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C00002 00002	%contex[e86,jmc]	Formalization of contexts
C00003 00003
C00004 00004	\noindent {\bf Introduction.}
C00013 00005	\noindent{\bf Remarks}
C00017 00006	\noindent The first version of contex.tex[e86,jmc] is dated 1986 July 5.
C00018 00007	Notes:
C00023 ENDMK
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%contex[e86,jmc]	Formalization of contexts
%
%CONTEX[S76,JMC] 24-Jul-76	CONTEXT DEPENDENT SEMANTICS
%ideas[e84,jmc]	
%contex[f84,jmc]		Ideas about context
%contex[s85,jmc]		Notes on contexts
%The best: contex[w86,jmc]		Contexts and mental situations
%notes[s86,jmc]		Amarel model and contexts
%genera[w86,jmc]		Generality in AI, for old Turing lecture
\input memo.tex[let,jmc]
\centerline{draft draft draft}
\vskip .5in
\centerline{\bf NOTES ON FORMALIZED CONTEXTS}
\vskip .1in
\centerline{John McCarthy, Stanford University}
\vskip .3in
%

\noindent {\bf Abstract:} Getting a general database of
common sense knowledge and expressing it in logic requires
formalizing the notion of context.  Since no context is absolutely
general, any context must be {\it elaboration tolerant} and we
discuss this notion.  Another formalism that seems useful involves
{\it entering} and {\it leaving} contexts; this is a generalization
of natural deduction.

\vskip .5in
%
\noindent {\bf Introduction.}

	These notes are part of the ``logicist'' approach to AI, which
involves expressing what an intelligent program knows as sentences
in logic and having it decide what to do by logical inference.  It
has already turned out that the required logical inference cannot merely be
deduction.  Deduction must be supplemented by nonmonotonic reasoning.

	A major goal of the logicist approach is to create a database
of common sense knowledge expressed as logical sentences.  Such a database
should be usable by any program needing the knowledge.  Therefore, the
knowledge should be as general as human knowledge of the phenomenon
in question.  No-one has succeeded in expressing common sense knowledge
in a really general way.  There are always so many special assumptions
that human common sense is required to keep track of the domain of
applicability of the knowledge, which usually turns out to be too
narrow as soon as the problem is much different from those previously
studied.

	This is a fundamental difficulty, and some philosophers and even
AI researchers seem to regard it as a reason for despair.  A common
reaction among AI researchers is to advocate some other formalism
than logic whose lack of generality is disguised by its unfamiliarity.
We propose trying to solve the problem by formalizing the notion
of context within mathematical logic.

	In the following discussion we will discuss logical counterparts
of certain English sentences.  However, our goal is artificial intelligence
and not linguistics.  The English sentences are there to inform the
reader and are used in ways that we hope are obvious in the (informal)
context of this paper.  We are not trying to represent the meaning
of the English sentence in general.  The meaning of the logical
predicates, functions and constants will ultimately be characterized by
the axioms to which we subject them.

%	Now it further seems that we need to formalize the notion of
%context and relations among contexts and between sentences and contexts.
%This should have been expected, because every formal mathematical
%theory applied to the real world is preceded in its exposition by
%natural language prose describing what is assumed and describing the
%connections between the terms of the theory and the real world.
%These natural language explanations, while they describe what
%is assumed by the theory, in turn depend on assumptions and
%terminology common to the author and the reader.  If computer
%programs are to behave intelligently, they will also require
%some way of expressing the contexts.  Some ideas for this are
%described in these notes.

	Consider the sentence ``The cat is on the mat''.  When
expressed in logic it is often formalized $On(Cat,Mat)$.  Here
we're using the convention that capitalized words denote constants.
  However,
when we need to consider change and use the situation calculus,
we are inclined to use $On(Cat,Mat,s)$, so that we can also consider
the truth-value of (say) $On(Cat,Mat,Result(e,s))$, where $Result(e,s)$
is the situation that results from the event $e$ occurring in situation
$s$.  Even this is not specific enough when it is not obvious which
cat and which mat.  Our goal is to have it both ways --- to be able
to ``enter'' a context and say $On(Cat,Mat)$, but also to be able
to consider it equivalent to
``John McCarthy's youngest cat was on
John McCarthy's front door mat on July 12, 1986 at 17:35
Pacific Daylight time'' in a less specified context.
Of course, even that assumes a context in which
a particular John McCarthy is identified.  We don't pursue the
most general context that assumes nothing, because we regard that
as unattainable.
%
\vskip .3in
\noindent {\bf Formalized contexts.}

	We begin by introducing some abstract objects called
contexts denoted by the letter $c$ with appropriate subscripts
and other decorations.  Denoting propositions by $p$, a basic
relation is
%
$$Holds(p,c),$$
%
asserting that the proposition $p$ holds in the context $c$.
%
\vskip .3in
\noindent {\bf What is a context?}

	Contexts are abstract entities with the following properties.

	1. A context involves a set of
conventions and assumptions.  The conventions peculiar to a given context may
be definitional, i.e. they may involve what certain terms mean.
The assumptions
may be temporal and/or spatial, i.e. they may prescribe that
the sentences refer to a particular time and place.

	2. The set of assumptions in a context are only partially
known to the user of the context.  Indeed there may be an infinite
set of assumptions.

	3. Contexts may be related by specialization.
$c1 ≤ c2$ means that $c1$ is a more specialized context than $c2$.
We have

$$c1 ≤ c2 ⊃ (∀p.Holds(p,c2) ⊃ Holds(p,c1)).$$

	4. Some contexts make assumptions about time and/or place.
Thus $Holds(on(A,B),c1)$ will be meaningful if $c1$ assumes a time.

	5. If $c0$ doesn't assume a time $t$ and  $c1$ is like $c0$
except that it does, we'll have $c1 ≤ c0$ and something like
$Holds(on(A,B),assumetime(t,c0))$.

	6. Contexts may depend on parameters.
\noindent{\bf Remarks}

	We return to ``The cat is on the mat'' represented
$On(Cat,Mat)$
Since this representation is context dependent, the
corresponding sentence is $Holds(On(Cat,Mat),C1)$, where $C1$
is the assumed context.  However, this formula presumes a particular
cat and mat are referred to --- determined by the context.
%
	In a more general context, we may use the formula

$$Holds(On(OwnedBy(John,Cat),At(At(Front(OwnedBy(John,House),Door),Mat)),C2),$$
%
which specifies that the cat is owned by John and the mat is at the
front door of the house owned by John.  Here we are using the convention
that the concept of John's cat is designated by a modifier (in this
case $OwnedBy(John,\-)$ applied to the concept denoted by $Cat$.  The
concept of the mat at the front door of John's house is denoted similarly.
We note that $C2$ is still not a very general context because it isn't
said which John is meant.  Moreover, there is a presumption that John
has only one cat and one front door.

	The two formulas are equivalent provided the contexts $C1$ and
$C2$ have the right relation to one another.  What is this relation?

\noindent THIS IS TO BE DISCUSSED.
\noindent ***

\noindent Meanings

	It seems useful to have more detailed equivalences than of
formulas.  It happens that an expression in one context has the
same value as another expression in different context.  Thus $Cat$
in $C1$ has the same value as $OwnedBy(John,Cat)$ in $C2$.  However,
the relation is actually stronger than that.  $Cat$ in $C1$ even has
the same meaning as $OwnedBy(John,Cat)$ in $C2$ and can therefore
be substituted in certain positions that are otherwise opaque.  Formalizing
this gives
%
$$Meaning(Cat,C1) = Meaning(OwnedBy(John,Cat),C2).$$
%
and even
%
$$Holds(Knows(Tom,On(Cat,Mat)),C1)≡Holds(Knows(Tom,
On(OwnedBy(John,Cat),At(At(Front(OwnedBy(John,House),Door),Mat))),C2).$$
\noindent The first version of contex.tex[e86,jmc] is dated 1986 July 5.

\noindent This version \TeX ed on \jmcdate\space at \theTime.
\vfill\eject\end
Notes:

	Predicates on contexts:

	$deftime(c)$ means that $c$ has a definite time.  Similarly
we have $defplace(c)$.  Other predicates besides these are required
to make meaningful certain sentences.  For example, what is required
of $c$ to make $Holds(On(Cat,Mat),c)$ meaningful?

	Do we have sentences that quantify over contexts?  Do these
hold in outer contexts?

	There may also be ``general'' contexts.  In such a general
context, $Holds(Chase(Dogs,Cats),c)$ can be meaningful.  We need
rules that go from such general contexts to specific contexts so
that we can conclude that a particular dog is likely to chase
a particular cat.  Are $Chase(Dogs,Cats)$ and $Fear(Cats,Dogs)$ the
correct notations for these assertions.

There is a general context that is appropriate for the beginning
of a book, i.e. no special information is assumed other than an
understanding of the language being used and common sense knowledge.
That is the context of the common sense database or at least of the
introduction to the common sense database.

Oct 16

Give a good example of the relation between the English language
general context (alternatively the common sense database context)
and the context of a specific conversation or of a specific instance
of problem solving.

In many human contexts, some time-dependent conditions are expressed
in a time-dependent way and others not.  I live in Stanford, California
and at 1:15pm my class starts.

At the beginning of a conversation, context is set via Gricean
implicatures.

Nov 12

Meaning(Scalpel,C2) = Meaning(Give(Scalpel17),C1),

but maybe it's

Meaning(Say(Scalpel),C2) = Meaning(Say(Give(Scalpel17)),C1)

or

Say-Meaning(Scalpel,C2) = Say-Meaning(Give(Scalpel17),C1).

Perhaps

C2 = <Conversants-Performing-Operation>(C1).

Specialization of contexts involves both assumptions and abbreviations.
Entering the context in which I am at home isn't exactly an assumption;
perhaps it needs to be called specialization, but then maybe they are
all specialization.

Generalization of contexts requires that theories have structure
beyond logical equivalence.

Nov 13

examples:

	world of Sherlock Holmes

	world of formalized missionaries and cannibals, i.e. imagine
an English discussion of m&c where the 16 state model is already
implicitly accepted.  Shooting a cannibal is now not a possible action.

Nov 21
expected(p,c) is weaker than  holds(p,c)

∃ generalization  c'  of  c  such that  ¬holds(p,c')

1987 Mar 24

The contexts we have been considering are explicit contexts; there
are also implicit contexts.  Namely if I say, ``it is necessary to
put block A on block B'' and you reply ``necessary for whom'', then
it seems that I was assuming an implicit context in my remark.  Better
examples can be found with fewer extraneous features.