Polish G-spaces, the generalized model theory and complexity
aa r X i v : . [ m a t h . L O ] O c t Polish G -spaces, the generalized model theory andcomplexity A. Ivanov , B. Majcher-Iwanow
Abstract.
Given Polish space Y and a continuous language L we studythe corresponding logic Iso ( Y )-space Y L . We build a framework of gener-alized model theory towards analysis of Borel/algorithmic complexity ofsubsets of Y kL × ( Iso ( Y )) l . Keywords:
Polish G-spaces, Continuous logic, Generalized model theory.
Let ( Y , d ) be a Polish space and Iso ( Y ) be the corresponding isometry group endowedwith the pointwise convergence topology. Then Iso ( Y ) is a Polish group.For any countable continuous signature L the set Y L of all continuous metric L -structures on ( Y , d ) can be considered as a Polish Iso ( Y )-space. We call this action logic . It is known that the logic action is universal for Borel reducibility of orbitequivalence relations of Polish G -spaces with closed G ≤ Iso ( Y ) [15], [30]. Moreoverby a result of J.Melleray in [40] every Polish group G can be realised as the automor-phism group of a continuous metric structure on an appropriate Polish space ( Y , d )and the structure is approximately ultrahomogeneous.On the other hand typical notions naturally arising for logic actions can be appliedin the general case of a Polish G -space X with G as above. If we consider G togetherwith a family of grey subgroups, then distinguishing an appropriate family B of greysubsets of X we arrive at the situation very similar to the logic space Y L (see [2],13], [37], [38] for non-Archimedian G and [30] for the general case). For example wecan treat elements of B as continuous formulas. Then many theorems of traditionalmodel theory can be generalized to topological statements concerning spaces with nice topologies (i.e. defined by B ). This approach is called the generalized modeltheory , [3].The aim of this paper is to demonstrate that the tools of the generalized modeltheory nicely work for some other aspects of logic actions. Our basic concern isas follows. Viewing the logic space Y L as a Polish space and using the recipe ofgeneralized model theory we distinguish some subsets of Y kL × ( Iso ( Y )) l and thenstudy Borel/algorithmic complexity of them.We usually fix a countable dense subset S Y of Y and study subsets of Y L whichare invariant with respect to isometries stabilizing S Y setwise. This is the approachwhere a structure on Y (say M ) is considered together with its presentation over S Y , i.e. together with the set Diag ( M, S Y ) = { ( φ, q ) : M | = φ < q, where q ∈ [0 , ∩ Q and φ is a continuoussentence with parameters from S Y } . The best example of this situation is the logic space U L over the bounded Urysohnmetric space U where distinguishing the countable counterpart Q U of U (see Section3 of [30]) we study Iso ( Q U )-invariant subsets of U L .We demonstrate in Section 2 that this setting appears as a part of generalizedmodel theory studied in [3] and [30]. Moreover our methods also work in the Hilbertspace case and in the measure algebras case.In Section 2.4 we consider other examples suggested by topological notions involv-ing nice topologies. Since they correspond to natural logic constructions we view thismaterial as a further development of generalized model theory.In Section 3 we show that our approach gives a framework to computable membersof Y L and their computable indexations. We will see that it supports standardapproaches both to computable model theory and to effective metric spaces. Moreoverit is suited to the general setting of computable Polish group actions presented in therecent paper [41].In Section 4 we examine our approach in the cases of separable categoricity andultrahomogeneity. In particular in Section 4.1 we find a Borel subset SC of Y L whichis Iso ( S Y )-invariant and can be viewed as the set of all presentations over S Y ofseparably categorical structures on Y . Moreover in Section 4.2 we study complexityof the index set of computable members of SC .The paper is self-contained. We address it to logicians and do not assume anyspecial background. We believe that the ideas presented in it can be helpful in modeltheory, descriptive set theory and computability theory. in this paper we do not use the Urysohn space U ; we only use the ball of it of diameter 1 anddenote it by U Logic space of continuous structures
Our paper belongs to a field of modern logic which can be situated between InvariantDescriptive Set Theory ([4], [27]) and Continuous Model Theory ([6], [10], [12]), seealso [7], [13] - [15], [44]. The definitions below basically correspond to these sources. A Polish space (group) is a separable, completely metrizable topological space(group). Sometimes we extend the corresponding metric to tuples by d (( x , ..., x n ) , ( y , ..., y n )) = max ( d ( x , y ) , ..., d ( x n , y n )) . If a Polish group G continuously acts on a Polish space X , then we say that X is a Polish G -space . We say that a subset of X is invariant if it is G -invariant.Let ( Y , d ) be a Polish space and Iso ( Y ) be the corresponding isometry groupendowed with the pointwise convergence toplogy. Then Iso ( Y ) is a Polish group. Acompatible left-invariant metric can be obtained as follows: fix a countable dense set S = { s i : i ∈ { , , ... }} and then define for two isometries α and β of Y ρ S ( α, β ) = ∞ X i =1 − i min (1 , d ( α ( s i ) , β ( s i ))) . We will study closed subgroups of
Iso ( Y ). We fix a dense countable set Υ ⊂ Iso ( Y ).In any closed subgroups of Iso ( Y , d ) we distinguish the base consisting of all sets ofthe form N σ,q = { α : ρ S ( α, σ ) < q } , σ ∈ Υ and q ∈ Q . We may assume that Υ is asubgroup of Iso ( Y ). To get this it is enough to replace Υ by G = h Υ i . We now fix a countable continuous signature L = { d, R , ..., R k , ..., F , ..., F l , ... } . Let us recall that a metric L -structure is a complete metric space ( M, d ) with d bounded by 1, along with a family of uniformly continuous operations on M anda family of predicates R i , i.e. uniformly continuous maps from appropriate M k i to[0 , L assigns to each predicate symbol R i a continuitymodulus γ i : [0 , → [0 ,
1] so that any metric structure M of the signature L satisfiesthe property that if d ( x j , x ′ j ) < γ i ( ε ) with 1 ≤ j ≤ k i , then the inequality | R i ( x , ..., x j , ..., x k i ) − R i ( x , ..., x ′ j , ..., x k i ) | < ε. holds for the corresponding predicate of M . It happens very often that γ i coincideswith id . In this case we do not mention the appropriate modulus. We also fix conti-nuity moduli for functional symbols. 3ote that each countable structure can be considered as a complete metric struc-ture with the discrete { , } -metric.Atomic formulas are the expressions of the form R i ( t , ..., t r ), d ( t , t ), where t i are simply classical terms (built from functional L -symbols). We define formulas tobe expressions built from 0,1 and atomic formulas by applications of the followingfunctions: x/ x ˙ − y = max ( x − y,
0) , min ( x, y ) , max ( x, y ) , | x − y | , ¬ ( x ) = 1 − x , x ˙+ y = min ( x + y,
1) , sup x and inf x . Statements concerning metric structures are usually formulated in the form φ = 0 , where φ is a formula. Sometimes statements are called conditions ; we will use bothnames. A theory is a set of statements without free variables (here sup x and inf x play the role of quantifiers).We often extend the set of formulas by the application of truncated products by positive rational numbers. This means that when q · x is greater than 1, thetruncated product of q and x is 1. Since the context is always clear, we preserve thesame notation q · x . The continuous logic after this extension does not differ from thebasic case.It is worth noting that the choice of the set of connectives guarantees that for anycontinuous relational structure M , any formula φ is a γ -uniform continuous functionfrom the appropriate power of M to [0 , γ ( ε ) is of the form1 n · min { γ ′ ( ε ) : γ ′ is a continuity modulus of an L -symbol appearing in the formula } , where the number n only depends on the complexity of φ. This follows from the fact that when φ and φ have continuity moduli γ and γ respectively, then the formula f ( φ , φ ) obtained by applying a binary connective f ,has a continuity modulus of the form min ( γ ( x ) , γ ( x )).It is observed in Appendix A of [12] that instead of continuity moduli one canconsider inverse continuity moduli . Slightly modifying that place in [12] we defineit as follows. Definition 1.1
A continuous monotone function δ : [0 , → [0 , with δ (0) = 0 isan inverse continuity modulus of a map F (¯ x ) : X n → [0 , if for any ¯ a , ¯ b from X n , | F (¯ a ) − F (¯ b ) | ≤ δ ( d (¯ a, ¯ b )) . The choice of the connectives above guarantees that the following statement holds(see [30]).
Lemma 1.2
For any continuous relational structure M , where each n -ary relationhas n · id as an inverse continuity modulus, any formula φ admits an inverse continuitymodulus which is of the form k · id , where k depends on the complexity of φ . emark 1.3 By Lemma 4.1 of [10] each n -ary functional symbol F can be replacedby the predicate D F (¯ x, y ) = d ( F (¯ x ) , y ). It is clear that the continuity moduli withrespect to variables from ¯ x are the same and id works as a continuity modulus for y .Thus we may always assume that L is relational.For a continuous structure M defined on ( Y , d ) let Aut ( M ) be the subgroup of Iso ( Y ) consisting of all isometries preserving the values of atomic formulas. It iseasy to see that Aut ( M ) is a closed subgroup with respect to the topology on Iso ( Y )defined above.For every c , ..., c n ∈ M and A ⊆ M we define the n -type tp (¯ c/A ) of ¯ c over A asthe set of all ¯ x -conditions with parameters from A which are satisfied by ¯ c in M . Let S n ( T A ) be the set of all n -types over A of the expansion of the theory T by constantsfrom A . There are two natural topologies on this set. The logic topology is definedby the basis consisting of sets of types of the form [ φ (¯ x ) < ε ], i.e. types containingsome φ (¯ x ) ≤ ε ′ with ε ′ < ε . The logic topology is compact.The d -topology is defined by the metric d ( p, q ) = inf { d (¯ c, ¯ b ) | there is a model M with M | = p (¯ c ) ∧ q (¯ b ) } . By Propositions 8.7 and 8.8 of [6] the d -topology is finer than the logic topology and( S n ( T A ) , d ) is a complete space.The following notion is helpful when we study some concrete examples, for examplethe Urysohn space. A relational continuous structure M is approximately ultraho-mogeneous if for any n -tuples ( a , .., a n ) and ( b , ..., b n ) with the same quantifier-freetype (i.e. with the same values of predicates for corresponding subtuples) and any ε > g ∈ Aut ( M ) such that max { d ( g ( a j ) , b j ) : 1 ≤ j ≤ n } ≤ ε. As we already mentioned any Polish group can be chosen as the automorphism groupof a continuous metric structure which is approximately ultrahomogeneous.The bounded Urysohn space U (see Section 2.3) is ultrahomogeneous in the tra-ditional sense: any partial isomorphism between two tuples extends to an automor-phism of the structure [47]. Note that this obviously implies that U is approximatelyultrahomogeneous.We will use the continuous version of L ω ω from [8] (see also [10]. We remind thereader that continuous L ω ω -formulas are defined by the standard procedure appliedto countable conjunctions and disjunctions (see [8]). Each continuous infinite formuladepends on finitely many free variables. The main demand is the existence of conti-nuity moduli of such formulas. It is usually assumed that a continuity modulus δ φ,x satisfies the equality δ φ,x ( ε ) = sup { δ φ,x ( ε ′ ) : 0 < ε ′ < ε } and δ V Φ ,x ( ε ) = sup { δ ′ V Φ ,x ( ε ′ ) : 0 < ε ′ < ε } , where δ ′ V Φ ,x = inf { δ φ,x : φ ∈ Φ } . .3 Logic action Fix a countable continuous signature L = { d, R , ..., R k , ..., F , ..., F l , ... } and a Polish space ( Y , d ). Let S be a dense countable subset of Y . Let seq ( S ) = { ¯ s i : i ∈ ω } be the set (and an enumeration) of all finite sequences (tuples) from S . Letus define the space of metric L -structures on ( Y , d ). Using the recipe as in the caseof Iso ( Y ) we introduce a metric on the set of L -structures as follows. Enumerate alltuples of the form ( ε, j, ¯ s ), where ε ∈ { , } and when ε = 0, ¯ s is a tuple from seq ( S )of the length of the arity of R j , and for ε = 1, ¯ s is a tuple from seq ( S ) of the lengthof the arity of F j . For metric L -structures M and N let δ seq ( S ) ( M, N ) = ∞ X i =1 { − i | R Mj (¯ s ) − R Nj (¯ s ) | : i is the number of ( ε, j, ¯ s ) } . Since the predicates and functions are uniformly continuous (with respect to moduliof L ) and S is dense in Y , we see that δ seq ( S ) is a complete metric. Moreover by anappropriate choice of rational values for R j (¯ s ) we find a countable dense subset ofmetric structures on Y , i.e. the space obtained is Polish. We denote it by Y L . It isclear that Iso ( Y ) acts on Y L continuously. Thus we consider Y L as an Iso ( Y )-spaceand call it the space of the logic action on Y . Remark 1.4
It is worth noting that in this definition a structure on Y (say M ) isidentified with its presentation Diag ( M, S ), see Introduction.It is convenient to consider the following basis of the topology of Y L . Fix a finitesublanguage L ′ ⊂ L , a finite subset S ′ ⊂ S , a finite tuple q , ..., q t ∈ Q ∩ [0 ,
1] anda rational ε ∈ [0 ,
1] with 1 − ε < /
2. Consider a diagram D of L ′ on S ′ of someinequalities of the form d ( F j (¯ s ) , s ′ ) > ε , d ( F j (¯ s ) , s ′ ) < − ε, | R j (¯ s ) − q i | > ε , | R j (¯ s ) − q i | < − ε, with ¯ s ∈ seq ( S ′ ) , s ′ ∈ S ′ . (i.e. in the case of relations we consider negations of statements of the form: | R j (¯ s ) − q i | ≤ ε , | R j (¯ s ) − q i | ≥ − ε ). The set of metric L -structures realizing D is anopen set of the topology of Y L and the family of sets of this form is a basis ofthis topology. Compactness theorem for continuous logic (see [12]) shows that thetopology is compact. We will call it logic too.If in Remark 1.4 one relax the conditions on formulas φ used in Diag ( M, S ) (forexample allowing φ to be from some L ω ω -fragments) the topology can become richer(and the basis should be corrected). Moreover by the continuous version of the Lopez-Escobar theorem ([10], [15]) every Polish group action arises as an action of someclosed G ≤ Iso ( Y ) on the space of separable continuous structures of some L ω ω -sentence, [15]. This possibility will be discussed in the next section. resp. 2 − i d ( F Mj (¯ s ) , F Nj (¯ s )) when ε = 1 Good and nice topologies
In Section 2.2 we give the main concepts of the generalized model theory. They arebased on the notion of a grey subset introduced in [9]. The corresponding preliminariesare given in Section 2.1. In Section 2.3 we describe the most important examples of thesituation. In Section 2.4 we demonstrate several applications of our approach. Theyconcern complexity of some subsets of the logic space. In a sense this section explainsthe reason why the questions of complexity are considered under the framework ofgood/nice topologies (of Section 2.2).
The notion of grey subsets was introduced in [9]. It has become very fruitful, see [10],[15], [30] and [14].A function φ from a space X to [ −∞ , + ∞ ] is upper (lower) semi-continuous if the set φ For any continuous formula φ (¯ v ) of the language L there is a nat-ural number n such that for any tuple ¯ a ∈ S and ε ∈ [0 , , the subset M od ( φ, ¯ a, < ε ) = { M : M | = φ (¯ a ) < ε } ( or M od ( φ, ¯ a, > ε ) = { M : M | = φ (¯ a ) > ε } )of the space Y L of L -structures, belongs to Σ n . When G is a Polish group, then a grey subset H ⊑ G is called a grey subgroup if H (1) = 0 , ∀ g ∈ G ( H ( g ) = H ( g − )) and ∀ g, g ′ ∈ G ( H ( gg ′ ) ≤ H ( g ) + H ( g ′ )) . This is equivalent to Definition 2.5 from [9]. It is worth noting that by Lemma 2.6 of[9] an open grey subgroup is clopen. 7f H is a grey subgroup, then for every g ∈ G we define the grey coset Hg andthe grey conjugate H g as follows: Hg ( h ) = H ( hg − ) H g ( h ) = H ( ghg − ) . Observe that if H is open, then Hg is an open grey subset and H g is an open greysubgroup. Definition 2.2 Let X be a continuous G -space. A grey subset φ ⊑ X is called invariant with respect to a grey subgroup H ⊑ G if for any g ∈ G and x ∈ X wehave φ ( g ( x )) ≤ φ ( x ) ˙+ H ( g ) . Since H ( g ) = H ( g − ), the inequality from the definition is equivalent to φ ( x ) ≤ φ ( g ( x )) ˙+ H ( g ). Remark 2.3 (see Section 2.1 of [30]). It is clear that for every continuous structure M (defined on Y ) any continuous formula φ (¯ x ) defines a clopen grey subset of M | ¯ x | .Moreover note that when φ (¯ x, ¯ c ) is a continuous formula with parameters ¯ c ∈ M and δ is a linear inverse continuous modulus for φ (¯ x, ¯ y ) (see Definition 1.1), then φ isinvariant with respect to the open grey subgroup H δ, ¯ c ⊑ Aut ( M ) defined by H δ, ¯ c ( g ) = δ ( d (( c , . . . , c n ) , ( g ( c )) , . . . , g ( c n ))), where g ∈ Aut ( M ) , i.e. φ ( g (¯ a ) , ¯ c ) ≤ φ (¯ a, ¯ c ) + H δ, ¯ c ( g ) . In the space of continuous L -structures Y L this remark has the follows version(see Lemma 2.2 in [30]). Lemma 2.4 Let δ be an inverse continuity modulus for φ (¯ x ) , which is linear. Thegrey subset defined by φ (¯ c ) ⊑ Y L is invariant with respect to the grey stabiliser H δ, ¯ c ⊑ Iso ( Y ) defined as follows. H δ, ¯ c ( g ) = δ ( d (( c , . . . , c n ) , ( g ( c )) , . . . , g ( c n ))) , where g ∈ Iso ( Y ) . In this section we consider a certain class of Polish G -spaces. To describe it we needthe following definition. Definition 2.5 A family U of open grey subsets of a Polish space X with a topology τ is called a grey basis of τ if the family { φ In Remark 2.9 of [30] it is observed that for every Polish group G thereis a a countable G < G and a countable family of open grey subsets R satisfyingthese assumptions.We will see below that if the space ( Y , d ) is good enough (for example the boundedUrysohn space) and S is a dense countable subset of Y , then the family R of greysubsets of G = Iso ( Y ) can be chosen among grey cosets of the form ρ ( g ) = q · d (¯ b, g (¯ a )) , where q ∈ Q + and¯ a, ¯ b are tuples from S which are isometric in Y . If the metric is bounded by 1 we mean the truncated multiplication by q in the formulaabove.When we fix G , R and consider a Polish G -space ( X , d ) we also distinguish acountable grey basis U of the topology of X . Let τ be the corresponding topology.The approach of generalized model theory of H. Becker from [3] suggests thatalong with the d -topology τ we shall consider some special topology on X which iscalled nice . In the case of Polish G -spaces this idea has been realized in [30] withusing continuous logic. Since we do not need the corresponding material in exactform we introduce the following very general definition. Definition 2.7 Let R be a grey basis of G consisting of cosets of open grey subgroupsof G which also belong to R . Assume that the subfamily of R of all open grey subgroupsis closed under max and truncated multiplication by numbers from Q + .We say that a family B of Borel grey subsets of the G -space ( X , τ ) is a goodbasis with respect to R if:(i) B is countable and generates the topology finer than τ ;(ii) for each φ ∈ B there exists an open grey subgroup H ∈ R such that φ is H -invariant. It will be usually assumed that all constant functions q , q ∈ Q ∩ [0 , 1] are in B . Definition 2.8 A topology t on X is R - good for the G -space h X , τ i if the followingconditions are satisfied.(a) The topology t is Polish, t is finer than τ and the G -action remains continuous ith respect to t .(b) There exists a grey basis B of t which is good with respect to R . Nice bases and nice topologies introduced in [30] are good. We remind the readerthat a good basis B with respect to R is nice if the following additional propertieshold: (iii) for all φ , φ ∈ B , the functions ¬ φ , min ( φ , φ ), max ( φ , φ ), | φ − φ | , φ ˙ − φ φ ˙+ φ belong to B ;(iv) for all φ ∈ B and q ∈ Q + the truncated product q · φ belongs to B ;(v) for all φ ∈ B and open grey subsets ρ ∈ R the Vaught transforms (seeSection 2.1 in [30]) φ ∗ ρ , φ ∆ ρ belong B .In the situation of standard examples (see Section 2.3) the property that thebasis is good is straightforward. It is much more difficult to prove that the basis isnice. Theorem 3.2 from [30] is an example of a result of this kind. The followingtheorem gives existence of nice topologies. This is Theorem 2.12 in [30]. Note thatthe assumptions on the grey basis R follow from the conditions of/before Remark 2.6. Theorem 2.9 Let G be a Polish group and R be a countable grey basis satisfying theassumptions of Definition 2.7 and the following closure property:for every grey subgroup H ∈ R and every g ∈ G if Hg ∈ R , then H g ∈ R .Let h X , τ i be a Polish G -space and F be a countable family of Borel grey subsets of X generating a topology finer than τ such that for any φ ∈ F there is a grey subgroup H ∈ R such that φ is invariant with respect to H .Then there is an R -nice topology for G -space h X , τ i such that F consists of opengrey subsets. In Section 2.4 we describe possible applications of statements of this kind. In this section we give basic examples of good bases and topologies on some logicspaces. The following definition is taken from [7]. Definition 2.10 Let ( M , d ) be a Polish metric structure with universe M . We saythat a (classical) countable structure N is a countable approximating substruc-ture of M if the following conditions are satisfied: • The universe N of N is a dense countable subset of ( M , d ) . • Any automorphism of N extends to a (necessarily unique) automorphism of M ,and Aut ( N ) is dense in Aut ( M ) . Let G be a dense countable subgroup of Aut ( N ). We may consider it as a sub-group of Aut ( M ). 10 amily R M ( G ) . Let R be the family of all clopen grey subgroups of Aut ( M )of the (truncated) form H q, ¯ s : g → q · d ( g (¯ s ) , ¯ s ) , where ¯ s ⊂ N, and q ∈ Q + . It is clear that R is closed under conjugacy by elements of G . Consider the closureof R under the function max and define R M ( G ) to be the family of all G -cosetsof grey subgroups from max ( R ). Then R M ( G ) is countable and the family of all( H q, ¯ s ) 1. The space U is interpreted as M above and Q U will be our N . It is shown in Section 6.1 of [7] that there is an embedding of Q U into U so that:(i) Q U is an approximating substructure of U : it is dense in U ; any isometry of Q U extends to an isometry of U and Iso ( Q U ) is dense in Iso ( U );(ii) for any ε > 0, any partial isometry h of Q U with domain { a , ..., a n } and anyisometry g of U such that d ( g ( a i ) , h ( a i )) < ε for all i , there is an isometry ˆ h of Q U that extends h and is such that for all x ∈ U , d (ˆ h ( x ) , g ( x )) < ε .11et G be a dense countable subgroup of Iso ( Q U ). By (i) we may consider itas a subgroup of Iso ( U ). We now define R U ( G ) by the recipe above. As we alreadyknow G and R U ( G ) satisfy all the conditions of/before Remark 2.6 and in particular R U ( G ) satisfies the conditions of Theorem 2.9.Let L be a relational language of a continuous signature as above. Let L be acountable fragment of L ω ω and let B L be the family of all grey subsets defined bycontinuous L -sentences (with parameters from Q U ) as above. We already know that B L is a good basis. It is proved in [30] (see Theorem 3.2) that this basis is nice. Theorem 2.11 The family B L is a R U ( G ) -nice basis. (B) A separable Hilbert space. We follow [7] and [43]. Let us consider thecomplex Hilbert space l ( N ). Let Q denote the algebraic closure of Q , and considerthe countable subset Q l of l ( N ) of all sequences with finite support and coordinatesfrom Q . It is shown in Section 6.2 of [7] (with use Section 7 of [43]), that it is anapproximating substructure of l ( N ). In particular we have another pair playing therole of ( M , N ). Since l ( N ) is unbounded, the authors of [7] consider instead its closedunit ball, equipped with functions x → αx for | α | ≤ x, y ) → x + y , from which l ( N ) can be recovered. According Remark 1.3 we will consider a relational languagefor this structure.The automorphism group of the unit ball is U ( l ( N )), the unitary group of thewhole complex Hilbert space l ( N ). The topology of pointwise convergence is thestrong operator topology.Let G be a dense countable subgroup of U ( Q l ). We may consider it as a subgroupof U ( l ( N )). We now apply the procedure of R M ( G ) and B L . As a result we obtainthe family R H ( G ) and a grey basis defined on U ( l ( N )).Let L be a relational language of a continuous signature extending the language ofthe unit ball and satisfying the assumptions above and let L be a countable fragmentof L ω ω . Let B L be the corresponding family of all grey subsets of the logic space l ( N ) L . This is a good basis with respect to R H ( G ). (C) The measure algebra on [0 , . Denote by λ the Lebesgue measure on theunit interval [0 , Aut ([0 , , λ ) as the automor-phism group of the Polish metric structure( M ALG, , , ∧ , ∨ , ¬ , d ) , where MALG denotes the measure algebra on [0 , 1] and d ( A, B ) = λ ( A ∆ B ) (see[Kec95]). The approximating substructure is the countable measure algebra A gen-erated by dyadic intervals. This is observed in Section 6.3 od [7]. Exactly as in thecase of U and l ( N ) one can define a family of open grey subgroups of Aut ([0 , , λ ),say R Aut ( G ), and a good bases of the corresponding logic spaces with respect to R Aut ( G ). 12 emark 2.12 The basic continuous metric structures which appear in (A) - (C) , i.e. U , the unit ball of l ( N ) and MALG, are ultahomogeneous structures in the classicalsense: any partial isomorphism between two tuples extends to an automorphism ofthe structure. This is in particular mentioned in Section 3.1 of [5]. Iso ( U ) . Applications Given a Polish space Y let F ( Y ) denote the set of closed subsets of Y . The Effrosstructure on F ( Y ) is the Borel space with respect to the σ -algebra generated by thesets C U = { D ∈ F ( Y ) : D ∩ U = ∅} , for open U ⊆ Y . For various Y this space serves for analysis of Borel complexity offamilies of closed subsets (see [33] and [44] some recent results). It is convenient to usethe fact that there is a sequence of Kuratowski-Ryll-Nardzewski selectors s n : F ( Y ) → Y , n ∈ ω , which are Borel functions such that for every non-empty F ∈ F ( Y ) theset { s n ( F ); n ∈ ω } is dense in F .Given a Polish group G and a continuous (or Borel) action of G on a Polish space Y one can consider the Borel space F ( Y ) m × F ( G ) n . In the situation when Y and G have grey bases B and R respectively which satisfythe conditions of Theorem 2.9 one can consider Y m with respect to the good topologyinduced by B (say t ). Then many natural Borel subsets of Y m × G n can be viewedas elements of F (( Y , t )) m × F ( G ) n . By Theorem 2.2 of [15] for any Polish group G and any standard Borel G -space X there is a continuous group monomorphism Φ : G → Iso ( U ) and a Borel Φ-equivariantinjection f : X → U L . We only need here that the language L is countable relationalwith 1-Lipschitz symbols of unbounded arity. As a result all Polish groups can beconsidered as elements of F ( Iso ( U )), all Polish spaces are elements of F ( U L ) andPolish G -spaces are pairs from F ( U L ) × F ( Iso ( U )) . Let R U ( G ) and B L be grey bases defined in Section 2.3 in the case of Iso ( U ) and U .Let t be the corresponding nice topology. The following proposition is a version of awell-known fact. Proposition 2.13 (1) The following relations from ( F ( Iso ( U ))) , ( F ( U L )) , ( F ( U L , t )) , ( F ( Iso ( U ))) , F ( Iso ( U )) × F ( U L ) × F ( U L ) and F ( Iso ( U )) × F ( U L , t ) × F ( U L , t ) (undernatural interpretations) are Borel: { ( A, B ) : A ⊆ B } , { ( A, B, C ) : AB ⊆ C } . 2) The closed subgroups of Iso ( U ) form a Borel set U ( Iso ( U )) in F ( Iso ( U )) .(3) The Polish G -spaces form a Borel set in F ( Iso ( U )) × F ( U L ) × F ( U L ) and closed G -subspaces of ( U L , t ) form a Borel set in F ( Iso ( U )) × F ( U L , t ) × F ( U L , t ) .Proof. Statement (2) is well-known: see Section 3.2 of [44]. Moreover statements(1) and (2) are variants of Lemmas 2.4 and 2.5 from [33] which were proved for S ∞ .It is also mentioned in [33] that they hold in general. Statement (3) follows from (1)and (2). We only mention here that a Polish G -space is viewed as a triple consistingof G , the subspace and the graph of the action. (cid:3) In model theory a theory T is a model companion of T if T is model completeand every model of T embeds into a model of T and vice versa. One of the definitionsof model completeness states that any formula is equivalent to an existential one (ora universal one).In the case of the logic space U L theories are identified with t -closed invariantsubsets. It is convenient to fix an enumeration of the sets B L ( Q ) = { ( φ ) Let X and X be closed invariant subsets of ( U L , t ) . We say that X is a companion of X if τ -closures of X and X coincide and any element of B L is τ -clopen on X . Theorem 2.15 The set of pairs ( X , X ) of Iso ( U ) -invariant members of F ( U L , t ) with the condition that X is a companion of X is Borel.Proof. Applying Proposition 2.13 we consider pairs ( X , X ) of t -closed invariantsubsets as elements of the corresponding Borel set of triples ( Iso ( U ) , X , X ). UsingKuratowski-Ryll-Nardzewski selectors the condition that τ -closures of X and X arethe same can be written as follows:( ∀ A k ∈ B o L ( Q )) ∀ i ∃ j ∃ l ( s i ( X ) ∈ A k → s j ( X ) ∈ A k ) ∧ ( s i ( X ) ∈ A k → s l ( X ) ∈ A k ) . Now note that for any two t -closed A and B the condition A ∩ X ⊆ B ∩ X isequivalent to the formula: ∀ j ( s j ( X ) ∈ A → s j ( X ) ∈ B ) . In particular this condition is Borel. We can now express that any element of B L ( Q )is τ -clopen on X as follows: ∀ i ( ∀ B l ∈ B L ( Q ))( ∃ A k ∈ B o L ( Q ))( ∃ A m ∈ B o L ( Q ))( s i ( X ) ∈ B l → s i ( X ) ∈ A k ∧ ( A k ∩ X ⊆ B l ∩ X )) ∧ ( s i ( X ) B l → s i ( X ) ∈ A m ) ∧ ( A m ∩ X ⊆ X \ B l )) . (cid:3) emark 2.16 The theorem above is a counterpart of the statement that identifyingtheories a language L with closed subsets of the compact space of complete L -theoriesthe binary relation to be a model companion is Borel. Although the authors have notfound it in literature, it is true and possibly is folklore.Theorem 2.15 confirms that the approach of good/nice topologies is useful. Itprovides a topological tool for a general property from logic (model companions). If X is of the form Y L then it makes sense to study complexity of sets of indices ofcomputable structures of natural model-theoretic classes. In the case of first orderstructures this approach is traditional, see [1], [22], [23] and [28]. In Section 3.1 wegive an appropriate generalization and in Section 3.2 we illustrate it in the case of U L for relational L .These ideas were already presented by the authors in Section 5 of preprint [29].We have discovered that they are closely related to the approach of the recent paperof A.G. Melnikov and A. Montalb´an [41]. In fact the main concern below is to realizethe situation of Sections 2.1 - 2.2 of [41] in cases (A) - (C) of Section 2.3. Havingthis we arrive in a field where the results of [41] work.It is worth noting here that the approach of Section 3.1 can be applied when oneconsiders computable presentations of Polish spaces, see [42] and [48]. Some detailsare given in Remark 3.2. In a sense this is the easiest case. The approach of [41] alsoworks here. Consider the situation of Section 2.2. Let G be a Polish group and R be a distin-guished countable family of clopen grey cosets which is a grey basis of G : • the family { ρ Having this assumption we arive at the case that ( G, R Q ) is a com-putably presented ω -continuous domain , see [19] and [20]. In fact our assump-tions are slightly stronger. Moreover in A1 - A4 below we will make them muchstronger. • Let ( X , τ ) be a Polish ( G, G , R )-space together with a distinguished countable G -invariant grey basis U (see Definition 2.5) of clopen grey subsets which isclosed under max and truncated multiplying by positive rational numbers. • Let U + Q = { σ It is worth noting that when we have a recursively presented Polishspace in the sense of the book of Moschovakis [42] (Section 3), then a basis of theform U Q as above (in fact U + Q ) can be naturally defined. Indeed, let us recall thata recursive presentation of a Polish space ( X , d ) is any sequence S X = { x i : i ∈ ω } which is a dense subset of X satisfying the condition that ( i, j, m, k )-relations d ( x i , x j ) ≤ mk + 1 and d ( x i , x j ) < mk + 1are recursive. If in this case for all i we define grey subsets φ i ( x ) = d ( x, x i ), then allballs ( φ i ) We assume that under our 1-1-enumerationsof the families R Q and U Q the sets of indices of U + Q , R + Q and the set of rational cones V + Q = { H We assume that under our 1-1-enumerations of the families R Q and U Q thebinary relation to be in the pair σ We also assume that the following relation is computable: Inv ( V, U ) ⇔ ( V ∈ V + Q ) ∧ ( U ∈ U + Q ) ∧ ( U is V -invariant ) .By invariantness we mean the property that U is presented as { x ∈ X : φ ( x ) < r } ,the set V is presented as { g ∈ G : H ( g ) < s } and φ is an H -invariant grey subset (inparticular the inequality φ ( g ( x )) < r + s holds for x ∈ U and g ∈ V ). A4. We assume that there is an algorithm deciding the problem whether for anatural number i and for a basic set of the form σ We say that an element x ∈ X is computable if the relation Sat x ( U ) ⇔ ( U ∈ U Q ) ∧ ( x ∈ U ) is computable. In the case of the logic action of S ∞ , when x is a structure on ω and all H and φ are two-valued, this notion is obviously equivalent to the notion of a computablestructure.We will denote by Sat x ( U Q ) the set { C : C ∈ U Q and Sat x ( C ) holds } . Remark 3.4 In [41] computable topological spaces are considered under so called formal inclusion ≪ (it corresponds to terms ”approximation” or ”way-below” in othersources). In Definition 2.2 of [41] it is defined for computable Polish metric spaces,but in fact this relation can be defined in more general situations. Axioms (F1) - (F4)17iven in [41] after Definition 2.2 describe the field of applications of this notion. Itis always assumed in [41] that ≪ is computably enumerable. In our framework thisrelation can be defined as follows: σ The following relations belong to Π :(1) { e : the function ϕ e is a characteristic function of a subset of U Q } ;(2) { ( e, e ′ ) : there is a computable element x ∈ X such that the function ϕ e is a char-acteristic function of the set Sat x ( U Q ) and the function ϕ e ′ realizes the correspondingfunction κ defined in Lemma 3.5 } ;(3) { ( e, e ′ ) : there is an element g ∈ G such that the function ϕ e is a character-istic function of the subset { N ∈ R Q : g ∈ N } and the function ϕ e ′ realizes thecorresponding function κ defined as in Lemma 3.5 } .Proof. (1) Obvious. Here and below we use the fact that a function is computableif and only if its graph is computably enumerable.(2) Under A1 and A4 the corresponding definition can be described as follows:(” e is a characteristic function of a subset of U Q ”) ∧ ( ∀ n )(( ϕ e ′ ( n ) ∈ U + Q ) ∧ ( ϕ e ′ ( n ) = ∅ ) ∧ ( ϕ e ( ϕ e ′ ( n )) = 1) ∧ ( diam ( ϕ e ′ ( n )) < − n )) ∧ ( ∀ d )( ∃ n )(( ”every element U ′ of the finite subset of U Q with the canonicalindex d satisfies ϕ e ( U ′ ) = 1”) ↔ ( ” ϕ e ′ ( n ) is contained in any element U ′ of the finite subset of U Q with the canonical index d ”)).18he last part of the conjunction ensures that the intersection of any finite subfamilyof U Q of cones U ′ with ϕ e ( U ′ ) = 1 contains a closed cone of the form φ ≤ r of suffi-ciently small diameter. Now the existence of the corresponding x follows by Cantor’sintersection theorem for complete spaces.(3) is similar to (2). (cid:3) We say that e is an index of a computable element x ∈ X if ϕ e is a characteristicfunction of Sat x ( U Q ). We now have the following straightforward proposition. Proposition 3.7 The set of indices of computable elements of X belongs to Σ . In this section we show that the computability assumptions A1 - A4 given in Section3.1 are satisfied in the case of good graded bases presented in (A) - (C) of Section2.3. In fact we only consider case (A) . Cases (B) , (C) are similar. (D) Computable presentation of the logic space over U . Let L be a relationallanguage satisfying assumptions of Section 2.3. Let us consider the space U L and thefamily of grey cosets R U ( G ) defined in Section 2.3 (A) . The latter will be interpretedas R of Section 3.1.To define the grey basis U of Section 3.1 we use the recipe of the definition ofthe basis of the topology of Y L in Section 1.3. For a finite sublanguage L ′ ⊂ L , afinite subset S ′ ⊂ Q U and a finite tuple q , ..., q t ∈ Q ∩ [0 , 1] consider the maximum max of some grey subsets of the form | R j (¯ s ) − q i | , 1 ˙ −| R j (¯ s ) − q i | , with ¯ s ∈ seq ( S ′ ) , s ′ ∈ S ′ . When σ is this maximum, the inequality σ < ε corresponds to a basic open set of thetopology of Y L as in Section 1.3.Let L be the fragment of all first order continuous formulas. Let B be the nicebasis corresponding to L (see Theorem 2.11). It is worth noting that the grey basis U is a subfamily of the family of all grey subsets from B . Moreover U correspondsto quantifier free L -formulas.To verify that the Iso ( U )-space U L and the bases R , U satisfy the computabilityconditions of Section 3.1 (in particular A1 - A4 ), we need the following proposition. Proposition 3.8 The elementary theory of the structure Q U in the binary languageof inequalities d ( x, x ′ ) ≤ ( or ≥ ) q , where q ∈ Q ∩ [0 , , extended by all constants from Q U is decidable.Proof. It is well known that Q can be identified with the natural numbers sothat the ordering of the rational numbers becomes a computable relation. Thus thelanguage in the formulation can be considered as a computable one.19t is noticed in [35] that the first order structure Q U is universal ultrahomogeneousin the language d ( x, x ′ ) = q , where q ∈ Q ∩ [0 , . This obviously implies that Q U is a universal ultrahomogeneous first-order struc-ture in the language of inequalities as in the statement of the proposition. So onecan present Q U in this language as a Fra¨ıss´e limit of an effective sequence of finitestructures. Enumerating elements of structures from this sequence and describingdistances between them, we obtain an effective set of axioms of the form d ( c, c ′ ) ≤ ( or ≥ ) q , where c, c ′ ∈ Q U and q ∈ Q ∩ [0 , . We also add all standard ∀∃ -axioms stating that the age of Q U is an amalgamationclass. The obtained axiomatization describes a complete theory having eliminationof quantifiers. (cid:3) Corollary 3.9 The structure Q U under the language of binary relations d ( x, y ) ≤ ( or ≥ ) q , where q ∈ Q ∩ [0 , , has a presentation on ω so that all relations first-order definable in Q U , are decidable. This obviously follows from Proposition 3.8. Let us fix such a presentation. Coding R Q , cones of grey cosets. Let H q, ¯ s : g → q · d ( g (¯ s ) , ¯ s ) , where ¯ s ⊂ Q U , and q ∈ Q + . be a grey subgroup and g ∈ G take ¯ s ′ to ¯ s .Then we can code the ∗ q ′ -cone of the grey coset H q, ¯ s g : g → q · d ( g (¯ s ′ ) , ¯ s ) , by the number of the tuple ( q, ¯ s, ¯ s ′ , q ′ , ∗ ), where ¯ s , ¯ s ′ are identified with the corre-sponding tuples from ω with respect to the presentation of Corollary 3.9 and ∗ isone of the symbols <, ≤ , >, ≥ . Note that the tuples ¯ s , ¯ s ′ have the same quantifierfree diagram (which is determined by a finite subdiagram). By Corollary 3.9 the setof all tuple ( q, ¯ s, ¯ s ′ , q ′ , ∗ ) of this form is computable and by ultrahomogeneity of thestructure from this corollary they code all possible cones.To see that the relation of inclusion between cones of this form is decidable notethat ( q, ¯ s, ¯ s ′ , q ′ , ∗ ) defines a subset of the cone of ( q , ¯ s , ¯ s ′ , q ′ , ∗ ) if for everytuple ¯ s ′′ ¯ s ′′ of the same quantifier free type with ¯ s ′ ¯ s ′ which also satisfiesthe ∗ -inequality between q · d (¯ s ′′ , ¯ s ) and q ′ , the corresponding ∗ -inequalitybetween q · d (¯ s ′′ , ¯ s ) and q ′ holds. 20ndeed by ultrahomogeneity this exactly states that if for an automorphism g the ∗ -inequality between q · d ( g (¯ s ′ ) , ¯ s ) and q ′ holds, then the corresponding ∗ -inequalitybetween q · d ( g (¯ s ′ ) , ¯ s ) and q ′ also holds. Thus to decide the inclusion problembetween these cones it suffices to formulate the statement above as a formula (withparameters ¯ s ′ , ¯ s , ¯ s ′ , ¯ s )) and to verify if it holds in the structure Q U . Cones of grey subgroups (i.e. the set V Q ) are distinguished in the set of codesof R Q by the computable subset of tuples as above with ¯ s = ¯ s ′ . Coding U Q . Since we interpret elements of B by first order L -formulas with pa-rameters from Q U and without free variables, it is obvious that both B and U can becoded in ω so that the operations of connectives are defined by computable functions.Moreover U is a decidable subset of B . Thus the elements of the grey basis U arecoded as a computable set. Now all cones of the form σ Let U be of the form σ Let σ be a max -formula of the previous paragraphs which defines anelement of U . To compute diam ( σ 1. Let I be the (finite) subset of such i . Then diam ( σ The structure ( U , s ) s ∈ Q U of the expansion of the bounded Urysohnspace by constants from Q U has decidable continuous theory.The same statement holds for structures ( M , s ) s ∈ N where ( M , d ) ∈ { l ( N ) , M ALG } and N is the corresponding countable approximating substructure see Section 2.3, (B) and (C) ). roof. To prove the theorem we use Corollary 9.11 of [11]. We only consider thecase of ( U , s ) s ∈ Q U . The remaining cases are similar.Let T Q U be the set of the standard axioms of U (with rational ε and δ , see Section 5in [47]) together with all quantifier free axioms describing distances between constantsfrom Q U . We claim that the set T Q U is computable. Since the set of all standardaxioms of U is computable (see [47]), it suffices to check that the set of all axioms ofthe form d ( c, c ′ ) = q , where c, c ′ ∈ Q U and q ∈ Q ∩ [0 , , is computable. This follows from the fact that the elementary (not continuous) theoryof the structure Q U in the language of binary relations together with all constants c ∈ Q U is decidable, Proposition 3.8.Note that T Q U axiomatizes the continuous theory of a single continuous structure,i.e. the corresponding continuous theory is complete. Indeed, otherwise there isa separable continuous structure M | = T Q U such that for some tuple ¯ s ∈ Q U thestructures ( U , ¯ s ) and the reduct of M , say M ′ , to the signature ( d, ¯ s ), do not satisfythe same inequalities of the form φ (¯ s ) ≤ ( < ) q or φ (¯ s ) ≥ ( > ) q where q ∈ Q ∩ [0 , . On the other hand since U is separably categorical (see Section 4) and ultrahomoge-neous, the structures M ′ and ( U , ¯ s ) are isomorphic, contradicting the previous sen-tence.By Corollaries 9.8 and 9.11 of [11] there is an algorithm which for every continuoussentence φ (¯ s ) computes its value in U . (cid:3) Remark 3.12 It is worth noting that when we apply Proposition 3.8 we only needcomputability of the set of axioms of the form d ( c, c ′ ) = q , where c, c ′ ∈ Q U and q ∈ Q ∩ [0 , , This can be shown as in the proof of Proposition 3.8. Moreover the correspondingargument works in the cases of l ( N ) and M ALG . In this section we fix a countable continuous signature L = { d, R , ..., R k , ..., F , ..., F l , ... } , a Polish space ( Y , d ). a countable dense subset S Y of Y and study subsets of Y L which are invariant with respect to isometries stabilising S Y setwise. Viewing thelogic space Y L as a Polish space one can consider Borel/algorithmic complexity ofsome natural subsets of Y L of this kind. This approach differs from the one of23ection 2.4. It corresponds to considering a structure on Y (say M ) together with its presentation over S Y , i.e. the set Diag ( M, S Y ) = { ( φ, q ) : M | = φ < q, where q ∈ [0 , ∩ Q and φ is a continuoussentence with parameters from S Y } . It is natural in the cases of examples (A) - (C) of Section 2.3 and the correspondingcomputable presentations as in Section 3. Moreover it corresponds to the approachof computable model theory.We will concentrate on separable categoricity.A theory T is separably categorical if any two separable models of T are iso-morphic. A useful reformulation of this notion is given in Theorem 4.1. Since wewill only use this theorem below all necessary facts concerning separable categoricity(together with the proof of Theorem 4.1) are given in Appendix. We preserve all the assumptions of Section 1 on the space ( Y , d ). For simplicity weassume that all L -symbols are of continuity modulus id . Simplifying notation we put S = S Y . We reformulate separable categoricity as follows. Theorem 4.1 Let M be a non-compact, separable, continuous, metric structure on ( Y , d ) . The structure M is separably categorical if and only if for any n and ε thereare finitely many conditions φ i (¯ x ) ≤ δ i , i ∈ I , so that any n -tuple of M satisfies oneof these conditions and the following property holds:for any i ∈ I , and any a , ..., a n ∈ M realizing φ i (¯ x ) ≤ δ i and any finite setof formulas ∆( x , ..., x n , x n +1 ) realized in M and containing φ i (¯ x ) ≤ δ i ,there is a tuple b , ..., b n , b n +1 realizing ∆ such that max i ≤ n d ( a i , b i ) < ε . We now introduce a class of structures which is justified by its formulation. Ifwe assume that all parameters appearing in it can be taken from S we arrive at thefollowing definition. Definition 4.2 Let SC S be the set of all L -structures M on Y with the followingcondition: for every n and rational ε there is a finite set F of tuples ¯ a i from S together withconditions φ i (¯ x ) ≤ δ i ( i ∈ I and all δ i are rational) with φ Mi (¯ a i ) ≤ δ i , i ∈ I , and thefollowing properties • any n -tuple ¯ a from S satisfies in M one of these φ i (¯ x ) ≤ δ i • when φ Mi (¯ a ) ≤ δ i and ¯ c is an ( n + 1)-tuple from S with c , ..., c n satisfying φ i (¯ x ) ≤ δ i in M ,for any finite set ∆ of L -formulas φ (¯ y ), | ¯ y | = n + 1 with φ M (¯ c ) = 0there is an ( n + 1)-tuple ¯ b = ( b , . . . b n +1 ) ∈ S so that max j ≤ n ( d ( a j , b j )) ≤ ε and φ M (¯ b ) = 0 for all formulas φ ∈ ∆.24ote that Theorem 4.1 implies that if M is a separably categorical structure on Y , there is a dense set S ′ ⊆ Y so that M belongs to the corresponding set of L -structures SC S ′ . To see this we just extend S to some countable S ′ which satisfies theproperty of Definition 4.2 in which we additionally require that a , ..., a n ∈ S ′ . Thusthe following statement becomes interesting. Proposition 4.3 The subset SC S ⊂ Y L is Iso ( S ) -invariant and Borel.Proof. It is clear that SC S is Iso ( S )-invariant. To see that SC S is a Borel subsetsof Y L it suffices to note that given rational ε > 0, finitely many formulas φ i (¯ x ), i ∈ I ,with | ¯ x | = n + 1, and an n -tuple ¯ a from S the set of L -structures M on Y with theproperty thatthere is an ( n +1)-tuple ¯ b ∈ S so that max j ≤ n ( d ( a j , b j )) ≤ ε and φ Mi (¯ b ) = 0for all i ∈ I ,is a Borel subset of Y L . The latter follows from Lemma 2.1, which in particular saysthat any set of L -structures of the form { M : M | = max ( max j ≤ n ( d ( a j , b j ) ˙ − ε ) , max i ∈ I ( φ i (¯ b ))) = 0 } is a Borel subset of Y L . (cid:3) The proof above demonstrates that SC S is of Borel level ω .In Section 4.3 we will discuss the conjecture that any separably categorical con-tinuous L -structure on Y is homeomorphic to a structure from SC S .Since SC S is a subset of the standard space Y L we do need to specify grey bases R , U as in Section 3.2. To be definite one can generate R by grey stabilizers and U by atomic formulas. This issue becomes important in the next section when weconsider computable members of SC S . The following proposition is an effective version of Proposition 4.3 in the case (A) ofSection 2.3. We work in the effective presentation given in Section 3.2 ( D ). Proposition 4.4 Let SC Q U be the Iso ( Q U ) -invariant Borel subset of U L defined as inSection 4.1.Then the subset of indices of computable structures from SC Q U is hyperarithmetical.Proof. Under the framework of Sections 3.1 and 3.2 ( D ) the following statementholds.The set of all pairs ( i, j ) where j is an index of a cone from ( B ) Q and i isan index of a computable structure from this cone, is hyperarithmeticalof level ω . 25his is an effective version of Proposition 2.1. It follows from Lemma 3.6 by standardarguments. Note that as we have shown in Section 3.2 ( D ) all assumptions of Lemma3.6 are satisfied under the circumstances of our proposition.It remains to verify that the definition of SC Q U defines a hyperarithmetical subsetof indices of computable structures. This is straghtforward (similar to the proof ofProposition 4.3). (cid:3) We conjecture that when ( Y , d ) is as in the cases (A) - (C) of Section 2.3 and thedense subset S is chosen as the corresponding approximating substructure, then anyseparably categorical structure from Y L is homeomorphic to an element of SC S . Inthis section we connect it with countable dense homogeneity.A separable space X is countable dense homogeneous (CDH) if given any twocountable dense subsets D and E of X there is a homeomorphism f : X → X suchthat f ( D ) = E (see [25]). It is known that the unbounded Urysohn space and thespaces l are CDH (and they are homeomorphic).In [17] J. Dijkstra introduced Lipschitz CDH as follows. Definition 4.5 A metric space ( X, d ) is called Lipschitz countable dense homoge-neous if given ε and A , B countable dense subsets of X there is a homeomorphism h : X → X such that f ( A ) = B and − ε < d ( h ( x ) , h ( y )) d ( x, y )) < ε for all x, y ∈ X. He has proved in [17] that every separable Banach space is Lipschitz CDH. More-over it is shown in [36] that the unbounded Urysohn space is also Lipschitz CDH.Regarding the property SC S this notion seems very helpful. Indeed, as we havealready noted Theorem 4.1 implies that if M is a separably categorical structureon Y , there is a dense set S ′ ⊆ Y so that M belongs to the corresponding Borelset of L -structures SC S ′ . To support the conjecture of this subsection we need ahomeomorphism which takes S ′ onto S and takes M into SC S . However the lattercondition is not easy to control. If in the definition of the class SC S we restrict ourselves by only quantifier free formulaswe arrive at a definition of a subset of SC S which we denote by SCU S . The approachof Sections 4.1 and 4.2 works in this case too. As in Section 4.3 one can conjecturethat when ( Y , d ) is as in the cases (A) - (C) of Section 2.3 and the dense subset S is chosen as the corresponding approximating substructure, then any separablycategorical ultrahomogeneous structure from Y L is homeomorphic to an element of SCU S . 26t is worth noting here that since any Polish group can be realized as the auto-morphism group of an approximately ultrahomogeneous structure it makes sense tostudy Polish groups by description of the corresponding classes of approximately ul-trahomogeneous structures and to study the complexity of these classes. For examplewe do not know if the class of approximately ultrahomogeneous L -structures on Y isa Borel subset of Y L . In this section we give a different example of complexity of a subclass of Y L for aPolish space ( Y , d ). It somehow corresponds to the result of M.Malicki [39] thatthe set of all Polish groups admitting compatible complete left-invariant metrics iscoanalytic non-Borel as a subset of a standard Borel space of Polish groups.The following statement is Lemma 9.1 of [10].Let G be the automorphism group of a continuous L -structure M onthe space ( Y , d ). Then the group G admits a compatible complete left-invariant metric (i.e. Aut ( M ) is CLI) if and only if each L ω ω -elementaryembedding of M into itself is surjective.What is the complexity of the class of structures from this proposition? We havesome remark concerning this question. Proposition 4.6 Let L be a countable fragment of L ω ω . The subset of Y L consistingof structures M admitting proper L -elementary embeddings into itself is analytic. Iso ( S ) -invariant and coanalytic.Proof. Consider the extension of L by a unary function f . All expansions of L -structures satisfying the property that f is an isometry which preserves the values L -formulas, form a Borel subset of the (Polish) space of all L ∪ { f } -structures on Y .If s ∈ S = S Y and ε ∈ Q ∩ [0 , 1] then the condition that f ( S ) does not intersectthe ε -ball of s is open. Thus the set of L ∪ { f } -structures with a proper embedding f into itself, is Borel. The rest is easy. (cid:3) We need the following definition. Definition 5.1 Let A ⊆ M . A predicate P : M n → [0 , is definable in M over A ifthere is a sequence ( φ k ( x ) : k ≥ of L ( A ) -formulas such that predicates interpreting φ k ( x ) in M converge to P ( x ) uniformly in M n . Let p (¯ x ) be a type of a theory T . It is called principal if for every model M | = T ,the predicate dist (¯ x, p ( M )) is definable over ∅ .By Theorem 12.10 of [6] a complete theory T is separably categorical if and only iffor each n > 0, every n -type p is principal. Another property equivalent to separable27ategoricity states that for each n > 0, the metric space ( S n ( T ) , d ) is compact. Inparticular for every n and every ε there is a finite family of principal n -types p , ..., p m so that their ε -neighbourhoods cover S n ( T ).In the classical first order logic a countable structure M is ω -categorical if andonly if Aut ( M ) is an oligomorphic permutation group, i.e. for every n , Aut ( M ) hasfinitely many orbits on M n . In continuous logic we have the following modification. Definition 5.2 An isometric action of a group G on a metric space ( X , d ) is said tobe approximately oligomorphic if for every n ≥ and ε > there is a finite set F ⊂ X n such that G · F = { g ¯ x : g ∈ G and ¯ x ∈ F } is ε -dense in ( X n , d ) . Assuming that G is the automorphism group of a non-compact separable continu-ous metric structure M , G is approximately oligomorphic if and only if the structure M is separably categorical (C. Ward Henson, see Theorem 4.25 in [45]). It is alsoknown that separably categorical structures are approximately homogeneous inthe following sense: if n -tuples ¯ a and ¯ c have the same types (i.e. the same values φ (¯ a ) = φ (¯ b ) for all L -formulas φ ) then for every c n +1 and ε > b , ..., b n , b n +1 of the same type with ¯ c, c n +1 , so that d ( a i , b i ) ≤ ε for i ≤ n . In fact forany n -tuples ¯ a and ¯ b there is an automorphism α of M such that d ( α (¯ c ) , ¯ a ) ≤ d ( tp (¯ a ) , tp (¯ c )) + ε. (i.e M is strongly ω -near-homogeneous in the sense of Corollary 12.11 of [6]).To prove Theorem 4.1 we start with the following observation. Lemma 5.3 Let M be a non-compact, separable, continuous, metric structure on ( Y , d ) . The structure M is separably categorical if and only if for any n and ε thereare finitely many conditions φ i (¯ x ) ≤ δ i , i ∈ I , so that any n -tuple of M satisfies oneof these conditions and the following property holds:for any i ∈ I , any a , ..., a n ∈ M realising φ i (¯ x ) ≤ δ i and any type p ( x , ..., x n , x n +1 ) realized in M and containing φ i (¯ x ) ≤ δ i , there is a tuple b , ..., b n , b n +1 realizing p such that max i ≤ n d ( a i , b i ) < ε .Proof. By Theorem 12.10 of [6] a complete theory T is separably categorical ifand only if for each n > 0, every n -type is principal. An equivalent condition statesthat for each n > 0, the metric space ( S n ( T ) , d ) is compact. In particular for every n and every ε there is a finite family of principal n -types p , ..., p m so that their ε/ S n ( T ).Thus when M is separably categorical, given n and ε , we find appropriate p i , i ∈ I ,define P i (¯ x ) = dist (¯ x, p i ( M )), the corresponding definable predicates and n -conditions φ i (¯ x ) ≤ δ i describing the corresponding ε/ p i . The rest followsby strong ω -near-homogeneity. 28o see the converse assume that M satisfies the property from the formulation. Tosee that G = Aut ( M ) is approximately oligomorphic take any n and ε and find finitelymany conditions φ i (¯ x ) ≤ δ i , i ∈ I , satisfying the property from the formulation for n and ε/ 4. Choose ¯ a i with φ i (¯ a i ) ≤ δ i and let F = { ¯ a i : i ∈ I } . To see that G · F is ε -dense we only need to show that if ¯ a satisies φ i (¯ x ) ≤ δ i , then there is an automorphismwhich takes ¯ a to the ε -neighbourhood of ¯ a i . This is verified by ”back-and-forth” asfollows. Let ( ε k ) be an infinite sequence of positive real numbers whose sum is lessthan ε/ 4. At every step l (assuming that l ≥ n ) we build a finite elementary map α l and l -tuples ¯ c l and ¯ d l so that • ¯ c n = ¯ a and ¯ d n = ¯ a i ; • for l > n , α l takes ¯ c l to ¯ d l • for l > n + 1, the first l − c l (resp. ¯ d l ) are at distance less than ε l away from the corresponding coordinates of ¯ c l − (resp. ¯ d l − ); • the sets S { ¯ c l : l ∈ ω } and S { ¯ d l : l ∈ ω } are dense in M .In fact we additionally arrange that for even l , ¯ c l +1 extends ¯ c l and for odd l ¯ d l +1 extends ¯ d l . In particular the type of ¯ c l +1 always extends the type of ¯ c l . At the ( n + 1)-th step we find finitely many conditions φ ′ j (¯ x ) ≤ δ ′ j , j ∈ J , so that any ( n + 1)-tupleof M satisfies one of these conditions and for any j ∈ J , any a ′ , ..., a ′ n +1 ∈ M realising φ ′ j (¯ x ) ≤ δ ′ j and any type p ( x , ..., x n +1 , x n +2 ) realized in M and containing φ ′ j (¯ x ) ≤ δ j ,there is a tuple b , ..., b n +1 , b n +2 realizing p such that max t ≤ n +1 d ( a ′ t , b t ) < ε n +1 . Nowby the choice of i for any extension of ¯ a = ¯ c n to an ( n + 1)-tuple ¯ c n +1 we can finda tuple ¯ d n +1 realizing tp (¯ c n +1 ) so that the first n coordinates of ¯ d n +1 are at distanceless than ε/ d n = ¯ a i . If n is even wechoose such ¯ c n +1 and ¯ d n +1 ; if n is odd we replace the roles of ¯ c n +1 and ¯ d n +1 . For thenext step we fix the condition φ ′ j (¯ x ) ≤ δ ′ j satisfied by ¯ c n +1 and ¯ d n +1 .The ( l +1)-th step is as follows. Assume that l is even (the odd case is symmetric).Extend ¯ c l to an appropriate ¯ c l +1 (aiming to density of S { ¯ c l : l ∈ ω } ). There are finitelymany conditions φ ′′ k (¯ x ) ≤ δ ′′ k , k ∈ K , so that any ( l + 1)-tuple of M satisfies one ofthese conditions and for any k ∈ K , any a ′ , ..., a ′ l +1 ∈ M realising φ ′′ k (¯ x ) ≤ δ ′′ k andany type p ( x , ..., x l +1 , x l +2 ) realized in M and containing φ ′′ k (¯ x ) ≤ δ ′′ k , there is a tuple b , ..., b l +1 , b l +2 realizing p such that max t ≤ l +1 d ( a ′ t , b t ) < ε l +1 . We find the conditionsatisfied by ¯ c l +1 and a tuple ¯ d l +1 realizing tp (¯ c l +1 ) so that the first l coordinates of¯ d l +1 are at distance less than ε l away from the corresponding coordinates of ¯ d l .As a result for every k we obtain Cauchy sequences of k -restrictions of ¯ c l -s and¯ d l -s. For k = n their limits are not distant from ¯ a and ¯ a i more than ε/ 2. Moreoverthe limits lim { ¯ c l } and lim { ¯ d l } are dense subsets of Y and realize the same type. Thisdefines the required automorphism of M . (cid:3) Proof of Theorem 4.1. It suffices to show that the condition of the formulationimplies the corresponding condition of Lemma 5.3. Given n and ε take the family φ i (¯ x ) ≤ δ i , i ∈ I , satisfying the condition of the theorem for n and ε/ 2. Let p (¯ x, x n +1 )be a type with φ i (¯ x ) ≤ δ i and a , ..., a n be as in the formulation.29et ( ε k ) be an infinite sequence of positive real numbers whose sum is less than ε/ n + 1 and ε / ψ (¯ x, x n +1 ) ≤ τ ,belongs to p and for any c , ..., c n , c n +1 ∈ M realising ψ (¯ x, x n +1 ) ≤ τ , and any finitesubset ∆ ⊂ p containing ψ (¯ x, x n +1 ) ≤ τ there is a tuple c ′ , ..., c ′ n , c ′ n +1 realizing ∆, suchthat max i ≤ n +1 d ( c i , c ′ i ) < ε / 2. 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q , σ ≤ q , σ ≥ q can beenumerated so that all natural relations between them (in particular relations from A2 ) are computable. For example if S ′ is a finite subset of Q U and cones σ