Coalgebraic Semantics for Derivations in Logic Programming

Every variable-free logic program induces a P P -coalgebra on the set of atomic formulae in the program. The coalgebra p sends an atomic formula A to the set of the sets of atomic formulae in the antecedent of each clause for which A is the head. In an earlier paper, we identified a variable-free logic program with a P P -coalgebra on Set and showed that, if C(P P ) is the cofree comonad on P P , then given a logic program P qua P P -coalgebra, the corresponding C(P P )-coalgebra structure describes the parallel and-or derivation trees of P. In this paper, we extend that analysis to arbitrary logic programs. That requires a subtle analysis of lax natural transformations between Poset-valued functors on a Lawvere theory, of locally ordered endofunctors and comonads on locally ordered categories, and of coalgebras, oplax maps of coalgebras, and the relationships between such for locally ordered endofunctors and the cofree comonads on them.