Uncertainty ad Incompleteness: Breaking the Symmetry of Defeaible Reasoning * Piero P. Bonissone David A. Cyrluk James W. Goodwint Jonathan Stillma Artifcial Intelligence Program General Electric Corporate Research and Development Schenectady, New York 12301 Arpanet: bonissone@crd.ge.com cyrluk@crd.ge.com stillma@crd.ge.com Abstract Two majo difculties in using default logics are their intractability and the problem of selecting among multiple extensions. We propose an ap proach toheseproblemsbaedonintegrating non onooniceaoingihlauible eaningbaed on triangular norms. A previously propedsystem frreasoningwithuncertainty (RUM) perfomsun certain monotonic inferences on an acyclic graph. We have eteded RUM toallownonmonotonic in feences and cycle within nonmonotonic rles. By restricting the sie ad compleity of the nonmon tonic cycles e can still perform efcient inferences. Uncertanty meaures provide a bai for deciding between multiple defalts. Diferent algorithms and heuristics for fnding the optimal defaults are dis cussed. 1 Introduction 1.1 Moiaion The management of uncertain inforation in first generation expert systems, when addressed at all, ha lagely been left to ad hc methods. This has been efective onl becase oeational expert sys tems normally asme that knowledge is complete, ecise, ad unvaying. Thisfundamental asump tion is a principal sorce of the limitation of any Th k a ial supported by the Defense Advaced Reeac Pojede Agency (DARPA) under USAF /Rome Air Development Center contract F3002-85 C-033. Views ad conclwions contained in ths pape Me those of the authors &d should not be interpreted a repre senting the ofca oinon or policy of DARPA or the U.S. Goverent. !Cur ently with Knowledge Analysis, Belmont, Masachuetts. 34 diagnosticsyste tosingle fault diagnoses, and the limitationofcasifcationsystemstotime-invariant phenomena. The maagement of incomplete information ha aso lacked a clear focus, a some reearchers have attemptedto fnd its solution by defning new non monotoniclogics,byaugmentingcasicallogicwih default rules of inference, by seaching for minimal models via fuctional optimiation, or by concen tratig only on the instruments, Le. TMSs, rather than the theory reuired to handle this problem. In the pat, a subset of the athos have co tributed to the developmet of individual theories for eaoning with uncertainy and incompleteness. Bonisone ha proposed RUM, a system for rea soning with uncertaint whose underlying theory i anchored on the semantics of many-valued log ics (Bon87]. This system proides a representation layer to capture structural and numerical informa tion about the uncetainty, an infeence layer to provide a selection of truth-functional tiagular norm baed calculi [Bon87], and a control layer to focus the reaoning on subsets of the KB, to (proce duraly) reslveignoranceandconfict, and tomai tai the integrity of the inference bae via a belief revision ysem. RUM, however, does not provide anydeclaraiverepresentation tohandle incomplete inforation. Goodwin (Goo87] and Brown [BGB87] hae pro vided sch a representation by deelopig theoies baed on nnmnnic deedenc networks and algebraic euations over boolean lattices, respec tiely. These approaches, however, hae negleced the aspect of uncerain information. Another otivation is the eistence of a new class of robles, referred to a dnamic clasi fcain blems [BW88], which cannot be prop-