Counting Answers to Existential Positive Queries

Abstract

Existential positive formulas form a fragment of first-order logic that includes and is semantically equivalent to unions of conjunctive queries, one of the most important and well-studied classes of queries in database theory. We consider the complexity of counting the number of answers to existential positive formulas on finite structures and give a trichotomy theorem on query classes, in the setting of bounded arity. This theorem generalizes and unifies several known results on the complexity of conjunctive queries and unions of conjunctive queries.

The logic of counting query answers

“…Then there is a constant k such that for all cores ψ of pp-formulas in Φ`we have twpψq ď k and twpcontractpψqq`1 ď k. With the results of Section V it follows that qawpψq ď 2k for such cores ψ. As discussed prior to the statement of Theorem 4.4, we can express, for each φ P Φ, the function |φp¨q| as a polynomial in unknowns |ψp¨q| with ψ P Φ` [18]. We can then substitute every term |ψ 1 p¨q| by a 7PP-formula of width at most 2k with the results of Section V. This yields a 7PP-formula of width at most 2k.…”

“…Previous work showed that, for bounded arity Φ, countpΦq is fixed-parameter tractable if Φ satisfies the tractability condition; interreducible with the parameterized clique problem if Φ satisfies the contraction condition but not the tractability condition; and as hard as the parameterized counting clique problem otherwise (see [4] for a precise statement). 3 Note that in previous articles [4], [18] this graph was defined in terms of the core of the given formula.…”

“…(In general, the mentioned polynomial makes use of negative integers; this is a reason why our definition of 7logic allows arbitrary integers, and not just natural numbers.) For more information, we refer the reader to [18]. only if there exists k ě 1 such that each formula in Φ has a 7EP-representation φ 1 having 7-widthpφ 1 q ď k. Theorem 4.5: There exists an algorithm that, given an epformula φ, outputs a 7EP-representation ψ of φ of minimum width.…”

“…Proof. As each φ i pS i q is connected and liberal, the result [18,Theorem 5.14] ensures that there exists a structure C 1 such that the values |φ i pC 1 q| are pairwise different, and where |φpC 1 q| ą 0 for all pp-formulas φ. By taking a sufficiently large power P of C 1 , we may obtain that for the structure C " C 1PˆA , the values c i " |φ i pCq| are pairwise different.…”

Decomposing Quantified Conjunctive (or Disjunctive) Formulas

“…Another direction is to study the complexity of counting answers to formulas; in this problem, one is given a first-order formula and a finite structure, and the goal is to count the number of assignments, to the free variables of the formula, that satisfy the formula on the structure. There have been recent results on this direction in the context of primitive positive formulas [28,23] and existential positive formulas [24].…”