And some of the problems we consider turn out to be decidable or to have unknown decidability status. Therefore their work had no practical e ect on areas of math outside of logic. To explain these group theoretic problems without proofs we will be using groups that. M is a tm and m halts on input w proof is by reduction from atm. In this paper, we study algorithmic problems for automaton semigroups and automaton groups related to freeness and finiteness. We use language theory to study the rational subset prob lem for groups. What are the most attractive turing undecidable problems in. Precisely, starting from an initial configuration of the turing machine, we explicitly construct an element of the group such that the turing machine stops if, and only if, this element is of finite order. What are the most attractive turing undecidable problems in mathematics. In a similar way well talk about other decision problems, ultimately talking about some underlying language. Sanchit sir is taking live sessions on unacademy plus for gate 2020 link for subscribing to the course is. Group theory math 1, summer 2014 george melvin university of california, berkeley july 8, 2014 corrected version abstract these are notes for the rst half of the upper division course abstract algebra math 1. Kim and roush consider two algorithmic problems for modules over a ring r. We are starting to see a line of reasoning we can use to find unsolvable problems.
And from this follows the undecidability of various problems about 2d cellular automata see note below and spin systems. Every complete undecidable theory is an essentially undecidable theory, as is formal arithmetic cf. A decision problem p is called undecidable if the language l of all yes instances to p is not decidable. Transpositions are special permutations that only interchange two entries. Thus, in this setup, the border line between decidability and undecidability is known. Essentiallyundecidable theory encyclopedia of mathematics. Blondel, emmanuel jeandel, pascal koiran, and natacha portier abstract. For undecidability in axiomatic mathematics, see list of statements undecidable in zfc.
If you can figure out a systematic way an algorithm to answer the question correctly. Most of the problems are nontrivial and come from a variety of published research articles. Ultimately, the proofs of undecidability rely on a reduction to the halting problem for turing machines. We have reached a contradiction, so as long as nothing else is questionable our assumption must be wrong. The essence of reducing one problem to another is the existence of a function from. 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. So must show how a tm that decides halttm can be used to decide atm. Let denote an equilateral triangle in the plane with origin as the centroid. Show that we can build a tm that uses m as a subroutine in order to recognize l.
Generic complexity of undecidable problems springerlink. The purpose of the paper is to show that the elementary theory of groups is undecidable. In the course of this study, we also exhibit some connections between the algebraic structure of automaton semigroups and their dynamics on the boundary. Undecidability in group theory, analysis, and topology. Pdf decidable and undecidable problems about quantum. 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. Try to show that the unsolvability of that problem entails the unsolvability of other problems.
Problems in group theory dover books on mathematics. Ams proceedings of the american mathematical society. In turn, problems about groups are transformed into problems about turing machines. Dec 07, 2015 decidable and undecidable problems on context free grammars. 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. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. Even before the resolution of hilberts tenth problem, some problems in group theory were proved undecidable. 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. 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. Pdf strongly, super, and absolutely undecidable problems. A permutation is said to be even if it has an even number of inversions, and odd if it has an odd number of inversions. We do not pretend that our list of undecidable problems is complete in any sense. An outgrowth of the conference on decision problems in group theory, university of california at irvine, calif.
This lecture will discuss how undecidability in group theory led to undecidability in topology, and how undecidability in number theory led to undecidability in analysis. Decidable and undecidable problems related to completely 0simple. First, we show that it is undecidable to check whether the group gener. 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. In the case of deterministic nite automata, problems like equivalence can be solved even in polynomial time. Only about 20 years later were examples of undecidable decision problems found elsewhere in mathematics, namely in group theory. In the 1930s, church and turing proved independently that there are decision problems that are undecidable. Then nhas a complement in gif and only if n5 g solution assume that n has a complement h in g. It turns out that some classical undecidable problems are, in fact, strongly. Let gbe a nite group and g the intersection of all maximal subgroups of g.
An algorithmicallyundecidable logical theory, all consistent extensions of which are also undecidable see undecidability. Somewhere in there, undecidable problems fail in your case, the length restriction of solution candidates is probably not fulfilled, as. I think the answer is yes if you believe that zfc is true. In computability theory, an undecidable problem is a problem whose language is not a recursive set. If problem p reduces to problem q, and p is undecidable, then q is undecidable. An elementary theory is an essentiallyundecidable theory if and only if every model of it has an undecidable elementary theory.
This is a good compilation of problems in group theory. Undecidability of the word problem for onerelator inverse. What is the difference between decidable and undecidable. Pdf decidable and undecidable problems about quantum automata. Decision problems and the burnside problem in group theory. Decidable and undecidable problems on context free grammars. Aug 30, 2016 heres probably the oldest known example. The problems cover all aspects of the elementary theory, starting from subgroups, commutators up to representations and linear groups. 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.
The goal of this survey article is to demonstrate that undecidable decision problems arise naturally in many branches of mathematics. Decidability and undecidability stanford university. Nov 07, 2017 even before the resolution of hilberts tenth problem, some problems in group theory were proved undecidable. This lecture will discuss how undecidability in group theory led to undecidability in topology, and how undecidability in number. Every complete undecidable theory is an essentiallyundecidable theory, as is formal arithmetic cf. Two notions of undecidability there are two common settings in which one speaks of undecidability. Group theory and its application to physical problems. Morton hamermesh group theory and its application to physical problems addisonwesley publishing company inc. 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. 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 mathemat ics. Kim and roush consider two algorithmic problems for modules over a. Undecidability in group theory, topology, and analysis. First, we show that it is undecidable to check whether the group generated by a given invertible.
What are the most attractive turing undecidable problems. Groups are sets equipped with an operation like multiplication, addition, or composition that satisfies certain basic properties. We recommend the survey by poonen poo14 for many undecidable problems in mathematics. Is there any reasonably updated list of representative examples of finitely presented groups with undecidable word problem. Notes on group theory 5 here is an example of geometric nature. The unidirectional word problem for abelian semigroups is decidable and, hence, so are the three subproblems described above. 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. For another survey of undecidable problems, see dav77. Reduction proof by contradiction and construction assume mbis a tm that decides lb. As the building blocks of abstract algebra, groups are so general and fundamental that they arise in nearly every branch of mathematics and the sciences. Examples are the properties of being trivial, finite, abelian, nilpotent. If all instances of the decision problem were provable or disprovable in zfc, then you can decide the decision problem for a given instance just by simultaneously searching for a proof and a disproof of the instance in zfc.
First, we show that it is undecidable to check whether the group. Some examples already appear on the wikipedia page. 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. Many, if not most, undecidable problems in mathematics can be posed as word problems. The strengthened version is already quite close to the group case.
The study of algorithmic problems in group theory has a long tradition. Basic papers on undecidable propositions, unsolvable problems and computable functions dover books on mathematics on free shipping on qualified orders. For every turing machine, we construct an automaton group that simulates it. The word problem for groups is undecidable and, hence, so are the three more extensive problems. I am looking for an undecidable problem that i could give as an easy example in a presentation to the general public. Basic papers on undecidable propositions, unsolvable problems and computable functions dover books on mathematics. 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 simple machine models, such as nite automata or pushdown automata, many decision problems are solvable. By representative i mean avoiding obvious redundancy, i. The criterion for selection of a problem in this survey is simply that the author finds it entertaining. To explain these grouptheoretic problems without proofs we will be using groups that. The strategy of the proof, as for hilberts tenth problem, is to build a group g such that solving the word problem for g is at least as hard as solving the halting problem. Finding unsolvable problems we can use the fact that l d. More informally, such problems cannot be solved in general by computers.
The working mathematician, unless he is studying the foundations of mathema tics, usually does not find it necessary to make explicit references to axioms of set theory except perhaps to invoke the axiom of choice or the continuum hypothesis. The problems cover all aspects of the elementary theory, starting from subgroups, commutators up. Genericcase complexity, decision problems in group theory and. Or, given a string of zeros and ones, is it a palindrome. Construct a machine description m1 such that lm1 lm. Hence, a sentence is valid in g if and only if it is satisfied in every system g, which is a group in the sense of modern algebra see 1. We will see this used extensively in the upcoming weeks. If the turing machine is universal, the corresponding automaton group has an undecidable order problem. Paul schupp told me once that millers paper contains ideas that are really subtle from the point of view of computability theory having in mind that most undecidability results in group theory just consist in just encoding some undecidability feature and contain no original idea from the point of view of computability. First, we show that it is undecidable to check whether the group generated by a given invertible automaton has a. It is also undecidable whether two sets of relations specify the same group or semigroup. There are thousands of examples, so please post here only the most attractive, best examples.
The nonzero complex numbers c is a group under multiplication. 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. Algorithmic problems in group theory drops schloss dagstuhl. Dec 07, 2016 sanchit sir is taking live sessions on unacademy plus for gate 2020 link for subscribing to the course is. An example of an easy to understand undecidable problem. 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. Note that there are uncountably many undecidable problems. 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.
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