Propagation: A New Operation in a Framework for Abstract Interpretation of Logic Programs

call and success substitutions but also with a current state ASc which describes correctly the part of CS local to the clause in question. In Fig. 1, assuming Bj is to be executed, that means that all clauses on the path between Bj and the root (H *--BI,..., Bz, G ~-AI,..., Ak,...) are adorned with an AS~ but also the clauses which can be involved in solving calls to the left of calls on the path to the root, i.e. clauses in the AND-OR subgraph for BI,..., in the ANDOR subgraph for At, .... All these local abstract states ASi together make up a global state AS. In the new framework we have AS I : Prop(AS) where both AS ~ and AS are global states and Prop is a function which reexecutes all operations affecting AS until a stable state is reached. In particular, if O1,..., O,~, On+l is the sequence of operations leading to CS ~, then their abstract counterparts, say AO1,..., AO,~, AO,~+I, are reexecuted until a stable state is reached. All these operations AO~ are weak operations, without propagation, which by themselves need not to be safe. The point is that when the global propagation ceases, a safe approximation is reached, i.e~ all local states AS~ correctly describe state CS ~ and this for any CS represented by AS. The operations which need to be considered for reexecution are : All operations involved in reexecuting calls to the left of the path from the