(1) The tree for S contains an answer S ← |**; or**
(2) For each node N **in the tree for** S:
(a) The underlying subgoal of the selected literal of N **is marked as complete;**
or
(b) The underlying subgoal of the selected literal of N is in S **and there are**
no applicable New Subgoal, Program Clause Resolution, **Positive** Return (Definition 4.5), Negative Return, Delaying **(Definition 4.7)**
or Oracle Resolution **(Definition 4.11) operations for** N.
Once a set of literals is determined to be completely evaluated, a Completion operation marks the trees for each literal (Definition 4.2). If a subgoal S **is completed**
due to condition 1 holding, we say that S is early completed**. If condition 1 does**
not hold, condition 2a of the above definition prevents the Completion **operation**
from being applied to one of a set of trees if certain other operations are applicable to those trees. This notion of completion is incremental in the sense **that once a** set S **of mutually dependent subgoals is fully evaluated, the derivation need not be** concerned with the trees for S **apart from any answers they contain. In an actual** implementation resources for such trees can be reclaimed.

In certain cases the propagation of conditional answers through resolution (Definition 4.3) can lead to a set of unsupported answers **— conditional answers that**
are false in the well founded model (see, e.g., Example 1 of [Swift et al. **2009]).**16 Intuitively, these answers, which have positive mutual dependencies, correspond to an unfounded set, but their technical definition is based on the form of conditional answers. Definition 4.13 Let F be an SLG(O) forest, and Answer **be an atom that occurs** in the head of some answer in a tree with root S. Then Answer **is supported in** F if and only if:

(1) S **is not completely evaluated; or**
(2) there exists an answer node Answer′ ← Delays| in S such that Answer′subsumes Answer and for every positive literal L ∈ Delays, L **is supported in**
F.
We are now able to characterize the last two operations of SLG(O)**: one allows**
the completion of trees, and the other removes unsupported answers.

Definition 4.14 (SLG(O) Operations - 4) Let Kd = (O, Od,P
d) be a doubled Hybrid MKNF knowledge base. Given a forest Fn of an SLG(O**) evaluation**
of Kd, Fn+1 **may also be produced by one of the following operations.**

(8) Completion: Given a completely evaluated set S **of literals (Definition 4.12),**
mark the trees for all literals in S **as complete.**
(9) Answer Completion: Given a set of unsupported answers UA**, create a**
failure node as a child for each answer Ans **∈ UA**.