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2.1 Logics and games We assume familiarity with first-order logic [MATH] . All our vocabularies are finite and relational, and all structures are finite. For a structure [MATH] , we write [MATH] to denote its universe, and we often write [MATH] and [MATH] interchangeably to mean the number of elements in the universe. ... |
For a fixed positive integer [MATH] , we write [MATH] to denote the fragment of first-order logic in which every formula has at most [MATH] |
variables, free or bound. We also write [MATH] for the existential positive fragment of [MATH] . This consists of those formulas of [MATH] formed using only the positive Boolean connectives [MATH] and [MATH] , and existential quantification. [MATH] is the extension of first-order logic with |
counting quantifiers . For each natural number [MATH] , we have a quantifier [MATH] where [MATH] if, and only if, there are at least [MATH] distinct elements [MATH] such that [MATH] . While the extension of first-order logic with counting quantifiers is no more expressive than [MATH] |
itself, the presence of these quantifiers does affect the number of variables that are necessary to express a query. Let [MATH] denote the [MATH] -variable fragment of [MATH] in which no more than [MATH] |
variables appear, free or bound. For two structures [MATH] and [MATH] , we write [MATH] to denote that they are not distinguished by any sentence of [MATH] . All that we need to know about [MATH] is that for every formula [MATH] of [MATH] there is a [MATH] such that if [MATH] then [MATH] if, and only if, [MATH] . We al... |
and [MATH] by the [MATH] -pebble bijective game Both versions of the game are played on a pair of structures [MATH] and [MATH] by two players, Spoiler and Duplicator, using [MATH] |
pairs of pebbles [MATH] . In a game position, some (or all) of the pebbles [MATH] are placed on elements of [MATH] while the matching pebbles among [MATH] are placed on elements of [MATH] . Where it causes no confusion, we do not distinguish notationally between the pebble [MATH] (or [MATH] and the element on which it ... |
existential [MATH] -pebble game , at each move Spoiler chooses a pebble [MATH] (which might or might not already be on an element of [MATH] ) and places it on any element of [MATH] Duplicator has to respond by placing [MATH] on an element of [MATH] . If the resulting partial map from [MATH] to [MATH] |
given by [MATH] is not a partial homomorphism, then Spoiler has won the game. In the [MATH] -pebble bijective game Spoiler chooses a pair of pebbles [MATH] and Duplicator has to respond by giving a bijection [MATH] which agrees with the map [MATH] for all [MATH] . Spoiler chooses a pair [MATH] on which to place the peb... |
; and [MATH] if, and only if, Duplicator has a winning strategy in the [MATH] -pebble bijective game played on [MATH] and [MATH] |
For undirected graphs, the relation [MATH] has a simple combinatorial characterization in terms of vertex refinement (see ). For any graph [MATH] , there is a coarsest partition [MATH] of the vertices of [MATH] such that for each [MATH] there exists [MATH] such that each [MATH] has exactly [MATH] neighbours in [MATH] .... |
such that [MATH] and [MATH] for all [MATH] and [MATH] All classes of structures we consider in this paper are assumed to be closed under isomorphism. Let [MATH] be such a class of structures and for any [MATH] , let [MATH] denote the structures in [MATH] with at most [MATH] elements. The |
counting width of [MATH] is the function [MATH] where [MATH] is the smallest value such that for any [MATH] and any [MATH] , we have [MATH] . Note that [MATH] . Because [MATH] |
whenever [MATH] and [MATH] have different numbers of elements, [MATH] is also the smallest value such that [MATH] is a union of [MATH] -classes. In particular, it follows that the counting width of [MATH] is the same as that of its complement. For [MATH] , we say that two disjoint classes [MATH] and [MATH] are [MATH] -... |
and [MATH] we have [MATH] . Then, if we define [MATH] to be the set that contains, for every [MATH] , all structures of size [MATH] that are [MATH] -equivalent to some structure in [MATH] , it witnesses the second condition. In the other direction, if for some [MATH] , we have [MATH] and [MATH] and [MATH] , then any [M... |
must also contain [MATH] 2.2 Interpretations Consider two vocabularies [MATH] and [MATH] . A [MATH] -ary [MATH] -interpretation of [MATH] in [MATH] is a sequence of first-order formulas in vocabulary [MATH] consisting of: (i) a formula [MATH] ; (ii) a formula [MATH] ; (iii) for each relation symbol [MATH] of arity [MAT... |
such that: (i) [MATH] is surjective onto [MATH] ; (ii) [MATH] if, and only if, [MATH] ; (iii) [MATH] if, and only if, [MATH] and (iv) [MATH] if, and only if, [MATH] . Note that an interpretation [MATH] associates [MATH] -structure with [MATH] only if [MATH] defines an equivalence relation on [MATH] that is a congruence... |
For a class of structures [MATH] and an interpretation [MATH] we write [MATH] to denote the class [MATH] . We mainly use interpretations to define reductions between classes of structures. These allow us to transfer bounds on separability, by the following lemma. |
Lemma 1 Let [MATH] be a [MATH] -bounded interpretation of dimension [MATH] and let [MATH] be the maximum number of variables appearing in any formula of [MATH] . If [MATH] and [MATH] are two disjoint classes of structures such that [MATH] |
and [MATH] are [MATH] -separable, then [MATH] and [MATH] are [MATH] -separable. Proof. Let [MATH] and [MATH] be two structures. Then, since [MATH] and [MATH] have size at most [MATH] , there is a formula [MATH] such that [MATH] and [MATH] We compose [MATH] with the interpretation [MATH] to obtain [MATH] . That is to sa... |
accounts for any other variables that may appear in the formulas of [MATH] When we wish to define a reduction from a class [MATH] by a first-order interpretation, it suffices to give an interpretation [MATH] for all structures in [MATH] with at least two elements (or, indeed, at least [MATH] elements for any fixed [MAT... |
So, fix a value [MATH] , and let [MATH] be the least integer such that [MATH] . In a structure [MATH] with at least two elements, we say that a [MATH] -tuple of elements [MATH] |
codes an integer [MATH] if [MATH] is the binary representation of [MATH] and for all [MATH] we have [MATH] if, and only if, [MATH] . For each [MATH] , we can clearly define a formula [MATH] with [MATH] free variables that defines those tuples that code [MATH] . Now, for any formula [MATH] let [MATH] be the formula [MAT... |
[EQUATION] In other words, [MATH] picks out those [MATH] tuples [MATH] where [MATH] satisfies [MATH] and [MATH] codes an integer in [MATH] and [MATH] identifies distinct tuples which have the same [MATH] and the same integer [MATH] . An interpretation using these can be seen to yield a structure with [MATH] disjoint co... |
The Basic Gap Construction The problems 3SAT and 3XOR both ask to decide if a formula consisting of the conjunction of Boolean constraints each on exactly three Boolean variables is satisfiable. In 3SAT the constraints are disjunctions of literals on three distinct variables. In 3XOR the constraints are parities of thr... |
: the class of satisfiable instances cannot be separated in [MATH] , for bounded [MATH] , from the class of unsatisfiable ones. Our aim is to show that this result can be strengthened to show that the class of satisfiable instances is not [MATH] -separable from the class of instances that are highly unsatisfiable , mea... |
that establishes this for any [MATH] , with a lower bound on the value of [MATH] that is linear in the number of variables in the system. Then we use this construction to get one for 3SAT for any [MATH] , also for a value of [MATH] that is linear in the number of variables. In both cases, the constants [MATH] and [MATH... |
3.1 Systems of constraints Let [MATH] be a finite set of relations over a finite domain [MATH] also called a constraint language . Let [MATH] be a collection (multi-set) of constraints, each of the form [MATH] , where [MATH] is a [MATH] -ary relation in [MATH] , and [MATH] are [MATH] distinct [MATH] -valued variables f... |
constraints [MATH] from [MATH] . Note that, as we are counting the number of satisfied constraints, multiplicities matter and this is why we have multi-sets rather than sets of constraints. |
We think of a system [MATH] over the constraint language [MATH] as a finite structure in two ways. In the first encoding, the universe is the disjoint union of [MATH] |
and [MATH] . The vocabulary includes binary relations [MATH] such that [MATH] holds if the constraint [MATH] has arity [MATH] or more and [MATH] is the [MATH] th variable in [MATH] . The vocabulary also includes a unary relation [MATH] for each relation [MATH] in [MATH] such that [MATH] holds if [MATH] is an [MATH] -co... |
The constraint language [MATH] is also encoded as a finite structure in two ways. In the first encoding the domain is [MATH] , where [MATH] is the maximal arity of a relation in [MATH] . The relations [MATH] are interpreted by the projections: [MATH] holds for [MATH] |
and [MATH] if, and only if, [MATH] and [MATH] . The relations [MATH] are interpreted by the relation [MATH] itself as a unary relation over the universe: [MATH] holds if [MATH] is the arity of [MATH] and [MATH] belongs to [MATH] . In the second encoding, the universe is just [MATH] , and the relation symbol [MATH] is i... |
It is easily seen that, in both encodings as finite structures, a system [MATH] over [MATH] is satisfiable if, and only if, there is a homomorphism from the structure that encodes [MATH] to the structure that encodes [MATH] . We say that the system is [MATH] -locally satisfiable if [MATH] |
For 3SAT , the constraint language is denoted [MATH] . It has domain [MATH] and the relations are the eight relations [MATH] defined by the eight possible clauses on three variables. For 3XOR , the constraint language is denoted [MATH] . It also has domain [MATH] and the relations are the two relations [MATH] |
defined by the two possible linear equations [MATH] with three variables over [MATH] . Accordingly, 3XOR instances [MATH] can be identified with systems of linear equations [MATH] over [MATH] . In the following, [MATH] and [MATH] are referred to as the left-hand side matrix of [MATH] and right-hand side vector of [MATH... |
It is probably useful to spell out, in simple words, what it means for 3SAT or 3XOR instance to be [MATH] -locally satisfiable. Intuitively, what this means is that every set of less than [MATH] variables induces a satisfiable subformula and that, in addition, at least one of the satisfying assignments that exists can ... |
second encoding; in the first encoding one has to consider sets of less than [MATH] variables and clauses, and then the correspondence with satisfying assignments with the extension property is not as direct. However, what is true and useful (and easy to see) is that if a 3SAT or 3XOR instance is [MATH] -locally satisf... |
3.2 Gap construction We now focus on 3XOR and hence on systems of linear equations over [MATH] A starting point for us is the following construction which allows us to convert any [MATH] -locally satisfiable system of equations into a pair of systems that are [MATH] -indistinguishable. See , Prop. 32] for a related con... |
that satisfiability of systems of linear equations over [MATH] is not invariant under [MATH] for any [MATH] For any instance [MATH] of 3XOR we define another instance [MATH] of 3XOR which has two variables [MATH] |
and [MATH] for each variable [MATH] of [MATH] . For each equation [MATH] in [MATH] , we have eight equations in [MATH] given by the eight possible values of [MATH] |
in [MATH] . If [MATH] is the system [MATH] , then the homogeneous companion of [MATH] is the system [MATH] , which we denote [MATH] . Note that the system [MATH] is satisfiable for any [MATH] by setting each variable [MATH] to [MATH] . We show that, despite this, as long as [MATH] is locally satisfiable, [MATH] |
is hard to distinguish from its homogeneous companion [MATH] Lemma 2 For every 3XOR instance [MATH] and every integer [MATH] if [MATH] is [MATH] -locally satisfiable, then [MATH] |
Proof. We describe a strategy for Duplicator in the [MATH] -pebble bijective game played on [MATH] and [MATH] , given a strategy in the existential [MATH] -pebble game on [MATH] and [MATH] |
Suppose we have a position in the existential [MATH] -pebble game on [MATH] and [MATH] with pebbles on [MATH] , for some [MATH] in [MATH] , and corresponding pebbles on [MATH] |
in [MATH] . Suppose further that this is a winning position for Duplicator, i.e. she has a strategy to play forever from this position. Then, we claim that the position in the bijective game where the pebbles in [MATH] are on [MATH] for some [MATH] and the matching pebbles in [MATH] are on [MATH] is a winning position ... |
if, and only if, [MATH] is an equation in [MATH] if, and only if, [MATH] is an equation in [MATH] , but this last equation is [MATH] . Thus, the map from [MATH] |
to [MATH] is a partial isomorphism. To see that Duplicator can maintain the condition, suppose Spoiler moves the pebbles on [MATH] . By assumption, Duplicator has a response in the existential game whenever Spoiler moves the pebble from [MATH] to [MATH] . This response defines a function [MATH] from the variables in [M... |
As far as the degree of satisfiability is concerned, the construction preserves a gap in the following quantifiable terms: Lemma 3 |
For every 3XOR instance [MATH] and every [MATH] , the following hold: 1. if [MATH] is [MATH] -satisfiable, then [MATH] is [MATH] -satisfiable, |
2. if [MATH] is not [MATH] -satisfiable, then [MATH] is not [MATH] -satisfiable. Proof. For proving 1, let [MATH] be an assignment of values to the variables of [MATH] that satisfies at least [MATH] of the [MATH] equations in [MATH] . Define the assignment [MATH] on the variables of [MATH] by [MATH] . For each equation... |
satisfied by [MATH] , all eight equations arising from [MATH] are satisfied by [MATH] and so [MATH] satisfies at least [MATH] of the [MATH] equations in [MATH] |
For proving 2, suppose [MATH] is an assignment of values in [MATH] to the variables [MATH] in [MATH] . Let [MATH] be the assignment defined by [MATH] . We claim that if [MATH] is an equation [MATH] in [MATH] that is not satisfied by [MATH] then at least four of the eight equations in [MATH] |
arising from [MATH] are falsified by [MATH] . To see this, consider two cases. First, suppose that [MATH] for some [MATH] . Without loss of generality, we assume [MATH] . Then consider the four pairs of equations |
[EQUATION] obtained by taking the four possible values of [MATH] and [MATH] Since [MATH] , if one equation in a pair is satisfied by [MATH] the other is necessarily falsified. Thus, at least four equations are falsified. For the second case, suppose that for each [MATH] occurring in [MATH] we have [MATH] . But then, si... |
falsifies at least four of the equations arising from [MATH] Now, suppose that [MATH] satisifes at least [MATH] of the [MATH] equations in [MATH] . We claim that [MATH] satisfies at least [MATH] equations in [MATH] . Suppose for contradiction that [MATH] |
falsifies a proportion [MATH] of the equations. By the above argument, then [MATH] falsifies at least [MATH] of the equations in [MATH] . But [MATH] |
contradicting the assumption that [MATH] satisfies at least [MATH] equations. The extreme cases of Lemma are given by [MATH] for point , and [MATH] with [MATH] for point . Indeed, every 3XOR instance is [MATH] -satisfiable, as witnessed by the all-zero assignment, or the all-one assignment, whichever satisfies more equ... |
Lemma 4 For every two reals [MATH] and [MATH] there exists an integer [MATH] such that for every sufficiently large integer [MATH] and every matrix [MATH] , where [MATH] and each row of [MATH] has exactly three ones, if [MATH] is chosen uniformly at random in [MATH] then, with probability at least [MATH] both 3XOR inst... |
Proof. Fix [MATH] and [MATH] and let [MATH] be any integer bigger than [MATH] . Let [MATH] be sufficiently large, let [MATH] , and let [MATH] be any matrix with [MATH] and [MATH] that has exactly three ones in each row. For each [MATH] , the instance [MATH] has one variable [MATH] |
for each [MATH] and one equation [MATH] for each [MATH] , where [MATH] are the three columns of [MATH] that have ones in row [MATH] . The instance [MATH] has three variables [MATH] for each [MATH] and eight equations [MATH] for each [MATH] |
For each assignment [MATH] for the variables of [MATH] and each [MATH] , let [MATH] be the indicator random variable for the event that [MATH] ; i.e., for the event that [MATH] |
satisfies the equation [MATH] . The probability of this event is [MATH] , and all such events, as [MATH] ranges over [MATH] , are mutually independent. Thus, setting [MATH] , we have that [MATH] is a binomial random variable with expectation [MATH] . By Hoeffding’s inequality, the probability that [MATH] is at most [MA... |
Similarly, for each assignment [MATH] for the variables of [MATH] and each [MATH] , let [MATH] be the fraction of equations of [MATH] among those that come from [MATH] that are satisfied by [MATH] i.e., precisely, [MATH] is [MATH] -th of the number of triples [MATH] for which the equality [MATH] holds. We claim that th... |
for some [MATH] . In case 1), either all eight equations that come from [MATH] are satisfied, or none is, and each possibility happens with probability [MATH] according to the outcome of the random choice of [MATH] . The expectation of [MATH] is thus [MATH] in this case. In case 2), exactly half of the eight equations ... |
[MATH] are satisfied, and which half depends on the outcome of the random choice of [MATH] . The expectation of [MATH] is thus [MATH] |
also in this case. This shows that the expectation of [MATH] is [MATH] in either case. Moreover, the random variables [MATH] , as |
[MATH] ranges over [MATH] , are mutually independent. Thus, setting [MATH] , we have that [MATH] is the average of [MATH] independent random variables with range in [MATH] . By Hoeffding’s inequality, the probability that [MATH] is at most [MATH] . In particular, the probability that |
[MATH] is at most [MATH] . By the union bound, the probability that some [MATH] satisfies [MATH] is at most [MATH] Since [MATH] and [MATH] , twice [MATH] |
is at most [MATH] and is less than [MATH] for all sufficiently large values of [MATH] . Thus, for any large enough [MATH] , the probability that both [MATH] and [MATH] are at most [MATH] -satisfiable is at least [MATH] |
The next step is to show that an appropriate choice of the matrix [MATH] will give a locally satisfiable instance [MATH] for any right-hand side [MATH] . Entirely analogous claims have been known and proved in the context of the proof complexity of propositional resolution; indeed, our proof builds on the methods for r... |
, and their relationship to existential pebble games from In the proof, we need the notion of a graph [MATH] that is a bipartite unique-neighbour expander graph with parameters [MATH] where [MATH] and [MATH] are integer parameters with [MATH] and [MATH] is a positive real number. What this means is that [MATH] is a bip... |
with [MATH] and [MATH] vertices respectively; each [MATH] has exactly [MATH] neighbours in [MATH] ; and for every [MATH] with [MATH] we have [MATH] , where [MATH] denotes the set of vertices in [MATH] that are unique neighbours of [MATH] ; i.e., they are neighbours of a single vertex in [MATH] |
Lemma 5 For every integer [MATH] there is a real [MATH] such that for every sufficiently large integer [MATH] there is a matrix [MATH] where [MATH] , such that each row of [MATH] has exactly three ones and, for every vector [MATH] , the 3XOR instance [MATH] |
is [MATH] -locally satisfiable for [MATH] Proof. Fix an integer [MATH] and reals [MATH] and [MATH] , and let [MATH] be sufficiently large that for every [MATH] there exists a graph [MATH] that is a bipartite unique-neighbour expander graph with parameters [MATH] . For the existence of such graphs with these parameters ... |
29 , Chapter 4] . Let [MATH] be the incidence matrix of [MATH] , where [MATH] and [MATH] are the two sides of [MATH] , for [MATH] . For each [MATH] , the 3XOR instance [MATH] has one variable [MATH] |
for each [MATH] , and one equation [MATH] for each [MATH] where [MATH] are the three neighbours of [MATH] in [MATH] We claim that every choice of [MATH] gives that [MATH] |
is [MATH] -locally satisfiable for [MATH] with [MATH] Claim 6 For every [MATH] , every set of at most [MATH] equations from [MATH] is satisfiable. |
Proof. For each [MATH] , let [MATH] be the set of equations that are indexed by vertices in [MATH] , and let [MATH] be the set of variables that appear in [MATH] . We prove, by induction on [MATH] , that if [MATH] and [MATH] , then there exists an assignment that sets all the variables in [MATH] and that satisfies all ... |
may assign some of the variables of the equation [MATH] , but not all, since [MATH] is not a neighbour of any vertex in [MATH] . Let [MATH] be the unique extension of [MATH] that first sets all the variables in [MATH] to [MATH] , and then sets [MATH] to the unique value that satisfies the equation [MATH] . This assignm... |
Claim 7 For every [MATH] and [MATH] , the instance [MATH] is [MATH] -locally satisfiable. Proof. If [MATH] is satisfiable, then Duplicator certainly has a winning strategy and there is nothing to prove. Assume then that [MATH] is unsatisfiable and let [MATH] be a minimally unsatisfiable subsystem; a subset of the equat... |
with [MATH] with [MATH] , where [MATH] stands for the negative literal [MATH] if [MATH] and the positive literal [MATH] if [MATH] . Let [MATH] be the 3CNF formula that is the union of all the [MATH] as [MATH] ranges over [MATH] . Observe that [MATH] is an unsatisfiable 3CNF. We intend to apply Theorem 5.9 from |
to it. Let [MATH] be the collection of all Boolean functions [MATH] defined by [EQUATION] for [MATH] . Each function in [MATH] is sensitive in the sense of Definition 5.5 from |
, and compatible with [MATH] in the sense of Definition 5.3 from . Moreover, if [MATH] is the set of functions that corresponds to the minimally unsatisfiable subsystem [MATH] of [MATH] , then its cardinality [MATH] satisfies [MATH] by Claim . It follows that the expansion [MATH] in the sense of Definition 5.8 from |
is at least [MATH] . By Theorem 5.9 in , every resolution refutation of [MATH] requires width at least [MATH] , and hence at least [MATH] since [MATH] . By Theorem 2 in |
, Duplicator has a winning strategy for the existential [MATH] -pebble game played on the structures [MATH] and the constraint language [MATH] of 3SAT , in the second encoding discussed in Section 3.1 . We use this winning strategy to design a winning strategy for Duplicator in the existential [MATH] -pebble game playe... |
While playing the game on [MATH] , Duplicator plays the game on [MATH] on the side and keeps the invariant that each pebbled variable in the game on [MATH] is also pebbled in the side game, and each pebbled equation in the game on [MATH] has its three variables pebbled in the side game. Whenever a new variable is pebbl... |
This completes the proof of Lemma We can now prove our first two gap theorems. Theorem 8 For every real [MATH] , if [MATH] is the collection of 3XOR instances that are satisfiable and [MATH] is the collection of 3XOR instances that are not [MATH] -satisfiable, then [MATH] and [MATH] are not [MATH] -separable for any [M... |
Proof. By combining Lemma with Lemma , there is a family of systems [MATH] with [MATH] variables and equations such that [MATH] is not [MATH] -satisfiable but [MATH] is [MATH] -locally satisfiable. Let [MATH] and [MATH] . Then [MATH] by Lemma . Moreover, by the first part of Lemma , the instance [MATH] is satisfiable w... |
Theorem 9 For every real [MATH] , if [MATH] is the collection of 3SAT instances that are satisfiable and [MATH] is the collection of 3SAT instances that are not [MATH] -satisfiable, then [MATH] and [MATH] are not [MATH] -separable for any [MATH] such that [MATH] |
Proof. Consider the reduction [MATH] from 3XOR to 3SAT that translates each equation into a conjunction of four clauses. Thus [MATH] becomes four clauses [MATH] |
with [MATH] and [MATH] , where [MATH] stands for the negative literal [MATH] if [MATH] and the positive literal [MATH] if [MATH] . This is easily defined in first-order logic. As the set of variables in [MATH] is the same as in [MATH] , it is linearly bounded. We claim that applying [MATH] to Theorem with [MATH] reset ... |
gives the theorem through Lemma . First, it is clear that if [MATH] is a 3XOR instance that is satisfiable, then [MATH] is also satisfiable. Now, suppose that [MATH] is a system of [MATH] equations that is not [MATH] -satisfiable, and let [MATH] be an assignment of truth values to the variables [MATH] |
of [MATH] . Applied to [MATH] , the assignment [MATH] falsifies at least [MATH] of the equations. For each equation, [MATH] must falsify at least one of the four corresponding clauses in [MATH] . Thus, [MATH] falsifies at least [MATH] |
clauses in [MATH] and so satisfies at most [MATH] of the [MATH] clauses. To formulate the consequences of these two theorems for [MATH] |
definability, it is useful to introduce some terminology. Say that a term [MATH] of [MATH] [MATH] -approximates MAX 3XOR , where [MATH] , if whenever [MATH] is an instance of 3XOR in which a maximum of [MATH] clauses are simultaneously satisfiable, then the interpretation of [MATH] of [MATH] in [MATH] is a value such t... |
[MATH] -approximating MAX 3SAT is defined similarly. Corollary 10 For any [MATH] 1. there is no term of [MATH] that [MATH] -approximates MAX 3XOR ; and |
2. there is no term of [MATH] that [MATH] -approximates MAX 3SAT Proof. If there were such a term in case (1), we would obtain an [MATH] |
sentence defining a class [MATH] of counting width bounded by a constant which separates the class of 3XOR instances that are satisfiable from those that are not [MATH] -satisfiable, constradicting Theorem . The analogous situation holds in case (2) and Theorem |
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