The complexity of quantified constraints using the algebraic formulation

The complexity of quantified constraints using the algebraic formulation

Abstract

Let be an idempotent algebra on a finite domain. We combine results of Chen [11], Zhuk [24] and Carvalho et al. [7] to argue that if satisfies the polynomially generated powers property (PGP), then QCSP is in NP. We then use the result of Zhuk to prove a converse, that if satisfies the exponentially generated powers property (EGP), then QCSP is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying the moral correctness of what we term the Chen Conjecture (see [12]).

We examine in closer detail the situation for domains of size three. Over any finite domain, the only type of PGP that can occur is switchability. Switchability was introduced by Chen in [11] as a generalisation of the already-known Collapsibility [9]. For three-element domain algebras that are Switchable, we prove that for every finite subset of Inv, Pol is Collapsible. The significance of this is that, for QCSP on finite structures (over three-element domain), all QCSP tractability explained by Switchability is already explained by Collapsibility.

Finally, we present a three-element domain complexity classification vignette, using known as well as derived results.

Quantified Constraints, Computational Complexity, Universal Algebra, Constraint Satisfaction
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C. Carvalho et al..

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F.2.2 Nonnumerical Algorithms and Problems

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John Q. Open and Joan R. Acces \EventNoEds2 \EventLongTitle42nd Conference on Very Important Topics (CVIT 2016) \EventShortTitleCVIT 2016 \EventAcronymCVIT \EventYear2016 \EventDateDecember 24–27, 2016 \EventLocationLittle Whinging, United Kingdom \EventLogo \SeriesVolume42 \ArticleNo23

1 Introduction

A large body of work exists from the past twenty years on applications of universal algebra to the computational complexity of constraint satisfaction problems (CSPs) and a number of celebrated results have been obtained through this method. One considers the problem CSP in which it is asked whether an input sentence holds on , where is primitive positive, that is using only , and . The CSP is one of a wide class of model-checking problems obtained from restrictions of first-order logic. For almost every one of these classes, we can give a complexity classification [18]: the two outstanding classes are CSPs and its popular extension quantified CSPs (QCSPs) for positive Horn sentences – where is also present – which is used in Artificial Intelligence to model non-monotone reasoning or uncertainty [15].

The outstanding conjecture in the area is that all finite-domain CSPs are either in P or are NP-complete, something surprising given these CSPs appear to form a large microcosm of NP, and NP itself is unlikely to have this dichotomy property. This Feder-Vardi conjecture [16], given more concretely in the algebraic language in [6], remains unsettled, but is now known for large classes of structures. It is well-known that the complexity classification for QCSPs embeds the classification for CSPs: if is with the addition of a new isolated element not appearing in any relations, then CSP and QCSP are polynomially equivalent. Thus the classification for QCSPs may be considered a project at least as hard as that for CSPs. The following is the merger of Conjectures 6 and 7 in [12] which we call the Chen Conjecture.

Conjecture 1 (Chen Conjecture).

Let be a finite relational structure expanded with all constants. If Pol has PGP, then QCSP is in NP; otherwise QCSP is Pspace-complete.

In [12], Conjecture 6 gives the NP membership and Conjecture 7 the Pspace-completeness. We now know from [24] and [7] that the NP membership of Conjecture 6 is indeed true. The most interesting result of this paper is Theorem 1 below, but note that we permit infinite signatures (languages) although our domains remain finite. This aspect of our work will be discussed in detail later. {theorem}[Revised Chen Conjecture] Let be an idempotent algebra on a finite domain . If satisfies PGP, then QCSP is in NP. Otherwise, QCSP is co-NP-hard. Zhuk has previously proved [24] that only the cases PGP and EGP may occur, as well even in the non-idempotent case. With infinite languages, the NP-membership for Theorem 1 is no longer immediate from [7], but requires a little extra work. We are also able to refute the following form.

Conjecture 2 (Alternative Chen Conjecture).

Let be an idempotent algebra on a finite domain . If satisfies PGP, then for every finite subset , QCSP is in NP. Otherwise, there exists a finite subset so that QCSP is co-NP-hard.

In proving Theorem 1 we are saying that the complexity of QCSPs, with all constants included, is classified modulo the complexity of CSPs. {corollary} Let be an idempotent algebra on a finite domain . Either QCSP is co-NP-hard or QCSP has the same complexity as CSP. In this manner, our result follows in the footsteps of the similar result for the Valued CSP, which has also had its complexity classified modulo the CSP, as culminated in the paper [17].

For a finite-domain algebra we associate a function , giving the cardinality of the minimal generating sets of the sequence as , respectively. A subset of is a generating set for exactly if, for every , there exists a -ary term operation of and so that , …, . We may say has the -GP if for all . The question then arises as to the growth rate of and specifically regarding the behaviours constant, logarithmic, linear, polynomial and exponential. Wiegold proved in [23] that if is a finite semigroup then is either linear or exponential, with the former prevailing precisely when is a monoid. This dichotomy classification may be seen as a gap theorem because no growth rates intermediate between linear and exponential may occur. We say enjoys the polynomially generated powers property (PGP) if there exists a polynomial so that and the exponentially generated powers property (EGP) if there exists a constant so that where .

In Hubie Chen’s [11], a new link between algebra and QCSP was discovered. Chen’s previous work in QCSP tractability largely involved the special notion of Collapsibility [9], but in [11] this was extended to a computationally effective version of the PGP. For a finite-domain, idempotent algebra , -collapsibility may be seen as that special form of the PGP in which the generating set for is constituted of all tuples in which at least of these elements are equal. -switchability may be seen as another special form of the PGP in which the generating set for is constituted of all tuples in which there exists , for , so that

where , , …, . Thus, are the indices where the tuple switches value. Note that these are not the original definitions, which we will see shortly, but they are proved equivalent to the original definitions (at least for finite signatures) in [7]. Moreover, these are the definitions that we will use. We say that is collapsible (switchable) if there exists such that it is -collapsible (-switchable). We note that Zhuk uses this definition of switchability in [24] in which he proved that the only kind of PGP for finite-domain algebras is switchability.

Let us capitalise Collapsibility and Switchability to indicate Chen’s original definitions from [11] are used, following an example for arithmetic versus Arithmetic by Raymond Smullyan in [22]. There is the potential for confusion at the start of the sentence but, as was the case with Smullyan, the two will transpire to be interchangeable throughout our discourse. It is straightforward to see that -Switchability implies -switchability and -Collapsibility implies -collapsibility. The converses, for finite signatures, also hold, but this requires rather more work [7]. For any finite algebra, -Collapsibility implies -Switchability. and for any -element algebra, -Switchability implies -Collapsibility. Chen originally introduced Switchability because he found a -element algebra that enjoyed the PGP but was not Collapsible [11]. He went on to prove that Switchability of implies that the corresponding QCSP is in P, what one might informally state as QCSP in P, where can be seen as the structure over the same domain as whose relations are precisely those that are preserved by (invariant under) all the operations of . However, the QCSP was traditionally defined only on finite sets of relations (else the question arises as to encoding), thus a more formal definition might be that, for any finite subset of , QCSP is in P. What we prove in this paper is that, as far as the QCSP is concerned, Switchability on a three-element algebra is something of a mirage. What we mean by this is that when is Switchable, for all finite subsets of Inv, already Pol is Collapsible. Thus, for QCSP complexity for three-element structures, we do not need the additional notion of Switchability to explain tractability, as Collapsibility will already suffice. Since these notions were originally introduced in connection with the QCSP this is particularly surprising. Note that the parameter of Collapsibility is unbounded over these increasing finite subsets while the parameter of Switchability clearly remains bounded. In some way we are suggesting that Switchability itself might be seen as a limit phenomenon of Collapsibility.

1.1 Infinite languages

Our use of infinite languages (i.e. signatures, since we work on a finite domain) is the only controversial part of our discourse and merits special discussion. We wish to argue that a necessary corollary of the algebraic approach to (Q)CSP is a reconciliation with infinite languages. The traditional approach to consider arbitrary finite subsets of Inv is unsatisfactory in the sense that choosing this way to escape the – naturally infinite – set Inv is as arbitrary as the choice of encoding required for infinite languages. However, the difficulty in that choice is of course the reason why this route is often eschewed. The first possibility that comes to mind for encoding a relation in Inv is probably to list its tuples, while the second is likely to be to describe the relation in some kind of “simple” logic. Both these possibilities are discussed in [14], for the Boolean domain, where the “simple” logic is the propositional calculus. For larger domains, this would be equivalent to quantifier-free propositions over equality with constants. Both Conjunctive Normal Form (CNF) and Disjunctive Normal Form (DNF) representations are considered in [14] and a similar discussion in [3] exposes the advantages of the DNF encoding. The point here is that testing non-emptiness of a relation encoded in CNF may already be NP-hard, while for DNF this will be tractable. Since DNF has some benign properties, we might consider it a “nice, simple” logic while for “simple” logic we encompass all quantifier-free sentences, that include DNF and CNF as special cases. The reason we describe this as “simple” logic is to compare against something stronger, say all first-order sentences over equality with constants. Here recognising non-emptiness becomes Pspace-hard and since QCSPs already sit in Pspace, this complexity is unreasonable.

For the QCSP over infinite languages Inv, Chen and Mayr [13] have declared for our first, tuple-listing, encoding. In this paper we will choose the “simple” logic encoding, occasionally giving more refined results for its “nice, simple” restriction to DNF. Our choice of the “simple” logic encoding over the tuple-listing encoding will ultimately be justified by the (Revised) Chen Conjecture holding for “simple” logic yet failing for tuple-listings. Note that our demonstration of the (Revised) Chen Conjecture for infinite languages with the “simple” logic encoding does not resolve the original Chen Conjecture for finite languages with constants because QCSP could conceivably have higher complexity than QCSP due to a succinct representation of relations in . Indeed, this belies one justification for the preferential study of finite subsets of , since for finite signature we can then say QCSP and QCSP must have the same complexity. Note that for finite relational bases of , QCSP and QCSP must have the same complexity. Further, we do not know of any concrete finite with constants, so that QCSP and QCSP have different complexity.

Let us consider examples of our encodings. For the domain , we may give a binary relation either by the tuples or by the “simple” logic formula . For the domain , we may give the ternary (not-all-equal) relation by the tuples or by the “simple” logic formula . In both of these examples, the simple formula is also in DNF.

Nota Bene. The results of this paper apply for the “simple” logic encoding as well as the “nice, simple” encoding in DNF except where specifically stated otherwise. These exceptions are Proposition 2 and Corollary 3.2 (which uses the “nice, simple” DNF) and Proposition 4 (which uses the tuple encoding).

1.2 Related work

This paper is the merger of [20, 19], neither of which was submitted for publication, considerably extended.

2 Preliminaries

Let . A -ary polymorphism of a relational structure is a homomorphism from to . Let Pol be the set of polymorphisms of and let Inv be the set of relations on which are invariant under (each of) the operations of some finite algebra . Pol is an object known in Universal Algebra as a clone, which is a set of operations containing all projections and closed under composition (superposition). A term operation of an algebra is an operation which is a member of the clone generated by .

We will conflate sets of operations over the same domain and algebras just as we do sets of relations over the same domain and constraint languages (relational structures). Indeed, the only technical difference between such objects is the movement away from an ordered signature, which is not something we will ever need. A reduct of a relational structure is a relational structure over the same domain obtained by forgetting some of the relations. If is some finite subset of Inv, then we may view a being a finite reduct of the structure (associated with) Inv.

A -ary operation over is a projection if , for some . When are strict subsets of so that , then a -ary operation on is said to be -projective if there exists so that if (respectively, ), then (respectively, ).

We recall QCSP, where is some structure on a finite-domain, is a decision problem with input , a pH-sentence (i.e. using just , , and ) involving (a finite set of) relations of , encoded in propositional logic with equality and constants. The yes-instances are those for which . If the input sentence is restricted to have alternation then the corresponding problem is designated -CSP.

2.1 Games, adversaries and reactive composition

We now recall some terminology due to Chen [9, 11], for his natural adaptation of the model checking game to the context of pH-sentences. We shall not need to explicitly play these games but only to handle strategies for the existential player. This will enable us to give the original definitions for Collapsibility and Switchability. An adversary of length is an -ary relation over . When is precisely the set for some non-empty subsets of , we speak of a rectangular adversary (we will sometimes specify this as a tuple rather than a product). Let be a pH-sentence with universal variables and quantifier-free part . We write and say that the existential player has a winning strategy in the -game against adversary iff there exists a set of Skolem functions such that for any assignment of the universally quantified variables of to , where , the map is a homomorphism from (the canonical database) to , where

(Here, denotes the set of universal variables preceding and the restriction of to .) Clearly, iff the existential player has a winning strategy in the -game against the so-called full (rectangular) adversary (which we will denote hereafter by ). We say that an adversary of length dominates an adversary of length when . Note that and implies . We will also consider sets of adversaries of the same length, denoted by uppercase Greek letters as in (here the length is ); and, sequences thereof, which we denote with bold uppercase Greek letters as in . We will write to denote that holds for every adversary in .

Let be a -ary operation of and be adversaries of length . We say that is reactively composable from the adversaries via , and we write iff there exist partial functions for every in and every in such that, for every tuple in adversary the following holds.

  • for every in , the values are defined and the tuple is in adversary ; and,

  • for every in , .

We write if there exists a -ary operation such that Reactive composition allows to interpolate complete Skolem functions from partial ones. {theorem}[[11, Theorem 7.6]] Let be a pH-sentence with universal variables. Let be an adversary and a set of adversaries, both of length .

If and then . As a concrete example of an interesting sequence of adversaries, consider the adversaries for the notion of -Collapsibility. Let be some fixed integer. For in , let be the set of all rectangular adversaries of length with co-ordinates that are the set and all the others that are the fixed singleton . For , let be the union of for all in . Let be the sequence of adversaries . We will define a structure to be -Collapsible from source iff for every and for all pH-sentence with universal variables, implies .

For -Switchability, the adversaries will be of the form which contains all tuples which have no more than switches.

For rectangular adversaries, such as , reactive composition is rather simpler than in the definition above, becoming just (ordinary) composition, as follows. is composable from the adversaries via if , where and each . Reactive composition plays a key role in the proof of our main theorem but its use appears only in other papers that we will cite. Ordinary composition is the only type of reactive composition that will be used in this paper.

3 The Chen Conjecture

3.1 NP-membership

We need to revisit the main result of [7] to show that it holds not just for finite signatures but for infinite signatures also. In its original the following theorem discussed “projective sequences of adversaries, none of which are degenerate”. This includes Switching adversaries and we give it in this latter form. We furthermore remove some parts of the theorem that are not currently relevant to us. {theorem}[In abstracto [7]] Let be the sequence of the set of all (-)Switching -ary adversaries over the domain of , a finite structure. The following are equivalent.

  1. For every , for every pH-sentence with universal variables, implies .

  2. For every , generates .

{corollary}

[In abstracto levavi] Let be the sequence of the set of all (-)Switching -ary adversaries over the domain of , a finite-domain structure with an infinite signature. The following are equivalent.

  1. For every , for every pH-sentence with universal variables, implies .

  2. For every , generates .

Proof.

We know from Theorem 3.1 that the following are equivalent:

  1. For every finite-signature reduct of and , for every pH-sentence with universal variables, implies .

  2. For every finite-signature reduct of and every , generates .

Since it is clear that both and , it remains to argue that and .

[.] By contraposition, if fails then it fails on some specific pH-sentence which only mentions a finite number of relations of . Thus also fails on some finite reduct of mentioning these relations.

[.] Let be given. Consider some chain of finite reducts of so that each is a reduct of for and every relation of appears in some . We can assume from that generates , for each . But since the number of tuples and operations from to witnessing generation in is finite, the sequence of operations witnessing these must have an infinitely recurring element as tends to infinity. One such recurring element we call and this witnesses generation in . ∎

Note that in above we did not need to argue uniformly across the different and it is enough to find an infinitely recurring operation for each of these individually.

The following result is essentially a corollary of the works of Chen and Zhuk [11, 24] via [7]. {theorem} Let be an idempotent algebra on a finite domain . If satisfies PGP, then QCSP reduces to a polynomial number of instances of CSP and is in NP.

Proof.

We know from Theorem 7 in [24] that is Switchable, whereupon we apply Corollary 3.1, . By considering instances whose universal variables involve only the polynomial number of tuples from the Switching Adversary, one can see that QCSP reduces to a polynomial number of instances of CSP and is therefore in NP. Further details of the NP algorithm are given in Corollary 38 of [7] but the argument here follows exactly Section 7 from [11], in which it was originally proved that Switchability yields the corresponding QCSP in NP. ∎

Note that Chen’s original definition of Switchability, based on adversaries and reactive composability, plays a key role in the NP membership algorithm in Theorem 3.1. It is the result from [7] that is required to reconcile the two definitions of switchability as equivalent, and indeed Corollary 3.1 is needed in this process for infinite signatures. If we were to use just our definition of switchability then it is only possible to prove, à la Proposition 3.3 in [11], that the bounded alternation -CSP is in NP. Thus, using just the methods from [11] and [24], we can not prove the Revised Chen Conjecture, but rather some bounded alternation (re)revision.

3.2 co-NP-hardness

Suppose there exist strict subsets of so that , define the relation defined by

where . Strictly speaking, the and are parameters of but we dispense with adding them to the notation since they will be fixed at any point in which we invoke the . The purpose of the relations is to encode co-NP-hardness through the complement of the problem (monotone) -not-all-equal-satisfiability (3NAESAT). Let us introduce also the important relations defined by

where . {lemma} The relation is pp-definable in .

Proof.

We will argue that is definable by the conjunction of instances of that each consider the ways in which two variables may be chosen from each of the , i.e. or or (where is infix for ). We need to show that this conjunction entails (the converse is trivial). We will assume for contradiction that is satisfiable but not. In the first instance of of some atom must be true, and it will be of the form or or . Once we have settled on one of these three, , then we immediately satisfy of the conjunctions of , leaving unsatisfied. Now, we can not evaluate true any of the others among without contradicting our assumption. Thus we are now down to looking at variables with subscript other than and in this fashion we have made the space one smaller, in total . Now, we will need to evaluate in some other atom of the form or or , for . Once we have settled on one of these three then we immediately satisfy of the conjunctions remaining of , leaving still unsatisfied. Iterating this thinking, we arrive at a situation in which clauses are unsatisfied after we have gone through all subscripts, which is a contradiction. ∎

{theorem}

Let be an idempotent algebra on a finite domain . If satisfies EGP, then QCSP is co-NP-hard.

Proof.

We know from Lemma 11 in [24] that there exist strict subsets of so that and the relation is in , for each . From Lemma 3.2, we know also that is in , for each .

We will next argue that enjoys a relatively small specification in DNF (at least, polynomial in ). We first give such a specification for .

which is constant in size when is fixed. Now it is clear from the definition that the size of is polynomial in .

We will now give a very simple reduction from the complement of 3NAESAT to QCSP. 3NAESAT is well-known to be NP-complete [21] and our result will follow.

Take an instance of 3NAESAT which is the existential quantification of a conjunction of atoms . Thus is the universal quantification of a disjunction of atoms . We build our instance of QCSP from by transforming the quantifier-free part to .

( implies .) From an assignment to the universal variables of to elements of , consider elements according to

  • implies ,

  • implies , and

  • implies we don’t care, so w.l.o.g. say .

The disjunct that is satisfied in the quantifier-free part of now gives the corresponding disjunct that will be satisfied in .

( implies .) From an assignment to the universal variables of to elements of , consider elements according to

  • implies is some arbitrarily chosen element in , and

  • implies is some arbitrarily chosen element in .

The disjunct that is satisfied in now gives the corresponding disjunct that will be satisfied in the quantifier-free part of . ∎

The demonstration of co-NP-hardness in the previous theorem was inspired by a similar proof in [2]. Note that an alternative proof that is in is furnished by the observation that it is preserved by all -projections (see [24]). We note surprisingly that co-NP-hardness in Theorem 3.2 is optimal, in the sense that some (but not all!) of the cases just proced co-NP-hard are also in co-NP.

Proposition 1.

Let strict subsets of so that and . Then QCSP is in co-NP.

Proof.

Assume , i.e. (note that the proof is trivial otherwise). Let be an input to QCSP. We will now seek to eliminate atoms () from . Suppose has an atom . If is universally quantified, then is false (since ). Otherwise, either the atom may be eliminated with the variable since does not appear in a non-equality relation; or is false because there is another atom for ; or may be removed by substitution of into all non-equality instances of relations involving . This preprocessing procedure is polynomial and we will assume w.l.o.g. that contains no atoms . We now argue that is a yes-instance iff is a yes-instance, where is built from by instantiating all existentially quantified variables as any . The universal can be evaluated in co-NP (one may prefer to imagine the complement as an existential to be evaluated in NP) and the result follows. ∎

In fact, this being an algebraic paper, we can even do better. Let signify a set of relations on a finite domain but not necessarily itself finite. For convenience, we will assume the set of relations of is closed under all co-ordinate projections and instantiations of constants. Call existentially trivial if there exists an element (which we call a canon) such that for each -ary relation of and each , and for every , whenever then also . We want to expand this class to almost existentially trivial by permitting conjunctions of the form or with relations that are existentially trivial. {lemma} Let be strict subsets of so that and . The set of relations pp-definable in is almost existentially trivial.

Proof.

Consider a formula with a pp-definition in . We assume that only free variables appear in equalities since otherwise we can remove these equalities by substitution. Now existential quantifiers can be removed and their variables instantiated as the canon . Indeed, their atoms may now be removed since they will always be satisfied. Thus we are left with a conjunction of equalities and atoms , and the result follows. ∎

Proposition 2.

If is comprised exclusively of relations that are almost existentially trivial, then QCSP is in co-NP under the DNF encoding.

Proof.

The argument here is quite similar to that of Proposition 1 except that there is some additional preprocessing to find out variables that are forced in some relation to being a single constant or pairs of variables within a relation that are forced to be equal. In the first instance that some variable is forced to be constant in a -ary relation, we should replace with the -ary relation with the requisite forcing. In the second instance that a pair of variables are forced equal then we replace again the -ary relation with a -ary relation as well as an equality. Note that projecting a relation to a single or two co-ordinates can be done in polynomial time because the relations are encoded in DNF. After following these rules to their conclusion one obtains a conjunction of equalities together with relations that are existentially trivial. Now is the time to propagate variables to remove equalities (or find that there is no solution). Finally, when only existentially trivial relations are left, all remaining existential variables may be evaluated to the canon . ∎

{corollary}

Let be strict subsets of so that and . Then QCSP is in co-NP under the DNF encoding. This last result, together with its supporting proposition, is the only time we seem to require the “nice, simple” DNF encoding, rather than arbitrary propositional logic. We do not require DNF for Proposition 1 as we have just a single relation in the signature for each arity and this is easy to keep track of. We note that the set of relations is not maximal with the property that with the constants it forms a co-clone of existentially trivial relations. One may add, for example, .

The following, together with our previous results, gives the refutation of the Alternative Chen Conjecture.

Proposition 3.

Let strict subsets of so that and . Then, for each finite signature reduct of , QCSP is in NL.

Proof.

We will assume contains all constants (since we prove this case gives a QCSP in NL, it naturally follows that the same holds without constants). Take so that, for each , . Recall from Lemma 3.2 that is pp-definable in . We will prove that the structure given by admits a -ary near-unanimity operation as a polymorphism, whereupon it follows that admits the same near-unanimity polymorphism. We choose so that all tuples whose map is not automatically defined by the near-unanimity criterion map to some arbitrary . To see this, imagine that this were not a polymorphism. Then some tuples in would be mapped to some tuple not in which must be a tuple of elements from . Note that column-wise this map may only come from -tuples that have instances of the same element. By the pigeonhole principle, the tuple must appear as one of the tuples in and this is clearly a contradiction.

It follows from [9] that QCSP reduces to a polynomially bounded ensemble of instances CSP, and the result follows. ∎

3.3 The question of the tuple encoding

Proposition 4.

Let and . Then, QCSP is in P under the tuple encoding.

Proof.

Consider an instance of this QCSP of size involving relation but no relation for . The number of tuples in is . Following Proposition 1 together with its proof, we may assume that the instance is strictly universally quantified over a conjunction of atoms (involving also constants). Now, a universally quantified conjunction is true iff the conjunction of its universally quantified atoms is true. We can further say that there are at most atoms each of which involves at most variables. Therefore there is an exhaustive algorithm that takes at most steps with is . ∎

The proof of Proposition 4 suggests an alternative proof of Proposition 3, but placing the corresponding QCSP in P instead of NL. Proposition 4 shows that Chen’s Conjecture fails for the tuple encoding in the sense that it provides a language , expanded with constants, so that Pol has EGP, yet QCSP is in P under the tuple encoding. However, it does not imply that the algebraic approach to QCSP violates Chen’s Conjecture under the tuple encoding. This is because is not of the form Inv for some idempotent algebra . For this stronger result, we would need to prove QCSP is in P under the tuple encoding.

4 Switchability, Collapsability and the three-element case

An algebra is a G-set if its domain is not one-element and every of its operation is of the form where and is a permutation on A. An algebra contains a G-set as a factor if some homomorphic image of a subalgebra of is a G-set. A Gap Algebra [9] is a three-element idempotent algebra that omits a G-set as a factor and is not Collapsible.

Our first task is the deduction of the following theorem, whose lengthy proof appears in Appendix A. For each of the following two theorems, and are chosen such that are strict subsets of , and .

{theorem}

Suppose is a Gap Algebra that is not -projective. Then, for every finite subset of of Inv, Pol is Collapsible.

Our second task is the deduction of the following theorem, whose lengthy proof appears in Appendix B.

{theorem}

Suppose is a -element idempotent algebra that is not -projective, containing a -element G-set as a subalgebra. Then, is Collapsible.

{corollary}

Suppose is a -element idempotent algebra that is not EGP, i.e. is Switchable. Then, for every finite subset of of Inv, Pol is Collapsible.

Proof.

Recall Lemma 11 in [24] that has EGP iff there exists and such that are strict subsets of , , and all operations of are -projective.

If does not contain a G-set as a factor, then is a Gap Algebra and the result follows from Theorem 4. Otherwise, contains a G-set as a factor. If contains a G-set as a homomorphic image then has EGP from [11]. Else, since is -element, contains a -element G-set as a subalgebra and we are in the situation of Theorem 4. ∎

5 A three-element vignette

We would love to be able to improve Theorem 1 to describe the boundary between those cases that are co-NP-complete and those that are Pspace-complete, if indeed such a result is true. However, even in the three-element case this appears challenging, but we are able to provide a variant vignette, whose proof appears in Appendix C. {theorem} Let be an idempotent algebra on a -element domain. Either

  • -CSP is in NP, for all ; or

  • -CSP is co-NP-complete, for all ; or

  • -CSP is -hard, for some .

Note that the trichotomy of Theorem 5 does not hold for QCSP along the same boundary for, respectively, NP, co-NP-complete and Pspace-complete. For the semilattice-without-unit it is known that -CSP is co-NP-complete, for all , while QCSP is Pspace-complete [4].

6 Discussion

The major contribution of this paper is its discussion of the Chen Conjecture with two infinite-signature variants one of which is proved to hold (with encoding in “simple logic”) and one of which fails (with the tuple listing).

In addition to this, the contribution is largely mathematical, examining the relationship between Switchability and Collapsibility in the three-element case. However, this mathematical study uncovers something of importance to the computer scientist who is not reconciled to infinite signatures! Since here it demonstrates that all three-element domain NP-memberships that may be shown by Switchability, may already be shown by Collapsibility.

The work associated with Theorem 4 is distinctly non-trivial and involves a new method, whereas the work associated with Theorem 4 uses known methods and involves mostly turning the handle with these. Similarly, the work involved with the three element vignette uses known methods on top of our earlier new results.

The Chen Conjecture in its original form remains open. As does the general question (for arbitrary finite domains) as to whether, if is Switchable, all finite subsets of Inv are so that Pol is Collapsible. However, to now prove the Chen Conjecture it is sufficient to prove, for any finite expanded with all constants such that Pol has EGP, that there exists polynomially (in ) computable pp-definitions (over ) of the relations (where and are suitably chosen to witness EGP). A first step towards this is to establish whether there are even polynomially sized pp-definitions of these .

The appearance of a co-NP-complete QCSP is likely to be an anomaly of our introduction of infinite signatures. Such a QCSP is unlikely to exist with a finite signature (at least, nothing like this is hitherto known). Indeed, its presence might be used as an argument against the acceptance of infinite signatures, if it is interpreted as an aberration. For the reader in this mind, we ask to please review the earlier paean to infinite signatures.

Acknowledgements

We thank Hubie Chen and Michał Wrona for many useful discussions, as well as two anonymous referees for their advising on a previous draft.

Appendix A: Switchability and Collapsibility of Gap Algebras

Let be a -ary idempotent operation on domain . We say is a generalised Hubie-pol on if, for each , ( in the th position). When this is called a Hubie-pol in and gives -Collapsibility from source . In general, a generalised Hubie-pol does not bestow Collapsibility (e.g. Chen’s -ary Switchable operation , below). The name Hubie operation was used in [7] for Hubie-pol and the fact that this leads to Collapsibility is noted in [9].

For this appendix is an idempotent algebra on a -element domain . Assume has precisely two subalgebras on domains and and contains the idempotent semilattice-without-unit operation which maps all tuples off the diagonal to . Thus, is a Gap Algebra as defined in [11]. Note that the presence of removes the possibility to have a -set as a factor. We say that is -projective if for each -ary in there exists so that, if then and if then . Let us now further assume that is not -projective. This rules out the Gap Algebras that have EGP and we now know that is Switchable [11]. We will now consider the -ary operation defined by Chen in [11]. Let be the idempotent operation satisfying

Chen proved that is -Switchable but not -Collapsible, for any [11]. Let be a -ary operation in that is not -projective. Violation of -projectivity in means that for each either

  • there is and so that , or

  • or and there is so that .

Note that we can rule out the latter possibility and further assume , by replacing if necessary by the -ary . Thus, we may assume that (*) for each there is and so that .

We wish to partition the co-ordinates of into those for which violation of -projectivity, on words in :

  • happens with to but never to .

  • happens with to but never to .

  • happens on both to and to .

Note that Classes and are both non-empty (Class can be empty). This is because if Class were empty then would be a Hubie-pol in and if Class were empty we would similarly have a Hubie-pol in . We will write -tuples with vertical bars to indicate the split between these classes. Suppose there exists a so that . Then we can identify all the variables in one among Class or Class to obtain a new function for which one of these classes is of size one. Note that if, e.g., Class is made singleton, this process may move variables previously in Class into Class , but never to Class .

Thus we may assume that either Class or Class is singleton or, for all over , . Indeed, these singleton cases are dual and thus w.l.o.g. we need only prove one of them. Recall the global assumptions are in force for the remainder of the paper.

.1 Properties of Gap Algebras that are Switchable

{lemma}

Any algebra over containing and is either Collapsible or has binary term operations and so that

  • and , and

  • and .

Proof.

Consider a tuple over that witnesses the breaking of -projectivity for some Class variable from to ; so . Let be with the s substituted by and the s substituted by . If, for each such over that witnesses the breaking of -projectivity for each Class variable, we find , then is a Hubie-pol in . Thus, for some such