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A LOGICAL AI APPROACH TO CONTEXT

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

jmc@cs.stanford.edu

http://www-formal.stanford.edu/jmc/

1996 Feb 6, 12:09 p.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 ﬁrst main idea is to make

contexts ﬁrst 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 diﬀerent 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].

Introduction

As requested by Johan van Benthem, this is a brief introduction to the logical

formalism for context being explored by John McCarthy and Saˇsa Buvaˇc at

Stanford University. It is motivated by the need to use contexts as ﬁrst order

objects for artiﬁcial intelligence. I hope the description is suitable for com-

parison with other approaches to context that often have other motivations.

2 Features of the Formalism

Here are some features of our formalizations.

- We oﬀer no deﬁnition of context. There are mathematical context
structures of diﬀerent properties, some of which are useful. Asking
what a context is is like asking what a group element is. See section 4

for more on this.

- Sentences about propositions and contexts are built up from a formula
`ist(c, p)`

which is to be understood as asserting that the proposition pis true in the context c. When we have entered the context c, we can

write

c :

p.

(1)

- 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. - Reasoning and communicating in context permits taking only limited
phenomena into account. Treating contexts as objects permits stating
the limitations explicitly within the formalism.

- Statements about contexts are themselves in contexts.
- 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 deﬁnitions always fail—and usually in uninteresting ways.

Humans and machines must start at middle levels of the conceptual

world and both specialize and generalize.

- 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.

- Because of the possibility of transcendence, the use of contexts as ob-
jects is not just a matter of eﬃciency. Any given set of sentences
including contexts can always be ﬂattened (at the cost of lengthening)

to eliminate explicit contexts. However, the resulting ﬂat theory can

no longer be transcended within the formalism, because it is not an

object that can be referred to as a whole.

- 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 diﬀerent 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

ﬁnite when the set of sentences that can express these propositions will

be inﬁnite. 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 individual concept in acontext c as discussed in [McC79]. This presents problems beyond

those presented by propositions, because in general the space of values

of individual concepts will depend on some outer context.

3 Applications

Here are some applications of the logical theory of contexts.

- 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 number 3

scalpel”. See [Buv96]. These applications require associating contexts

with sentences or parts of sentences.

- Deﬁning 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
simple theory of

`above(x, y)`

as the transitive closure of`on(x, y)`

to anouter 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 onand 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.

- Deﬁning 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 a 32 state space, because all that

is relevant in this context is the numbers of missionaries, cannibals and

boats on each bank of the river.

- Relating databases with diﬀerent 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 cAF for the Air Force

database, another cGE for the GE database, and a third context c0 that

needs to relate information from both. Lifting formulas in the context

true in c0 relate information in the diﬀerent databases to the context

in which reasoning is done, , e.g. they tell about the relation of the

prices listed in cAF and cGE to the inclusion or not of spare parts.

- Buvaˇc and McCarthy have also discussed using context to combine
aspects of plans generated by diﬀerent 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.

4 Desiderata for a Mathematical Logic of Con-text

The simplest approach to a logic of context is to treat

`ist(c, p)`

as a modaloperator with p quantiﬁer free. Saˇsa 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 itwill be a ﬁrst 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 thatit 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 insome 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 levelcontext, 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 ﬁrst

identiﬁed 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 deﬁnition 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)`

ratherthan

`ist(c, p)`

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

5 Remarks

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 ’eﬃciency’ 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 deﬁning 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 diﬀerent

contexts has been more important than computational eﬃciency.

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

eral other papers on context on his Web page http://www-formal.stanford.edu/buvac/.References

[BBM95] Saˇsa Buvaˇc, Vanja Buvaˇc, and Ian A. Mason. Metamathematics

of contexts. Fundamenta Informaticae, 23(3), 1995.

[Buv95] Saˇsa Buvaˇc.

Saˇsa buvaˇc’s web page, 1995.

formal.stanford.edu/buvac/.

[Buv96] Saˇsa Buvaˇc. Resolving lexical ambiguity using a formal theory

of context. In Semantic Ambiguity and Underspeciﬁcation. CSLI

Lecture Notes, Center for Studies in Language and Information,

Stanford, CA, 1996.

[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 Saˇsa Buvaˇc. Formalizing Context (Expanded

Notes). Technical Note STAN-CS-TN-94-13, Stanford University,

1994.

[McC79] John McCarthy. First order theories of individual concepts and

propositions. In Donald Michie, editor, Machine Intelligence, vol-

ume 9. Edinburgh University Press, Edinburgh, 1979. Reprinted

in [McC90].

[McC90] John McCarthy. Formalizing Common Sense: Papers by John Mc-

Carthy. Ablex Publishing Corporation, 355 Chestnut Street, Nor-

wood, NJ 07648, 1990.

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

Available on http://www-formal.stanford.edu/jmc/.

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

In Stephen Muggleton, editor, Machine Intelligence 15. Oxford

University Press, 1996.

to appear, available on http://www-

formal.stanford.edu/jmc/.

/@steam.stanford.edu:/u/jmc/f95/context.tex: begun 1995 Sep 22, latexed 1996 Feb 6 at 12:09 p.m.