We assume that there is a countable collection of variables X and a countable domainDof values. A variable-value pair (x, a), representing the assignment of valuea∈ Dto variablex∈ X, is known as a point. Aflat pattern (or simply a pattern)P =AP, ρPis a subsetAP ofX ×Dequipped with a (partial) function ρP from the pairs of points (x, a),(y, b) of P such that x = y to {negative, positive}. ThusP consists of a set of variable-value assignments (x, a) together with a set of negative and positive edges representing the compatibility of pairs of assignments. In figures we represent negative edges by dashed lines, positive edges by solid lines and points corresponding to assignments to the same variable are grouped into ovals. Three patternsP1,P2,P3 are shown in Fig.1.
Fig. 1.Examples of the occurrence of a pattern in another pattern:P1→P2,P2→ P1,P1→P3,P2→P3.
128 D. A. Cohen et al.
We give a recursive definition of connectedness. Two points (x, a),(y, b) in a patternP areconnected ifx=y orρP((x, a),(y, b))∈ {negative, positive} or if (x, a),(y, b) are both connected to some point (z, c) of P. Clearly, each pattern has a decomposition into connected components according to this definition of connectedness.
Acompletely specified binary CSP instance(or simply aninstance) is a pat- tern I = AI, ρI in which the function ρI is total, i.e. the compatibility of each pair of variable-value assignments (to distinct variables) is specified. Given an instance I on n variables, a solution to I is a clique of positive edges of sizen, which corresponds to a pairwise-compatible assignment of values to vari- ables. The question associated with an instance is the existence of a solution.
An instanceIisarc consistent if for all points (x, a) ofIand all variablesy=x ofI, (x, a) has a support at y, i.e.∃b∈ Dsuch that {(x, a),(y, b)} is a positive edge in I.
A patternP =AP, ρPoccursin patternQ=AQ, ρQif there is a mapping f fromAPtoAQwhich respects variables, maps negative edges to negative edges and positive edges to positive edges, i.e.
1. f(x, a) = (u, c) andf(x, b) = (v, d) implies thatu=v.
2. f(x, a) = (u, c), f(y, b) = (v, d) and ρP((x, a),(y, b)) ∈ {negative, positive} implies thatu=v andρP((x, a),(y, b)) =ρQ((u, c),(v, d)).
We use the notationP →Qto denote thatP occurs in patternQ(andP Q if it does not). It is easy to see from its definition that occurrence is transitive:
P → Q and Q → R implies P → R. We consider two patterns P, Q to be equivalent ifP →Q andQ →P: we write P ≈Q. For example, patterns P1 and P2 in Fig.1 are equivalent; we notably have P1 → P2 since (x, a), (y, b) can both map to (z, c). Clearly, we have P2 → P3, and then, by transitivity, P1 → P3. For simplicity of presentation, throughout this paper, we will talk about patterns rather than equivalence classes of patterns.
Each patternP defines a corresponding set of instances in whichP does not occur. For example, the pattern P3 of Fig.1 defines a set of instances which includes all binary CSP instances with Boolean domains, since if P3→I then the points (v, d), (v, e), (v, f) must map to three distinct values for the same variable inI, due to the positive and negative edges inP3.
Note that in previous work, it has sometimes been convenient to assume that when P occurs inQ, distinct variables ofP map to distinct variables ofQ [11,15,19]. However, to establish a Galois connection for flat patterns, we require a looser definition of occurrence in which two or more variables ofP may map to the same variable inQ. To impose the stricter definition of occurrence (inducing an injective mapping of variables of P), it suffices, for each pair of distinct variables x, y, to add two new points (x, a), (y, b) toAP and an extra dummy positive edge between points (x, a), (y, b) inP; this preventsx, ymapping to the same variable in Q (and only changes the semantics ofP in a trivial way). A more elegant solution (in order to impose an injective mapping of variables) is to augment the patterns with a not-equal-to relation between variables which is possible in the framework of augmented patterns discussed in Sect.6.
Galois Connections for Patterns 129 We consider setsS of patterns. These sets will usually be finite, indeed, often a singleton. When forbidden, a setSof patterns defines a set of instances (those sets of instances in which none of the patterns in S occurs). Such sets T of instances are hereditary in the sense that (I∈T)∧(I ⊆I) =⇒(I∈T), where I ⊆Imeans (AI ⊆AI)∧(ρI =ρI|AI). Many, but not all, classes of interest are hereditary. For example, for any k, the set of instances whose tree-width is bounded by k is hereditary. On the other hand, the set of instances which is arc-consistent is not hereditary, since a value which has a support at another variable in an instance I will not necessarily have a support in I ⊂ I. Thus forbidden flat patterns alone cannot express any class of instances which requires arc consistency (or a higher level of consistency) [36]. Nevertheless, we will see in Sect.6how a combination of augmented patterns and filters on instances provides a very expressive language in which to define classes on instances, allowing us to express such classes of instances.
In order to obtain a Galois connection we consider sets of generic instances, where a generic instance can be viewed as a partially-specified instance and is, in fact, again just a pattern. However, the lattice structure on sets of patterns is different depending on whether we view these patterns as partially-specified instances or as forbidden sub-instances. When defining tractability of sets of generic instances we filter instances keeping only those that are completely spec- ified.
Definition 1. A setT of generic instances istractableif there is a polynomial- time algorithm which decides all completely-specified instances inT. A setS of forbidden patterns is tractable if the corresponding set of instances in which none of the patterns in S occur is tractable.
To define lattices of (sets of) instances and (sets of) patterns, we also require the following operation on patterns: ifP, Q are patterns, thenP+Qis a single pattern consisting of (copies of) the two patternsPandQ(without any common points and without any edges between P andQ). We call this thejuxtaposition of the two patterns P and Q. Observe that P +P ≈ P (since P +P → P follows from the definition of occurrence which allows us to map the two copies of P to P). If S1, S2 are sets of patterns, then S1+S2 is the set of patterns {P+Q|P ∈S1∧Q∈S2}.
We also require another operation on pairs of patterns, which can be seen as the greatest lower bound of the two patterns. If P, Q are patterns, thenP×Q is a single pattern built by forming the juxtaposition of all patternsRsuch that (R→P)∧(R→Q). We say that such patternsRarecommon factorsofP and Q. We only include patterns Rwhich are maximal in the sense that there is no R ≈R such thatR →R and (R →P)∧(R → Q). Observe that including only maximalR, ensures that we haveP×P≈P. The operation×is illustrated in Fig.2. In this example, the patternsPandQhave only two maximal common
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P Q
P×Q
Fig. 2.The operationP×Q.
factors (modulo the equivalence relation≈) andP ×Q is the juxtaposition of these two common factors. Note that P1 and P2 (shown in Fig.1) are both common factors ofP andQ, but sinceP1≈P2 we only need to include one of these patterns inP×Q. IfS1, S2are sets of patterns, thenS1×S2is the set of patterns{P×Q|P ∈S1∧Q∈S2}.
The following lemmas provide a logical interpretation of the + and×oper- ations on patterns.
Lemma 1. For all patternsP1, P2, I, we haveP1+P2Iif and only if (P1 I∨P2I)
Proof. For all patternsP1, P2, I,P1+P2→Iif and only if (P1→I∧P2→I) by the definition ofP1+P2. By contraposition, for all patternsP1, P2, I,P1+P2I if and only if (P1I∨P2I).
Lemma 2. For all patternsP, I1, I2,P I1×I2 if and only if(P I1∨P I2).
Galois Connections for Patterns 131 Proof. By contraposition, it suffices to show that P → I1×I2 if and only if P → I1∧P →I2. If P →I1∧P →I2, then P is a common factor of I1 and I2 and hence P → I1×I2. On the other hand, if P → I1×I2, then, due to the lack of edges between the connected components of I1×I2,P must be the juxtaposition of patterns P1, . . . , Pr where for each i = 1, . . . , r, Pi → Ri for some Ri which is one of the connected components ofI1×I2. Each connected componentRiofI1×I2satisfiesRi →Ri for some common factorRiofI1and I2. By transitivity of the occurrence relation and by definition ofI1×I2, we have Pi→I1andPi→I2 (fori= 1, . . . , r) and hence P→I1 andP →I2.