J McCarthy-a logical AI approach to context

A LOGICAL AI APPROACH TO CONTEXT

John McCarthy

Computer Science Department

Stanford University

Stanford,CA94305

jmc@e1d6670d763231126edb1118

e1d6670d763231126edb1118/jmc/

1996Feb6,12:09p.m.

Abstract

Logical AI develops computer programs that represent what they know about the world primarily by logical formulas and decide what

to do primarily by logical reasoning—including nonmonotonic logical

reasoning.It is convenient to use logical sentences and terms whose

meaning depends on context.The reasons for this are similar to what

causes human language to use context dependent meanings.This note

gives elements of some of the formalisms to which we have been led.

Fuller treatments are in[McC93],[Guh91]and[MB94]and the refer-

ences cited in the Web page[Buv95].The?rst main idea is to make

contexts?rst class objects in the logic and use the formula ist(c,p)

to assert that the proposition p is true in the context c.A second

idea is to formalize how propositions true in one context transform

when they are moved to di?erent but related contexts.An ability to

transcend the outermost context is needed to give computer programs

the ability to reason about the totality of all they have thought about

so far[McC96].

1Introduction

As requested by Johan van Benthem,this is a brief introduction to the logical formalism for context being explored by John McCarthy and Saˇs a Buvaˇc at

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Stanford University.It is motivated by the need to use contexts as?rst order objects for arti?cial intelligence.I hope the description is suitable for com-parison with other approaches to context that often have other motivations. 2Features of the Formalism

Here are some features of our formalizations.

1.We o?er no de?nition of context.There are mathematical context

structures of di?erent properties,some of which are useful.Asking what a context is is like asking what a group element is.See section4 for more on this.

2.Sentences about propositions and contexts are built up from a formula

ist(c,p)which is to be understood as asserting that the proposition p is true in the context c.When we have entered the context c,we can write

c:p.(1)

3.Once a program has inferred a sentence q from p,it can leave the

context c and have ist(c,q).This generalizes natural deduction.

4.Reasoning and communicating in context permits taking only limited

phenomena into account.Treating contexts as objects permits stating the limitations explicitly within the formalism.

5.Statements about contexts are themselves in contexts.

6.There is no universal context.This is a fact of epistemology(both of

the physical world and the mathematical world).It is always possible to generalize the concepts one has used up to the present.Attempts at ultimate de?nitions always fail—and usually in uninteresting ways.

Humans and machines must start at middle levels of the conceptual world and both specialize and generalize.

7.We can deal with this phenomenon in our formalism by ensuring that

it is always possible to transcend the outermost context used so far.

Thus a robot designed in this way is not stuck with the concepts it has been given.

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8.Because of the possibility of transcendence,the use of contexts as ob-

jects is not just a matter of e?ciency.Any given set of sentences including contexts can always be?attened(at the cost of lengthening) to eliminate explicit contexts.However,the resulting?at theory can no longer be transcended within the formalism,because it is not an object that can be referred to as a whole.

9.There is often a theory associated with a context—the set of sentences

true in the context.However,two contexts with the same theory need not be the same,because they may have di?erent relations with other contexts.Not all useful contexts will be closed under logical inference.

10.We advocate using propositions as discussed in[McC79]for the objects

true in contexts rather than logical or natural language sentences.This has the advantage that the set of propositions true in a context may be ?nite when the set of sentences that can express these propositions will be in?nite.However,our present applications of context would work equally well if sentences were used.Buvaˇc and Mason[BBM95]treat ist(c,p)as a modal logic formula in a propositional theory.

11.Besides the truth of propositions in contexts,we consider the value

value(c,exp)of a term exp representing an inpidual concept in a context c as discussed in[McC79].This presents problems beyond those presented by propositions,because in general the space of values of inpidual concepts will depend on some outer context.

3Applications

Here are some applications of the logical theory of contexts.

1.Conventional linguistic applications like the referents of pronouns can

be treated using contexts as objects,but formalized contexts are also useful for more complex anaphora.For example,we need to relate the surgeon’s“Scalpel”to the sentence“Please hand me a number3 scalpel”.See[Buv96].These applications require associating contexts with sentences or parts of sentences.

2.De?ning a theory in a narrow context in a way that permits it to be

lifted to a richer outer context and applied.[McC93]discusses lifting a

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simple theory of above(x,y)as the transitive closure of on(x,y)to an outer situation calculus context that uses on(x,y,s)and above(x,y,s).

A key formula of that paper is

c:(?xys)(on(x,y,s)≡ist(context-of-situation(s),on(x,y))),

(2)

which relates the three argument situation calculus predicate on(x,y,s) and the two element predicate on(x,y)of the specialized theory of on and above.The use of contexts to implement“microtheories”in Cyc is described in[Guh91].This allowed people entering knowledge about some phenomenon,e.g.automobiles,to do it in a limited context,but leave open the ability to use the knowledge in a larger context.

3.De?ning a narrow context for a problem and importing facts that per-

mit the problem to be solved by considering only a small set of pos-sibilities.For example,in formulating the missionaries and cannibals problem a person or program must take a number of common sense facts into account,but ends up with a32state space,because all that is relevant in this context is the numbers of missionaries,cannibals and boats on each bank of the river.

4.Relating databases with di?erent conventions[MB94].Imagine that

the Airforce and the General Electric Company have databases both of which include prices for the jet engines that the company sells the Airforce.However,suppose the databases don’t agree on what the price covers,e.g.spare parts.We use one context c AF for the Air Force database,another c GE for the GE database,and a third context c0that needs to relate information from both.Lifting formulas in the context true in c0relate information in the di?erent databases to the context in which reasoning is done,,e.g.they tell about the relation of the prices listed in c AF and c GE to the inclusion or not of spare parts. 5.Buvaˇc and McCarthy have also discussed using context to combine

aspects of plans generated by di?erent planners not originally designed to work together—or plans originally intended to work together but which have drifted apart in the course of separate development.

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4Desiderata for a Mathematical Logic of Con-text

The simplest approach to a logic of context is to treat ist(c,p)as a modal operator with p quanti?er free.Saˇs a Buvaˇc and Ian Mason[BBM95]did this.However,the applications to natural language,to databases and to formalizing common sense knowledge and reasoning require a lot more.Here are some desiderata for a formal theory.1

?truths(c)is the set of p such that ist(c,p).In some formalizations it will be a?rst class object.In any case we can think about it in the metatheory.

?The simplest possibility for truths(c)for a particular context c is that it is an arbitrary set of propositions,i.e.not required to be closed under some logical operations.

?The second possibility is that truths(c)is closed under deduction in some logical system—perhaps the theory of contexts.

?truths(c)may be the set of propositions true about some subject mat-ter.We can assert propositions about this set of proposition without knowing what sentences are in it.

?Associated with at least some contexts is a domain domain(c).As with truths(c),domain(c)may be an object,presumably in a higher level context,or it may be only in the metalanguage.

The variety of potential applications of contexts as objects suggests look-ing at contexts as mathematics looks at group elements.Groups were?rst identi?ed as sets of transformations closed under certain operations.How-ever,it was noticed that the integers with addition as an operation,the non-zero rationals with multiplication as an operation and many others had the same algebraic property.This motivated the de?nition of abstract group around the turn of the century.In such a theory,formulas express relations among contexts would be primary rather than the propositions true in the contexts.Thus the theory would emphasize specializes(c1,c2,time)rather than ist(c,p).

1Just so Johan doesn’t get o?too easily in keeping his promise to make one.

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5Remarks

Johan van Benthem asked for the following in soliciting this essay and John Perry’s.

My proposal is the following.I would like to invite the two Johns to send me a rough outline of their contribution.It would

be good if you could bring out(1)what the notion of context is

and what it does according to you:in both cases,I think you

want it to achieve’e?ciency’and’portability’of information,

(2)what is involved in the dynamics of changing contexts,

perhaps with attendant changes in linguistic formulation(add or

drop variables,etcetera).I would then like to comment on this,

adding some thoughts on possible logical formalizations,empha-

sizing the interplay between what is said in a formula and what

remains implicit in the models where it gets evaluated.

I have rejected the idea of de?ning what a context is,but I hope I have

given some idea of what they do.The example relating the three argument

on and the two argument on should provide a basis for comments.In the formulation of the ideas,the ability to combine formulas arising in di?erent contexts has been more important than computational e?ciency.

[McC93]and[MB94]have additional references.Also Saˇs a Buvaˇc has sev-

eral other papers on context on his Web page e1d6670d763231126edb1118/buvac/. References

[BBM95]Saˇs a Buvaˇc,Vanja Buvaˇc,and Ian A.Mason.Metamathematics of contexts.Fundamenta Informaticae,23(3),1995.

[Buv95]Saˇs a Buvaˇc.Saˇs a buvaˇc’s web page,1995.e1d6670d763231126edb1118/buvac/.

[Buv96]Saˇs a Buvaˇc.Resolving lexical ambiguity using a formal theory of context.In Semantic Ambiguity and Underspeci?cation.CSLI

Lecture Notes,Center for Studies in Language and Information,

Stanford,CA,1996.

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[Guh91]R.V.Guha.Contexts:A Formalization and Some Applications.

PhD thesis,Stanford University,1991.Also published as techni-

cal report STAN-CS-91-1399-Thesis,and MCC Technical Report

Number ACT-CYC-423-91.

[MB94]John McCarthy and Saˇs a Buvaˇc.Formalizing Context(Expanded Notes).Technical Note STAN-CS-TN-94-13,Stanford University,

1994.

[McC79]John McCarthy.First order theories of inpidual concepts and propositions.In Donald Michie,editor,Machine Intelligence,vol-

ume9.Edinburgh University Press,Edinburgh,1979.Reprinted

in[McC90].

[McC90]John McCarthy.Formalizing Common Sense:Papers by John Mc-Carthy.Ablex Publishing Corporation,355Chestnut Street,Nor-

wood,NJ07648,1990.

[McC93]John McCarthy.Notes on formalizing context.In IJCAI-93,1993.

Available on e1d6670d763231126edb1118/jmc/.

[McC96]John McCarthy.Making robots conscious of their mental states.

In Stephen Muggleton,editor,Machine Intelligence15.Oxford

University Press,1996.to appear,available on http://www-

e1d6670d763231126edb1118/jmc/.

/@e1d6670d763231126edb1118:/u/jmc/f95/context.tex:begun1995Sep22,latexed1996Feb6at12:09p.m.

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