    # Finite Relation Algebras with Normal Representations

One of the traditional applications of relation algebras is to provide a setting for infinite-domain constraint satisfaction problems. Complexity classification for these computational problems has been one of the major open research challenges of this application field. The past decade has brought significant progress on the theory of constraint satisfaction, both over finite and infinite domains. This progress has been achieved independently from the relation algebra approach. The present article translates the recent findings into the traditional relation algebra setting, and points out a series of open problems at the interface between model theory and the theory of relation algebras.

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## 1. Introduction

One of the fundamental computational problems for a relation algebra is the network satisfaction problem for , which is to determine for a given -network whether it is satisfiable in some representation of (for definitions, see Sections 2 and 3). Robin Hirsch named in 1995 the Really Big Complexity Problem (RBCP) for relation algebras, which is to ‘clearly map out which (finite) relation algebras are tractable and which are intractable’ [Hir96]. For example, for the Point Algebra the network satisfaction problem is in P and for Allen’s Interval Algebra it is NP-hard. One of the standard methods to show that the network satisfaction problem for a finite relation algebra is in P is via establishing local consistency. The question whether the network satisfaction problem for can be solved by local consistency methods is another question that has been studied intensively for finite relation algebras (see [BJ17] for a survey on the second question).

If has a fully universal square representation (we follow the terminology of Hirsch [Hir96]) then the network satisfaction problem for can be formulated as a constraint satisfaction problem (CSP) for a countably infinite structure. The complexity of constraint satisfaction is a research direction that has seen quite some progress in the past years. The dichotomy conjecture of Feder and Vardi from 1993 states that every CSP for a finite structure is in P or NP-hard; the tractability conjecture [BKJ05] is a stronger conjecture that predicts precisely which CSPs are in P and which are NP-hard. Two independent proofs of these conjectures appeared in 2017 [Bul17, Zhu17], based on concepts and tools from universal algebra. An earlier result of Barto and Kozik [BK09] gives an exact characterisation of those finite-domain CSPs that can be solved by local consistency methods.

Usually, the network satisfaction problem for a finite relation algebra cannot be formulated as a CSP for a finite structure. However, suprisingly often it can be formulated as a CSP for a countably infinite -categorical structure . For an important subclass of -categorical structures we have a tractability conjecture, too. The condition that supposedly characterises containment in P can be formulated in many non-trivially equivalent ways [BKO17, BP16, BOP17] and has been confirmed in numerous special cases, see for instance the articles [BK08, BMPP16, KP17, BJP17, BMM18, BM16] and the references therein.

In the light of the recent advances in constraint satisfaction, both over finite and infinite domains, we revisit the RBCP and discuss the current state of the art. In particular, we observe that if has a normal representation (again, we follow the terminology of Hirsch [Hir96]), then the network satisfaction problem for falls into the scope of the infinite-domain tractability conjecture. We also show that there is an algorithm that decides for a given finite relation algebra with a fully universal square representation whether has a normal representation. (In other words, there is an algorithm that decides for a given whether the class of atomic -networks has the amalgamation property.) The scope of the tractability conjecture is larger, though. We describe an example of a finite relation algebra which has an -categorical fully universal square representation (and a polynomial-time tractable network satisfaction problem) which is not normal, but which does fall into the scope of the conjecture.

Whether the infinite-domain tractability conjecture might contribute to the resolution of the RBCP in general remains open; we present several questions in Section 7 whose answer would shed some light on this question. These questions concern the existence of -categorical fully universal square representations and are of independent interest, and in my view they are central to the theory of representable finite relation algebras.

## 2. Relation Algebras

A proper relation algebra is a set together with a set of binary relations over such that

1. ;

2. If and are from , then ;

3. ;

4. ;

5. If , then ;

6. If , then ;

7. If and are from , then ; where

 R1∘R2:={(x,z)|∃y((x,y)∈R1∧(y,z)∈R2)}.

We want to point out that in this standard definition of proper relation algebras it is not required that denotes . However, in most examples, indeed denotes ; in this case we say that the proper relation algebra is square. The inclusion-wise minimal non-empty elements of are called the basic relations of the proper relation algebra.

###### Example 1 (The Point Algebra).

Let be the set of rational numbers, and consider

 R={∅,=,<,>,≤,≥,≠,Q2}.

Those relations form a proper relation algebra (with the basic relations , and where denotes ) which is known under the name point algebra. ∎

The relation algebra associated to is the algebra with the domain and the signature obtained from in the obvious way. An abstract relation algebra is a -algebra that satisfies some of the laws that hold for the respective operators in a proper relation algebra. We do not need the precise definition of an abstract relation algebra in this article since we deal exclusively with representable relation algebras: a representation of an abstract relation algebra is a relational structure whose signature is ; that is, the elements of the relation algebra are the relation symbols of . Each relation symbol is associated to a binary relation over such that the set of relations of induces a proper relation algebra, and the map is an isomorphism with respect to the operations (and constants) . In this case, we also say that is the abstract relation algebra of . An abstract relation algebra that has a representation is called representable. For , we write as a shortcut for the partial order defined by . The minimal elements of with respect to are called the atoms of . In every representation of , the atoms denote the basic relations of the representation. We mention that there are abstract finite relation algebras that are not representable [Lyn50], and that the question whether a finite relation algebra is representable is undecidable [HH01].

###### Example 2.

The (abstract) point algebra is a relation algebra with 8 elements and 3 atoms, , , and , and can be described as follows. The values of the composition operator for the atoms of the point algebra are shown in the table of Figure 1. Note that this table determines the full composition table. The inverse of is , and denotes which is its own inverse. This fully determines the relation algebra.

The proper relation algebra with domain presented in Example 1 is a representation of the point algebra. ∎

## 3. The network satisfaction problem

Let be a finite relation algebra with domain . An -network consists of a finite set of nodes and a function .

A network is called

• atomic if the image of only contains atoms of and if

 f(a,c)≤f(a,b)∘f(b,c) for all a,b,c∈V (1)

(here we follow again the definitions in [Hir96]);

• satisfiable in , for a representation of , if there exists a map (where denotes the domain of ) such that for all

 (s(x),s(y))∈f(x,y)B;
• satisfiable if is satisfiable in some representation of .

The (general) network satisfaction problem for a finite relation algebra is the computational problem to decide whether a given -network is satisfiable. There are finite relation algebras where this problem is undecidable [Hir99]. A representation of is called

• fully universal if every atomic -network is satisfiable in ;

• square if its relations form a proper relation algebra that is square.

The point algebra is an example of a relation algebra with a fully universal square representation. Note that if has a fully universal representation, then the network satisfaction problem for is decidable in NP: for a given network , simply select for each an atom with , replace by , and then exhaustively check condition (1). Also note that a finite relation algebra has a fully universal representation if and only if the so-called path-consistency procedure decides satisfiability of atomic -networks (see, e.g., [BJ17, HLR13]).

However, not all finite relation algebras have a fully universal representation. An example of a relation algebra with 4 atoms which has a representation with seven elements but where path consistency of atomic networks does not imply consistency, called , has been given in [LKRL08]. A representation of with domain is given by the basic relations where , for . In fact, every representation of is isomorphic to this representation. Let be the network with , , , , for all , and for all . Then is atomic but not satisfiable.

## 4. Constraint Satisfaction Problems

Let be a structure with a (finite or infinite) domain and a finite relational signature . Then the constraint satisfaction problem for is the computational problem of deciding whether a finite -structure homomorphically maps to . Note that if is a square representation of , then the input can be viewed as an -network . The nodes of are the elements of . To define for variables of the network, let be a list of all elements such that . Then define ; if , then . Observe that has a homomorphism to if and only if is satisfiable in (here we use the assumption that is a square representation).

Conversely, when is an -network, then we view as the -structure whose domain are the nodes of , and where if and only if . Again, has a homomorphism to if and only if is satisfiable in .

###### Proposition 3.

Let be a fully universal square representation of a finite relation algebra . Then the network satisfaction problem for equals the constraint satisfaction problem for (up to the translation between -networks and finite -structures presented above).

###### Proof.

We have to show that a network is satisfiable if and only if it has a homomorphism to . Clearly, if has a homomorphism to then it is satisfiable in , and hence satisfiable. For the other direction, suppose that the -network is satisfiable in some representation of . Then there exists for each an atomic such that and such that the network obtained from by replacing by satisfies (1); hence, is atomic and satisfiable in since is fully universal. Hence, is satisfiable in , too. ∎

For general infinite structures a systematic understanding of the computational complexity of is a hopeless endeavour [BG08]. However, if is a first-order reduct of a finitely bounded homogeneous structure (the definitions can be found below), then the universal-algebraic tractability conjecture for finite-domain CSPs can be generalised. This condition is sufficiently general so that it includes fully universal square representations of almost all the concrete finite relation algebras studied in the literature, and the condition also captures the class of finite-domain CSPs. As we will see, the concepts of finite boundedness and homogeneity are conditions that have already been studied in the relation algebra literature.

### 4.1. Finite boundedness

Let be a relational signature, and let be a set of -structures. Then denotes the class of all finite -structures such that no structure in embeds into . For a -structure we write for the class of all finite -structures that embed into . We say that is finitely bounded if has a finite relational signature and there exists a finite set of finite -structures such that . A simple example of a finitely bounded structure is . It is easy to see that the constraint satisfaction problem of a finitely bounded structure is in NP.

###### Proposition 4.

Let be a finite relation algebra with a fully universal square representation . Then is finitely bounded.

###### Proof sketch.

Besides some bounds of size at most two that make sure that the atomic relations partition , it suffices to include appropriate three-element structures into that can be read off from the composition table of . ∎

### 4.2. Homogeneity

A relational structure is homogeneous (or ultra-homogeneous [Hod97]) if every isomorphism between finite substructures of can be extended to an automorphism of . A simple example of a homogeneous structure is .

A representation of a finite relation algebra is called normal if it is square, fully universal, and homogeneous [Hir96]. The following is an immediate consequence of Proposition 3 and Proposition 4.

###### Corollary 5.

Let be a finite relation algebra with a normal representation . Then the network satisfaction problem for equals the constraint satisfaction problem for a finitely bounded homogeneous structure.

A versatile tool to construct homogeneous structures from classes of finite structures is amalgamation à la Fraïssé. We present it for the special case of relational structures; this is all that is needed here. An embedding of into is an isomorphism between and a substructure of . An amalgamation diagram is a pair where are -structures such that there exists a substructure of both and such that all common elements of and are elements of . We say that is a 2-point amalgamation diagram if . A -structure is an amalgam of over if for there are embeddings of to such that for all . In the context of relation algebras , the amalgamation property can also be formulated with atomic -networks, in which case it has been called the patchwork property [HLR13]; we stick with the model-theoretic terminology here since it is older and well-established.

###### Definition 1.

An isomorphism-closed class of -structures has the amalgamation property if every amalgamation diagram of structures in has an amalgam in . A class of finite -structures that contains at most countably many non-isomorphic structures, has the amalgamation property, and is closed under taking induced substructures and isomorphisms is called an amalgamation class.

Note that since we only look at relational structures here (and since we allow structures to have an empty domain), the amalgamation property of implies the joint embedding property (JEP) for , which says that for any two structures there exists a structure that embeds both and .

###### Theorem 6 (Fraïssé [Fra54, Fra86]; see [Hod97]).

Let be an amalgamation class. Then there is a homogeneous and at most countable -structure whose age equals . The structure is unique up to isomorphism, and called the Fraïssé-limit of .

The following is a well-known example of a finite relation algebra which has a fully universal square representation, but not a normal one.

###### Example 7.

The left linear point algebra (see [Hir97, Dün05]) is a relation algebra with four atoms, denoted by , , , and . Here we imagine that ‘’ signifies that is earlier in time than . The idea is that at every point in time the past is linearly ordered; the future, however, is not yet determined and might branch into different worlds; incomparability of time points and is denoted by . We might also think of as is to the left of if we draw points in the plane, and this motivates the name left linear point algebra. The composition operator on those four basic relations is given in Figure 2. The inverse of is , denotes , and is its own inverse, and the relation algebra is uniquely given by this data.

It is well known (for details, see [Bod04]) that the left linear point algebra has a fully universal square representation. On the other hand, the networks drawn in Figure 3 show the failure of amalgamation.

An algorithm to test whether a finite relation algebra has a normal representation can be found in Section 6. Figure 3. Example showing that atomic networks for the left linear point algebra do not have the amalgamation property. A directed edge from x to y signifies x

### 4.3. The infinite-domain dichotomy conjecture

The infinite-domain dichotomy conjecture applies to a class which is larger than the class of homogeneous finitely bounded structures. To introduce this class we need the concept of first-order reducts.

Suppose that two relational structures and have the same domain, that the signature of a structure is a subset of the signature of , and that for all common relation symbols . Then we call a reduct of , and an expansion of . In other words, is obtained from by dropping some of the relations. A first-order reduct of is a reduct of the expansion of by all relations that are first-order definable in . The CSP for a first-order reduct of a finitely bounded homogeneous structure is in NP (see [Bod12]). An example of a structure which is not homogeneous, but a reduct of finitely bounded homogeneous structure is the representation of the left-linear point algebra (Example 7) given in [Bod04].

###### Conjecture 8 (Infinite-domain dichotomy conjecture).

Let be a first-order reduct of a finitely bounded homogeneous structure. Then is either in P or NP-complete.

Hence, the infinite-domain dichotomy conjecture implies the RBCP for finite relation algebras with a normal representation. In Section 5 we will see a more specific conjecture that characterises the NP-complete cases and the cases that are in P.

## 5. The Infinite-Domain Tractability Conjecture

To state the infinite-domain tractability conjecture, we need a couple of concepts that are most naturally introduced for the class of all -categorical structures. A theory is called -categorical if all its countably infinite models are isomorphic. A structure is called -categorical if its first-order theory is -categorical. Note that finite structures are -categorical since their first-order theories do not have countably infinite models. Homogeneous structures with finite relational signature are -categorical. This follows from a very useful characterisation of -categoricity given by Engeler, Svenonius, and Ryll-Nardzewski (Theorem 9). The set of all automorphisms of is denoted by . The orbit of a -tuple under is the set . Orbits of pairs (i.e., -tuples) are also called orbitals.

###### Theorem 9 (see [Hod97]).

A countable structure is -categorical if and only if has only finitely many orbits of -tuples, for all .

The following is an easy consequence of Theorem 9.

###### Proposition 10.

First-order reducts of -categorical structures are -categorical.

First-order reducts of homogeneous structures, on the other hand, need not be homogeneous. An example of an -categorical structure which is not homogeneous is the -categorical representation of the left linear point algebra given in [Bod04] (see Example 7). Note that every -categorical structure , and more generally every structure with finitely many orbitals, gives rise to a finite relation algebra, namely the relation algebra associated to the unions of orbitals of (see [BJ17]); we refer to this relation algebra as the orbital relation algebra of .

We first present a condition that implies that an -categorical structure has an NP-hard constraint satisfaction problem (Section 5.1). The tractability conjecture says that every reduct of a finitely bounded homogeneous structure that does not satisfy this condition is NP-complete. We then present an equivalent characterisation of the condition due to Barto and Pinsker (Section 5.2), and then yet another condition due to Barto, Opršal, and Pinsker, which was later shown to be equivalent (Section 5.3).

### 5.1. The original formulation of the conjecture

Let be an -categorical structure. Then is called

• a core if all endomorphisms of (i.e., homomorphisms from to ) are embeddings (i.e., are injective and also preserve the complement of each relation).

• model complete if all self-embeddings of are elementary, i.e., preserve all first-order formulas.

Clearly, if is a representation of a finite relation algebra , then is a core. However, not all representations of finite relation algebras are model complete. A simple example is the orbital relation algebra of the structure where denotes the non-negative rationals: its representation with domain has self-embeddings that do not preserve the orbital .

Let be a relational signature. A -formula is called primitive positive if it is of the form where is of the form or of the form for of arity . The variables can be free or from . Clearly, primitive positive formulas are preserved by homomorphisms.

###### Theorem 11 ([Bod07, Bhm10]).

Every -categorical structure is homomorphically equivalent to a model-complete core , which is unique up to isomorphism, and again -categorical. All orbits of -tuples are primitive positive definable in .

The (up to isomorphism unique) structure from Theorem 11 is called the model-complete core of . Let and be structures, let , and let be a surjection. Then is called a primitive positive interpretation if the pre-image under of , of the equality relation on , and of all relations of is primitive positive definable in . In this case we also say that interprets primitively positively. The complete graph with three vertices (but without loops) is denoted by .

###### Theorem 12 ([Bod08]).

Let be an -categorical structure. If the model-complete core of has an expansion by finitely many constants so that the resulting structure interprets primitively positively, then is NP-hard.

We can now state the infinite-domain tractability conjecture.

###### Conjecture 13.

Let be a first-order reduct of a finitely bounded homogeneous structure. If does not satisfy the condition from Theorem 12 then is in P.

This conjecture has been verified in numerous special cases (see, for instance, the articles [BK08, BMPP16, KP17, BJP17, BMM18, BM16]), including the class of finite-structures [Bul17, Zhu17].

### 5.2. The theorem of Barto and Pinsker

The tractability conjecture has a fundamentally different, but equivalent formulation: instead of the non-existence of a hardness-condition, we require the existence of a polymorphism satisfying a certain identity; the concept of polymorphisms is fundamental to the resolution of the Feder-Vardi conjecture in both [Bul17] and [Zhu17].

###### Definition 2.

A polymorphism of a structure is a homomorphism from to , for some . We write for the set of all polymorphisms of .

An operation is called

• Siggers if it satisfies

 f(x,y,x,z,y,z)=f(z,z,y,y,x,x)

for all ;

• pseudo-Siggers modulo if

 e1(f(x,y,x,z,y,z))=e2(f(z,z,y,y,x,x))

for all .

###### Theorem 14 ([Bp16]).

Let be an -categorical model-complete core. Then either

• can be expanded by finitely many constants so that the resulting structure interprets primitively positively, or

• has a pseudo-Siggers polymorphism modulo endomorphisms of .

### 5.3. The wonderland conjecture

A weaker condition that implies that an -categorical structure has an NP-hard CSP has been presented in  [BOP17]. For reducts of homogeneous structures with finite signature, however, the two conditions are equivalent [BKO17]. Hence, we obtain yet another different but equivalent formulation of the tractability conjecture. The advantage of the new formulation is that it does not require that the structure is a model-complete core.

Let be a countable structure. A map is called minor-preserving if for every of arity and all -ary projections we have where denotes composition of functions. The set is equipped with a natural complete ultrametric (see, e.g., [BS16]). To define , suppose that . For we define if and have different arity; otherwise, if both have arity , then

 d(f,g):=2−min{n∈N∣∃s∈{1,…,n}k:f(s)≠g(s)}.
###### Theorem 15 (of [Bop17]).

Let be -categorical. Suppose that has a uniformly continuous minor-preserving map to . Then is NP-complete.

We mention that there are -categorical structures where the condition from Theorem 15 applies, but not the condition from Theorem 12 [BKO17].

###### Theorem 16 (of [Bko+17]).

If is a reduct of a homogeneous structure with finite relational signature, then the conditions given in Theorem 12 and in Theorem 15 are equivalent.

## 6. Testing the Existence of Normal Representations

In this section we present an algorithm that tests whether a given finite relation algebra has a normal representation. This follows from a model-theoretic result that seems to be folklore, namely that testing the amalgamation property for a class of structures that has the JEP and a signature of maximal arity two which is given by a finite set of forbidden substructures is decidable. We are not aware of a proof of this in the literature.

###### Theorem 17.

There is an algorithm that decides for a given finite relation algebra which has a fully universal square representation whether also has a normal representation.

###### Proof.

First observe that the class of all atomic -networks, viewed as -structures, has the JEP: if and are atomic networks, then they are satisfiable in since is fully universal, and hence embed into when viewed as structures. Since is square the substructure of induced by the union of the images of and is an atomic network, too, and it embeds and .

Let be the number of atoms of . It clearly suffices to show the following claim, since the condition given there can be effectively checked exhaustively.

Claim. has the AP if and only if all 2-point amalgamation diagrams of size at most amalgamate.

So suppose that is an amalgamation diagram without amalgam. Let be a maximal substructure of that contains such that has an amalgam. Let be a maximal substructure of that contains such that has an amalgam. Then for some ; let be a substructure of that extends by one element, and let . Then is a 2-point amalgamation diagram without an amalgam. Let . Let and . For each there exists an element such that the network with , , fails the atomicity property (1). Let be the substructure of induced by and be the substructure of induced by . Then the amalgamation diagram has no amalgam, and has size at most . ∎

## 7. Conclusion and Open Problems

Hirsch’s Really Big Complexity Problem (RBCP) for finite relation algebras remains really big. However, the network satisfaction problem of every finite relation algebra known to the author can be formulated as the CSP of a structure that falls into the scope of the infinite-domain tractability conjecture. Most of the classical examples even have a normal representation, and therefore the RBCP for those is implied by the infinite-domain tractability conjecture (Corollary 5). We presented an algorithm that tests whether a given finite relation algebra has a normal representation. Figure 4. Subclasses of finite representable relation algebras. Membership of relation algebras from D to the innermost box A is decidable (Theorem 17). Example 7 separates Box A and Box B. The finite relation algebra from [Hir99] separates Box D and E. Box B falls into the scope of the infinite-domain tractability conjecture. Boxes C and D might also fall into the scope of this conjecture (see Problem (1) and Problem (2)).

To better understand the RBCP in general, or at least for finite relation algebras with fully universal square representation, we need a better understanding of representations of finite relation algebras with good model-theoretic properties. We mention some concrete open questions; also see Figure 4.

1. Is there a finite relation algebra with a fully universal square representation, but without an -categorical fully universal square representation?

2. Is there a finite relation algebra with an -categorical fully universal square representation but without a fully universal square representation which is not a first-order reduct of a finitely bounded homogeneous structure?

3. Find a finite relation algebra such that there is no -categorical structure such that the general network satisfaction problem for equals the constraint satisfaction problem for . (Note that we do not insist on being a representation of .)

4. Find a finite relation algebra with an -categorical fully universal square representation which is not the orbital relation algebra of an -categorical structure.