Random Subgroups of Rationals
Ziyuan Gao, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, Alexander, Melnikov, Karen Seidel, Frank Stephan

TL;DR
This paper explores the algorithmic randomness of subgroups of rationals, analyzing their logical properties and learnability, revealing decidability results and limitations in syntactic identification of their finitely generated subgroups.
Contribution
It introduces a notion of algorithmic randomness for rational subgroups and studies their model-theoretic, recursion-theoretic, and learnability properties, connecting generator complexity with subgroup learnability.
Findings
The theory of the subgroup coincides with that of integers and is decidable.
There exists a generating sequence with a co-recursively enumerable word problem.
Finitely generated subgroups are learnable in the limit but not syntactically identifiable.
Abstract
This paper introduces and studies a notion of \emph{algorithmic randomness} for subgroups of rationals. Given a randomly generated additive subgroup of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of ; second, what learnability properties can one extract from and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for with respect to any generating sequence for is not even semi-decidable, one can build a generating sequence such that the word problem for with respect to is co-recursively enumerable (assuming that the set of generators of is limit-recursive). In regard to the second question,…
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Department of Mathematics, National University of Singapore, [email protected] School of Computing, National University of Singapore, [email protected] Department of Computer Science, University of Auckland, New [email protected] Department of Mathematics, National University of Singapore, [email protected]
Institute of Natural and Mathematical Sciences, Massey University, New [email protected] Hasso Plattner Institute, University of Potsdam, [email protected]
Department of Mathematics, National University of Singapore, [email protected]
\CopyrightZiyuan Gao, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, Alexander Melnikov, Karen Seidel, and Frank Stephan\fundingSanjay Jain was supported in part by NUS grant C252-000-087-001. Furthermore, Ziyuan Gao, Sanjay Jain and Frank Stephan have been supported in part by the Singapore Ministry of Education Academic Research Fund Tier 2 grant MOE2016-T2-1-019 / R146-000-234-112. Bakhadyr Khoussainov was supported by the Marsden fund of Royal Society of New Zealand. Karen Seidel was supported by the German Research Foundation (DFG) under Grant KO 4635/1-1 (SCL) and by the Marsden fund of Royal Society of New Zealand.
Acknowledgements.
The authors would like to thank Philipp Schlicht and Tin Lok Wong for helpful discussions, as well as thank Timo Kötzing for pointers to the literature.\EventEditors \EventNoEds3 \EventLongTitle \EventShortTitle \EventAcronym \EventYear2019 \EventDate \EventLocation \EventLogo \SeriesVolume \ArticleNo
Random Subgroups of Rationals
Ziyuan Gao
,
Sanjay Jain
,
Bakhadyr Khoussainov
,
Wei Li
,
Alexander Melnikov
,
Karen Seidel
and
Frank Stephan
Abstract.
This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup of rationals, two main questions are addressed: first, what are the model-theoretic and recursion-theoretic properties of ; second, what learnability properties can one extract from and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for with respect to any generating sequence for is not even semi-decidable, one can build a generating sequence such that the word problem for with respect to is co-recursively enumerable (assuming that the set of generators of is limit-recursive). In regard to the second question, it is proven that there is a generating sequence for such that every non-trivial finitely generated subgroup of is recursively enumerable and the class of all such subgroups of 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 is limit-recursive). On the other hand, the class of non-trivial finitely generated subgroups of cannot be syntactically identified in the limit with respect to any generating sequence for . 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.
Key words and phrases:
Martin-Löf randomness, subgroups of rationals, finitely generated subgroups of rationals, learning in the limit, behaviourally correct learning
1991 Mathematics Subject Classification:
\ccsdesc[500]Theory of computation Inductive inference, Theory of computation Pseudorandomness and derandomization
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 [math] 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 .
Theorem 3.1**.**
[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.
Theorem 3.2**.**
[3*]** Let be any subgroup of . Then there is an integer , as well as a sequence with such that . *
Definition 3.3**.**
Let be a real in the Cantor space, i.e. an infinite sequence of [math]’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.
Lemma 3.4**.**
If is Martin-Löf random, then is finite for every , where is defined as in Definition 3.3. In other words, the group is not infinitely divisible by any prime.
Proof 3.5**.**
This is an easy observation, as in no Martin-Löf random w.r.t the Lebesgue measure only finitely many [math]’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.
Definition 3.6**.**
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.
Lemma 3.7**.**
If is the distribution on assigning [math] probability , probability , probability and probability , then with probability holds for some Martin-Löf random .
Proof 3.8**.**
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 3.9**.**
Suppose is Martin-Löf random. Then there does not exist any strictly increasing recursive enumeration such that for each , there is some with .
Proof 3.10**.**
Suppose that such an enumeration did exist. We show that this contradicts the Martin-Löf randomness of . By Theorem 3.1, 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 [math] and the -th occurrence of [math] is immediately succeeded by [math], then set . Else, let be the largest for which either or contains at least occurrences of [math], and set
[TABLE]
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.
Theorem 3.11**.**
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 3.12**.**
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 3.3. As does not end with infinitely many ’s, can be computed in finitely many steps for every and .
- . Let .
- . Check for every whether . If let . Replace all occurring in with by some respective integer, for which existence we argue below, such that
[TABLE]
stays the same or enlarges if equals the first entries of instead of . Further, let
[TABLE]
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
[TABLE]
and would be .
We proceed by showing that there is always such an integer .
Claim 1**.**
For every in step it is possible to alter finitely many entries of to obtain such that .
Proof 3.13** (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
[TABLE]
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 2**.**
For every there is some such that for all we have , with .
Proof 3.14** (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 3**.**
The sequence generates .
Proof 3.15** (Proof of the Claim.).**
Let and as in Definition 3.3. 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 4**.**
Equality in is co-r.e.
Proof 3.16** (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.
Corollary 3.17**.**
There exists a co-r.e. random subgroup of the rational numbers.
Remark 3.18**.**
*Proposition 3.9 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 *
[TABLE]
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 3.9 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.
Theorem 3.19**.**
If is Martin-Löf random, then equality modulo 1 on is r.e. with respect to some generating sequence.
Proof 3.20**.**
*The construction of the generating sequence follows the construction of in the proof of Theorem 3.11 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 *
[TABLE]
As we have to preserve equality modulo 1 and each prime occurs at most once as basis of a denominator, we may use [math] 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].
Theorem 3.21**.**
([29], as cited in [12]) Let be a prime number and be a group. For all , and elements , define and the following predicate :
[TABLE]
Define the parameters and as follows.
[TABLE]
(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 .
Theorem 3.22**.**
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 3.23**.**
Define the predicate and the parameters and as in Theorem 3.21. Let be a prime number and suppose . By Theorem 3.21, it suffices to show that the three parameters and coincide on and . First, cannot be a subgroup of or when and since by Theorem 3.2, 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 [math] 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
[TABLE]
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.
Corollary 3.24**.**
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 4.1**.**
[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 4.1 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 .
Theorem 4.2**.**
*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 4.3**.**
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 3.11. 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
[TABLE]
Let , then there are and coprime, such that is generated by . Let be such that all prime factors of or are less or equal to . We let be such that all having powers of a prime below as denominator have stabilized up to stage and the exponents occurring in the prime factorizations of and are .
We may assume that only the -th component for some has to be replaced by some integer . Thus, for all we want to make sure
[TABLE]
For this, it suffices to show that . By the Chinese Remainder Theorem there exists some integer such that for all we have . With this there is some integer such that
[TABLE]
*Because we obtain that divided by is an integer and moreover is a factor of . All integer-multiples of are members of . In a nutshell, enumerating and all elements of for yields the set of all representations of elements of w.r.t. . *
Remark 4.4**.**
The statement of Theorem 4.2 excludes the trivial subgroup because for any generating sequence for , cannot be r.e. To see this, suppose, by way of contradiction, that were r.e. Given any , set , and for all , if and [math] otherwise, and if and [math] otherwise. Then . Thus if were r.e., then equality with respect to would also be r.e., which, as was shown earlier, is impossible.
We note that there cannot be any generating sequence for such that there are finitely generated subgroups of with r.e. and co-r.e.
Theorem 4.5**.**
Let be Martin-Löf random. Let be any generating sequence for . Then for any finitely generated subgroups and of , one of the following holds: (i) both and are r.e., (ii) both and are co-r.e., or (iii) at least one of and is neither r.e. nor co-r.e.
Proof 4.6**.**
Fix any generating sequence for . Assume, by way of contradiction, that for some and , where and , is r.e. and is co-r.e; without loss of generality, assume that . It will be shown that this implies the existence of a strictly increasing recursive enumeration such that for all . For all , let be the first found such that for all and there is some such that the following conditions are satisfied.
- (1)
. 2. (2)
For all , .
The Martin-Löf randomness of implies that there are arbitrarily large primes with . For each such that , there is some with , and so . Moreover, for all , since and , one has . Hence is defined for all . Furthermore, suppose some prime and satisfy Conditions 1 and 2. Condition 1 implies that , and so
[TABLE]
Condition 2 implies that , and so
[TABLE]
It follows from (1) and (2) that if for some relatively prime integers and with , then , and . Consequently, , as required. But the existence of a strictly increasing recursive enumeration such that for all contradicts Proposition 3.9.
Notation 4.7**.**
*Let be Martin-Löf random and let be any generating sequence of . For any subgroup of , denotes the set of all representations of elements of with respect to , that is, . Furthermore, define \mathcal{F}_{\beta}:=\{F_{\beta}\mid\mbox{FG_{R}}\}. *
Theorem 4.8**.**
Let be Martin-Löf random. Then there is a generating sequence of such that is r.e. for every non-trivial finitely generated subgroup of and is -learnable.
Proof 4.9**.**
We will reuse the generating sequence for constructed in the proof of Theorem 4.2. For all , let denote the -th approximation to the -st element of . Define a learner on any text as follows. Let be the length of the text segment seen so far. First, let be all the positive integers such that for every , there is some for which . If no such exists, then just outputs a default index, say an r.e. index for the set of representations for . Otherwise, uses as its current guess for the numerator of the target subgroup’s generating element. Next, define an approximation to the denominator of the target subgroup for every as follows. Consider every element of of the shape , where (1) , (2) is the only non-zero coordinate of the element and it occurs in the -st position, (3) for some , and (4) is the smallest number such that (as before, is the only non-zero coordinate and it occurs in the -st position). Let be the product of all factors such that and satisfy items 1 to 4; if there is no such factor, then set . outputs an index such that enumerates all such that for some and , .
It will be verified that is indeed a behaviourally correct learner for . Let be any finitely generated subgroup of , where and are relatively prime natural numbers, and let be any text for . Since every integer in is a multiple of and must contain , it follows that after seeing a sufficiently long segment of , will always correctly guess that the numerator of the target subgroup’s generating element is equal to .
Suppose for some positive integers and primes with . For all , contains an element of the shape , where, if the -st coordinate of this element is the only non-zero entry, then for some with . Consequently, for sufficiently large , divides whenever ( may be divisible by other prime powers as well). Let be such that for all , for some . Fix such that
- i.
for all and all , if for some and prime , then (in other words, all entries of that are equal to for some and have stabilised at stage ); in particular, for all ; 2. ii.
* for all prime powers that are factors of either or .*
First, it will be shown that if . Fix any and . By the choice of , divides . Suppose for some positive integers and primes that do not divide or . Consider any such that
[TABLE]
for some . It may be assumed without loss of generality that for any , . Let be such that for all , for some ; by the preceding assumption, for all . We show that
[TABLE]
this will establish that .
The following relation will be established:
[TABLE]
It suffices to show that for every , ; this fact, combined with (3), will establish (5).
Pick any ; without loss of generality, assume that . By the Martin-Löf randomness of , it may be assumed that every -st entry of (for any ) is either equal to for some or equal to some integer such that the -st coordinate of is changed exactly once from some value (where ) to ; in addition, for any two terms of of the shape and , where and , . Thus and for some and .
**Case 1:: **
. Then
[TABLE]
By Conditions i and ii, as well as by the choice of (as given in the proof of Theorem 4.2), every prime power factor of must divide ; in particular, divides . Furthermore, by (3) and the following two facts: (a) and (b) does not divide for all with , one has . Therefore
[TABLE]
**Case 2:: **
. By Condition i, ; suppose for some , so that . As in Case 1,
[TABLE]
Conditions i and ii, together with the choice of , imply that divides . Thus, as before,
[TABLE]
This establishes (5). Now if , then there must be a least such that . But since contains , where the -st position is the only non-zero entry and , must then be a factor of , a contradiction. Hence .
*Furthermore, since for sufficiently large and for any given , there is some with whenever , one also has that . Thus , as required. *
The next result shows, in contrast to Theorem 4.8, that if is Martin-Löf random, then, given any generating sequence for such that is r.e. for every non-trivial finitely generated subgroup of , the class is not explanatorily learnable.
Theorem 4.10**.**
Let be Martin-Löf random. Suppose is a generating sequence for such that for any non-trivial finitely generated subgroup of , is r.e. Then is not -learnable.
Proof 4.11**.**
Assume, by way of contradiction, that such a learner did exist. By Proposition 4.1, one could then find a locking sequence for on the set of representations of with respect to . We show that this implies the existence of a strictly increasing recursive enumeration such that for all , . The enumeration is defined as follows. For each , let be the first found such that for all and there is a sequence satisfying the following conditions.
- (1)
. 2. (2)
For all , .
Note that Condition 2 is semi-decidable because is r.e. By the Martin-Löf randomness of , there exist infinitely many such that . For each such , since must explanatorily learn , and , there exists some such that . Furthermore, for each , one has
[TABLE]
Thus is defined for all . It remains to show that for all , . To see this, one first observes that by the locking sequence property of , if is the sequence found together with satisfying Conditions 1 and 2, then implies that there exists some with ; in other words, . By Condition 2, and therefore must be of the shape for some that is coprime to . Consequently, , as required. But by Proposition 3.9, the existence of the enumeration would contradict the fact that is Martin-Löf random. Hence cannot be explanatorily learnable.
The next theorem considers the learnability of the set of representations of any finitely generated subgroup of the quotient group with respect to the generating sequence for constructed in the proof of Theorem 3.19. Slightly abusing the notation defined in Notation 4.7, for any generating sequence for , will denote the set of representations of any subgroup of with respect to , and will denote \{F_{\beta}\mid\mbox{FG_{R}/\mathbb{Z}}\}.
Theorem 4.12**.**
Suppose is Martin-Löf random. Let be the quotient group of by . Then there is a generating sequence for such that is r.e. for all finitely generated subgroups of and is -learnable.
Proof 4.13**.**
We will use the fact that any finitely generated subgroup of is finite333This may be seen as follows. Suppose for some relatively prime . Then for all , there are and with such that . (see, for example, [26, page 106]). Let be the generating sequence for constructed in the proof of Theorem 3.19; as was shown in the proof of this theorem, equality is r.e. with respect to , that is, is r.e. Then for any finitely generated subgroup of with elements , if is a representation for for all , then is r.e. Define a learner on any text as follows. On input , outputs an r.e. index for the closure under equality of all , that is, . Let be any finitely generated subgroup of , and suppose is fed with a text for . By construction, always conjectures a set that is contained in . Furthermore, since is finite, there is a sufficiently large such that for all , contains some with . Thus, as always conjectures a set that is closed under equality with respect to , it follows that for all , on will conjecture .
As in the case of the collection of non-trivial finitely generated subgroups of , the class is not explanatorily learnable with respect to any generating sequence for . The proof is entirely analogous to that of Theorem 4.10.
Theorem 4.14**.**
Let be Martin-Löf random. Suppose is a generating sequence for such that for any finitely generated subgroup of , is r.e. Then is not -learnable.
A natural question is whether the learnability or non-learnability of a class of representations for a collection of subgroups of is independent of the choice of the generating sequence for . We have seen in Theorem 4.10, for example, that the non explanatory learnability of the class of non-trivial finitely generated subgroups of holds for any generating sequence for such that is r.e. whenever is a finitely generated subgroup. The next theorem gives a positive learnability result that is to some extent independent of the choice of the generating sequence: for any generating sequence for such that equality with respect to is -recursive and is r.e. whenever is a finitely generated subgroup of , the class is explanatorily learnable relative to oracle .
Theorem 4.15**.**
Let be Martin-Löf random. Then for any generating sequence for such that equality with respect to is -recursive (in other words, the set is -recursive) and is r.e. for all finitely generated subgroups of , is -learnable.
Proof 4.16**.**
Let be any generating sequence for satisfying the hypothesis of the theorem. Define an learner as follows. On input , where for all , oracle is first used to determine a representation of a generator for the subgroup generated by . This can be done in a recursive fashion. We first identify the indices (if any) such that and (here 0 denotes any zero vector); if no such index exists, then let be any representation of [math]. Set . Having defined , use oracle to determine relatively prime integers such that ; without loss of generality, assume that . Then search for some with , and set . Assuming inductively that represents a generator for the subgroup generated by , one deduces from the relations and that
[TABLE]
Hence . Since , it follows from the induction hypothesis that for all , . Thus, setting , generates the subgroup generated by .
* now outputs the least (if any such exists) such that the following hold:*
- (1)
. 2. (2)
For all , there is some integer such that .
If there is no satisfying all of the above conditions, then outputs a default index, say [math]. Suppose is fed with a text for the set of representations of some finitely generated subgroup . Then will identify a generator such that in the limit; Condition 1 ensures that in the limit, will conjecture a set such that , while Condition 2 ensures that in the limit, will not overgeneralise, that is, it will not output a set containing elements not in . Hence explanatorily learns relative to oracle .
We recall from Theorem 3.19 that there is a generating sequence for such that equality modulo with respect to is r.e.; in other words, the set is r.e. The next result considers the learnability of a class that is in some sense “orthogonal” to the class : the class of all sets of representations of with respect to any generating sequence for such that is r.e. Equivalently, we ask whether the collection of all r.e. sets of pairs for which equality modulo holds with respect to any given generating sequence for can be learnt; it turns out that this class is not even behaviourally correctly learnable. In the statement and proof of the next theorem, for any generating sequence for , let denote the set .
Theorem 4.17**.**
Let be Martin-Löf random. Let be the collection of all generating sequences for such that is r.e., and define . Then is not -learnable.
Proof 4.18**.**
Assume, by way of contradiction, that has a behaviourally correct learner . Using a standard type of argument in inductive inference, we will build a limiting recursive generating sequence for and a text for such that is r.e. and on outputs some wrong conjecture for infinitely often, that is, there are infinitely many for which .
The basic construction of follows the proof of Theorem 3.19, the main difference being that at various stages of the construction, one searches for some sequence to extend the current text segment such that on conjectures some r.e. set containing a pair such that and are both of the shape , where and are both longer than any ; one may then ensure that is a wrong conjecture by setting the -th position of to [math] and the -th position of to for some fixed prime with . The constructions of and are given in more detail below. The approximations of and at stage will be denoted by and respectively. Fix some prime such that .
- (1)
Set and . 2. (2)
At stage , let be a generating sequence for that extends a prefix of and is defined as in Theorem 3.19, so that equality modulo with respect to is r.e. In other words, one builds in increasing segments by searching at every stage a new element such that , where is the -th approximation to , and adding as a new term to the current approximation of . Furthermore, for every such that the -th term of the current approximation of does not belong to , the -th term of is permanently set to [math]. Note that an r.e. index for can be uniformly computed from . Now search for some such that enumerates a pair satisfying the following:
- (a)
. 2. (b)
* and .* 3. (c)
For all , and . 4. (d)
The first (resp. ) terms of (resp. ) are equal to [math] while the last term of (resp. ) is equal to .
Since is r.e. by construction, must behaviourally correctly learn . Moreover, since every term of is either an element of of the shape for some or equal to [math], Proposition 3.9 implies that has infinitely many terms equal to [math]. Hence such and must eventually be found.
Without loss of generality, assume that . Now let be the sequence of length such that the -st position of is equal to for the least such that for some minimum , and does not contain , the -th position of is and the terms of between the -th and -st positions inclusive are all equal to [math], and the terms of between the -st and the -st positions inclusive are equal to the respective terms of the -st approximation of ; furthermore, all the terms of are corrected up the -st approximation, that is, every term of belongs to and is either equal to [math] or is of the shape for some . Let be a string whose range consists of all such that , and set .
Set and (more precisely, for each , and ). Arguing as in the proof of Theorem 3.19, the set is r.e.; in addition, the range of is precisely equal to . On the other hand, by construction on infinitely often outputs an r.e. index for some set not equal to . Hence cannot be a behaviourally correct learner for .
In contrast to Theorem 4.17, we present a positive learnability result for the collection of all co-r.e. sets of pairs of representations of for which equality holds with respect to any generating sequence for . In the statement and proof of the next theorem, given any generating sequence for such that equality with respect to is co-r.e., will denote the set .
Theorem 4.19**.**
Let be Martin-Löf random. Let be the collection of all generating sequences for such that is co-r.e., and define . Then is explanatorily learnable relative to oracle using co-r.e. indices. That is to say, there is a -recursive learner such that for any and any text for , on will output an r.e. index for in the limit.
Proof 4.20**.**
Define a -recursive learner for as follows. On input , first guesses the minimum such that the -st term of the generating sequence is non-zero; it takes to be the smallest such that (where 0 denotes the zero vector of length ; denotes the vector of length whose first coordinates are [math] and whose last coordinate is ); if no such exists, then outputs a default index, say an r.e. index for . Based on the -st term of the generating sequence and , calculates some of the remaining terms of this sequence as a multiple of . For each such that there are , and with , , and , the -st term of the generating sequence is . Let be all the numbers such that for each , has determined a rational number for which the -st term of the generating sequence equals (in particular, there is a with ). finds all pairs for which all non-zero positions of and belong to and . outputs the least index satisfying the following conditions (if such a exists).
- (1)
. 2. (2)
For all , .
If no such exists, then conjectures .
Suppose is presented with a text for some , where is a generating sequence for such that equality is co-r.e. with respect to . Suppose . Then on will find in the limit the least number such that (since for all , ). By Condition 1, will, in the limit, always conjecture a set contained in . Furthermore, for all and every , there are integers with and such that , and therefore must contain . Since , one has
[TABLE]
and thus by Condition 2, will, in the limit, always conjecture a set containing . will therefore converge to the least index satisfying , as required.
5. Random Subrings of Rationals and Random Joins of Prüfer Groups
We have seen in Section 3 that any Martin-Löf random sequence gives rise to a random subgroup of rationals such that for some generating sequence for , equality with respect to is co-r.e. and another such that the set of representations of any non-trivial finitely generated subgroup of with respect to is r.e. The present section will define other random structures with similar properties in an entirely analogous manner.
We begin by defining random subrings of rationals based on Martin-Löf random sequences. First, one observes that for every subring of , there is a set of primes such that consists of all fractions with an integer and a product of prime powers , for some .444To see this, suppose is non-trivial (otherwise the statement is immediate); then must contain [math] as well as , and therefore by induction contains all integers. Let be the set of primes such that has a multiplicative inverse in . Then for all and all integers , one has . Conversely, let and be relatively prime integers such that and . Let and be integers with ; then . Thus for every prime factor of , and so . Let be any Martin-Löf random sequence that is Turing reducible to , and let be the subring of such that for all , iff . By the preceding observation, consists of all fractions such that is any integer and is any product of prime powers with for all and . By analogy to the definition of a generating sequence for , a generating sequence for is any infinite sequence such that . All the earlier definitions that applied to will be adapted, mutatis mutandis, to the subring .
Theorem 5.1**.**
Let be Martin-Löf random w.r.t the Lebesgue measure on . Then there is a generating sequence for such that
- (i)
equality with respect to is co-r.e.; 2. (ii)
for any finitely generated subgroup of , the set of representations of with respect to is co-r.e.; 3. (iii)
the class of all sets of representations of finitely generated subgroups of with respect to is explanatorily learnable using co-r.e. indices.
Proof 5.2**.**
We follow the construction of in the proof of Theorem 3.11 with a few modifications. Fix some prime such that ; the Martin-Löf randomness of implies that such a exists. As before, denotes the -th approximation of ; without loss of generality, assume that for all , . The -st term of will be denoted by , while the -th approximation of will be denoted by . For any , the -st term of will be denoted by . The construction of proceeds in stages. For any sequence , and , denotes the sequence obtained from by replacing its -st term with .
- (1)
Stage [math]. Set . 2. (2)
Stage .
- (a)
*Compute in succession until the least is found such that for some minimum , and is not a term of . (The Martin-Löf randomness of implies that such and exist.) Set . For each such that or contains a term equal to , and for to , if is not a term of , set . (This step ensures that for all , eventually contains all terms of the shape , where .) Then go to Step 2.b. * 2. (b)
Check for every whether the -st term of equals for some and such that . Suppose there is a least such , say . Then the -st term of is replaced with for some that is large enough so that for all occurring in and all inequalities over the range with respect to are preserved; in other words, for all pairs such that , one also has the relation , where . Set , then go to Step 2.c. 3. (c)
*Repeat Step 2.b until no term of is equal to for some and with , then go to Stage . *
Set . Then by Step 2.a, for every and , contains a term equal to . Since, as was observed earlier, every is of the shape for some and , is a generating sequence for . It remains to verify that satisfies (i), (ii) and (iii).
(i) It suffices to show that is equal to the r.e. set , where . Suppose . Since is limiting-recursive, there is an large enough so that whenever , and the value of for every in the domain of or has stabilised, i.e. for all . It follows that .
Now suppose for some , so that . By Steps 2.b and 2.c in the construction of , one has for all . In particular, if is the least number such that for all , whenever , then , which is equivalent to . Therefore .
(ii) Let be any finitely generated subgroup of that is generated by for some relatively prime integers and with . Note that every term of is equal to or is of the shape for some and , and that is a one-one sequence. Hence there is a least such that for all , ; furthermore, the Martin-Löf randomness of implies that can be chosen so that contains a term of the shape for some with . Fix some such that for all , and whenever . We claim that the complement of the set of representations of with respect to , denoted by , is equal to the r.e. set
[TABLE]
That the latter set contains follows from the fact that for all , there is an such that whenever , and for all .
Now consider any and such that and . It will be shown that . Consider all such that for each , there is some with . It may be assumed without loss of generality that for all (for if , then any difference between the value of and would have no effect on whether ). By Steps 2.b and 2.c in the construction of , there are with for all such that . Without loss of generality, assume that for all . Then . Since , and , there is an such that equals for some that is coprime to and for the largest such that is a term of (such a term exists by the choice of ). Thus is of the shape for some relatively prime integers and such that divides . Since contains a term such that (where, as stated at the beginning of the proof, and are relatively prime integers with such that is a generator of ), it follows that and therefore . Consequently, .
(iii) We first observe the following. Suppose is any text for , where for some relatively prime integers with .
- (I)
Since , the value of can be determined in the limit from by taking the greatest common divisor of all elements such that . 2. (II)
For every prime power factor of , there is some such that , and therefore . 3. (III)
There is a least such that for all , is of the shape for some prime and with ; in particular, . 4. (IV)
*By (ii), there is a minimum such that . *
Define a learner as follows. On input , determines the greatest common divisor of the set of all such that (if no such exists, then just sets ). Next, identifies all such that for all , and it approximates by determining the least such that and setting the approximation to be . For each , let be the current approximation of , where . then takes to be its current guess for a generator of , where and for each , is the largest number such that . Having determined a guess for , finds the least such that the -th approximation of
[TABLE]
denoted by , satisfies . If no such exists, then outputs a co-r.e. index for ; otherwise, outputs a co-r.e. index for . By (I), (II) and (III), on will correctly identify a generator for in the limit. Furthermore, defining as in (IV), if , then contains some element in ; thus, for all , will reject and conjecture as the correct hypothesis in the limit.
Remark 5.3**.**
The explanatory learner in the proof of (iii) of Theorem 5.1 is also conservative in the sense that for any two text initial segments and for any , where , only if (we assume that ’s hypothesis space is some fixed numbering of co-r.e. subsets of ).
We recall that for any prime , a Prüfer -group (denoted by ) may be defined as the quotient of the group of all rational numbers whose denominator is a power of by . Regarding as a subgroup of , we define a random join of Prüfer groups based on any given Martin-Löf random sequence as follows. As before, suppose is Martin-Löf random and is Turing reducible to . Then the subgroup is defined to be the join of all such that for all , the -th bit of is . In other words, consists of every fraction (modulo ) whose denominator is a product of finitely many powers of primes belonging to . The next result is the analogue of Theorem 5.1 for ; the proof is entirely similar to that of Theorem 5.1.
Theorem 5.4**.**
Let be Martin-Löf random w.r.t the Lebesgue measure on . Then there is a generating sequence for such that
- (i)
equality with respect to is co-r.e.; 2. (ii)
for any finitely generated subgroup of , the set of representations of with respect to is co-r.e.; 3. (iii)
the class of all sets of representations of finitely generated subgroups of with respect to is explanatorily learnable using co-r.e. indices.
By adapting the proofs of Theorems 3.19, 4.12 and 4.10, one obtains an almost “symmetrical” version of Theorem 5.4 for .
Theorem 5.5**.**
Let be Martin-Löf random w.r.t the Lebesgue measure on . Then there is a generating sequence for such that
- (i)
equality with respect to is r.e.; 2. (ii)
for any finitely generated subgroup of , the set of representations of with respect to is r.e.; 3. (iii)
the class of all sets of representations of finitely generated subgroups of with respect to is -learnable but not -learnable.
6. Conclusion and Possible Future Research
This paper introduced a method of constructing random subgroups of rationals, whereby Martin-Löf random binary sequences are directly encoded into the generators of the group. It was shown that if the Martin-Löf random sequence associated to a randomly generated subgroup is limit-recursive, then one can build a generating sequence for such that the word problem for is co-r.e. with respect to , as well as another generating sequence such that the word problem for with respect to is r.e. We also showed that every non-trivial finitely generated subgroup of has an r.e. representation with respect to a suitably chosen generating sequence for ; moreover, the class of all such r.e. representations is behaviourally correctly learnable but never explanatorily learnable.
A question deserving further attention is the extent to which the choice of the generating sequence for a randomly generated subgroup of rationals influences the learnability of its finitely generated subgroups; in particular, is there a generating sequence for such that every non-trivial finitely generated subgroup of has an r.e. representation with respect to and the class of all such representations with respect to is not even behaviourally correctly learnable? We also did not extend the definition of algorithmic randomness to all Abelian groups; we suspect that such a general definition might be out of reach of current methods due to the fact that the isomorphism types of even rank groups (subgroups of ) are still unknown.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] Janis M. Bārzdiņs̆. Two theorems on the limiting synthesis of functions. In Janis M. Bārzdiņs̆, editor, Theory of Algorithms and Programs I , volume 210 of Proceedings of the Latvian State University , pages 82–88. Latvian State University, Riga, 1974. In Russian.
- 2[2] Reinhold Baer. Abelian groups without elements of finite order. Duke Mathematical Journal , 3(1):68–122, 1937.
- 3[3] Ross A. Beaumont and Herbert S. Zuckerman. A characterization of the subgroups of the additive rationals. Pacific Journal of Mathematics , 1(2):169–177, 1951.
- 4[4] Lenore Blum and Manuel Blum. Toward a mathematical theory of inductive inference. Information and Control , 28:125–155, 1975.
- 5[5] John Case and Carl Smith. Comparison of identification criteria for machine inductive inference. Theoretical Computer Science , 25:193–220, 1983.
- 6[6] Rodney G. Downey and Denis R. Hirschfeldt. Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York, 2010.
- 7[7] Jerome A. Feldman. Some decidability results on grammatical inference and complexity. Information and Control 20(3):244–262, 1972.
- 8[8] Mark Fulk. A study of inductive inference machines. Ph.D. Thesis, SUNY/Buffalo, 1985.
