In the present paper, i will discuss some of these connections between group theory and semigroup theory, and i will also discuss some rather surprising contrasts between the theories. Many problems in combinatorial group theory are undecidable. To explain these group theoretic problems without proofs we will be using groups that. Theorem fix a decision problem and axiom system a such that. For this article we need only the facts that the word problem of a wordhyperbolic group is solvable in linear time and the word problem of an automatic group is solvable in quadratic time.
Silva we showed in 2016 that the problem to solve equations with idempotent variables in free inverse monoids is exptimecomplete. These problems may be partially decidable but they will never be decidable. The class of orderable groups is a quasivariety with. In group theory, a branch of abstract algebra, the whitehead problem is the following question. Decidable and undecidable problems in theory of computation. Since such a group exists, a nitely presented quandle with undecidable word problem must also exist. Usually for the word problem the group is fixed and the input is just a word over the generators.
In 1936 turing went to princeton as a visiting graduate student. Nov 07, 2017 even before the resolution of hilberts tenth problem, some problems in group theory were proved undecidable. Nov 20, 2019 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. Discreteness is undecidable michael kapovich november 15, 2015 abstract we prove that the discreteness problem for 2generated nonelementary subgroups of slp2. 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. Decidable and undecidable problems related to completely 0simple semigroups t. This is a prolongation of the padic valuation on q. Genericcase complexity, decision problems in group theory and. Negative solution of hilberts 10th problem this conjecture was very important because if it were true then the negative solution of h10 problem would immediately follow from it. This follows from the fact that there is a finitely presented group with unsolvable word problem. Undecidability of the word problem for groups matteo acclavio.
Some results concerning the decision problem in group theory are given by presburger o, szmielew 9, and tarski 12. Magnus invented his famous magnus embedding, which became a very e cient instrument in the theory of solvable groups. Pdf unsolvability of the endomorphic reducibility problem. In 1977, paris and harrington proved that the parisharrington principle, a version of the ramsey theorem, is undecidable in the axiomatization of arithmetic given by the peano axioms but can be proven to be true. Algorithmic problems in group theory drops schloss dagstuhl. Decidable and undecidable problems related to completely 0. Encoding programs as natural numbers, suppose that there were. The word problem for the free group fn is decidable. We use golodshafarevich construction and immune sets from the classical recursion theory to build dehn monsters. Jun 25, 2012 in this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group g is the algorithmic problem of deciding whether two words in the generators represent the same element.
It is easy to see that, given two distinct finite presentations ttx and tt2 such that gffigf, then the decision problems. In number theory it is known to be undecidable whether diophantine. 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. It turns out that some classical undecidable problems are, in fact, strongly. A related undecidable problem is whether two integer matrices can be multiplied together in some sequence to yield the zero matrix. Fundamental to combinatorial group theory is the concept of a free group.
Undecidability in group theory, topology, and analysis. Clearly a solution to the membership problem, in a given group, implies a solution to the word problem. Our analysis allows us to conclude that the security and reliability of a wellknown authentication scheme in group. This is a survey of some problems in geometric group theory which i. Only about 20 years later were examples of undecidable decision problems found elsewhere in mathematics, namely in group theory. Mar 29, 2021 if x contains a finitely presented group with unsolvable word problem, then the theory of x is undecidable.
Trotter january 20, 1996 abstract the undecidable problems of the title are concerned with the question. Pdf modeltheoretic and algorithmic questions in group theory. The word problem for finitely presented quandles is undecidable. At the end of this exposition, we explain, among other things, how to analyze the security of a particular zeroknowledge, i. About decidable and undecidable fragments volker diekert university of stuttgart in a joint paper with florent martin, g eraud s enizergues, and pedro v. Gromov indicated that \most nitely presented groups are wordhyperbolic. Exponential diophantine equations finally, a group of american mathematicians martin davis, hillary putnam and. Suppose g is a group in the class x and suppose that g has finite presentation. Since modular lattices have an undecidable equational theory fre80, the same is true for modular residuated lattices. Pdf this work is a survey on decidable and undecidable problems in matrix theory. Szmielews proof of the decidability of the elementary theory of abelian groups is.
Formally, an undecidable problem is a decision problem for which one cannot construct a single algorithm that will always provide a correct answer in finite time. He stayed at princeton for two years and completed his phd under church. Undecidability in group theory, analysis, and topology. In the problems below, g, h, k, and n generally denote groups. Our main result is that already its quasiidentities are undecidable. What are the most attractive turing undecidable problems in. In combinatorics it is known in general to be undecidable whether a given set of tiles can cover the plane see page 19. Theorems with the greatest impact on group theory as a whole. Tarskis proof of undecidability for the elementary theory of abelian cancellation semigroups is presented in detail. Equational theory an overview sciencedirect topics. It also follows that whether a matrix semigroup is a group is undecidable for 4x4 integer matrices. In this case as derek says having a decidable word problem is presentation independent. Undecidable undecidable undecidable undecidable undecidable lecture 17. S n denotes the group of all permutations of a set with n elementsthe so called symmetric group.
A particularly simple class of problems in algebra because of their linearity, are module problems. More precisely, if a is a finite set of generators for g then the word problem is the membership problem for the formal language of all words in a. The general idea of generic behavior in the context of group theory was introduced by m. In computability theory, an undecidable problem is a type of computational problem that.
Decision problems in group theory 1 introduction in 1911, dehn formulated three fundamental decision problems for groups. 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. Undecidable problems about polynomials around hilberts 10th problem anton sadovnikov. As with the word problem, the membership problem is in general undecidable. An infinite extension of the elementary theory of abelian groups is constructed, which is proved to be decidable, while the elementary theory of its finite models is shown to be undecidable. Knot theory analysis inequalities complex analysis integration undecidability in group theory, topology, and analysis bjorn poonen rademacher lecture 2 november 7, 2017. In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations present isomorphic groups the isomorphism problem was identified by max dehn in 1911 as one of three fundamental decision problems in group theory. The set with an operation, to be a group, have to satisfy a four of axioms. It is also undecidable whether two sets of relations specify the same group or semigroup. In 1973, the whitehead problem in group theory was shown to be undecidable, in the first sense of the term, in standard set theory. In 1977, paris and harrington proved that the parisharrington principle, a version of the ramsey theorem, is undecidable in the axiomatization of arithmetic given by the peano axioms but can be proven. This will subsume the main results of 14 and 18, since. The logic obtained from fl by adding the modularity axiom is undecidable. The problems studied are simply formulated, however most of them are.
In this paper we provide a positive answer to the problem stated at the very beginning of the paper, i. This paper is motivated by the following basic question in the theory of discrete subgroups of lie groups. Generic complexity of undecidable problems springerlink. The word problem for finitely presented quandles is. Dehn proved that there are algorithms to decide each of these problems when g is the fundamental group of a closed 2manifold. Another reason for proposing this theorem is that finding classes of groups which have soluble word problem is more than just a fun game. A nitely presented group with undecidable conjugacy problem was rst constructed in. Pdf decidable and undecidable problems about quantum.
Let g be a noncommutative finitely presented 2generator group with undecidable word problem. In 25 turing also showed that the halting problem for turing machines is undecidable, and as a corollary, he arrived at the undecidability of the decision problem for rstorder logic. Therefore their work had no practical e ect on areas of math outside of logic. Proving undecidability acceptance language a tm m is a tm description and m accepts input w we proved atm is undecidable last class. Undecidability everywhere wang tiles undecidability. The aim of this note is to construct a finitely generated f. He showed in 55,54 that the diophantine problem is undecidable in any nonabelian free metabelian group and in any nonabelian free nilpotent group of nilpotency class at least 9. For example the triviality problem takes as input a presentation and asks if it presents the trivial group. 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.
Examples are word problems for groups and semigroups see page 1141, and. Is every abelian group a with ext 1 a, z 0 a free abelian group shelah 1974 proved that whiteheads problem is independent of zfc, the standard axioms of set theory. Undecidability hence appears as a genuine quantum property here. 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.
Table 1 decidable and undecidable problems for probabilistic and quantum automata. This result was later generalized by rices theorem. In particular, any nitely presented group g with undecidable word problem gives rise to a nitely presented quandle q g with undecidable word problem. In fact, the word problem is so bad in these groups that there is no any algorithmic way to produce an in nite set of pairwise distinct elements of g. We will also show that for any field, the theory of a group algebra of a torsion free hyperbolic group is undecidable and for a field of zero characteristic even the diophantine problem is undecidable. We shall describe a class ii of such decision problems and a construction a such that if p is a problem in q, then ap will be a finitely presented group in which p is recursively undecidable. Decidability and undecidability of theories of abelian groups with. Then nhas a complement in gif and only if n5 g solution assume that n has a complement h in g.
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