Regular Signed Resolution Applied to Annotated Logic Programs

R&R‐resolution [21 is an SLD-style proof procedure for a restricted form of annotated logic programs
Computing annotated logic programs coincides with regular signed resolution, the restriction of signed resolution to regular signs* , as shown in [1]. It follows that annotated logic programs can be computed by a single rule of inference which performs set-intersection. This is done by R&R‐resolution. It is ‘SLD-style’: A proof starts with a query, each deduction step produces a new query (but possibly with a non-empty head, a so-called proof-query) and the proof ends when the query is empty. A proof query is a clause Q<σ>← Q1<↑μ1>,...,Qn<↑μn> where σ is the intersection of a positive regular and a negative regular set (σ is not necessarily regular). R&R‐resolution is quasi-regular in the sense that each proof can be rearranged such that it is a regular signed proof
For example, consider the program (with underlying truth value space FOUR) (1) p<↑V> ← q(X)<↑V> (2) q(a)<↑t> ← (3) q(b)<↑f> ← and the query q(X)<↑b>. There is no regular signed proof starting with the query, but there is such an R&R‐proof
We also tackle the problem of instantiating the program clauses with truth values. It turns out that we can restrict to so-called minimal valuations, which exist, if the underlying truth value space is a distributive lattice that fulfills the descending chain condition. Often there are just a few minimal valuations, such that the enormous search space can be substantially reduced. The implementation REGULA [2] performs R&R‐resolution and restricts the search space using minimal valuations as mentioned above