Logical Foundations of Well-Founded Semantics

We propose a solution to a long-standing problem in the foundations of well-founded semantics (WFS) for logic programs. The problem addressed is this: which (non-modal) logic can be considered adequate for well-founded semantics in the sense that its minimal models (appropriately defined) coincide with the partial stable models of a logic program? We approach this problem by identifying the HT 2 frames previously proposed by Cabalar to capture WFS as structures of a kind used by Došen to characterise a family of logics weaker than intuitionistic and minimal logic. We define a notion of minimal, total HT 2 model which we call partial equilibrium model. Since for normal logic programs these models coincide with partial stable models, we propose the resulting partial equilibrium logic as a logical foundation for well-founded semantics. In addition we axiomatise the logic of HT -models and prove that it captures the strong equivalence of theories in partial equilibrium logic.