Transforming Prioritized Defaults and Specificity into Parallel Defaults

We show how to transform any set of priori­ tized propositional defaults into an equivalent set of parallel (i.e. , unprioritized) defaults, in circumscription. We give an algorithm to im­ plement the transform. We show how to use the transform algorithm as a generator of a whole family of inferencing algorithms for cir­ cumscription. The method is to employ the transform algorithm as a front end to any in­ ferencing algorithm, e.g . , one of the previously available, that handles the parallel (empty) case of prioritization. Our algorithms provide not just coverage of a new expressive class, but also alternatives to previous algorithms for im­ plementing the previously covered class (lay­ ered) of prioritization. In particular, we give a new query-answering algorithm for prioritized cirumscription which is sound and complete for the full expressive class of unrestricted finite prioritization par­ tial orders, for propositional defaults (or mini­ mized predicates). By contrast, previous algo­ rithms required that the prioritization partial order be layered, i .e . , structured similar to the system of rank in the military. Our algorithm enables, for the first time, the implementation of the most useful class of pri­ oritization: non-layered prioritization partial orders. Default inheritance, for example, typi­ cally requires non-layered prioritization to rep­ resent specificity adequately. Our algorithm enables not only the implementation of de­ fault inheritance (and specificity) within pri­ oritized circumscription, but also the exten­ sion and combination of default inheritance with other kinds of prioritized default reason­ ing, e.g . : with stratified logic programs with negation-as-failure. Such logic programs are previously known to be representable equiv­ alently as layered-priority predicate circum­ scriptions. Worst-case, the transform increases the num­ ber of defaults exponentially. We discuss how inferencing is practically implementable nevertheless in two kinds of situations: general expressiveness but small numbers of defaults, or expressive special cases with larger numbers of defaults. One such expressive special case is non-top-heaviness of the prioritization partial order. In addition to its direct implementation, the transform can also be exploited analytically to generate special case algorithms, e.g . , a tractable transform for a class within default inheritance (detailed in another, forthcoming paper) . We discuss other aspects o f the significance of the fundamental result . One can view the transform as reducing n degrees of partially or­ dered belief confidence to just 2 degrees of con­ fidence: for-sure and ( unprioritized) default. Ordinary, parallel default reasoning, e.g. , in parallel circumscription or Poole's Theorist, can be viewed in these terms as reducing 2 degrees of confidence to just 1 degree of con­ fidence: that of the non-monotonic theory's conclusions. The expressive reduction's com­ putational complexity suggests that prioritiza­ tion is valuable for its expressive conciseness, just as defaults are for theirs. For Reiter's Default Logic and Poole's Theo­ rist , the transform implies how to extend those formalisms so as to equip them with a concept of prioritization that is exactly equivalent to that in circumscription. This provides an in­ teresting alternative to Brewka's approach to equipping them with prioritization-type prece­ dence. A longer version of this paper, including the proof of the central transform result, will soon be available as an IBM Research Report.