Hybrid programs: Symmetrically combining natively discrete and continuous truth-values

The iterates T _P ^t (I) of the one-step consequence operator T _P of a finite or infinite propositional normal logic program P applied to Herbrand interpretation I constitute a function t → T _P ^t (I) from natural numbers to Herbrand interpretations. Without loss of generality, altering clause p←q1,.qm ¬r1,. ¬rn of P to dp/dt = p ⊕ (q1,. qm, ¬r1,.¬rn) amounts to regarding P as a system of first-order differential equations, where the mapping t → T _P ^t (I) is a projection of the flow of the system with initial condition t ₀ → I. The aim of this shift in viewpoint is to seamlessly combine logic program clauses with conventional first-order ordinary differential equations involving e.g. realvalued functions of a real variable. This is rigorously enabled by differentiation of functions that are morphisms in the category CONV of convergence spaces. The form of differentiation we describe is a conservative extension of differentiation of functions between familiar spaces associated with ordinary analysis. In particular, we can integrate logic programs over continuous time. In that case, stable models semantics provides the natural means to have an ordinary normal program be equivalent to its differential version.