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CIRCUMSCRIPTION—A FORM OF
Computer Science Department
Stanford, CA 94305
Humans and intelligent computer programs must often jump to
the conclusion that the objects they can determine to have certain
properties or relations are the only objects that do. Circumscription
formalizes such conjectural reasoning.
INTRODUCTION. THE QUALIFICATIONPROBLEM
(McCarthy 1959)1 proposed a program with “common sense” that would
represent what it knows (mainly) by sentences in a suitable logical language.
It would decide what to do by deducing a conclusion that it should perform
a certain act. Performing the act would create a new situation, and it would
again decide what to do. This requires representing both knowledge about
the particular situation and general common sense knowledge as sentences
The “qualiﬁcation problem”, immediately arose in representing general
common sense knowledge.
It seemed that in order to fully represent the
conditions for the successful performance of an action, an impractical and
implausible number of qualiﬁcations would have to be included in the sen-
tences expressing them. For example, the successful use of a boat to cross a
river requires, if the boat is a rowboat, that the oars and rowlocks be present
and unbroken, and that they ﬁt each other. Many other qualiﬁcations can
be added, making the rules for using a rowboat almost impossible to apply,
and yet anyone will still be able to think of additional requirements not yet
Circumscription is a rule of conjecture that can be used by a person or
program for “jumping to certain conclusions”. Namely, the objects that can
be shown to have a certain property P by reasoning from certain facts A are
all the objects that satisfy P . More generally, circumscription can be used to
conjecture that the tuples < x, y..., z > that can be shown to satisfy a relation
P (x, y, ..., z) are all the tuples satisfying this relation. Thus we circumscribe
the set of relevant tuples.
We can postulate that a boat can be used to cross a river unless “some-
thing” prevents it. Then circumscription may be used to conjecture that the
only entities that can prevent the use of the boat are those whose existence
follows from the facts at hand. If no lack of oars or other circumstance pre-
venting boat use is deducible, then the boat is concluded to be usable. The
correctness of this conclusion depends on our having “taken into account”
all relevant facts when we made the circumscription.
Circumscription formalizes several processes of human informal reasoning.
For example, common sense reasoning is ordinarily ready to jump to the
conclusion that a tool can be used for its intended purpose unless something
prevents its use. Considered purely extensionally, such a statement conveys
no information; it seems merely to assert that a tool can be used for its
intended purpose unless it can’t. Heuristically, the statement is not just a
tautologous disjunction; it suggests forming a plan to use the tool.
Even when a program does not reach its conclusions by manipulating
sentences in a formal language, we can often proﬁtably analyze its behavior
by considering it to believe certain sentences when it is in certain states, and
we can study how these ascribed beliefs change with time. See (McCarthy
1979a). When we do such analyses, we again discover that successful people
and programs must jump to such conclusions.
2 THE NEED FOR NONMONOTONIC REA-SONING
We cannot get circumscriptive reasoning capability by adding sentences to
an axiomatization or by adding an ordinary rule of inference to mathematical
logic. This is because the well known systems of mathematical logic have the
following monotonicity property. If a sentence q follows from a collection A
of sentences and A ⊂ B, then q follows from B. In the notation of proof
theory: if A (cid:96) q and A ⊂ B, then B (cid:96) q. Indeed a proof from the premisses
A is a sequence of sentences each of which is a either a premiss, an axiom or
follows from a subset of the sentences occurring earlier in the proof by one
of the rules of inference. Therefore, a proof from A can also serve as a proof
from B. The semantic notion of entailment is also monotonic; we say that
A entails q (written A |= q) if q is true in all models of A. But if A |= q and
A ⊂ B, then every model of B is also a model of A, which shows that B |= q.
Circumscription is a formalized rule of conjecture that can be used along
with the rules of inference of ﬁrst order logic. Predicate circumscription
assumes that entities satisfy a given predicate only if they have to on the basis
of a collection of facts. Domain circumscription conjectures that the “known”
entities are all there are. It turns out that domain circumscription, previously
called minimal inference, can be subsumed under predicate circumscription.
We will argue using examples that humans use such “nonmonotonic”
reasoning and that it is required for intelligent behavior. The default case
reasoning of many computer programs (Reiter 1980) and the use of THNOT
in MICROPLANNER (Sussman, et. al. 1971) programs are also examples of
nonmonotonic reasoning, but possibly of a diﬀerent kind from those discussed
in this paper. (Hewitt 1972) gives the basic ideas of the PLANNER approach.
The result of applying circumscription to a collection A of facts is a
sentence schema that asserts that the only tuples satisfying a predicate
P (x, ..., z) are those whose doing so follows from the sentences of A. Since
adding more sentences to A might make P applicable to more tuples, cir-
cumscription is not monotonic. Conclusions derived from circumscription
are conjectures that A includes all the relevant facts and that the objects
whose existence follows from A are all the relevant objects.
A heuristic program might use circumscription in various ways. Suppose
it circumscribes some facts and makes a plan on the basis of the conclusions
reached. It might immediately carry out the plan, or be more cautious and
look for additional facts that might require modifying it.
Before introducing the formalism, we informally discuss a well known
problem whose solution seems to involve such nonmonotonic reasoning.
3 MISSIONARIES AND CANNIBALS
The Missionaries and Cannibals puzzle, much used in AI, contains more than
enough detail to illustrate many of the issues. “Three missionaries and three
cannibals come to a river. A rowboat that seats two is available.
cannibals ever outnumber the missionaries on either bank of the river, the
missionaries will be eaten. How shall they cross the river?”
Obviously the puzzler is expected to devise a strategy of rowing the boat
back and forth that gets them all across and avoids the disaster.
Amarel (1971) considered several representations of the problem and dis-
cussed criteria whereby the following representation is preferred for purposes
of AI, because it leads to the smallest state space that must be explored to
ﬁnd the solution. A state is a triple comprising the numbers of missionaries,
cannibals and boats on the starting bank of the river. The initial state is
331, the desired ﬁnal state is 000, and one solution is given by the sequence
We are not presently concerned with the heuristics of the problem but
rather with the correctness of the reasoning that goes from the English state-
ment of the problem to Amarel’s state space representation. A generally
intelligent computer program should be able to carry out this reasoning. Of
course, there are the well known diﬃculties in making computers understand
English, but suppose the English sentences describing the problem have al-
ready been rather directly translated into ﬁrst order logic. The correctness
of Amarel’s representation is not an ordinary logical consequence of these
sentences for two further reasons.
First, nothing has been stated about the properties of boats or even the
fact that rowing across the river doesn’t change the numbers of missionaries
or cannibals or the capacity of the boat. Indeed it hasn’t been stated that
situations change as a result of action. These facts follow from common sense
knowledge, so let us imagine that common sense knowledge, or at least the
relevant part of it, is also expressed in ﬁrst order logic.
The second reason we can’t deduce the propriety of Amarel’s represen-
tation is deeper. Imagine giving someone the problem, and after he puzzles
for a while, he suggests going upstream half a mile and crossing on a bridge.
“What bridge”, you say. “No bridge is mentioned in the statement of the
problem.” And this dunce replies, “Well, they don’t say there isn’t a bridge”.
You look at the English and even at the translation of the English into ﬁrst
order logic, and you must admit that “they don’t say” there is no bridge. So
you modify the problem to exclude bridges and pose it again, and the dunce
proposes a helicopter, and after you exclude that, he proposes a winged horse
or that the others hang onto the outside of the boat while two row.
You now see that while a dunce, he is an inventive dunce. Despairing
of getting him to accept the problem in the proper puzzler’s spirit, you tell
him the solution. To your further annoyance, he attacks your solution on the
grounds that the boat might have a leak or lack oars. After you rectify that
omission from the statement of the problem, he suggests that a sea monster
may swim up the river and may swallow the boat. Again you are frustrated,
and you look for a mode of reasoning that will settle his hash once and for
Circumscription is one candidate for accomplishing this.
In spite of our irritation with the dunce, it would be cheating to put into
the statement of the problem that there is no other way to cross the river
than using the boat and that nothing can go wrong with the boat. A human
doesn’t need such an ad hoc narrowing of the problem, and indeed the only
watertight way to do it might amount to specifying the Amarel representation
in English. Rather we want to avoid the excessive qualiﬁcation and get the
Amarel representation by common sense reasoning as humans ordinarily do.
It will allow
us to conjecture that no relevant objects exist in certain categories except
those whose existence follows from the statement of the problem and common
sense knowledge. When we circumscribe the ﬁrst order logic statement of
the problem together with the common sense facts about boats etc., we will
be able to conclude that there is no bridge or helicopter. “Aha”, you say,
“but there won’t be any oars either”. No, we get out of that as follows: It is
a part of common knowledge that a boat can be used to cross a river unless
there is something wrong with it or something else prevents using it, and if
our facts don’t require that there be something that prevents crossing the
river, circumscription will generate the conjecture that there isn’t. The price
is introducing as entities in our language the “somethings” that may prevent
the use of the boat.
If the statement of the problem were extended to mention a bridge, then
the circumscription of the problem statement would no longer permit showing
the non-existence of a bridge, i.e. a conclusion that can be drawn from
a smaller collection of facts can no longer be drawn from a larger. This
nonmonotonic character of circumscription is just what we want for this
kind of problem. The statement, “There is a bridge a mile upstream, and the
boat has a leak.” doesn’t contradict the text of the problem, but its addition
invalidates the Amarel representation.
In the usual sort of puzzle, there is a convention that there are no ad-
ditional objects beyond those mentioned in the puzzle or whose existence is
deducible from the puzzle and common sense knowledge. The convention
can be explicated as applying circumscription to the puzzle statement and a
certain part of common sense knowledge. However, if one really were sitting
by a river bank and these six people came by and posed their problem, one
wouldn’t take the circumscription for granted, but one would consider the
result of circumscription as a hypothesis. In puzzles, circumscription seems
to be a rule of inference, while in life it is a rule of conjecture.
Some have suggested that the diﬃculties might be avoided by introducing
probabilities. They suggest that the existence of a bridge is improbable. The
whole situation involving cannibals with the postulated properties cannot be
regarded as having a probability, so it is hard to take seriously the conditional
probability of a bridge given the hypotheses. More to the point, we mentally
propose to ourselves the normal non-bridge non-sea-monster interpretation
bef ore considering these extraneous possibilities, let alone their probabili-
ties, i.e. we usually don’t even introduce the sample space in which these
possibilities are assigned whatever probabilities one might consider them to
have. Therefore, regardless of our knowledge of probabilities, we need a way
of formulating the normal situation from the statement of the facts, and non-
monotonic reasoning seems to be required. The same considerations seem to
apply to fuzzy logic.
Using circumscription requires that common sense knowledge be expressed
in a form that says a boat can be used to cross rivers unless there is some-
thing that prevents its use.
In particular, it looks like we must introduce
into our ontology (the things that exist) a category that includes something
wrong with a boat or a category that includes something that may prevent its
use. Incidentally, once we have decided to admit something wrong with the
boat, we are inclined to admit a lack of oars as such a something and to ask
questions like, “Is a lack of oars all that is wrong with the boat?”.
Some philosophers and scientists may be reluctant to introduce such
things, but since ordinary language allows “something wrong with the boat”
we shouldn’t be hasty in excluding it. Making a suitable formalism is likely
to be technically diﬃcult as well as philosophically problematical, but we
We challenge anyone who thinks he can avoid such entities to express in
his favorite formalism, “Besides leakiness, there is something else wrong with
the boat”. A good solution would avoid counterfactuals as this one does.
Circumscription may help understand natural language, because if the
use of natural language involves something like circumscription, it is un-
derstandable that the expression of general common sense facts in natural
language will be diﬃcult without some form of nonmonotonic reasoning.
4 THE FORMALISM OF CIRCUMSCRIP-
Let A be a sentence of ﬁrst order logic containing a predicate symbol P (x1, . . . , xn)which we will write P (¯x). We write A(Φ) for the result of replacing all oc-
currences of P in A by the predicate expression Φ. (As well as predicate
symbols, suitable λ-expressions are allowed as predicate expressions).
Deﬁnition. The circumscription of P in A(P ) is the sentence schema
A(Φ) ∧ ∀¯x.(Φ(¯x) ⊃ P (¯x)) ⊃ ∀¯x.(P (¯x) ⊃ Φ(¯x)).
(1) can be regarded as asserting that the only tuples (¯x) that satisfy P
are those that have to — assuming the sentence A. Namely, (1) contains a
predicate parameter Φ for which we may subsitute an arbitrary predicate ex-
pression. (If we were using second order logic, there would be a quantiﬁer ∀Φ
in front of (1).) Since (1) is an implication, we can assume both conjuncts on
the left, and (1) lets us conclude the sentence on the right. The ﬁrst conjunct
A(Φ) expresses the assumption that Φ satisﬁes the conditions satisﬁed by P,
and the second ∀¯x.(Φ(¯x) ⊃ P (¯x)) expresses the assumption that the entities
satisfying Φ are a subset of those that satisfy P . The conclusion asserts the
converse of the second conjunct which tells us that in this case, Φ and P
We write A (cid:96)P q if the sentence q can be obtained by deduction from the
result of circumscribing P in A. As we shall see (cid:96)P is a nonmonotonic form
of inference, which we shall call circumscriptive inference.
A slight generalization allows circumscribing several predicates jointly;
thus jointly circumscribing P and Q in A(P, Q) leads to
A(Φ, Ψ) ∧ ∀¯x.(Φ(¯x) ⊃ P (¯x)) ∧ ∀¯y.(Ψ(¯y) ⊃ Q(¯y))
⊃ ∀¯x.(P (¯x) ⊃ Φ(¯x)) ∧ ∀¯y.(Q(¯y) ⊃ Ψ(¯y))
in which we can simultaneously substitute for Φ and Ψ. The relation A (cid:96)P,Q q
is deﬁned in a corresponding way. Although we don’t give examples of joint
circumscription in this paper, we believe it will be important in some AI
Consider the following examples:
Example 1. In the blocks world, the sentence A may be
isblock A ∧ isblock B ∧ isblock C
asserting that A, B and C are blocks. Circumscribing isblock in (2) gives
Φ(A) ∧ Φ(B) ∧ Φ(C) ∧ ∀x.(Φ(x) ⊃ isblock x) ⊃ ∀x.(isblock x ⊃ Φ(x)).
If we now substitute
Φ(x) ≡ (x = A ∨ x = B ∨ x = C)
into (3) and use (2), the left side of the implication is seen to be true, and
∀x.(isblock x ⊃ (x = A ∨ x = B ∨ x = C)),
which asserts that the only blocks are A, B and C, i.e.
just those objects
that (2) requires to be blocks. This example is rather trivial, because (2)
provides no way of generating new blocks from old ones. However, it shows
that circumscriptive inference is nonmonotonic since if we adjoin isblock D
to (2), we will no longer be able to infer (5).
Example 2. Circumscribing the disjunction
isblock A ∨ isblock B
(Φ(A) ∨ Φ(B)) ∧ ∀x.(Φ(x) ⊃ isblockx) ⊃ ∀x.(isblock x ⊃ Φ(x)).
We may then substitute successively Φ(x) ≡ (x = A) and Φ(x) ≡ (x = B),
and these give respectively
(A = A ∨ A = B) ∧ ∀x.(x = A ⊃ isblock x) ⊃ ∀x.(isblock x ⊃ x = A), (8)
which simpliﬁes to
which simpliﬁes to
(9), (11) and (6) yield
isblock A ⊃ ∀x.(isblock x ⊃ x = A)
(B = A ∨ B = B) ∧ ∀x.(x = B ⊃ isblock x) ⊃ ∀x.(isblock x ⊃ x = B), (10)
isblock B ⊃ ∀x.(isblock x ⊃ x = B).
∀x.(isblock x ⊃ x = A) ∨ ∀x.(isblock x ⊃ x = B),
which asserts that either A is the only block or B is the only block.
Example 3. Consider the following algebraic axioms for natural numbers,
i.e., non-negative integers, appropriate when we aren’t supposing that natural
numbers are the only objects.
isnatnum 0 ∧ ∀x.(isnatnum x ⊃ isnatnum succ x).
Circumscribing isnatnum in (13) yields
Φ(0)∧∀x.(Φ(x) ⊃ Φ(succ x))∧∀x.(Φ(x) ⊃ isnatnum x) ⊃ ∀x.(isnatnum x ⊃ Φ(x)).(14)
(14) asserts that the only natural numbers are those objects that (13) forces
to be natural numbers, and this is essentially the usual axiom schema of
induction. We can get closer to the usual schema by substituting Φ(x) ≡
Ψ(x) ∧ isnatnum x. This and (13) make the second conjunct drop out giving
Ψ(0) ∧ ∀x.(Ψ(x) ⊃ Ψ(succ x)) ⊃ ∀x.(isnatnum x ⊃ Ψ(x)).
Example 4. Returning to the blocks world, suppose we have a predicate
on(x, y, s) asserting that block x is on block y in situation s. Suppose we have
above(x, y, s) which asserts that block x is above block y
in situation s. We may write
on(x, y, s) ⊃
above(x, y, s))
above(x, y, s) ∧
above(y, z, s) ⊃
above(x, z, s)),
i.e. above is a transitive relation. Circumscribing above in (16)∧(17) gives
on(x, y, s) ⊃ Φ(x, y, s))
∧∀xyzs.(Φ(x, y, s) ∧ Φ(y, z, s) ⊃ Φ(x, z, s))
∧∀xys.(Φ(x, y, s) ⊃
above(x, y, s))
above(x, y, s) ⊃ Φ(x, y, s))
which tells us that above is the transitive closure of on.
In the preceding two examples, the schemas produced by circumscription
play the role of axiom schemas rather than being just conjectures.
5 DOMAIN CIRCUMSCRIPTION
The form of circumscription described in this paper generalizes an earlier ver-
sion called minimal inference. Minimal inference has a semantic counterpart
called minimal entailment, and both are discussed in (McCarthy 1977) and
more extensively in (Davis 1980). The general idea of minimal entailment
is that a sentence q is minimally entailed by an axiom A, written A |=m q,
if q is true in all minimal models of A, where one model if is considered
less than another if they agree on common elements, but the domain of the
larger many contain elements not in the domain of the smaller. We shall
call the earlier form domain circumscription to contrast it with the predicate
circumscription discussed in this paper.
The domain circumscription of the sentence A is the sentence
Axiom(Φ) ∧ AΦ ⊃ ∀x.Φ(x),
where AΦ is the relativization of A with respect to Φ and is formed by
replacing each universal quantiﬁer ∀x. in A by ∀x.Φ(x) ⊃ and each existential
quantiﬁer ∃x. by ∃x.Φ(x)∧. Axiom(Φ) is the conjunction of sentences Φ(a)
for each constant a and sentences ∀x.(Φ(x) ⊃ Φ(f (x))) for each function
symbol f and the corresponding sentences for functions of higher arities.
Domain circumscription can be reduced to predicate circumscription by
relativizing A with respect to a new one place predicate called (say) all, then
circumscribing all in Aall ∧ Axiom(all), thus getting
Axiom(Φ) ∧ AΦ ∧ ∀x.(Φ(x) ⊃
all(x)) ⊃ ∀x.(
all(x) ⊃ Φ(x)).
Now we justify our using the name all by adding the axiom ∀x.
all(x) so that
(20) then simpliﬁes precisely to (19).
In the case of the natural numbers, the domain circumscription of true,
the identically true sentence, again leads to the axiom schema of induction.
Here Axiom does all the work, because it asserts that 0 is in the domain and
that the domain is closed under the successor operation.
6 THE MODEL THEORY OF PREDICATE
This treatment is similar to Davis’s (1980) treatment of domain circumscrip-
tion. Pat Hayes (1979) pointed out that the same ideas would work.
The intuitive idea of circumscription is saying that a tuple ¯x satisﬁes the
It has to satisfy P if this follows from the
predicate P only if it has to.
sentence A. The model-theoretic counterpart of circumscription is minimal
entailment. A sentence q is minimally entailed by A, if q is true in all minimal
models of A, where a model is minimal if as few as possible tuples ¯x satisfy
the predicate P. More formally, this works out as follows.
Deﬁnition. Let M (A) and N (A) be models of the sentence A. We say that
M is a submodel of N in P, writing M ≤P N , if M and N have the same
domain, all other predicate symbols in A besides P have the same extensions
in M and N , but the extension of P in M is included in its extension in N .
Deﬁnition. A model M of A is called minimal in P if M (cid:48) ≤P M only if
M (cid:48) = M . As discussed by Davis (1980), minimal models don’t always exist.
Deﬁnition. We say that A minimally entails q with respect to P , written
A |=p q provided q is true in all models of A that are minimal in P .
Theorem. Any instance of the circumscription of P in A is true in all models
of A minimal in P , i.e. is minimally entailed by A in P .
Proof. Let M be a model of A minimal in P . Let P (cid:48) be a predicate satisfying
the left side of (1) when substituted for Φ. By the second conjunct of the
left side P is an extension of P (cid:48). If the right side of (1) were not satisﬁed,
P would be a proper extension of P (cid:48). In that case, we could get a proper
submodel M (cid:48) of M by letting M (cid:48) agree with M on all predicates except P
and agree with P (cid:48) on P . This would contradict the assumed minimality of
Corollary. If A (cid:96)P q, then A |=P q.
While we have discussed minimal entailment in a single predicate P , the
relation <P,Q, models minimal in P and Q, and |=P,Q have corresponding
properties and a corresponding relation to the syntactic notion (cid:96)P,Q men-
7 MORE ON BLOCKS
prevents(z, move(x, y), s) ⊃
on(x, y, result(move(x, y), s)))
states that unless something prevents it, x is on y in the situation that results
from the action
We now list various “things” that may prevent this action.
∀xys.(¬isblock x ∨ ¬isblock y ⊃ prevents(N ON BLOCK,
move(x, y), s))
clear(x, s) ∨ ¬
clear(y, s) ⊃ prevents(COV ERED,
move(x, y), s))
prevents(weightx, move(x, y), s)).
Let us now suppose that a heuristic program would like to move block A
onto block C in a situation s0. The program should conjecture from (21) that
move(A, C) would have the desired eﬀect, so it must try to estab-
prevents(z, move(A, C), s0). The predicate λz.
prevents(z, move(A, C), s0)can be circumscribed in the conjunction of the sentences resulting from spe-
cializing (22), (23) and (24), and this gives
(¬isblock A ∨ ¬isblock C ⊃ Φ(N ON BLOCK))
clear(A, s0) ∨ ¬
clear(C, s0) ⊃ Φ(COV ERED))
∧(tooheavyA ⊃ Φ(weightA))
prevents(z, move(A, C), s0))
prevents(z, move(A, C), s0) ⊃ Φ(z))
which says that the only things that can prevent the move are the phenomena
described in (22), (23) and (24). Whether (25) is true depends on how good
the program was in ﬁnding all the relevant statements. Since the program
wants to show that nothing prevents the move, it must set ∀z.(Φ(z) ≡ f alse),
after which (25) simpliﬁes to
(isblock A ∧ isblock B ∧
clear(A, s0) ∧
clear(B, s0) ∧ ¬tooheavyA
prevents(z, move(A, C), s0).
We suppose that the premisses of this implication are to be obtained as
on(A, B, s0). Circumscription of λx y.
on(x,y,s0)in this assertion gives
Φ(A, B) ∧ ∀xy.(Φ(x, y) ⊃
on(x, y, s0)) ⊃ ∀xy.(
on(x, y, s0) ⊃ Φ(x, y)), (27)
and taking Φ(x, y) ≡ x = A ∧ y = B yields
on(x, y, s0) ⊃ x = A ∧ y = B).
clear(x, s) ≡ ∀y.¬
on(y, x, s))
as the deﬁnition of clear yields the second two desired premisses.
tooheavy(x)might be explicitly present or it might also be conjec- tured by a circumscription assuming that if x were too heavy, the facts would
Circumscription may also be convenient for asserting that when a block
is moved, everything that cannot be proved to move stays where it was. In
the simple blocks world, the eﬀect of this can easily be achieved by an axiom
that states that all blocks except the one that is moved stay put. However, if
there are various sentences that say (for example) that one block is attached
to another, circumscription may express the heuristic situation better than
8 REMARKS AND ACKNOWLEDGEMENTS1. Circumscription is not a “nonmonotonic logic”. It is a form of nonmono-
tonic reasoning augmenting ordinary ﬁrst order logic. Of course, sentence
schemata are not properly handled by most present general purpose resolu-
tion theorem provers. Even ﬁxed schemata of mathematical induction when
used for proving programs correct usually require human intervention or spe-
cial heuristics, while here the program would have to use new schemata pro-
duced by circumscription. In (McCarthy 1979b) we treat some modalities in
ﬁrst order logic instead of in modal logic. In our opinion, it is better to avoid
modifying the logic if at all possible, because there are many temptations to
modify the logic, and it would be very diﬃcult to keep them compatible.
be on a block y only if this is explicitly stated, i.e. the default is that x is
not on y. Then for each individual block x, we may be able to conclude that
it isn’t on block A, but we will not be able to conclude, as circumscription
would allow, that there are no blocks on A. That would require a separate
default statement that a block is clear unless something is stated to be on it.
I thank for their help. Without it, circumscribing a disjunction, as in the
second example in Section 4, would lead to a contradiction.
logic. The program would sometimes apply circumscription to certain pred-
icates in sentences. In particular, when it wants to perform an action that
might be prevented by something, it circumscribes the prevention predicate
in a sentence A representing the information being taken into account.
Clearly the program will have to include domain dependent heuristics for
deciding what circumscriptions to make and when to take them back.
will appear as conjuncts on the left side of the implication unchanged. There-
fore, the original versions of these facts can be used in proving the left side.
circumscription of this formula can be taken to be
∀x.(A(x) ∧ (x ⊂ a) ⊃ (a ⊂ x)).
If a occurs in A(a) only in expressions of the form z ∈ a, then its mathemati-
cal properties should be analogous to those of predicate circumscription. We
have not explored what happens if formulas like a ∈ z occur.
can be axiomatized using the relation on or the relation above considered
in section 4 or also in terms of the heights and horizontal positions of the
blocks. Since the results of circumscription will diﬀer according to which
representation is chosen, we see that the choice of representation has episte-
mological consequences if circumscription is admitted as a rule of conjecture.
Choosing the set of predicates in terms of which to axiomatize a set of facts,
such as those about blocks, is like choosing a co-ordinate system in physics
or geography. As discussed in (McCarthy 1979a), certain concepts are de-
ﬁnable only relative to a theory. What theory admits the most useful kinds
of circumscription may be an important criterion in the choice of predicates.
It may also be possible to make some statements about a domain like the
blocks world in a form that does not depend on the language used.
00524, in part by the IBM 1979 Distinguished Faculty Program at the T. J.
Watson Research Center, and in part by the Center for Advanced Study in
the Behavioral Sciences.
Amarel, Saul (1971). On Representation of Problems of Reasoning about
Actions, in D. Michie (ed.), Machine Intelligence 3, Edinburgh University
Press, pp. 131–171.
Chandra, Ashok (1979). Personal conversation, August.
Davis, Martin (1980). Notes on the Mathematics of Non-Monotonic Reason-
ing, Artiﬁcial Intelligence 13 (1, 2), pp. 73–80.
Hayes, Patrick (1979). Personal conversation, September.
Hewitt, Carl (1972). Description and Theoretical Analysis (Using Schemata)
of PLANNER: a Language for Proving Theorems and Manipulating Models
in a Robot, MIT AI Laboratory TR-258.
McCarthy, John (1959). Programs with Common Sense, Proceedings of the
Teddington Conference on the Mechanization of Thought Processes, London:
Her Majesty’s Stationery Oﬃce. (Reprinted in this volume, pp. 000–000).
McCarthy, John and Patrick Hayes (1969)2. Some Philosophical Problems
from the Standpoint of Artiﬁcial Intelligence, in B. Meltzer and D. Michie
(eds), Machine Intelligence 4, Edinburgh University.
(Reprinted in B. L.
Webber and N. J. Nilsson (eds.), Readings in Artiﬁcial Intelligence, Tioga,
1981, pp. 431–450; also in M. J. Ginsberg (ed.), Readings in Nonmonotonic
Reasoning, Morgan Kaufmann, 1987, pp. 26–45. Reprinted in (McCarthy
McCarthy, John (1977). Epistemological Problems of Artiﬁcial Intelligence,
Proceedings of the Fifth International Joint Conference on Artiﬁcial Intel-
ligence, M.I.T., Cambridge, Mass. (Reprinted in B. L. Webber and N. J.
Nilsson (eds.), Readings in Artiﬁcial Intelligence, Tioga, 1981, pp. 459–465;
also in M. J. Ginsberg (ed.), Readings in Nonmonotonic Reasoning, Morgan
Kaufmann, 1987, pp. 46–52. Reprinted in (McCarthy 1990).
McCarthy, John (1979a). Ascribing Mental Qualities to Machines3 , Philo-
sophical Perspectives in Artiﬁcial Intelligence, Martin Ringle, ed., Humani-
ties Press. Reprinted in (McCarthy 1990).
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Propositions4 in Michie, Donald (ed.) Machine Intelligence 9, Ellis Horwood.
Reprinted in (McCarthy 1990).
McCarthy, John (1990). Formalizing Common Sense, Ablex.
Reiter, Raymond (1980). A Logic for Default Reasoning, Artiﬁcial Intelli-
gence 13 (1, 2), pp. 81–132.
Sussman, G.J., T. Winograd, and E. Charniak (1971). Micro-Planner Ref-
erence Manual, AI Memo 203, M.I.T. AI Lab.
Circumscription and the nonmonotonic reasoning formalisms of McDer-
mott and Doyle (1980) and Reiter (1980) diﬀer along two dimensions. First,
circumscription is concerned with minimal models, and they are concerned
with arbitrary models. It appears that these approaches solve somewhat dif-
ferent though overlapping classes of problems, and each has its uses. The
other diﬀerence is that the reasoning of both other formalisms involves mod-
els directly, while the syntactic formulation of circumscription uses axiom
schemata. Consequently, their systems are incompletely formal unless the
metamathematics is also formalized, and this hasn’t yet been done.
However, schemata are applicable to other formalisms than circumscrip-
tion. Suppose, for example, that we have some axioms about trains and their
presence on tracks, and we wish to express the fact that if a train may be
present, it is unsafe to cross the tracks. In the McDermott-Doyle formalism,
this might be expressed
on(train, tracks) ⊃ ¬saf e-to-
where the properties of the predicate on are supposed expressed in a formula
that we may call Axiom(on). The M in (1) stands for “is possible”. We
propose to replace (1) and Axiom(on) by the schema
Axiom(Φ) ∧ Φ(train, tracks) ⊃ ¬saf e-to-
where Φ is a predicate parameter that can be replaced by any predicate
expression that can be written in the language being used. If we can ﬁnd
a Φ that makes the left hand side of (2) provable, then we can be sure
that Axiom(on) together with
on(train, tracks) has a model assuming that
Axiom(on) is consistent. Therefore, the schema (2) is essentially a conse-
quence of the McDermott-Doyle formula (1). The converse isn’t true. A
predicate symbol may have a model without there being an explicit formula
realizing it. I believe, however, that the schema is usable in all cases where
the McDermott-Doyle or Reiter formalisms can be practically applied, and,
in particular, to all the examples in their papers.
(If one wants a counter-example to the usability of the schema, one might
look at the membership relation of set theory with the ﬁnitely axiomatized
G¨odel-Bernays set theory as the axiom. Instantiating Φ in this case would
amount to giving an internal model of set theory, and this is possible only in
a stronger theory).
It appears that such use of schemata amounts to importing part of the
model theory of a subject into the theory itself.
It looks useful and even
essential for common sense reasoning, but its logical properties are not obvi-
We can also go frankly to second order logic and write
∀Φ.(Axiom(Φ) ∧ Φ(train, tracks) ⊃ ¬saf e-to-
Second order reasoning, which might be in set theory or a formalism
admitting concepts as objects rather than in second order logic, seems to
have the advantage that some of the predicate and function symbols may be
left ﬁxed and others imitated by predicate parameters. This allows us to say
something like, “For any interpretation of P and Q satisfying the axiom A, if
there is an interpretation in which R and S satisfy the additional axiom A(cid:48),
then it is unsafe to cross the tracks”. This may be needed to express common
sense nonmonotonic reasoning, and it seems more powerful than any of the
above-mentioned nonmonotonic formalisms including circumscription.
The train example is a nonnormal default in Reiter’s sense, because we
cannot conclude that the train is on the tracks in the absence of evidence to
the contrary. Indeed, suppose that we want to wait for and catch a train at
a station across the tracks. If there might be a train coming we will take a
bridge rather than a shortcut across the tracks, but we don’t want to jump
to the conclusion that there is a train, because then we would think we were
too late and give up trying to catch it. The statement can be reformulated
as a normal default by writing
cross(tracks) ⊃ ¬saf e-to-
but this is unlikely to be equivalent in all cases and the nonnormal expression
seems to express better the common sense facts.
Like normal defaults, circumscription doesn’t deal with possibility di-
rectly, and a circumscriptive treatment of the train problem would involve
circumscribing saf e-to-
cross(tracks) in the set of axioms. It therefore might
not be completely satisfactory.
Addendum to References
McDermott, Drew and Jon Doyle (1980). Nonmonotonic Logic I, Artiﬁcial
Intelligence 13 (1, 2), pp. 41–72.
Reiter, Raymond (1980). A Logic for Default Reasoning, Artiﬁcial Intelli-
gence 13 (1, 2), pp. 81–132.