From: AAAI-82 Proceedings. Copyright ©1982, AAAI (www.aaai.org). All rights reserved.
CIRCUMSCRIPTIVE
IGNORANCE
Kurt Konolige
Artificial Intelligence
Center
SRI International
ABSTRACT
the puzzle is being circumscribed;
In formal
systems
that reason about knowledge,
inferring that an agent actually does not know a particular
fact can be problematic.
Collins [l] has shown that there
are many different modes of reasoning that a subject can
use to show that he is ignorant
of something;
some of
these, for example, involve the subject reasoning about the
limitations
of his own information-gathering
and memory
abilities. This paper will consider a single type of inference
about ignorance, which we call circumscriptive
ignorance.
We present
a partial
formalization
of circumscriptive
ignorance and apply it to the Wise Man Puzzle.’
1. Circumscriptive
to an agent
will be called
ignorance.
Circumscriptive
Man
solution
part
of a problem
of the
Missionaries
and
only a single
river;
this,
invoke
using
limitations
formalization
For
Puzzle,
is available
to solve the puzzle.
the first-order
an unstated
abstraction.
Cannibals
boat
one cannot
Sudan
there
is a limited
amount
or strategies
available
for
is often
problem
a helicopter
has
axiomatizations
of
(Goad
However,
there
statements
to capture
that
that
this is the only knowledge
the initial
situation;
puzzle it should
that
in an adequate
such as
unstated
of this puzzle),
be possible
possible-world
McCarthy
procedures,
[6],
and
no formalization
including
first-order
semantics
Sato
for
[7]).
in a modal
of using a modal
and the ability to
decision
developed
for modal
we outline a modal
procedures,
logic.
In
formalization
the
of
ignorance.
The
it is
modal
of languages
Pr
for Fool,
agents
a fictitious
shall
Sp.
use
is a propositional
agent
by the choice of propositional
0 E Sp is a reserved
name
whose knowledge is common
knowledge.
For a particular
language K4 is the propositional
choice of Pr and Sp, the
calculus over Pr, together
with a set of indexed unary modal operators
[S], S E Sp.
The intended
meaning of [S](Y is that agent S knows a.
The axiom schemata
of the
to prove that each wise man is
where
for K4 are
(Al)
All propositional
(A4
WV
(A4
(A5)
[SIC-~
[Sla3 PI w
[wm~wlaw~
Pl~xolcsl~
Q and ,f3 denote
an arbitrary
‘This short note describes current research collected in 141. The work
presented
here was supported
by grant N0014-80-C-0296
from the
Office of Naval Research.
we
parameterized
and
wise men have about
ignorant of the color of his own spot in the initial situation.
In effect, the knowledge
that is available to the agents in
logic
modal logic based on Sato’s K4 [7], which includes an
axiomatization
of common knowledge.2
K4 is a family
is at least one
of the puzzle
formalization
in
2. The Modal Logic K14
from the
each man can see his neighbors’
spots and knows from the king that there
white spot.
It is an unstated
condition
for the Wise
especially
formalized
has been
circumscriptive
letters
for a full statement
knowledge
a
[a],
have been
few sections
the
on resources.
3 below
common
that
next
that
across
In puzzles that involve reasoning about the knowledge
there are often unstated
conditions
on
agents possess,
the initial information
given an agent, as well as on the
information
he can acquire. In the Wise Man Puzzle (see
Section
been
by circumscriptive
in the
[5] has investigated
of problem
schema
people
of
the
essential
brigade
McCarthy
a circumscription
but
example,
it is important
to ferry
ignorance,
Puzzle,
knowledge
reasoning
logic of knowledge.
The advantages
formalization
are clarity of expression
Ignorance
one would say
(or -p), then S does not know whether p is true. Proving
ignorance based on a limitation on the knowledge available
use inference
The premise
that
information,
resources,
informally
“The only facts that agent S knows about proposition p
are F.” If from the facts F it is not possible for S to infer p
agent.
arbitrary
Axioms
tautologies
7
sentences,
Al-A4
2For simplici t y we use K4 rather than Sato’s
which deals explicitly with time.
202
(1)
and S denotes
give the system
more complicated
S4
KT4,
for each modality
axiom:
knows.
[S], while A5 is the common
knowledge
then so is K14.
what any fool knows, any fool knows everyone
The two rules of inference are modus ponens and
necessitation
(from
(Y, infer [Slay).
with reasonable
by analogy with
Kripke
procedures
In K4 and other modal logics of knowledge, an agent’s
as a theory,
that is, as a set of
knowledge
is described
formulas
that
contain
rules of inference.
[S](r
for which
that
modus
the term
th .e axioms
To see this,
ponens
agent’s
in which
all instances
no proper
itself.
axioms
axioms
(i.e.,
of
is a Hilbert
(A8)
of K4.
We write
are allowed; an agent’s
we can assert formulas
~+$9
iY4, i.e., cy>/?’ is a theorem
h-4
bY
circumscriptive
itself
of
ignorance,
the proper
axioms
that
knowledge
in an agent’s
if p is in the a-theory
to
sufficient
of the a-theory
theory.
the schemata
paper.
of
The
represent
but there
take
be used to derive
The extended
terms,
[a] is intended
language
to mean
is called K14.
this in K4
of K4,
is, [cr]/3 holds just in case p is in the a-theory
the notion
The
of provability
is explicitly
axiomatization
of K4.
introduced
of [cr] is problematic,
sufficient
set of axioms
use of the intended
K4.
The axioms
for KI4
interpretation
that
are simply
W)
[4P,
where +$
(A7)
-[4A
for KI4
decidable;
axioms
further,
(2) only
form
it is obvious
the
If we
translates
into
For example,
if
q1 or 42, assert
[a][S]p,
interesting
properties
related
to its
is complete:
Thus KI4
(with
it has no consistent
circumscription
atoms
given by the theorems
of K4.
The Wise Man Puzzle can be stated as follows [6]:
A king wishing to know which of his three wise men is
in
the wisest,
(2)
set if K4
that
is decidable,
white
dots
on each of their
at least one spot is white,
the color of his own spot.
announces
that
his spot
foreheads,
and asks each
After
is white,
a while
reasoning
as follows:
“Suppose my spot were black.
The second
wisest of us would then see a black and a white and would
are the same as K4.
a recursive
paints
tells them that
the wisest
if K4
this
theory.
from
of K4.
3. The Wise Man Puzzle
reason
The
either
operators
extension.
to determine
The rules of inference
of [LL] comes
of this
nature.
For every atom of the form
it or its negation
is provable
in KI4.
of the language K14 whose atoms are all
only) has a single model,
together
.
axioms
in the remainder
operator,
he knows
circumscription
with the schemata
where Q ,&.P
of the logical
to the a-theory
of an agent’s
p is that
proper
by making
those of K4,
(3)
it
of [CL] as provability
of K14
of Al-A5
A10 and All are modus
quality
of its meaning
NY14 has several
of K4 are
However, a
can be obtained
all instances
A6 and A7 directly
circumscriptive
[a]@, either
Every subset
Thus
since
that
cr to be a knowledge
about
into K14.
involves formalizing
not only which sentences
provable,
but also which sentences
are not.
could
with cr = [S]q1v[S]q2.
In informal
the a-theory
is more in
a = [Sl%[aI P'lc ks out the agent’s theory for which q is
the only proper axiom. More complicated
statements
are
possible; for example, to say that the only knowledge S has
K4 is extended
To express circumscriptive
ignorance,
modal operators
of new unary
indexed
bY a family
These are called circumscriptive
by sentences
of K4.
modalities,
and are written
as [cy], where Q is a sentence
of K4.
/3 an instance
of K4 are present;
a circumsciption
If we look at a particular
is also no way to express
axiomatization
language
circumscriptive
restriction
th eory for an agent, where the only proper axiom is cr, it
to derive proofs of certain formulas within
is impossible
that theory;
itself.
of A6 that
modal
ponens and necessitation,
respectively,
for K4. However,
there is no similar characterization
of A7. We will use
since there is no way of limiting
could conceivably
characterization
[cr]p,
A8 and A9 ensure
theory
of the
of K4.
not
1s
in [A].
system
form [S]P).~
We define the a-theory of K4, for a fixed
sentence cr, as the set of formulas p for which cr>/!? is a
theorem
are detailed
W) bla
w4 bl(P~7D@lPx47)
(All) blP~b][S]P
.
Q for
the axiomatization
K4
computational
those given by
be given as follows:
and
We shall use
by A4.
in K4.
between
and K4
An alternative
are provable,
theory to mean the set of formulas
theory
allows proper
that
of K4
is implemented
which [S]cy can be proven
There is a difference
an agent’s
note
[3] for S4. These
the style of typical
and are closed u nder the
(Y is an axiom
Sato [7] gives a proof that K4 is decidable,
and decision procedures
properties
can be found
is
that
if his spot
were black,
the least
wise would
see two black spots and would conclude that his spot is
white on the basis of the king’s assurance. He would have
announced
3This difference becomes apparent in the rules of inference: necessitation is not included as a rule of inference in an agent’s theory, since
it would allow the derivation
of [O]p
from [S]p.
simplify
it by now, so my spot
this puzzle
by having
in turn what color his spot is, starting
203
must
be white.”
We
the king ask each wise man
with the least wise.
To formalize
the
puzzle,
we use the
language
K14
with Sp = {O,S1,&,S3}
and Pr = {pl,p2,p3}.
pi is the
sentence asserting that Si has a white spot on his forehead.
A handy abbreviation
is [Sl]pk
[S]pv[S]-p,
knows whether or not p is true. Then the following
suffice
for the
initial
about
time):
worrying
conditions
of the
puzzle
4. Conclusion
This
i.e., S
axioms
formalized
(without
It then
Wise
ciyl)
pl Ap2Ap3
(w2)
[o](PlvP2vP3)
brief
in a modal
becomes
Man
where
knowledge
that
about
at least
the basis of common
knowledge
own spot.
in K14,
provable
that
it suffices
procedure
sentence,
S1 does
not
know
looks for a K4-model
and indeed finds one.
in K4, and -[a][S,jjp,
under
[l]
is provable
the decision
a> [Sll]pl.
-[lS1npl,
axioms
for this situation
utility
of K14
this modification;
hearing
Goad,
C.,
of a Theory
in Proceedings
“A Formal
needs
of Human
Plaus-
of Theoretical
Stanford,
[4]
California
Issues
B.
(June
Methuen
Ii.,
“Modal
(February
1982).
McCarthy,
J.,
Stanford
M. J., An Introduction
Ltd.,
Logics for Belief,”
Artificial
Uni-
1976).
and Company
“Circumscription-A
Reasoning,”
for Situations
note,
(December
G. E. and Cresswell,
Konolige,
notes
Representation
unpublished
Knowledge,”
Hughes,
London,
unpublished
Form
Intelligence
McCarthy,
J. et al., “On the Model Theory
(5) [B]ledge,”
A!emo AIM-312 , Stanford University,
of Non13 (1980).
of KnowStanford
(1978).
[7]
Sato,
M., .4 Study
of Kripke-type
Modal Logics by Gentzen’s
Institute
a = ~2A~3A~7A~8A[S3]pl
would
work
Massachusetts
[3]
S2 does not know
Sl’s reply,
further
of other
K14
Cambridge,
versity,
does not know the color of his spot.
able to deduce that his own spot is white.
are WlLW3 and
that
R. (eds.),
Involving
in the decidable
theory K4, it is
-((Y > fS2jp2) has a model; thus S2
the color of his spot after
obvious
and Schank,
Monotonic
S3 knows that
could reason
of knowledge
, Nash-Webber,
[5]
In the final situation,
In this way agents
1975).
along with
Again, by reasoning
possible to show that
facts.
Processing
are WI-W3,
Q = ~2/\~3A~5A[s,]p,A[s2]p3.
various
of K14 occurs if the
operator
are taken from
Language
[a]
i.e., S1
(~6)
bw2nP2-us2nP2,
to derive
of the state
A., “Fragments
1968.
where
K14.
in the
in Natural
The decision
of the negation
of this
Hence a >[rS1npl
is not
is a theorem of K14 by
by W4, that
Collins,
ible Reasoning,”
The second situation
is similar, except that S2 has
heard Sl’s reply to the king that he does not know his
The
The
of
be
References
to Modal Logic,
color.
called
nonknowledge
logic.
it is no longer
does not know the color of his own spot.
spot’s
idea
could
to be done on this problem.
the color of his
by A6 and A7 to apply
implies,
However,
be decidable
to prove, through
or its negation
for K4 to the sentence
This in turn
this
by an agent
(W2 and W3) and his own
it is possible
To see if [~][S,l]p~
procedure
A6.
situation,
needed
the limitations
agents.
one
observations.
use of W4,
within
K14 itself, ra.ther than K4.
spot is white; W3 says that everyone knows that each can
see the spots of the others.
W4 is the circumscription
axiom: it says that S1 knows whether pl holds solely on
the
of knowledge
to infer
An interesting
modification
sentences
in the circumscriptive
& = W2AW3A[Sl]pzA[Sl]ps.
it’s common
In this initial
introduced
the
showing
how it
logic
possible
Puzzle
knowledge
that
has
ignorance,
is not limited to this puzzle, however; it should be possible
within this logic to do general reasoning about the state of
(4
IV2 says
note
circumscriptive
Kyoto,
and so S3 is
The axioms here
A[S3]p2.
It is possible to prove in K4 that (~>[&]ps,
and so W9
simply asserts that S3 knows the color of his spot.
204
for h4athematical
Japan,
July 1976.
Sequential
Sciences,
Models for Some
Method, Research
Kyoto
University,