Beans y páginas JSF

One of the main tasks of TMSs is to handle contradictions or inconsistencies. Classically, logical contradictions are well formed formulae (wff) of the form it a -.Jt although a

contradiction may refer to the presence of both 7t and —iJt in a set of theorems derivable from some set of premises without actually deriving Jt a —iJt. Any set of premises from which a

contradiction can be derived is said to be inconsistent. The use of contradictions in logic is limited to the proof of propositions whose negation would otherwise lead to a contradiction through the use of reductio ad absurdum (RAA):

11 This approach is full of danger as shown in §1.1.1.1 when looking at NML-I [McDermott and Doyle In order to force the JTMS into giving npi this intuitive reading additional justifications must be added linking n_,n and nn and a contradiction must be set up between nn and n„.

2.3.2 Contradictions and Retractions

if T, a |- P and T, a |----if) then T |— itx.

The reasoning for this rule is based on the intuition that propositions can take only one truth value in both formal interpretations and in the world itself. In itself RAA says nothing about belief revision, i.e. how assertions and assumptions should be managed for if a is already asserted to be true (along with T) then RAA only leads to more contradiction. TMSs provide facilities for doing just this kind of management of beliefs.

In terms of problem solving and search, it is normally the case that a system is trying to satisfy some goal state through the application of operators that transform the initial state, subject to some set of constraints. This gives rise to the notion of contradiction in a broader sense. A contradiction is a proposition or set of propositions T that is judged should not occur or be worthy of further exploration. This can be done without the use of formal logical machinery by simply excluding such states from further computation12. Such states may (although not necessarily) contain a logical contradiction but it may be difficult and counter­ productive to derive it as such. Thus such non-logical contradictions as "free market capitalism contradicts socialist notions of equal pay for all" is not of the form of a logical contradiction but provides a guide for revising beliefs: if one believes both in free market capitalism and equal pay for all then one may consider revising ones beliefs.

This distinction between logical contradiction as multiple truth assignment to a single proposition, and non-logical contradiction as a guide for eliminating states or revising beliefs, is apparent in the way different TMSs define and deal with contradictions. In the LTMS a network is represented by a set of clauses asserted to be true (potentially to a lesser or greater degree), and a contradiction is defined to be an assignment of values to nodes that does not satisfy every clause. In fact the LTMS will generate a logical contradiction because of the way it performs limited inference (resolution) to derive values. Given a clause (((P or Q). false) (P. true) (Q. true)) and a labelling (V) that assigns values to nodes such that V(P or Q) = true, V(P) = false, the LTMS will have already made V(Q) = true so that the

12 If however one subscribes to the law of excluded middle (n v ->«) and double negation (Jt <-> -,-at) (not something that Intuitionist or Constructivist logicians would hold with) then implicit in saying P is contradicto­

ry is the assertion that - C is the case.

2.3.2 Contradictions and Retractions

assertion ((Q. false)) will assign Q two values.

In the case of the J- and ATMS contradictions are user-defined. In the JTMS individual nodes are marked as contradictions and a labelling becomes inconsistent when a contradiction is labelled in. In the ATMS contradictions are represented by a single node nj^ and contradictory propositions or sets of propositions T are indicated by having them support n i , T — »s l uI 13.

Despite the syntactic differences in the representation of contradictions, all styles of TMS approach retraction in the same way. Backtracking takes place to identify some class of justification that supports the contradiction. Some automatic procedure is used to remove the contradiction or where necessary the user is informed of the contradiction and asked to remove some justification when there is no clear candidate. In the case of the LTMS the backtracking proceeds to nodes whose values have been externally justified and the lowest ranked of these is removed. If two assertions are tied then the user is asked to make a choice | Me Allester 1980 pi 1). Similarly the JTMS recursively traces the supporting justifications14 of the contradiction to the underlying assumptions and premises. A random choice of which node to retract is made from the set of assumptions13. If no assumptions exist then the user is informed and expected to provide a resolution. In both cases TM is called to propagate any changes arising from the retraction.

The process of backtracking and refutation is less obvious in the ATMS. In fact no reference is made to either in de Kleer’s papers16. The justifications supporting nj_ generate the set of no-goods (contradictory environments) and because the environments consist exactly of the assumptions that characterise it there is no need to actually backtrack to

11 In actual fact the JTMS can use the same scheme as the ATMS with mxles previously marked ¡is contrad­ ictions used to support n ^ . Dependency directed backtracking suuls at a lower point hut is otherwise un­ changed.

14 In the JTMS any node that is in has an associated supporting justification that is valid (i.e. all the in-nodes are in and the out-nodes o u t - these are the supporting nodes). An assumption is a node whose supporting justification has a non-empty nudist. A premise is one that has no supporting nodes itself.

' ' It is easy to propose ad hoc criteria for which assumption to revoke in terms of number of other attached dependencies or nodes in the dependency but these have no sound theoretical hacking. More dependencies may indicate stability or generality, more nixies may represent specialisation and the number of justilications a nixie supports may indicate its criticality or entrenchment.

2.3.2 Contradictions and Retractions

discover them. Nor does any explicit retraction actually take place: no justifications are added to or removed from the dependency network. However it is necessary to remove all no-goods and their supersets from all node labels (retraction of environments). Rather than checking each label it is only necessary to check for where the environment is known to occur ("forward-tracking"). If {A B C) is known to be inconsistent then only the nodes in c = desc(A)r\desc(B)r\desc(C) need be checked17. Furthermore if {A B C) is a subset of an existing no-good {A B C D | then c can be refined even further and only c = [desc(A)r\desc(B)rvJesc(C)]/desc(D) need be checked. If a non-monotonic ATMS is being used it would be necessary to invoke TM to recalculate labels rather than simply performing subset tests. In this case it is even more relevant to use forward-tracking as the cost of recalculation is higher than for simple testing.

From this it can be seen that the process of retracting contradictions is inherently cyclic through the process of backtracking and recalculation. It is this cycle that can lead to much work being done as several passes are needed to remove a set of contradictions. It is this same circularity that leads to the computational problems of default logics as we shall see in Chapter 5.

The important point to make in this section is that the user needs to have the ability to specifically exclude particular interpretations through the avoidance of user defined "contradictions". Additionally, the retraction process often needs to call on the user to make choices about the refutation of assumptions. These refutations can be seen as meta-level operations in that they change the underlying network (and the theory that it instantiates) that is being interpreted. This is not actually the case in the ATMS as only the labelling is changed. However, as we shall see in the next section, this process takes place through a secondary interpretation of the ATMS labelling when particular contexts are constructed from a particular set of assumptions or environment. Reichgelt [ 19XX) is supporting this claim when he proposes an architecture based on an inference engine, a knowledge base and a knowledge base manager. The knowledge base is equipped with TM facilities and it is the

17 The function desc returns the descendants of a node as defined in §3.2.1.

2.3.2 Contradictions and Retractions

role of the knowledge base manager to propose assumptions and instantiations of defaults. In the case of TMSs the user (be it a human or machine reasoner) needs to have access to the designation and removal of contradictions