Two-element structures modulo primitive positive constructability

Primitive positive constructions have been introduced in recent work of Barto, Opršal, and Pinsker to study the computational complexity of constraint satisfaction problems. Let Pfin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {P}}_{\mathrm {fin}}$$\end{document} be the poset which arises from ordering all finite relational structures by pp-constructability. This poset is infinite, but we do not know whether it is uncountable. In this article, we give a complete description of the restriction PBoole\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {P}}_{\mathrm {Boole}}$$\end{document} of Pfin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {P}}_{\mathrm {fin}}$$\end{document} to relational structures on a two-element set. We use PBoole\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {P}}_{\mathrm {Boole}}$$\end{document} to present the various complexity regimes of Boolean constraint satisfaction problems that were described by Allender, Bauland, Immerman, Schnoor and Vollmer.


Introduction
Varieties play a central role in universal algebra.In 1974, Neumann [10] defined the notion of interpretability between varieties, which has been studied intensively, e.g., by Garcia and Taylor [6].The corresponding lattice basically corresponds to the homomorphism order of clones.
Recently, Barto, Opršal, and Pinsker [3] introduced minor-preserving maps, a weakening of the notion of a clone homomorphism.We study the poset that arises from ordering clones on a finite domain with respect to the existence of minor-preserving maps.It can be characterised in three very different, but equivalent, ways.One of the characterisations is in terms of primitive positive constructions for relational structures.Primitive positive constructions are also motivated by the study of the computational complexity of constraint satisfaction problems (CSPs).They preserve the complexity of the CSPs in the following sense: if A and B are finite structures such that A pp-constructs B then CSP(B) has a polynomial-time reduction to CSP(A).Barto, Opršal, and Pinsker [3] proved that A pp-constructs B if and only if it exists a minorpreserving map from Pol(A) to Pol(B) (Theorem 2.6).
Let P fin be the poset which arises from ordering all finite relational structures by pp-constructability.It follows from Bulatov's universal-algebraic proof [5] of the H-coloring dichotomy theorem of Hell and Nešetřil [8] that the poset P fin , restricted to all finite undirected graphs, just has three elements: K 3 (the clique with three vertices), K 2 (the graph consisting of a single edge), and the graph with one vertex and a loop.On the other hand, the cardinality of P fin is not known; it is clear that it has infinite descending chains (already for two-element structures), but it is not known whether it is uncountable.
In this article we study the restriction of P fin to all two-element structures.We call this subposet P Boole ; it turns out that it is a countably infinite lattice.We provide a description of P Boole in Theorem 3.22; in particular, we show that it has 3 atoms, one coatom, infinite descending chains, and a planar Hasse diagram.Our poset P Boole can be used to formulate a refinement of Schaefer's theorem [13] that matches the known results about the complexity of Boolean constraint satisfaction problems [1,2].

The PP-Constructability Poset
As already anticipated in the introduction, P fin can be defined in three different ways.In this section we introduce two of them.The third equivalent description relates the elements of P fin with classes of algebras closed not only under the classical operators H, S, and P, but also under so-called reflections; but this will not be relevant for the purposes of this article, so we refer the interested reader to [3].

PP-Constructions
Let τ be a relational signature.Two relational τ -structures A and B are homomorphically equivalent if there exists a homomorphism from A to B and viceversa.A primitive positive formula (over τ ) is a first-order formula which only uses relation symbols in τ , equality, conjunction and existential quantification.When A is a τ -structure and φ(x 1 , . . ., x n ) is a τ -formula with n free-variables x 1 , . . ., x n then {(a 1 , . . ., a n ) | A |= φ(a 1 , . . ., a n )} is called the the relation defined by φ.If φ is primitive positive, then this relation is said to be pp-definable in A. Given two relational structures A and B on the same domain A = B (but with possibly different signatures), we say that A pp-defines B if every relation in B is pp-definable in A. We say that B is a pp-power of A if it is isomorphic to a structure with domain A n , where n ≥ 1, whose relations are pp-definable from A (a k-ary relation on A n is regarded as a kn-ary relation on A).

Definition 2.1 ([3]
).Let A and B be relational structures.We say that A pp-constructs B, in symbols A B, if B is homomorphically equivalent to a pp-power of A.
The following result from [3] asserts that pp-constructability preserves the complexity of CSPs: Proposition 2.2.Let A and B be relational structures.If A pp-constructs B then CSP(B) is log-space reducible to CSP(A).
Since pp-constructability is a reflexive and transitive relation on the class of relational structures [3], the relation ≡ defined by This article is dedicated to the subposet P Boole of P fin , whose universe is the set of all pp-constructability types of relational structures on {0, 1}.

Minor-Preserving Maps
Another approach to the pp-constructability poset involves a weakening of the notion of clone homomorphism and certain identities called height 1 identities.Definition 2.3.Let τ be a functional signature.An identity is said to be a height 1 identity if it is of the form where f, g are function symbols in τ and π and σ are mappings, π : {1, . . ., n} → {1, . . ., r} and σ : {1, . . ., m} → {1, . . ., r}.
In other words, we require that there is exactly one occurrence of a function symbol on both sides of the equality.The use of nested terms is forbidden.Identities of the form f (x 1 , . . ., x n ) ≈ y are forbidden as well (identities of this form are often called linear or of height at most 1 ).
A height 1 condition is a finite set Σ of height 1 identities.We say that a set of functions F (for instance a clone) satisfies a height 1 condition Σ, and we write F |= Σ, if for each function symbol f appearing in Σ, there exists a function f F ∈ F of the corresponding arity such that every identity in Σ becomes a true statement when the symbols of Σ are instantiated by their counterparts in F .
We introduce some height 1 conditions that we are going to use later.
Definition 2.4.Let f be a k-ary operation symbol.
• The set of height 1 identities is called quasi near-unanimity condition (QNU(k)).
• The QNU(3) condition is also called quasi majority condition.
• The set of height 1 identities (k = 3) is called quasi minority condition.
A k-ary function f is a quasi near-unanimity operation if {f } satisfies the quasi near-unanimity condition QNU(k).A quasi majority and a quasi minority operation is defined in the same way.
We write A m B if there exists a minor-preserving map ξ : A → B, and we denote by ≡ m the equivalence relation where Again, we denote by A the ≡ m -class of A and we write A m B if and only if A m B. The connection between pp-constructability and minor-preserving maps is given by the following theorem.

Theorem 2.6 ([3]).
Let A and B be finite relational structures and let A and B be their polymorphism clones.Then the following are equivalent: 3. Every height 1 condition that holds in A also holds in B.

B ∈ ERP A.
We refer to [3] for the definitions involved in Item 4 of the statement; we just mention that ERP A contains the universal-algebraic variety HSP A.
Note that Theorem 2.6 provides an important tool to prove that two elements are distinct in our poset: if A B, then there is a height 1 condition which is satisfied in A but not in B. Also note that every operation clone on a finite set is the polymorphism clone of a finite relational structure.Therefore, the poset ({C | C is an operation clone on a finite set}; m ) is isomorphic to P fin .In fact, both posets will be called P fin and we will use the same symbol both for the pp-constructability order between structures and for the minor-preserving order between clones.

Post's Lattice
The set of clones on the Boolean set {0, 1} was first investigated by Post [12] in 1941.This set has countably many elements and forms a lattice with respect to the inclusion order.Since we built on this result, we dedicate a section to Post's lattice in order to fix some notation.Note that if C ⊆ D, then it follows that C D via the identity mapping.
We label the clones of Post's lattice by generators: if f 1 , . . ., f n are operations on {0, 1}, then [f 1 , . . ., f n ] denotes the clone generated by f 1 , . . ., f n .As usual, we may apply functions componentwise, i.e., if f is a k-ary map, and t 1 , . . ., t k ∈ {0, 1} m , then f (t 1 , . . ., t k ) denotes the m-tuple In the description of Post's lattice, we use the following operations.
For n = 3 we obtain the majority operation • The minority operation m(x, y, z) := x ⊕ y ⊕ z.
Post's lattice has 7 atoms, 5 coatoms and it is countably infinite because of the presence of some infinite descending chains; see Figure 1.

The Lattice P Boole
We consider the order defined in Section 2.2 restricted to the class of Boolean clones.Note that this is a coarser order than the usual inclusion order on the class of Boolean clones.In this section we are going to describe systematically the poset

Collapses
In this section we prove that certain clones on {0, 1} are in the same ≡-class, i.e., represent the same element in P Boole .We start with the observation that each clone collapses with its dual.Proof.To prove that C D, define ξ(f The same argument can be used to prove that D C. Proposition 3.2.Let C be any clone and D be a clone with a constant operation.Then C D.
Proof.Note that D contains a constant operation g n for every arity n.The map ξ : C → D that sends every n-ary operation to g n is minor-preserving.
It follows that the top-element in P Boole is the class of clones that contain a constant operation.The next proposition is about the bottom element.

Proof. We only have to prove that [c]
[∅].Then the map that fixes the projections and maps c(π Note that with the collapses we have reported so far we can make some observations on the number of atoms in P Boole .We already pointed out that [0] and [1] are not atoms in P Boole , since [0] = [1] is the top-element in P Boole .Furthermore, we have that Another case of collapse is that the clones [∨, ∧] and [d 3 , p] represent the same element in P Boole .We consider the binary relations ≤ := {(0, 0), (0, 1), (1, 1)} and B 2 := {(0, 1), (1, 0), (1, 1)} and define (following the notation in [2]): For the other inequality it suffices to prove that (B 2 , ≤) is homomorphically equivalent to a pp-power of D STCON .We consider the relational structure A with domain {0, 1} 2 and relations defined by Note that A is indeed a pp-power of D STCON .We define the map as follows: Let g : (B 2 , ≤) → A be a map such that g(0) := (0, 1) and g(1) := (1, 0).It is easy to check that both f and g are homomorphisms.This proves that A is homomorphically equivalent to (B 2 , ≤).
Recall that the idempotent reduct of a clone C is the clone C id that consists of all idempotent operations in C. Proof.Since C contains no constant operations, for any operation f in C either f (x, . . ., x) ≈ x holds or f (x, . . ., x) ≈ c(x) holds.We claim that there exists a minor-preserving map ξ : D → D id .We define ξ : D → D id as follows: for an n-ary operation f ∈ D By definition ξ(f ) ∈ D id .We claim that ξ is minor preserving: if f is idempotent, then ξ is the identity, and the claim trivially holds; in the other case, the claim follows by the definition of negation:

Separations
Recall that if A B then there is a height 1 condition Σ which is satisfied by some operations in A but by none of the operations in B. In this case we say that Σ is a witness for A B. We will use the following height 1 conditions.Definition 3.7.The following set of height 1 identities t 0 (x, y, z) ≈ t 0 (x, x, x) t n (x, y, z) ≈ t n (z, z, z) Proof.Define the operations: 3 (x, y, z) := p(z, x, y) t we have that t [∧] 1 (x, y, z) does not depend on the second argument.Moreover, t 0 (x, x, z) ≈ t 1 (x, x, z) implies that t [∧] 1 (x, y, z) does not depend on the third argument.Hence, we conclude that t [∧] 1 (x, y, z) = x.From the identity t 1 (x, z, z) ≈ t 2 (x, z, z) and using t Similarly, we obtain also that t 4 (x, y, z) = x.This is in contradiction with the identity t 4 (x, y, z) ≈ t 4 (z, z, z).Hence, we conclude that [∧] does not satisfy QJ(4).
The following structures are useful in the next proposition: where for all a, b, c, d ∈ {0, 1}: These structures are the relational counterparts of the atoms of P Boole in the sense that Proposition 3.9.The following holds: Proof.(1) By definition, d 3 is a quasi majority operation.Let f be any Boolean quasi majority operation.Then it is easy to check that f does not preserve R 110 and thus f / ∈ [∧] = Pol(D HORNSAT ).Hence, the quasi majority condition is a witness for We claim that the height 1 identity f (x, y) ≈ f (y, x) is a witness for [∧] [d 3 ].This identity is clearly satisfied by ∧.Let f be any Boolean binary commutative operation.Then f preserves neither R 00 nor R 11 .Hence, f / ∈ [d 3 ] = Pol(D 2SAT ) and thus the claim is proved.
In order to prove this fact, we introduce the following relational structures, also known as blockers [11]: Blockers are the relational counterparts of the clones considered in the chain (C1), because the same chain can be rewritten as: We use the QNU identities to prove that the order is strict: in fact, Pol(B n−1 ) satisfies QNU(n) but Pol(B n ) does not.Proposition 3.12.For any natural number n > 2, the quasi near-unanimity condition QNU(n) is a witness for Pol(B n−1 ) Pol(B n ).
Proof.Let f be an n-ary quasi near-unanimity operation.Suppose for contradiction that f is in Pol(B n ).Note that f is idempotent since it has to preserve the unary relations {0} and {1}.In the following (n × n)-matrix every column is an element of B n .Then we get a contradiction since, applying f row-wise, we obtain the missing n-tuple (0, . . ., 0).Proof.Since [d 3 , q] = Pol(B 2 ), the argument is essentially the same as the one of Proposition 3.16.

The Final Picture
Putting all the results of the previous section together, we display an order diagram of P Boole .We then use this diagram to revisit the complexity of Boolean CSPs.In Figure 3 we indicate for each element of P Boole the corresponding complexity class.
Theorem 3.22.The pp-constructability poset restricted to the case of Boolean clones is the lattice P Boole in Figure 3. and Schaefer's theorem [13] implies that the CSP of every other Boolean structure is in P. Following [1], we describe the complexity of Boolean CSPs within P. Combining Theorem 3.22 with the main result in [1] we obtain the following.The same complexity results can be reached using general results collected in the survey article [2].

Concluding Remarks and Open Problems
In this article we completely described P Boole , i.e., the pp-constructability poset restricted to structures over a two-element set; equivalently, we studied the poset that arises from ordering Boolean clones with respect to the existence of minorpreserving maps.
The natural next step is to study the pp-constructability poset on larger classes of finite structures.Janov and Mučnik [9] showed that there are continuum many clones over a three-element set, but all the clones considered in their proof have a constant operation, so they only correspond to a single element in our poset.Uncountably many idempotent clones over a three element set have been described by Zhuk [14].We have not yet been able to separate them with height 1 conditions.Hence, there is still hope that P fin has only countably many elements.While it is easy to construct infinite antichains in P fin , we also do not know whether P fin contains infinite ascending chains.
It is easy to see that P fin is a meet-semilattice: if C and D are clones on a finite set, then C × D is a clone that projects both to C and to D via minorpreserving maps, and all other clones with this property have a minor-preserving map to C × D. However, we do not know whether P fin is a lattice.It is known that K 3 pp-constructs (even pp-interprets; see, e.g., [4]) all finite structures.Hence, K 3 is the bottom element in P fin .Moreover, it can be shown that P fin has no atoms and that [m, q] is the only coatom in P fin .Note that for every finite set A with at least two elements, the class [m, q] contains the clone of all idempotent operations on A. One can see that when studying P fin we can focus on idempotent clones.However, the study of the entire poset P fin is ongoing and will be the topic of a future publication.
is an equivalence relation.The equivalence classes of ≡ are called the ppconstructability types and we write A for the pp-constructability type of A. For any two relational structures A and B we write A B if and only if A B. The poset P fin := ({A | A is a finite relational structure}; ) is called the pp-constructability poset.

First, we prove
that certain Boolean clones are in the same ≡-class.Later, in Section 3.2 we prove that certain Boolean clones lie in different ≡-classes and, for each separation, we provide a height 1 condition as a witness.We write C | D if C D and D C. Sometimes it will be useful to refer to relational descriptions of the clones: recall from Theorem 2.6 that if A = Pol(A) and B = Pol(B), then A B if and only if A B. Finally, in Section 3.3, we display an order diagram of P Boole .

Proposition 3 . 3 .
Let [∅] be the set of projections and let [c] be the clone generated by the Boolean negation.Then [∅] ≡ [c].

Lemma 3 . 5 .
Let C be a Boolean clone with no constant operations.Let D := [C ∪ {c}] be the clone generated by C and the Boolean negation c.Then we have D ≡ D id .

3 .
[d 3 , m] | [∧].Proof.(1) and (2) follow immediately from Proposition 3.9: the quasi minority condition is a witness for [d 3 , m] [d 3 ] and the quasi majority condition is a witness for [d 3 , m] [m].Concerning (3), it follows from Proposition 3.9 that the quasi majority condition is a witness for [d 3 , m] We claim that the quasi minority condition is a witness for [m] [d 3 , p].In fact, let f be any Boolean quasi minority operation.Then f does not preserve B 2 since the missing tuple (0, 0) can be obtained by applying f to tuples in B 2 .Hence, f / ∈ [d 3 , p] = Pol(B 2 , ≤) and thus the claim follows.Corollary 3.17.Let C be a Boolean clone such that [p] ⊆ C ⊆ [d 3 , p].Then C | [m].Proof.Let C be as in the hypothesis.Let us suppose that [m] C. Then we get [m] C [d 3 , p], contradicting Proposition 3.16.Let us suppose now that C [m].Then we get [∧] ≺ [p] C [m], contradicting Proposition 3.9.Corollary 3.18.[m] [d 3 , q].