On the Relation Between Truth Maintenance and Autoepistemic Logic

Ci rcumscr ip t i on on the one hand and autoepis-temic and defaul t logics on the other seem tohave qui te different characterist ics as formalsystems, which makes i t d i f f icu l t to comparethem as formal iza t ions of defeasible connmon-sense reasoning. In th is paper we accomplishtwo tasks: (1) we extend the or ig ina l semanticsof autoepistemic logic to a language which in-cludes variables quant i f ied in to the context ofthe autoepistemic operator , and (2) we showt h a t a certain class of autoepistemic theories inthe extended language has a m in ima l -mode l se-mant ics corresponding to c i rcumscr ip t ion . Weconclude tha t all of the f i rst-order consequencesof paral le l predicate c i rcumscr ip t ion can be ob-ta ined f r om this class of autoepistemic theories.T h e correspondence we construct also shedsl i gh t on the prob lemat ic t rea tment of equal i tyi n c i rcumscr ip t ion . 1 I n t r o d u c t i o n T h e re lat ions between the ma jo r nonmonoton ic logic for-mal isms of AI — defaul t logic, autoepistemic logic, andc i rcumscr ip t ion — is of some impor tance, since all ofthese logics have been proposed as formal isms for vari-ous types of commonsense reasoning. The basic formalequivalence of defaul t and autoepistemic logic has al-ready been shown (see [Konol ige, 1987]), but the relat ionbetween c i rcumscr ip t ion and defaul t or autoepistemiclogic remains obscure. Mos t l y th is is a consequence ofthe dif ferent foundat ions of these logics: c i rcumscr ipt ionis based on a m in ima l -mode l semantics (see [Lifschitz,1985]), wh i le the others use more prooftheoret ic tech-niques (defau l t logic [Reiter, 1980]) or an epistemic op-erator (autoepistemic logic [Moore, 1985]). In t r y i n g to express autoepistemic or default, logic inc i r cumscr ip t i on , researchers have found the basic prob-lem to be t ha t a m in ima l -mode l or even prefered-model *This research was supported by the Office of NavalResearch under Contract No. N00014-85-C-0251, by sub-contract f rom Stanford University under the Defense Ad-vanced Research Projects Administrat ion under ContractNo. N00039-84-C-0211, and by a gift from the System De-velopment Foundation.semantics s imply does not have the capabi l i ty of rep-resenting the requisite proof-theoret ic or epistemic con-cepts (see [Shoham, 1987]). We agree w i th this assess-ment, and say noth ing further about it here. On the other hand, there have been several results onexpressing c i rcumscr ipt ion in default logic. These resultsare summarized in [Ether ington, 1986]; they apply to therestricted case of predicate c i rcumscr ipt ion w i th no fixedpredicates and w i th a f in i te, fixed domain . From a model-theoretic po in t of v iew, the predicatec i rcumscr ipt ion Ci rcum(,4; P; Z) of a f irst-order sentenceA picks out those models of A in which the extension ofthe predicate P is m in ima l . The comparison is acrossmodels w i th the same domain and denotat ion func t ion ,but which might differ in the extensions of the predicatesZ. A l l predicates other than P and Z are fixed, that is,cannot vary in a comparison of models. It was recentlyshown (see [de Kleer and Konol ige, 1989]) tha t fixedpredicates are inessential in predicate c i rcumscr ip t ion,that is, there is a simple t ranslat ion f rom any c i rcum-scr ipt ion w i th fixed predicates to one w i thou t . Hencefixed predicates no longer present an obstacle to rep-resenting circumscript ions in default or autoepistemic