On computable aspects of algebraic and definable closure
OOn computable aspects of algebraic and definable closure
Nathanael Ackerman
Harvard UniversityCambridge, MA 02138, USA [email protected]
Cameron Freer
Massachusetts Institute of TechnologyCambridge, MA 02139, USA [email protected]
Rehana Patel
African Institute for Mathematical SciencesM’bour–Thi`es, Senegal [email protected]
Abstract
We investigate the computability of algebraic closure and definable closure withrespect to a collection of formulas. We show that for a computable collection of formulasof quantifier rank at most n , in any given computable structure, both algebraic anddefinable closure with respect to that collection are Σ n +2 sets. We further show thatthese bounds are tight. keywords: algebraic closure, definable closure, computable model theory In this paper we study the computability-theoretic content of two model-theoretic concepts:algebraic closure and definable closure. These notions are fundamental to model theory,and have been studied explicitly in various contexts [1, 3, 7].In order to study the computable model theory of these notions, we consider iteratively-defined algebraic and definable closure operators with respect to a specified set of formulas,and focus on certain associated sets. For a set of formulas Φ and a structure N , wedefine sets acl Φ , N and dcl Φ , N , which capture the information contained in the operatorsfor algebraic and definable closure in N with respect to formulas in Φ. We also define setsACL n and DCL n , which describe the computable information that is already present inthe first step of this iterative process, for first-order formulas of quantifier rank at most n .The paper is organized as follows. In Section 2, we provide definitions of ACL n , DCL n ,acl Φ , N , and dcl Φ , N and establish some basic relationships among them. Section 3 givesupper bounds on the computability-theoretic strength of these objects in the quantifier-free case — namely, of ACL , DCL , acl Φ , N , and dcl Φ , N where Φ is a computable set ofquantifier-free formulas. Section 4 gives corresponding lower bounds on these objects, whichestablish tightness of the upper bounds; for ACL and DCL , tightness is achieved viastructures that are model-theoretically “nice”, namely, are ℵ -categorical or of finite Morleyrank. Finally, in Section 5 we use these results to provide bounds on the computational1 a r X i v : . [ m a t h . L O ] J a n trength of ACL n , DCL n , acl Φ , N and dcl Φ , N for arbitrary n and computable collections Φof formulas of quantifier rank n .Parts of this paper appeared in an extended abstract [2] presented at LFCS 2020. In this section we introduce some terminology and notation, and define the main objects ofstudy of this paper: ACL n , DCL n , acl Φ , N , and dcl Φ , N . We prove some basic relationshipsamong them, and discuss their connection with standard model-theoretic notions ofalgebraic and definable closure.For standard notions from computability theory, see, e.g., [8]. We write { e } ( n ) torepresent the output of the e th Turing machine run on input n , if it converges, and in thiscase write { e } ( n ) ↓ . Define W e := { n ∈ N : { e } ( n ) ↓} and Fin := { e ∈ N : W e is finite } .Recall that Fin is Σ -complete [8, Theorem 4.3.2].In this paper we will focus on computable languages that are relational. Note thatthis leads to no loss of generality due to the standard fact that computable languageswith function or constant symbols can be interpreted computably in relational languageswhere there is a relation for the graph of each function. For the definitions of languages,first-order formulas, and structures, see [6].In the context of algebraic and definable closure, we will often consider formulas with aspecified partition of their free variables, which we write with a semicolon, e.g., ϕ ( x ; y ).When we refer to a set of formulas, we mean a set of formulas with specified variablepartitions.We will work with many-sorted languages and structures; for more details, see [9, § L be a (many-sorted) language, let N be an L -structure, and suppose that a is a tupleof elements of N . We say that the type of a is (cid:81) i ≤ n X i when a ∈ (cid:81) i ≤ n ( X i ) N , where eachof X , . . . , X n − is a sort of L . The type of a tuple of variables is the product of the sortsof its constituent variables (in order). The type of a relation symbol is defined to be thetype of the tuple of its free variables, and similarly for formulas. We write ( ∀ x : X ) and( ∃ x : X ) to quantify over a tuple of variables x of type X (which includes the special caseof a single variable of a given sort).If one wanted to avoid the use of many-sorted languages, one could instead encodeeach sort using a unary relation symbol — and indeed this would not affect most of ourresults. However, in Section 4 we are interested in how model-theoretically complicated thestructures we build are, and the single-sorted version of the construction in Proposition 4.1would no longer yield an ℵ -categorical structure.A graph is a pair ( V, E ) where V is a set of vertices and E is a symmetric irreflexivebinary relation on V . A chain in a graph is a cycle-free connected component of the grapheach of whose vertices has degree 1 or 2; hence a chain either is finite with at least twovertices, or is infinite on one side (an N -chain ), or is infinite on both sides (a Z -chain ).The order of a chain is the number of vertices in the chain.Similarly, a digraph is a pair ( V, E ) where V is a set of vertices and E is an asymmetricbinary relation on V , i.e., no vertices have self-loops and any two vertices have an edgein at most one direction. A path in a digraph is a connected component of the graph,containing at least one edge, in which each vertex has in-degree at most 1 and out-degree2t most 1, and having a (necessarily unique) vertex with in-degree 0. A vertex of a path is initial if it has in-degree 0 and final if it has out-degree 0. Hence a path either is finitewith a unique initial and unique final vertex, or is infinite with a unique initial vertex (an N -path ). The order of a path is the number of vertices in the path.We now define computable languages and structures. Definition 2.1.
Suppose L = (cid:0) ( X j ) j ∈ J , ( R i ) i ∈ I ) is a language, where I, J ∈ N ∪ { N } and ( X j ) j ∈ J and ( R i ) i ∈ I are collections of sorts and relation symbols, respectively. Let ty L : I → J <ω be such that for all i ∈ I , we have ty L ( i ) = ( j , . . . , j n − ) where the type of R i is (cid:81) k Definition 2.2. Let L be a language, let ϕ ( x ; y ) be a first-order L -formula, and let N bean L -structure. Suppose a ∈ N has the same type as x . The formula ϕ ( x ; y ) is algebraicat a if (cid:8) b ∈ N : N | = ϕ ( a ; b ) (cid:9) is finite (possibly empty), and definable at a if this set is a singleton. We now describe several sets that capture the information contained in a single step ofthe process of determining algebraic or definable closure. Definition 2.3. • CL := (cid:110) ( c, ϕ ( x ; y ) , a, k ) : c ∈ CompStr , ϕ ( x ; y ) is a first-order L c -formula, a ∈ M c has the same type as x , and k ∈ N ∪{∞} is such that (cid:12)(cid:12) { b ∈ M c : M c | = ϕ ( a ; b ) } (cid:12)(cid:12) = k (cid:111) . • ACL := (cid:8) ( c, ϕ ( x ; y ) , a ) : there exists k ∈ N with ( c, ϕ ( x ; y ) , a, k ) ∈ CL (cid:9) . • DCL := (cid:8) ( c, ϕ ( x ; y ) , a ) : ( c, ϕ ( x ; y ) , a, ∈ CL (cid:9) . • For Y ∈ { CL , ACL , DCL } and n ∈ N let Y n := { t ∈ Y : the second coordinate of t is a Boolean combination of Σ n - formulas } . For Y ∈ { CL , ACL , DCL } ∪ { CL n , ACL n , DCL n } n ∈ N and c ∈ CompStr , let Y c := { u : ( c ) ∧ u ∈ Y } , i.e., select those elements of Y whose first coordinate is c , and then remove this firstcoordinate. Note that CompStr is a Π set. Hence the sets CL , ACL , DCL are already complicatedfrom the computability-theoretic perspective. As such, when we consider the complexity ofwhether formulas are algebraic or definable at various tuples, we will consider the questionof how complex CL c , ACL c , DCL c can be, when c is a code for a structure. The next threelemmas connect these sets. Lemma 2.4. Uniformly in the parameters c ∈ CompStr and n ∈ N , the set (cid:8) ( ϕ ( x ; y ) , a, k ) ∈ CL cn : k ∈ N , k ≥ (cid:9) is Σ in DCL cn .Proof. Suppose ϕ ( x ; y ) is a Boolean combination of Σ n -formulas of L c , and let k ≥ 1. Foreach j < k , choose a tuple of new variables z j of the same type as y . Define the formulaΥ ϕ ( x ; y ) ,k := (cid:94) i Uniformly in the parameters c ∈ CompStr and n ∈ N , the set (cid:8)(cid:0) ϕ ( x ; y ) , a, k (cid:1) ∈ CL cn : k = 0 (cid:9) is Σ in DCL cn .Proof. Suppose ϕ ( x ; y ) is a Boolean combination of Σ n -formulas of L c . Let z be a tuple ofvariables having the same type as y and disjoint from x y . Define the formulaΨ ϕ ( x ; y ) ( x z, y ) := ϕ ( x, y ) ∨ ( y = z ) . Note that Ψ ϕ ( x ; y ) ( x z, y ) is also a Boolean combination of Σ n -formulas of L c .Now suppose b and b are distinct tuples of elements of M c having the same type as z . Then the following are equivalent: 4 (cid:0) Ψ ϕ ( x ; y ) ( x z ; y ) , a b (cid:1) ∈ DCL cn and (cid:0) Ψ ϕ ( x ; y ) ( x z ; y ) , a b (cid:1) ∈ DCL cn ; • (cid:8) b : M c | = ϕ ( a ; b ) (cid:9) = ∅ , i.e., ( ϕ ( x ; y ) , a, ∈ CL cn .The result is then immediate. Lemma 2.6. Uniformly in the parameters c ∈ CompStr and n ∈ N , there are computablereductions in both directions between ACL cn (cid:96) DCL cn and CL cn .Proof. It is immediate from the definitions that DCL cn is computable from CL cn . Further,ACL cn is computable from CL cn asACL cn = (cid:8) ( ϕ ( x ; y ) , a ) : there exists k with ( ϕ ( x ; y ) , a, k ) ∈ CL cn and k (cid:54) = ∞ (cid:9) and as ( ϕ ( x ; y ) , a, k ) ∈ CL cn holds for a unique k ∈ N ∪ {∞} .Lemmas 2.4 and 2.5 together tell us that the set (cid:8) ( ϕ ( x ; y ) , a, k ) ∈ CL cn : k ∈ N (cid:9) is computably enumerable from DCL cn . But ( ϕ ( x ; y ) , a, ∞ ) ∈ CL cn if and only if ( ϕ ( x ; y ) , a ) (cid:54)∈ ACL cn . Therefore when ϕ ( x ; y ) is a Boolean combination of Σ n -formulas, and given a ∈ M c , we can compute from ACL cn whether or not ( ϕ ( x ; y ) , a, ∞ ) ∈ CL cn . Further, if( ϕ ( x ; y ) , a, ∞ ) (cid:54)∈ CL cn , then we can compute from DCL cn the (unique) value of k such that( ϕ ( x ; y ) , a, k ) ∈ CL cn . Hence CL cn is computable from ACL cn (cid:96) DCL cn .Note that by Lemma 2.6 we are justified, from a computability-theoretic perspective,in restricting our attention to ACL and DCL (and their variants), as opposed to CL.We next define a closure operator with respect to a collection of formulas. We will useit to study computable aspects of algebraic and definable closure. (See [6, § Definition 2.7. Let L be a language, let Φ be a set of first-order L -formulas, and let N be an L -structure. Suppose B ⊆ N and S ⊆ N ∪ {∞} . Define cl n Φ , N ( B, S ) for n ∈ N byinduction as follows. • cl , N ( B, S ) := B , • cl , N ( B, S ) := B ∪ (cid:8) b ∈ N : there exists ϕ ∈ Φ and a tuple a from B with (cid:12)(cid:12) { d : N | = ϕ ( a ; d ) } (cid:12)(cid:12) ∈ S such that for some b ∈ N with b ∈ b, we have N | = ϕ ( a ; b ) (cid:9) , • cl n +1Φ , N ( B, S ) := cl ,c (cid:0) cl n Φ , N ( B, S ) (cid:1) .Let cl Φ , N ( B, S ) := (cid:83) i ∈ N cl i Φ , N ( B, S ) . Φ , N ( · , S ), it suffices to restrict our attention to the case where theargument is a finite subset of N , since for any B ⊆ N we havecl Φ , N ( B, S ) = (cid:91)(cid:8) cl Φ , N ( B , S ) : B is a finite subset of B (cid:9) . There are two instances of S for which the operator cl Φ , N ( · , S ) is especially importantmodel-theoretically. The algebraic and definable closure operators in N (with respect toΦ) are given, respectively, by acl Φ , N ( · ) := cl Φ , N ( · , N )and dcl Φ , N ( · ) := cl Φ , N ( · , { } ) . The standard model-theoretic notions of first-order algebraic and definable closure in N are acl L ω,ω ( L ) , N ( · ) and dcl L ω,ω ( L ) , N ( · ), respectively. In these two cases, when Φ = L ω,ω ( L )and S is either N or { } , we have cl Φ , N ( · , S ) = cl , N ( · , S ), i.e., the first step of theiterative process in Definition 2.7 is already idempotent. But this is not the case for everyset Φ of formulas, and so to obtain a closure operator, we need the full iterative process.Note that a key computability-theoretic distinction is whether or not S is finite, andindeed one can easily check that all the upper and lower bounds proved in this paper fordcl Φ , N ( · ) also hold for cl Φ , N ( · , S ) for any finite S .In order to study the computability-theoretic content of the algebraic and definableclosure operators, we will consider the following encodings of their respective graphs. Definition 2.8. Let L be a language, let Φ be a set of first-order L -formulas, and let N be an L -structure. Define acl Φ , N := (cid:8) ( a, A ) : a ∈ acl Φ , N ( A ) and A is a finite subset of N (cid:9) , dcl Φ , N := (cid:8) ( a, A ) : a ∈ dcl Φ , N ( A ) and A is a finite subset of N (cid:9) . For c ∈ CompStr, write cl Φ ,c ( B, S ) to denote cl Φ , M c ( B, S ), and similarly with acl Φ ,c ( B ),acl Φ ,c , dcl Φ ,c ( B ), and dcl Φ ,c .As can be seen from Definition 2.7, the set cl Φ ,c ( B, S ) is closely related to CL c via therelation Z S on M c defined by Z S := (cid:91) ψ ∈ Φ (cid:8) ( a, b ) : M c | = ψ ( a ; b ) and ( ψ ( x ; y ) , a, k ) ∈ CL c with k ∈ S (cid:9) . For example, suppose every formula in Φ has just two free variables, and let U S be thetransitive closure in M c of Z S . Thenacl Φ ,c ( B, S ) = (cid:8) b ∈ M c : there exists a ∈ B for which U N ( a, b ) holds (cid:9) and dcl Φ ,c ( B, S ) = (cid:8) b ∈ M c : there exists a ∈ B for which U { } ( a, b ) holds (cid:9) . Upper Bounds for Quantifier-Free Formulas We now provide straightforward upper bounds on the complexity of ACL c , DCL c , acl Φ ,c ,and dcl Φ ,c for c ∈ CompStr and Φ a computable set of quantifier-free first-order L c -formulas. Proposition 3.1. Uniformly in the parameter c ∈ CompStr , the set ACL c is a Σ set.Proof. Uniformly in c ∈ CompStr, a quantifier-free L c -formula ϕ ( x ; y ), and tuple a ∈M c of the same type as x , we can computably find an e ∈ N such that W e equals (cid:8) b ∈ M c : M c | = ϕ ( a ; b ) (cid:9) (where the tuples b of this set are encoded in N in a standardway).Further, ( ϕ ( x ; y ) , a ) ∈ ACL c if and only if (cid:8) b ∈ M c : M c | = ϕ ( a ; b ) (cid:9) is finite. ThereforeACL c is Σ , as Fin is Σ . Proposition 3.2. Uniformly in the parameter c ∈ CompStr , the set DCL c is the inter-section of a Π set and a Σ set (in particular, it is a ∆ set).Proof. Uniformly in c ∈ CompStr, the set of all pairs ( ϕ ( x ; y ) , a ) such that M c | = ( ∀ y , y ) (cid:0) ( ϕ ( a, y ) ∧ ϕ ( a, y )) → ( y = y ) (cid:1) holds is a Π set. Likewise, uniformly in c ∈ CompStr, the set of all pairs ( ϕ ( x ; y ) , a ) suchthat there exists b with M c | = ϕ ( a ; b ) is a Σ set.As a consequence, DCL c is computable from 000 (cid:48) . Proposition 3.3. Uniformly in the parameter c ∈ CompStr and an encoding of a com-putable set Φ of quantifier-free first-order L c -formulas, the set acl Φ ,c is Σ in ACL c . Inparticular, acl Φ ,c is a Σ set.Proof. Let A ⊆ M c be a finite set. Note that b ∈ acl Φ ,c ( A ) if and only if there is a finitesequence b , . . . , b n − ∈ M c where b = b n − such that for each i < n , there exists a formula ϕ i ( x ; y ) ∈ Φ, a tuple a i with entries from A ∪ { b j } j
Uniformly in the parameter c ∈ CompStr and an encoding of a com-putable set Φ of quantifier-free first-order L c -formulas, the set dcl Φ ,c is Σ in DCL c . Inparticular, dcl Φ ,c is a Σ set.Proof. Let A ⊆ M c be a finite set. Note that b ∈ dcl Φ ,c ( A ) if and only if there is a finitesequence b , . . . , b n − ∈ M c where b = b n − such that for each i < n , there exists a formula ϕ i ( x ; y ) ∈ Φ, a tuple a i with entries from A ∪ { b j } j
There is a parameter c ∈ CompStr such that the following hold.(a) L c consists of, for each i ∈ N , a sort X i and unary relation symbols U i and V i of sort X i . Each of the U i and V i is instantiated by a single element of M c .(b) For each ordinal α , the theory T c has ( | α + 1 | ω ) -many models of size ℵ α . In particular, T c is ℵ -categorical.(c) (cid:8) e : ( X e ) N is finite for every N | = T c (cid:9) ≡ Fin .(d) ACL c ≡ Fin . In particular, ACL c is a Σ -complete set.Proof. Let (cid:0) ( e i , n i ) (cid:1) i ∈ N be a computable enumeration without repetition of (cid:8) ( e, n ) : e, n ∈ N and { e } ( n ) ↓ (cid:9) . Note that for each (cid:96) ∈ N ∪ {∞} , there are infinitely many programs that halt on exactly (cid:96) -many inputs, and so there are infinitely many e ∈ N that are equal to e i for exactly (cid:96) -many i .Let c ∈ CompStr be a code such that L c is as in (a), and M c satisfies the following.8 The underlying set of M c is N ∪ ( { , } × N ), • ( U i ) M c = { (0 , i ) } and ( V i ) M c = { (1 , i ) } for i ∈ N , and • i ∈ ( X e i ) M c for i ∈ N .Each model of T c is determined up to isomorphism by the number of elements in theinstantiation of each sort. Consider a model of T c of size ℵ α . For each j ∈ N it has ℵ -many sorts whose instantiations are of size j . It also has ℵ -many whose instantiationsare infinite, each of which may have size ℵ β for arbitrary β ≤ α . Hence (b) holds.Note that ( X e ) M c = { i : e i = e } ∪ { (0 , e ) , (1 , e ) } . So for any countable N | = T c we have (cid:12)(cid:12) ( X e ) N (cid:12)(cid:12) = | W e | + 2 . Hence Fin is 1-equivalent to the set (cid:8) e : ( X e ) N is finite (cid:9) , which is equal to the set (cid:8) e : ( X e ) N is finite for every N | = T c (cid:9) , proving (c).Because all relation symbols in L c are unary, any definable set is the product of definablesets that are each contained in the instantiation of a single sort. Further, given a countable N | = T c , a finite A ⊆ N , and an i ∈ N , the definable sets (with parameters from A ) in( X i ) N are Boolean combinations of { ( U i ) N , ( V i ) N } ∪ (cid:8) { a } : a ∈ A ∩ ( X i ) N (cid:9) .Hence ACL c is 1-equivalent to (cid:8) e : ( X e ) M c is finite (cid:9) as well, establishing (d).We now show that the upper bound in Proposition 3.2 is tight. Proposition 4.2. There is a parameter c ∈ CompStr such that the following hold.(a) The language L c has one sort and one binary relation symbol E .(b) The structure M c has underlying set N and is a countable saturated model of T c .(c) For each ordinal α , the theory T c has ( | α + ω | ) -many models of size ℵ α , and hasfinite Morley rank.(d) There is a computable array (cid:0) U k,(cid:96) (cid:1) k,(cid:96) ∈ N of subsets of N such that each countablemodel of T c is isomorphic to the restriction of M c to the underlying set U k,(cid:96) forexactly one pair ( k, (cid:96) ) .(e) If N ∼ = M c then uniformly in N we can compute (cid:48) from the set (cid:8) a : (cid:12)(cid:12) { b : N | = E ( a ; b ) } (cid:12)(cid:12) = 1 (cid:9) . (f ) The set (cid:8) a : ( E ( x ; y ) , a ) ∈ DCL c (cid:9) has Turing degree (cid:48) . roof. Let g : N → { , } be the characteristic function of 000 (cid:48) , i.e., such that g ( n ) = 1 ifand only if n ∈ (cid:48) . As 000 (cid:48) is a ∆ set, there is some computable function f : N × N → { , } such that lim s →∞ f ( n, s ) = g ( n ) for all n ∈ N .We will construct M c in the language specified in (a) so as to satisfy the followingaxioms. • ( ∀ x ) ¬ E ( x, x ) • ( ∀ x, y ) ( E ( x, y ) → E ( y, x )) • ( ∀ x )( ∃ y ) E ( x, y ) • ( ∀ x )( ∃ ≤ y ) E ( x, y )These axioms specify that ( N , E M c ) is a graph that is the union of chains. In fact, wewill construct M c so as to have infinitely many chains of certain finite orders, infinitelymany N -chains, and infinitely many Z -chains.For n ∈ N , let p n denote the n th prime number. We now construct M c with underlyingset N , in stages.Stage 0:Let { N i } i ∈ N ∪ { Z i } i ∈ N ∪ { F } be a uniformly computable partition of N into infinite sets.For each i ∈ N , let the induced subgraph on N i be an N -chain, and let the inducedsubgraph on Z i be a Z -chain. The only other edges will be between elements of F (to bedetermined at later stages).Stage 2 s + 1:Let a s be the least element of F that is not yet part of an edge. Create a finite chain oforder ( p s ) f ( s,s ) consisting of a s and other elements of F not yet in any edge.Stage 2 s + 2:For each n ≤ s , we have two cases, based on the values of f : • If f ( n, s ) = f ( n, s + 1), do nothing. • Otherwise, if f ( n, s ) (cid:54) = f ( n, s + 1), consider the (unique) chain whose order so far is( p n ) k for some positive k . Extend this chain by (cid:0) ( p n ) k +1 − ( p n ) k (cid:1) -many elements of F which are not yet in any edge, to obtain a chain that has order ( p n ) (cid:96) + f ( n,s +1) forsome (cid:96) ∈ N .The resulting graph is computable, as every vertex participates in at least one edge,and whether or not there is an edge between a given pair of vertices is determined by thefirst stage at which each vertex of the pair becomes part of some edge.Observe that every element of F is part of a chain of elements of F whose order issome positive power of a prime, which moreover is the only chain in M c whose order is apower of that prime. 10very model of T c is determined up to isomorphism by the number of N -chains and thenumber of Z -chains in it. In a model of size ℵ α , there must be either ℵ α -many N -chainsand 0, 1, . . . , ℵ , . . . , or ℵ α -many Z -chains, or vice-versa. The countable saturated modelsof T c are those with ℵ -many N -chains and ℵ -many Z -chains, and since M c has ℵ -many N -chains and ℵ -many Z -chains, condition (b) holds. None of these N -chains or Z -chainsare first-order definable, and so condition (c) holds.For condition (d), let U k,(cid:96) := (cid:83) i Let c ∈ CompStr be the parameter constructed in the proof of Proposi-tion 4.1. Then there is a computable set Ξ of quantifier-free first-order L c -formulas suchthat if N ∼ = M c , then uniformly in N we can compute Fin from acl Ξ , N via a -reductionrelative to N . In particular, for computable such N , the set acl Ξ , N is Σ -complete.Proof. For each sort X i in L c , let ξ i ( x, y ) be the L c -formula that asserts that x and y are each of sort X i . Let Ξ := { ξ i ( x ; y ) } i ∈ N . Suppose N ∼ = M c . Then there exists anisomorphism τ : N → M c that is computable in N . Let A ⊆ N be finite and b ∈ N . Notethat b ∈ acl Ξ , N ( A ) if and only if there is some a ∈ A and i ∈ N for which (cid:0) ξ i ( x ; y ) , τ ( a ) (cid:1) ∈ ACL c and N | = ξ i ( a ; b ) . By the choice of the code c , for each i ∈ N , the unique element of ( V i ) N is in acl Ξ , N (cid:0) ( U i ) N (cid:1) if and only if X i is finite, establishing the proposition.We now build a structure that shows that the bound in Proposition 3.4 is also tight. Proposition 4.4. There is a parameter c ∈ CompStr such that L c contains a ternaryrelation symbol F and, letting Γ := { F ( x, y ; z ) } , if N ∼ = M c then uniformly in N we cancompute Fin from dcl Γ , N via a -reduction relative to N . In particular, for computablesuch N , the set dcl Γ , N is Σ -complete.Proof. Let L be the (one-sorted) language consisting of unary relation symbols A , B , C , D , H , a binary relation symbol E , and a ternary relation symbol F .We first define a computable L -structure Y such that dcl Γ , Y is computable. Write Y for its underlying set, and write (cid:63) for a distinguished element of Y . The relation D Y is thesingleton { (cid:63) } , and the other unary relations A Y , B Y , C Y , and H Y partition Y \ { (cid:63) } intodisjoint infinite sets. Let { a i } i ∈ N and { b i } i ∈ N be enumerations of the elements of A Y and B Y , respectively. 11he pair ( Y, E Y ) is a graph whose non-trivial connected components are finite chains L i , for i ∈ N , with the following properties. For each i ∈ N , the chain L i has order i + 2.The degree-1 vertices of L i are a i and b i , and its remaining vertices satisfy H (chosencomputably). Every element of H Y is in one such L i , and no elements of C Y or D Y are.We will define ( Y, F Y ) later.Observe that the graph ( A Y ∪ B Y ∪ H Y , E Y ) is rigid. Furthermore, for any graph P that is isomorphic to ( A Y ∪ B Y ∪ H Y , E Y ), the unique such isomorphism is computableuniformly in P .We will eventually use Y to build a computable L -structure M c having the sameunderlying set Y , satisfying the statement of the Proposition. The instantiations of A , B , C , D , H , and E on M c and Y will agree. They will also agree on F restricted to Y \ { (cid:63) } .We will encode Fin in M c via the behavior of F on triples that include (cid:63) .We now define F Y . The first coordinate of any F -triple in Y will satisfy either A or C .It will be useful to think of F Y as a collection, indexed by the first axis of F Y , of binaryrelations on Y : for r ∈ Y , write F r to denote the relation (cid:8) ( s, t ) ∈ Y × Y : Y | = F ( r, s, t ) (cid:9) . For r ∈ Y , the pair ( Y, F r ) will be a digraph whose edge set is either empty or forms asingle path with initial vertex satisfying A . For such a path, if F r is finite, then the finalvertex of the path will satisfy B ; all vertices of the path that are neither initial nor finalwill satisfy C .Partition C Y into sets { P i, ∞ } i ∈ N ∪ { P i,k } i,k ∈ N where for each i ∈ N , the set P i, ∞ isinfinite and the set P i,k has size k . For i ∈ N , enumerate P i, ∞ by { r i, ∞ ,j : j ∈ N } .For r ∈ Y \ ( A Y ∪ (cid:83) i ∈ N P i, ∞ ), let F r be empty.For i ∈ N , let ( Y, F a i ) have a single non-trivial connected component, namely a single N -path with its initial vertex equal to a i and vertex set { a i } ∪ P i, ∞ , with ( a i , r i, ∞ , ) ∈ F a i and ( r i, ∞ ,j , r i, ∞ ,j +1 ) ∈ F a i for j ∈ N .For each i, k ∈ N , let ( Y, F r i, ∞ ,k ) have a single non-trivial connected component, namelya path of order k + 2 with initial vertex a i , final vertex b i , and vertex set { a i , b i } ∪ P i,k .Note that for all i, k ∈ N , we have r i, ∞ ,k ∈ dcl Γ , Y ( { a i } ), and further, (cid:12)(cid:12)(cid:8) t : Y | = F ( r i, ∞ ,k , a i ; t ) (cid:9)(cid:12)(cid:12) = 1 . We say that P i,k witnesses that b i ∈ dcl Γ , Y ( { a i } ). This completes the definition of Y .We are now ready to define M c , a computable structure that has the same underlyingset as Y and that agrees with Y on Y \ { (cid:63) } .As in the proof of Proposition 4.1, let (cid:0) ( e i , n i ) (cid:1) i ∈ N be a computable enumerationwithout repetition of (cid:8) ( e, n ) : e, n ∈ N and { e } ( n ) ↓ (cid:9) . Let M c | = F ( r, s, t ) with (cid:63) ∈ { r, s, t } hold if and only if for some i ∈ N and k ≤ n i ,( r, s, t ) = ( r e i , ∞ ,k , a e i , (cid:63) ) . Consequently, for i ∈ N and k ≤ n i we have (cid:12)(cid:12)(cid:8) t : M c | = F ( r e i , ∞ ,k , a e i ; t ) (cid:9)(cid:12)(cid:12) = 2 , P e i ,k does not witness that b e i is in dcl Γ ,c ( { a e i } ).On the other hand, for i ∈ N , if for all h ∈ N with e h = e i we have k > n h then thepath P e i ,k still witnesses that b e i is in dcl Γ ,c ( { a e i } ).Let (cid:96) ∈ N . By construction, we have b (cid:96) ∈ dcl Γ ,c ( { a (cid:96) } ) if and only if this fact iswitnessed by P (cid:96),j for some j ∈ N . By the above, there is some j such that P (cid:96),j witnesses b (cid:96) ∈ dcl Γ ,c ( { a (cid:96) } ) if and only if sup { n : { (cid:96) } ( n ) ↓} is finite.Hence Fin, a Σ -complete set, is 1-reducible to (cid:8) ( a (cid:96) , b (cid:96) ) : b (cid:96) ∈ dcl Γ ,c ( { a (cid:96) } ) (cid:9) relative to N , as desired.In Propositions 3.3 and 3.4, we provided upper bounds on the difficulty of computingacl Φ ,c from ACL c , and of computing dcl Φ ,c from DCL c , for Φ a computable set of quantifier-free first-order L c -formulas. We now show that in general, merely knowing acl Φ ,c and dcl Φ ,c will not lower the difficulty of computing even the Φ-fiber of ACL c or DCL c . We do so byproviding examples where the Φ-fibers of ACL c and of DCL c are maximally complicatedbut acl Φ ,c and dcl Φ ,c are trivial. Proposition 4.5. There are c , c ∈ CompStr such that the following hold.(a) The (one-sorted) language L c = L c contains a ternary relation symbol F and aunary relation symbol U .(b) M c and M c have the same underlying set M and agree on all unary relations.(c) Let ψ ( x, y, z ) := F ( x, y, z ) ∧¬ F ( x, z, y ) , and write Ψ = { ψ ( x, y ; z ) } . For any A ⊆ M , acl Ψ ,c ( A ) = dcl Ψ ,c ( A ) = (cid:40) M if A ∩ U (cid:54) = ∅ , and ∅ if A ∩ U = ∅ . (d) If N ∼ = M c then uniformly in N , the set Fin is -reducible to the set (cid:8) ( u, a ) : u ∈ U and { b : N | = ψ ( u, a ; b ) } is finite (cid:9) , relative to N . In particular, Fin ≤ ACL c , and so ACL c is a Σ -complete set.(e) If N ∼ = M c then uniformly in N we can compute (cid:48) from the set (cid:8) ( u, a ) : u ∈ U and (cid:12)(cid:12) { b : N | = ψ ( u, a ; b ) } (cid:12)(cid:12) = 1 (cid:9) . In particular DCL c is Turing equivalent to (cid:48) .Proof. Let L (cid:48) be the (one-sorted) language consisting of unary relation symbols U , A , B , C , D , H , a binary relation symbol E , and a ternary relation symbol F . Let K be thesublanguage of L (cid:48) consisting of the relation symbols A , B , C , D , H and E .We begin by defining a computable L (cid:48) -structure Z . The reduct of the structure Z to K will be the same as the reduct to K of the structure Y in the proof of Proposi-tion 4.4 (in particular, the underlying set of Z is also Y ). This will imply that the graph13 A Z ∪ B Z ∪ H Z , E Z ) is rigid, and that for any graph P that is isomorphic to ( A Z ∪ B Z ∪ H Z , E Z ), the unique such isomorphism is computable uniformly in P .The unary relation U Z consists of three elements u , u , u where u , u ∈ C Z and D Z = { u } . It remains to describe F Z .We will eventually build computable L (cid:48) -structures M c and M c , each with underlyingset Y , which agree with Z on the unary and binary relations, and are such that F Z ⊆ F M cj for j ∈ { , } . For j ∈ { , } we will construct M c j such that if ( r, s, t ) ∈ F M cj , then r ∈ U Z .We now describe F Z . For any ( r, s, t ) ∈ F Z we will have r ∈ U Z . Define, for i ∈ { , , } , the binary relations F i := (cid:8) ( s, t ) ∈ Y × Y : Z | = F ( u i , s, t ) (cid:9) . For each i , the structure ( Y, F i ) will be a digraph; further if ( s, t ) ∈ F i and { s, t } ∩ U Z (cid:54) = ∅ then s = u i and t = u k , where k ≡ i + 1 (mod 3). In particular, for each i there is a single F i -edge in U Z and no other F i -edge involves an element of U Z .To complete the description of F Z , we now describe each F i outside U Z . Let the digraph( Y \ U Z , F ) be any computable infinite path, and let F be such that for s, t ∈ Y \ U Z wehave Z | = F ( s, t ) ↔ F ( t, s ). The digraph ( Y \ U Z , F ) has no edges, i.e., F = { ( u , u ) } .This completes the definition of Z .Now we move towards defining M c i for i ∈ { , } . Suppose G is a computable bipartitegraph with underlying set A Z ∪ B Z and underlying partition { A Z , B Z } , in the single-sortedlanguage consisting of a single binary relation symbol G . For such a G , define Z ( G ) to bethe L (cid:48) -structure with underlying set Y that agrees with Z on all unary relations and E ,and for which F Z ( G ) = F Z ∪ (cid:8) ( u , s, t ) : ( s, t ) ∈ G G (cid:9) . Let f be any computable function which takes a code for computable bipartite graphs G with underlying partition { A Z , B Z } and returns a code for Z ( G ). Similarly define d to beany computable function such that d ( G ) is a code for G .It is straightforward to check that for any computable bipartite graph G with underlyingpartition { A Z , B Z } , we have that Z ( G ) is computable andacl Ψ , Z = acl Ψ , Z ( G ) , dcl Ψ , Z = dcl Ψ , Z ( G ) . In particular, if G and G are such graphs then (a), (b) and (c) hold for c = f ( G ) and c = f ( G ).Observe that from the set of pairs of the form ( ψ ( x, y ; z ) , ( u , b )) in ACL f ( G )0 , we cancompute those of the form ( G ( x ; y ) , a ) in ACL d ( G )0 . Likewise, from the set of pairs ofthe form ( ψ ( x, y ; z ) , ( u , b )) in DCL f ( G )0 we can compute those of the form ( G ( x ; y ) , a ) inDCL d ( G )0 .To finish the proof, we now choose G and G . For i ∈ N , let a i be the unique elementof A Z in a chain of order i + 2 in ( Y, E Z ).Let G be any computable bipartite graph with underlying partition { A Z , B Z } suchthat for each e ∈ N , the vertex a e is adjacent to (cid:12)(cid:12) { n : { e } ( n ) ↓} (cid:12)(cid:12) -many elements in B Z . If14 is any computable structure isomorphic to f ( G ), then Fin is 1-reducible to (cid:8) ( u, a ) : u ∈ U and { b : N | = F ( u, a ; b ) } is finite (cid:9) relative to N . Hence Fin is 1-reducible to ACL d ( G )0 , and so (d) holds.Let G be any computable bipartite graph with partition { A Z , B Z } such that for each e ∈ N , the vertex a e is adjacent a single element of B Z if { e } (0) ↓ , and to no elementsotherwise. Then if N is any computable structure isomorphic to f ( G ), we can computeFin from (cid:8) ( u, a ) : u ∈ U and |{ b : N | = F ( u, a ; b ) }| = 1 (cid:9) . Hence DCL d ( G )0 can compute 000 (cid:48) , and so (e) holds, completing the proof. Σ n -Formulas In Sections 3 and 4 we studied, for c ∈ CompStr, the complexity of ACL c and DCL c , andof acl Φ ,c and dcl Φ ,c where Φ is a computable set of quantifier-free first-order L c -formulas.We now study the complexity of ACL cn and DCL cn , for arbitrary n ∈ N , and of acl Φ ,c anddcl Φ ,c where Φ is a computable set of first-order L c -formulas of quantifier rank at most n .Morleyization is a technique for translating a structure in a given language to a newstructure, in a new language, that has quantifier elimination but the same definable sets.This is done by introducing new relation symbols to take the place of existing formulas.The following lemma is a computable version of this standard method. The proof isstraightforward. Lemma 5.1. Let L be a computable language and A a computable L -structure. For each n ∈ N there is a computable language K n and a ( n ) -computable K n -structure A n such thatthe following hold. • L ⊆ K n ⊆ K n +1 . • A is the reduct of A n to the language L . • For each first-order K n -formula ϕ there is a first-order L -formula ψ ϕ (of the sametype as ϕ ) such that A n | = ( ∀ x , . . . , x k − ) ϕ ( x , . . . , x k − ) ↔ ψ ϕ ( x , . . . , x k − ) , where k is the number of free variables of ϕ . • For each first-order L -formula ψ , if ψ is a Boolean combination of Σ n -formulas thenthere is a first-order quantifier-free K n -formula ϕ ψ (of the same type as ψ ) such that A n | = ( ∀ x , . . . , x k − ) ψ ( x , . . . , x k − ) ↔ ϕ ψ ( x , . . . , x k − ) , where k is the number of free variables of ψ . We now use Lemma 5.1 to extend our earlier results about algebraic and definableclosure for quantifier-free formulas to formulas of higher quantifier rank; this comes at theexpense of greater computability-theoretic complexity.15 orollary 5.2. Let n ∈ N . Uniformly in c ∈ CompStr , we have that(a) ACL cn is a Σ n +2 set, and(b) DCL cn is a ∆ n +2 set.Further, uniformly in c ∈ CompStr and in a computable collection Φ of first-order L c -formulas of quantifier rank at most n , we have that(c) acl Φ ,c is a Σ n +2 set, and(d) dcl Φ ,c is a Σ n +2 set.Proof. By Lemma 5.1, we know that ACL n and DCL n are equivalent to the relativization,to the class of structures computable in 000 ( n ) , of ACL and DCL , respectively. Thereforeby Propositions 3.1 and 3.2, ACL cn is a Σ (000 ( n ) ) set and DCL cn is a ∆ (000 ( n ) ) set and so (a)and (b) hold.Similarly, (c) and (d) hold by Propositions 3.3 and 3.4.In Theorem 5.6 we will show that these bounds are tight. Towards this fact, we willneed a definition and the technical results in Lemma 5.4 and Proposition 5.5 below.Let ( N , Succ) be the digraph with underlying set N where Succ is the graph of thesuccessor function on N , i.e., such that Succ( k, (cid:96) ) holds precisely when (cid:96) = k + 1. Definition 5.3. Suppose that L is a language containing a sort N and a relation symbol S of type N × N . Let A be an L -structure. We call ( N A , S A ) a copy of N when there isan isomorphism between ( N A , S A ) and ( N , Succ) .Note that any such isomorphism is necessarily unique. Given (cid:96) ∈ N , we write (cid:98) (cid:96) todenote the corresponding element of N A under this isomorphism. Lemma 5.4. Let L be a language containing a sort N and a relation symbol S of type N × N (and possibly other sorts and relation symbols). Let A be an L -structure such that ( N A , S A ) is a copy of N . Let k ∈ N and let γ ( x, y ) be an L -formula that is a Booleancombination of Σ k -formulas, where x is of some type X , and y has sort N .Suppose that A | = ( ∀ x : X )( ∃ ≤ y : N )( ∃ z : N ) S ( y, z ) ∧ (cid:0) γ ( x, y ) ↔ ¬ γ ( x, z ) (cid:1) . Let f : X A × N → { True , False } be the function where A | = γ ( a, (cid:98) (cid:96) ) if and only if f ( a, (cid:96) ) = True . Note that lim (cid:96) →∞ f ( a, (cid:96) ) exists for all a ∈ X A .There is a first-order L -formula γ (cid:48) ( x ) , where x is of type X , such that γ (cid:48) is a Booleancombination of Σ k +1 -formulas and for all a ∈ X A , A | = γ (cid:48) ( a ) if and only if lim (cid:96) →∞ f ( a, (cid:96) ) = True . roof. Define the formula γ (cid:48) by γ (cid:48) ( x ) := (cid:2) ( ∀ y : N ) γ ( x, y ) (cid:3) ∨ (cid:2) ( ∃ y, z : N ) (cid:0) S ( y, z ) ∧ ¬ γ ( x, y ) ∧ γ ( x, z ) (cid:1)(cid:3) . Clearly γ (cid:48) is a Boolean combination of Σ k +1 -formulas and has the desired property. Proposition 5.5. Let n ∈ N . Let L be a language containing a sort N and a relationsymbol S of type N × N (and possibly other sorts and relation symbols). Suppose A is an L -structure that is computable in ( n ) and such that ( N A , S A ) is a computable copy of N .Then there is a computable language L + and a computable L + -structure A + with the sameunderlying set as A such that for every quantifier-free first-order L -formula η in which S does not occur, there is a first-order L + -formula ϕ η that is a Boolean combination of Σ n -formulas such that η A = ( ϕ η ) A + .Proof. We begin by defining, for relation symbols in L other than S , certain auxiliaryfunctions.For R a relation symbol in L other than S , let X be its type. For every k ∈ N suchthat 0 ≤ k ≤ n , inductively define the 000 ( n − k ) -computable function f R,k : X A × N k →{ True , False } satisfying the following, for all a ∈ X A . • f R, ( a ) = True if and only if A | = R ( a ). • Suppose k ≥ (cid:96) , . . . , (cid:96) k − ) ∈ N k − . There is at most one (cid:96) k − ∈ N forwhich f R,k ( a, (cid:96) , . . . , (cid:96) k − , (cid:96) k − ) (cid:54) = f R,k ( a, (cid:96) , . . . , (cid:96) k − , (cid:96) k − + 1) . Further, f R,k − ( a, (cid:96) , . . . , (cid:96) k − ) = lim (cid:96) k − →∞ f R,k ( a, (cid:96) , . . . , (cid:96) k − , (cid:96) k − ) . Next, we define the computable language L + as follows. • L + has the same sorts as L . • For each relation symbol R ∈ L other than S , there is a relation symbol R + ∈ L + oftype X × N n , where X is the type of R . The language L + also contains a relationsymbol S of type N × N . These are the only relation symbols in L + .Now define the computable L + -structure A + as follows. • A + has the same underlying set as A , and sorts are instantiated on the same sets in A + as in A . • S A + = S A . • For each R ∈ L other than S , each tuple a ∈ X A + where X is the type of R , andany (cid:96) , . . . , (cid:96) n − ∈ N , we have 17 + | = R + ( a, (cid:98) (cid:96) , . . . , (cid:100) (cid:96) n − ) if and only if f R,n ( a, (cid:96) , . . . , (cid:96) n − ) = True . (Recall that for (cid:96) ∈ N , we have defined (cid:98) (cid:96) ∈ N A + to be the (cid:96) th element of the copy of N .)We now build, for each relation symbol R ∈ L other than S , an L + -formula ϕ R , asfollows. First apply Lemma 5.4 (with k = 0) to A + and the L + -formula γ ( xy · · · y n − , y n − ) := R + ( x, y , . . . , y n − )(where x has type X and each y i has type N ) to obtain an L + -formula γ (cid:48) ( xy · · · y n − )that is a Boolean combination of Σ -formulas. Next apply Lemma 5.4 again (with k = 1)to A + and the L + -formula γ ( xy · · · y n − , y n − ) := γ (cid:48) ( xy · · · y n − )to obtain an L + -formula γ (cid:48) ( xy · · · y n − ) that is a Boolean combination of Σ -formulas.Proceed in this way for k = 2 , . . . , n − 1, to obtain an L + -formula ϕ R ( x ) := γ (cid:48) n − ( x ) thatis a Boolean combination of Σ n -formulas for which R A = ( ϕ R ) A + .We can now extend the definition of ϕ ψ to quantifier-free formulas ψ by induction,where ϕ ¬ ψ is ¬ ϕ ψ , where ϕ ψ ∧ ψ is ϕ ψ ∧ ϕ ψ , and where ϕ ψ ∨ ψ is ϕ ψ ∨ ϕ ψ .Combining Proposition 5.5 with results from Section 4, we obtain the following. Theorem 5.6. For each n ∈ N , the following hold.(a) There exists a ∈ CompStr such that ACL an is a Σ n +2 -complete set.(b) There exists b ∈ CompStr such that DCL bn ≡ T ( n +1) .(c) There is a computable set Φ of first-order L a -formulas, all of quantifier rank at most n such that acl Φ ,a is a Σ n +2 -complete set, where a ∈ CompStr is as in (a).(d) There exists d ∈ CompStr and a computable set Θ of first-order L d -formulas, all ofquantifier rank at most n , such that dcl Θ ,d is a Σ n +2 -complete set.Proof. We first prove (a) and (c). Let P be the relativization to the oracle 000 ( n ) of thecomputable structure M c from the statement of Proposition 4.1, and call its language L .Consider the set Ξ of quantifier-free formulas from Proposition 4.3. Let L ∗ be the languagethat extends L by a new sort N and a relation symbol S of type N × N . Consider the L ∗ -structure P ∗ whose restriction to L is P and such that ( N P ∗ , S P ∗ ) is a computable copyof N (instantiated on the new set of elements N P ∗ ). Let a ∈ CompStr be such that M a isthe computable structure ( P ∗ ) + obtained from Proposition 5.5 (when A = P ∗ ), and letΦ := (cid:8) ϕ η : η ∈ Ξ (cid:9) be the corresponding set of first-order ( L a ) + -formulas, each partitionedin the same way as in Ξ. Then ACL an and acl Φ ,a are Σ n +2 -complete sets, establishing (a)and (c).Towards (b), let Q be the relativization to 000 ( n ) of the structure M c from Proposition 4.2.Consider the structure Q ∗ obtained from Q by augmenting it by ( N Q ∗ , S Q ∗ ), a newcomputable copy of N , as in the proof of (a) and (c) above. Let b ∈ CompStr be such that M b is the computable structure ( Q ∗ ) + obtained by applying Proposition 5.5 to Q ∗ . ThenDCL bn ≡ T ( n +1) . 18e now prove (d). Let M c and F ( x, y, z ) be as in Proposition 4.4, and let R be therelativization of M c to 000 ( n ) . Consider the structure R ∗ obtained by augmenting R by acomputable copy of N , as above. Let d ∈ CompStr be such that M d is the computablestructure ( R ∗ ) + obtained by applying Proposition 5.5 to R ∗ , and let Θ := { ϕ F ( x, y ; z ) } .Then dcl Θ ,d is a Σ n +2 -complete set.Note that the structures constructed in Theorem 5.6 (a) and (b) do not obviouslyhave the nice model-theoretic properties ( ℵ -categoricity or finite Morley rank) that thoseconstructed in Propositions 4.1 and 4.2 do, because the application of Proposition 5.5encodes a copy of N in a way that makes their theories more elaborate. Nor is it obviousthat the structures constructed in Theorem 5.6 (c) and (d) have these nice model-theoreticproperties, because they derive from the structures constructed in Propositions 4.3 and 4.4,which themselves do not obviously have these properties. This leads us to the followingquestion. Question 1. Is there some c ∈ CompStr such that ACL cn is Σ n +2 -complete or DCL cn ≡ T ( n +1) , and M c is nice model-theoretically (e.g., ℵ -categorical, strongly minimal, stable,etc.)?Similarly, is there some c ∈ CompStr and computable set Ψ of first-order L c -formulas,all of quantifier rank at most n , such that either acl Ψ ,c or dcl Ψ ,c is a Σ n +2 -complete setand M c is nice model-theoretically? Acknowledgements The authors would like to thank Sergei Artemov, Valentina Harizanov, Anil Nerode, andthe anonymous referees for helpful comments. The third author’s work on this paper waspartially supported by the National Science Foundation under Grant No. DMS-1928930while she was in residence at the Mathematical Sciences Research Institute in Berkeley,California, during the Fall 2020 semester. References [1] N. Ackerman, C. Freer, and R. Patel, Invariant measures concentrated on countablestructures , Forum Math. 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