Researchers with an interest in turing machines, for example, have tackled the issue of the halting problem, looking at when computer programs stop. This paper tackles three algorithmic problems for probabilistic automata on finite words. Thanks for contributing an answer to mathematics stack exchange. What is the difference between decidable and undecidable. Decidable undecidable undecidable undecidable undecidable note. Alan turing, turing machines and stronger next, decidable problems and undecidable problems in pdas and vpas are explained.
A problem that cannot be solved for all cases by any algorithm whatsoeverequivalently, whose associated language cannot be recognized by a. Undecidable definition is not capable of being decided. The problem is known to be undecidable but semidecidable. This can be shown by reducing the halting problem to l for the halting problem instance n, y, create a new machine m for the l problem.
These problems may be partially decidable but they will never be decidable. The associated language is called a decidable language. Reduction of decidable and undecidable problems closed ask question asked 4 years, 1 month ago. A problem is called partially decidable, semidecidable, solvable, or provable if a is a recursively enumerable set. Determining whether or not a function f is total is undecidable. Decidable and undecidable problems computer action team. An undecidable problem is a question that cannot be resolved with the use of one algorithm.
We prove that this problem is decidable or undecidable depending on whether. Undecidable languages are not recursive languages, but sometimes, they may be recursively enumerable. Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines the notion of logical validity. And some of the problems we consider turn out to be decidable or to have unknown decidability status. Because this is impossible, tm h must not exist, so atm is undecidable. A undecidable problem is a decision problemthat are not decidable a decision problem is any arbitrary yesorno question on an infinite set of inputs. Posts correspondence problem but were still stuck with problems about turing machines only. Undecidable problems showed these problems for fa and cfgs decidable.
A proven undecidable problem the idea of the proof is to feed output, reversed, back into the input example. Posts correspondence problem pcp is an example of a problem that does not mention tms in its statement, yet is undecidable. A function or program f is said to be total if fx is defined for all x or similarly, if fx halts for all x. I think the post correspondence problem is a very good example of a simple undecideable problem that is also relatively unknown given a finite set of string tuples a, bba x ab, aa y bba, bb z the problem is to determine if there is a finite sequence of these tuples, allowing for repetition, such that the concatenation of the first half is equal to the concatenation of second half. For an undecidable language, there is no turing machine which accepts the language and makes a decision for every input string w tm can make decision for some input string though. What are the most attractive turing undecidable problems. Decidable problem article about decidable problem by the. Undecidable problems the problems for which we cant construct an algorithm that can answer the problem correctly in finite time are termed as undecidable problems. A decision problem p is called undecidable if the language l of all yes instances to p is not decidable. Partially decidable problems and any other problems that are not decidable are referred to as undecidable. Assume for the sake of contradiction that this language is decidable.
Not all properties of programs are functional some properties of programs are decidable because they are not about the function the. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. We use a pda to show that satisfy0,1,2,3butnot4 is contextfree. Decidable a language l is turing recognizable if some turing machine recognizes it. The emptiness problem asks, given some probability 0. Pdf decidable and undecidable problems about quantum. Two notions of undecidability there are two common settings in which one speaks of undecidability. A decision problem p is decidable if the language l of all yes instances to p is decidable for a decidable language, for each input string, the tm halts either at the accept or the reject state as depicted in the following. Undecidable definition of undecidable by merriamwebster. A valuable collection both for original source material as well as historical formulations of current problems. L m is a tm and l m empty so we can show that l is not decidable, because a tm u that receive l as a input need to test all elements over 1 and then decide to accept in case of m rejected all of them, so it. What are the most attractive turing undecidable problems in mathematics there are thousands of examples, so please post here only the most attractive, best examples.
One of the most wellknown examples of undecidable problems is the halting problem. If b were not undecidable, than we could use the solution to b to decide a a contradiction since a is undecidable. This means there are problems and possibly, or probably, an infinite number of problems that are undecidable, and not related to the halting. Partially decidable problems and any other problems that are not decidable are called undecidable. Rices theorem can be used to show that whether the language accepted by a turing machine is contextfree, regular, or even finite, are undecidable problems. Combining this with the proof from the last paragraph, we have shown. I chose to speak about undecidable problems because we are within a year of the centenary. An instance of the halting problem asks whether turning machine n halts on input y. The halting problem can be used to show that other problems are undecidable. If you can figure out a systematic way an algorithm to answer the question correctly. More undecidable problems rices theorem posts correspondence problem some real problems. Partially decidable semidecidable and totally not decidable. Decidable and undecidable problems about quantum automata article pdf available in siam journal on computing 346.
We will show that a decider for this language can decide the halting problem. Turing recognizable recursively enumerable re a language l is turing decidable if some turing machine decides it. Undecidable problems creating web pages in your account. Examples and counterexamples every contextsensitive grammar is recursive. Sat type problems using both universal and existential quantifiers. But avoid asking for help, clarification, or responding to other answers.
Let l be any language with property p, and let m l be a tm that accepts l. In all the undecidable decision problems we present, the source of the unde cidability can be traced back to a single undecidable decision problem, namely the halting problem, or equivalently the membership problem for listable sets see sections 3. If a reduces to b, and b is decidable, then a must also be decidable, since a solution to b provides a solution to a. Decidable and undecidable problems for probabilistic and quantum automata.
It boils to to arguing that there is a pda which accepts a substring. For another survey of undecidable problems, see dav77. There are even case, where we are sure that a problem is decidable without knowning. A problem is undecidable if it is not decidable, that is if there is no algorithm for. Given a decider m, you can learn whether or not a string w. A language is called decidable or recursive if there is a turing machine which accepts and halts on every input string w. In a similar way well talk about other decision problems, ultimately talking about some underlying language. Are there problems that cannot be if there is some turing machine that accepts every string in l and rejects every. Decidability and undecidability stanford university.
Decidable and undecidable problems table toc january 29, 2018 anup patel resources, toc table to check decidable and undecidable property of all grammar regular, cfl, dcfl, csl, recursive, recursive enumerable. Some examples already appear on the wikipedia page. Tm would be decidable but turing s theorem says a tm is not decidable if l and l are both recognizable, then l is decidable. Problem reduction in the universal tm halting problem we proved that the halting problem is undecidable, translating this into the question of whether a certain language l is undecidable. Decidable problems represent problem using language dfa q0. If a reduces to b, and a is undecidable, then b must also be undecidable. Recursive languages correspond to decidable problems. I suppose for a proof by contradiction that halt tm is decidable. The only subset that is not decidable in 1 is the empty set. Thanks for contributing an answer to computer science stack exchange. This is a subject of interest in mathematics and computer programming, where the undecidable problem has significant implications. In the case of deterministic nite automata, problems like equivalence can be solved even in polynomial time.
Wecandefine%at%least%three%different%time%complexities,%namely,%bestcase,worstcase,andaverage. Cisc462, fall 2018, decidability and undecidability 1 decidability and undecidability decidable problems from language theory for simple machine models, such as nite automata or pushdown automata, many decision problems are solvable. The idea is to look at each oddeven portion which can be easily controlled by the pda and accept the string if at least one of the pairs is not of the form. From pcp, we can prove many other nontm problems undecidable. The essence of reducing one problem to another is the existence of a function from one. We prove that this problem is decidable or undecidable depending on. Completeness results for undecidable bisimilarity problems article pdf available in electronic notes in theoretical computer science 98. To show that the halting problem is not decidable it is necessary to show that there. Decidable and undecidable problems about quantum automata. Here again we have an algorithm, and a decidable problem.
An example of an easy to understand undecidable problem. Because of this, it is traditional to define the decision problem equivalently as. Although it might take a staggeringly long time, m will eventually accept or reject w. The halting problem and other non decidable problems the problems in the set nph are called nphard e. Computer science stack exchange is a question and answer site for students, researchers and practitioners of computer science. The decidable problems are those that lie within turing degree 0. Also known as totally decidable problem, algorithmically solvable, recursively solvable. Researchers with an interest in turing machines, for example, have tackled the issue of the halting problem, looking at when computer programs stop, versus running infinitely. Are there languages that are not decidable by any turing machine tm. Or, given a string of zeros and ones, is it a palindrome.
Since we know atm is undecidable, we can show a new language b is undecidable if a machine that can decide b could be used to build a machine that can decide atm. Pdf completeness results for undecidable bisimilarity. Definition of undecidable problem, possibly with links to more information and implementations. A m is a tm that accepts w tm will show that a is not decidable ie, undecidable tm. Decidable and undecidable problems on context free grammars. See also undecidable problem, np, npcomplete, solvable, tractable, computable. The problems that would be decidable with an oracle for the halting problem would be the turing jump 0. Decidable languages a language l is called decidable iff there is a decider m such that. 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. Decidable and undecidable problems turing machine pdf bitbin. A problem is semidecidable if there is an algorithm that says yes.
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