Steps toward Formalizing Context ■ The importance of contextual reasoning is emphasized by

Steps toward FormalizingContextVarol Akman and Mehmet Surav

sThe importance of contextual reasoning is em-explicit using a formal approach and wouldphasized by various researchers in AI. (A partiallist includes John McCarthy and his group, R. V.be used as such by the program.

Guha, Yoav Shoham, Giuseppe Attardi and MariaThe main motivation for studying formalSimi, and Fausto Giunchiglia and his group.)contexts is to resolve the problem of generali-Here, we survey the problem of formalizing con-ty in AI, as introduced by McCarthy (1987).text and explore what is needed for an acceptableMcCarthy believes that AI programs sufferaccount of this abstract notion.

from a lack of generality. A seemingly minoraddition (as in the MYCINexample) to the par-ticular, predetermined possibilities that a pro-gram is required to handle often necessitatesa partial redesign and rewrite of the program.T

he issue of context arises in various ar-Explicitly represented contexts would helpeas of AI, including knowledge repre-because a program would then make its asser-sentation, natural language processing,tion about a certain context.

and intelligent information retrieval. Al-A more general objection to the implicitthough the word contextis frequently used inrepresentation of context—unless the repre-descriptions, explanations, and analyses ofsentation is part of a program that is not mis-computer programs in these areas, its mean-sion critical (in a broad sense of theing is frequently left to the reader’s under-term)—can be given as follows: Assume thatstanding; that is, it is used in an implicit andwe write longer (more involved in terms ofintuitive manner.1

complexity) or shorter (simpler in terms ofAn example of how contexts may help incomplexity) axioms depending on which im-AI is found in McCarthy’s (constructive) criti-plicit context we are in. The problem is thatcism (McCarthy 1984) of MYCIN(Shortliffelong axioms are often longer than is conve-1976), a program for advising physicians onnient in daily situations. Thus, we find ittreating bacterial infections of the blood andhandy to utter “this is clever,” leaving any ex-meningitis. When MYCINis told that the pa-planation about whether we are talking abouttient hasChlorae Vibrioin his intestines, ita horse or a mathematical argument to thewould immediately recommend two weeks ofcontext of talk. However, shorter axiomstetracycline treatment and nothing else.might invite just the opposite of a principleWhile this would indeed do away with theof charity from an adversary. To quote Mc-bacteria, the patient would perish long beforeCarthy (1987, p. 1034):

that due to diarrhea. A “contextual” versionConsider axiomatizing on so as to drawof mycin should know about the context of aappropriate consequences from the in-treatment and would realize that any pre-formation expressed in the sentence,scription must be made in the light of the fact“The book is on the table.” The [adver-that there is alarming dehydration. Thus, insary] may propose to haggle about thethe contextual MYCIN, the circumstances sur-precise meaning of on, inventingrounding a patient would have to be made

difficulties about what can be between

Copyright © 1996, American Association for Artificial Intelligence. All rights reserved. 0738-4602-1996 / $2.00

I wish honorablegentlemenwould havethe fairnessto give the entire contextof what I didsay, and notpick out detachedwords (R. Cobden[1849], quoted in Oxford English Dictionary[1978], p. 902).

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formalcontexts

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the book and the table, or about howmuch gravity there has to be in a space-craft in order to use the word on andwhether centrifugal force counts.

Although our aim in this article is to offer areview of recent formalizations ofcontext—those that can be used for automat-ed reasoning—we first identify the role ofcontext in various fields of AI. We also con-sider some (logic-based) attempts toward for-malizing context. The focus of this discussionis McCarthy’s (1993) proposal that, in ourview, is the groundwork for all other logicistformalizations.

The approach that we pursue in our ownline of research is inspired by situation theory(cf. Barwise and Perry [1983] and especiallyDevlin [1991]) and is detailed in Barwise(1986). The essence of Barwise’s proposal isreviewed, but objectivity and prudence dic-tate that we do not review our own workhere. Therefore, we refer the interested readerto two recent papers that detail our stand-point (Akman and Surav 1995; Surav and Ak-man 1995).2

Some Useful Definitions

According to theOxford English Dictionary,theterm contexttypically has two primary mean-ings:3(1) the words around a word, phrase,statement, and so on, often used to help ex-plain (fix) the meaning and (2) the generalconditions (circumstances) in which anevent, action, and so on, takes place. Clearly,the first definition is closely related to linguis-tic meaning and linguists’ use of the term,whereas the second—more gen-eral—definition is closer to a desirable ac-count of context in AI.4In The Dictionary ofPhilosophy(Angeles 1981, p. 47), the sameterm is defined, better reflecting the seconddefinition:

context(L. contexere, “to weave togeth-er,” from con, “with,” and texere, “toweave”: The sum total of meanings (as-sociations, ideas, assumptions, precon-ceptions, etc.) that (a) are intimately re-lated to a thing, (b) provide the originsfor, and (c) influence our attitudes, per-spectives, judgments, and knowledge ofthat thing.

Similarly, in Collins Cobuild English Lan-guage Dictionary(Collins 1987), the prevalentmeanings of the term include the following:First, the context of something consists of theideas, situations, events, or information thatrelate to it and make it possible to understandit fully. Second, if something is seen in con-

text or if it is put into context, it is consid-ered with all the factors that are related to itrather than considered on its own, so that itcan properly be understood. Third, if a re-mark, statement, and so on, are taken orquoted out of context, it is only consideredon its own, and the circumstances in which itwas said are ignored. Therefore, it seems tomean something different from the intendedmeaning.

The Role of Context

In this section, we discuss context as it relatesto natural language, categorization, intelli-gent information retrieval, and knowledgerepresentation and reasoning.

Context in Natural Language

Context is a crucial factor in communi-cation.5Ordinary observation proves its im-portance: Just consider the confusion that re-sults from the lack of contextual informationwhen, for example, you join a scheduledmeeting half an hour late. Without the cluesof the original context, you might find ithard to make sense of the ongoing discus-sion. In any case, participants would realizethat they cannot assume a lot about yourbackground knowledge and give you a quickrundown of the conversations so far. This isessentially the view of Clark and Carlson(1981), who regard context as informationthat is available to a person for interactionwith a particular process on a given occasion.Their intrinsic context is an attempt to cap-ture the information available to a processthat is potentially necessary for its success.The intrinsic context for grasping what aspeaker means on some occasion is the (limit-ed) totality of the knowledge, beliefs, andsuppositions that are shared by the speakerand the listener (that is, the commonground).

Leech (1981, p. 66) gives another particu-larly attractive quasidefinition, as follows:[W]e may say that the specification ofcontext (whether linguistic or non-lin-guistic) has the effect of narrowing downthe communicative possibilities of themessage as it exists in abstraction fromcontext.

Thus, context is seen as having a so-calleddisambiguating function (among others). Toquote Leech (1981, p. 67) once again, “Theeffect of context is to attach a certain proba-bility to each sense (the complete ruling-outof a sense being the limiting case of nil prob-ability).” Consider the following simple (pos-

sibly trivial for human beings) segment ofconversation (Barwise 1987a):

A (a woman, talking to B): I am aphilosopher.

B (talking to C and referring to A): She isa philosopher.

C (talking to A): So, you are a philoso-pher.

Context eliminates certain ambiguities ormultiple meanings in the message. In the pre-vious segment, one of the first context-de-pendent words is philosopher.The meaning ofthis word is determined using the context ofconversation. Although this segment is in-sufficient to carry the proper connotation ofthis word, our common understanding selectsan appropriate meaning from a set of possiblemeanings.6

In the previous example, the indexicals (forexample, I, you) can be bound to appropriatepersons only with the help of context. For ex-ample, the sentences uttered by A and B havethe same content, but we can only say thisusing some circumstantial information andconventions about the conversation. This cir-cumstantial information might be formalizedthrough context. To quote Recanati (1993, p.235), “[T]he meaning of a word like ‘I’ is afunction that takes us from a context of utter-ance to the semantic value of the word inthat context, which semantic value (the refer-ence of ‘I’) is what the word contributes tothe proposition expressed by the utterance.”This view was made popular by Kaplan’s(19) seminal work on the logic of demon-stratives.

Another function of context arises whenwe deal with quantifiers in logic or naturallanguage semantics. The range and interpre-tation of quantifiers depend on the context.For example, the quantifier allusually doesnot apply to all objects, only to those of aparticular kind in a particular domain, deter-mined by the contextual factors. Another ex-ample might be the interpretation of themeaning of many.In an automobile factory,10 automobiles might not qualify as many,but if a person owns 10 automobiles, itcounts as many. Clearly, even the last inter-pretation about a person with 10 automobilesis context dependent. One might proposethat manycan only be interpreted as a ratio,which, too, has a contextual dependency onthe ratio. In a class of students, half the stu-dents cannot be considered as many, enoughto cancel a midterm exam, but surely must beregarded as many in an influenza epidemic.Context might be used to fill the missing

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LibraryTourism SchoolPublishing CompanyRector's ResidenceEngineering BuildingFigure 1. A Partial View of the Bilkent Campus.

parameters in natural language utterances.Consider an utterance of the sentence, “CarlLewis is running.” Here, the time and theplace of the running action are determinedby the context. For example, if we are watch-ing a competing Lewis on television at the1992 Barcelona Olympic Games, then thetime and the place of the utterance are differ-ent from what we would get if we watch himpractice from our window.

Some relations stated in natural languagenecessarily need a context for disambigua-tion. Consider an utterance of the sentence,“The engineering building is to the left of thelibrary.” In the context of the Bilkent campus(figure 1), if we are viewing the buildingsfrom the publishing company, the utteranceis true, but if we are in the tourism school,the utterance is false. More interestingly, if weare looking from the rector’s residence, thisutterance must be considered neither true norfalse: The library is behind the engineeringbuilding. Thus, for natural languages, a flesh-ing-out strategy, that is, converting every-thing into decontextualized eternal sentences(Quine 1969), cannot be employed becausewe do not always have full and precise infor-mation about the relevant circumstances.

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Several studies in computational linguisticsfocused on the semantics of coherent multi-sentence discourse (or text).7The essentialidea is that in discourse, each new sentence sshould be interpreted in the context providedby the sentences preceding it. As a result ofthis interpretation, the context is enrichedwith the contribution made by s. (For exam-ple, an important aspect of this enrichment isthat elements are introduced that can serve asantecedents to anaphoric expressions follow-ing s.) Emphasizing the representation andinterpretation of discourse in context, dis-course representation theory (van Eijck andKamp 1996; Kamp and Reyle 1993) hasinfluenced much subsequent work in compu-tational linguistics.

To interpret extended discourse, some oth-er researchers regard discourse as a hierarchi-cally organized set of segments. The expecta-tion is that each segment displays some sortof local coherence; that is, it can be viewed asstressing the same point or describing thesame state of affairs. Grosz and Sidner (1986)outline a particularly illuminating model ofthis segmentation process, complete with anintentional constitution of discourse avail-able in the segments.

The use of general world knowledge (forexample, knowledge about causality and ev-eryday reasoning) is also fundamental for dis-cerning much discourse. Early work bySchank and Rieger (1974) used such knowl-edge for computer understanding of naturallanguage. More recent technical contribu-tions by Charniak (1988) and Hobbs et al.(1993) can be seen as two formal attempts inthis regard: (1) generate expectations that arematched into plausible interpretations of thediscourse and (2) construct an (abduction-based) argument that explains why the cur-rent sentence is true.

Finally, it is worth noting that context haslong been a key issue in social studies of lan-guage, that is, how human beings use lan-guage to build the social and cultural organi-zations that they inhabit. Lyons (1995, p.292), thinking that this is natural, affirmsthat “in the construction of a satisfactory the-ory of context, the linguist’s account of theinterpretation of utterances must of necessitydraw upon, and will in turn contribute to,the theories and findings of social sciences ingeneral: notably of psychology, anthropology,and sociology.” The reader is also referred toGoodwin and Duranti (1992) (and other arti-cles in the same volume). They consider con-text a key concept in ethnographically orient-ed studies of language use and claim that this

notion “stands at the cutting edge of muchcontemporary research into the relationshipbetween language, culture, and social organi-zation, as well as into the study of how lan-guage is structured in the way it is” (Goodwinand Duranti 1992, p. 32).

Context in Categorization

Categorization is one of the basic mental pro-cesses in cognition (Rosch 1978). We, as hu-man beings, can categorize various types ofobjects, events, and states of affairs, and ourcategorizations depend on the circumstanceand perspective. Consider the following sce-nario:

In Springfield (the hometown of BartSimpson), there are three barbers work-ing for money and a man who does notwork for money (because he has anotherjob) but serves the community by shav-ing senior citizens on Sundays. If welook at the situation from a common-sense perspective, there are four barbersin town, but from, say, the mayor’s pointof view, there are only three (licensed,tax-paying, and so on) barbers.

Here, it is clear that context (or perspective)plays an important part in the correctclassification.

Barwise and Seligman (1992) use naturalregularities to study the role of context in cat-egorization. An example regularity from Selig-man (1993) is, “Swans are white.” This is atypical natural regularity in the sense that itis both reliable and fallible. Natural regulari-ties are reliable because they are needed to ex-plain successful representation, knowledge,truth, and correct reference. They are falliblebecause they are needed to account for misin-terpretation, error, false statements, and de-feasible reference. Swans are, in general,white; thus, the regularity is reliable and ex-plains a fact. There might be exceptions suchas the Australian swans—they are usuallyblack—but it does not mean that the regulari-ty does not hold. Here, the fundamentalproblem with isolating the essential proper-ties of a regularity is that any statement ofthem depends on some context of evaluation;that is, we should evaluate this regularity for,say, the European swans.

There is a correlation between nonmono-tonic reasoning and the role of context-de-pendent factors in natural regularities. Al-though natural regularities are typicallyconsidered in philosophical discussions, theyintuitively correspond to material implicationin logic, and the effect of contextual factors issimilar to the effect of nonmonotonicity. In

logic, implication and nonmonotonicity areusually studied in a syntactic fashion, and thereasons behind the abnormalities are typical-ly omitted from the discussion.

If we could completely describe all the con-textual factors, then the problem would goaway, and we would not require extra ma-chinery. However, we must always include aso forthto cover the unexpected contextualfactors; in many cases, it is simply impossibleto state all the relevant ones (Tin and Akman1992). Still, we must somehow be able to dealwith them, which explains the introductionof the notion of context; using this notion incategorization should be useful.

Context in Intelligent Information Retrieval

A formal notion of context might be useful ininformation retrieval because it can increasethe performance by providing a frameworkfor well-defined queries and intelligent textmatching. Given the explicit context, a querymight be better described, and thus, the recalland precision might be enhanced. In thissense, we find the work of Hearst (1994) use-ful because she emphasizes the importance ofcontext in full-text information access.

Traditional methods of information retrievaluse statistical methods to find the similaritiesbetween the documents and the relevance ofthe documents to the query. In this respect, aformal context means that the query will bebetter described because it will contain moreinformation than just a few keywords in thesearch. Inclusion of the context of the queryalso allows us to run more sophisticated meth-ods to measure the relevance.

Various syntactic approaches can measurethe relevance of a term to a document. Untilrecently, the only respectable methods werethe statistical methods that are based on thefrequency of occurrence. Lately, psychologi-cal, epistemic, and semantic considerationsare beginning to flourish (Froehlich 1994).For example, Park (1994) studies the contri-butions of relevance to improving informa-tion retrieval in public libraries. According toher, the search criteria for any query shouldbe set according to the users’ criteria of rele-vance. Because different users exhibit differ-ent relevance criteria, the query formation isa dynamic task.

The essential work on relevance is owed toSperber and Wilson (1986), who mainly con-sider the psychological relevance of a propo-sition to a context. Their assumption is thatpeople have intuitions of relevance; that is,they can consistently distinguish relevant

from irrelevant information. However, theseintuitions are not easy to elicit or use as evi-dence because the ordinary language notionof relevance comes along with a fuzzy (vari-able) meaning. Moreover, intuitions of rele-vance are relative to contexts, and there is noway to control exactly which context some-one will have in mind at a given moment.Despite these difficulties, Sperber and Wilsonintend to invoke intuitions of relevance. Ac-cording to them, a proposition is relevant toa context if it interacts in a certain way withthe (context’s) existing assumptions aboutthe world, that is, if it has some contextualeffects. These contextual effects include (1)contextual implication,a new assumption canbe used together with the existing rules in thecontext to generate new assumptions; (2)strengthening, a new assumption can strength-en some of the existing assumptions; and (3)contradictionor elimination, a new assumptionmight change or eliminate some of the exist-ing assumptions of the context.

Sperber and Wilson talk about degrees ofrelevance. Clearly, one piece of informationmight be more relevant to a particular con-text than another. To compare the relevanceof pieces of information, they consider themental processing effort, that is, the length ofthe chain of reasoning and the amount of en-cyclopedic information involved, and so on.Finally, they propose their celebrated rele-vance maxim(Sperber and Wilson 1986),which has two parts: (1) An assumption is rel-evant in a context to the extent that its con-textual effects in this context are large. (2) Anassumption is irrelevant in a context to theextent that the effort required to process it inthis context is large.

Harter (1992) uses the theoretical frame-work of Sperber and Wilson to interpret psy-chological relevance in relation to informa-tion retrieval. According to him, reading anew bibliographic citation (the setting here isthat of a user, accepting or rejecting a biblio-graphic document retrieved by a library infor-mation system) can cause a user to create anew context. A set of cognitive changes takeplace in that context; the citation and thecontext influence each other to give rise tonew ideas. In other words, a retrieved citation(viewed as a psychological stimulus) is rele-vant to a user if it leads to cognitive changesin the user.

Using knowledge about data to integratedisparate sources can be considered a moresophisticated extension of information re-trieval (Goh, Madnick, and Siegel 1995). Al-though networking technologies make phys-

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A formal notion of context might be useful in informationretrieval because it can

increase theperformanceby providing a frameworkfor

well-definedqueries andintelligenttext

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ical connectivity (and, hence, access to as-sorted data) feasible, much of these data arenot perceived as meaningful because of thelack of information regarding the context ofthe data. This research aims at the develop-ment of a formal theory of context inter-change (Reddy and Gupta 1995) using (1)context definitions (for example, definingthe semantics, organization, and content ofdata) and (2) context characteristics (for ex-ample, data quality, security) and, thus, sug-gests a solution to the problem of semanticinteroperation between (semantically) het-erogeneous environments.

Context in Knowledge

Representation and Reasoning

When we state something, we do so in a con-text. For example, 37 degrees centigrade ishigh in the context of a weather report butnormal in the context of a medical diagnosis.In the context of Newtonian mechanics, timeis ethereal, but in the context of general rela-tivity, this is hardly the case. The examplescan be continued. The main point is that ifwe are to reason in a commonsense way, wehave to use certain contexts.

The importance of the notion of contexthas been realized by philosophers for cen-turies.8Early on, philosophers recognizedthat a causal connection between two eventsis only relative to a certain background and,thus, only in certain contexts. McCarthy(1987) was the first researcher to realize thatthe introduction of a formal notion of con-text is required for generality in AI.

According to McCarthy, there is simply nogeneral context in which all the stated ax-ioms always hold, and everything is mean-ingful. When one writes an axiom, it holds ina certain context, and one can always presentanother (more general) context in which theaxiom fails. McCarthy formalizes relativizedtruth within a context using a special predi-cate holds(p,c), which states that propositionpholds in context c.9

If we compare the two approaches, namely,(1) using holdsand (2) adding a context param-eter to each function and predicate, we mustprefer using holds because it allows us to usethe notion of context uniformly as first-classcitizens.10A problem with this approach (usingholds) is that if we are to live in a first-orderworld (the world of first-order logic [FOL]), wehave to reify pin holds(p,c). Alternative(modal) approaches to reifying assertions areinvestigated in Buvac˘, Buvac˘, and Mason(1995); Nayak (1994); and Shoham (1991).

Among the advantages gained as a result

of the use of contexts are the following:11

Economy of representation:Differentcontexts can circumscribe (in a nontechnicalsense) the parts of the knowledge base thatare accessible in different ways, allowing therepresentation of many knowledge bases in asingle structure.

Efficiency of reasoning:By factoring out apossibly large knowledge base, contextsmight permit more competent reasoningabout the real, intended scope.

Allowance for inconsistent knowledgebases: The knowledge base might be parti-tioned according to the context of its use. Inthis way, we can accommodate contradictinginformation in the same knowledge base aslong as we treat such information carefully.Resolving of lexical ambiguity:By usingcontext, the task of choosing the correct in-terpretation of lexical ambiguity is made easi-er.12However, some argue that although acontext formalism can represent lexical ambi-guity, additional knowledge is needed to per-form the resolution (Buvac˘ 1996b).

Flexible entailment:Context might affectthe entailment relation. For example, in aparticular context, entailment might warranta closed-world assumption, whereas in someother context, this assumption needs to bedropped (the classical case).

Being the largest commonsense knowl-edge-building attempt, CYC(Lenat 1995;Guha and Lenat 1990), has crucial pointerson reasoning with an explicit notion of con-text (Guha 1991). Some aspects of the repre-sentation of knowledge that are influenced bycontextual factors include the following:

Language:The language (that is, the predi-cates, functions, and categories) used for rep-resentation should be appropriate for theirintended domain. For example, MYCINandONCOCIN—two renowned medical diagnosisprograms—overlap significantly in their do-mains; however, ONCOCINhas some conceptof time, whereas MYCINdoes not.

Granularity and accuracy:As with vocab-ulary, the application area, thus context, de-termines the granularity and accuracy of thetheory.

Assumptions: The assumptions that a giv-en task permits often lead to a simplificationof the vocabulary. If we try to continue thissimplification for large domains, at onepoint, the assumptions become unstable.Thus, either we should use a highly expres-sive vocabulary or distribute the assumptionsto different tasks.

CYCresearchers identify two approaches tobuilding large commonsense knowledge bases

and reasoning with them. First, the straight-forward way that a knowledge base buildermight choose is the introduction of an ex-tremely expressive and powerful vocabulary.This approach increases the complexity of theproblem because such a vocabulary causesdifficulties in truth maintenance and pro-duces large search spaces. The second way(Guha 1991) is to make the context depen-dence of a theory explicit. In this approach,assertions (axioms, statements) are not uni-versally true; they are only true in a context.An assertion in one context might be avail-able for use in a different context by perform-ing a relative decontextualization. In CYC, theuses of context include the following:

A general theory of some topic: A theoryof mechanics, a theory of weather in Alaba-ma, a theory of what to look for when buyingdresses, and so on: Such contexts are calledmicrotheories(Guha 1991). Different mi-crotheories make different assumptions andsimplifications about the world. For any top-ic, there might be different microtheories ofthe topic, at varying levels of detail.

A basis for problem solving: For somedifficult problems, we can form a particularcontext. We collect all related assumptions,rules, and so on, in that context (called theproblem-solving context [PSC] in CYC[Guhaand Lenat 1990]) and can process a group ofrelated queries in a relatively small searchspace. Such contexts must be created dynami-cally and be disposed of afterward.

Context-dependent representation of ut-terances:Naturally, we can use anaphoricand indefinite statements without completelydecontextualizing them. For example, thewords the personmight be used in a discoursewithout identifying him/her exactly.

Formalizations in Logic

The notion of context was first introduced toAI in a logicist framework by McCarthy in his1986 Turing Award paper (McCarthy 1987).13McCarthy published his recent ideas on con-text in McCarthy (1995, 1994, 1993). Othernotable works on formalizing context are S.Buvac˘ et al. (Buvac˘, Buvac˘, and Mason 1995;Buvac˘ and Mason 1993), Attardi and Simi (At-tardi and Simi 1995, 1994), F. Giunchiglia et al.(Giunchiglia and Serafini 1994; Giunchiglia1993), Guha (1991), and Shoham (1991). Wereviewed McCarthy’s (1987) early ideas in theprevious section. In this section, we evaluatethe other logicist formalizations, starting withMcCarthy’s more recent proposal.

McCarthy on Contexts

McCarthy (1993) states three reasons for in-troducing the formal notion of context.14First, the use of context allows simple axiom-atizations. To explain, he states that axiomsfor static blocks world situations can be liftedto more general contexts—those in which thesituation changes.15Second, contexts allowus to use a specific vocabulary of, and infor-mation about, a circumstance. An examplemight be the context of a (coded) conversa-tion in which particular terms have particularmeanings that they would not have in dailylanguage in general.16 Third, McCarthy pro-posed a mechanism by which we can build AIsystems that are never permanently stuckwith the concepts they use at a given time be-cause they can always transcend the contextthey are in.

The third goal brings about two problems:First is when to transcend a context. Eitherthe system must be smart enough to do so, orwe must instruct it about when to transcendone or more levels up. Second is where totranscend. This problem can be clarified if weare prepared to accept that formulas are al-ways considered to be asserted within a con-text.

The basic relation relating contexts andpropositions is ist(c, p). It asserts that proposi-tion pis true in context c. Then, the main for-mulas are sentences of the formc′: ist(c, p) .

In other words, pis true in context c, whichitself is asserted in an outer context c′.To give an example of the use of ist,c0: ist(context-of(“Sherlock Holmes sto-ries”), “Holmes is a detective”)

asserts that it is true in the context of Sher-lock Holmes stories that Holmes is a detec-tive. Here, c0is considered to be the outer con-text. However, in the context context-of(“Sherlock Holmes stories”), Holmes’s moth-er’s maiden name does not have a value.

Two key properties of context are as fol-lows: First, contexts are abstract objects. Somecontexts will be rich objects just like the situ-ations in situation calculus.17Some contextswill not be as rich and might be fully de-scribed, for example, simple microtheories(Guha 1991). Second, contexts are first-classcitizens: We can use contexts in our formulasin the same way we use other objects.

Relations and Functions Involving Contexts

There are some relations working betweencontexts. The most notable one is Յ, which

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The notion ofcontext wasfirst

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in a logicistframework by McCarthyin his 1986Turing Awardpaper.

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defines a partial ordering over contexts. Be-tween two contexts, we might consider amore general than relation (c1Յc2), meaningthat the second context contains all the in-formation of the first context and probablymore. Using Յ, we can lift a fact from a con-text to one of its supercontexts using the fol-lowing nonmonotonic rule:᭙c1᭙c2᭙p(c1Յc2) ٙist(c1, p) ٙ¬ ab1(c1, c2, p) →ist(c2, p) .

Here, c2is a supercontext of c1, pis a proposi-tion of c1,ab1 is an abnormality predicate,and ¬ ab1(c1, c2, p) is used to support non-monotonicity. Analogously, we can state asimilar lifting rule between a context and oneof its subcontexts:

᭙c1᭙c2᭙p(c1Յc2) ٙist(c¬ ab2(c2, p) ٙ1, c2, p) →ist(c1, p) .

The difference between the abnormality rela-tions is crucial: ab1 represents the abnormali-ty in generalizing to a supercontext, whereasab2 corresponds to the abnormality in spe-cializing to a subcontext.

Here are some examples of functions oncontexts that we might want to define:value(c, t) is a function that returns thevalue of term tin context c:

value(context-of(“Sherlock Holmes sto-ries”), “number of wives of Holmes”) = 0.This example states that Holmes has no wifein the context of Sherlock Holmes stories.specialize-time(t, c) is a context related to cin which the time is specialized to the value t:c0: ist(specialize-time(t, c), at(JMC, Stan-ford))

states that at time tin context c, JMC (JohnMcCarthy) is at Stanford University. Insteadof specializing on time, we can also specializeon location, speaker, situation, subject mat-ter, and so on.

The formal theory of context can be usedto model inference in the style of deduction.Thus, assuming(p, c) is another context likecontext cin which proposition pis assumed.Using this function, we might dynamicallycreate a context containing the axioms wedesire. The new context validates the follow-ing rules (McCarthy and Buvac˘ 1994):

Importation:This is the rule c: p→q5assuming(c, p) : q.

Discharge: This is the rule assuming(c, p) :q5c: p→q.

When we take contexts in this natural de-duction sense (as suggested in McCarthy[1987]), the operations of entering and leav-ing a context might be useful and shortenthe proofs involving contexts. In this case,

ist(c, p)will be analogous to c→p, and theoperation of entering ccan be taken as as-suming(p, c). Then, entering cand inferring pwill be equivalent to ist(c, p) in the outer con-text.

Lifting

Here are some of the things we can

do with lifting. (By liftinga predicate fromone context to another, we mean transferringthe predicate to the other context with ap-propriate changes [when necessary].)

Transfer a formula verbatim: If two con-texts are using the same terminology for aconcept in an axiom, lifting is a naturalchoice. For example, the following lifting rulestates that we can use the axioms related tothe on(x, y) of above-theorycontext in general-blocks-worldcontext without any change:c0: ᭙x᭙yist(above-theory, on(x, y)) →ist(general-blocks-world, on(x, y)) .Change the arity of a predicate:In differ-ent contexts, the same predicate might take adifferent number of arguments. McCarthy’sexample for this is on,which takes two argu-ments in above-theorycontext and three argu-ments in a context cin which onhas a thirdargument denoting the situation. The liftingrule is

c0: ᭙x᭙y᭙sist(above-theory, on(x, y)) →ist(context-of(s), on(x, y, s)) ,

where context-of is a function returning thecontext associated with the situation sinwhich the usual above-theoryaxioms hold.Change the name of a predicate:Similarto the case with arities, we can change thename of a predicate by lifting rules. For exam-ple, we can translate onto üzerinde when wemove from above-theoryto turkish-above-theory:c0: ᭙x᭙yist(above-theory,on(x, y)) →ist(turkish-above-theory,üzerinde(x, y)) .

Other Issues

McCarthy proposed relative

decontextualization as a way to do the workof eternal sentences—the mythical class em-bracing those sentences that express the sameproposition no matter what world the utter-ance takes place in (Quine 1969) (assumingthat the world in question is linguisticallysimilar to ours). McCarthy feels strongly thateternal sentences do not exist. His proposedmechanism depends on the premise thatwhen several contexts occur in a discussion,there is a common context above all of theminto which all terms and predicates can belifted. (However, the outermost context doesnot exist.) Sentences in this context are rela-tively eternal. A similar idea is used in thePSCs of CYC.

Another place where context might be use-

ful is the representation of mental states (Mc-Carthy 1993). McCarthy proposes a schemein which mental states can be thought of asouter sentences; for example,believe(Jon, publication(AAAI) = AIMag, because…) ,

where the ellipsis denotes the reasons forJon’s belief that AI Magazineis a publicationof AAAI. The point of representing mentalstates with such sentences is that the groundsfor having a belief can be included. The ad-vantage gained by this is twofold: In a belief-revision system, when we are required to dobelief revision, incorporating the reasons forhaving a belief simplifies our work. However,when we use beliefs as usual (that is, no beliefrevision is required), we simply enter the re-lated context and assert them. For example,in an outer context, the sentence about AIMagazine, with reasons, is asserted. In an in-ner context, the simpler sentencepublication(AAAI) = AIMagwould suffice be-cause we have already committed ourselves toreasoning with this last proposition.

Guha on Contexts

Guha (1993, 1991) finds an essential use forformal contexts in implementing his so-called microtheories. Microtheoriesare theoriesof limited domains. Intuitively, microtheoriesare the context’s way of seeing the world andare considered to have the following two ba-sic properties: (1) a set of axioms is related toeach microtheory and (2) a vocabulary tellsus the syntax and semantics of each predicateand each function specific to the microtheo-ry. Similar to McCarthy’s conception, mi-crotheories are interrelated through liftingrules stated in an outer context.

Guha suggests several ways of using con-texts effectively in reasoning, including thefollowing:

First, contexts might be useful in puttingtogether a set of related axioms. In this way,contexts are used as a means for referring to agroup of related assertions (closed under en-tailment) about which something can be said.Second, contexts can be used as a mecha-nism for combining different theories. If theassertions in one context were not automati-cally available in other contexts, the systemmight as well be a set of disconnected knowl-edge bases. Therefore, by using lifting rules,different microtheories can be integrated.

Third, using contexts, we might have mul-tiple models of a task. For example, for thetask of finding out what to do in case of fire,we can offer different models for a workplaceand for a house. In a workplace, the first

thing to do might be to take away a file ofdocuments, whereas in a house, childrenmust be saved first.

Lifting rules might be used to transfer factsfrom one context (source) to another context(target). In the target context, the scope ofquantifiers, the interpretation of objects, andeven the vocabulary can change. Therefore,when we state a lifting rule, we must take allthe possible outcomes into account. In thecase of natural language, the problem be-comes more complicated because indexicalsand demonstratives come into play. Liftingrules should definitely be nonmonotonic.Guha uses default reasoning in the statementof lifting rules. His intuitions about the gen-eral lifting rules are as follows:

Default coreference: Although there aredifferences among contexts, it can be expect-ed that there will be similarities and overlap.As a result, a significant number of terms indifferent contexts refer to (mean) the samething. Such terms can be lifted from one con-text to another without any modification.Similarly, we can expect overlap in many for-mulas, which can be lifted from one contextto another without any change. Therefore, itwill be a great simplification if we assumethat a lifting operation will not require anymodification, unless it is explicitly stated thatthere should be a change.

Compositional lifting: Between contexts,there might be differences in vocabulariesboth in the words used and in the intendeddenotations of these words. In this case, spec-ifying lifting rules for individual predicatesshould be enough for the system to use theserules in the lifting of formulas involving thepredicates.

Although Guha’s proposal accommodatesany level of nesting with context, in CYC,there are two levels: (1) microtheories and (2)the default outer level. The lifting rules andgeneral facts are stated in the outer level, anda problem is solved by the construction of aPSC under this level, unless the problem is lo-cal to a microtheory.

S. Buvac˘ et al. on Contexts

Buvac˘ and Mason (1993) (and a more recentwork, Buvac˘, Buvac˘, and Mason [1995]) ap-proach context from a mathematical view-point. They investigate the logical propertiesof contexts. They use the modality ist(c, p) todenote context-dependent truth and extendthe classical propositional logic to what theycall the propositional logic of context.(Thequantificational logic of context is treated inBuvac˘ [1996a].) In their proposal, each con-

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Buvac˘ andMason … approach context from a

mathematicalviewpoint.

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Giunchiglia… takes acontext to be a theory of the world

that encodes an agent’sperspective

of it and that is used during a given reasoning process.

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text is considered to have its own vocabu-lary—a set of propositional atoms which aredefined (or meaningful) in that context.S. Buvac˘ and Mason discuss the syntax andsemantics of a general propositional languageof context and give a Hilbert-style (Gallier1987, p. 79) proof system for this language.The key contribution of their approach is pro-viding a model theory for contexts. Twomain results are the soundness and complete-ness proofs of this system. They also providesoundness and completeness results for vari-ous extensions of the general system andprove that their logic is decidable.

The formal system is defined by the axioms(PL, K, and ⌬) and inference rules (MP, Enter,Exit) given below:

(PL): 5c φ(meaning: a formula φis prov-able in context c[with a fixed vocabu-lary] provided φis an instance of a tau-tology).

(K): 5cist(c1, φ→ϕ) →(ist(c1, φ) →ist(c1,ϕ)) (meaning: every context is closedwith respect to logical consequence).(⌬): 5cist(c1, ist(c2, φ) ٚϕ) →ist(c1,ist(c2, φ)) ٚist(c1, ϕ) (meaning: everycontext is aware of what is true in everyother context).

(MP): From 5cφand 5cφ→ϕ, infer 5cϕ.

(Enter): From 5c,ist(c, φ), infer 5c

φ;

(Exit): From 5cφ, infer 5c′ist(c, φ).

This system has the following two features(Buvac˘ and Mason 1993):

First, a context is modeled by a set of par-tial truth assignments that describe the possi-ble states of affairs in the context. The istmodality is interpreted as validity: ist(c, p) istrue if and only if the propositional atom pistrue in all the truth assignments associatedwith context c.

Second, the nature of particular contexts isitself context dependent. S. Buvac˘ and Ma-son’s example is Tweety, which has differentinterpretations when it is considered in anonmonotonic-reasoning– literature contextand when it is considered in the context ofTweety and Sylvester. This observation leadsus to consider a context as a sequence of indi-vidual contexts rather than a solitary context.In S. Buvac˘ and Mason’s terminology, such aproperty is known as nonflatnessof the sys-tem. The acceptance of a sequence of con-texts respects the intuition that what holds ina particular context can depend on how thiscontext is reached.18

S. Buvac˘ and Mason show that the accep-tance of the outermost context simplifies the

metamathematics of the contexts. They firstassume that there is no outermost contextand build a proof system on this assumption.Then, they show that introducing the outer-most context only simplifies the way they aredealing with nonflatness.

F. Giunchiglia and Others on Contexts

Giunchiglia (1993) takes a context to be atheory of the world that encodes an agent’sperspective of it and that is used during a giv-en reasoning process. A context is necessarilypartial and approximate. Contexts are not sit-uations (of situation calculus) because a situa-tionis the complete state of the world at agiven instant.

In formalizing context, Giunchiglia’s pointof departure is partitioned databases (cf.Giunchiglia and Weyhrauch 1988) for originsof this work). Each partition, Ai, can have dif-ferent vocabulary. For example, A1supportsarithmetic operations, but A2might supportlogical operations. With this approach, thenotion of well-formedness can be localizedand can be distinct for each partition Ai. Informal terms, a context c⌬iis a triple , where Liis the language of the context,Aiis the axioms of the context, and ⌬iis theinference mechanism of the context. Underthis definition, linking (bridge) rules are ofthe form / , where Aiis a for-mula in ci, and Ajis the newly derived formu-la incj(also called ajustified assumption).Giunchiglia offers the following to show theuse of bridge rules:

First, the usual modus ponens (MP) can berepresented as / .Second, a multicontextual version of MP isrepresented as / .Third, McCarthy’s ist formula (asserted inc′) becomes / . (If we canprove Ain context c, then we can prove incontext c′that we can prove Ain c.)

The first rule allows us to derive Binside acontext just because we have derived A→Band Ain the same context. MulticontextualMP allows us to derive Bin context ckjust be-cause we have A→Bderived in context ciand Aderived in context cj.If these threecontexts are assumed to represent the beliefsof three agents, then it is seen that Bis notasserted as a result of deduction in ckbut,rather, as a consequence of dissemination ofresults from ciand cj.

In a related work, Giunchiglia and Serafini(1994) formalize multilanguage systems of thesort described here and propose them as an al-ternative to modal logic. (Multilanguage sys-

tems allow a hierarchy of first-order languages,each language containing names for the lan-guage below.) They then offer technical, epis-temological, and implementation motivationsto justify their proposal. Two useful applica-tions of multilanguage systems can be foundin Giunchiglia, Traverso, and Giunchiglia(1992) and Giunchiglia and Serafini (1991).

Attardi and Simi on Contexts

Attardi and Simi (1995) offer a viewpoint rep-resentation that primarily depends on theview of context in a natural deduction sense.According to Attardi and Simi, contexts aresets of reified sentences of the FOL.

The main purpose of Attardi and Simi is topresent a formalization of the notion of view-point as a construct meant for expressing va-rieties of relativized truth. The formalizationis done in a logic that extends the FOLthrough an axiomatization of provability andwith the proper reflection rules.19

The basic relation in the formalization isin(′A′, vp), where Ais a sentence provablefrom viewpoint vpby means of natural de-duction techniques. Viewpoints denote setsof sentences that represent the axioms of atheory. Viewpoints are defined as a set ofreified metalevel sentences.

Because viewpoints are defined as sets ofreified sentences, operations between view-points are carried out with metalevel rules,for example,

ᎏᎏin(′B′, vpʜ{′A′}).in(′A→B′, vp)

This operation corresponds to the followingin classical logic:ᎏᎏvpʜ{A} 5B.vp5A→B

The effective use of viewpoints in doinguseful proofs requires a connection betweenthe metalevel and the object-level rules. Thefollowing rules make this connection:vp15in(′A′, vp2) / vp1ʜvp25A .(reflection)vp5CA / 5in(′A′, vp) .(reification)

The notation 5Cstands for “classically deriv-able” or “derivable without using the reflec-tion rules.”

Attardi and Simi cite a wide range of exam-ples using the viewpoints. For example, basedon viewpoints, the notions of belief, knowl-edge, truth, and situation can be formalizedas follows:

Belief: The belief of an agent gis captured

by means of “in” sentences, using vp(g) as theviewpoint corresponding to the set of as-sumptions of the agent. Thus,Bel(g, A) = in(A, vp(g)) ,and by the reflection rulein(A, vp(g)) →(vp(g) →A) ,

we can use the beliefs of an agent.

Truth:Truth is captured as provability in aspecial theory, viz., the real world (RW). Ide-ally, everything that is true should be deriv-able in this theory, and truth can be definedas

True(A) = in(A, RW) .

Knowledge:Attardi and Simi view knowl-edge as true belief:

K(g, A) = Bel(g, A) ٙTrue(A) .

Clearly, all the properties typically ascribedto knowledge can be derived, for example, K(g, A) →A.

Situations in the manner of Barwise andPerry: Attardi and Simi take situations as setsof basic facts (Barwise and Etchemendy 1987)and use an approach similar to that of belief.Thus, they define a basic relationHolds(A, s) = in(A, vp(s)) ,

where vp(s) is the set of facts holding in a si-tuation s. (See The Situation-Theoretic Ap-proach.)

Shoham on Contexts

Shoham (1991) uses the alternative notationpcto denote that assertion pholds in contextc. According to him, every assertion is mean-ingful in every context, but the same asser-tion might have different truth values in dif-ferent contexts.20Thus, his approach isdifferent from the approaches of McCarthy,Guha, and S. Buvac˘ et al.

Shoham describes a propositional languagedepending on his more general than relation(ʛ

и

). The relation defines a weak partial order-ing between contexts; not every pair of con-texts is comparable under it. Is there a mostgeneral (or most specific) context? Mathemat-ically, this question corresponds to, “Is there

an upper (or lower) bound on ʛ

и

?” InShoham’s proposal, the question is not an-swered, but when the system is analyzed, theexistence of the most general and the mostspecific contexts is considered.21

The language Shoham describes is similar

to FOL, but his relations ʛ

и, ٚи, ٙи, and ¬иworkover contexts. Here, xٙ

иyis defined as thegreatest lower bound on to ʛ

иxand ywith respect

(if it exists). Similarly, xٚиyis defined asa least upper bound of the contexts xand it exists). When defined, ¬

иy(if

xis the contextArticles

Attardi

and Simi …offer a viewpoint representationthat

primarily de-pends on theview of context in a natural deductionsense.

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Abstract situations are the constructsthat are moreamenable

to

mathematicalmanipulation.An abstractsituationisdefined as a

… set of infons.

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that is not comparable to xunder ʛи.22

A con-text set is and-closedif it is closed under con-junction, or-closedif it is closed under dis-junction, and-or-closedif it is both, not-closedif it is closed under negation, and simplyclosedif it is all three. From these definitions,we see that if an or-closed context set con-tains both xand ¬

иxfor some x, then the con-text set contains the most general context,that is, the tautological context.Similarly, un-der the same condition, an and-closed con-text set contains the most specific context,that is, the contradictory context.

What should be the logical status of pc?Shoham takes it to be an assertion and intro-duces a simple language for discussing con-texts and assertions about them. Basically,

given a context set Cwith the partial order ʛи

and a propositional language ᑦ, the set ofwell-formed formulas is the smallest set Ssuch that the following is true:

First, if c1, c2⑀C, then c1ʛи

c2⑀S.Second, if p⑀ᑦ, then p⑀S.

Third, if s⑀Sand c⑀C, then scFourth, if s1, s2⑀S, then s1ٙи⑀S.

s2

⑀Sand ¬ s1⑀S.

Shoham’s purpose is not really to offer theright semantics for pc; he is more interested inidentifying some options for achieving thisultimate goal and investigating the interac-tion between modal operators (for example,the knowledge operator K in the logic ofknowledge and belief) and context. Some in-teresting proposals are given in this directionto investigate the notion of contextualknowledge, that is, the meaning of Kcp—hisnotation for “p is known in context c.”

The Situation-Theoretic

Approach

The standard reference on situation theory isDevlin (1991). The original work of Barwiseand Perry (1983) is still worthy of study andremains an elegant philosophical argumentfor introducing situations. Unfortunately, thenotation and (sometimes) the terminology ofBarwise and Perry are rather outdated. Ac-cordingly, we use Devlin’s notation and ter-minology in the following discussion.

According to situation theory, infonsarethe basic informational units (discrete itemsof information). They are denoted as ͗͗P, a1,… , an, i͘͘, where Pis an n-place relation; a1,… , anare objects appropriate for the respec-tive argument places of P; and iis the polarity(1 or 0) indicating whether the relation doesor does not hold.

Situations are first-class citizens of the the-

ory and are defined intensionally. A situationis considered a structured part of the realitythat an agent manages to pick out or individ-uate. Situations and infons are related by thesupports relation:

ssupports α(denoted s2α) means thatαis an infon that is true of s.

For example, a situation sin which Bobhugs Carol would be described by s2͗͗hugs,Bob, Carol, l, t, 1͘͘,where l and ttogether givethe spatiotemporal coordinates of this hug-ging action.

Abstract situations are the constructs thatare more amenable to mathematical manipu-lation. An abstract situationis defined as a(possibly not-well-founded [Barwise andEtchemendy 1987] set of infons. Given a realsituation s, the set {α| s 2α} is the corre-sponding abstract situation.

One of the important ideas behind situa-tion theory is the scheme ofindividuation, away of carving the world into uniformities.As constructs that link the scheme of individ-uation to the technical framework of the the-ory, types are important features of situationtheory. Just as individuals, temporal loca-tions, spatial locations, relations, and situa-tions, types are also uniformities that are dis-criminated by agents. Relations can havetheir argument places filled either with indi-viduals, situations, locations, and other rela-tions or with types of individuals, situations,locations, and relations. Some basic types areTIM(the type of a temporal location), LOC(the type of a spatial location), IND(the typeof an individual), and SIT(the type of a situa-tion).

In situation theory, for each type T,aninfinite collection of basic parameters T1, T2,… is introduced. For example IND3parameter. We use the notations иl, иis anIND

t

, aи, sи, andso on, to denote parameters of type LOC,TIM, IND, SIT, and so on, respectively. Some-times, rather than parameters ranging over allindividuals, we need parameters that rangeover a more restricted class, namely, restrictedparameters, for example,иr

1 = aии, bи, 1͘͘ .a

и↑͗͗kicking, a

b

и= IND3 ↑͗͗man, IND3, 1͘͘ .= IND2 ↑͗͗football,IND2, 1͘͘.In this case, иr1 ranges over all men kickingfootballs.

We use the term parametric infonto empha-size that in a particular infon, one or moreparameters occurs free. Infons that have nofree parameters are called parameter free.Relat-ed to parametric infons, an anchor is a con-

struct by which we can assign values to pa-rameters. Formally, an anchor for a set Aofbasic parameters is a function defined on Athat assigns to each parameter Tiin Aan ob-ject of type T. Therefore, if fis an anchor forA,and Tiis a parameter in A,then

͗͗of-type, f(Ti), T, 1͘͘.

For example, if fanchors aиto the type IND3

individual Sullivan, we write

f(aи) = Sullivanto denote this anchoring.

Let sbe a given situation. If x

иis a parame-ter, and Iis a set of infons (involving x

и), thenthere is a type

[xи| s2I] .This is the type of all those objects to which xиcan be anchored in s, such that the conditionsimposed by Iobtain. We refer to this process

of obtaining a type from a parameter x

и, a situ-ation x

иs, and a set Iof infons as type abstraction.is known as the abstraction parameter,and sis known as the grounding situation.

In situation theory, the flow of informationis realized through constraints, represented asS0˜S1 .

Here, S0and S1are situation types. Cognitive-ly, if this relation holds, then it is a fact thatif S0is realized (that is, there is a real situations0: S0), then so is S1(that is, there is a real sit-uation s1: S˜S1). For example, with the con-straint Ssf, we might represent the regular-ity “smoke means fire,” provided that wehave

Ss

= [sи| sи 2͗͗smoke-present, иl, иt, 1͘͘] .Sf= [sи| sи2͗͗fire-present,иl, иt

, 1͘͘] .This constraint is read as “Ssinvolves Sf” andrepresents a fact (that is, a factual, parameter-free infon):

͗͗involves, Ss, Sf, 1͘͘ .

Attunement to this constraint is what enablesan intelligent agent that sees smoke in a situ-ation to realize that there is a fire.

Barwise on Contexts

Barwise’s ideas on circumstance, thus on con-text, are best articulated in his work on con-ditionals and circumstantial information(Barwise 1986). Situations represent a way ofmodeling contexts. In fact, in Barwise(1987b), the author expounds why a contextis a situation. Briefly, he proposes that adefinite relationship exists between situationsand what are known as context sequencesinpossible world semantics (Akman and Tin1990). In possible world semantics, given asentence s, the person pwho uttered the sen-

tence, the spatiotemporal location of the ut-terance l, and t, the object o thatpis referring

to, and so on, are all lumped together into asequence c= . Basically, crepre-sents various contextual elements that play arole in obtaining the propositional content ofany particular use of s. Barwise claims that cis nothing more than a representation of asituation, the portion of the world that is es-sentially needed (relevant) to determine thecontent of the utterance of s. Thus, he claimsthat by admitting situations, one no longerneeds ad hoc devices such as c.

The Missing Pollen Let us consider Claire(Barwise’s then nine-month-old daughter).Barwise knows that if Claire rubs her eyes,then she is sleepy, expressed by the condi-tional statement, “If Claire rubs her eyes,then she is sleepy.”

For months, this was a sound piece of (con-ditional) knowledge that Barwise and his wifeused to understand Claire and learn whenthey should put her to bed. However, in earlysummer, this knowledge began to fail them.Combined with other symptoms, Barwise andhis wife eventually figured out that Claire wasallergic to something. They called it pollen Xbecause they did not know its precise identi-ty; so, pollen X could also cause Claire to rubher eyes.

Barwise formalizes the problem stated inthis example as follows: Briefly, with con-straint C= [S˜S′], a real situation scontainsinformation relative to such an actual con-straint Cif s: S. Clearly, scan contain variouspieces of information relative to C,but themost general proposition that s contains, rel-ative to C,is that s′is realized, where s′: S′.Thus, we can represent this conditional in-formation with the following parametric con-straint C:

S= [sи| sи2͗͗rubs, Claire, eyes,иl, иt, 1͘͘].S′= [sи| sи2͗͗sleepy, Claire,иl, иt, 1͘͘] .C= [S˜S′] .

Before pollen X was present, this constraint

represented a reasonable account. However,when pollen X arrived, the constraint becameinadequate and required revision. Barwisepoints out two alternatives to deal with theproblem:

First, from [if φ, then ϕ], infer [if φand β,then ϕ].

Second, from [if φ, then ϕ], infer [if β, thenif φ, then ϕ].

Here, βcorresponds to the additional back-ground conditions.

Barwise chooses the second way, modifiesinvolves, and makes the background assump-

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In

situation theory, we

represent implicationswith

constraints.

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Logic vs. Situation Theory (S.T.)Modal TreatmentNatural DeductionParadox FreeCircularityMc931LogicNoYesNoNoGu912LogicNoYesNoNoSh913LogicYesNoYesNoGi934LogicNoYesYesNoBM935LogicYesYesYesNoAS956LogicNoYesYesYesBa867S.T.NoNo?YesTable 1. Comparison of Approaches toward Formalizing Context.

Notes

1. Mc93 = Notes on formalizing context (McCarthy 1993).

2. Gu91 = Contexts: A formalization and some applications (Guha 1991).3. Sh91 = Varieties of context (Shoham 1991).4. Gi93 = Contextual reasoning (Giunchiglia 1993).

5. BM93 = Propositional logic of context (Buvac˘ and Mason 1993).6. AS95 = A formalization of viewpoints (Attardi and Simi 1993).7. Ba86 = Conditionals and conditional information (Barwise 1986).

tions explicit by introducing a third parame-ter Β:23

S0˜S1| B.

With the new involves, the missing pollen ex-ample can be solved with the introduction of

a B, which supports the following:

и, 0͘͘.͗͗exists, pollen X,иl, tи| sи2͗͗rubs, Claire, eyes,ии, 1͘͘] .S= [sl, t

и| sи2͗͗sleepy, Claire,ии, 1͘͘] .S′= [sl, tи| sи2͗͗exists, pollen X,ии, 0͘͘] .B= [sl, tC= [S˜S′| B] .

In the beginning, it was winter, and there

was no pollen. The context, call it c1, must bea situation type that supports

и, 0͘͘c2͗͗exists, pollen X,иl, t

1

In cA, we have the following infons sup-ported:

͗͗corresponds,Iи, A, 1͗͗͘͘philosopher,Iи, 1͘͘,where corresponds is a function that associatesan indexical to a person, and utterancesabout being a philosopher are represented

и, 1͘͘.with infons of type ͗͗philosopher,xи, A, 1͗͗͘͘corresponds,She

и, 1͘͘. ͗͗philosopher,She

и, A,1͗͗͘͘corresponds,You

и, 1͘͘. ͗͗philosopher,YoucCsupportscBsupports

(and possibly other things related to Claire,

rubbing one’s eyes, and so on). Using contextc1as the grounding situation, we do not vio-late the background condition Bof constraintCand, thus, can conclude that “Claire issleepy.”

Later, in summer, the new context, c2, sup-ports the infon

и, 1͘͘,c2͗͗exists, pollen X,иl, t

2

Now, it is a trivial matter to observe that Iи,

и, and Sheиall collapse to Abecause the an-You

choringf(Iи) = Aи) = Af(You

и) = Af(She

does the job. Consequently, the utterance ofAmight be decontextualized as ͗͗philosopher,A, 1͘͘.

The Obligatory Tweety Example As westated before, in situation theory, we repre-sent implications with constraints. Whilestating the constraints, we can use back-ground conditions to add nonmonotonicity:

ؒ| sؒ2͗͗bird,xи, 1͘͘] .S= [sؒ| sؒ2͗͗flies, xؒ, 1͘͘] .S1= [sؒ| sؒ2͗͗penguin,xؒ, 0ٙ͘͘sؒ2B= [s

͗͗present, Air, 1͘͘] .C= [S0˜S1| B] .

The constraint Cstates that every bird flies

0

and when we use c2as the grounding situa-tion, we are faced with an inconsistency be-tween Band c2. Therefore, Cbecomes void in

the new context of the talk, and the conclu-sion “Claire is sleepy” cannot be reached.I Am a Philosopher We will prove that thecontent of all three sentences (given earlier) isthe same; that is, Ais a philosopher.

We have three contexts associated witheach individual in the conversation: cA, cB,andcC,respectively. We represent the indexi-и, and Sheи.cals with special parameters Iи, You

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unless it is a penguin, or there is no air. Here,the important contribution of the situation-theoretic account is that the environmentalfactors can easily be included in the reason-ing phase by suitably varying B.

Conclusion

The comparison of the previous approaches issummarized in table 1, where the first rowmarks the language of formalization. Not allthe pertinent works (by an author or a groupof authors) are listed; instead, an exemplarypublication was selected as a starting point.Except for Barwise, all the previous approach-es are stated in a more or less logicist frame-work. Among these, Shoham, and S. Buvac˘and Mason propose context as a modal opera-tor; S. Buvac˘ and Mason further consider con-textual reasoning in a natural deductionsense and allow operations of entering andexiting contexts. F. Giunchiglia and cowork-ers propose their multilanguage systems as atrue alternative to modal logic.

Among the previous approaches, Mc-Carthy’s and Guha’s are not paradox free,whereas S. Buvac˘ and Mason’s and Attardiand Simi’s approaches are paradox free.Shoham’s approach must be paradox free: Itis basically propositional modal logic. We donot know whether Barwise’s (hence our) ap-proach is paradox free or not. However, in athought-provoking work (Barwise andEtchemendy 1987), situation theory is shownto be powerful enough to deal with circulari-ty. The last row of the table reflects this pow-er. (It must be noted that an important por-tion of Attardi and Simi’s work also focuseson evading paradoxes.)

Clearly, other tables can be made to high-light the motivations, expressiveness, andcomplexity of particular approaches.24How-ever, in the end, applicability is the sole ba-sis of ratification for any approach. Al-though the idea of formalizing contextseems to have caught on and produced goodtheoretical outcomes, the area of innovativeapplications remains relatively unex-plored—evidently where further research oncontext should converge.

Acknowledgments

A preliminary version of this article was sub-mitted to AI Magazinein October 1994. Itthen underwent a revision, and we are grate-ful to the editor and an anonymous refereefor thoughtful remarks and suggestions. Forthe penultimate version of the article, webenefited from the very detailed and con-

structive criticism of two anonymous refereesand the editor emeritus, and once again, weare thankful. As usual, responsibility for thecontents remains with us.

Notes

1. In an early influential paper, Clark and Carlson(1981) state that context has become a favorite wordin the vocabulary of cognitive psychologists andthat it has appeared in the titles of a vast numberof articles. They then complain that the denotationof the word has become murkier as its uses havebeen extended in many directions and deliver thenow-widespread opinion that context has becomesome sort of a “conceptual garbage can.”

2. The two workshop proceedings in which our pa-pers appear can also be an excellent starting pointto get a taste of current research in contextual rea-soning.

3. The Oxford English Dictionaryalso gives several oth-er meanings, most of which are not currently used. 4. This quest for a more general meaning is in thespirit of the following observation of McCarthy(19, p. 180): “Almost all previous discussion ofcontext has been in connection with natural lan-guage, and the present paper relies heavily on ex-amples from natural language. However, I believethe main AI uses of formalized context will not bein connection with communication but in connec-tion with reasoning about the effects of actions di-rected to achieving goals. It’s just that natural lan-guage examples come to mind more readily.”5. Linguists have talked about context for over acentury and have many interesting results. Much,much more would need to be said about context inlinguistics but because we cannot hope to do jus-tice to such a great body of work in this review, weare content with a brief, superficial appraisal.6. According to the Merriam-Webster Dictionary(Merriam-Webster 1974), the word philosopher canstand for any of the following: a reflective thinker(scholar); a student of, or specialist in, philosophy;one whose philosophical perspective enables him/her to meet trouble calmly. Most probably, it is thesecond meaning the reader typically thinks of.7. Establishing coherence is an integral part of theinterpretation task. Allen (1995, p. 465) explains itnicely: “A discourse is coherent if you can easily de-termine how the sentences in the discourse are re-lated to each other. A discourse consisting of unre-lated sentences would be very unnatural. Tounderstand a discourse, you must identify howeach sentence relates to the others and to the dis-course as a whole. It is this assumption of coher-ence that drives the interpretation process.”8. Cf. Wittgenstein (1984, p. 166): “‘He measuredhim with a hostile glance and said …’. The readerof the narrative understands this; he has no doubtin his mind. Now you say: ‘Very well, he suppliesthe meaning, he guesses it.’—Generally speaking:no. Generally speaking he supplies nothing, guess-es nothing.—But it is also possible that the hostileglance and the words later prove to have been pre-

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tense, or that the reader is kept in doubt whetherthey are so or not, and so that he really does guessat a possible interpretation.—But then the mainthing he guesses at is a context. He says to himselffor example: The two men who are here so hostileto one another are in reality friends, etc. etc.”9. In McCarthy’s newer work (1993), holds(p,c) is re-named ist(c, p).

10. This preference for holds is understandable be-cause “[f]ormalizing commonsense reasoning needscontexts as objects, in order to match human abili-ty to consider context explicitly” (McCarthy 19,p. 180). Furthermore, adding a context parameterto, say, each predicate would be unnatural. Consid-er the following example about the predicate at,adapted from McCarthy and Buvac˘ (1994). In onecontext, atcan be taken in the sense of being regu-larly at a place; in another, it can mean physicalpresence at a certain instant. Programs based onformalized contexts would extract the appropriatemeaning automatically. Otherwise, a need arises touse two different predicates, for example, at.

1andat211. In identifying the given computational advan-tages of the notion, we heavily depended onShoham (1991, pp. 395–396).

12. A good example (Shoham 1991) is the use of theword glasses: The appropriate meaning would bedifferent in the context of a wine-and-cheese partyand in the context of a visit to an ophthalmologist.13. Contexts were not mentioned in McCarthy’sTuring award talk in 1971.

14. A newer and expanded treatment is available inMcCarthy and Buvac˘ (1994). The most recent refer-ence is McCarthy (1995), although this report pre-sents only a brief introduction.

15. The word lifting,used frequently in this article,is discussed later.

16. A more concrete use (from computer science)can be identified if we form an analogy with pro-gramming language and database concepts. Mc-Carthy’s approach corresponds to the use of localvariables and local functions in programming lan-guages and views in database systems. In each case,the meaning of a term depends on the context inwhich it is used.

17. A rich object cannot be defined or completelydescribed. A system can be given facts about a richobject but never the complete description.

18. The formal system presented earlier implicitlyassumes flatness (every context looks the same re-gardless of which context it is being studied from)and would have to be modified to deal withnonflatness, which is done by replacing the rulesEnter and Exit with a single rule C(Buvac˘, Buvac˘,and Mason 1995).

19. The reader is referred to Attardi and Simi (1991)for an early sketch of the theory of viewpoints.This theory is basically an extension of first-orderlogic (FOL), adjoined by a carefully formulatedreflective theory where mixed object-level and meta-level statements are allowed.

20. Shoham (1991) notes that “a general treat-ment of contexts may indeed wish to exempt con-texts from the obligation to interpret every asser-tion” (p. 400).

21. Shoham gives a particularly nice account of thephrase more general than (Shoham 1991, p. 398):“[W]hether one of the two contexts human beingsin general and human beings in conditions whereinfluenza viruses are present is more general than theother depends on what we mean by each one. Ifthe first context includes some information aboutpeople and the second context includes that infor-mation plus further information about viruses,then the former is more general. If the first includesall information about people and the second somesubset of it that has to do with viruses, then the lat-ter is more general. Otherwise, the two are non-comparable.”

22. The phrase not comparable means that each con-text contains some axioms that are not containedin the other context; cf. the preceding note.23. Although these alternatives are equivalent froma logical point of view, the second is more appro-priate to reflect the intuitions behind the back-ground conditions. In the first case, the rule “if φ,then ϕ“ is directly modified to use background con-ditions, whereas in the second case, it is nottouched but is evaluated only when the back-ground conditions hold. The introduction of back-ground conditions for rules corresponds to a non-monotonic reasoning mechanism.

24. We refer the reader to Massacci (1996) for somecomplexity results regarding contextual reasoning.

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Varol Akmanis a full professorof computer engineering atBilkent University, Ankara,Turkey. His current research areasinclude, in addition to formalcontexts, computational aspectsof situation theory and situationsemantics and, in general, AI andphilosophy. From 1980 to 1985,

Akman was a Fulbright scholar at Rensselaer Poly-technic Institute, Troy, New York, where he re-ceived a Ph.D. in computer engineering. Prior tojoining Bilkent in 1988, he held a senior researcherposition with the Centrum voor Wiskunde en In-formatica, Amsterdam, The Netherlands.

Mehmet Surav received his B.S.from the Department of Comput-er Engineering and InformationScience, Bilkent University,Ankara, Turkey, in 1992. He re-ceived his M.S. in computer engi-neering and information sciencefrom the same institution in1994. He is currently working for

NewNet Inc., Shelton, Connecticut, as a softwareengineer. His areas of interest include telecommu-nication network protocols, object-oriented systemdesign, and representation of context.