm that can answer either yes or no. A language for which membership can be decided by an algorithm that halts on all inputs in a finite number of steps.

Decidable problem is also called as totally decidable problem, algorithmically solvable, recursively solvable. Undecidable Problem Semi-dedidable or Totally not decidable A problem that cannot be solved for all cases by any algorithm whatsoever. Equivalent Language cannot be recognized by a Turing machine that halts for all inputs. The following problems are undecidable problems: Halting Problem: A halting problem is undecidable problem. There is no general method or algorithm which can solve the halting problem for all possible inputs.

Totality Problem Is the complement of L G1 context-free? Undecidable problems are two types: Partially decidable Semi-decidable and Totally not decidable.

Semi decidable: A problem is semi-decidable if there is an algorithm that says yes. Totally not decidable Not partially decidable : A problem is not decidable if we can prove that there is no algorithm that will deliver an answer. Tags : Computer Sc. Theory of Computation. Oct 13 Computer Sc. Member since Feb Related Posts.Context-free languages have many applications in programming languagesin particular, most arithmetic expressions are generated by context-free grammars.

Different context-free grammars can generate the same context-free language. Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar by comparing multiple grammars that describe the language.

The set of all context-free languages is identical to the set of languages accepted by pushdown automatawhich makes these languages amenable to parsing. Further, for a given CFG, there is a direct way to produce a pushdown automaton for the grammar and thereby the corresponding languagethough going the other way producing a grammar given an automaton is not as direct.

This language is not regular. This set is context-free, since the union of two context-free languages is always context-free. Determining an instance of the membership problem ; i. Context-free recognition for Chomsky normal form grammars was shown by Leslie G. Valiant to be reducible to boolean matrix multiplicationthus inheriting its complexity upper bound of O n 2. Practical uses of context-free languages require also to produce a derivation tree that exhibits the structure that the grammar associates with the given string.

The process of producing this tree is called parsing. Known parsers have a time complexity that is cubic in the size of the string that is parsed. Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata PDA. A special subclass of context-free languages are the deterministic context-free languages which are defined as the set of languages accepted by a deterministic pushdown automaton and can be parsed by a LR k parser.

See also parsing expression grammar as an alternative approach to grammar and parser. The class of context-free languages is closed under the following operations.

That is, if L and P are context-free languages, the following languages are context-free as well:. The context-free languages are not closed under intersection. In formal language theory, questions about regular languages are usually decidable, but ones about context-free languages are often not.

It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language with a different grammar. The following problems are undecidable for arbitrarily given context-free grammars A and B:. According to Hopcroft, Motwani, Ullman[24] many of the fundamental closure and un decidability properties of context-free languages were shown in the paper of Bar-HillelPerles, and Shamir [25].

To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages [25] or a number of other methods, such as Ogden's lemma or Parikh's theorem. From Wikipedia, the free encyclopedia. Main article: Parsing. See Matrix multiplication Algorithms for efficient matrix multiplication and Coppersmith—Winograd algorithm for bound improvements since then.

The grammar for B is analogous. General context-free recognition in less than cubic time Technical report. Carnegie Mellon University. Valiant Journal of Computer and System Sciences.Prerequisite — Turing Machine.

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 to To find the solution of this problem, we can easily devise an algorithm that can enumerate all the prime numbers in this range.

Now talking about Decidability in terms of a Turing machine, a problem is said to be a Decidable problem if there exists a corresponding Turing machine which halts on every input with an answer- yes or no. It is also important to know that these problems are termed as Turing Decidable since a Turing machine always halts on every input, accepting or rejecting it.

Semi- Decidable Problems — Semi-Decidable problems are those for which a Turing machine halts on the input accepted by it but it can either halt or loop forever on the input which is rejected by the Turing Machine.

Such problems are termed as Turing Recognisable problems. These problems may be partially decidable but they will never be decidable. That is there will always be a condition that will lead the Turing Machine into an infinite loop without providing an answer at all.

If we feed this problem to a Turing machine to find such a solution which gives a contradiction then a Turing Machine might run forever, to find the suitable values of n, a, b and c. But we are always unsure whether a contradiction exists or not and hence we term this problem as an Undecidable Problem. Read next articles — DecidabilityUndecidability and Reducibility.

This article is contributed by Aishwarya Agarwal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below. Writing code in comment? Please use ide. Prerequisite — Turing Machine A problem is said to be Decidable if we can always construct a corresponding algorithm that can answer the problem correctly.

Load Comments.The course is designed to provide basic understanding of theory of automata, formal languages, turing machines and computational complexity. The following notes are compiled by Hari Prasad Pokhrel who has been teaching in various Engineering Colleges in Nepal since long time. We would like to thank him for his hard effort in compiling the notes of all subjects and helping for making educational resources accessible easily.

Click on the corresponding link to read online or download the notes. The following two links consist of tutorial questions along with solutions to some problems. I m not reaching in Undecidibility chapter?? Is it not Included or what!! N thaks for Ur Kind Work.

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Related posts. April 5, at am. April 6, at pm. Leave a Reply Cancel reply Your email address will not be published.In computability theory and computational complexity theoryan undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer.

The halting problem is an example: it can be proven that there is no algorithm that correctly determines whether arbitrary programs eventually halt when run. A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as the set of inputs for which the problem returns yes.

These inputs can be natural numbers, but also other values of some other kind, such as strings of a formal language.

Thus, a decision problem informally phrased in terms of a formal language is also equivalent to a set of natural numbers. To keep the formal definition simple, it is phrased in terms of subsets of the natural numbers.

Formally, a decision problem is a subset of the natural numbers. 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. A problem is called partially decidable, semi-decidablesolvable, or provable if A is a recursively enumerable set.

This means that there exists an algorithm that halts eventually when the answer is yes but may run for ever if the answer is no. Partially decidable problems and any other problems that are not decidable are called undecidable.

In computability theorythe halting problem is a decision problem which can be stated as follows:. Alan Turing proved in that a general algorithm running on a Turing machine that solves the halting problem for all possible program-input pairs necessarily cannot exist.

Hence, the halting problem is undecidable for Turing machines. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that an axiomatization of the natural numbers that is both complete and sound is impossible. The "sound" part is the weakening: it means that we require the axiomatic system in question to prove only true statements about natural numbers.

Since soundness implies consistencythis weaker form can be seen as a corollary of the strong form. The weaker form of the theorem can be proved from the undecidability of the halting problem as follows. Assume that we have a sound and hence consistent and complete axiomatization of all true first-order logic statements about natural numbers.

## Decidable and Undecidable Problems Table | TOC

Then we can build an algorithm that enumerates all these statements. This means that there is an algorithm N n that, given a natural number ncomputes a true first-order logic statement about natural numbers, and that for all true statements, there is at least one n such that N n yields that statement.

Now suppose we want to decide if the algorithm with representation a halts on input i. We know that this statement can be expressed with a first-order logic statement, say H ai. So if we iterate over all n until we either find H ai or its negation, we will always halt, and furthermore, the answer it gives us will be true by soundness.

2007 matrix fuse boxThis means that this gives us an algorithm to decide the halting problem. Since we know that there cannot be such an algorithm, it follows that the assumption that there is a consistent and complete axiomatization of all true first-order logic statements about natural numbers must be false.

Undecidable problems can be related to different topics, such as logicabstract machines or topology. Since there are uncountably many undecidable problems, [1] any list, even one of infinite lengthis necessarily incomplete.

There are two distinct senses of the word "undecidable" in contemporary use. The second sense is used in relation to computability theory and applies not to statements but to decision problemswhich are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set.

The connection between these two is that if a decision problem is undecidable in the recursion theoretical sense then there is no consistent, effective formal system which proves for every question A in the problem either "the answer to A is yes" or "the answer to A is no".

Sezione ifla document delivery and resource sharingBecause of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense. The usage of "independent" is also ambiguous, however.

It can mean just "not provable", leaving open whether an independent statement might be refuted. Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means.

Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement.

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