Decidable and undecidable problems in matrix theory pdf

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 2. List of undecidable problems wikipedia republished wiki 2. Basic papers on undecidable propositions, unsolvable problems and computable functions dover books on mathematics. Trotter january 20, 1996 abstract the undecidable problems of the title are concerned with the question. Decidable and undecidable problems related to completely 0simple semigroups t. 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. For emphasis, the equivalent term totally decidable problem is sometimes used. Partially decidable semi decidable and totally not decidable. 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. The method to prove undecidabilities is the one found by paterson pat in 1970 to prove that the mortality of finitely generated matrix monoids is undecidable. The halting problem for turing machines is definitely undecidable. If a problem is not semidecidable and not decidable. On the decidability of membership in matrixexponential.

See also undecidable problem, np, npcomplete, solvable, tractable, computable. Are there problems that cannot be if there is some turing machine that accepts every string in l and rejects every. Relationship between nphard and undecidable problems. L1 known not to be in d l1 in d but l1 not in d r l2 new language whose if l2 in d so l2 not in d decidability we are trying to determine to show l 2 undecidable choose a language l 1. And some of the problems we consider turn out to be decidable or to have unknown decidability status. Decidable and undecidable problems in matrix theory 1997. We consider the decidability of the membership problem for matrixexponential semigroups. Given a decider m, you can learn whether or not a string w. Dragan, kent state university 1 chapter 4 decidability contents decidable languages decidable problems concerning regular languages decidable problems concerning contextfree languages the halting problem the diagonalization method the halting problem is undecidable a turing. An example of an easy to understand undecidable problem. 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. Complexity of a decidable problem solution complexity is a measure of the number of elementary operations, say, additions, to decide a decidable problem.

Some examples already appear on the wikipedia page. Turing, and post cover unsolvable problems from the theory of abstract computing machines, mathematical logic, and algebra, and material by kleene and post includes initiation of the classification theory. Cubitt, david perezgarcia november 24, 2011 abstract this note is intended to foster a discussion about the extent to which typical problems arising in quantum information theory are algorithmically decidable in principle rather than in practice. This paper tackles three algorithmic problems for probabilistic automata on finite words. 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. Pdf this work is a survey on decidable and undecidable problems in matrix theory. This implies that its undecidable, but isnt implied by it is undecidable. Undecidable languages are not recursive languages, but sometimes, they may be. The emptiness problem asks, given some probability 0. Decidability and undecidability stanford university. The wikipedia article on nphardness addresses part of your question in the second paragraph of the examples section it lists the halting problem as an example of an undecidable problem that is nphard because of how a turing machine may be transformed into truth value assignments. Examples of undecidable problems for 2generator matrix.

A problem that cannot be solved for all cases by any algorithm whatsoeverequivalently, whose associated language cannot. Two notions of undecidability there are two common settings in which one speaks of undecidability. Pdf decidable and undecidable problems about quantum. Pdf decidable and undecidable problems in matrix theory. A problem is nphard if an oracle for it would make solving npcomplete problems easy i. Decidable undecidable undecidable undecidable undecidable note. However, your ta is saying that you can make the strictly stronger statement that the set is not even semidecidable. On the positive side, it is known that it is decidable for two matrices and for instance for an arbitrary number of upper. A decision problem p is called undecidable if the language l of all yes instances to p is not decidable.

Iterated function systems are used as tools to define. A simple way to see this is to assume that it is decida. In this paper we obtain general results for undecidable first order decision problems about groups that is, problems about elements in a particular group, such as the word and conjugacy problems. Using that pcp is known to be undecidable for seven dominos, we obtain that matrix mortality is undecidable for sets of fifteen matrices. These problems may be partially decidable but they will never be decidable. Are problems in quantum information theory 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. Three problems on the decidability and complexity of stability. For another survey of undecidable problems, see dav77. I suppose for a proof by contradiction that halt tm is decidable. In the case of deterministic nite automata, problems like equivalence can be solved even in polynomial time. 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. A problem is semi decidable if there is an algorithm that says yes. The associated language is called a decidable language.

What are the most attractive turing undecidable problems. 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. N and square matrices a 1, a k, c, all of the same dimension and with real algebraic entries, decide whether c is contained in the semigroup generated by the matrix exponentials exp a i t, where i. 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. In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yesorno answer. 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. This work is a survey on decidable and undecidable problems in matrix theory. What is the difference between decidable and undecidable. That is there will always be a condition that will lead the turing machine into an infinite loop without providing an answer. Has the halting problem of turing machine been proven to be decidable. For every matrix x, the sequence x k admits a subsequence that is a decidable and undecidable quantum automata problems 1471 cauchy sequence, by compactness. Decidable and undecidable problems related to completely 0. This problem can be seen as a continuous analog of babai et. Decidable languages a language l is called decidable iff there is a decider m such that.

List of undecidable problems from wikipedia, the free encyclopedia in computability theory, an undecidable problem is a type of computational problem that requires a yesno answer, but where there cannot possibly be any computer program that always gives the correct answer. Undecidable problems in fractal geometry simant dube department ofmathematics, statistics, and computing science, university of new england, armidale nsw 2351, australia abstract. Although it might take a staggeringly long time, m will eventually accept or reject w. One can trade the number of matrices with their dimension and show that matrix mortality is undecidable as well for two matrices. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. Are there languages that are not decidable by any turing machine tm. On the positive side, it is known that it is decidable for two matrices and for instance for an.

With correct knowledge and ample experience, this question becomes very easy to solve. By inspecting the dfas transitions to see if there is any path to a final state. Decidable languagea decision problem p is said to be decidable i. Definition of undecidable problem, possibly with links to more information and implementations. In this paper, a relationship between the classical theory of computation and fractal geometry is established.

Identifying languages or problems as decidable, undecidable or partially decidable is a very common question in gate. A language is called decidable or recursive if there is a turing machine which accepts and halts on every input string w. The algorithmic theory of matrix groups and semigroups is a staple of computational algebra 3 with numerous applications to automata theory and program analysis 7, 10,11,18,19,26 and has been. The problems studied are simply formulated, however most of them are undecidable. Also known as totally decidable problem, algorithmically solvable, recursively solvable. Decidability and undecidability in toc geeksforgeeks. If you can figure out a systematic way an algorithm to answer the question correctly.

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. Contextfree languages are decidable theorem 9 every contextfree language is decidable. To relate the solutions of two problems if a solution to a problem b can be used to give a solution to a problem a, it seems that a cannot be harder than b e. An nphard is a problem that is at least as hard as any npcomplete problem therefore an undecidable problem can be nphard. The problems studied are simply formulated, however most of them are.