Conjunctive Partial Deduction: Foundations, Control, Algorithms, and Experiments

ion Operators We now specify the abstraction operators Absγ,L, deciding which conjunctions are added to the global tree in order to ensure coveredness for bodies of newly derived resultants. Ensuring coveredness is basically simple: add to the global tree all (unchanged) bodies of produced resultants as new, “to be partially deduced” conjunctions. However, this strategy leads usually to non-termination, and thus, the need for abstraction arises. For an element B in bodies(U(P, QL)), the abstraction operator Absγ,L should consider whether adding B to the global level may endanger termination. To this end, Absγ,L should detect whether B is (in some sense) bigger than another label already occurring in SeqβL , since adding B might then lead to some systematic growing behaviour resulting in non-termination. According to Definition 3.2, abstraction allows two operations: conjunctions can be split and generalised. There are many ways this can be done and the concrete way will (usually) directly rely on the relation detecting growing behaviour. Since we aim at removing shared, but unnecessary variables from conjunctions, there is no point in keeping atoms together that do not share variables. We will therefore always break up conjunctions into maximal connected subparts (conjunctions) and abstraction will only consider these. In other words, resultant bodies will be automatically split into such connected chunks and it will be the latter that are considered by the abstraction operator proper. Global termination then follows in a way similar to the local one. Definition 3.12. (maximal connected subconjunctions) Given a conjunction Q = A1 ∧ . . . ∧ An, where A1, . . . , An are atoms, we define the binary relation ↓Q over the atoms in Q as follows: Ai ↓Q Aj iff vars(Ai) ∩ vars(Aj ) 6= ∅. By ⇓Q we denote the reflexive and transitive closure of ↓Q. The maximal connected subconjunctions of Q, denoted by mcs(Q), are defined to be the multiset of conjunctions {Q1, . . . , Qm} such that 1. Q =r Q1 ∧ . . . ∧Qm, 2. Ai ⇓Q Aj iff Ai and Aj occur in the same Qk and 3. for every Qk there exists a sequence of indices j1 < j2 < . . . < jl such that Qk = Aj1 ∧ . . . ∧ Ajl . For two conjunctions Q = A1 ∧ . . .∧An and Q = A1 ∧ . . .∧A ′ n where Ai and A ′ i have the same predicate symbols for all i, a most specific generalisation msg(Q, Q) exists, which is unique modulo variable renaming. Given a conjunction Q = A1 ∧ . . .∧Ak any conjunction Q = Ai1 ∧ . . .∧Aij such that 1 ≤ i1 < . . . < ij ≤ k is called an ordered subconjunction of Q. We now extend the definition of homeomorphic embedding to conjunctions. This notion is closely related to those of “variable-chained sequence” and “block” of atoms used in [64, 65].