Equivalence of the frame and halting problems chris fields. Tripakis in folk theorems on the determinization and. In this chapter we will look at three undecidable decision problems. May 27, 2017 one of the challenges i found in teaching ap computer science principles is the wording of some of the standards makes it difficult to define what sort of tangible experiences my students should have with the content. It turns out that some classical undecidable problems are, in fact, strongly undecidable, i. Undecidable problems for recursively enumerable languages. Undecidability and intractability in theoretical physics. Posts correspondence problem pcp is an example of a problem that does not mention tms in its statement, yet is undecidable. Propositions, unsolvable problems and computable functions.
Finding unsolvable problems we can use the fact that l d. Hilberts entscheidungsproblem is mathematics decidable. As in the traditional literature, undecidability of a problem p in the library is often established by constructing a manyone reduction from an undecidable problem to p. Important undecidable problems 26 solving undecidable problems undecidable means there is no program that 1. Handwritten theory of computation notes pdf toc notes pdf. Decidable and undecidable problems computer action team. Undecidable problem semidedidable or totally not decidable a problem that cannot be solved for all cases by any algorithm whatsoever. Undecidable problems in unreliable computations sciencedirect. This is often done via an intermediate step, where a ram machine with a single register is used. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. Apr 10, 2014 this implies that the selfapplicability problem is undecidable, and so is the halting problem.
So must show how a tm that decides halttm can be used to decide atm. In all the undecidable decision problems we present, the source of the undecidability 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. The essential idea behind the model is to use a timed automaton control automaton to describe the release or arrival patterns of the tasks. Undecidable undecidable undecidable undecidable undecidable lecture 17. Ia problem is not algorithmically solvable if every tm loops on at least one input. Larcheywendling a library of undecidable problems in coq lola 2018 july 12 1.
Basic papers on undecidable propositions, unsolvable problems and computable functions, authorm. Given two programs pi and p2, do both problems halt. The post correspondence problem due to emil post is another undecidable problem that turns out to be a very helpful tool for proving problems in logic or in formal language theory to be undecidable. The value 1 problem is the special case of the isolation problem when. Undecidable problems in fractal geometry download pdf.
The rst undecidable problems found by church, post and turing were either native to mathematical logic or dependent on a xed model of computation. If problem p reduces to problem q, and p is undecidable, then q is undecidable. It is suggested that such problems are common in physical models, and some other potential examples are discussed. We prove that this problem is decidable or undecidable depending. Important undecidable problems 25 impossibility of vulnerability detection lecture 18. Simant dube department of mathematics, statistics, and computing science, university of new. More formally, an undecidable problem is a problem whose language is not a. Growth rate of functions, the classes of p and np, quantum computation. Undecidable problem 1 undecidable problem in computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct a single algorithm that always leads to a correct yesorno answer. From pcp, we can prove many other nontm problems undecidable. Pdf undecidable problems for propositional calculi with.
S on input m, w, an encoding of a tm m and a string w. The domino problem of the hyperbolic plane is undecidable. Undecidable problems in fractal geometry by simant dube. Undecidable verification problems for programs with unreliable channels. This work is a survey on decidable and undecidable problems in matrix theory. A problempcan be shown undecidable by giving an undecidable problemqand a computable function reducingqtop. Ichapter 4 discusses problems that are solvableunsolvable by tms. Undecidable problems about timed automata springerlink. Towards a library of formalised undecidable problems in coq. Examples of undecidable problems for 2generator matrix. M is a tm and m halts on input w proof is by reduction from atm. If the cyk algorithm produces a parse tree, accept. One is hilberts tenth problem, another is posts correspondence problem and the third is the halting problem. Indian institute of technology, kharagpur math 204.
Pdf we study schedulability problems of timed systems with nonuniformly recurring computation tasks. For any of these problems, in principle we can compute a speci c ifor which y. Arguments are presented by john lucas, david chalmers, william reinhardt and timothy. Show that we can build a tm that uses m as a subroutine in order to recognize l. Additionally, in section 4, w e also pro v decidabilit y for problems where ariables o ccur at most once. We then reformulate the frame problem as a quantum decision problem, and show that it is undecidable by any finite quantum computer. On the complexity of undecidable problems in automata theory. Because a tm is undecidable, contradiction is shown. Pdf decidable and undecidable problems about quantum. Specifically, we construct families of translationallyinvariant, nearestneighbour hamiltonians on a 2d square lattice of dlevel quantum systems d constant, for which determining whether the system is gapped or gapless is an undecidable problem.
Table 1 decidable and undecidable problems for probabilistic and quantum automata. Some general results about hierarchies of undecidable problems in automata theory are given, and studies are described which show how properties of sets. We discuss some basic undecidable problems for contextfree languages, starting from valid and invalid computations of tms. Basic papers on undecidable propositions, unsolvable problems and computable functions dover books on mathematics on amazon. Still in the sixth section, we give a detailed description and a counting of the tiles. While termination is decidable for lossy counter machines, structural termination termination for every input is undecidable. Theproblemofdeciding whetheragivencfgisambiguousis undecidable. Decidability and undecidability stanford university. Basic papers on undecidable propositions, unsolvable problems. L contains two different strings of the same length all these problems are undecidable. Highly undecidable problems about recognizability by.
Undecidable problems about turing machines free download as powerpoint presentation. Generic complexity of undecidable problems springerlink. Emil post stated in 1944 thath10 begs for an unsolvability proof 26. In particular, since group theory is at the core of large chunks of modern mathematics and physics, there are a lot of problems within those areas that. Posts correspondence problem but were still stuck with problems about turing machines only. Definition of posts correspondence problem, the modified pcp, other undecidable problems.
The corresponding stability problem is then undecidable. Finland abstract we show that the following two problems are undecidable. Because the answer is a contradiction, such a program cannot exist the problem is therefore undecidable this is called the halting problem. Three problems on the decidability and complexity of stability. In combinatorics it is known in general to be undecidable whether a given set of tiles can cover the plane see page 19. Pdf absolutely undecidable problems and the power of. Request pdf undecidable problems about timed automata we solve some decision problems for timed automata which were recently raised by s. In fact, it turns out that eqcfg is not decidable, that is, the equivalence problem for contextfree grammars is undecidable. A good example of this is the standard on undecidable problems. Jul 06, 2010 both these problems are known to be undecidable 11,4,3. We reduce from atm by designing a pda that accepts all strings that are not computation strings.
An immediate consequence of our result is the undecidability of the existence of arbitrarily large structures selfassembled using tiles from a given tile set. Nov 09, 2017 decidable, or prove from scratch that the problem is hard undecidable e. Gilman in 7, where several semialgorithms for the discreteness problem in psl2. 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. The posts correspondence problem is undecidable when the alphabet has at least two elements. It is undecidable whether a pda accepts every string. One of the first problems suspected to be undecidable, in the second sense of the term, was the word problem for groups, first posed by max dehn in 1911, which asks if there is a finitely presented group for which no algorithm exists to determine whether two words are equivalent. Pdf decidable and undecidable problems in schedulability. We show that in the case of timed buchi automata accepting infinite timed words some of these problems are. Iknowing which problems are unsolvable is important and pretty cool.
The halting problem is undecidable although it is a fundamental problem in software and hardware design. This paper discusses the implications of godels incompleteness theorems on epistemology, especially on the question of computability of mathematical reasoning. We will construct an algorithm a to solve the halting problem. Assume that there is an algorithm b to solve the bothhalt problem. Is it decidable whether a given turing machine a has at least 481 states. An instance of the post correspondence problem for short, pcp is given by two sequences u u.
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. There are many wellknown examples of undecidable problems. A related undecidable problem is whether two integer matrices can be multiplied together in some sequence to yield the zero matrix. Totality for pdas is undecidable we now prove our earlier claim that totality for contextfree languages is undecidable. H10, to the contrary, can be stated to every mathematician and its formulation is independent from a model of computation. Undecidable problems about timed automata request pdf. Undecidable problems in unreliable computations core. For more information on the structure of the library, the synthetic approach, and included problems see publications below, our wiki, look at the slides or the recording of. A dfa is an abstract machine that solves a pattern matching problem.
Convert p 2s input instance to p 1s input instance s. Undecidable theories pdf download full pdf read book page. Lecture notes on theory of computation module bput. A proven undecidable problem the strategy is to design an oversight program that, given an input program, arrives at a contradiction. The undecidability of intuitionistic linear logic yannick forster and dominique larcheywendling lola 2018 july 12 computer sci ence saarl and univer sity y. There are well known reductions of the halting problem for turing machines tm to the post correspondence problem pcp, and of pcp to the intersection problem for contextfree grammars cfi.
In computability theory and computational complexity theory, an undecidable problem is a. Proofs of algorithmic undecidability are either direct or indirect. Reduce the halting problem onto the posts correspondence problem. Find a string over the alphabet a that can be composed in two different ways. For a recursively enumerable language l it is undecidable to determine whether l is finite. Decidable and undecidable problems turing machine pdf bitbin. Sep 25, 2006 we show that the problem of the minimization of the number of clocks of a timed automaton is undecidable. Because the answer is a contradiction, such a program cannot exist the problem is therefore undecidable this is called the halting problem, and the contradiction proved by alan turing. Ho w ev er, since sim ultaneous rigid euni cation is decidable for one equation, e pro v e in section 4 decidabilit y for one secondorder ariable problems. Iwhy try to write a program to solve a problem when someones already told you it cant be done. Decidable and undecidable problems about quantum automata. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. Undecidable problems about turing machines computability.
As original source for the undecidability results the book refers to barhillel etal. Download pdf download citation view references request permissions export to collabratec alerts. Home cheriton school of computer science university of. About the emptiness problem, we provide a new simple undecidability proof and prove that it is undecidable for automata with as few as two probabilistic transitions. Iwhy try to write a program to solve a problem when someones already told you it. Pdf decidable and undecidable problems in matrix theory. It is also undecidable whether the shuffle of two timed regular languages is timed regular. Construct p using previous description that has a match i.
Equivalent language cannot be recognized by a turing machine that halts for all inputs. Download the ios download the android app other related materials. Generic complexity of undecidable problems myasnikov. Highly undecidable problems for in nite computations. We show that the spectral gap problem is undecidable.
Context a problem is decidable if some turing machine decides solves the. We need the fact that a pda can decide whether one con. Our goal is to prove this problem about strings to be undecidable, and then use its undecidability to prove other problems undecidable. Notice that almost all undecidability proofs rely on the undecidability of the post correspondence problem which is complete for the class of recursively enumerable problems, i. Mar 17, 2003 undecidable verification problems for programs with unreliable channels inform. It is also undecidable whether two sets of relations specify the same group or semigroup.
To say that a problem is undecidable means that there is no way, even given unlimited resources and an infinite amount of time, that the problem can be decided by algorithmic means. Posts correspondence problem pcp involves strings rather than tm. 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. A direct proof, such as the one outlined above, normally uses diagonalization in some form. Here are some examples of decision problems involving turing machines. Important undecidable problems 28 recap if you can simulate a turing machine with programming language pl, it is a universal programming language there is no algorithm for deciding halts for p. Davis published 1966 mathematics, computer science many cyberinformaticians would agree that, had it not been for. Undecidable problems we will now discuss the notion of undecidability. 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. The posts correspondence problem is an undecidable decision problem that was introduced by emil leon post in 1946.
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