Generic Expression Hardness Results for Primitive Positive Formula Comparison

Abstract

We study the expression complexity of two basic problems involving the comparison of primitive positive formulas: equivalence and containment. In particular, we study the complexity of these problems relative to finite relational structures. We present two generic hardness results for the studied problems, and discuss evidence that they are optimal and yield, for each of the problems, a complexity trichotomy.set of relations that are definable by a primitive positive formula forms a robust algebraic object known as a relational clone; a known Galois correspondence associates, in a bijective manner, each such relational clone with a clone, a set of operations with certain closure properties. This correspondence provides a way to pass from a relational structure B to an algebra A B whose set of operations is the mentioned clone, in such a way that two structures having the same algebra have the same complexity (for each of the mentioned problems). In a previous paper by the present authors [6], we developed this correspondence and presented some basic complexity results for the problems at hand, including a classification of the complexity of the problems on all two-element structures.Our hardness results. Our first hardness result (Section 3) yields that for any structure B whose associated algebra A B gives rise to a variety V(A B ) that admits the unary type, both the equivalence and containment problems are Π p 2 -hard. Note that this is the maximal complexity possible for these problems, as the problems are contained in the class Π p 2 . The condition of admitting the unary type originates from tame congruence theory, a theory developed to understand the structure of finite algebras [13]. We observe that this result implies a dichotomy in the complexity of the studied problems under the G-set conjecture for the constraint satisfaction problem (CSP), a conjecture put forth by Bulatov, Jeavons, and Krokhin [7] which predicts exactly where the tractability/intractability dichotomy lies for the CSP. (Recall that the CSP can be formulated as the problem of deciding, given a structure B and a primitive positive sentence φ, whether or not φ holds on B.) In particular, under the G-set conjecture, the structures not obeying the described condition have equivalence and containment problems in coNP. The resolution of the G-set conjecture, on which there has been focused and steady progress over the past decade [9,14,10,3], would thus, in combination with our hardness result, yield a coNP/Π p 2 -complete dichotomy for the equivalence and containment problems. In fact, our hardness result already unconditionally implies dichotomies for our problems for all classes of structures where the G-set conjecture has already been established, including the class of three-element structures [9], and the class of conservative structures [8].One formulation of the G-set conjecture is that, for a structure B whose associated algebra A B is idempotent, the absence of the unary type in the variety generated by A B implies that the CSP o...

Testing Assignments to Constraint Satisfaction Problems

“…Hence, to establish the main result of this section, Theorem 6.6, it suffices to show that ∃CSP(A) is not sublinear-query testable with one-sided error when V(Alg(A)) is not congruence modular. The results in this section make use of ideas developed in [8] and [14].…”

“…The structure P 2 is defined on signature {R} and has R P 2 = {(b, c, c ′ ) ∈ B P × C P × C P | (c, c ′ ) ∈ α P b }. The definition of P 2 comes from [8]. In forming conjunctive queries over this signature {R} each variable has a sort (first or second) associated with each variable; an atom R(x, y, y ′ ) may be formed if x is of the first sort and y and y ′ are of the second sort.…”

“…, x m ), ℓ, w), f be an input to the first problem. We make use of a construction in the literature (introduced in [8] and also employed in [14]) which allows us to create, from the circuit T (x 1 , . .…”

“…Note that this theorem makes use of a construction from [8,Theorem 7] and shares elements in common with the proof of [14,Theorem 13].…”

“…, y ℓ ′ denote the quantified variables of the second sort in φ. We use the translation of [8,Theorem 7] to obtain a formula φ ′ (x ′ 1 , . .…”