Contexts for Nonmonotonic RMSes

A new kind of RMS, based on a close merge of TMS and ATMS, is proposed. It uses the TMS graph and interpretation and the ATMS multiple context labelling procedure. In order to fil l in the problems of the ATMS environments in presence of nonmonotonic inferences, a new kind of environment, able to take into account hypotheses that do not hold, is defined. These environments can inherit formulas that hold as in the ATMS context lattice. The dependency graph can be interpreted with regard to these environments; so every node can be labelled. Furthermore, this leads to consider several possible interpretations of a query. Reason maintenance systems (RMS) are aimed at managing a knowledge base considering different kinds of reasoning. Such a system is connected to a reasoner (or problem solver) which communicates every inference made. The RMS has in charge the maintenance of the reasoner's current belief base. RMSes developed so far focussed on nonmonotonic reasoning or multiple contexts reasoning. They record each inference in a justification that relates nodes representing propositional formulas plus a special atom (J_) representing contradiction. A justification (< { i ] , . . . i n } {o 1 . . o m }> : c) is made of an IN-list ( { i 1 . . i n } ) and an OUT-list ({o1,...om}). Such a justification is said to be valid if and only if all the nodes in the IN-list are known to hold while those in the OUT-list are not; a node, in turn, is known to hold if and only if it is the consequent (c) of a valid justification. The recursion of the definition is stopped by nodes without justification and by the axioms that are nodes with a justification containing empty INand OUT-lists. Jon Doyle's TMS [Doyle, 1979] proceeds by labelling the nodes of the graph with IN and OUT tags which reflect whether they are known to hold or not. A labelling respecting the constraints stated above is an admissble labelling and a labelling which labels the node OUT is a consistent labelling. The TMS algorithm finds a (weakly) founded labelling, i.e. a consistent admissible labelling which relies on no circular argument. The main work of the TMS occurs when it receives a new justification. It then has to integrate the justification in the graph and, if this changes the validity of the formula, it must propagate this validity: all the nodes that could be IN-ed because of the justified node and all those which could be OUT-ed are examined and updated. If an inconsistency occurs following the addition of a justification, the system backtracks on the justifications in order to invalidate a hypothesis — a formula inferred non monotonically — which supports the inconsistency. Figure 1. A dependency graph is here represented as a boolean circuit where or-gates are nodes and and-gates arc justifications where the nodes in the IN-list come directly while nodes in the OUT-list come through a not-gate. Nodes that have a justification whose !Nand OUT-lists are empty (e.g. D) represent true formulas because they do not need to be inferred. White nodes and justifications are considered valid while hatched ones are invalid. Of course, the value propagation satisfies the rules implied by the circuit components, So, the formulas in the base are ensured to have a valid justification (i.e. corresponding to a valid inference). Johan De Kleer's as sumption-based TMS [De Kleer, 1986; 1988] is rather different. This system considers only monotonic inferences (with only IN-list; <{ i1 , . . . in }>: c), but it deals with several contexts at a time. It considers initial formulas called hypotheses; so, the user can generate and test hypotheses with great efficiency. A set of hypotheses is called an environment and the set of all the environments constitutes a complete lattice structured by the "includes'' relation (cf. Figure 2). Instead of labelling absolutely a node (with IN or OUT tags), each node is assigned a label consisting of the set of environments under which it is known to hold An environment is consistent if JL is not known to hold in it and the computed labels are minimal in the sense that they do not contain comparable environments. After each inference, the system computes the set of environments that support the inference, inserts it in the label of the inferred node and propagates it through the graph. Then, in order to know if a formula is valid, it compares the current hypothesis set with the label of the node. 300 Automated Reasoning Figure 2. The environment lattice constructed with the hypotheses A, B and C in which the environment {B, C} is known as inconsistent. As a summary, the TMS handles nonmonotonic inferences and is able to maintain the set of deduced formulas with regard to an axiom set The ATMS, for its part, cannot accept nonmonotonic inferences, but is able to consider several contexts simultaneously. Merging both systems is needed in order to dispose of a RMS able to deal with nonmonotonicity in multiple contexts. This is the purpose of our context-propagation TMS (CP-TMS), The first section shows the problem of doing it by extending the ATMS. Section 2 sketches the ideas underlying the CP-TMS. Its construction spreads through sections 3, 4> 5 and 6 by defining the valuations that environments represent, the interpretations that extend valuations to all the formulas, the labels tied to the formulas and their properties of completeness, correctness, minimality and consistency. Section 7 shows that queries can exploit labels in several ways. The last sections are dedicated to the description of a partial implementation (§8), a discussion of some shortcomings of the system (§9) and several solutions to these problems (§10). l . A T M S and n o n m o n o t o n i c i t y Using an ATMS in order to deal with nonmonotonic reasoning seems attractive. In fact, the introduction of nonmonotonic inferences in the ATMS does not fit well. The main advantage of the ATMS is its use of the context lattice structure in order to infer that if a formula is valid in some environment, it will be under all its supersets. This is the strict definition of monotonicity. So, nonmonotonic inferences lead to important problems in the ATMS (cf. exemple 1), Example 1. Deciding under which environment the inference <{A}{B}>:D is valid is not possible. At first sight, {A} is a good candidate because A holds in it while B does not. But, since, in that case, D will be inherited from {A} to {A, B} (cf. Figure 2), in which B holds, {A} is not an adequate environment. Several authors proposed a special use of the ATMS or similar systems in order to solve these problems [Dressier, 1989; De Kleer, 1988; Giordano and Martelli, 1990]. Here is Oskar Dressier's solution: for each node N, an hypothesis Out(N) can be created whose label represents the set of environments under which N does not hold. Inconsistency between N and Out(N) is dealt with by adding a constraint, and completeness is achieved with the help of a special While this approach works well, it suffers from some drawbacks: nonmonotonicity is achieved by multiplying entities. This leads to a "monotonization" of the reasoning and a multiplication of inconsistent contexts (leading, in turn, to intensive hyperresolution). Moreover, additional out-hypotheses are not all significant element for the user. Since such an approach consists in adding an interface level on the ATMS, it has the advantage to not modify it significantly. But, while it is conceptually simple, it reveals to be a bit cumbersome to work,