Semantics of Constraints
Integrity constraints are closed formulas that specify the properties that the intended interpretations of a rule set are supposed to have. These formulas are simply checked against the intended models of the ruleset. Therefore, the semantics of integrity constraints is fully defined by the notion of satisfaction in the models of rulesets.
Constraints can be full first-order formulas even in case of LP rulesets because the notion of satisfaction in a concrete model is not subject to the syntactic restrictions of LP languages.
While semantics can be defined for very general classes of constraints and queries, this does not mean that the problem of constraint satisfaction/query answering is decidable or has efficient evaluation techniques for such general classes.