Symmetric Datalog and Constraint Satisfaction Problems in Logspace

Dualities for Constraint Satisfaction Problems

“…The following analog of Theorems 15 and 27 holds for symmetric Datalog: for a core structure B, if co-CSP(B) is definable in symmetric Datalog then the variety var(A B ) omits types 1, 2, 4, and 5 (i.e., it admits only type 3) [48]. It is shown in [19] that definability of co-CSP(B) in symmetric Datalog is equivalent to definability in a certain fragment of second order logic (this parallels a result in [16]). It would be interesting to find a convenient pebble game and an appropriate notion of duality for symmetric Datalog, in the spirit of Theorems 10 and 18.…”

“…A restriction of linear Datalog, symmetric Datalog, has been recently introduced in [19]. A linear Datalog program is called symmetric, if, for every rule of the form t 0 : − t 1 , t 2 , .…”

“…We say that co-CSP(B) is definable in symmetric Datalog if it is accepted by a symmetric Datalog program. In broad terms, symmetric Datalog for CSP is to LOGSPACE what linear Datalog for CSP is to NL: if co-CSP(B) is definable in symmetric Datalog then CSP(B) is in LOGSPACE, and, for all problems CSP(B) that are known to be in LOGSPACE, co-CSP(B) is definable in symmetric Datalog [19]. In particular, this holds for all Boolean problems CSP(B) in LOGSPACE.…”

Two new homomorphism dualities and lattice operations

“…In fact, all problems CSP(B) that are known to be tractable can be solved via Datalog, via the "few subpowers property" [24], or via a combination of the two. Furthermore, for every problem CSP(B) that is currently known to belong to NL or to LOGSPACE, the complement of CSP(B) can be defined in linear Datalog and symmetric Datalog, respectively (see [6,9,13]). The algebraic approach to the CSP was recently linked with non-definability in the above fragments of Datalog [29].…”

Caterpillar Duality for Constraint Satisfaction Problems

“…In fact, all problems CSP(B) that are known to be tractable can be solved via Datalog, or via the "few subpowers property" [18], or via a combination of the two. Furthermore, for every problem CSP(B) that is currently known to belong to NL or to LOGSPACE, the complement of CSP(B) can be defined in linear Datalog and symmetric Datalog, respectively (see [4,6,9]). The algebraic approach to the CSP was recently linked with non-definability in the above fragments of Datalog [23].…”