Random Subgroups of Rationals

This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup (G,+) of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of (G,+); second, what learnability properties can one extract from G and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of (G,+) coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for G with respect to any generating sequence for G is not even semi-decidable, one can build a generating sequence β such that the word problem for G with respect to β is co-recursively enumerable (assuming that the set of generators of G is limit-recursive). In regard to the second question, it is proven that there is a generating sequence β for G such that every non-trivial finitely generated subgroup of G is recursively enumerable and the class of all such subgroups of G is behaviourally correctly learnable, that is, every non-trivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of G is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of G cannot be syntactically identified in the limit with respect to any generating sequence for G. The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.

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1 Introduction

The concept of algorithmic randomness, particularly for strings and infinite sequences, has been extensively studied in recursion theory and theoretical computer science [6, 16, 19]. It has also been applied in a wide variety of disciplines, including formal language and automata theory [15]

, machine learning

[31], and recently even quantum theory [20]. An interesting and long open question is whether the well-established notions of randomness for infinite sequences have analogues for infinite structures such as graphs and groups. Intuitively, it might be reasonable to expect that a collection of random infinite structures possesses the following characteristics: (1) randomness should be an isomorphism invariant property; in particular, random structures should not be computable; (2) the collection of random structures (of any type of algebraic structure) should have cardinality equal to that of the continuum. The standard random infinite graph thus does not qualify as an algorithmically random structure; in particular, it is isomorphic to a computable graph and has a countable categorical theory. Very recently, Khoussainov [13, 14] defined algorithmic randomness for infinite structures that are akin to graphs, trees and finitely generated structures.

This paper addresses the following three open questions in algorithmic randomness: (A) is there a reasonable way to define algorithmically random structures for standard algebraic structures such as groups; (B) can one define algorithmically randomness for groups that are not necessarily finitely generated; (C) what are the model-theoretic properties of algorithmically random structures? The main contribution of the present paper is to answer these three questions positively for a fundamental and familiar algebraic structure, the additive group of rationals, denoted . Prior to this work, question (A) was answered for structures such as finitely generated universal algebras, connected graphs, trees of bounded degree and monoids [13]. Concerning question (C), it is still unknown whether the first order theory of algorithmically random graphs (or trees) is decidable. In fact, it is not even known whether any two algorithmically random graphs (of the same bounded degree) are elementarily equivalent [13].

As mentioned earlier, one goal of this work is to formulate a notion of randomness for subgroups of

. This is a fairly natural class of groups to consider, given that the isomorphism types of its subgroups have been completely classified, as opposed to the current limited state of knowledge about the isomorphism types of even rank

groups. As has been known since the work of Baer [2], the subgroups of coincide, up to isomorphism, with the torsion-free Abelian groups of rank . Moreover, the group is robust enough that it has uncountably many algorithmically random subgroups (according to our definition of algorithmically random subgroups of ), which contrasts with the fact that there is a unique standard random graph up to isomorphism. At the same time, the algorithmically random subgroups of are not too different from one other in the sense that they are all elementarily equivalent (a fact that will be proven later), which is similar to the case of standard random graphs being elementarily equivalent.

The properties of the subgroups of were first systematically studied by Baer [2] and then later by Beaumont and Zuckerman [3]. Later, the group was studied in the context of automatic structures [30]. An early definition of a random group is due to Gromov [10]

. According to this definition, random groups are those obtained by first fixing a set of generators, and then randomly choosing (according to some probability distribution) the relators specifying the quotient group. An alternative definition of a general random infinite structure was proposed by Khoussainov

[13, 14]; this definition is based on the notion of a branching class, which is in turn used to define Martin-Löf tests for infinite structures entirely in analogy to the definition of a Martin-Löf test for sequences. An infinite structure is then said to be Martin-Löf random if it passes every Martin-Löf test in the preceding sense. The existence of a branching class of groups, and thus of continuunm many Martin-Löf random groups, was only recently established [11].

Like Gromov’s definition of a random group, the one adopted in the present work is syntactic, in contrast to the semantic and algebraic definition due to Khoussainov. However, rather than selecting the relators at random according to a prescribed probability distribution for a fixed set of generators, our approach is to directly encode a Martin-Löf random binary sequence into the generators of the subgroup. More specifically, we fix any binary sequence , and define the group as that generated by all rationals of the shape , where denotes the -st prime and is the number of ones occurring between the -th and -st occurrences of zero in ; is the number of starting ones, and if there is no -st zero then is defined to be zero for all greater than and is generated by all with less than and all such that is any positive integer. is then said to be randomly generated if and only if is Martin-Löf random. In order to derive certain computability properties, it will always be assumed in the present paper that any Martin-Löf random sequence associated to a randomly generated subgroup of is also limit-recursive. It may be observed that no finitely generated subgroup of is randomly generated in the sense adopted here; this corresponds to the intuition that in any “random” infinite binary sequence , the fraction of zeroes in the first bits should tend to a number strictly smaller than one as grows to infinity. For a similar reason, no randomly generated subgroup is infinitely divisible by a prime, that is, there is no prime such that belongs to for all .

The first main part of this work is devoted to the study of the model-theoretic and recursion-theoretic properties of randomly generated subgroups of . It is shown that the theory of any randomly generated subgroup coincides with that of the integers with addition (denoted ), and is therefore decidable111For a proof of the decidability of the theory of , often known as Presburger Arithmetic, see [17, pages 81–84].. Next, we define the notion of a generating sequence for a randomly generated group ; this is an infinite sequence such that is generated by the terms of . We then consider the word problem for with respect to : in detail, this is the problem of determining, given any two finite integer sequence representations and of elements of with respect to , whether or not and represent the same element of . We show that the word problem for with respect to any generating sequence is never recursively enumerable (r.e.); on the other hand, one can construct a generating sequence for such that the corresponding word problem for is co-r.e. Moreover, one can build a generating sequence for such that the word problem for the quotient group of by with respect to is r.e.

The second main part of this paper investigates the learnability of non-trivial finitely generated subgroups of randomly generated subgroups of from positive examples, also known as learning from text. Stephan and Ventsov [27] examined the learnability of classes of substructures of algebraic structures; the study of more general classes of structures was undertaken in the work of Martin and Osherson [18, Chapter III]. The general objective is to understand how semantic knowledge of a class of concepts can be exploited to learn the class; in the context of the present problem, semantic knowledge refers to the properties of every finitely generated subgroup of any randomly generated subgroup of rationals, such as being generated by a single rational [2]. It may be noted that the present work considers learning of the actual representations of finitely generated subgroups, which are all isomorphic to each other, as opposed to learning their structures up to isomorphism, as is considered in the learning framework of Martin and Osherson [18]. Various positive learnability results are obtained: it will be proven, for example, that for any randomly generated subgroup of , there is a generating sequence for such that the set of representations of every non-trivial finitely generated subgroup of with respect to is r.e.; furthermore, the class of all such representations is behaviourally correctly learnable, that is, all these representations can be identified in the limit up to semantic equivalence. On the other hand, it will be seen that the class of all such representations can never be explanatorily learnable, or learnable in the limit. Similar results hold for the class of non-trivial finitely generated subgroups of the quotient group of by . Thus this facet of our work implies a connection between the limit-recursiveness of the set of generators of a randomly generated subgroup of and the learnability of its non-trivial finitely generated subgroups.

2 Preliminaries

Any unexplained recursion-theoretic notation may be found in [23, 25, 21]. For background on algorithmic randomness, we refer the reader to [6, 19]. We use to denote the set of all natural numbers and to denote the set of all integers. The -st prime will be denoted by . denotes the set of all finite sequences of integers. Throughout this paper, is a fixed acceptable programming system of all partial recursive functions and is a fixed acceptable numbering of all recursively enumerable (abbr. r.e.) sets of natural numbers. We will occasionally work with objects belonging to some countable class different from ; in such a case, by abuse of notation, we will use the same symbol to denote the set of objects obtained from by replacing each member with for some fixed bijection between and .

Given any set , denotes the set of all finite sequences of elements from . By we denote any fixed canonical indexing of all finite sets of natural numbers. Cantor’s pairing function is given by for all . The symbol denotes the diagonal halting problem, i.e., . The jump of , that is, the relativised halting problem , will be denoted by .

For and we write to denote the element in the -th position of . Further, denotes the sequence . Given a number and some fixed , , we denote by the finite sequence , where occurs exactly times. Moreover, we identify with the empty string . For any finite sequence we use to denote the length of . The concatenation of two sequences and is denoted by ; for convenience, and whenever there is no possibility of confusion, this is occasionally denoted by . For any sequence (infinite or otherwise) and , denotes the initial segment of of length . For any and ,

denotes the vector of length

whose first coordinates are and whose last coordinate is . Furthermore, given two vectors and of equal length, denotes the scalar product of and , that is, . For any and , denotes the vector obtained from by coordinatewise multiplication with , that is, . For any non-empty , denotes .

Cantor space, the set of all infinite binary sequences, will be denoted by . The set of finite binary strings will be denoted by . For any binary string , denotes the cylinder generated by , that is, the set of infinite binary sequences with prefix . For any , the open set generated by is . The Lebesgue measure on will be denoted by ; that is, for any binary string , . By the Carathéodory Theorem, this uniquely determines the Lebesgue measure on the Cantor space.

3 Randomly Generated Subgroups of Rationals

We first review some basic definitions and facts in algorithmic randomness which in our setting is always understood w.r.t the Lebesgue measure. An r.e. open set is an open set generated by an r.e. set of binary strings. Regarding as a subset of , one has an enumeration of all r.e. open sets. A uniformly r.e. sequence of open sets is given by a recursive function such that for each

. As infinite binary sequences may be viewed as characteristic functions of subsets of

, we will often use the term “set” interchangeably with “infinite binary sequence”; in particular, the subsequent definitions apply equally to subsets of and infinite binary sequences.

Martin-Löf [22] defined randomness based on tests. A Martin-Löf test is a uniformly r.e. sequence of open sets such that . A set fails the test if ; otherwise passes the test. is Martin-Löf random if passes each Martin-Löf test.

Schnorr [24] showed that Martin-Löf random sets can be described via martingales. A martingale is a function that satisfies for every the equality . For a martingale mg and a set , the martingale mg succeeds on if .

[24] For any set , is Martin-Löf random iff no r.e. martingale succeeds on .

The following characterisation of all subgroups of forms the basis of our definition of a random subgroup.

[3] Let be any subgroup of . Then there is an integer , as well as a sequence with such that .

Let be a real in the Cantor space, i.e. an infinite sequence of ’s and ’s. Then the group is the subgroup of the rational numbers generated by with for all , where for each , by we denote the -st prime and by the number of consecutive ’s in between the -th and -st zero in , with which we let count the number of starting ’s. If there is no -st zero, we let , meaning that for all the fraction is in .

Clearly, is always a subgroup of and if and only if the -th and -st zero in are consecutive. Thus, if ends with infinitely many zeros, then is isomorphic to . Moreover, there is a prime such that for all and for all , for short infinitely divides , if and only if ends with an infinite sequence of ’s.

If is Martin-Löf random, then is finite for every , where is defined as in Definition 3. In other words, the group is not infinitely divisible by any prime.

Proof.

This is an easy observation, as in no Martin-Löf random w.r.t the Lebesgue measure only finitely many ’s occur. ∎

A similar argument shows that for Martin-Löf random there are infinitely many primes occurring as basis of a denominator of a generator.

Fix a probability distribution on the natural numbers and let

be a sequence of iid random variables taking values in

with distribution for all . Denote by the subgroup of generated by , where denotes the -st prime.

The so obtained random group might follow a more uniform process.

If is the distribution on assigning probability , probability , probability and probability , then with probability holds for some Martin-Löf random .

Proof.

This follows immediately, as the set of ML-randoms has measure with respect to the Lebesgue measure. From , , , , we obtain an infinite binary sequence by recursively appending in step to the already established initial segment of , starting with the empty string. By definition the Lebesgue measure assigns probability to having the (intermediate) subsequence in . This is exactly the probability of the event . ∎

A generating sequence for is an infinite sequence such that . We will often deal with generating sequences rather than minimal generating sets for , mainly due to the fact that if the terms of a sequence are carefully chosen based on a limiting recursive programme for (so that itself is limiting recursive), then, as will be seen later, the set of representations of elements of with respect to can have certain desirable computability properties, such as equality being co-r.e.

Proposition .

Suppose is Martin-Löf random. Then there does not exist any strictly increasing recursive enumeration such that for each , there is some with .

Proof.

Suppose that such an enumeration did exist. We show that this contradicts the Martin-Löf randomness of . By Theorem 3, it suffices to show that there is a recursive martingale mg succeeding on . Define mg as follows. For any , if there is some such that contains at least occurrences of and the -th occurrence of is immediately succeeded by , then set . Else, let be the largest for which either or contains at least occurrences of , and set

It may be directly verified that mg satisfies the martingale equality for all . Furthermore, grows to infinity with and so mg succeeds on , contradicting the fact that is Martin-Löf random. ∎

If is Martin-Löf random, then is co-r.e., meaning that is recursive and there is a generating sequence with respect to which equality is co-r.e.

Proof.

For a fixed generating sequence of there is an epimorphism from the set of finite sequences of integers to by identifying with . We call a representation of w.r.t.  or .

Obviously, for any generating sequence of addition is recursive as only the components of the representations have to be added as integers.

In order to prove that equality is co-r.e., we construct a specific generating sequence . Based on the result of the computation of after steps, we are going to define finite sequences of rational numbers recursively, such that and inequality on , interpreted as representations w.r.t. , is decided and extends the inequalities on , even though they originate from an interpretation as representations according to . With this in the limit we obtain a generating sequence of , meaning that for every there is some such that for all the -th element of is the same as the -th element of , which we denote by . Further, generates and for this generating sequence equality will be co-r.e.

In the following we write for according to , i.e. the number of ’s between the -th and -st zero in , as introduced in Definition 3. As does not end with infinitely many ’s, can be computed in finitely many steps for every and .

  1. . Let .

  2. . Check for every whether . If let . Replace all occurring in with by some respective integer, for which existence we argue below, such that

    stays the same or enlarges if equals the first entries of instead of . Further, let

    where is minimal such that is an element of and does not yet occur in . If there is no such , let .

For example, if the tape after stage started with , after steps contained and , then in we would have to replace by an integer such that for arbitrary integers between and we have

and would be .

We proceed by showing that there is always such an integer .

Claim.

For every in step it is possible to alter finitely many entries of to obtain such that .

Proof of the Claim..

Let . It suffices to show that one entry can be replaced in this desired way. As the argument does not depend on the position, we further assume that it is the last entry. For all we want to prevent

This is a linear equation having zero or one solution in . As there are only finitely many choices for the pair , an integer not fulfilling any of these equations can be found in a computable way. ∎

We continue by proving that the entries of the stabilize, such that in the limit we obtain a sequence of elements of .

Claim.

For every there is some such that for all we have , with .

Proof of the Claim..

Let . If there is such that the entry had to be changed, then is an integer and thus, it will never be changed lateron. In case this does not happen, we obtain for all and therefore . ∎

By the next claim the just constructed sequence generates the random group.

Claim.

The sequence generates .

Proof of the Claim..

Let and as in Definition 3. We argue that there is some with . Let be the position of the -st zero in the Martin-Löf random . Then there is such that after computation steps is not changed any more. Thus, after at most additional steps all generators of having one of the first primes as denominator are in the range of . ∎

Finally, we observe that w.r.t. the generating sequence all pairs of unequal elements of can be recursively enumerated.

Claim.

Equality in is co-r.e.

Proof of the Claim..

We run the algorithm generating and in step return all elements of the finite set . As inequalities w.r.t yield inequalities w.r.t. , we only enumerate correct information. Further, for every two elements of fix representations w.r.t. and large enough such that not more than the first of the occur in these representations, all of these have stabilized up to stage and all coefficients in the representations take values between and . Then if and only if the tuple of their representations is in . ∎

This finishes the proof of the theorem. ∎

As there are -recursive Martin-Löf random reals, we obtain the following corollary.

There exists a co-r.e. random subgroup of the rational numbers.

Proposition 3 implies, in particular, that if is Martin-Löf random, then there cannot exist any generating sequence for with respect to which equality of members of is r.e. Indeed, suppose that such a generating sequence did exist, so that is r.e. Fix any such that (since , such a must exist). Then there is a strictly increasing recursive enumeration such that for all , is the first found for which the following hold: (i) whenever ; (ii) there are and relatively prime positive integers with and such that for some , . Note that

The Martin-Löf randomness of implies that contains infinitely many terms of the form with , and relatively prime and positive, and . Thus is defined for all , and by Proposition 3 this contradicts the Martin-Löf randomness of .

Further, a variation of the algorithm yields that equality of the proper rational part is r.e. on random groups.

If is Martin-Löf random, then equality modulo 1 on is r.e. with respect to some generating sequence.

Proof.

The construction of the generating sequence follows the construction of in the proof of Theorem 3 with the main difference that in step instead of making sure that in case of replacements no already enumerated inequalities are destroyed, we have to make sure that all equalities modulo that have been established in the first steps are preserved. Formally, this reads as with

As we have to preserve equality modulo 1 and each prime occurs at most once as basis of a denominator, we may use to replace the prime power fraction(s) if necessary. The rest of the proof works the same way. ∎

The next main result is concerned with the model-theoretic properties of random subgroups of rationals. We recall that two structures (in the model-theoretic sense) and with the same set of non-logical symbols are elementarily equivalent (denoted ) iff they satisfy the same first-order sentences over ; the theory of a structure (denoted ) is the set of all first-order sentences (over the set of non-logical symbols of ) that are satisfied by . The reader is referred to [17] for more background on model theory. We will prove a result that may appear a bit surprising: even though Martin-Löf random subgroups of (viewed as classes of integer sequence representations) are not computable, any such subgroup is elementarily equivalent to - the additive group of integers - and thus has a decidable theory. In other words, the incomputability of a random subgroup of rationals, at least according to the notion of “randomness” adopted in the present work, has little or no bearing on the decidability of its first-order properties. We begin by showing that the theory of any subgroup of rationals reduces to that of the subgroup of generated by the set of all rationals either equal to or of the shape , where is a prime infinitely dividing and . Our proof of this fact rests on a sufficient criterion due to Szmielew [29] for the elementary equivalence of two groups; this result will be stated as it appears in [12].

([29], as cited in [12]) Let be a prime number and be a group. For all , and elements , define and the following predicate :

the images of in the factor group are
such that are linearly independent in .

Define the parameters and as follows.

(Here and is the -th power of the primary cyclic group on elements, that is, it consists of all elements such that .) Then any two groups and are elementarily equivalent iff , and for all primes and all .

The definition of a pure subgroup will not be used in the proof of the subsequent theorem; it will be observed that if is a subgroup of the rationals, then for and , it cannot contain as a subgroup in any case, so that .

Let be a subgroup of . Then , where denotes the set of all primes infinitely dividing and for a set of primes we write for the subgroup of generated by .

Proof.

Define the predicate and the parameters and as in Theorem 3. Let be a prime number and suppose . By Theorem 3, it suffices to show that the three parameters and coincide on and . First, cannot be a subgroup of or when and since by Theorem 3, no non-trivial subgroup of any subgroup of rationals can be torsion222We recall that a group is torsion iff for every , there is some such that is equal to the identity element of .; thus and are both equal to for as well as . For a similar reason, for every and , and therefore and . Furthermore, may be regarded as a vector space over the field , and holds iff the dimension of the -vector space (denoted ) is at least . It follows that

Similarly, is a -vector space and . Thus it suffices to show that .

Case 1: . Then . It follows that ; the same argument shows that .

Case 2: . Then there is some non-zero such that . It may be assumed without loss of generality that because if for some non-zero integers and , then, taking , is a subgroup of that is isomorphic to such that . Assuming , there is a fixed integer such that is generated (as a subgroup of ) by rationals of the shape , where () is prime and . As before, it may be assumed without loss of generality that . It will be shown that each such generator is congruent to an integer modulo . Fix a generator of the shape . Let and be integers such that . Then . It follows that is isomorphic to and so . Using the case assumption that , one also has that , and so the same argument as before yields . ∎

Note that is undecidable; in contrast, for Martin-Löf random we have , so the promised corollary follows.

Let be Martin-Löf random. Then and have the same theories.

One may ask whether this still holds for richer structures. This is not the case, as for example the theory of is different from , as in the latter is a satisfying assignment for the formula . There does not exist an with this property for a ML-random .

4 Learning Finitely Generated Subgroups of a Random Subgroup of Rationals

In this section, we investigate the learnability of non-trivial finitely generated subgroups of any group generated by a Martin-Löf random sequence such that . More specifically, we will examine for any given the set of representations of elements of any non-trivial finitely generated subgroup of with respect to a fixed generating sequence for such that all are r.e., and consider the learnability of the class of all such sets of representations.

We will consider learning from texts, where a text is an infinite sequence that contains all elements of for the to be learnt and may contain the symbol , which indicates a pause in the data presentation and thus no new information. For any text and , denotes the -st term of and denotes the finite sequence , i.e., the initial segment of length of ; denotes the set of non-pause elements occurring in . A learner is a recursive function mapping into ; the symbol permits to abstain from conjecturing at any stage. A learner is fed successively with growing initial segments of the text and it produces a sequence of conjectures , which are interpreted with respect to a fixed hypothesis space. In the present paper, we stick to the standard hypothesis space, a fixed Gödel numbering of all r.e. subsets of . In our setting from the generator of we can immediately derive an index for and therefore in the proofs we argue for learning and . The learner is said to behaviourally correctly (denoted ) learn the representation of a finitely generated subgroup with respect to a fixed generating sequence for iff on every text for , the sequence of conjectures output by the learner converges to a correct hypothesis; in other words, the learner almost always outputs an r.e. index for [7, 5, 1]. If almost all of the learner’s hypotheses on the given text are equal in addition to being correct, then the learner is said to explanatorily (denoted ) learn (or it learns in the limit) [9].

A useful notion that captures the idea of the learner converging on a given text is that of a locking sequence, or more generally that of a stabilising sequence. A sequence is called a stabilising sequence [8] for a learner on some set if and for all , . A sequence is called a locking sequence [4] for a learner on some set if is a stabilising sequence for on and .

The following proposition due to Blum and Blum [4] will be occasionally useful.

Proposition .

[4] If a learner explanatorily learns some set , then there exists a locking sequence for on . Furthermore, all stabilising sequences for on are also locking sequences for on .

Clearly, also a -version of Proposition 4 holds.

It is not clear in the first place whether or not every finitely generated subgroup of a randomly generated subgroup of can even be represented as an r.e. set. This will be clarified in the next series of results. We recall that a finitely generated subgroup of is any subgroup of that has some finite generating set , which means that every element of can be written as a linear combination of finitely many elements of and the inverses of elements of . is trivial if it is equal to ; otherwise it non-trivial. Furthermore, if is a subgroup of , then any finitely generated subgroup of is cyclic, that is, for some and with (see, for example, [28, Theorem 8.1]). The latter fact will be used freely throughout this paper. For any generating sequence for and any finitely generated subgroup of , the set of representations of elements of with respect to will be denoted by .

Let be Martin-Löf random. Then there is a generating sequence of such that for every non-trivial finitely generated subgroup of the set is r.e.

Proof.

We denote the set of all non-trivial finitely generated subgroups of by and modify the construction of the generating sequence in the proof of Theorem 3. In contrast we show that for every there is some such that for every in step we can assure that replacements do not violate the property to represent an element of , i.e. it is possible to change entries of to obtain , such that we have , where