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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.

  1. 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.

  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 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.

  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.

  8. 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.

  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 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 a

    context 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.

  10. 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.

  11. 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 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.

  12. 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.

  13. 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.

  14. 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 modal

    operator 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 it

    will 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 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 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) rather

    than 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.

    http://www-

    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.