Citation Context

...e to argue the closure property under 9 AP-reducibility (without the projection operation). This property is a key to the subsequent lemmas and propositions. This line of study was lately explored in =-=[2]-=-. To pursue notational succinctness, we use the following notations in the rest of this paper. For any index i ∈ [k] and any bit c ∈ {0, 1}, let the notation fxi=c denote the function g satisfying tha...

Citation Context

...max(z, t)(NAND(x, y) ∧ IMP(z, x) ∧ IMP(t, y)). Thus {NEQ, IMP,OR,NAND} ⊆ 〈R,R′〉max, and it suffices to show that 〈NEQ, IMP,OR,NAND〉max = II2. 14 The rest of the proof is derived from that of Lemma 15 =-=[11]-=-, only it does not have to deal with weights. Let R(x1, . . . , xn) be any relation. For each I ⊆ [n] with aI ∈ R introduce a new variable zI . Consider the relation given by Q = ∧ I⊆[n],aI∈R ∧ i∈I...

Citation Context

...d(Γ ∪ Γpin) unless NP=RP. Here Γpin = {{(0)}, {(1)}} and a relation is called affine if it is an affine subspace of Fk2. Theorem 2 can be seen as an extension of the following result of Bulatov et al =-=[BDGJ12]-=-, which we also rely on in the proof: Lemma 6. [BDGJ12, Theorem 16] LetF be a finite set of signatures. IfF is not a subset of WeightedNEQ-conj then for any finite subset S of arity-one signatures the...

Citation Context

... connection exists for partial co-clones (closed only under intersections) and the approximation complexity of counting CSPs [7, 10]. Certain analogs of co-clones for weighted CSPs were introduced in =-=[8, 12]-=- Finally the connection from [7, 10] was strengthened by introducing the max-quantication construction that also preserves the approximation complexity of counting CSPs [9]. In that paper we only est...