
TL;DR
This paper explores extensions of algorithmic randomness to measures and quantum states, discusses group theory's relation to logic with new oligomorphic group results, and covers metric spaces, Scott rank, and interpretability.
Contribution
It introduces new results on oligomorphic groups and extends concepts of algorithmic randomness to measures and quantum states.
Findings
New results on oligomorphic groups
Extensions of algorithmic randomness to measures and quantum states
Insights into metric spaces, Scott rank, and interpretability
Abstract
Some notions from algorithmic randomness are extended to measures and to quantum states. There is a lot on group theory and its relation to logic. This includes some new results on oligomorphic groups. There's also metric spaces and Scott rank, and interpretability.
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Taxonomy
TopicsAdvanced Algebra and Logic · Geometric and Algebraic Topology · Advanced Topology and Set Theory
Logic Blog 2018
Editor: André Nies
The Logic Blog is a shared platform for
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rapidly announcing results and questions related to logic
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putting up results and their proofs for further research
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parking results for later use
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getting feedback before submission to a journal
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foster collaboration.
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How does the Logic Blog work?
Writing and editing. The source files are in a shared dropbox. Ask André () in order to gain access.
Citing. Postings can be cited. An example of a citation is:
H. Towsner, Computability of Ergodic Convergence. In André Nies (editor), Logic Blog, 2012, Part 1, Section 1, available at \urlhttp://arxiv.org/abs/1302.3686.
The logic blog, once it is on arXiv, produces citations on Google Scholar.
Contents
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3 Nies and Schlicht: the normaliser of a finite permutation group
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4.4 -categorical structures with essentially finite language
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5 Kassabov and Nies: supershort first order descriptions in certain classes of finite groups
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7 Nies and Schneider: Concrete presentations, isomorphism, and descriptions
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7.3 Describing simple Lie algebras over by a first-order sentence with an additional predicate
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8 Nies, Schlicht and Tent:Bi-interpretations for -categorical structures and theories
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8.5 Bi-interpretability of -categorical theories is given by a condition
Part I Computability theory
1. Nies and Stephan: randomness and -triviality for measures
1.1. A randomness notion for measures
We consider algorithmically defined randomness notions for finite measures on Cantor space (usually probability measures). We use the letters etc for finite measures, with reserved to the uniform measure. Letters denote binary strings, elements of , . So .
This research interacts with a recent attempt to define ML-randomness for quantum states corresponding to infinitely many qubits [17]. (Probability measures correspond to the quantum states where the matrix is diagonal for each .) Here is the main definition, which was discussed during a meeting on effective dynamical systems in Toulouse March 2018, but is implicit in the earlier preprint [17]. We now have a paper on this [18].
Definition 1.1**.**
A measure is called Martin-Löf absolutely continuous (ML-a.c. for short) if for each ML-test .
It suffices to only consider descending ML-tests, because we can replace by the ML-test , and of course implies . So we can change the passing condition to .
Also, just as for bit sequences, it suffices to only consider the usual universal ML-test . So Martin-Löf a.c. ness is a property of measures.
Since is the set of non-MLR bit sequences, we obtain
is Martin-Löf a.c. .
It follows that we can actually restrict the definition to any descending universal ML-test, such as in the notation of [16, Ch. 3].
Recall that a *Solovay test * is a sequence of uniformly sets such that . A bit sequence passes such a test if . We say that a measure passes such a test if . For , we let denote the probability measure that is concentrated on .
Fact 1.2**.**
- (i)
The uniform measure is Martin-Löf a.c.
- (ii)
* is Martin-Löf a.c. iff is ML-random.*
- (iii)
Let , for a sequence of reals in with .
* is Martin-Löf a.c. iff all the sets are ML-random.*
Proof.
(i) and (ii) are immediate.
(iii) : If for a ML-test then , do is not Martin-Löf a.c. .
: given a ML-test , note that the pass this test as a Solovay test. Hence for each , there is such that for each and . This implies that . ∎
The well known fact that ML-tests are equivalent to Solovay tests generalises to measures. We use the following variant for measures of a result by Tejas Bhojraj that he proved in the quantum setting.
Fact 1.3**.**
A measure is Martin-Löf a.c. iff passes each Solovay test.
Proof.
Each ML-test is a Solovay test. So the implication from right to left is immediate. For the implication from left to right, suppose that is a Solovay test and . We define a ML-test that fails at level . Let denotes the clopen set given by strings in of length . By a minor modification of the standard proof (e.g. [16, Prop. 3.2.19]), let be the open set generated by strings such that
for many .
As in the standard proof one shows that . Then (after thinning) can be turned it into a ML-test.
Given we pick sufficiently large so that for some set of size we have for each . We show that . Let range over strings of length . We have
by definition of . Since , this implies
.
Since this shows as required. ∎
For a measure and string with let be the localisation: . Clearly if is ML-a.c. then so is .
A set of probability measures is called convex if for implies that the convex combination , where the are reals in summing up to . The extreme points of are the ones that can only be written as convex combinations of length 1 of elements of .
Proposition 1.4**.**
The Martin-Löf a.c. probability measures form a convex set. Its extreme points are the Dirac measures.
Proof.
For convexity, suppose is a descending ML-test. Then
for each ,
and hence .
If is a Dirac measure then it is an extreme point. Conversely, if is not Dirac there is a least number such that the decomposition
is nontrivial. Hence is not an extreme point. ∎
1.2. Initial segment complexity of a measure as a -average
Let be the -average of all the over all strings of length . In a similar way we define .
Fact 1.5**.**
, and therefore
Proof.
Suppose is chosen so that for each we have (we can in fact ensure with the right universal machine, see [16, Ch 2]).
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This does it. ∎
We say that has complex initial segments if . The analog of Levin-Schnorr fails for measures in both directions.
Example 1.6**.**
There is a Martin-Löf a.c. measure such that
.
Proof.
We let where is ML-random and for a sequence of reals in that add up to , and a sufficiently fast growing sequence . Such a is Martin-Löf a.c. by Fact 1.2.
For we have
.
So if we ensure that we are good. For instance, we can let and . ∎
We falsify the converse implication by the following.
Theorem 1.7**.**
There are a random and a non-random such that, for all , .
Proof.
Let be a low Martin-Löf random set. There is strictly growing function such that the complement of the image of is a recursively enumerable set and for all . Note that this function exists, as is low and Martin-Löf random and so, for all , the maximal such that can be found in the limit.
Now let . By a result of Miller and Yu [14, Cor. 3.2], there is a Martin-Löf random such that there exist infinitely many with . For this set , let
.
Note that , as one can enumerate the set until there are, up to , only many places not enumerated and then one can reconstruct from and and the last bits of . As is Martin-Löf random, and so,
.
The definitions of give . This shows that for almost all .
However, the set is not Martin-Löf random, as there are infinitely many with . Now can be computed from and , as one needs only to enumerate until the nonelements of below are found and they allow to see where the zeroes have to be inserted into the string in order to obtain . Note furthermore, that for almost all and thus for infinitely many , so cannot be Martin-Löf random. ∎
Note that the measure has only two equal-weighted atoms and furthermore satisfies that one of these atoms is not Martin-Löf random. So every component of a universal Martin-Löf test has at least -measure . On the other hand, for almost all by the preceding result. Thus one has the following corollary.
Corollary 1.8**.**
There is a measure with complex initial segments which is not Martin-Löf a.c.
Proposition 1.9**.**
Suppose that is a measure such that for infinitely many . Then is Martin-Löf a.c.
Proof.
Suppose that is not Martin-Löf a.c. So there is a ML-test and such that for each . If is a string of length such that then
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To see this let be the machine that on a pair of auxiliary inputs gives a description of length for each such (so the descriptions for different are prefix free). It follows that for as above
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Now view as given by an enumeration of strings, and choose large enough so that , where denotes the open set given by the strings in this enumeration of length at most . Let be a constant such that for each of length . We have
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The last inequality holds because
and
.
Now given let . By the above, for large enough we have . So is not strongly Chaitin random. ∎
Question 1.10**.**
In analogy to the case of bit strings, does strong Chaitin randomness of a measure imply Martin-Löf a.c. ness relative to ?
If the measure has an atom but is not Dirac then function is not bounded from below by for any . The reason is that when then for this atom, the function is not bounded by any constant and therefore it can go arbitrarily low; this would then make the average to be below for any given at infinitely many .
A fan is a prefix-free set such that . Note that is necessarily finite. The -average length of is
.
Generalising the above, we let
.
We say that has complex initial segments in the strong sense if
for each fan . To be done: does this imply Martin-Löf a.c. ?
1.3. Connection to ML-randomness of measures in
A natural probability measure on the space of probability measures on Cantor space has been introduced implicitly in Mauldin and Monticino [13], and in Quinn Culver’s thesis [4] in the context of computability, where he shows that this measure is computable. Let be the closed set of representations of probability measures; namely, consists of those such that for each string . is the unique measure on such that for each string and , we have
.
That is, we choose at random w.r.t. the uniformly distribution in the interval ), and the choices made at different strings are independent.
Proposition 1.11**.**
Every probability measure that is ML-random wrt to is Martin-Löf a.c. .
The proof is based on two facts. For be open, for the duration of this proof let range over and let
.
Fact 1.12**.**
.
Proof.
Clearly for each we have
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Further, for because there is a -preserving transformation of such that . Therefore .
If are incompatible then . Now it suffices to write where the strings are incompatible, so that . ∎
Fact 1.13**.**
Let and let be a ML-test such that there is with . Then is not ML-random w.r.t. .
Proof.
Observe that by the foregoing fact
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Let which is uniformly in . Fix such that ; then is a ML-test w.r.t. that succeeds on . ∎
Culver shows that each ML-random for is non-atomic. So by Fact 1.2 the converse of Prop. 1.11 fails: not every Martin-Löf a.c. is ML-random with respect to .
1.4. SMB theorem
We recall some notation from the 2017 Logic Blog, Section 6.2, adapting some letter uses. denotes the space of one-sided infinite sequences of symbols in . We can assume that this is the sample space, so that . By we denote their joint distribution. A dynamics on is given by the shift operator , which erases the first symbol of a sequence. A measure on is -invariant if for each measurable .
We consider the r.v.
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(recall that is w.r.t. base ).
Recall that is ergodic if every integrable function with is constant -a.s. An equivalent condition that is easier to check is the following: for ,
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For ergodic , the entropy is defined as , where
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One notes that so that the limit exists. Also note that .
The following says that in the ergodic case, -a.s. the empirical entropy equals the entropy of the measure.
Theorem 1.14** (SMB theorem).**
Let be an ergodic invariant measure for the shift operator on the space . Then for -a.e. we have .
If is computable, then the conclusion holds for -ML-random by results of Hochman (implicit) [8] and Hoyrup [10]. Recent work of A. Day extends this to spaces other than and amenable group actions. Here we keep the space but change the type of object. We say that a measure is -Martin-Löf a.c. if for each -ML test . Here is a special case of Conjecture 6.3 in 2017 Logic Blog where the states when restricted to the matrix algebra are diagonal. Enough patience will suffice.
Conjecture 1.15** (Effective SMB theorem for measures).**
Let be a computable ergodic invariant measure for the shift operator on the space . Suppose the measure is -Martin-Löf a.c. . Then .
1.5. -triviality for measures
Definition 1.16**.**
A measure is called -trivial if .
For Dirac measures this is the same as saying that is -trivial in the usual sense.
Proposition 1.17**.**
Suppose is -trivial. Then has atoms. In fact, is concentrated on its atoms.
Proof.
For each there is (in fact ) such that for each there are at most strings of length with . Since is non-atomic, there is so that for each of length we have . Note that there is a constant such that for each . Then we have that in the - average , the of length such that have total measure at most , and each . So the average is at least up to a constant. ∎
Proposition 1.18**.**
For each order function there is a non-atomic measure such that .
In fact, for each nondecreasing unbounded function which is approximable from above there is a non-atomic measure such that .
Proof.
There is a recursively enumerable set such that, for all , has up to and up to a constant non-elements. One let be the measure such that in the case that all ones in are not in and otherwise, here is the number of non-elements of below . One can see that when then can be computed from and the string which describes the bits at the non-elements of . Thus . It follows that , as the -average of strings with is at most plus a constant. ∎
REMARK. Note that when is a recursive order function or an order function which is approximable from above then there is a further order function which is approximable from above such that for all ; one just chooses . Thus one can bring the above result into the form that for all recursive order functions there is a measure satisfying .
The -trivial measure form a convex set. However it is not closed under infinite convex sums. One takes finite sets which pointwise converge to and let the fall sufficiently slowly so that at level there is still measure on and therefore the corresponding -average grows like the squareroot of , and not like .
In more detail, let and . All sets are finite and thus -trivial. Furthermore, the sum of all is .
Let . Then for almost all and thus the average grows faster than . So the measure is not -trivial.
We call a measure low for if for each
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Thus we form the -average over all oracles. Clearly if is low for as a set then is low for . Merkle and Yu have shown that is low for . So lowness for does not imply -triviality. It would still be interesting to relate lowness for with -triviality in the case of measures.
2. Yu: A note on
Let be the least ordinal that cannot be presented by a -well ordering over and be the one relative to . Define
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The following result must be well known but I have not found a reference.
Proposition 2.1**.**
* is but neither nor .*
Proof.
if and only if every -singleton coding a well ordering of is bounded by a -singleton coding a well ordering of if and only if there is a real coding a well ordering of such that contains all the -singletons and every real in is . So is a -set.
Since every nonempty -set contains a -member, we have that is not . Now suppose that is . Then let be a real computing all the -reals. Then every -random is -random. By the assumption, is a -set. If contains an -random real, then is not empty. So by Shoenfield’s absoluteness, is not empty and contains a real . But every real must be -random, a contradiction. ∎
Part II Group theory and its connections with logic
3. Nies and Schlicht: the normaliser of a finite permutation group
Let be a group. The group of inner automorphisms forms a normal subgroup of . The quotient group is called the group of outer automorphisms, denoted by . For instance, has 2 elements. For more examples, note that since is invariant, there is a canonical surjection with kernel containing . In the case of equality holds, so that .
It is well-known that no cyclic group of odd order is of the form for any group . On the other hand, every finite group is the outer automorphism group of some group which can be chosen to be fundamental group of a closed hyperbolic 3-manifold (a result of Sayadoshi Kojima). See [2] for background.
Here is a simple (and known) fact. Given a finite group with domain , we think of as embedded into via the left regular representation where . (E.g. for , we have .) Let denote the normaliser of in .
Proposition 3.1**.**
There is a canonical surjection mapping to , thereby showing that is isomorphic to .
Proof.
A canonical map is defined by
if .
Clearly is an automorphism of for each . To check that is a homomorphism, note that for
.
Now . So the equation above implies .
is a surjection: iff (where we identify with ).
Finally, iff for each iff is inner. ∎
In fact we don’t need that is finite.
4. Kaplan, Nies, Schlicht and others: closed subgroups of
We make some remarks on closed subgroups of . These are the automorphism groups of structures with domain . We in particular consider the following kind. A closed subgroup of is called oligomorphic if it has only finitely many -orbits, for each . These are the automorphism groups of the -categorical structures with domain . We say that a topological group is quasi-oligomorphic if it is in a topological group isomorphism with an oligomorphic group.
4.1. The centre
Let be a closed subgroup of . For , by we denote the number of orbits of the natural action of on ; such orbits will be called -orbits. (The parameter is denoted in [3].) For let denote the number of -orbits containing a pair of the form (which only depends on the 1-orbit of ). Suppose that and let represent the -orbits.
Fact 4.1**.**
. In particular, if is -transitive then the size of the centre is at most .
Proof.
Write . For any , and , if then and are in different -orbits. Hence .
Consider now the natural left action (Cartesian product of sets). If are distinct then for some and , so that . Hence . This shows the required bound. ∎
Greg Cherlin suggested another way to prove that for oligomorphic the centre is finite (but without an explicit bound on its size). We may suppose for an -categorical structure with domain . Note that is definable as a binary relation on iff is invariant under the natural action of on , i.e., . Since there are only finitely many definable binary relations, is finite.
We consider examples of oligomorphic groups with a nontrivial centre.
-
For any finite abelian group , the natural action of on yields an 1-transitive oligomorphic group with centre . The number of -orbits is at least because implies that . (In fact it is .)
-
Let be the structure with one equivalence relation that has all classes of size ; say, for
.
Write . We have , where denotes the unrestricted wreath product. Here is viewed with its natural action on ; so is an extension of by with acting by , for . (Note that is the automorphism group of the structure where the individual equivalence classes are now distinct unary predicates. is the normaliser of in .) The centre consists of the identity and the automorphism that maps each element to the other one in its class. The centre of is trivial.
- Let be the structure with equivalence relations such that each -class has size 2 and each -class has size 4. Then the automorphism group of a single -class is , and hence . As before, the centre consists of the identity and the automorphism that switches each element in its class. We have .
Similarly, for each there is an oligomorphic group with a chain of higher centres.
4.2. The central quotient
The main purpose in this section is to show that for oligomorphic , the central quotient is quasi-oligomorphic. Some facts needed along the way hold in more generality.
Suppose a group acts on a set and . Write for the orbit equivalence relation of the subaction of . Note that acts naturally on via (where is the class of ). Since elements of act as the identity, acts on .
Suppose now initially acts faithfully on a set , say . Let and let act on by the usual diagonal action. Let .
Fact 4.2**.**
The action of on is faithful.
To see this, suppose . So grab such that
.
Let . Then , because any element of such that satisfies , so that .
We now switch to topological setting. Given a Polish group with a faithful action , say for a countable set , we obtain a monomorphism given by . A Polish group action is continuous iff it is separately continuous (i.e. when one argument is fixed). In the case of an action on countable (with the discrete topology), the latter condition means that
- (a)
for each , the set is open.
So is continuous iff is continuous.
Definition 4.3**.**
We say that a faithful action is strongly continuous if the embedding is topological.
Equivalently, the action is continuous, and for each neighbourhood of , also is open, namely,
- (b)
for each neighbourhood of , there is finite set such that implies .
Since is Polish, strong continuity of the action implies that is topologically isomorphic via to a closed subgroup of (see e.g. [5, Prop. 2.2.1]).
Now consider the case that and is a closed subgroup of . Since is closed, is naturally a Polish group via the quotient topology: for , the subgroup is declared to be open iff is open in . (See e.g. [5, Prop. 2.2.10].)
Let as above. Suppose that is infinite (e.g. when is finite), so through the action above we obtain an (algebraic) embedding of into (which can be identified with ).
Claim 4.4**.**
Suppose that is a closed subgroup of that acts 1-transitively on . Suppose that is finite. Then is a topological embedding.
Proof.
We check the conditions (a) and (b) above.
(a) Suppose that . Then iff there are and such that and . Since is finite this condition is open.
(b) An open neighbourhood of has the form wherewhere is open and . By definition of the topology on , we may assume that for some finite (as usual is the pointwise stabiliser). Let . Consider , where .
Suppose that . This means that for each , there is a such that . Since acts faithfully and 1-transitively on , for each , and each , implies that . Therefore given another pair , . Let be this unique witness. Then for each , hence and therefore . ∎
Theorem 4.5**.**
Let be oligomorphic. The central quotient is quasi-oligomorphic (i.e. homeomorphic to an oligomorphic group).
Proof.
It is known (as pointed out to us by Todor Tsankov) that we may assume is 1-transitive. To see this, one shows that there is an open subgroup such that the left translation action of on the left cosets of is faithful and oligomorphic, and the corresponding embedding into a copy of is topological. To get , let represent the 1-orbits of . Let be the pointwise stabiliser of . If then there is and such that . So , and hence . So the action is faithful. The rest is routine using (a) and (b) above.
Now we can apply Claim 4.4, recalling that is finite. ∎
For any closed subgroup of the higher centres are normal, so their orbit equivalence relations are -invariant. If is oligomorphic they are definable in . Hence the progression of higher centres has to stop at a finite stage for each oligomorphic group .
4.3. Conjugacy
We show that conjugacy of oligomorphic groups is smooth. For a closed subgroup of , let denote the orbit equivalence structure. For each this structure has a -ary relation symbol, denoting the orbit relation on -tuples. (One could require here that the tuples have distinct elements.)
The following fact holds for non-Archimedean groups in general.
Fact 4.6**.**
Let be closed subgroups of .
* are conjugate via via .*
Proof.
Immediate.
Let be the canonical structure for ; namely there are many -ary relation symbols, denoting the -orbits. Let be the structure in the same language where the -equivalence classes of are named so that is an isomorphism . Clearly and . Further, . ∎
Proposition 4.7**.**
Conjugacy of oligomorphic groups is smooth.
Proof.
The map is Borel because we can in a Borel way find a countable dense subgroup of , which of course has the same orbits. Now apply Fact 4.6. For countable structures in a fixed language, mapping to its theory is Borel. Since the theory can be seen as a real, for -categorical structures, this shows smoothness. ∎
For corresponding structures with , conjugacy of via means that and have the same definable subsets. To see this, consider the case that is the canonical structure for .
We note the following topological variation of Prop 3.1. The notation is introduced above.
Proposition 4.8**.**
(i) equals the normaliser of in .
(ii) If is oligomorphic then as a topological group is profinite.
Proof.
(i) Let , . Clearly maps -orbits to -orbits, so is a renumbering of the named -orbits in . Therefore .
Let be an -orbit, and let . If and , choose such that . So is contained in an -orbit . By a similar argument, is contained in an -orbit. Therefore is an -orbit. Hence .
(ii). Let be the number of -orbits of . Define a continuous homomorphism by if is the finite permutation describing the way permutes -orbits (numbered in some way). Clearly equals the kernel of , and is therefore a topological embedding. Since the range is compact, its inverse is also continuous. A closed subgroup of a profinite group is again profinite. ∎
The converse of (ii) may fail: can be profinite, and even trivial, without being oligomorphic. For instance, there is a countable maximal-closed subgroup of , e.g. , the automorphism group of the structure , with the ternary function (Kaplan and Simon). This structure is not -categorical.
We don’t know at present whether every profinite group occurs that way.
4.4. -categorical structures with essentially finite language
One says that a structure has essentially finite language if is interdefinable with a structure over a language in a finite signature. (Interdefinable means same domain and same definable relations.). We present a basic fact that can be used to obtain an oligomorphic group that is not isomorphic to the automorphism group of such an -categorical structure.
Lemma 4.9**.**
The following are equivalent for a countable structure .
- (i)
* is interdefinable with a structure in finite language with maximum arity , and quantifier elimination.*
- (ii)
* is -categorical, and for each , each -orbit of is given by its projections to -orbits.*
Proof.
(ii) implies (i): Let , and let be the orbit structure of . Thus, is like above but has an -ary predicate for each -orbit. (Note that is a Fraisse limit. is a reduct of , and as in the finite case above its automorphism group is the normaliser of .)
(i) implies (ii): Clearly is -categorical, as there are only finitely many types for each . Each formula in variables is a Boolean combination of q-free formulas in variables. If describes an -orbit we can assume it is a conjunction of such formulas. A formula in variables describes a finite union of -orbits. Hence the -orbit is given by its projections: if two tuples have are in the same projection orbits then both or none satisfy .
∎
5. Kassabov and Nies: supershort first order descriptions in certain classes of finite groups
Nies and Tent [19] showed that every finite simple group has a first-order description (in the usual language of group theory) of length . This result is near optimal for the whole class of finite simple groups because of the cyclic groups, using a counting argument together with the prime number theorem. We show that shorter descriptions can be obtained for certain natural classes of finite simple groups. This works for instance when the groups in the class have presentations of length and the diameter of the corresponding Cayley graph is also . For instance, by this method the alternating groups can be described in length .
The following definition is from Nies and Tent [19].
Definition 5.1**.**
Let be an unbounded function. We say that an infinite class of finite -structures is -compressible if for each structure in , there is a sentence in such that and describes .
For notational convenience, we will use the definition
[TABLE]
Theorem 5.2** ([19], Thm. 1.2).**
The class of finite simple groups is -compressible.
The first-order formulas for generation developped in [19] will be used in the context of presentations with Cayley graphs of small diameter.
Lemma 5.3** ([19], proof of Lemma 2.4).**
For each positive integers , there exists a first-order formula of length in the language of groups such that for each group , for some word in of length at most .
Proof.
Let
[TABLE]
For let
[TABLE]
Then has length , and if and only if can be written as a product, of length at most , of ’s and their inverses. ∎
Lemma 5.4**.**
Suppose that a finite simple group has a presentation
* of length .*
Also suppose that the diameter of the Cayley graph is bounded by , that is, each has the form for some free group word of length at most .
There is a sentence of length describing the structure .
Proof.
Let be the formula
Replacing the by new constant symbols, the models of the sentence thus obtained are the nontrivial quotients of . Since is simple, this sentence describes . ∎
Lemma 5.5**.**
Suppose is a generating set of containing a 3-cycle, say . Then the Cayley graph of with respect to has diameter .
Proof.
acts -transitively on the set of -cycles on by conjugation. Since the number of -cycles is , each -cycle can be expressed by a word of length in the generating set. Any even permutation can be written as a product of at most 3-cycles. ∎
Proposition 5.6**.**
The classes of finite alternating/symmetric groups and of finite symmetric groups are -compressible.
Proof.
We want to describe each , and we may assume . By [6, Cor 3.23], has a presentation of length . By construction, one of the generators of in the above presentation is a 3-cycle. Therefore the diameter of the Cayley graph is at most by Lemma 5.5. So we can apply Lemma 5.4 with . (Actually a more careful look at the generation set gives that all 3-cycles can be expressed as words of length and the diameter of the Cayley graphs is . )
The case of symmetric groups is similar. One uses a transposition instead of 3-cycle. We need to take into account that the symmetric groups are not simple. Since the only nontrivial quotient has size , it suffices to require in the description that the group has at least elements. ∎
Proposition 5.7**.**
Fix a prime power . The class of groups is -compressible.
Proof.
The argument is similar to the case of alternating groups. By [6, Thm. A and Thm. 6.1], has a presentation of length
.
The generating set for this presentation contains a generating set for with diameter in and a generating set of with diameter in . Thus, every elementary matrix in can be expressed as a word of length at most . Finally a row reduction argument gives at any matrix in is a product of at most elementary matrices, which implies that the diameter of the Cayley graph is at most (this bound can be improved to by more careful examination of all element in generating set). By Lemma 5.4 the groups can be described by sentence of length . ∎
5.1. Rank 1 groups
The result in [6, Thm. 4.36] gives a bound for both length of presentation and diameter for groups such as and . Since the size of these groups is polynomial in , this doesn’t help to get descriptions shorter than the ones in Theorem 5.2.
If we fix the characteristic and allow descriptions in second order logic, something can be done. Recall that second order logic allows quantification over relations and functions of arbitrary arity.
Proposition 5.8**.**
Fix a prime . The field has a second-order description of length
Proof.
For the first-order sentence from [19, Section 4] describing says that the structure is a field of characteristic such that for all elements we have and there is some with . Now in the second order version introduce function symbols such that and . Thus . Given these we can express that in length using the binary expansion of . ∎
By the biinterpretability method described in [19, Section 5], short descriptions of the fields imply short descriptions of the finite simple groups defined over them. For Suzuki and Ree groups, we have and , respectively.
Theorem 5.9**.**
The classes of Suzuki groups and of small Ree groups are compressible in second order logic.
Part III Metric spaces and descriptive set theory
6. Nies and Schlicht: Scott relation in Polish metric spaces
For tuples in a Polish metric space, the Scott relation at level is defined as usual: for each challenge on the left side there is response on the right side so that the enumerated metric spaces and are isometric; similar for the sides interchanged.
Proposition 6.1**.**
There is a computable Polish metric space and a computable sequence of distinct elements of such that the set
[TABLE]
is -complete.
Proof.
Note that is clearly a set. To prove that it is -complete, we first fix some notation. If and , let denote the -th slice of and its projection to the first coordinate. For , we say that is universal for a point class on if every set in occurs as a slice.
Claim 6.2**.**
There is a -universal set such that is -complete.
Proof.
Let be universal for -subsets of . Let be a computable bijection and the induced bijection. It is easy to see that is universal for -subsets of . Since the projection of to is universal for -subsets of , it is -complete. Since has the same -degree as it follows that the projection is -complete as well. ∎
We fix a -universal set as in the previous claim. Then is -universal. It now follows that the equivalence relation on defined by is -complete as a set, since and is -complete.
Question 6.3**.**
Show that is -complete as an equivalence relation.
Claim 6.4**.**
We can associate in a computable way to each a Polish space of diameter at most and some with distance set .
Proof.
We first define an auxiliary Polish metric space . Let . Let for and let be the standard ultrametric on . We define
[TABLE]
To see that is a metric on , first note that by the ultrametric inequality, is at most the maximum of and . So if or then
[TABLE]
We can hence assume that and additionally that by symmetry between and . Again by the ultrametric inequality for . In both cases, we have
[TABLE]
By our assumption , we further have
[TABLE]
Hence satisfies the triangle inequality.
We now define the required spaces as subspaces of . By identifying with the set of irrational numbers in , we let be a closed set with . Note that we can obtain computably in , assuming that our universal sets are constructed in the usual way. Now let and let be an element that is identified with . It is clear that has the required distance set in . ∎
Now let and the metric on given by the metrics and if and for some . Since player II wins if and only if , it is now easy to see from the previous two claims that is -complete. ∎
Part IV Model theory and definability
7. Nies and Schneider: Concrete presentations, isomorphism, and descriptions
7.1. Summary, mostly in layman’s terms.
Mathematical structures are usually given by concrete presentations. A computer scientist might think of a graph as a concrete object stored in a computer, for instance an adjacency list, which is a list of all the vertices and all the edges. For another example, a set of generators together with a set of relators on them present a group.
What really counts is the essence of the structure, the structure “up to isomorphism”: think of the shape of the graph, or of the abstract group. Two concrete presentations that yield the same abstract structure are called isomorphic. Being concrete, the presentations can be used as input to some kind of computation. The question arises:
Question 7.1**.**
How hard is it to tell whether two presentations are isomorphic?
It is still unknown whether one can decide efficiently that two concretely presented finite graphs are isomorphic (though Babai has recently shown that the decision problem is in pseudopolynomial time).
Related questions are the following. Given a reference class of infinite structures,
Question 7.2**.**
which structures are determined within the class by their first order theory?
For the class of finitely generated groups, this property is called quasi axiomatisable (QA); see the last two chapters of the by now venerable survey [15]. For instance, abelian groups are QA. Even better,
Question 7.3**.**
which structures can be described within the class by a single sentence in first-order logic?
In the same context, this property is called quasi finitely axiomatisable, or QFA [15]. Abelian groups are never QFA, but other very common groups are, e.g. the Heisenberg group or the Baumslag-Solitär group .
Even a description by the full first-order theory would necessarily only determine the essence of the structure (technically: isomorphic concrete structures are elementarily equivalent). It is interesting to study these questions especially in the setting of topological algebra and Lie algebras, where not much has been done so far. The point is that first-order logic can only indirectly address the topology, because that is given by subsets.
7.2. Some more detail for mathematicians.
As mentioned, we have to distinguish between concrete presentations of a structure, and the abstract structure “up to isomorphism”. Consider a finite presentation of a group:
.
This describes the concrete group , where is the normal subgroup generated by . Given two finite presentations, it is undecidable in general whether they describe isomorphic groups (Rabin).
A finite presentation of a group or Lie algebra, say, is a description of a concrete structure. We can also describe a structure merely up to isomorphism. If we want to do this, we have to pick some language from mathematical logic and a corresponding satisfaction relation. First-order logic has the additional advantage that it doesn’t look beyond the immediate structure as given by the elements and the relations among them (for instance, subsets of the structure are not allowed). This is a severe restriction, given that for instance in group theory, one frequently studies things like maximal subgroups etc. In first order logic, we can talk about particular ones if they are definable, e.g., the centre or the centraliser of an element. But we can’t quantify over the whole lot.
If the structure is finite, we can look for a short sentence, relative to the size of the structure, describing it; e.g. Nies and Tent [19] do this for (classes of) finite groups.
If the structure is infinite, we need the external information given by the reference class, because we can not describe it by a single first-order sentence. The class needs to at least prescribe the cardinality of the structure. For instance, we can describe the ordering of the rationals by a single sentence within the countable structures. A f.g. group is called QFA (for quasi-finitely axiomatisable) if, within the class of f.g. groups, it can be described by a single sentence in the language of group theory,
To reiterate, given a class of concretely presented structures of the same signature, there are two interrelated types of question
- (a)
How complicated is the isomorphism relation between structures in ?
- (b)
Which structures can up to isomorphism be described within by their theory?
- (c)
Or even by single a first-order sentence?
As for (a), the intuition may be that if a concrete structure has a complicated equivalence class under the isomorphism relation, it is hard to describe abstractly. The trivial upper bound for isomorphism is (assuming the class itself is arithmetical). On the other hand, elementary equivalence is easier, namely hyperarithmetical, and in fact .
Question (b) is interesting in particular if some classification of structures in is known; for instance, we will below consider the simple Lie algebras over . In this case, one would try to prove that each structure in the class has a description by a first-order sentence. This means that the classification can be expressed in first-order language for the right signature, given the reference class (which may be not first-order axiomatisable). In this case one would hope that all the sentences have a common bound on the number of quantifier. If so, this makes the isomorphism relation arithmetical: two structures are non-isomorphic iff there is a sentence at the given level of complexity that holds in one but not the other.
7.3. Describing simple Lie algebras over by a first-order sentence with an additional predicate
We describe simple Lie algebras over by a first-order sentence in the language with and the equivalence relation that vectors and span the same subspace.
There is a formula expressing that generate as a vector space:
.
Let denote the free associative algebra in generators over ; it is generated as a -vector space by all the words in the generators. The usual commutator in is denoted . With this commutator, becomes a Lie algebra. The free Lie algebra in n variables over is the Lie subalgebra of generated by .”
Each finite-dimensional Lie algebra is finitely presented, because the multiplication table on the basis elements gives a finite presentation. So we have relators of the form where the are complex coefficients. (The situation is analogous to the case of finite groups.)
Using Cartan’s classification, Serre proved that each semisimple Lie algebra over is finitely presented where coefficients are integers in . This can be seen from the Cartan table; see Humphreys [11, Section 18.1] for the presentation, and also note that the and there generate . Then there is depending on the dimension of such that only commutators of depth up to in the Lie generators are needed to generate as a vector space. Thus the generators in satisfy the formula saying that each commutator of depth in those generators is a linear combination of the commutators of depth .
To describe a simple within the Lie algebras over , we express that is non-trivial and that there are satisfying the Serre relations, the formula , and, using the formulas , that the commutators of depth in the generate as a vector space. Since is simple, this sentence describes .
To obtain a description in the language of Lie algebras we would need to define in terms of the Lie operations. First one would show that is invariant under automorphisms in the finite dimensional case.
8. Nies, Schlicht and Tent:Bi-interpretations for -categorical structures and theories
We discuss bi-interpretations of pairs of -categorical theories. We begin with structures rather than theories, because definability is easier to grasp. Definability will always mean without parameters.
8.1. Interpretations of structures
Suppose that are first-order languages in countable signatures. Interpretations via first-order formulas of -structures in -structures are formally defined, for instance, in Hodges [9, Section 5.3]. Informally, an -structure is interpretable in a -structure if the elements of can be represented by tuples in a definable -ary relation on , in such a way that equality of becomes a -definable equivalence relation on , and the other atomic relations on are also definable.
We think of the interpretation of in as a decoding function . It decodes from using first-order formulas, so that is an -structure. Each -formula corresponds to a -formula which is the saturation under of a -formula . We write .
For a structure , recall that has a sort for each definable equivalence relation on and definable -closed , and besides the inherited ones has definable relations , between and , given by
[TABLE]
For instance if , we have the relation that is the equivalence class of .
Clearly acts on . The -orbits on a sort have the form where is an -orbit of . So if is oligomorphic, there are only finitely many such -orbits.
Example 8.1**.**
Let be an equivalence relation with all classes of size . Take unary predicates partitioning the domain, and let be the structure where each -class has exactly one element in and one in , for some . Then has 4 orbits, the sort only has two orbits. For the first, above can be either or . **
Throughout we will have -categorical structures with . There are various equivalent views of expressing interpretation of in .
- (a)
for some interpretation , as above
- (b)
A map with range contained in single sort, 111An alternative definition (CITE EVANS) allows the range to be a subset of finitely many sorts. sending relations -definable in to relations -definable in . This map extends canonically to a map .
- (c)
There exists a topological homomorphism such that the range of is oligomorphic.
(a), (b) are merely reformulations of each other. For (a) to (c), let which has oligomorphic range by remarks above.
For (c) to (a) see Hodges [9, Section 7.4].
8.2. Bi-interpretations of structures
There are several equivalent formulations. Fix structures , .
(a) , and some isomorphisms and are definable in , in , respectively. (Thus, is described by a formula with free variables, where is the product of the dimensions of the two interpretations.) If the structures are -categorical, we can let be the restriction of to the sort on which is defined.
(b) , and the maps given by , and analogous for , are definable in the respective structure.
Note that for some sort .
A bi-interpretation introduces a matching of orbits. Suppose is -dimensional, and is -dimensional.
Fact 8.2**.**
For each -orbit of , is an -orbit of (under the action of , on the sort which contains the range of .
Proof.
As usual let . For simplicity first let . Recall from Ahlbrandt/Ziegler [1] (detail in \hrefhttp://wwwf.imperial.ac.uk/ dmevans/Bonn2013_DE.pdf David Evans’ 2013 notes, Thm. 2.9) that the “dual” is a topological isomorphism, where
[TABLE]
So as ranges over , ranges over . Let , then as required. More generally, for each we have . ∎
8.3. Bi-interpretations of theories
We can also formulate biinterpretability for complete theories , easiest in countable languages. Note that theories can be seen as infinite bit sequences and hence the set of theories carries the usual Cantor space topology. The complete theories form a closed set. To be -categorical is an arithmetical property of theories, because by Ryll-Nardzewski this property is equivalent to saying that for each , the Boolean algebra of formulas with at most free variables modulo -equivalence is finite.
To fix some notation, the sorts in models of have the forms , resp, where is an -ary definable relation, is -ary, and , are definable equivalence relations. each are -ary, where . Given as above, we express that for an arbitrary model of and , we have , and evaluated in induces an isomorphism of and (a structure with domain a sort of ); similarly, is an isomorphism of and . This can be expressed by two possibly infinite lists of sentences that have to be in , and in , respectively.
Remark 8.3**.**
Since ’s domain is a sort of and is -categorical, requiring that exists is actually redundant: can be chosen to be “”. This means that we apply the interpretation to the definable isomorphism , obtaining an isomorphism , i.e. . Clearly is invariant under the action on . Hence is -definable.
Fact 8.4**.**
Suppose , are bi-interpretable theories in the notation above. For each model of , letting , a model of , we have that are bi-interpretable as models.
Proof.
is the identity map on this concrete structure . we can therefore choose the same as , and is definable in . is the same as , hence definable in . ∎
Remark 8.5**.**
In the case of theories rather than structures, the matching of orbits in Fact 8.2 becomes a matching of types. Each -type of , i.e. an atomic formula with free variables , is given by a type of in the sense that is , and therefore by a type of whose saturation under the definable equivalence relation on is equivalent to (note there could be various such types ). Similarly for types of .
8.4. Isomorphism of groups and bi-interpretability
By isomorphisms of topological groups, we always mean topological isomorphisms. Two -categorical structures are bi-interpretable iff their automorphism groups are isomorphic; \hrefhttp://wwwf.imperial.ac.uk/ dmevans/Bonn2013_DE.pdf David Evans’ 2013 notes, Thm. 2.9. This was originally proved by Coquand.
Theorem 8.6**.**
Isomorphism of oligomorphic groups is Borel bi-reducible with bi-interpretability of -categorical theories.
Proof.
: From oligomorphic we can in a Borel way determine a countable dense subgroup . The canonical structures for and are equal. The canonical structure for can thus be Borel determined from . From we can Borel determine the theory . Then iff and are bi-interpretable by \hrefhttp://wwwf.imperial.ac.uk/ dmevans/Bonn2013_DE.pdf David Evans’ 2013 notes, Thm. 2.9.
: From a consistent theory in a countable signature, via the Henkin construction we can in a Borel way determine a model with domain . Let be the automorphism group of such a model, which is a closed subgroup of . Then for -categorical theories , we have that is bi-interpretable with iff is isomorphic to by \hrefhttp://wwwf.imperial.ac.uk/ dmevans/Bonn2013_DE.pdf David Evans’ 2013 notes, Thm. 2.9. ∎
8.5. Bi-interpretability of -categorical theories
is given by a condition
Theorem 8.7**.**
There is a relation which coincides with bi-interpretability on the set of -categorical theories. In particular, bi-interpretability of -categorical theories is and hence Borel.
Proof.
(a) The initial block of existential quantifiers in the condition states that there are (numbers that are codes for) sorts as in Subsection 8.2 in the languages of , and a potential isomorphism described by a formula (which determines an isomorphism as in Remark 8.3).
(b) To complete the interpretations of theories, it is sufficient to provide for each -type of (describing a -orbit in any model of ) a type of , in which case the orbit for goes to the orbit induced on the sort by . Similarly for and interchanged.
We now Turing compute from the join of and as oracles a tree whose maximal branching at each node is also bounded computably in . Any path on the tree will provide a bi-interpretation based on the given finite information in (a). It is in that there is such a branch by König’s Lemma, so the whole statement is as required.
The -th level of the tree provides matchings of the -types of with -types of , and -types of with -types of , according to Remark 8.5. From , we can compute how many -types there are, and how many possible matchings exists, so the branching bound is computable in the join of and . We require that the matchings are consistent with that the map defined by describes an isomorphism.
To see how to do this, we take a -type of . For simplicity of notation, assume that . So we are given an atomic formula for . A path of length in the tree provides sufficient information about . We want to show that in any bi-interpretation extending this path, maps the 2-orbit defined by in to the 2-orbit defined by in .
Starting from , we have (in the sense introduced in Subsection 8.2 above) where is evaluated on the sort of which contains the domain of . We can see as a formula with two blocks of free variables each, which is saturated under . Similarly, where is saturated under with blocks of free variables each. Let denote tuples of variables, and view the variables of as two blocks of variables. The condition for the theory has to satisfy when admitting this path of length onto the tree is
.
∎
By a straightforward modification of [12, Section 2], the set of countable -categorical theories is -complete with respect to continuous reductions. However, if we suitably change the topology we can make this set closed while retaining the same Borel sets. In this way the -categorical theories can be considered as points in a Polish space.
Corollary 8.8**.**
Bi-interpretability on the set of -categorical theories is Borel isomorphic to a -equivalence relation on a Polish space.
Proof.
By a well known fact from descriptive set theory e.g. [5, Corollary 4.2.4], there is a finer Polish topology with the same Borel sets in which the set of -categorical theories is closed. Then the condition above yields a description of bi-interpretability on this closed set. ∎
Recall that a Borel equivalence relation on a Polish space is called countable if every equivalence class is countable.
Theorem 8.9**.**
Isomorphism of oligomorphic groups is Borel reducible to a countable Borel equivalence relation.
Proof.
A Borel equivalence relation on a Polish space is called potentially if there is a finer Polish topology on with the same Borel sets in which is . By Hjorth and Kechris [7, Proposition 3.7], this condition is equivalent to being Borel reducible a equivalence relation on a Polish space.
By Theorem 8.6 and Corollary 8.8, isomorphism of oligomorphic groups is potentially . Nies, Schlicht and Tent (Oligomorphic groups are essentially countable, in preparation) showed this relation is Borel equivalent to the isomorphism relation on a Borel invariant set of models, and hence the orbit equivalence relation of a Borel action . It now suffices to apply another result of Hjorth and Kechris [7, Theorem 3.8]: if the orbit equivalence relation given by a Borel action of is potentially , then it is Borel reducible to a countable Borel equivalence relation.
∎
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