The complexity of positive first-order logic without equality∗

Florent MadelaineUniv Clermont1, EA2146, Laboratoire d’algorithmique et d’image de Clermont-Ferrand,

Aubiere, F-63170, France.fmadelaine@laic.u-clermont1.fr

Barnaby MartinEquipe de Logique Mathematique - CNRS UMR 7056, Universite Denis Diderot - Paris 7,

UFR de Mathematiques - case 7012, site Chevaleret 75205 Paris Cedex 13, France.barnabymartin@gmail.com

Abstract

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finitestructure B. This may be seen as a natural generalisation ofthe non-uniform quantified constraint satisfaction problemQCSP(B). We introduce surjective hyper-endomorphismsand use them in proving a Galois connection that charac-terises definability in positive equality-free FO. Through analgebraic method, we derive a complete complexity classifi-cation for our problems as B ranges over structures of sizeat most three. Specifically, each problem is either in L, isNP-complete, is co-NP-complete or is Pspace-complete.

1 Introduction

The evaluation problem under a logic L – here always afragment of first-order logic (FO) – takes as input a structure(model) B and a sentence ϕ ofL, and asks whether B |= ϕ.1

When L is the existential conjunctive positive fragment ofFO, {∃,∧}-FO, the evaluation problem is equivalent to themuch-studied constraint satisfaction problem (CSP). Sim-ilarly, when L is the (quantified) conjunctive positive frag-ment of FO, {∃,∀,∧}-FO, the evaluation problem is equiv-alent to the well-studied quantified constraint satisfactionproblem (QCSP). In this manner, the QCSP is the gen-eralisation of the CSP in which universal quantification isrestored to the mix. In both cases it is essentially irrele-vant whether or not equality is permitted in the sentences,

∗The second author acknowledges the support of the project ANRENUM (ANR-07-BLAN-0327).

1We resist the better known terminology of ‘model checking problem’because in the majority of this paper we consider the structure B to befixed.

as it may be propagated out by substitution. Much work hasbeen done on the parameterisation of these problems by thestructure B – that is, where B is fixed and only the sentenceis input. It is conjectured [8] that the ensuing problemsCSP(B) attain only the complexities P and NP-complete.This may appear surprising given that 1.) so many naturalNP problems may be expressed as CSPs (see, e.g., myriadexamples in [10]) and 2.) NP itself does not have this ‘di-chotomy’ property (assuming P 6= NP) [12]. While thisdichotomy conjecture remains open, it has been proved forcertain classes of B (e.g., for structures of size at most three[4] and for undirected graphs [9]). The like parameterisa-tion of the QCSP is also well-studied, and while no over-arching polychotomy has been conjectured, only the com-plexities P, NP-complete and Pspace-complete are knownto be attainable (for trichotomy results on certain classessee [3, 18], as well as the dichotomy for boolean structures,e.g., in [6]).

In previous work, [16, 15], we have studied the evalu-ation problem, parameterised by the structure, under var-ious fragments of FO obtained by restrictions on whichof the symbols of {∃,∀,∧,∨,¬,=, 6=} is permitted. Ofcourse, many of the ostensibly 27 such fragments may bediscarded as totally trivial or as repetitions through de Mor-gan duality. There are four fragments each equivalent tothe CSP and QCSP: these are {∃,∧}-FO, {∃,∧,=}-FO,{∀,∨}-FO, {∀,∨, 6=}-FO and {∃,∀,∧}-FO, {∃,∀,∧,=}-FO, {∃,∀,∨}-FO, {∃,∀,∨, 6=}-FO, respectively. Here,equivalent means that a complexity classification for oneyields a complexity classification for the other; but, thecomplexity classes need not be the same. For example,the class of problems given by fixing the structure under{∃,∧}-FO would display dichotomy between P and NP-complete iff the like class of problems under {∀,∨}-FOdisplays dichotomy between P and co-NP-complete. Var-

2009 24th Annual IEEE Symposium on Logic In Computer Science

1043-6871/09 $25.00 © 2009 IEEE

DOI 10.1109/LICS.2009.15

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ious complexity classifications are obtained in [16, 15] andit is observed that the only interesting fragment, other thanthe eight associated with CSP and QCSP, is {∃,∀,∧,∨}-FO.2 The evaluation problem over {∃,∀,∧,∨}-FO may beseen as the generalisation of the QCSP in which disjunc-tion is returned to the mix. Note that the absence of equal-ity is here important, as there is no general method for itsbeing propagated out by substitution. Indeed, we will seethat evaluating the related fragment {∃,∀,∧,∨,=}-FO isPspace-complete on any structure B of size at least two.

In this paper we initiate a study of the evaluation prob-lem for the fragment {∃,∀,∧,∨}-FO over a fixed relationalB – the problem we denote {∃,∀,∧,∨}-FO(B). We demon-strate at least that this class displays a complexity-theoreticrichness absent from those other fragments that are not as-sociated with the CSP or QCSP. It is possibly to be hoped,however, that a full classification for this class is not as re-sistant as that for the CSP or QCSP. We undertake ourstudy through the algebraic method that has been so fruitfulin the study of the CSP and QCSP (see [11, 4, 3, 5]). Tothis end, we define surjective hyper-endomorphisms and usethem to define a new Galois connection that characterisesdefinability under {∃,∀,∧,∨}-FO.3 We are able to prove acomplete complexity classification for {∃,∀,∧,∨}-FO(B)when B ranges over structures of size at most three. Onthe class of boolean structures we see dichotomy betweenL and Pspace-complete. On the class of structures ofsize three we see tetrachotomy between L, NP-complete,co-NP-complete and Pspace-complete. Some of the re-sults that appear in this paper had been obtained throughadhoc methods in [17] – although there the tetrachotomyextends only to digraphs and not arbitrary relational struc-tures. Also, little insight was provided as to the underlyingproperties of the classification. It is a pleasing consequenceof our algebraic approach that we can give quite simple ex-planation to the delineation our subclasses.

The paper is organised as follows. In Section 2, we in-troduce the preliminaries, including the relevant Galois con-nection together with the central notions of surjective hyper-endomorphism (she) and down-she-monoid. In Section 3,we begin by outlining conditions under which the problem

2For many of the other fragments the complexity classification is nearlytrivial. For example, this is true for {∃,∧,∨}-FO, {∀,∧,∨}-FO and{∃, ∀,∧,∨,¬}-FO (also for these classes with = or 6=). For others theclassification may be read through the Schaefer classification for booleanCSP and QCSP, because computational hardness is clear over fixed struc-tures of size at least three. For example, this is the case for {∃,∧, 6=}-FO,{∀,∨,=}-FO and {∃, ∀,∧, 6=}-FO, {∃, ∀,∨,=}-FO. Note that the con-sideration of 6= is not explicit in [16, 15]. Similarly, fragments involvingboth quantifiers and = or 6= are not explicitly considered. In both cases,the results may be read off from de Morgan duality together with standardSchaefer class results (for which we refer to [6]).

3While this Galois connection appears here for the first time, it doesfollow a general recipe as outlined, e.g., in [2]. Note that it is not clearthat the many different Galois connections associated with fragments ofFO can be proved in a straightforwardly uniform manner.

{∃,∀,∧,∨}-FO(B) either drops from or attains maximalcomplexity. We then proceed to classify the complexityof the problems {∃,∀,∧,∨}-FO(B), when B ranges over,firstly, boolean structures and, secondly, structures of sizethree. In the first instance a dichotomy – between L andPspace-complete – is obtained; in the second instance atetrachotomy – between L, NP-complete, co-NP-completeand Pspace-complete – is obtained. We conclude, in Sec-tion 4, with some final remarks. Central to our tetrachotomyresult is that the lattice of down-she-monoids in the three-element case has a certain structure. The proof that the lat-tice has this structure proceeds by a basic exhaustive analy-sis of cases. For reasons of space, this proof is omitted, butmay be found in an extended version of this paper [14].

2 Preliminaries

Throughout, let B be a finite structure, with domain B,over the finite relational signature σ. Let {∃,∀,∧,∨}-FOand {∃,∀,∧,∨,=}-FO be the positive fragments of first-order (FO) logic, without and with equality, respectively.An extensional relation is one that appears in the signatureσ. We will usually denote extensional relations of B by Rand other relations by S (or by some formula that definesthem). In {∃,∀,∧,∨}-FO the atomic formulae are exactlysubstitution instances of extensional relations. The problem{∃,∀,∧,∨}-FO(B) has:

• Input: a sentence ϕ ∈ {∃,∀,∧,∨}-FO.

• Question: does B |= ϕ?

The related problem {∃,∀,∧,∨,=}-FO(B) permits sen-tences ϕ that may involve equalities, in the obvious way.When B is of size one, the evaluation of any FO sentencemay be accomplished in L (essentially, the quantifiers areirrelevant and the problem amounts to the boolean sentencevalue problem, see [13]). In this case, it follows that both{∃,∀,∧,∨}-FO(B) and {∃,∀,∧,∨,=}-FO(B) are also inL.

Consider the set B and its power set P(B). A hyper-operation on B is a function f : B → P(B) \ {∅} (that theimage may not be the empty set corresponds to the hyper-operation being total, in the parlance of [1]). If the hyper-operation f has the additional property that

• for all y ∈ B, there exists x ∈ B such that y ∈ f(x),

then we designate (somewhat abusing terminology) f sur-jective. A surjective hyper-operation in which each elementis mapped to a singleton set is identified with a permuta-tion (bijection). A surjective hyper-endomorphism (she) ofB is a surjective hyper-operation f on B that satisfies, forall extensional relations R of B,

430

• if B |= R(x1, . . . , xi) then, for all y1 ∈ f(x1),. . . , yi ∈ f(xi), B |= R(y1, . . . , yi).

More generally, for r1, . . . , rk ∈ B, we say f is a shefrom (B, r1, . . . , rk) to (B, r′1, . . . , r′k) if f is a she of B andr′1 ∈ f(r1), . . . , r′k ∈ f(rk). A she may be identified witha surjective endomorphism if each element is mapped to asingleton set. On finite structures surjective endomorphismsare necessarily automorphisms.

For b1, . . . , b|B| an enumeration of the elements of B,let the quantifier-free formula ΦB(v1, . . . , v|B|) be a con-junction of the positive facts of B, where the variablesv1, . . . , v|B| correspond to the elements b1, . . . , b|B|. Thatis, for R an extensional relation of B, R(vλ1 , . . . , vλi

) ap-pears as an atom in ΦB iff B |= R(bλ1 , . . . , bλi

). For exam-ple, let K3 be the antireflexive 3-clique, that is the structurewith domain {0, 1, 2} and single binary relation

E := {(0, 1), (1, 0), (1, 2), (2, 1), (2, 0), (0, 2)}.

Then

ΦK3(v0, v1, v2) := E(v0, v1) ∧ E(v1, v0) ∧ E(v1, v2)∧E(v2, v1) ∧ E(v2, v0) ∧ E(v0, v2).

The existential sentence ∃v1, . . . , v|B| ΦB(v1, . . . , v|B|)is known as the canonical query of B. More gener-ally, for a (not necessarily distinct) l-tuple of elementsr := (r1, . . . , rl) ∈ Bl, define the quantifier-freeΦB(r)(v1, . . . , vl) to be the conjunction of the positive factsof r, where the variables v1, . . . , vl correspond to the el-ements r1, . . . , rl. That is, R(vλ1 , . . . , vλi) appears as anatom in ΦB(r) iff B |= R(rλ1 , . . . , rλi). For example,

ΦK3(0,0,2)(v0, v1, v2) := E(v0, v2) ∧ E(v2, v0)∧E(v1, v2) ∧ E(v2, v1).

We refer to elements in B as r, s, t (also x, y), orb1, . . . , b|B| when this is an enumeration. We reserve u, v, wto refer to variables in FO formulae.

2.1 Galois Connections

For a set F of surjective hyper-operations on the finitedomain B, let Inv(F ) be the set of relations on B of whicheach f ∈ F is a she (when these relations are viewed as astructure over B). We say that S ∈ Inv(F ) is invariant orpreserved by (the hyper-operations in) F . Let shE(B) be theset of shes of B. Let Aut(B) be the set of automorphisms ofB.

Let 〈B〉{∃,∀,∧,∨}-FO and 〈B〉{∃,∀,∧,∨,=}-FO be the sets ofrelations that may be defined on B in {∃,∀,∧,∨}-FO and{∃,∀,∧,∨,=}-FO, respectively.

Lemma 1. Let r := (r1, . . . , rk) be a k-tuple of elementsof B. There exists:

(i). a formula ηr(u1, . . . , uk) ∈ {∃,∀,∧,∨,=}-FO s.t.(B, r′1, . . . , r′k) |= ηr(u1, . . . , uk) iff there is an au-tomorphism from (B, r1, . . . , rk) to (B, r′1, . . . , r′k).

(ii). a formula θr(u1, . . . , uk) ∈ {∃,∀,∧,∨}-FO s.t.(B, r′1, . . . , r′k) |= θr(u1, . . . , uk) iff there is a shefrom (B, r1, . . . , rk) to (B, r′1, . . . , r′k).

Proof. For Part (i), let b1, . . . , b|B| an enumeration of theelements of B and ΦB(v1, . . . , v|B|) be the associated con-junction of positive facts. Set ηr(u1, . . . , uk) :=

∃v1, . . . , v|B| ΦB(v1, . . . , v|B|)∧∀v (v = v1 ∨ . . . ∨ v = v|B|)∧u1 = vλ1 ∧ . . . ∧ uk = vλk

,

where r1 = bλ1 , . . . , rk = bλk. The forward direction

follows since B is finite, so any surjective endomorphismis necessarily an automorphism. The backward directionfollows since all first-order formulae are preserved by auto-morphism.

[Part (ii).] This will require greater dexterity. Letr ∈ Bk, s := (b1, . . . , b|B|) be an enumeration of B andt ∈ B|B|. Recall that ΦB(r,s)(u1, . . . , uk, v1, . . . , v|B|)is a conjunction of the positive facts of (r, s), where thevariables (u,v) correspond to the elements (r, s). Sim-ilarly, ΦB(r,s,t)(u1, . . . , uk, v1, . . . , v|B|, w1, . . . , w|B|) isthe conjunction of the positive facts of (r, s, t), where thevariables (u,v,w) correspond to the elements (r, s, t). Setθr(u1, . . . , uk) :=

∃v1, . . . , v|B| ΦB(r,s)(u1, . . . , uk, v1, . . . , v|B|)∧

∀w1 . . . w|B|∨t∈B|B| ΦB(r,s,t)(u1, . . . , uk, v1, . . . , v|B|, w1, . . . , w|B|).

[Part (ii), backwards.] Suppose f is a she from(B, r1, . . . , rk) to (B′, r′1, . . . , r′k), where B′ := B (wewill wish to differentiate the two occurrences of B). Weaim to prove that B′ |= θr(r′1, . . . , r

′k). Choose arbi-

trary s′1 ∈ f(b1), . . . , s′|B| ∈ f(b|B|) as witnesses forv1, . . . , v|B|. Let t′ := (t′1, . . . , t

′|B|) ∈ B′|B| be any val-

uation of w1, . . . , w|B| and take arbitrary t1, . . . , t|B| s.t.t′1 ∈ f(t1), . . . , t′|B| ∈ f(t|B|) (here we use surjectivity).Let t := (t1, . . . , t|B|). It follows from the definition of shethat

B′ |= ΦB(r,s)(r′1, . . . , r′k, s

′1, . . . , s

′|B|)∧

ΦB(r,s,t)(r′1, . . . , r′k, s

′1, . . . , s

′|B|, t

′1, . . . , t

′|B|).

[Part (ii), forwards.] Assume that B′ |= θr(r′1, . . . , r′k),

where B′ := B. Let b′1, . . . , b′|B| be an enumeration of

B′ := B.4 Choose some witness elements s′1, . . . , s′|B| for

4One may imagine b1, . . . , b|B| and b′1, . . . , b′|B| to be the same enu-

meration, but this is not essential. In any case, we will wish to keep thedashes on the latter set to remind us they are in B′ and not B.

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v1, . . . , v|B| and a witness tuple t := (t1, . . . , t|B|) ∈ B|B|

s.t.

(†) B′ |= ΦB(r,s)(r′1, . . . , r′k, s

′1, . . . , s

′|B|)∧

ΦB(r,s,t)(r′1, . . . , r′k, s

′1, . . . , s

′|B|, b

′1, . . . , b

′|B|).

Consider the following partial hyper-operations from B →P(B′) \ {∅}.

1. fr given by fr(ri) := {r′i}, for 1 ≤ i ≤ k.

2. fs given by fs(bi) = {s′i}, for 1 ≤ i ≤ |B|.

3. ft given by b′i ∈ ft(bj) iff ti = bj , for 1 ≤ i, j ≤ |B|.

Let f := fr ∪ fs ∪ ft; f is a hyper-operation whose surjec-tivity is guaranteed by ft (note that totality is guaranteed byfs). That f is a she follows from the right-hand conjunct of(†).

Theorem 1. For a finite structure B we have

(i). 〈B〉{∃,∀,∧,∨,=}-FO = Inv(Aut(B)) and

(ii). 〈B〉{∃,∀,∧,∨}-FO = Inv(shE(B)).

Proof. Part (i) is well-known and may be proved in a simi-lar, albeit simpler, manner to Part (ii), which we now prove.

[ϕ(v) ∈ 〈B〉{∃,∀,∧,∨}-FO ⇒ ϕ(v) ∈ Inv(shE(B)).]This is proved by induction on the complexity of ϕ(v).

(Base Case.) ϕ(v) := R(v).5 Follows from the defini-tion of she.

(Inductive Step.) There are four subcases. We progressthrough them in a workmanlike fashion. Take f ∈ shE(B).ϕ(v) := ψ(v)∧ψ′(v).6 Let v := (v1, . . . , vl). Suppose

B |= ϕ(x1, . . . , xl); then both B |= ψ(x1, . . . , xl) and B |=ψ′(x1, . . . , xl). By Inductive Hypothesis (IH), for any y1 ∈f(x1), . . . , yl ∈ f(xl), both B |= ψ(y1, . . . , yl) and B |=ψ′(y1, . . . , yl), whence B |= ϕ(y1, . . . , yl).ϕ(v) := ψ(v) ∨ ψ′(v). Let v := (v1, . . . , vl). Suppose

B |= ϕ(x1, . . . , xl); then one of B |= ψ(x1, . . . , xl) or B |=ψ′(x1, . . . , xl); w.l.o.g. the former. By IH, for any y1 ∈f(x1), . . . , yl ∈ f(xl), B |= ψ(y1, . . . , yl), whence B |=ϕ(y1, . . . , yl).ϕ(v) := ∀w ψ(v, w). Let v := (v1, . . . , vl). Sup-

pose B |= ∀w ψ(x1, . . . , xl, w); then for each x′, B |=ψ(x1, . . . , xl, x

′). By IH, for any y1 ∈ f(x1), . . . , yl ∈f(xl), we have for all y′ (remember f is surjective), B |=ψ(y1, . . . , yl, y

′), whereupon B |= ∀w ψ(y1, . . . , yl, w).ϕ(v) := ∃w ψ(v, w). Let v := (v1, . . . , vl). Sup-

pose B |= ∃w ψ(x1, . . . , xl, w); then for some x′, B |=5The variables v may appear multiply inR and in any order. ThusR is

an instance of an extensional relation under substitution and permutationof positions.

6The presence of, e.g., v in ψ(v)∧ψ′(v) should not be taken as indi-cation that all v appear free in both ψ and ψ′.

ψ(x1, . . . , xl, x′). By IH, for any y1 ∈ f(x1), . . . , yl ∈

f(xl), y′ ∈ f(x′) (remember f(x′) can not be empty), B |=ψ(y1, . . . , yl, y

′), whereupon B |= ∃w ψ(y1, . . . , yl, w).[S ∈ Inv(shE(B)) ⇒ S ∈ 〈B〉{∃,∀,∧,∨}-FO.] Consider

the k-ary relation S ∈ Inv(shE(B)). Let r1, . . . , rm be thetuples of S. Set

θS(u1, . . . , uk) := θr1(u1, . . . , uk)∨. . .∨θrm(u1, . . . , uk).

Manifestly, θS(u1, . . . , uk) ∈ {∃,∀,∧,∨}-FO. Forri := (ri1, . . . , rik), note that (B, ri1, . . . , rik) |=θri(u1, . . . , uk) (the ‘identity’ she will be formally intro-duced in the next section). That θS(u1, . . . , uk) = Snow follows from Part (ii) of Lemma 1, since S ∈Inv(shE(B)).

In the following,≤L indicates the existence of a logspacemany-to-one reduction.

Theorem 2. Let B and B′ be finite structures over the samedomain B.

(i). If Aut(B) ⊆ Aut(B′) then {∃,∀,∧,∨,=}-FO(B′) ≤L

{∃,∀,∧,∨,=}-FO(B).

(ii). If shE(B) ⊆ shE(B′) then {∃,∀,∧,∨}-FO(B′) ≤L

{∃,∀,∧,∨}-FO(B).

Proof. Again, Part (i) is well-known and the proof is simi-lar to that of Part (ii), which we give. If shE(B) ⊆ shE(B′),then Inv(shE(B′)) ⊆ Inv(shE(B)). From Theorem 1, it fol-lows that 〈B′〉{∃,∀,∧,∨}-FO ⊆ 〈B〉{∃,∀,∧,∨}-FO. Recallingthat B′ contains only a finite number of extensional rela-tions, we may therefore effect a Logspace reduction from{∃,∀,∧,∨}-FO(B′) to {∃,∀,∧,∨}-FO(B) by straightfor-ward substitution of predicates.

2.2 Down-she-monoids

Consider a finite domain B. The identity hyper-operation idB is defined by x 7→ {x}. Given hyper-operations f and g, define the composition g◦f by x 7→ {z :∃y z ∈ g(y) ∧ y ∈ f(x)}. Finally, a hyper-operation f is asub-hyper-operation of g – denoted f ⊆ g – if f(x) ⊆ g(x),for all x. A set of surjective hyper-operations on a finite setB is a down-she-monoid, if it contains idB , and is closedunder composition and sub-hyper-operations (of course, notall sub-hyper-operations of a surjective hyper-operation aresurjective – we are only concerned with those that are). idB

is a she of all structures, and, if f and g are shes of B, thenso is g ◦ f . Further, if g is a she of B, then so is f forall (surjective) f ⊆ g. It follows that shE(B) is always adown-she-monoid. The down-she-monoids of B form a lat-tice under (set-theoretic) inclusion and, as per the Galois

432

connection of the previous section, classify the complexi-ties of {∃,∀,∧,∨}-FO(B). If F is a set of surjective hyper-operations on B, then let 〈F 〉 denote the minimal down-she-monoid containing the operations of F . If F is the sin-gleton {f}, then, by abuse of notation, we write 〈f〉 insteadof 〈{f}〉

For a surjective hyper-operation f , define its inverse f−1

by x 7→ {y : x ∈ f(y)}. Note that f−1 is also a surjectivehyper-operation and (f−1)−1 = f , though f ◦ f−1 = idB

only if f is a permutation. For a set of surjective hyper-operations F , let F−1 := {f−1 : f ∈ F}. If F is adown-she-monoid then so is F−1. We will see this alge-braic duality resonates with the de Morgan duality of ∃ and∀, and the complexity-theoretic duality of NP and co-NP.However, we resist discussing it further as it plays no directrole in the derivation of our results.

A permutation subgroup on a finite set B is a set of per-mutations of B closed under composition. It may easily beverified that such a set contains the identity and is closed un-der inverse. A permutation subgroup may be identified witha particular type of down-she-monoid in which all hyper-operations have only singleton sets in their range. The per-mutation subgroups form a lattice under inclusion whoseminimal element contains just the identity and whose maxi-mal element is the symmetric group S|B|. As per the Galoisconnection of the previous section, this lattice classifies thecomplexities of {∃,∀,∧,∨,=}-FO(B) – although we shallsee these are relatively uninteresting.

In the lattice of down-she-monoids, the minimal elementstill contains just idB , but the maximal element contains allsurjective hyper-operations. However, the lattice of permu-tation subgroups always appears as a sub-lattice within thelattice of down-she-monoids.

3 Classification results

We are now in a position to study the interplay betweenthe shes of a structure B and the complexity of the problem{∃,∀,∧,∨}-FO(B).

3.1 Shes inducing lower complexity

We begin by studying three classes of she, the pres-ence of any of which reduces the complexity of the prob-lem {∃,∀,∧,∨}-FO(B). Let B be a finite structure, withdistinct elements b, b′. We define the following surjectivehyper-operations from B to P(B) \ {∅}.

∀b(x) :={

B if x = b{x} otherwise.

∃b(x) := {x, b}

∀b∃b′(x) :={

B if x = b{b′} otherwise.

We call their classes ∀-, ∃- and ∀∃-hyper-operations, respec-tively. In Figure 1, four digraphs G1–G4 are drawn. For

0

0

1

1

2

2

G1 G2

0

0

1

1 2

2

G3 G4

Figure 1. Sample digraphs admitting ∀-, ∃-and ∀∃-hyper-operations as shes.

typographic reasons we will mark-up, e.g., the surjectivehyper-operation 0 7→ {0, 1}, 1 7→ {1} and 2 7→ {1, 2} as

0 011 12 12

. It may easily be verified that the down-she monoids

shE(G1)–shE(G4) are as follows.

shE(G1) shE(G2) shE(G3) shE(G4)〈 0 01

1 12 12

〉 〈 0 01 0122 2

〉 〈 0 21 0122 2

〉 〈 0 0121 12 012

〉

We see that G1, G2 and G3 admit the shes ∃1, ∀1 and ∀1∃2,respectively. G4 admits each of the shes ∀0, ∀2, ∃1, ∀0∃1

and ∀2∃1.

Remarks 1. We have not considered shes ∀b∃b, defined asabove but with b′ := b. The down-she-monoid 〈∀b∃b〉 iseasily seen to contain all surjective hyper-operations. It fol-lows that any structure B that has ∀b∃b as a she already hasall shes of the form ∀b′∃b′′ with b′ 6= b′′.

Note that the down-she-monoids 〈∀b∃b′〉 and〈{∀b,∃b′}〉 = 〈∀b ◦ ∃b′〉 = 〈∃b′ ◦ ∀b〉 do not in gen-eral coincide, though the first is always a subset of thefollowing three. Also, we note the identities ∃−1

b = ∀b,∀−1

b = ∃b and (∀b∃b′)−1 = ∀b′∃b.

We now give a series of three lemmas, one associ-ated with each of the surjective hyper-operations ∀b, ∃b

and ∀b∃b′ . They will ultimately be used in a form ofquantifier elimination that will diminish the complexity of{∃,∀,∧,∨}-FO(B), if B has one of these as a she.

Lemma 2. Let ϕ(u,v) be a formula of {∃,∀,∧,∨}-FO.Let B be a finite structure with ∀b as a she. Then

B |= ∀u ϕ(u,v) ⇐⇒ B |= ϕ(b,v).

433

Proof. The forward direction is trivial; we prove the back-ward. Consider the relation defined by the formula ϕ(u,v),where v := (v1, . . . , vk), of {∃,∀,∧,∨}-FO on B. ByTheorem 1, it is invariant under ∀b ∈ shE(B). For anyx1, . . . , xk ∈ B, assume B |= ϕ(b, x1, . . . , xk). Taking anarbitrary c ∈ B, and noting each xi ∈ ∀b(xi) and c ∈ ∀b(b),we derive B |= ϕ(c, x1, . . . , xk). The result follows.

Lemma 3. Let ϕ(u,v) be a formula of {∃,∀,∧,∨}-FO.Let B be a finite structure with ∃b as a she. Then

B |= ∃u ϕ(u,v) ⇐⇒ B |= ϕ(b,v).

Proof. The backward direction is trivial; we prove the for-ward. Consider the relation defined by the formula ϕ(u,v),where v := (v1, . . . , vk), of {∃,∀,∧,∨}-FO on B. ByTheorem 1, it is invariant under ∃b ∈ shE(B). Forany x1, . . . , xk ∈ B, and some c ∈ B, assume B |=ϕ(c, x1, . . . , xk). Noting each xi ∈ ∃b(xi) and b ∈ ∃b(c),we derive B |= ϕ(b, x1, . . . , xk). The result follows.

Lemma 4. Let ϕ(u,v) be a formula of {∃,∀,∧,∨}-FO,where the arity of v is k. Let B be a finite structure with∀b∃b′ as a she. For all c ∈ B and x := (x1, . . . , xk) ∈{b, b′}k,

B |= ϕ(b,x)(I)

=⇒ B |= ϕ(c,x)(II)=⇒ B |= ϕ(b′,x).

Proof. Consider the relation defined by the formula ϕ(u,v)of {∃,∀,∧,∨}-FO onB. By Theorem 1, it is invariant under∀b∃b′ ∈ shE(B). Take arbitrary c ∈ B. Noting that xi isfrom {b, b′}, we have xi ∈ ∀b∃b′(xi) and c ∈ ∀b∃b′(b).Part (I) follows. Now noting that b′ ∈ ∀b∃b′(c), Part (II)follows.

We are now ready to state how the presence of ∀-, ∃-or ∀∃-hyper-operations as shes of B can diminish the com-plexity of {∃,∀,∧,∨}-FO(B). In each case we proceed byquantifier elimination.

Theorem 3. If B has a ∀-operation as a she then{∃,∀,∧,∨}-FO(B) is in NP. If B has an ∃-operation asa she then {∃,∀,∧,∨}-FO(B) is in co-NP. If B has a ∀∃-operation as a she then {∃,∀,∧,∨}-FO(B) is in L.

Proof. Let ϕ be a sentence of {∃,∀,∧,∨}-FO, and letϕ[∀/b] (respectively, ϕ[∃/b] and ϕ[∀/b,∃/b′]) be ϕ with alluniversal variables substituted by b (respectively, existentialvariables substituted by b and universal variables substitutedby b and existential variables substituted by b′).

If B has a she ∀b, then consider a sentence ϕ ∈{∃,∀,∧,∨}-FO, w.l.o.g. in prenex form. It follows by re-peated application of Lemma 2 on ϕ – either from the out-ermost quantifier in, or from the innermost quantifier out– that B |= ϕ iff B |= ϕ[∀/b]. Similarly, if B has a she

∃b′ , then it follows by repeated application of Lemma 3 thatB |= ϕ iff B |= ϕ[∃/b′].

If B has a she ∀b∃b′ , then, again, assume the sentenceϕ ∈ {∃,∀,∧,∨}-FO to be in prenex form. It follows by re-peated application of Lemma 4 – from the outermost quan-tifier in – that B |= ϕ iff B |= ϕ[∀/b,∃/b′]. Note that, in thiscase, one can not move from the innermost quantifier outbecause this may involve the possibility of free variablestaking values from outside the set {b, b′}. The result nowfollows since evaluating ϕ[∀/b,∃/b′] on B is equivalent to aboolean sentence value problem, known to be in L [13].

Returning to the examples of Figure 1, we see that{∃,∀,∧,∨}-FO(G1) is in co-NP, {∃,∀,∧,∨}-FO(G2) is inNP and both {∃,∀,∧,∨}-FO(G3) and {∃,∀,∧,∨}-FO(G4)are in L.

3.2 Down-she-monoids of high complexity

Lemma 5. Let B, with |B| ≥ 2, be a structure s.t. shE(B)is a permutation subgroup. Then {∃,∀,∧,∨}-FO(B) isPspace-complete.

Proof. Let BNAE be the structure on B with a singleternary relation RNAE := B3 \ {(b, b, b) : b ∈ B}.{∃,∀,∧,∨}-FO(BNAE) is a generalisation of the problemQCSP(BNAE), well-known to be Pspace-complete (in thecase |B| = 2, this is quantified not-all-equal 3-satisfiability,see, e.g., [19]). shE(BNAE) is the symmetric group S|B|.The statement of the theorem now follows from Theorem 2,since shE(B) ⊆ shE(BNAE).

Corollary 1. For all B s.t. |B| ≥ 2, {∃,∀,∧,∨,=}-FO(B)is Pspace-complete.

Proof. {∃,∀,∧,∨,=}-FO(B) may be rephrased as theproblem {∃,∀,∧,∨}-FO(B′), where B′ is the structure Bexpanded with the graph of equality. Owing to the presenceof the graph of equality, shE(B′) must be a permutation sub-group, and the result follows from the previous lemma.

The following is a generalisation of Lemma 5.

Lemma 6. Let B be a structure whose universe admits thepartition B1, . . . , Bl (l ≥ 2). If all shes of B are sub-hyper-operations of some f of the form f(x) := Bi iffx ∈ Bπ(i), for π a permutation on the set {1, . . . , l}, then{∃,∀,∧,∨}-FO(B) is Pspace-complete.

Proof. Let K|B1|,...,|Bl| be the complete l-partite graphwith partitions of size |B1|, . . . , |Bl|. It may easily beverified that the shes of K|B1|,...,|Bl| are of the formof the lemma. Furthermore, K|B1|,...,|Bl| agrees withthe antireflexive l-clique Kl on all sentences of equality-free FO logic (for more detail on why this is, see,e.g., the Homomorphism Theorem of [7]), and certainly

434

0

0

1

1

2

2

G6G5

0 0

1

2

1

2

G7 G8

Figure 2. Further sample digraphs.

{∃,∀,∧,∨}-FO. Pspace-hardness of {∃,∀,∧,∨}-FO(Kl)follows from Lemma 5, and so Pspace-hardness of{∃,∀,∧,∨}-FO(K|B1|,...,|Bl|) follows a fortiori. Finally,Pspace-hardness of {∃,∀,∧,∨}-FO(B) now follows fromTheorem 2, since shE(B) ⊆ shE(K|B1|,...,|Bl|).

In Figure 2, four more digraphs G5–G8 are drawn. It mayeasily be verified that shE(G5)–shE(G8) are as follows.

shE(G5) shE(G6) shE(G7) shE(G8)〈 0 0

1 12 2

〉 〈 0 01 12 2

〉 〈 0 01 22 1

〉 〈 0 021 12 02

〉

It follows from Lemmas 5 and 6 that each of{∃,∀,∧,∨}-FO(G5), . . . , {∃,∀,∧,∨}-FO(G8) is Pspace-complete.

3.3 The boolean case

We consider the case |B| = 2, with the normalised do-main B := {0, 1}. It may easily be verified that there arefive down-she-monoids in this case, depicted as a lattice inFigure 3. The two elements of this lattice that represent thetwo subgroups of S2 are drawn in the middle and bottom.

Theorem 4 (Dichotomy). Let B be a boolean structure.

I. If either ∀0∃1 or ∀1∃0 (i.e., 0 011 1

or 0 01 01

) is a

she of B, then {∃,∀,∧,∨}-FO(B) is in L.

II. Otherwise, {∃,∀,∧,∨}-FO(B) is Pspace-complete.

Proof. shE(B) must be one of the five down-she-monoidsdepicted in Figure 3. If shE(B) contains one of ∀0∃1 or∀1∃0, then L membership follows from Theorem 3. Other-wise shE(B) is either 〈 0 0

1 1〉 or 〈 0 1

1 0〉; in both cases

the hardness result follows from Lemma 5.

⟨0 011 01

⟩L

⟨0 01 01

⟩L

<<zzzzzzzz *0 11 0

+Pspace− c

OO

⟨0 011 1

⟩L

bbDDDDDDDD

*0 01 1

+Pspace− c

bbEEEEEEEE

OO <<yyyyyyyy

Figure 3. The boolean lattice of down-she-monoids with their associated complexity.

Remark 1. In the boolean case, 〈∀1∃0〉 = 〈{∀1,∃0}〉 and〈∀0∃1〉 = 〈{∀0,∃1}〉.

3.4 The three-element case

We consider the case |B| = 3, with the normalised do-main B := {0, 1, 2}. In fact, we have the necessary ma-chinery to obtain a full classification theorem. We refer tothe six shes

∀0∃1 ∀1∃0 ∀0∃2 ∀2∃0 ∀1∃2 ∀2∃10 0121 12 1

0 01 0122 0

0 0121 22 2

0 01 02 012

0 21 0122 2

0 11 12 012

as L-shes. In Figure 4, the lattice of down-she-monoids ispartially drawn (all the remaining down-she-monoids con-tain an L-she). The classification theorem depends on thefact that the lattice is fully drawn on down-she-monoids thatdo not contain an L-she. We remind the reader of the fol-lowing six shes which also play a role in our classification.

∀0 ∀1 ∀2 ∃0 ∃1 ∃20 0121 12 2

0 01 0122 2

0 01 12 012

0 01 012 02

0 011 12 12

0 021 122 2

Theorem 5 (Tetrachotomy). Let B be a three-elementstructure.

I. If shE(B) contains any of the L-shes, then{∃,∀,∧,∨}-FO(B) is in L.

II. If shE(B) contains none of the L-shes, but containsone of ∀0, ∀1 or ∀2, then {∃,∀,∧,∨}-FO(B) is NP-complete.

435

⟨0 01 02 012

⟩ ⟨0 11 12 012

⟩ ⟨0 21 0122 2

⟩

⟨0 01 0122 0

⟩ ⟨0 0121 12 1

⟩ ⟨0 0121 22 2

⟩

⟨0 01 022 01

⟩ ⟨0 01 012 02

⟩oo

⟨0 0121 12 2

⟩//⟨

0 0121 22 1

⟩

⟨0 121 12 10

⟩ ⟨0 011 12 12

⟩oo

⟨0 01 12 2

⟩

��

��

������

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����

����

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�� ��666

6666

6666

6666

6666

6

��///

////

////

////

////

////

////

///

��+++

++++

++++

++++

++++

++++

++++

++++

++++

++++

+

[[66666666666666666666

OO CC��������������������

WW//////////////////////////////

GG GG������������������������������ ⟨0 01 0122 2

⟩//⟨

0 21 0122 0

⟩

⟨0 121 022 2

⟩ ⟨0 021 122 2

⟩oo

⟨0 01 12 012

⟩//⟨

0 11 02 012

⟩

⟨0 01 12 12

⟩

[[

44

⟨0 01 12 02

⟩

\\

66

��

⟨0 01 012 2

⟩

\\

EE

⟨0 01 122 2

⟩

YY

AA

⟨0 021 12 2

⟩

ZZ

EE

��

⟨0 011 12 2

⟩

]]

KK

⟨0 01 22 1

⟩

RR II

��

��

⟨0 21 12 0

⟩

UU JJ

��

��

⟨0 11 02 2

⟩

aa II

��

��

⟨0 11 22 0

⟩

��⟨0 11 22 0

,0 01 22 1

⟩

⟨0 01 122 12

⟩

��

⟨0 021 12 02

⟩

��

⟨0 011 012 2

⟩

��⟨0 121 02 0

⟩ ⟨0 11 022 1

⟩ ⟨0 21 22 10

⟩

Figure 4. Part of the lattice of down-she-monoids in the three-element case. All the remaining down-she-monoids contain one of the six L-shes (drawn here on the top two rows).8

436

III. If shE(B) contains none of the L-shes, but contains oneof ∃0, ∃1 or ∃2, then {∃,∀,∧,∨}-FO(B) is co-NP-complete.

IV. Otherwise, {∃,∀,∧,∨}-FO(B) is Pspace-complete.

Proof. The L case, Case I, follows from Theorem 3, as doesmembership of NP and co-NP, for Cases II and III, respec-tively.

For NP-hardness in Case II, consider the disjoint unionK2 ] K1 of the antireflexive 2- and 1-cliques, i.e. a di-graph (isomorphic to that) with vertices {r, s, t} and edges{(r, s), (s, r)}. shE(K2 ] K1) is either 〈 0 012

1 22 1

〉, 〈 0 21 0122 0

〉

or 〈 0 11 02 012

〉, depending on the vertex labelling. {∃,∧,∨}-

FO(K2) is NP-complete (by reduction from 3-not-all-equalsatisfiablity, setRNAE(u, v, w) := E(u, v)∨E(v, w)), andK2 ] K1 agrees with K2 on all sentences of {∃,∧,∨}-FO(see [16]). It follows that {∃,∧,∨}-FO(K2 ] K1) is NP-complete and that {∃,∀,∧,∨}-FO(K2 ] K1) is NP-hard.The result for NP-hardness now follows from Theorem 2,since shE(B) ⊆ shE(K2 ] K1), for one of the vertex la-bellings.

For a graph G, define its complement G over the samevertex set to have the complementary edge set (i.e. G |=E(x, y) iff G |=/ E(x, y)). It is a simple application ofde Morgan duality that {∃,∀,∧,∨}-FO(G) is in NP (resp.,is NP-complete) iff {∃,∀,∧,∨}-FO(G) is in co-NP (resp.,is co-NP-complete) – see [16]. A similar argument tothat in the previous paragraph, with the complement graphK2 ] K1 yields the co-NP-hardness result for Class III.

Finally, since all other down-she-monoids are of theform of either Lemmas 5 or 6, the Pspace-hardness resultsfollow for Class IV.

In Figure 4, Cases I, II and III may be seen as occupy-ing the top two rows, rightmost two columns and leftmosttwo columns, respectively. All the other down-she-monoidsdrawn fall under Case IV.

Casting our mind back to the digraphs G1 and G2 ofFigure 1, we can read from the previous theorem that{∃,∀,∧,∨}-FO(G1) and {∃,∀,∧,∨}-FO(G2) are co-NP-complete and NP-complete, respectively.

4 Final remarks

We have introduced the class of problems{∃,∀,∧,∨}-FO(B) as well as an algebraic frameworkin which to study their complexity. We hope that wehave adequately demonstrated that this class of problemsdisplays complexity-theoretic richness, while not being tooresistant to full classification in simple cases. The algebraicmethod used in our classification for the three-element casegives simple explanation where there previously was none

– if one were to look at the examples of Figures 1 and2, there is little obvious in their immediate structure thatbetrays their position in the classification.

We note that our positive algorithms, for membershipof NP, co-NP and – especially – L, are uniform, andare based on simple quantifier elimination. Perhaps itis to be hoped that a full classification for the problems{∃,∀,∧,∨}-FO(B) would make use only of versions ofquantifier elimination. In any case, we conjecture that thetetrachotomy of Theorem 5 extends to all structures B;though we know we would need more sophisticated classesof shes than those of Section 3.1 to prove this. We notealso that, unlike the situation with clones and the CSP, thedown-she-monoids associated with a finite domain are al-ways finite. This means that their lattice should be effec-tively computable for low domain sizes like four or five.

Acknowledgements

We are grateful for the assistance of Manuel Bodirskyand Ferdinand Borner in advancing our understanding ofvarious Galois connections. We are also grateful for sim-plifications suggested by a helpful, if sarcastic, anonymousreferee.

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