Double-black (DB) nodes have no place in red-black (RB) trees. So when DB nodes are formed, they are immediately removed. The removal of DB nodes that cause rotation and recoloring of other connected nodes poses greater challenges in the teaching and learning of RB trees. To ease this difficulty, this paper extends our previous work on the symbolic arithmetic algebraic (SA) method for removing DB nodes. The SA operations that are given as, Red + Black = Black; Black - Black = Red; Black + Black = DB; and DB – Black = Black removes DB nodes and rebalances black heights in RB trees. By extension, this paper projects three SA mathematical equations, namely, general symbolic arithmetic rule, ∆_([DB,r,p]); partial symbolic arithmetic rule1, ∂_([DB,p])^'; and partial symbolic arithmetic rule2, ∂_([r])^''. The removal of a DB node ultimately affects black heights in RB trees. To balance black heights using the SA equations, all the RB tree cases, namely, LR, RL, LL, and RR, were considered in this work; and the position of the nodes connected directly or indirectly to the DB node was also tested. In this study, to balance a RB tree, the issues considered w.r.t. the different cases of the RB tree were i) whether a DB node has an inner, outer, or both inner and outer black nephews; or ii) ) whether a DB node has an inner, outer or both inner and outer red nephews. The nephews r and x in this work are the children of the sibling s to a DB, and further up the tree, the parent p of a DB is their grandparent g. Thus, r and x have indirect relationships to a DB at the point of formation of the DB node. The novelty of the SA equations is in their effectiveness in the removal of DB that involves rotation of nodes as well as the recoloring of nodes along any simple path so as to balance black heights in a tree. Our SA methods assert when, where, and how to remove a DB node and the nodes to recolor. As shown in this work, the SA algorithms have been demonstrated to be faster in performance w.r.t. to the number of steps taken to balance a RB tree when compared to the traditional RB algorithm for DB removal. The simplified and systematic approach of the SA methods has enhanced student learning and understanding of DB node removal in RB trees.

First-order logic based data structure have knowledge representations in Prolog-like syntax. In an agent based system where beliefs or knowledge are in FOL ground fact notation, such representation can form the basis of agent beliefs and inter-agent communication. This paper presents a formal model of classification rules in first-order logic syntax. In the paper, we show how the conjunction of boolean [Passed, Failed] decision predicates are modelled as Passed(N) or Failed(N) formulas as well as their implementation as knowledge in agent oriented programming for the classification of students’ skills and recommendation of learning materials. The paper emphasizes logic based contextual reasoning for accurate diagnosis of students’ skills after a number of prior skills assessment. The essence is to ensure that students attain requisite skill competences before progressing to a higher level of learning.