Decidable and undecidable problems computer action team. A decommenting program can decomment any text file, including. A m is a tm that accepts w tm will show that a is not decidable ie, undecidable tm. Proving undecidability 22 next class examples of some problems we actually care about that are undecidable are there any problems that we dont know if they are decidable or undecidable. Undecidable problem semidedidable or totally not decidable a problem that cannot be solved for all cases by any algorithm whatsoever. The halting problem can be used to show that other problems are undecidable.
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 language is turingrecognizable if there exists a turing machine which halts in an accepting state i its input is in the language. Decidable undecidable undecidable undecidable undecidable note. This means there are problems and possibly, or probably, an infinite number of problems that are undecidable, and not related to the halting. An undecidable problem is a question that cannot be resolved with the use of one algorithm. From pcp, we can prove many other nontm problems undecidable. A decision problem p is called undecidable if the language l of all yes instances to p is not decidable. A decision problem a is called decidable or effectively solvable if a is a recursive set. This is a subject of interest in mathematics and computer programming, where the undecidable problem has significant implications. Well, i seem to have come out to a conclusionthat undecidable problems are a subset of np hard problemsthis is based on the following scenario all problems in np are decidable. Two notions of undecidability there are two common settings in which one speaks of undecidability. The decidable problems are those that lie within turing degree 0.
What is the difference between decidable and undecidable. Undecidable problems showed these problems for fa and cfgs decidable. The problems that would be decidable with an oracle for the halting problem would be the turing jump 0. The problem can be hard moreover, algorithmically undecidable on he whole set of inputs, but decidable moreover, effectively decidable for the almost all. If b were not undecidable, than we could use the solution to b to decide a a contradiction since a is undecidable. Pdf decidable and undecidable problems about quantum automata. A problem is undecidable if it is not decidable, that is if there is no algorithm. If a reduces to b, and a is undecidable, then b must also be undecidable. Decidable and undecidable languages wellesley college.
A problem is called partially decidable, semi decidable, solvable, or provable if a is a recursively enumerable set. Generic complexity of undecidable problems request pdf. In a similar way well talk about other decision problems, ultimately talking about some underlying language. Well allow informal descriptions as long as we are confident they can in principle be turned into tms consider adfa m,w. But avoid asking for help, clarification, or responding to other answers. 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. This can include developing algorithms that work for some values, discussing the specifics of the problem that make it impossible to treat effectively with an algorithm for all values, and related activities. More undecidable problems rices theorem posts correspondence problem some real problems. Find out whether the following problem is decidable.
The 5th postulate states that, given a straight line on a plane and a point on the same plane outside that line, there always exists one and only one straight line passing through that. Apr 30, 2020 far from being a topic of theoretical interest only, the undecidable problem can have important implications for the real world. A decidable language to show that a language is decidable, we have to describe an algorithm that decides it. 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. Decidable and undecidable problems about quantum automata article pdf available in siam journal on computing 346. Rices theorem and other undecidable problemsrices theorem rices theorem 1 iwe have shown that a number of related problems are undecidable. Decidable and undecidable languages computer science. Many, if not most, undecidable problems in mathematics can be posed as word problems. Well soon see examples of languages that are in re but. Decidable problems represent problem using language dfa q0.
Decidable and undecidable languages recursively enumerable. Details of the reduction process differ, depending on the problem. For emphasis, the equivalent term totally decidable problem is sometimes used. Partially decidable problems and any other problems that are not decidable are called undecidable. Or, given a string of zeros and ones, is it a palindrome. Equivalent language cannot be recognized by a turing machine that halts for all inputs. Examination of responses to undecidable problems suggested that the more similar the appearance of the pair of houses, the easier it was to detect the problem as undecidable. 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. I am looking for an undecidable problem that i could give as an easy example in a presentation to the general public.
Is it decidable whether a given turing machine a has at least 481 states. Undecidable semiassociative relation algebras maddux. Decidability and undecidability stanford university. 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. Undecidability in diagonalizable algebras shavrukov, v. The systems attempt to work through the problem can eat through resources, causing the system to freeze or creating system vulnerabilities. Children asked for extra information more often when problems were undecidable than when they were decidable. Relationship between nphard and undecidable problems. 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. 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. Pdf decidable and undecidable problems in matrix theory. There are some problems in np hard which are not undecidable decidable, and which i guess are np complete. The first undecidable problem does a tm accept a given input string. Can you guys list the problems that you are aware that are decidable for context free language and for deterministic context free languages.
By inspecting the dfas transitions to see if there is any path to a final state. Decidable languages a language l is called decidable iff there is a decider m such that. Though undecidable languages are not recursive languages, they may be subsets of turing recognizable languages. Pdf decidable and undecidable problems about quantum. Determining whether a finite set of upper triangular.
These problems may be partially decidable but they will never be decidable. Decidable and undecidable problems on context free grammars. Ntms are known to be no more powerful than tms in the sense that the set of problems decidable by ntms is identical to the set of problems decidable by tms, so clearly by this definition there can be no undecidable. For another survey of undecidable problems, see dav77. For a decidable language, for each input string, the tm halts either at the accept or the reject state as depicted in the following diagram. The proof that the halting problem is undecidable relativizes, that is, it still works if the turing machine is given access to an oracle. 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. In essence, similarly to how all npc problems are related, this whole class of undecidable problems is related. A problem that cannot be solved for all cases by any algorithm whatsoeverequivalently, whose associated language cannot.
L is decidable iff both l and l are recognizable there are languages over 0,1 that are not decidable assuming the churchturing thesis, this means there are problems that no computing device can solve we can prove this using a counting argument. Not all properties of programs are functional some properties of programs are decidable because they are not about the function the. Decidable and undecidable languages 329 selfreference is not a problem consider the following. A decommenting program can decomment any text file, including the decommenting program itself. A proven undecidable problem the idea of the proof is to feed output, reversed, back into the input example. If you can figure out a systematic way an algorithm to answer the question correctly. This paper tackles three algorithmic problems for probabilistic automata on finite words. I special halting problem k i general halting problem h i halting problem on empty tape h 0 imany more results of this type could be shown.
Pdf this work is a survey on decidable and undecidable problems in matrix theory. More undecidable problems the halting problem h f jm stops on wg is undecidable. Define and explain core examples of computational problems, include a, e, eq, halt tm for either dfa or tm explain what it means for one problem to reduce to another use reductions to prove undecidability or decidability decidable undecidable a dfa a tm e dfa a tm c eq dfa halt tm. But were still stuck with problems about turing machines only. 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. Undecidable problems we will now discuss the notion of undecidability. Decidable a language l is turing recognizable if some turing machine recognizes it. A java compiler can compile any java program, including one that specifies a java compiler. The associated language is called a decidable language. Identification of an undecidable problem can occur in the context of math and computer science research.
For example, some computer viruses present systems with undecidable problems. A problem that cannot be solved for all cases by any algorithm whatsoeverequivalently, whose associated language cannot be recognized by a turing machine that halts for all inputs. So i was thinking about various undecidable problems and it occured to me that all the ones that i could think of were reducible, or in fact proven undecidiable by reducing to the halting problem. Theory of computer science rices theorem and other. Lecture notes on theory of computation module bput.
Now we are ready to show that the halting problem is undecidable by means of a similar. Determining whether or not a function f is total is undecidable. One of the most wellknown examples of undecidable problems is the halting problem. 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. Decidable and undecidable problems in theory of computation. The essence of reducing one problem to another is the existence of a function from. The problem can be hard moreover, algorithmically undecidable on he whole set of inputs, but decidable moreover, effectively decidable for the almost all inputs. Once a problem is believed to be undecidable, researchers can apply a variety of tactics to disprove this theory. Aug 30, 2016 heres probably the oldest known example. Oct, 2019 undecidable problem semidedidable or totally not decidable a problem that cannot be solved for all cases by any algorithm whatsoever. The problems studied are simply formulated, however. Although it might take a staggeringly long time, m will eventually accept or reject w. 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. A language is called decidable or recursive if there is a turing machine which accepts and halts on every input string w.
But we almost always show that if there were an algorithm for our specific kind of problem, then we could use that algorithm as a subroutine to produce an algorithm that solves the halting problem. Some undecidable problems about the tracesubshift associated with a turing machine a problem is fpt with respect to parameters p, q if it is decidable in time parameterized complexity of synchronization and road coloring. Using halting problem to prove undecidable problems. In particular, the halting problem for turing machines with oracle access to the usual halting problem isnt solvable by turing machines with oracle access to the usual halting problem. Jan 29, 2018 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.
Well, i seem to have come out to a conclusionthat undecidable problems are a subset of np hard problems this is based on the following scenario all problems in np are decidable. I suppose for a proof by contradiction that halt tm is decidable. Iinstead, we prove a much more general result, rices theorem, which shows that a very. Turing recognizable recursively enumerable re a language l is turing decidable if some turing machine decides it. An example of an easy to understand undecidable problem. An equivalent definition of np is that it consists of all problems that are decidable not just verifiable in polynomial time by a nondeterministic turing machine. Decidable and undecidable languages 309 selfreference is not a problem consider the following. The emptiness problem asks, given some probability 0.
I mean easy in the sense that the mathematics behind it can be described, well, without mathematics, that is with analogies and intuition, avoiding technicalities. Undecidable relativizations of algebras of relations mikulas, szabolcs and marx, maarten, journal of symbolic logic, 1999. Sat type problems using both universal and existential quantifiers. Thanks for contributing an answer to mathematics stack exchange.
The halting problem and other non decidable problems the problems in the set nph are called nphard e. Given a decider m, you can learn whether or not a string w. A decision problem p is decidable if the language l of all yes instances to p is decidable. Here are some examples of decision problems involving turing machines. 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. And some of the problems we consider turn out to be decidable or to have unknown decidability status. There are some problems in np hard which are not undecidable decidable, and which i.
Chapter 4 decidability and undecidability nyu computer science. Posts correspondence problem pcp is an example of a problem that does not mention tms in its statement, yet is undecidable. Ntms are known to be no more powerful than tms in the sense that the set of problems decidable by ntms is identical to the set of problems decidable by tms, so clearly by this definition there can be no undecidable problems in np. If the algorithm deciding h gives the answer \no i. A language is turingdecidable if it halits in an accepting state for every input in the language, and halts in a rejecting state for every other input. Also known as totally decidable problem, algorithmically solvable, recursively solvable.
Definition of undecidable problem, possibly with links to more information and implementations. It is undecidable to determine, given cfgs g 1 and g 2, whether lg 1. I did get some info on list of undecidable problems on stack overflow and wiki, but not related to cfg or dcfg as such in detail. Decidable undecidable undecidable undecidable undecidable undecidable lecture 17. Researchers with an interest in turing machines, for example, have tackled the issue of the halting problem, looking at when. Decidable article about decidable by the free dictionary. In each of these, have access to genie which can answer.
Undecidable languages are not recursive languages, but sometimes, they may be. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. We have shown that a cfl is decidable and a cfg can be simulated by a tm. A problem is called partially decidable, semidecidable, solvable, or provable if a is a recursively enumerable set. A decidable variety that is finitely undecidable jeong, joohee, journal of symbolic logic, 1999. There exists a tm that accepts yes instances, but might reject or.