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%contex[w89,jmc]		Reifying context - for paper for Thomason

\section{Reifying Context}

	The formula $holds(p,c)$ asserts that the proposition $p$
holds in context $c$.  It is used to express explicitly the
dependence of assertions on context.  The relation $c1 ≤ c2$ asserts
that the context $c2$ is more general than the context $c1$.

	Formalizing common sense reasoning needs contexts as objects,
in order to match the ability of human reasoning to consider context
explicitly.  The proposed database of general common sense knowledge
will make assertions in a general context called $C0$.  However, $C0$
cannot be maximally general, i.e. it will involve unstated
presuppositions.  Indeed we claim that there can be no
maximally general context.  Every context involves unstated presuppositions
both linguistic and factual.  Sometimes the reasoning system will
have to transcend $C0$, and tools will have to be provided to do
this.  For example, if Boyle's law of the dependence of the volume
of a sample of gas on pressure were built into $C0$, discovery of
its dependence on temperature, would have to trigger generalization
that might lead to the perfect gas law.

	The following ideas about how the formalization might
proceed are tentative.  They are depend on using formalized
nonmonotonic reasoning which is also new.  In particular, there
will be nonmonotonic ``inheritance rules'' that allow default
inference from $holds(p,c)$ to $holds(p,c')$, where $c'$ is
either more general or less general than $c$.

	Almost all previous discussion of context has been in
connection with natural language, and the present paper
relies heavily on examples from natural language.  However, I
believe the main AI uses of formalized context will not be in
connection with communication but in connection with reasoning
about the effects of actions directed to achieving goals.  It's
just that natural language examples come to mind more readily.

	Here is an example of intended usage.

	Consider
%
$$holds(at(he,inside(car)),c17).$$
%
Let us suppose that this sentence is intended to assert that a
particular person is in a particular car on a particular occasion,
i.e. the sentence is not just being used as a
linguistic example but is meant seriously.  A corresponding
English sentence is ``He's in the car'' where who he is and which
car and when is determined by the context in which the sentence
is uttered.  Suppose, for simplicity, that the sentence is said
by one person to another in a situation in which the car is
visible to the speaker but not to the hearer and the time at
which the the subject is asserted to be on the mat is the same
time at which the sentence is uttered.

	In our formal language $c17$ has to carry the information about
who he is, which car and when.

	Now suppose that the same fact is to be conveyed as in
example 1, but the context is a certain Stanford Computer Science
Department 1980s context.  Thus familiarity with cars is
presupposed, but no particular person, car or occasion is
presupposed.  The meanings of certain names is presupposed, however.
We can call that context (say) $c5$.  We might then have the sentence
%
$$holds(at(``Timothy McCarthy'',inside((\iota x)(iscar(x)∧
belongs(x,``John McCarthy'')))),c5).$$
%
	A yet more general context might not identify a
specific John McCarthy, so that the sentence itself would need
more information.  What would constitute an adequate identification
might also be context dependent.

	Here are some of the properties formalized contexts might have.

	1. In the above example, we will have $c17 ≤ c5$, i.e. $c5$ is
more general than $c17$.
There will be nonmonotonic rules like
%
$$(∀ c1\ c2\ p)(c1 ≤ c2) ∧ holds(p,c1) ∧ ¬ab1(p,c1,c2) ⊃ holds(p,c2)$$
%
and
%
$$(∀ c1\ c2\ p)(c1 ≤ c2) ∧ holds(p,c2) ∧ ¬ab2(p,c1,c2) ⊃ holds(p,c1).$$
%
Thus there is nonmonotonic inheritance both up and down in the generality
hierarchy.

	2. There are functions forming new contexts by specialization.
We could have something like
%
$$c19 = specialize({he = Timothy McCarthy, belongs(car, John McCarthy)},c5).$$
We will have $c19 ≤ c5$.

	3. Besides $holds(p,c)$, we may have $value(term,c)$, where
$term$ is a term.  The domain in which $term$ takes values is defined
in some outer context.

	4. Some presuppositions of a context are linguistic and some
are factual.  In the above example, who the names refer to are
linguistic.  The properties of people and cars are factual, e.g. it
is presumed that people fit into cars.

	5. We may want meanings as abstract objects.  Thus we might
have
%
$$meaning(he,c17) = meaning(``Timothy McCarthy'',c5).$$

	6. Contexts are ``rich'' entities not to be fully described.
Thus the ``normal English language context'' contains factual assumptions
and linguistic conventions that a particular English speaker may not
know.  Moreover, even assumptions and conventions in a context that
may individually accessible cannot be exhaustively listed.  A person
or machine may know facts about a context but doesn't ``know the context''.

	7. Contexts should not be confused with the situations of the
situation calculus of (McCarthy and Hayes 1969).  Propositions about
situations can hold in a context.  For example, we may have
%
$$holds(Holds1(at(I,airport),result(drive-to(airport,result(walk-to(car),S0))),c1).$$
%
This can be interpreted as asserting that under the assumptions embodied
in context $c1$, a plan of walking to the car and then driving to the airport
would get the robot to the airport starting in situation $S0$.

	8. The context language can be made more like natural
language and more extensible if we introduce notions of entering
and leaving a context.  These will be analogous to the notions
of making and discharging assumptions in natural deduction systems,
but the notion seems to be more general.  Suppose we have $holds(p,c)$.
We then write

\noindent $enter c$.

\noindent This enables us to write $p$ instead of $holds(p,c)$.
If we subsequently infer $q$, we can replace it by $holds(q,c)$
and leave the context $c$.  $holds(q,c)$ will itself hold in
the outer context in which $holds(p,c)$ holds.  When a context
is entered, there need to be restrictions analogous to those
that apply in natural deduction when an assumption is made.

	One way in which this notion of entering and leaving
contexts is more general than natural deduction is that formulas like
$holds(p,c1)$ and (say) $holds(not\ p,c2)$ behave differently
from $c1 ⊃ p$ and $c2 ⊃ ¬p$ which are their natural deduction
analogs.  For example, if $c1$ is associated with the time 5pm
and $c2$ is associated with the time 6pm and $p$ is $at(I, office)$,
then $holds(p,c1) ∧ holds(not\ p,c2)$ might be used to infer that
I left the office between 5pm and 6pm.  $(c1 ⊃ p) ∧ (c2 ⊃ ¬p)$
cannot be used in this way; in fact it is equivalent to
$¬c1 ∨ ¬c2$.

	9. The expession $Holds(p,c)$ (note the caps) represents
the proposition that $p$ holds in $c$.  Since it is a proposition,
we can assert $holds(Holds(p,c),c')$.

	10. Propositions will be combined by functional analogs of 
the Boolean operators as discussed in (McCarthy 197xx).  As discussed
in that paper, treating propositions involving quantification is
necessary, but it is difficult to determine the right formalization.

	11. The major goals of research into formalizing context
should be to determine the rules that relate contexts to their
generalizations and specializations.  Many of these rules will
involve nonmonotonic reasoning.

	1. Consider
%
$$holds(on(cat,mat),c17).$$
%
and suppose that this sentence is intended to assert that a particular
cat is on a particular mat on a particular occasion, i.e.
the sentence is not just being used as a linguistic example but is meant
seriously.  A corresponding English sentence is ``The cat is on the mat'',
where the particular cat and mat and the time is determined by the
context in which the sentence is uttered.  Suppose, for simplicity, that
the sentence is said by one person to another in a situation in which the
cat is visible to the speaker but not to the hearer and the time at which
the cat is asserted to be on the mat is the same time at which the sentence
is uttered.

	In our formal language $c17$ has to carry the information about
which cat, which mat and when.

	2. Now suppose that the same fact is to be conveyed as in example
1, but the context

*****
	8. In natural language parts of sentences have their own contexts
specializing the context of the sentence as a whole.  I'm not sure that
we need this facility for AI purposes, but it might take the following
form.  Consider the sentence ``Timothy tied his shoes''.
There is a default convention that ``his'' in this sentence refers to
Timothy.  Suppose we represent the sentence by
%
$$hold(p,c)$$
%
where $p$ is
%
$$tied(belong(Timothy,shoes),result(ties(Timothy,belong(he,shoes)),s)).$$
%
We would need a notation for the context of $he$ in this
expression relative to the context $c$.  I don't see a really
good notation, but something like the following would be
required.
%
$$c7 = context('he,'p,c).$$
%
We would then have
%
$$meaning(he,c7) = meaning(Timothy,c).$$

	One ordinarily thinks of context in connection with
communication.  Here, however, we are mainly concerned with using
context in the memory of an AI system.  It roughly corresponds to
the fact that in a person's unspoken thinking as well as in his
speech and writing, there are presuppositions and linguistic
conventions of various kinds.

	The problem of context arises when we undertake to design
a language for a general purpose database of commonsense
knowledge.  As soon as we consider putting a particular axiom in
the database, we can think of conditions under which it isn't
quite true.  There seems to be no end to the qualifications that
have to be made.  The problem is especially acute when we have to
decide how many and which arguments to give the predicate and
function symbols.

	Moreover, as soon as we commit to a particular language,
the system seems to be limited in what it can reason about.
Moreover, it is hard to use the language to express facts about
its own limitations.

	We propose to try to fix these problems by introducing
contexts as formal objects, i.e. by reifying context.
  The basic
predicate is $holds$, where
%
$$holds(p,c)$$
%
means that the proposition or sentence $p$ holds in the context
$c$.  (The present considerations do not require us to decide
what kind of propositions or sentences $hold$ applies to).

	The rules relating what holds in different but related
context will be nonmonotonic, so that succes in treating contexts as
formal objects will depend on using circumscription and other
forms of nonmonotonic reasoning.
%The Qualification Problem and 

	It is now well accepted among AI researchers that AI
systems dealing with the common sense informatic situation
require formalized nonmonotonic reasoning.  Some of the relevant
research is discussed in (Lifschitz 1988b).  However, it seems
that building a general common sense database may require further
modifications to mathematical logic.  At least it will be
necessary to change the way common sense facts are represented by
sentences.

	First of all, the context problem arises when
constructing a general database of facts about the common sense
world.  Every axiom one considers holds only in a certain
context.  A critic can always invent some intellectual or
physical circumstance in which the axiom is too narrow and can be
transcended by people.  Some researchers have supposed that it is
only necessary to be patient and that the axioms can be
sufficiently qualified.  Some of the qualification can be done by
introducing abnormalities.  Consider the following axiom that
might be proposed for using a boat.
%
$$\eqalign{(∀ l1 l2 person& water{-}body boat s)\cr
(¬ab(aspect1(l1,l2,water-body,&boat,person,s))\cr
location(person,s) &= boat\cr
∧ location(boat,s) &= l1\cr
 ∧ shore-point(l1,&water-body)\cr
 ∧ shore-point(l2,&water-body)\cr
∧ navigable(water-body&, boat,s)\cr
⊃ location(boat,result(act&(person,propel-boat(l1,l2)),s)) = l2)}.$$
%
It asserts in the situation calculus that a boat may be used to
go from point $l1$ to point $l2$ provided certain conditions are
met.  We have not put in anything about the person and other things
in the boat remaining remaining there, because we expect to get this
as a consequence of the common sense law of inertia.


	The reader will readily see that the conditionals in this
axiom scheme other than the abnormality condtion are trivial
preconditions, e.g., that the boat is at the location $l1$.  Thus,
the axiom requires further qualification.  It isn't completely
obvious that this cannot be
done by suitable conditions on 
%
$$ab(aspect1(l1,l2,water-body,boat,person,s)),$$
%
but this doesn't seem easy, and we propose to explore a different idea.

	The idea is that this sentence is true only in a certain context,
and we propose to reify contexts, i.e. to introduce variables ranging
over contexts.

	The main predicate will be
%
$$holds(p,c)$$
%
asserting that the proposition $p$ holds in context $c$.  It is often
convenient to consider the value of an individual concept as
represented by the expression
%
$$value(exp,c)$$,
%
where $exp$ is an expression representing an individual concept.
Propositions and individual concepts are discussed in (McCarthy
1979b).  There is an additional difficulty with individual
concepts in that we may want to
regard their domains of values as also
context dependent.

	We propose to use formalized contexts to do the same
work that context does in natural language and more.  The ``more''
refers to the fact that much our non-linguistic thinking is
also context dependent.

%from notebook 1988 March 1 - June 20

	Our formalization will have the following properties.

	1. Contexts are among the sorts of individuals in a (first order)
language.

	2. The basic predicate is $holds$, where $holds(p,c)$ asserts
that proposition $p$ holds in context $c$.  (Those familiar with
situation calculus should avoid being misled by a similarity of notation
into assuming that contexts are just a variant of situations).  That
they are different will become apparent.  The similarity is related to
the fact that there may sometimes be a context associated with a
situation.

	3. Contexts are ``rich'' entities not to be fully described.
Thus the ``normal English language context'' contains factual assumptions
and linguistic conventions that a particular English speaker may not
know.  Moreover, even assumptions and conventions in a context that
may individually accessible cannot be exhaustively listed.  A person
or machine may know facts about a context but doesn't ``know the context''.

	4. As an example of intended usage consider the formula
$holds(on(cat,mat),c17)$.  Suppose it plays the role of asserting
that a particular cat is on a particular mat on a particular occasion.
$c17$ carries information identifying $cat$ with a particular cat
and also determines the mat and the occasion.  The occasion may
be real or fictitious or hypothetical, and the nature of the occasion
is part of $c17$.  It is a feature of $c17$ that the time doesn't
have to be specified in the proposition $on(cat,mat)$, although $c17$
may require some other kinds of propositions to include times.

	5. We may even regard the use of predicate calculus notation
as a convention of $c17$.  Thus we might have
%
%$$holds({\rm ``The\ cat\ is\ on\ the\ mat.''},c18),$$

\centerline{$holds$(``The cat is on the mat.'',$c18$),}
%
where $c18$ is related to $c17$ in certain ways.

	6. The above-mentioned $c17$ would be a rather specialized
context.  Assertions made in $c17$, i.e. sentences of the form
$holds(p,c17)$ are connected to less specialized contexts by
sentences of our language of contexts.  For example, we might have
%
$$meaning(cat,c17) = meaning(belongingto({\rm John\ McCarthy},cat),c9),$$
%
where $c9$ is a context in which ``John McCarthy'' is identified but
the reference to $cat$ as a particular beast is not.

	7. There is no ``most general context''.  The reason for this
is that the context language needs to allow the possibility of  creating
new contexts generalizing certain aspects of the most general previous
contexts.

	A convenient example is provided by a variant of the John
Searle's ``Chinese room'' story in which the man keeps all the
information about how to respond to sequences of Chinese
characters in his head.  Part of Searle's confusion about who or
what knows Chinese in the situation is the usual identification
of a personality with the body it occupies.  This is a convention
of English and other natural languages and does no harm as long
as there is only one personality per body.  Avoiding confusion in
the Chinese room puzzle requires distinguishing the personality
carrying on the dialog which does know Chinese from that of the
man in the normal sense which doesn't.  If it were actually
feasible and common to carry out conversations at some
significant speed by interpreting a description of a personality,
English would presumably have occasion to make the distinction.

	An AI system capable of being told about multiple
personalities in one body would have to be able to generalize the
context of its internal language to one in which the personality
is distinguished from the body.

	8. We give one example of a rule relating
contexts in which certain sentences are explicitly indexed by
times and a more specialized context in which a time is implicit.
%
$$∀ptc(holds(p,spectime(c,t)) ≡ holds(attime(t,p),c)).$$
%

	9. The context language can be made more like natural
language and more extensible if we introduce notions of entering
and leaving a context.  These will be analogous to the notions
of making and discharging assumptions in natural deduction systems,
but the notion seems to be more general.  Suppose we have $holds(p,c)$.
We then write

\noindent $enter c$.

\noindent This enables us to write $p$ instead of $holds(p,c)$.
If we subsequently infer $q$, we can replace it by $holds(q,c)$
and leave the context $c$.  $holds(q,c)$ will itself hold in
the outer context in which $holds(p,c)$ holds.  When a context
is entered, there need to be restrictions analogous to those
that apply in natural deduction when an assumption is made.

	One way in which this notion of entering and leaving
contexts is more general than natural deduction is that formulas like
$holds(p,c1)$ and (say) $holds(not\ p,c2)$ behave differently
from $c1 ⊃ p$ and $c2 ⊃ ¬p$ which are their natural deduction
analogs.  For example, if $c1$ is associated with the time 5pm
and $c2$ is associated with the time 6pm and $p$ is $at(I, office)$,
then $holds(p,c1) ∧ holds(not\ p,c2)$ might be used to infer that
I left the office between 5pm and 6pm.  $(c1 ⊃ p) ∧ (c2 ⊃ ¬p)$
cannot be used in this way; in fact it is equivalent to
$¬c1 ∨ ¬c2$.

	10. Perhaps it will be convenient to start the reasoning
done by the program by entering some rather general context.

	11. The major goals of research into formaling context
should be to determine the rules that relate contexts to their
generalizations and specializations.  Many of these rules will
involve nonmonotonic reasoning.