Lindenbaum-algebraic semantics of logic programs
I have shown that the notion of derivability from a program can be analyzed as a preorder which leads to a certain algebraic structure constructed by equivalence classes of formulas. This algebra can be considered as imposing constraints on the possible truth value assignments of models of the program.
In the case of positive programs a distributive lattice was obtained as the underlying algebraic structure. Although derivability from logic programs with strong negation was shown to be sound with respect to DeMorgan algebras those algebras did not prove to be the general model structure looked for. This is because there is no contraposition law for conditional derivability with respect to strong negation. Only for contrapositionally complete programs the construction of the Lindenbaum algebra proved to be adequate. In the general case it remains an open question what is the appropriate algebraic structure constituting the Lindenbaum algebra.
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