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CIRCUMSCRIPTION—A FORM OF

NONMONOTONIC REASONING

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

jmc@cs.stanford.edu

http://www-formal.stanford.edu/jmc/

Abstract

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

1http://www-formal.stanford.edu/jmc/mcc59.html

again decide what to do. This requires representing both knowledge about

the particular situation and general common sense knowledge as sentences

of logic.

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

stated.

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

If the

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

(331,220,321,300,311,110,221,020,031,010,021,000).

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

all.

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

must try.

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-

TION

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)

(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

must coincide.

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

applications.

Consider the following examples:

Example 1. In the blocks world, the sentence A may be

isblock A ∧ isblock B ∧ isblock C

(2)

asserting that A, B and C are blocks. Circumscribing isblock in (2) gives

the schema

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

this gives

∀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).

(3)

(4)

(5)

Example 2. Circumscribing the disjunction

leads to

isblock A ∨ isblock B

(6)

(Φ(A) ∨ Φ(B)) ∧ ∀x.(Φ(x) ⊃ isblockx) ⊃ ∀x.(isblock x ⊃ Φ(x)).

(7)

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

and

which simpliﬁes to

(9), (11) and (6) yield

isblock A ⊃ ∀x.(isblock x ⊃ x = A)

(9)

(B = A ∨ B = B) ∧ ∀x.(x = B ⊃ isblock x) ⊃ ∀x.(isblock x ⊃ x = B), (10)

isblock B ⊃ ∀x.(isblock x ⊃ x = B).

(11)

∀x.(isblock x ⊃ x = A) ∨ ∀x.(isblock x ⊃ x = B),

(12)

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).

(13)

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)).

(15)

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

another predicate `above(x, y, s)`

which asserts that block x is above block y

in situation s. We may write

∀xys.(`on(x, y, s)`

⊃ `above(x, y, s)`

)

(16)

and

∀xyzs.(`above(x, y, s)`

∧ `above(y, z, s)`

⊃ `above(x, z, s)`

),

(17)

i.e. above is a transitive relation. Circumscribing above in (16)∧(17) gives

∀xys.(`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)`

)

⊃ ∀xys.(`above(x, y, s)`

⊃ Φ(x, y, s))

(18)

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),

(19)

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)).

(20)

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

CIRCUMSCRIPTION

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

M .

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-

tioned earlier.

7 MORE ON BLOCKS

The axiom

∀xys.(∀z.¬`prevents(z, move(x, y), s)`

⊃ `on(x, y, result(move(x, y), s))`

)

(21)

states that unless something prevents it, x is on y in the situation that results

from the action `move(x, y)`

.

We now list various “things” that may prevent this action.

∀xys.(¬isblock x ∨ ¬isblock y ⊃ prevents(N ON BLOCK, `move(x, y)`

, s))

∀xys.(¬`clear(x, s)`

∨ ¬`clear(y, s)`

⊃ prevents(COV ERED, `move(x, y)`

, s))

(22)

(23)

∀xys.(tooheavyx ⊃ `prevents(weightx, move(x, y), s)`

).

(24)

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

the action `move(A, C)`

would have the desired eﬀect, so it must try to estab-

lish ∀z.¬`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))

∧∀z.(Φ(z) ⊃ `prevents(z, move(A, C), s0)`

)

⊃ ∀z.(`prevents(z, move(A, C), s0)`

⊃ Φ(z))

(25)

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

⊃ ∀z.¬`prevents(z, move(A, C), s0)`

.

(26)

We suppose that the premisses of this implication are to be obtained as

follows:

- isblock A and isblock B are explicitly asserted.
- Suppose that the only onness assertion explicitly given for situation
s0 is
`on(A, B, s0)`

. Circumscription of λx y.`on(x,y,s0)`

in this assertion givesΦ(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

∀xy.(

`on(x, y, s0)`

⊃ x = A ∧ y = B).Using

∀xs.(

`clear(x, s)`

≡ ∀y.¬`on(y, x, s)`

)as the deﬁnition of clear yields the second two desired premisses.

(28)

(29)

- ¬
`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 wouldestablish it.

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

an axiom.

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.

- The default case reasoning provided in many systems is less general
than circumscription. Suppose, for example, that a block x is considered to
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.

- The conjunct ∀¯x.(Φ(¯x) ⊃ P (¯x)) in the premiss of (1) is the result
of suggestions by Ashok Chandra (1979) and Patrick Hayes (1979) whom
I thank for their help. Without it, circumscribing a disjunction, as in the

second example in Section 4, would lead to a contradiction.

- The most direct way of using circumscription in AI is in a heuristic
reasoning program that represents much of what it believes by sentences of
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.

- In circumscription it does no harm to take irrelevant facts into account.
If these facts do not contain the predicate symbol being circumscribed, they
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 can be used in other formalisms than ﬁrst order logic.
Suppose for example that a set a satisﬁes a formula A(a) of set theory. The
circumscription of this formula can be taken to be

∀x.(A(x) ∧ (x ⊂ a) ⊃ (a ⊂ x)).

(30)

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.

- The results of circumscription depend on the set of predicates used
to express the facts. For example, the same facts about the blocks world
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.

- This investigation was supported in part by ARPA Contract MDA-
903-76-C-0206, ARPA Order No. 2494, in part by NSF Grant MCS 78-
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.

9 References

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

1990).

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).

2http://www-formal.stanford.edu/jmc/mcchay69.html

3http://www-formal.stanford.edu/jmc/ascribing.html

McCarthy, John (1979b). First Order Theories of Individual Concepts and

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 OTHER NONMONOTONIC FORMALISMS
ADDENDUM:
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

(1)

M

`on(train, tracks)`

⊃ ¬saf e-to-`cross(tracks)`

,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

(2)

Axiom(Φ) ∧ Φ(train, tracks) ⊃ ¬saf e-to-

`cross(tracks)`

,4http://www-formal.stanford.edu/jmc/concepts.html

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 thatAxiom(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-

ous.

We can also go frankly to second order logic and write

∀Φ.(Axiom(Φ) ∧ Φ(train, tracks) ⊃ ¬saf e-to-

`cross(tracks)`

).(31)

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

M¬saf e-to-

`cross(tracks)`

⊃ ¬saf e-to-`cross(tracks)`

,(32)

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 mightnot 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.