Smooth Approximations and Relational Width Collapses
aa r X i v : . [ c s . L O ] F e b SMOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES
ANTOINE MOTTET, TOM ´AˇS NAGY, MICHAEL PINSKER, AND MICHA L WRONA
Abstract.
We prove that relational structures admitting specific polymorphisms (namely,canonical pseudo-WNU operations of all arities n ≥
3) have low relational width. This im-plies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPsstudied in the literature. Moreover, we obtain a characterization of bounded width for first-orderreducts of unary structures and a characterization of MMSNP sentences that are equivalent toa Datalog program, answering a question posed by Bienvenu et al. . In particular, the boundedwidth hierarchy collapses in those cases as well. Introduction
Local consistency checking is an algorithmic technique that is central in computer science.Intuitively speaking, it consists in propagating local information through a structure so as toinfer global information (consider, e.g., computing the transitive closure of a relation as derivingglobal information from local one). Local consistency checking has a prominent role in the areaof constraint satisfaction, where one is given a set of variables V and constraints and one has tofind a satisfying assignment h : V → D for the constraints. In this setting, the local consistencyalgorithm can be used to decrease the size of the search space efficiently or even to correctly solvesome constraint satisfaction problems in polynomial time (for example, 2-SAT or Horn-SAT).However, the use of local consistency methods is not limited to constraint satisfaction. Indeed,local consistency checking is also used for such problems as the graph isomorphism problem, whereit is is known as the Weisfeiler-Leman algorithm. Again, the technique can be used to deriveimplied constraints that an isomorphism between two graphs has to satisfy so as to narrow downthe search space, but local consistency is in fact powerful enough to solve the graph isomorphismproblem over any non-trivial minor-closed class of graphs [35]. Notably, the best algorithm forgraph isomorphism to date also uses local consistency as a subroutine [3]. Finally, local consistencycan be used to solve games involved in formal verification such as parity games and mean-payoffgames [15].One of the reasons for the ubiquity of local consistency is that its underlying principles canbe described in many different languages, such as the language of category theory [1], in the lan-guage of finite model theory (by Spoiler-Duplicator games [37] or by homomorphism duality [2]),and logical definability (in Datalog, or infinitary logics with bounded number of variables). Forconstraint satisfaction problems over a finite template, the power of local consistency checkingcan additionally be characterised algebraically. More precisely, there are conditions on the set of polymorphisms of a template A such that local consistency correctly solves its constraint satisfac-tion problem CSP( A ) if, and only if, the polymorphisms of A satisfy these conditions. Moreover,whenever local consistency correctly solves CSP( A ), where A is finite, then in fact only a veryrestricted form of local consistency checking is needed [4]. This fact is known as the collapse ofthe bounded width hierarchy , and it has strong consequences both for complexity and logic. Onthe one hand, the collapse gives efficient algorithms that are able to solve all the CSPs that are Antoine Mottet has received funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation programme (grant agreement No 771005).Tom´aˇs Nagy has received funding from the Austrian Science Fund (FWF) through project No P32337 andthrough Lise Meitner Grant No M 2555-N35 and from the Czech Science Foundation (grant No 18-20123S).Michael Pinsker has received funding from the Austrian Science Fund (FWF) through project No P32337 andfrom the Czech Science Foundation (grant No 18-20123S).Micha l Wrona is partially supported by National Science Centre, Poland grant number 2020/37/B/ST6/01179. solvable by local consistency methods, and in fact this gives a polynomial-time algorithm solvinginstances of the uniform CSP . On the other hand, this collapse induces collapses in all the areasmentioned at the beginning of this paragraph.Many natural problems from computer science can only be phrased as CSPs where the templateis infinite. This is the case for linear programming, some reasoning problems in artificial intelligencesuch as ontology-mediated data access, or even problems as simple to formulate as the digraphacyclicity problem. In order to understand the power of local consistency in more generality itis thus necessary to consider its use for infinite-domain CSPs. Infinite-domain CSPs with an ω -categorical template form a very general class of problems for which the algebraic approach fromthe finite case can be extended, and numerous results in the recent years have shown the powerof this approach. An algebraic characterisation of local consistency checking for infinite-domainCSPs is, however, missing. In fact, the negative results of [18], refined in [34], show that no purelyalgebraic description of local consistency is possible for CSPs with ω -categorical templates; this iseven the case for temporal CSPs [19]. These negative results are to be compared with the recentresult by Mottet and Pinsker [42] that did provide an algebraic description of local consistency forseveral subclasses of ω -categorical templates.In the finite, the algebraic characterisation of local consistency relies on a set of algebraictools whose development eventually led to the solutions of the Feder-Vardi dichotomy conjecture.Bulatov’s proof of the Feder-Vardi conjecture [29] builds on his theory of edge-colored algebras,that were also used in his characterisation of bounded width [28]; Zhuk’s proof [47, 48] relieson the concept of absorption, which was developed by Barto and Kozik in their effort to provethe bounded width conjecture [5, 7]. Comparable algebraic tools, or a general theory, are at themoment missing in the theory of infinite-domain CSPs, even with an ω -categorical template. Themost general results obtained so far use canonical operations , which behave like operations onfinite sets, and for which it is sometimes possible to mimic the universal-algebraic approach tofinite-domain CSPs. Canonical operations alone do not seem to be sufficient in full generality anda characterisation of their applicability is also missing, but on the positive side their applicabilitycovers a vast majority of the results that were proved in the area. The application of canonicaloperations to approach the question of local consistency for infinite-domain CSPs has only beenstarted recently [17, 42, 46].1.1. Results.
In the present paper, we focus on applying the theory of canonical functions to studythe power of local consistency checking for constraint satisfaction problems over ω -categoricaltemplates. Our objective is two-fold: on the one hand, we wish to obtain generic sufficientconditions that imply that local consistency solves a given CSP, and on the other hand we wishto understand the amount of locality needed for local consistency to solve the CSP, as measuredby the so-called relational width .In order to solve the first objective, we build on recent work by Mottet and Pinsker [42] andexpand the use of their smooth approximations to fully suit equational (non-)affineness , whichis roughly the algebraic situation imposed by local consistency solvability. The main technicalcontribution is a new loop lemma that exploits deep algebraic tools from the finite [6] and, assumingthe use of canonical functions is unfruitful, allows to obtain the existence of polymorphisms of everyarity n ≥ Theorem 1.1.
The Datalog-rewritability problem for MMSNP is decidable, and is 2NExpTime-complete.
In order to solve the second objective, we prove that sufficiently locally consistent instancesof a given CSP can be turned into locally consistent instances of a finite-domain CSP. If thefinite-domain CSP has bounded width then it has relational width (2 ,
3) by [4], which allows us toobtain a collapse of bounded width for structures whose clone of canonical polymorphisms satisfy
MOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES 3 suitable identities, thus obtaining a similar collapse as in the finite case. In particular, it turnsout that the relational width of a structure then only depends on certain simple parameters of thestructure whose automorphism group is considered in the notion of canonicity.
Theorem 1.2.
Let k, ℓ ≥ , and let A be a first-order reduct of a k -homogeneous ℓ -bounded ω -categorical structure B . • If the clone of
Aut( B ) -canonical polymorphisms of A contains pseudo-WNUs modulo Aut( B ) of all arities n ≥ , then A has relational width (2 k, max(3 k, ℓ )) . • If the clone of
Aut( B ) -canonical polymorphisms of A contains pseudo-totally symmetricoperations modulo Aut( B ) of all arities, then A has relational width ( k, max( k + 1 , ℓ )) . Note that every finite structure A with domain { a , . . . , a n } is a first-order reduct of the struc-ture ( { a , . . . , a n } ; { a } , . . . , { a n } ), which is easily seen to be 1-homogeneous and 2-bounded. Thusthe width obtained in Theorem 1.2 coincides with the width given by Barto’s collapse resultfrom [4].As a corollary of Theorem 1.2, we obtain a collapse of the bounded width hierarchy for first-order reducts of the unary structures mentioned above, as well as of numerous other structuresstudied in the literature [20, 17, 16, 38]. Corollary 1.3.
Let A be a structure that has bounded width. If A is a first-order reduct of: • the universal homogeneous graph G or tournament T , or of a unary structure, then A hasrelational width at most (4 , ; • the universal homogeneous K n -free graph H n , where n ≥ , then at most (2 , n ) ; • ( N , =) , the countably infinite equivalence relation with infinitely many equivalence classes C ωω , or the random partial order P , then at most (2 , .Proof. A first-order reduct of G or T has bounded width if and only if the algebraic condition in thefirst item of Theorem 1.2 is satisfied [42]. Since both G and T are 2-homogeneous and 3-boundedour claim follows. First-order reducts of H n , ( N ; =) or C ωω have bounded width if and only ifthe condition in the second item of Theorem 1.2 is satisfied, by [16], [13] and [25], respectively.Since H n is 2-homogeneous and n -bounded, and since both ( N ; =) and C ωω are 2-homogeneous and3-bounded, the claimed bound follows.By appeal to Theorems 1.2 and 5.4 in the present paper our claim holds for first-order reductsof unary structures.Finally, a first-order reduct of P with bounded width is either homomorphically equivalent toa first-order reduct of ( Q ; < ) or it satisfies the algebraic condition in the second item of Theo-rem 1.2 [38]. In the latter case we are done by Theorem 1.2, in the former we appeal to thesyntactical characterization of first-order reducts of ( Q , < ). Indeed, such a structure has boundedwidth iff it is definable by a conjunction of so-called Ord-Horn clauses [19]. It then follows by [27]that a first-order reduct of ( Q ; < ) with bounded width has relational width (2 , P follows. (cid:3) Related results.
Local consistency for ω -categorical structures was studied for the first timein [12] where basic notions were introduced and some basic results provided. First-order reductsof certain k -homogeneous l -bounded structures with bounded width were characterized in [42, 19].A structure A has bounded strict width [32] if not only CSP( A ) is solvable by local consistency,but moreover every partial solution of a locally consistent instance can be extended to a totalsolution. The articles [46] and [45] give the upper bound (2 , ℓ ) on the relational width for someclasses of 2-homogeneous, ℓ -bounded structures under the stronger assumption of bounded strictwidth; it also follows from [46] that first-order reducts of H n with bounded width have relationalwidth at most (2 , n ).1.3. Organisation of the present article.
In Section 2 we provide the basic notions and def-initions. The reduction to the finite using canonical functions which leads to the collapse of thebounded width hierarchy is given in Section 3. We then extend the algebraic theory of smooth ap-proximations in Section 4 before applying it to first-order reducts of unary structures and MMSNPin Section 5.
ANTOINE MOTTET, TOM´AˇS NAGY, MICHAEL PINSKER, AND MICHA L WRONA Preliminaries
Structures and model-theoretic notions.
For sets
B, I , the orbit of a tuple b ∈ B I underthe action of a permutation group G on B is the set { α ( b ) | α ∈ G } . A countable structure B is ω -categorical if its automorphism group Aut( B ) is oligomorphic , i.e., for all n ≥
1, the numberof orbits of the action of Aut( B ) on n -tuples is finite. For ℓ ≥
1, we say that B is ℓ -bounded iffor every finite X , if all substructures Y of X of size at most ℓ embed in B , then X embeds in B .For k ≥
1, we say that B is k -homogeneous if for all tuples a, b of arbitrary finite length, if all k -subtuples of a and b are in the same orbit under Aut( B ), then a and b are in the same orbitunder Aut( B ). A first-order reduct of a structure B is a structure on the same domain whoserelations have a first-order definition in B .2.2. Polymorphisms, clones and identities. A polymorphism of a relational structure A is ahomomorphism from some finite power of A to A . The set of all polymorphisms of a structure A is denoted by Pol( A ); it is a function clone , i.e., a set of finitary operations on a fixed set whichcontains all projections and which is closed under arbitrary compositions.If C is a function clone, then we denote the domain of its functions by C ; we say that C acts on C . The clone C also naturally acts (componentwise) on C l for any l ≥
1, on any invariant subset S of C (by restriction), and on the classes of any invariant equivalence relation ∼ on an invariantsubset S of C (by its action on representatives of the classes). We write C y C l , C y S and C y S/ ∼ for these actions. Any action C y S/ ∼ is called a subfactor of C , and we also call thepair ( S, ∼ ) a subfactor. A subfactor ( S, ∼ ) is minimal if ∼ has at least two classes and no propersubset of S intersecting at least two ∼ -classes is invariant under C . For a clone C acting on a set X and Y ⊆ X we write h Y i C for the smallest C -invariant subset of X containing Y .For n ≥
1, a k -ary operation f defined on the domain C of a permutation group G is n -canonical with respect to G if for all a , . . . , a k ∈ C n and all α , . . . , α k ∈ G there exists β ∈ G such that f ( a , . . . , a k ) = β ◦ f ( α ( a ) , . . . , α k ( a k )). In particular, f induces an operation on the set C n / G of G -orbits of n -tuples. If all functions of a function clone C are n -canonical with respect to G ,then C acts on C n / G and we write C n / G for this action; if G is oligomorphic then C n / G is afunction clone on a finite set. A function is canonical with respect to a permutation group G if itis n -canonical with respect to G for all n ≥
1. We say that it is diagonally canonical if it satisfiesthe definition of canonicity in case α = · · · = α k .We write G C to denote the largest permutation group contained in a function clone C , and saythat C is oligomorphic if G C is oligomorphic. For n ≥
1, the n -canonical (canonical) part of C isthe clone of those functions of C which are n -canonical (canonical) with respect to G C . We write C can n and C can for these sets which form themselves function clones.For a set of functions F over the same fixed set C we write F for the set of those functions g such that for all finite subsets F of C , there exists a function in F which agrees with g on F . Wesay that f locally interpolates g modulo G , where f, g are k -ary functions and G is a permutationgroup all of which act on the same domain, if g ∈ { β ◦ f ( α , . . . , α k ) | β, α , . . . , α k ∈ G } . Similarly,we say that f diagonally interpolates g modulo G if f locally interpolates g with α = · · · = α k .If G is the automorphism group of a Ramsey structure in the sense of [11], then every functionon its domain locally (diagonally) interpolates a canonical (diagonally canonical) function modulo G [24, 22]. We say that a clone D locally interpolates a clone C modulo a permutation group G if for every g ∈ D there exists f ∈ C such that f locally interpolates g modulo G . A clone C is a model-complete core if its unary functions are locally interpolated by G C . A structure A is calleda model- complete core if its polymorphism clone is.A function f is idempotent if f ( x, . . . , x ) = x for all values x of its domain; a function clone isidempotent if all of its functions are. An operation w : B k → B is called a weak near-unanimity(WNU) operation if it satisfies the set of identities containing an equation for each pair of terms in { w ( x, . . . , x, y ) , . . . , w ( y, x, . . . , x ) } . It is called totally symmetric if w ( x , . . . , x k ) = w ( y , . . . , y k )whenever { x , . . . , x k } = { y , . . . , y k } . Each set of identities also has a pseudo -variant obtainedby composing each term appearing in the identities with a distinct unary function symbol. Forexample, a ternary pseudo WNU operation f satisfies the identities: e ◦ f ( y, x, x ) = e ◦ f ( x, y, x ), e ◦ f ( y, x, x ) = e ◦ f ( x, x, y ) and e ◦ f ( x, y, x ) = e ◦ f ( x, x, y ). If C is a function clone and MOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES 5 U ⊆ C is a set of unary functions, then C satisfies a set of pseudo-identities modulo U if itsatisfies the identities in such a way that the unary function symbols are assigned values in U .An arity-preserving map ξ : C → D between function clones is called a clone homomorphism ifit preserves projections, i.e., maps the i -th n -ary projection in C to the i -th n -ary projection in D and compositions, i.e., it satisfies ξ ( f ◦ ( g , . . . , g n )) = ξ ( f ) ◦ ( ξ ( g ) , . . . , ξ ( g n )) for all n, m ≥ n -ary f ∈ C and m -ary g , . . . , g n ∈ C . An arity-preserving map ξ is a minion homomorphism if it preserves compositions with projections, i.e., compositions where g , . . . , g n are projections.We say that a function clone C is equationally trivial if it has a clone homomorphism to the clone P of projections over the two-element domain, and equationally non-trivial otherwise. We alsosay that C is equationally affine if it has a clone homomorphism to an affine clone , i.e., a clone ofaffine maps over a finite module. It is known that a finite idempotent clone is either equationallyaffine or it contains WNU operations of all arities n ≥ A is an ω -categoricalmodel-complete core, then Pol( A ) can is either equationally affine, or it contains pseudo-WNUoperations modulo Aut( A ) of all arities n ≥ C , D are function clones and D has a finite domain, then a clone (or minion) homomorphism ξ : C → D is uniformly continuous if for all n ≥ B of C n such that ξ ( f ) = ξ ( g ) for all n -ary f, g ∈ C which agree on B .A first-order formula is called a primitive-positive (pp-)formula if it is built exclusively fromatomic formulae, existential quantifiers, and conjunction. A relation is pp-definable in a struc-ture B if it is first-order definable by a pp-formula; in that case, it is invariant under Pol( B ).Any ω -categorical model-complete core pp-defines all orbits of n -tuples with respect to its ownautomorphism group, for all n ≥ CSP, Relational Width, Minimality. A CSP instance over a set A is a pair I = ( V , C )where V is a finite set of variables, and C is a set of constraints C ⊆ A U , with U ⊆ V equipped witha total ordering of its elements ( U is the scope of C ). We say that I is an instance of CSP( A ) if thereexists a k -ary relation R of A such that for all f : U → A , f ∈ C ⇔ ( f ( u ) , . . . , f ( u k )) ∈ R . Givena constraint C ⊆ A U and K ⊆ U , the projection of C onto K is defined by C | K := { f | K : f ∈ C } .We now give a formal definition of a ( k, ℓ )-minimal instance. Definition 2.1.
We say that an instance I over V of CSP( A ) is ( k, l )-minimal with k ≤ ℓ if bothof the following hold: • every subset of at most ℓ variables in V is the scope of some constraint in I and • for every at most k -element subset of variables K ⊆ V and any two constraints C , C ∈ I whose scopes contain K , the projections of C and C onto K coincide.We say that an instance I of the CSP is non-trivial if it does not contain an empty constraint.Otherwise, I is trivial .Set k ≤ ℓ . Clearly not every instance I over variables V of CSP( A ) is ( k, ℓ )-minimal. However,every instance I is equivalent to a ( k, ℓ )-minimal instance I ′ (i.e., I and I ′ have the same set ofsolutions). In particular we have that if I ′ is trivial, then I has no solutions. Moreover, if A is ω -categorical, then I ′ can be computed in time polynomial in the size of I . Indeed, it is enoughto introduce a new constraint A L for every set L ⊆ V with at most ℓ elements to satisfy the firstcondition. Then the algorithm removes tuples (in fact, orbits of tuples with respect to Aut( A ))from the constraints in the instance as long as the second condition is not satisfied. Since A is ω -categorical and every relation in I is a union of a finite number of orbits of tuples with respectto Aut( A ) the algorithm terminates. Definition 2.2.
A relational structure A has relational width ( k, ℓ ) if every non-trivial ( k, ℓ )-minimal instance I of A has a solution. A has bounded width if it has relational width ( k, ℓ ) forsome natural numbers k ≤ ℓ . Theorem 2.3 ([7]) . Let I be a non-trivial (2 , -minimal CSP instance over a finite set. Supposethat the constraints of I are preserved by WNUs of all arities m ≥ . Then I has a solution. ANTOINE MOTTET, TOM´AˇS NAGY, MICHAEL PINSKER, AND MICHA L WRONA
Theorem 2.4 ([31, 32]) . Let I be a non-trivial (1 , -minimal CSP instance over a finite set.Suppose that the constraints of I are preserved by totally symmetric polymorphisms of all arities.Then I has a solution. Smooth Approximations.
We are going to apply the fundamental theorem of smoothapproximations [42] to lift an action of a function clone to a larger clone.
Definition 2.5 (Smooth approximations) . Let A be a set, n ≥
1, and let ∼ be an equivalencerelation on a subset S of A n . We say that an equivalence relation η on some set S ′ with S ⊆ S ′ approximates ∼ if the restriction of η to S is a (possibly non-proper) refinement of ∼ ; we call η an approximation of ∼ .For a permutation group G acting on A and leaving η as well as the ∼ -classes invariant, we saythat the approximation η is smooth if each equivalence class C of ∼ intersects some equivalenceclass C ′ of η such that C ∩ C ′ contains a G -orbit. Theorem 2.6 (The fundamental theorem of smooth approximations [42]) . Let C ⊆ D be functionclones on A , and let G be a permutation group on A such that D locally interpolates C modulo G . Let ∼ be a C -invariant equivalence relation on S ⊆ A with G -invariant classes and finiteindex, and η be a D -invariant smooth approximation of ∼ with respect to G . Then there exists auniformly continuous minion homomorphism from D to C y S/ ∼ . Collapses in the Relational Width Hierarchy
Definition 3.1.
Let I be a CSP instance over A with variables V . Let G be a permutation groupon A , let k ≥
1, and let O be the set of orbits of k -tuples under G . Let I G ,k be the followinginstance over O : • The variable set of I G ,k is the set (cid:0) V k (cid:1) of k -element subsets of V . Thus, every variable K of I G ,k is meant to take a value in O , and we consider that the values for K are K -orbits ,i.e., orbits of maps f : K → A under the natural action of G . • For every constraint C ⊆ A U in I , I G ,k contains the constraint C G ,k ⊆ O ( Uk ) defined by C G ,k = (cid:26) g : (cid:18) Uk (cid:19) → O | ∃ f ∈ C ∀ K ∈ (cid:18) Uk (cid:19) ( f | K ∈ g ( K )) (cid:27) . Note that the notation f | K ∈ g ( K ) makes sense precisely because g ( K ) is a K -orbit.Observe that if I is non-trivial, then so is I G ,k . Lemma 3.2.
Let ≤ a ≤ b . If I is ( ak, bk ) -minimal, then I G ,k is ( a, b ) -minimal.Proof. Let K , . . . , K b ∈ (cid:0) V k (cid:1) . Note that U := S i K i has size at most bk , and therefore there existsa constraint C ⊆ A U in I since I is ( ak, bk )-minimal. The scope of the associated constraint C G ,k is (cid:0) Uk (cid:1) , which contains K , . . . , K b .Let K = { K , . . . , K a } and let C G ,k ⊆ O ( Uk ) , D G ,k ⊆ O ( Wk ) be two constraints whose scopescontain K . Then S K is contained in the scope of the associated C ⊆ A U and D ⊆ A W andhas size at most ak , so that by ( ak, bk )-minimality of I , the projections of C and D onto S K coincide. Thus for every g ∈ C G ,k , there exists by definition an f ∈ C such that f | K i ∈ g ( K i ) forall i , and by the previous sentence there exists f ′ ∈ D such that f ′ | K i ∈ g ( K i ) for all i . Thus, g | K is in the projection of D G ,k to K . The argument is symmetric, showing that the projectionsof C G ,k and D G ,k to K coincide. (cid:3) Note that for every solution h of I , the map χ h : (cid:0) V k (cid:1) → O defined by K
7→ { αh | K | α ∈ G } defines a solution to I G ,k . The next lemma proves that every solution to I G ,k is of the form χ h forsome solution h of I , provided that I is ( k, ℓ )-minimal and that G = Aut( B ) for some ℓ -bounded k -homogeneous structure B . Lemma 3.3.
Let ≤ k < ℓ . Let B be ℓ -bounded and k -homogeneous, let A be a first-order reductof B , and let I be a ( k, ℓ ) -minimal instance of CSP( A ) . Then every solution to I Aut( B ) ,k lifts to asolution of I . MOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES 7
Proof.
Let h : (cid:0) V k (cid:1) → O be a solution to I Aut( B ) ,k . Recall that for any K ∈ (cid:0) V k (cid:1) , we view h ( K ) as a K -orbit, and one can therefore restrict h ( K ) to any L ⊆ K by setting h ( K ) | L := { f | L | f ∈ h ( K ) } .Note that since I is ( k, k )-minimal, we have h ( K ) | K ∩ K ′ = h ( K ′ ) | K ∩ K ′ for all K, K ′ ∈ (cid:0) V k (cid:1) .We now define an equivalence relation ∼ on V . Suppose first that k = 1. Then every orbit of B must be a singleton (for any orbit with two elements a, b , the pairs ( a, a ) and ( a, b ) are not inthe same orbit but their entries are, so that B is not 1-homogeneous). In that case, we identify O with the domain B itself, and set x ∼ y if and only if h ( { x } ) = h ( { y } ); that is, ∼ is essentiallythe kernel of h .Suppose next that k ≥
2, and set x ∼ y iff there is K ∈ (cid:0) V k (cid:1) containing x, y such that h ( K ) | { x,y } consists of constant maps. It can be seen that one could equivalently ask that this holds for all K containing x, y by 2-minimality, and that this is indeed an equivalence relation by (2 , I . Moreover, h descends to (cid:0) V / ∼ k (cid:1) : if K ′ = { [ v ] ∼ , . . . , [ v k ] ∼ } is a k -element set, define ˜ h ( K ′ ) := h ( { v , . . . , v k } ). The definition of ˜ h does not depend on the choice of representatives, by the verydefinition of ∼ .Define a finite structure C with domain V / ∼ in the signature of B as follows. Let K = { [ v ] ∼ , . . . , [ v k ] ∼ } . The orbit ˜ h ( K ) describes an atomic type on the elements of K ; one defines C such that its substructure induced by K has the same atomic type. This is a well-definedconstruction by the previous paragraphs.Finally, note that all substructures of C of size at most ℓ embed into B . Indeed, let L be an ℓ -element substructure of C , and let L ′ ⊆ V be an ℓ -element set containing one representativefor each element of L . By ( k, ℓ )-minimality of I , there exists C ⊆ A L ′ in I , and a corresponding C Aut( B ) ,k ⊆ I Aut( B ) ,k . Thus, h | ( L ′ k ) ∈ C Aut( B ) ,k , so that there exists g ∈ C such that for all K ∈ (cid:0) L ′ k (cid:1) , g | K ∈ h ( K ). Thus g corresponds to an embedding of every k -element substructureof L into B , and since B is k -homogeneous, g is an embedding of L into B . Finally, since B is ℓ -bounded, it follows that there exists an embedding e of C into B .It remains to check that the composition of e with the canonical projection V → V / ∼ is asolution to I , which is trivial since the relations of A are definable in B . (cid:3) Every operation f that is canonical with respect to a group G induces an operation on the setorbits of k -tuples under G , by definition. We denote this operation by f G ,k . Lemma 3.4.
Let f be a polymorphism of A that is canonical with respect to G . Every constraint C G ,k in I G ,k is preserved under f G ,k .Proof. Let f be an n -ary polymorphism of A that is canonical with respect to G , and let C ⊆ A U be a constraint in I . In particular since I is an instance of CSP( A ), C is preserved by f .Let g , . . . , g n ∈ C G ,k . By definition, for every i ∈ { , . . . , n } there is g ′ i ∈ C such that for all K ∈ (cid:0) Uk (cid:1) , g ′ i | K ∈ g i ( K ). Note that f ( g ′ | K , . . . , g ′ n | K ) = f ( g ′ , . . . , g ′ n ) | K , so that f ( g ′ , . . . , g ′ n ) | K ∈ f G ,k ( g ( K ) , . . . , g n ( K )). Since f ( g ′ , . . . , g ′ n ) ∈ C , it follows that f G ,k ( g , . . . , g n ) is in C G ,k . (cid:3) Finally, this allows us to prove Theorem 1.2 from the introduction.
Theorem 1.2.
Let k, ℓ ≥ , and let A be a first-order reduct of a k -homogeneous ℓ -bounded ω -categorical structure B . • If the clone of
Aut( B ) -canonical polymorphisms of A contains pseudo-WNUs modulo Aut( B ) of all arities n ≥ , then A has relational width (2 k, max(3 k, ℓ )) . • If the clone of
Aut( B ) -canonical polymorphisms of A contains pseudo-totally symmetricoperations modulo Aut( B ) of all arities, then A has relational width ( k, max( k + 1 , ℓ )) .Proof. Suppose that the assumption of the first item of Theorem 1.2 is satisfied. Let I be a non-trivial (2 k, max(3 k, ℓ ))-minimal instance of CSP( A ), and let I Aut( B ) ,k be the associated instanceof Definition 3.1. By Lemma 3.2, I Aut( B ) ,k is a (2 , I Aut( B ) ,k are preserved by WNUs of all arities m ≥ I Aut( B ) ,k admits a solution and since I is ( k, max(3 k, ℓ ))-minimal, this solutionlifts to a solution of I by Lemma 3.3. Thus, A has width (2 k, max(3 k, ℓ )). ANTOINE MOTTET, TOM´AˇS NAGY, MICHAEL PINSKER, AND MICHA L WRONA
Suppose now that the assumption in the second item is satisfied. By the same reasoning butusing Theorem 2.4 instead of Theorem 2.3, given a ( k, max( k + 1 , ℓ ))-minimal instance I , theassociated instance I Aut( B ) ,k is (1 , I is ( k, max( k +1 , ℓ ))-minimal, this solution lifts to a solution of I . (cid:3) The following example shows that for some structures under consideration, the bound on rela-tional width provided by the first item of Theorem 1.2 is tight.
Example 3.5.
Let ( A ; E ) be the universal homogeneous graph with edge relation E , and let B := ( A ; E, N, =), where N := ( V \ E ) ∩6 =. Consider the first-order reduct A := ( A ; E, N, R = , R = )of B , where R = := { ( a, b, c, d ) ∈ V | E ( a, b ) ∧ E ( c, d ) or N ( a, b ) ∧ N ( c, d ) } and R = := { ( a, b, c, d ) ∈ V | E ( a, b ) ∧ N ( c, d ) or N ( a, b ) ∧ E ( c, d ) } .Observe that A is preserved by a canonical ternary operation g that is a pseudo-majority moduloAut( B ). Hence, by Theorem 1.2 implies that A has relational width (4 , I = ( { , . . . , } , { C , C } ) where C = { f ∈ A { , , , } | ( f (1) , f (2) , f (3) , f (4)) ∈ R = } , C = { f ∈ A { , , , } | ( f (1) , f (2) , f (3) , f (4)) ∈ R = } is non-trivial, (3 , l )-minimal for every l ≥ , J = ( { , . . . , } , { D , D , D } ) where D = { f ∈ A { , , , } | ( f (1) , f (2) , f (3) , f (4)) ∈ R = } , D = { f ∈ A { , , , } | ( f (3) , f (4) , f (5) , f (6)) ∈ R = } and D = { f ∈ A { , , , } | ( f (1) , f (2) , f (5) , f (6)) ∈ R = } is non-trivial and has no solution. It follows thatthe exact relational width of A is (4 , A New Loop Lemma for Smooth Approximations
We refine the algebraic theory of smooth approximations from [42]. Building on deep algebraicresults from [6] on finite idempotent algebras that are equationally non-trivial, we lift some of thetheory from binary symmetric relations to cyclic relations of arbitrary arity.4.1.
The loop lemma.Definition 4.1.
The linkedness congruence of a binary relation R ⊆ A × B is the equivalencerelation λ R on proj (2) ( R ) defined by ( b, b ′ ) ∈ λ R iff there are k ≥ a , . . . , a k − ∈ A and b = b , . . . , b k = b ′ ∈ B such that ( a i , b i ) ∈ R and ( a i , b i +1 ) ∈ R for all i ∈ { , . . . , k − } . We saythat R is linked if it is non-empty and λ R relates any two elements of proj (2) ( R ).If A is a set and m ≥
2, then we call a relation R ⊆ A m cyclic if it is invariant under cyclicpermutations of the components of its tuples. The support of R is its projection on any argument.We apply the same terminology as above to any cyclic R , viewing R as a binary relation betweenproj (1 ,...,m − ( R ) and proj ( m ) ( R ).If R is invariant under an oligomorphic group action on A × B , then there is an upper boundon the length k to witness ( b, b ′ ) ∈ λ R , and therefore λ R is pp-definable from R ; in particular, itis invariant under any function clone acting on A × B and preserving R . Definition 4.2.
Let G be a permutation group on a set A . A pseudo-loop with respect to G is atuple of elements of A all of whose components belong to the same G -orbit [43, 9]. If G containsonly the identity function, then a pseudo-loop is called a loop .The proof of the following is essentially the proof of Theorem 4.2 in [6]. Since our statement isdifferent, we provide the proof for completeness. Theorem 4.3 (Consequence of the proof of Theorem 4.2 in [6]) . Let C be an idempotent functionclone on a finite domain that is equationally non-trivial. Then any C -invariant cyclic linkedrelation on its domain contains a loop.Proof. Let R as in the statement be given, and denote its arity by m ; we may assume m ≥ ≤ i ≤ m , we set R i := proj (1 ,...,i ) ( R ); moreover, we set R ( i,j ) := proj ( i,j ) ( R ) for all1 ≤ i, j ≤ m .We denote the support of R by A . Note that for all i ∈ { , . . . , m − } we have that R i +1 is linked when viewed as a binary relation between R i and A . We give the short and obvious MOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES 9 argument showing linkedness for the convenience of the reader. Let a, b ∈ A be arbitrary; sincethey are linked in R m , there exist k ≥ a = c , . . . , c k = b such that ( c j +1 , c j ) ∈ R m and ( c j +1 , c j +2 ) ∈ R m for all 0 ≤ j < k . For all such j , we have c j +1 ∈ R m − ; moreover,proj ( ℓ,...,m − (( c j +1 )) ∈ R m − ℓ for all 1 ≤ ℓ ≤ m − R . Hence, these elementsprove linkedness of a, b in R i +1 for all 1 ≤ i ≤ m − (1 ,i ) ( R ) is linked for all i ∈ { , . . . , m } .For an algebra C and a subset S of its domain we write S ⊳⊳ C if S is a minimal absorbingsubuniverse of C . Let A be the algebra associated with the clone C acting on A , and let R bethe subalgebra of A m with domain R . Similarly, we write R i for the subalgebra of A i on thedomain R i , for all 1 ≤ i ≤ m . Following the proof of Theorem 4.2 in [6], we prove by inductionon i ∈ { , . . . , m } that the following properties hold:(1) There exists I ⊳⊳ A such that I i ⊳⊳ R i .(2) For all I , . . . , I i ⊳⊳ A such that R i ∩ ( I × · · · × I i ) = ∅ , we have R i ∩ ( I × · · · × I i ) ⊳⊳ R i .The case i = 1 is trivial, since R = A . The case i = m gives us a constant tuple in R .We prove property (2) for i + 1. Let I , . . . , I i +1 ⊳⊳ A be such that R i +1 ∩ ( I × · · · × I i +1 ) = ∅ .By the induction hypothesis, R i ∩ ( I × · · · × I i ) ⊳⊳ R i . By the argument above, R i +1 is linked asa relation between R i and A . Thus, by elementary facts about minimal absorbing subuniverses,we have R i +1 ∩ ( I × · · · × I i +1 ) ⊳⊳ R i +1 .We prove property (1) for i + 1. Define a directed graph H on R i by setting H := { (( a , . . . , a i ) , ( a , . . . , a i +1 )) | ( a , . . . , a i +1 ) ∈ R i +1 } . Let I ⊳⊳ A be such that I i ⊳⊳ R i , which exists by the induction hypothesis. We show that: • I i is a subset of a (weak) connected component of H , • this connected component has algebraic length 1.For the first item, let X := { x | ∃ a , . . . , a i ∈ I (( a , . . . , a i , x ) ∈ R i +1 ) } , which is an absorbingsubuniverse of A . Let X ⊆ X be a minimal absorbing subuniverse of A . Then since ( I i × X ) ∩ R i +1 = ∅ property (2) gives us ( I i × X ) ⊆ R i +1 . Reiterating this idea, we find minimal absorbingsubuniverses X , . . . , X i of A such that for all 1 ≤ j ≤ i we have that I i − j +1 × X × · · · × X j is contained in R i +1 . Now pick an arbitrary tuple ( a , . . . , a i ) ∈ I i , and an arbitrary tuple( x , . . . , x i ) ∈ X × · · · × X i . Then there is a path in H from ( a , . . . , a i ) to ( x , . . . , x i ) by theabove, proving the first item.For the second item, let E := { ( x, y ) | ∃ c , . . . , c i ∈ I (( x, c , . . . , c i , y ) ∈ R i +1 ) } . Let V , V be the projection of E onto its first and second coordinate, respectively; then E is arelation between V and V . We have that E is an absorbing subuniverse of the algebra R (1 ,i +1) induced by R (1 ,i +1) in A . By assumption on R , we have that R (1 ,i +1) is linked. Therefore, E is linked, and V and V are absorbing subuniverses of A . Note that I ⊆ V and I ⊆ V . Let b ∈ I be arbitrary. Then there exist k ≥ c , . . . , c k +1 such that ( c j , c j +1 ) ∈ E for all0 ≤ j ≤ k and ( c j +2 , c j +1 ) ∈ E for all 0 ≤ j ≤ k − c = b = c k +1 . Theseelements can be assumed to lie in minimal absorbing subuniverses of A by the general theory ofabsorption. We then have, by property (2), ( c j , b, . . . , b, c j +1 ) ∈ R i +1 for all 0 ≤ j ≤ k and( c j +2 , b, . . . , b, c j +1 ) ∈ R i +1 for all 0 ≤ j ≤ k −
1. This gives a path of algebraic length 1 in H from ( b, . . . , b ) to itself. This proves the second item.By Theorem 3.6 in [6] there exists a loop in H which lies in a minimal absorbing subuniverse K of R i ; by the definition of H , this loop is a constant tuple ( a, . . . , a ). By projecting K on thefirst component, we obtain a minimal absorbing subuniverse J ⊳⊳ A ; since a ∈ J , we have that J i +1 ∩ R i +1 = ∅ . By property (2), we get that J i +1 ⊳⊳ R i +1 , so that property (1) holds. (cid:3) The following is a generalization of [42, Theorem 10] from binary symmetric relations to arbi-trary cyclic relations.
Proposition 4.4.
Let n ≥ , and let D be an oligomorphic function clone on a set A which is amodel-complete core. Let C ⊆ D can n be such that C n / G D is equationally non-trivial. Let ( S, ∼ ) be a minimal subfactor of the action C n with G D -invariant ∼ -classes. Then for every D -invariantcyclic relation R with support h S i D one of the following holds: (1) The linkedness congruence of R is a D -invariant approximation of ∼ . (2) R contains a pseudo-loop with respect to G D .Proof. Let R be given, and denote its arity by m . Assuming that (1) does not hold, we prove (2).Denote by O the set of orbits of n -tuples under the action of G D thereon. Let R ′ be the relationobtained by considering R as a relation on O , i.e., R ′ := { ( O , . . . , O m ) ∈ O m | R ∩ ( O × · · · × O m ) = ∅} . Thus, R ′ is an m -ary cyclic relation with support S ′ ⊆ O , and R ′ contains a loop if and only if R satisfies (2).By assumption, the action C n / G D is equationally non-trivial; moreover, it is idempotent since D is a model-complete core. Note also that R ′ , and in particular S ′ , are preserved by this action.It is therefore sufficient to show that R ′ is linked and apply Theorem 4.3.Recall that we consider R also as a binary relation between proj m − ( R ) and h S i D ; similary,we consider R ′ as a binary relation between proj m − ( R ′ ) and S ′ . By the oligomorphicity of D ,the linkedness congruence λ R of R is invariant under D . By our assumption that (1) does nothold, there exist c, d ∈ S which are not ∼ -equivalent and such that λ R ( c, d ) holds; otherwise, λ R would be an approximation of ∼ . This implies that the orbits O c , O d of c, d are related via λ R ′ .By the minimality of ( S, ∼ ), we have that h S i D = h{ c, d }i D . Since D is a model-complete core, itpreserves the G D -orbits, and it follows that any tuple in h S i D = h{ c, d }i D is λ R -related to a tuplein the orbit of c . Hence, λ R ′ = ( S ′ ) , and thus R ′ is linked. Theorem 4.3 therefore implies that R ′ contains a loop, and hence R contains a pseudo-loop with respect to G D , which is what we hadto show. (cid:3) Upgrading approximations.
The approximations provided by Proposition 4.4 are usefulin applications only if they can be made smooth at least. We make some trivial observations whenthis is the case, which will however be useless for unary structures because of the algebraicity.
Definition 4.5.
Let n, m ≥
1, let A be a set, let G be a permutation group acting on A , andlet R be an m -ary cyclic relation on A n . Any sequence of tuples in R linking, via λ R , two tuples b, c ∈ A n which are disjoint and in the same G -orbit is called a G -fork of R . The elements b, c arethen called the tips of the fork.Following the terminology from [42], an approximation η of ∼ is tasteless with respect to G if each ∼ -equivalence class C intersects an η -equivalence class C ′ such that C ∩ C ′ contains twodisjoint tuples in the same orbit under G . Lemma 4.6.
Let n ≥ , let A be a set, let G be a permutation group acting on A , and let R be acyclic relation on A n . If λ R is an approximation of an equivalence relation ∼ , and if R has two G -forks which have their respective tips in different ∼ -classes, then the approximation is tastelesswith respect to G .Proof. Trivial. (cid:3)
A group G acting on A has no algebraicity if for every k ≥ a , . . . , a k ∈ A , the finiteorbits of the stabiliser of G by a , . . . , a k are precisely { a } , . . . , { a k } . We now observe that adisjoint tuple plus no algebraicity gives us a fork. Definition 4.7.
Let n, m ≥
1. An m -tuple of n -tuples is disjoint if it is injective as an nm -tuple. Lemma 4.8.
Let n ≥ , let A be a set, let G be a permutation group acting on A , and let R be acyclic relation on A n . If G has no algebraicity, then any element which belongs to a disjoint tuplein R belongs to the tip of a G -fork.Proof. Trivial. (cid:3)
The following is a generalization of Lemma 12 in [42] from binary relations and functions torelations and functions of higher arity.
MOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES 11
Lemma 4.9.
Let n ≥ , and let D be an oligomorphic polymorphism clone on a set A that is amodel-complete core. Let ∼ be an equivalence relation on a set S ⊆ A n with G D -invariant classes.Let m ≥ , and let P be an m -ary relation on h S i D . Suppose that every m -ary D -invariant cyclicrelation R on h S i D which contains a tuple in P with components in at least two ∼ -classes containsa pseudo-loop with respect to G D .Then there exists an m -ary f ∈ D such that for all a , . . . , a m ∈ A n we have that if the tuple ( f ( a , . . . , a m ) , f ( a , . . . , a m , a ) , . . . , f ( a m , a , . . . , a m − )) is in P , then it intersects at most one ∼ -class.Proof. The proof is similar to the proof of Lemma 12 in [42]. Fix m ≥
2. Call an m -tuple( a , . . . , a m ) of elements of A n troublesome if there exists r ∈ D such that( b , . . . , b m ) := ( r ( a , . . . , a m ) , r ( a , . . . , a m , a ) , . . . , r ( a m , a , . . . , a m − ))is in P and has components in at least two ∼ -classes. For each such troublesome tuple, the smallest D -invariant relation containing the set { ( b , . . . , b m ) , ( b , . . . , b m , b ) , . . . , ( b m , b , . . . , b m − ) } is cyclic, contains a tuple in P with components in at least two ∼ -classes, and its support iscontained in h S i D . Hence, it contains a pseudo-loop with respect to G D by our assumptions. Thisimplies that there exists g ∈ D such that the entries of the tuple( g ( b , . . . , b m ) , . . . , g ( b m , b , . . . , b m − ))all belong to the same G D -orbit. The function h ( x , . . . , x m ) := g ( r ( x , . . . , x m ) , . . . , r ( x m , x , . . . , x m − ))thus has the property that( h ( a , . . . , a m ) , h ( a , . . . , a m , a ) , . . . , h ( a m , a , . . . , a m − ))all lie in the same G D -orbit. A standard compactness argument repeating this argument for alltroublesome tuples (see [42, Lemma 12]) yields f satisfying the required property; it is here thatwe need the assumption that D is a polymorphism clone, i.e., topologically closed. (cid:3) Applications: Collapses of the bounded width hierachies for some classes ofinfinite structures
We now apply the algebraic results of Section 4 and the theory of smooth approximations toobtain a characterisation of bounded width for CSPs of first-order reducts of unary structures( k = 2 , ℓ = 2) and for CSPs in MMSNP (where k and ℓ are arbitrarily large). Moreover, theresults of Section 3 then imply a collapse of the bounded width hierarchy for such CSPs.5.1. Unary Structures.
We will use the following fact which states that Pol( A ) locally interpo-lates Pol( A ) can . Lemma 5.1 (Proposition 6.5 in [17]) . Let A be a first-order expansion of a stabilized partition ( N ; V , . . . , V r ) . For every f ∈ Pol( A ) there exists g ∈ Pol( A ) can which is locally interpolated by f modulo Aut( A ) . We now show that in the presence of the functions of the preceding lemma, if Pol( A ) can isequationally affine, then so is its action on N / Aut( A ). Essentially the same result was obtainedin [17, Proposition 6.6] for the equationally trivial case. Proposition 5.2 (Proposition 6.6 in [17]) . Let A be a first-order expansion of a stabilized partition ( N ; V , . . . , V r ) , and assume it is a model-complete core. Suppose that Pol( A ) contains a binaryoperation whose restriction to V i is injective for all ≤ i ≤ r . Then the following are equivalent: • Pol( A ) can is equationally affine; • Pol( A ) can y N / Aut( A ) is equationally affine. Proof.
Trivially, if Pol( A ) can y N / Aut( A ) is equationally affine, then so is Pol( A ) can .For the other direction, assume that Pol( A ) can y N / Aut( A ) is not equationally affine. Thenfor almost all k ≥
3, Pol( A ) can contains a k -ary operation g k whose action on N / Aut( A ) is a WNUoperation. Fix for all such k ≥ f k of arity k whose restriction to V i is injective forall i ∈ { , . . . , r } , and consider the operation h k ( x , . . . , x k ) := f k ( g k ( x , . . . , x k ) , g k ( x , . . . , x k , x ) , . . . , g k ( x k , x , . . . , x k − )) . Then evaluating h ( a, b, . . . , b ) for arbitrary a, b ∈ N , all the arguments of f k belong to the sameset V i , by the fact that g k acts on N / Aut( A ) as a weak near-unanimity operation. Since A is anexpansion of ( N ; V , . . . , V r ), we obtain that h k ( a, b, . . . , b ) belongs to V i . The same is true forany permutation of the tuple ( a, b, . . . , b ), so that h k acts as a WNU operation on N / Aut( A ). Bythe injectivity of f k when restricted to V i , it also follows that h k acts as a WNU operation on N / Aut( A ), and hence it is a pseudo-WNU operation. Canonising these operations, we see thatPol( A ) can contains pseudo-WNU operations of all arities ≥
3, and hence it is not equationallyaffine. (cid:3)
The following will allow us to assume, in most proofs, the presence of functions in Pol( A ) whichare injective on every set in the stabilized partition. This is the analogue to the efforts to obtainbinary injections in [42]. Lemma 5.3 (Subset of the proof of Proposition 6.6 [17]) . Let A be a first-order expansion ofa stabilized partition ( N ; V , . . . , V r ) , and assume it is a model-complete core. If Pol( A ) has nocontinuous clone homomorphism to P , then it contains operations of all arities whose restrictionsto V i are injective for all ≤ i ≤ r .Proof. We show by induction that for all 1 ≤ j ≤ r , there exists a binary operation in Pol( A )whose restriction to each of V , . . . , V j is injective; higher arity functions with the same propertyare then obtained by nesting the binary operation. For the base case j = 1, observe that thedisequality relation = is preserved on V since A is a model-complete core; together with therestriction of Pol( A ) to V being equationally non-trivial, we then obtain an operation which actsas as an essential function on V . This in turn easily yields a function that acts as a binary injectionon V – see e.g. [13]. For the induction step, assuming the statement holds for 1 ≤ j < r , we showthe same for j + 1. By the induction hypothesis, there exist binary functions f, g ∈ Pol( A ) suchthat the restriction of f to each of the sets V , . . . , V j is injective, and the restriction of g to V j +1 is injective. If the restriction of f to V j +1 depends only on its first or only on its second variable,then it is injective in that variable since disequality is preserved on V j +1 , and hence either thefunction f ( g ( x, y ) , f ( x, y )) or the function f ( f ( x, y ) , g ( x, y )) has the desired property. If on theother hand the restriction of f to V j +1 depends on both variables, then the same argument as inthe base case yields a function which is injective on V j +1 , and this function is still injective oneach of the sets V , . . . , V j . (cid:3) Theorem 5.4.
Let A be a first-order reduct of a unary structure, and assume that A is a model-complete core. Then one of the following holds: • Pol( A ) can is not equationally affine, or equivalently, it contains pseudo-WNUs modulo Aut( A ) of all arities n ≥ ; • Pol( A ) has a uniformly continuous minion homomorphism to an affine clone. In the first case, A has relational width (4 ,
6) by Theorem 1.2, and in the second case it doesnot have bounded width by results from [21, 39]. Theorem 5.4 gives a characterization of boundedwidth for all first-order reducts of unary structures, since this class is closed under taking model-complete cores by Lemma 6.7 in [17].The two items of Theorem 5.4 are invariant under expansions of A by a finite number ofconstants. Thus, by Proposition 6.8 in [17], one can assume that A is a first-order expansion of( N ; V , . . . , V r ) where V , . . . , V r form a partition of N in which every set is either a singleton orinfinite. Such partitions were called stabilized partitions in [17], and we shall also call the structure( N ; V , . . . , V r ) a stabilized partition. MOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES 13
Proof of Theorem 5.4.
Let A as in Theorem 5.4 be given; by the remark preceding this proof, wemay without loss of generality assume that A is a first-order expansion of a stabilized partition( N ; V , . . . , V r ). Assume henceforth that Pol( A ) can is equationally affine; we show that Pol( A ) hasa uniformly continuous minion homomorphism to an affine clone.If Pol( A ) has a continuous clone homomorphism to P , then we are done. Assume therefore thecontrary; then by Lemma 5.3, Pol( A ) contains for all k ≥ k -ary operation whose restriction to V i is injective for all 1 ≤ i ≤ r . In particular, Proposition 5.2 applies, and thus Pol( A ) can y N / Aut( A )is equationally affine. Let ( S, ∼ ) be a minimal subfactor of Pol( A ) can such that Pol( A ) can acts onthe ∼ -classes as an affine clone; the fact that this exists is well-known (see, e.g., Proposition 3.1in [44]).Let R be any Pol( A )-invariant cyclic relation with support h S i Pol( A ) , containing a tuple withcomponents in pairwise distinct Aut( A )-orbits and which intersects at least two ∼ -classes. ByProposition 4.4, R either gives rise to a Pol( A )-invariant approximation of ∼ , or it contains apseudo-loop with respect to Aut( A ). In the first case, the presence of the tuple required aboveimplies smoothness of the approximation: if t ∈ R is such a tuple, c ∈ S appears in t , and d ∈ S belongs to the same Aut( A )-orbit as c , then there exists an element of Aut( A ) which sends c to d and fixes all other elements of t . Hence, c and d are linked in R , and the entire Aut( A )-orbit of c is contained in a class of the linkedness relation of R . Thus, Pol( A ) admits a uniformly continuousminion homomorphism to an affine clone by Theorem 2.6.Hence we may assume that for any R as above the second case holds. We are now goingto show that this leads to a contradiction, finishing the proof of Theorem 5.4. By Lemma 4.9applied with any m ≥ P the set of m -tuples with entries in pairwise distinct Aut( A )-orbits within h S i Pol( A ) , we obtain an m -ary function f ∈ Pol( A ) with the property that the tuple( f ( a , . . . , a m − ) , . . . , f ( a , . . . , a m − , a )) intersects at most one ∼ -class whenever it has entriesin pairwise distinct Aut( A )-orbits, for all a , . . . , a m − ∈ S . Let ( A , < ) be the expansion of A by alinear order that is convex with respect to the partition V , . . . , V r and dense and without endpointson every infinite set of the partition. The structure ( A , < ) can be seen to be a Ramsey structure ,since Aut( A , < ) is isomorphic as a permutation group to the action of the product Q ri =1 Aut( V i ; < ),and each of the groups of the product is either trivial or the automorphism group of a Ramseystructure [36]. By diagonal interpolation we may assume that f is diagonally canonical with respectto Aut( A , < ). Let a, a ′ ∈ A m be so that a i , a ′ i belong to the same orbit with respect to Aut( A ) forall 1 ≤ i ≤ m . Then there exists α ∈ Aut( A , < ) such that α ( a ) = α ( a ′ ), and hence f ( a ) and f ( a ′ )lie in the same Aut( A )-orbit by diagonal canonicity; hence f is 1-canonical with respect to Aut( A ).Applying Lemma 5.1, we obtain a canonical function g ∈ Pol( A ) can which acts like f on N / Aut( A ).The property of f stated above then implies for g that g ( a , . . . , a m − ) ∼ g ( a , . . . , a m − , a ) for all a , . . . , a m − ∈ S such that the values g ( a , . . . , a m − ) , . . . , g ( a m − , a , . . . , a m − ) lie in pairwisedistinct Aut( A )-orbits.By the choice of ( S, ∼ ) we have that Pol( A ) can acts on S/ ∼ by affine functions over a fi-nite module. We use the symbols + , · for the addition and multiplication in the correspondingring, and also + for the addition in the module and · for multiplication of elements of the mod-ule with elements of the ring. We denote by 1 the multiplicative identity of the ring, by − S/ ∼ , and we denote the identity element of its additive group by[ a ] ∼ . Pick an arbitrary element [ a ] ∼ = [ a ] ∼ from S/ ∼ , and let m ≥ m · [ a ] ∼ = [ a ] ∼ .Let g ∈ Pol( A ) can be the m -ary operation obtained in the preceding paragraph. If the values g ( a , . . . , a m − ) , . . . , g ( a m − , a , . . . , a m − ) lie in pairwise distinct Aut( A )-orbits, then computingindices modulo m we have that g ([ a ] ∼ , . . . , [ a m − ] ∼ ) , . . . , g ([ a m − ] ∼ , . . . , [ a m + m − ] ∼ ) are all equal.If on the other hand they do not, then g ([ a k ] ∼ , . . . , [ a k + m − ] ∼ ) = g ([ a k + j ] ∼ , . . . , [ a k + j + m − ] ∼ )for some 0 ≤ k < m and 1 ≤ j < m . Hence, in either case we may assume the latterequation holds. By assumption, g acts on S/ ∼ as an affine map, i.e., as a map of the form( x , . . . , x m − ) P m − i =0 c i · x i , where c , . . . , c m − are elements of the ring which sum up to 1. We compute (with indices to be read modulo m )[ a ] ∼ = g ([ a k + j ] ∼ , . . . , [ a k + j + m − ] ∼ ) + ( − · g ([ a k ] ∼ , . . . , [ a k + m − ] ∼ )= m − X i =0 c i · [ a k + j + i ] ∼ + ( − · m − X i =0 c i · [ a k + i ] ∼ = m − X i =0 c i · ( k + i + j ) · [ a ] ∼ + ( − · m − X i =0 c i · ( k + i ) · [ a ] ∼ = m − X i =0 c i ! · j · [ a ] ∼ = j · [ a ] ∼ . But j · [ a ] ∼ = [ a ] ∼ since the order of [ a ] ∼ equals m > j , a contradiction. (cid:3) MMSNP.
MMSNP is a fragment of existential second order logic that was discovered byFeder and Vardi in their seminal paper [32]. We prefer not to define the syntax of MMSNP, andrather define it using a correspondence between MMSNP sentences and certain coloring problems.We refer to [14] for a precise definition of all the terms employed here.Let τ be a relational signature, let σ be a unary signature whose relations are called the colors ,and let F be a finite set of finite connected ( τ ∪ σ )-structures whose vertices have exactly onecolor. We call F a colored obstruction set in the following. The problem FPP( F ) takes as inputa τ -structure G and asks whether there exists a ( τ ∪ σ )-expansion G ∗ of G whose vertices are allcolored with exactly one color and such that for every F ∈ F , there exists no homomorphism from F to G ∗ . The connection between MMSNP and FPP is shown in [40, Corollary 3.7]: every MMSNPsentence Φ is equivalent to a union FPP( F ) ∪ · · · ∪ FPP( F p ), in the sense that a τ -structure G satisfies Φ iff it is a yes-instance for one of the problems FPP( F i ). We say Φ is connected if it isequivalent to a single FPP( F ).Every set F as above has a strong normal form G such that FPP( F ) = FPP( G ). We say F is precolored if for every symbol M ∈ σ , there is an associated unary symbol P M ∈ τ , and moreoverif F contains for every M = M ′ a 1-element structure whose vertex belongs to P M and M ′ . Every F has a standard precoloration , obtained by enlarging τ with the necessary symbols and enlarging F with the associated obstructions.It was shown in [14, Definition 4.3] that for every set F in strong normal form, there exists an ω -categorical τ -structure A F such that for any finite τ -structure B , B is a yes-instance of FPP( F )iff there exists an injective homomorphism from B to A F , and such that: • If F is precolored, then the orbits of the elements of A F under Aut( A F ) correspond tothe colors of F and to the corresponding predicates in τ . In particular, the action ofPol( A F ) can1 on Aut( A F )-orbits of elements is idempotent [14, Proposition 7.1]. • Every f ∈ Pol( A F ) locally interpolates an operation g ∈ Pol( A F ) can1 , and there exists alinear order < on A F such that every f diagonally interpolates an operation f ′ that isdiagonally canonical with respect to Aut( A F , < ).The precise definition of A F is not needed for the arguments below, we only need to recall thefollowing proposition (proved in the case m = 2 in [14], since only small alterations to the proofare needed to prove the more general version, we omit it). Proposition 5.5.
Let F be a precolored obstruction set and in normal form. Let B have ahomomorphism to A F and let m ≥ . There exists an embedding e of { , . . . , m } × B , the disjointunion of m copies of B , into A F such that ( e ( i , a ) , . . . , e ( i m , a m )) and ( e ( j , b ) , . . . , e ( j m , b m )) are in the same orbit under Aut( A F , < ) provided that: • a k and b k are in the same color for all k ∈ { , . . . , m }• a k and a ℓ are in distinct colors for all k = ℓ , • { i , . . . , i m } = { j , . . . , j m } = { , . . . , m } . MOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES 15
We finally solve the Datalog-rewritability problem for MMSNP and prove that a connectedsentence Φ is equivalent to a Datalog-program iff the action of Pol( A F ) can1 on Aut( A F )-orbits ofelements is not equationally affine, where F is any strong normal form for Φ.The following proposition shows that for the question of Datalog-rewritability, one can reduceto the precolored case without loss of generality. The same proposition was shown in [14] for theP/NP-complete dichotomy, with P replacing affine clones in the statement. Proposition 5.6.
Let F be a colored obstruction set in strong normal form and let G be itsstandard precoloration. There is a uniformly continuous minion homomorphism from Pol( A G ) toan affine clone if and only if there is a uniformly continuous minion homomorphism from Pol( A F ) to an affine clone.Proof. It is shown in [14] that Pol( A G ) has a uniformly continuous minion homomorphism toPol( A F ) and that Pol( A F , =) has a uniformly continuous minion homomorphism to Pol( A G ).Thus, it suffices to show that if Pol( A F , =) has a uniformly continuous minion homomorphism toan affine clone, then so does Pol( A F ).Let p ≥ R and R be the relations defined by { ( x, y, z ) ∈ Z p | x + y + z = i mod p } for i ∈ { , } . For an arbitrary ω -categorical structure B , it is known that the existenceof a uniformly continuous minion homomorphism Pol( B ) to an affine clone is equivalent to theexistence of a p such that the relational structure ( Z p ; R , R ) has a pp-construction in B .Suppose that ( Z p ; R , R ) has a pp-construction in ( A Φ , =). Thus, there is n ≥ φ ( x , y , z ) , φ ( x , y , z ) defining relations S , S such that ( A n ; S , S ) and ( Z p ; R , R ) arehomomorphically equivalent; we take n to be minimal with the property that such pp-formulasexist. Since R and R are totally symmetric relations (i.e., the order of the entries in a tupledo not affect its membership into any of R or R ), we can assume that S and S are, too, andthat the formulas pp-defining them are syntactically invariant under permutation of the block ofvariables x , y , and z .We first claim that φ i does not contain any equality atom or any inequality atom x j = y j for j ∈{ , . . . , n } (so that by symmetry, also y j = z j and x j = z j do not appear). Let h : ( Z p ; R , R ) → ( A n ; S , S ) be a homomorphism. Since (0 , , ∈ R , we have that ( h (0) , h (0) , h (0)) satisfies φ ,and therefore the listed inequality atoms cannot appear. The same holds for φ , by considering( h (0) , h (0) , h (1)) and its permutations.In order to rule out equalities, we proceed as in [14]. Suppose that φ contains x i = x j for i = j .Then the entries i and j of h ( q ) are equal, for any q ∈ { , . . . , p − } , since every q belongs to thesupport of R . Thus, one can also add x i = x j to φ , since the structure defined by the modifiedformula still admits a homomorphism from ( Z p ; R , R ). By existentially quantifying x j , y j , z j in φ and φ , one obtains a pp-construction of some ( A n − ; S ′ , S ′ ) that is still homomorphicallyequivalent to ( Z p ; R , R ), a contradiction to the minimality of n . If φ contains x i = y j for j = i ,then it also contains y j = z i and z i = x j since we enforced that φ is syntactically symmetric. Bytransitivity, we obtain that x i = x j is implied by φ and we are back in the first case. Suppose nowthat φ contains x i = y i . Then the i th entry of h (0) and h ( q ) are equal, for all q ∈ { , . . . , p − } ,since for all q there exists r such that (0 , q, r ) ∈ R . Thus we can again reduce n by fixing the i thcoordinate.Let ψ i be the formula obtained from φ i by removing the possible inequality literals, and let T i be defined by ψ i in A F . We claim that ( A n ; T , T ) and ( A n ; S , S ) are homomorphicallyequivalent, which concludes the proof. Since φ i implies ψ i , we have that ( A n ; S , S ) is a (non-induced) substructure of ( A n ; T , T ), and therefore it homomorphically maps to ( A n ; T , T ) bythe identity map. For the other direction, we prove the result by compactness and show that everyfinite substructure B of ( A n ; T , T ) has a homomorphism to ( A n ; S , S ). Let b , . . . , b m be theelements of B . Let C be the τ -structure over precisely n · m elements { c ij | i, j } corresponding tothe entries of b ij , whose relations are pulled back from A Φ under the map π : c ij b ij . Note thatno structure from F has a homomorphism to C (otherwise, we would obtain a homomorphismto A F by composition with π ), and thus it admits an injective homomorphism g to A F . Weclaim that if ( b i , b j , b k ) ∈ T then ( g ( c i ) , g ( c j ) , g ( c k )) ∈ S . Indeed, suppose that ( b i , b j , b k )satisfies ψ . Then by construction ( g ( c i ) , g ( c j ) , g ( c k )) satisfies ψ . Moreover, by injectivity of g , we have g ( c ir ) = g ( c js ) as long as i = j or r = s . Consider any inequality atom in φ . By ourfirst claim, it is not of the form x r = y r , and therefore it is satisfied by ( g ( c i ) , g ( c j ) , g ( c k )). Thus,( g ( c i ) , g ( c j ) , g ( c k )) satisfies φ . The same reasoning for φ shows that g induces a homomorphism B → ( A n ; S , S ) by mapping b i to g ( c i ). (cid:3) The following theorem gives a characterization of Datalog-rewritability in terms of precolorednormal forms. The proof is similar to that of Theorem 5.4.
Theorem 5.7.
Let Φ be a connected MMSNP τ -sentence, let F be an equivalent colored obstruc-tion set and suppose that F is precolored and in strong normal form. The following are equivalent: (1) ¬ Φ is equivalent to a Datalog program; (2) Pol( A F ) does not have a uniformly continuous minion homomorphism to an affine clone; (3) The action of
Pol( A F ) can1 on Aut( A F ) -orbits of elements is not equationally affine; (4) A F has relational width ( k, max( k +1 , ℓ )) , where k and ℓ are such that A F is k -homogeneous ℓ -bounded.Proof. (1) implies (2) by general principles [39, 8].(2) implies (3). We do the proof by contraposition. The proof is essentially the same as in thecase of reducts of unary structures (Theorem 5.4). Suppose that Pol( A F ) can1 y A F / Aut( A F ) isequationally affine and let ( S, ∼ ) be a minimal module for this action.Let m ≥ R be an m -ary cyclic relation invariant under Pol( A F ) and containing atuple ( a , . . . , a m ) whose entries are pairwise distinct. By Proposition 4.4, either the linkednesscongruence of R defines an approximation of ∼ , or R contains a pseudoloop modulo Aut( A F ).In the first case, the approximation is smooth and we obtain a uniformly continuous minionhomomorphism from Pol( A F ) to a clone of affine maps. Any such clone admits a uniformlycontinuous minion homomorphism to Z p for some p , and by composition this gives us a uniformlycontinuous minion homomorphism Pol( A F ) → Z p .So let us assume that for all m ≥
2, every such relation R contains a pseudoloop. By applyingLemma 4.9, we obtain a polymorphism f such that for all a , . . . , a m , if f ( a , . . . , a m ) , . . . , f ( a m , a , . . . , a m − )are pairwise distinct, then they intersect at most one ∼ -class. As in the proof of Theorem 5.4,pick an arbitrary a ∈ S such that [ a ] ∼ is not the zero element of the module S/ ∼ . Let m ≥ O i be the orbit of i · [ a ] ∼ , for i ∈ { , , . . . , m − } . By Lemma 5.8, we obtain g ∈ Pol( A F ) can1 such that g ( O k , . . . , O k + m − ) ∼ g ( O j + k , . . . , O j + k + m − )for some k ∈ { , . . . , m − } and j ∈ { , . . . , m − } . The same computation as in Theorem 5.4then gives a contradiction and concludes the proof.(3) implies (4). First, note that A F is infinite, and therefore k ≥
2. Let I be a non-trivial( k, max( k + 1 , ℓ ))-minimal instance of A F . Let G be Aut( A F ). Consider the instance I G , as inDefinition 3.1. Thus, the variables of I G , are the same as the variables of I (up to the naturalbijection between V and (cid:0) V (cid:1) ) and the values for the variables are taken from the set of colors of F . By Lemma 3.2, I G , is (2 , h . Note that we cannot use Lemma 3.3 to obtain a solution to I , since we only considered I G , . Let B be the τ -structure described by I (i.e., B is the canonical database of I ). Let B ∗ be the ( τ ∪ σ )-expansion of B obtained by coloring the vertices of B according to h . Since I is( k, ℓ )-minimal, it can be seen that B ∗ does not contain any homomorphic copy of F ∈ F , so that B admits a homomorphism to A F , i.e., I has a solution in A F .(4) implies (1). Trivial. (cid:3) Lemma 5.8.
Let ( S, ∼ ) be a subfactor of Pol( A F ) can1 with Aut( A F ) -invariant ∼ -classes. Let m ≥ , and let f ∈ Pol( A F ) be as in Lemma 4.9: for all a , . . . , a m ∈ A F we have that if the entries ofthe tuple ( f ( a , . . . , a m ) , f ( a , . . . , a m , a ) , . . . , f ( a m , a , . . . , a m − )) all belong to different colors,then it intersects at most one ∼ -class. Let O , . . . , O m − ∈ S be pairwise distinct orbits under Aut( A F ) . There exists g ∈ Pol( A F ) can1 that is locally interpolated by f and that satisfies ( ⋆ ) g ( O k , . . . , O k + m − ) ∼ g ( O j + k , . . . , O j + k + m − ) MOOTH APPROXIMATIONS AND RELATIONAL WIDTH COLLAPSES 17 for some ≤ k < m and ≤ j < m .Proof. Recall that the expansion of A F by a generic linear order is a Ramsey structure [14]. Thus, f diagonally interpolates a function g ∈ Pol( A F ) with the same properties and which is diagonallycanonical with respect to Aut( A F , < ), and without loss of generality we can therefore assume that f is itself diagonally canonical.Let B := { , . . . , m − } × A F be the disjoint union of m copies of A F and let e be an embeddingof { , . . . , m − } × B into A Φ with the properties stated in Proposition 5.5. Let e i ( x ) := e ( i, x ),which is a self-embedding of A F . Consider f ′ ( x , . . . , x m − ) := f ( e x , . . . , e m − x m − ), and notethat f ′ is 1-canonical when restricted to m -tuples where all entries are in pairwise distinct orbits.Let g be obtained by canonising f ′ with respect to Aut( A F , < ). In particular g ∈ Pol( A F ) can1 and g ( O k , . . . , O k + m − ) and f ′ ( O k , . . . , O k + m − ) are in S and ∼ -equivalent for all k .As in the proof of Theorem 5.4, there are suitable 0 ≤ k < m and 1 ≤ j < m such that f ( e k O k , . . . , e k + m − O k + m − ) ∼ f ( e k + j O k + j , . . . , e k + j + m − O j + k + m − )holds, where indices are computed modulo m . Then g ( O k , . . . , O k + m − ) ∼ f ( e O k , . . . , e m − O k + m − ) ∼ f ( e k O k , . . . , e k + m − O k + m − )( ⋆ ) ∼ f ( e k + j O k + j , . . . , e k + j + m − O k + j + m − ) ∼ f ( e O k + j , . . . , e m − O k + j + m − )( ⋆ ) ∼ g ( O k + j , . . . , O k + j + m − ) , where the equivalences marked ( ⋆ ) hold by the fact that f is diagonally canonical with respect toAut( A F , < ) and by Proposition 5.5. (cid:3) Combining Proposition 5.6, Theorem 5.7, and known facts about MMSNP and normal forms [14],this allows us to obtain Theorem 1.1 from the introduction.
Theorem 1.1.
The Datalog-rewritability problem for MMSNP is decidable, and is 2NExpTime-complete.Proof.
Let Φ be an MMSNP sentence, which is equivalent to a disjunction Φ ∨· · ·∨ Φ p of connectedMMSNP sentences [14, Proposition 3.2]. Moreover, if p is minimal then ¬ Φ is equivalent toa Datalog program iff every ¬ Φ i is equivalent to a Datalog program (see, e.g., Proposition 3.3in [14], for a proof of a similar fact).By Theorem 4.3 in [14], one can compute for every Φ i a coloured obstruction set F i that is instrong normal form. Let G i be the standard precoloration of F i . By Proposition 5.6, one has auniformly continuous minion homomorphism from Pol( A G i ) to an affine clone iff one has one fromPol( A F i ) to an affine clone. Then, by Theorem 5.7, we get that deciding Datalog-rewritability for G i is equivalent to deciding whether Pol( A G i ) can1 y A G i / Aut( A G i ) is equationally non-affine, whichis known to be decidable in polynomial time since Pol( A G i ) can1 y A G i / Aut( A G i ) is idempotent.The computation of a strong normal form is costly and can be performed in 2-ExpSpace. Inorder to obtain a 2NExpTime-algorithm, we rather compute a normal form F i for Φ i (by Lemma3.1 in [14]), which can be done in doubly exponential-time. The consequence of not working witha strong normal form is that the clone Pol( A F i ) can1 y A F i / Aut( A F i ) is not a core; its core is theaction considered for the strong normal form. Deciding whether such a clone admits a minionhomomorphism to an affine clone is in NP [30, Corollary 6.8]. We obtain overall a 2NExpTimealgorithm. The complexity lower bound is Theorem 18 in [26]. (cid:3) References [1] Samson Abramsky, Anuj Dawar, and Pengming Wang. The pebbling comonad in finite model theory. In , pages 1–12. IEEE Computer Society, 2017. URL: https://doi.org/10.1109/LICS.2017.8005129 , doi:10.1109/LICS.2017.8005129 . [2] Albert Atserias, Andrei A. Bulatov, and V´ıctor Dalmau. On the power of k -consistency. In LarsArge, Christian Cachin, Tomasz Jurdzinski, and Andrzej Tarlecki, editors, Automata, Languages andProgramming, 34th International Colloquium, ICALP 2007, Wroclaw, Poland, July 9-13, 2007, Pro-ceedings , volume 4596 of
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Email address : [email protected] URL : Institut f¨ur Diskrete Mathematik und Geometrie, FG Algebra, TU Wien, Austria and Departmentof Algebra, Faculty of Mathematics and Physics, Charles University
Email address : [email protected] Institut f¨ur Diskrete Mathematik und Geometrie, FG Algebra, TU Wien, Austria, and Departmentof Algebra, Faculty of Mathematics and Physics, Charles University
Email address : [email protected] URL : http://dmg.tuwien.ac.at/pinsker/ Theoretical Computer Science Department, Jagiellonian University, Poland
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