TY - JOUR

T1 - The complexity of quantified constraints: Collapsibility, switchability and the algebraic formulation

AU - Carvalho, Catarina

AU - Madelaine, Florent

AU - Martin, Barnaby

AU - Zhuk, Dmitriy

N1 - © 2023 Association for Computing Machinery. This is the accepted manuscript version of an article which has been published in final form at https://doi.org/10.1145/3568397

PY - 2023/1/18

Y1 - 2023/1/18

N2 - Let A be an idempotent algebra on a finite domain. By mediating between results of Chen and Zhuk, we argue that if A satisfies the polynomially generated powers property (PGP) and B is a constraint language invariant under A (that is, in Inv(A)), then QCSP(B) is in NP. In doing this we study the special forms of PGP, switchability and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now the original Chen Conjecture is known to be false. Switchability was introduced by Chen as a generalisation of the already-known collapsibility. For three-element domain algebras A that are switchable and omit a G-set, we prove that, for every finite subset D of Inv(A), Pol(D) is collapsible. The significance of this is that, for QCSP on finite structures (over a three-element domain), all QCSP tractability (in P) explained by switchability is already explained by collapsibility.

AB - Let A be an idempotent algebra on a finite domain. By mediating between results of Chen and Zhuk, we argue that if A satisfies the polynomially generated powers property (PGP) and B is a constraint language invariant under A (that is, in Inv(A)), then QCSP(B) is in NP. In doing this we study the special forms of PGP, switchability and collapsibility, in detail, both algebraically and logically, addressing various questions such as decidability on the way. We then prove a complexity-theoretic converse in the case of infinite constraint languages encoded in propositional logic, that if Inv(A) satisfies the exponentially generated powers property (EGP), then QCSP(Inv(A)) is co-NP-hard. Since Zhuk proved that only PGP and EGP are possible, we derive a full dichotomy for the QCSP, justifying what we term the Revised Chen Conjecture. This result becomes more significant now the original Chen Conjecture is known to be false. Switchability was introduced by Chen as a generalisation of the already-known collapsibility. For three-element domain algebras A that are switchable and omit a G-set, we prove that, for every finite subset D of Inv(A), Pol(D) is collapsible. The significance of this is that, for QCSP on finite structures (over a three-element domain), all QCSP tractability (in P) explained by switchability is already explained by collapsibility.

KW - cs.CC

U2 - 10.1145/3568397

DO - 10.1145/3568397

M3 - Article

SN - 1529-3785

VL - 24

SP - 1

EP - 26

JO - ACM Transactions on Computational Logic (TOCL)

JF - ACM Transactions on Computational Logic (TOCL)

IS - 1

ER -