Semantics for the ATMS

In the JTMS the intuitive reading or semantics of the values V is clear and absolute. V(n) = in means that the proposition prop(n) associated with n is believed and V(n) = out means that it is not believed. Valuations are truth assignments over a network.

For the ATMS the values are not absolute but relative to a particular network and do not give a simple truth assignment to propositions. When designing a specific ATMS-IDN network the designer must specify the distinguished set of assumptions JA and the a-type dependency associated with that assumption. This gives the assumed nodes a fixed value and the belief or truth of other propositions or nodes is defined relative to them. So the values correspond to a set of possible worlds indexed by A where each possible world consists of a different interpretation of the same set of nodes.

3.4.3.1 Semantics for the ATMS

Having constructed a valuation consisting of a set of consistent possible worlds, that valuation can then be used to create a particular truth assignment corresponding to a single possible world. This is done by picking a set of assumptions M corresponding to a particular model and checking each node to see if it is true in that particular world. This is the secondary interpretation phase described in §2.1.2.

Although the summation functions above would seem to correspond with the informal algorithm given by de Kleer [1986a, p50-152], comparing the possible admissible valuations over cycles it can be seen that the ATMS-IDN does not produce the results sanctioned by the formal semantics [Reiter and de Kleer 1987, ECAI-TMS workshop]. In particular, given I ) = ( a — >b, b— >a[ one should get a valuation V = {(a {}) (b (})} but any valuation V' = {(a L) (b L)) will be admissible for the ATMS-IDN. To remove this anomaly it is sufficient to choose the minimal admissible valuation9.

The choice of minimal admissible valuation for the correct interpretation of ATMS networks introduces another problem: the derivation and recording of contradictions. The process of dealing with contradictions has two parts. First contradictions are generated by supporting -L with some dependency A — » _L. The resulting label of J_ are the no-good or inconsistent environments. Secondly, the no-goods are removed from each nodes label, thus preventing them from being derived under contradictory circumstances. However, this may result in the no-good being removed from J_.

For example, consider I) = ( — »(( a | ) a. a— >b, b— >-L}. If V is an admissible valuation then V(a) = {(A | }, V(b) = remove(( ( A )), V(±)) and V(_L) = V(b). I.e. b ’s label is [ (A | ) with any inconsistent environments removed, while the inconsistent environments are exactly those environments of b that are consistent - ( A ) is no-good iff it is not no-good. It is only when an environment first appears in the justification supporting ± that this loop is avoided.

There are several solutions to the problem of recording contradictions: either allow a self-supporting justification of no-goods and let admissible valuations be non-minimal over

1 This works for the ATMS and not the JTMS because of the tnonotonic nature of the cycles which means that minimality implies groundedness.

3.4.3.1 Semantics for the ATMS

_L; or use justifications to directly record contradictions. E.g. if | A B C) is determined to be no-good, say from d— fcl and V(d) = ({A B C ) ), then add { A B C } — > 1. to the network. This problem shows the difficulty of trying to treat a meta-level concept (e.g. contradictions) within the object framework. In both cases it is difficult to maintain the origin of the contradiction but remove its affect from the network.

It is obvious why the second is desired (maintaining a contradiction-free labelling is the purpose of the ATMS) but the first is equally necessary if changes to network are to be allowed. Not only may contradictions be explicitly removed by removing dependencies supporting -L, but they may also be removed implicitly by removing some of the other intermediate dependencies linking the assumptions to J.. Given it is not possible to maintain constant support for no-goods it becomes necessary to explicitly check the validity of no­ goods after the removal of any dependencies. For this reason the removal of justifications can be computationally expensive10.

What is harder to capture is the notion of entailment. As described above, I)a = 0 and so £(<N,I)>) = {<N,D>}. Given there is only a single admissible valuation for any ATMS- IDN network, the following theorem results:

Thm T Rl=<NiD> a iff T = 0 and a e ( (n, v) I ne N, v = V(n), V e TA(<N,I)>)) Proof Easy

We have returned to the problem of §3.3.1 of only having a single extension which we overcame by introducing assertional dependencies and which has now (unsurprisingly) reappeared when l)a = 0.

111 Dc Kleer suggests not actually removing justifications hut adding tut extra antecedent assumption to each dependency to represent its validity. To retract the dependency this assumption is simply removed from the problem solving context. However, given the complexity of calculating labels is exponential in the size of en­ vironments |Provan 1990| this will quickly lead to a combinatorial explosion given a network of some depth and interconnectedness.