A wide range of additional forward chaining applications could be realized with deductive databases, if their rule formalism, their immediate consequence operator, and their fixpoint iteration process would be more flexible. Deductive databases normally represent knowledge using stratified Datalog programs with default negation. But many practical applications of forward chaining require an extensible set of user–defined built–in predicates. Moreover, they often need function symbols for building complex data structures, and the stratified fixpoint iteration has to be extended by aggregation operations. We present an new language Datalog*, which extends Datalog by stratified meta–predicates (including default negation), function symbols, and user–defined built–in predicates, which are implemented and evaluated top–down in Prolog. All predicates are subject to the same backtracking mechanism. The bottom–up fixpoint iteration can aggregate the derived facts after each iteration based on user–defined Prolog predicates.

The word problem for a finite set of ground identities is known to be decidable in polynomial time using congruence closure, and this is also the case if some of the function symbols are assumed to be commutative or defined by certain shallow identities, called strongly shallow. We show that decidability in P is preserved if we add the assumption that certain function symbols f are extensional in the sense that f (s₁, . . . , sn) ≈ f (t₁, . . . , tn) implies s₁ ≈ t₁, . . . , sn ≈ tn. In addition, we investigate a variant of extensionality that is more appropriate for commutative function symbols, but which raises the complexity of the word problem to coNP.