Semantics of PFCSes.
A named relation of a system of constraints denotes a predicate over the cartesian product of the prime field; the number of factors of the cartesian product is the arity of the relation. The predicate holds exactly on the tuples of prime field elements that, when assigned to the parameters of the relation, satisfy all the constraints that define the relation. An equality constraint is satisfied when its evaluated left and right sides are equal. A relation application in the defining body of another one is satisfied when the predicate it denotes holds on the prime field elements that result from evaluating its argument expressions. There must be no recursion in the relation definitions for this to be well-defined. However, the body of a relation definition may include variables that are not among the formal parameters: these are regarded as existentially quantified, i.e. the predicate denoted by the relation holds on a tuple when there exist values for those extra variables that, together with the values of the tuple, satisfy all the constraints.
We formalize this as a deeply embedded semantics and as a shallowly embedded semantics.