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ELABORATION TOLERANCE

John McCarthy

Computer Science Department

Stanford University

Stanford, CA 94305

jmc@cs.stanford.edu

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

2003 Sep 29, 11:54 p.m.

Abstract

A formalism is elaboration tolerant to the extent that it is conve-

nient to modify a set of facts expressed in the formalism to take into

account new phenomena or changed circumstances. Representations

of information in natural language have good elaboration tolerance

when used with human background knowledge. Human-level AI will

require representations with much more elaboration tolerance than

those used by present AI programs, because human-level AI needs to

be able to take new phenomena into account.

The simplest kind of elaboration is the addition of new formulas.

We’ll call these additive elaborations. Next comes changing the values

of parameters. Adding new arguments to functions and predicates

represents more of a change. However, elaborations not expressible as

additions to the object language representation may be treatable as

additions at a meta-level expression of the facts.

Elaboration tolerance requires nonmonotonic reasoning. The elab-

orations that are tolerated depend on what aspects of the phenomenon

are treated nonmonotonically. Representing contexts as objects in a

logical formalism that can express relations among contexts should

also help.

We use the missionaries and cannibals problem and about 20 vari-

ants as our Drosophila in studying elaboration tolerance in logical AI.

The present version has only some parts of a situation calculus

formalization. However, the English language elaborations listed are

enough to serve as a challenge to logical AI formalisms claiming elab-

oration tolerance.

1 Introduction

In several papers, e.g. [McC88] and [McC89], I discussed the common sense

informatic situation and contrasted it with the information situation within

a formal scientific theory. In the latter, it is already decided what phenomena

to take into account. In the former, any information possessed by the agent

is available and potentially relevant to achieving its goals.

Elaboration

tolerance seems to be the key property of any formalism that can represent

information in the common sense informatic situation.

Elaboration tolerance 1 is the ability to accept changes to a person’s or a

computer program’s representation of facts about a subject without having

to start all over. Often the addition of a few sentences describing the change

suffices for humans and should suffice for computer programs.

Humans have considerable elaboration tolerance, and computers need it

to reach human-level AI. In this article we study elaboration tolerance in

terms of logical AI. However, researchers pursuing other AI methodologies

will also have to face the problem of elaboration tolerance; maybe they just

haven’t noticed it yet. The relation to belief revision will be discussed briefly

in section 8.

Humans represent information about the world in natural language and

use background knowledge not ordinarily expressed in natural language and

2 The combination of linguistic and

which is quite difficult to express.

non-linguistic knowledge is what gives us humans our elaboration tolerance.

Unfortunately, psychological research hasn’t yet led to enough understanding

of the background knowledge, so it is hard to study elaboration tolerance in

humans. However, it is easy to give plenty of examples of human elaboration

tolerance, e.g. those in this article.

1The concept was first mentioned in [McC88].

2The non-linguistic background knowledge has been emphasized in connection with

physical skills by Hubert Dreyfus and others [Dre92], but there is important non-linguistic

knowledge also when the skill is purely symbolic. Even though a mathematician or a stock

broker operates in a purely symbolic domain, he still cannot verbalize his full set of skills.

The Drosophila 3 of this article is the missionaries and cannibals problem

(MCP).

After describing the original MCP, we give a large number of elaborations.

Humans tolerate these elaborations in the sense that we can use the sentences

expressing one of the elaborations to get a modified problem. People will

agree on what the modified problem is and will agree on whether a proposed

solution is ok.

Then we consider logical formalizations of the original problem and dis-

cuss which elaborations different formalisms tolerate. Our goal—not achieved

in this article—is a formalism for describing problems logically that is as

elaboration tolerant as English and the associated background knowledge.

However, some logical languages are more elaboration tolerant than others.

2 The Original Missionaries and Cannibals

Problem

The missionaries and cannibals problem (abbreviated MCP):

Three missionaries and three cannibals come to a river

and find a boat that holds two. If the cannibals ever out-

number the missionaries on either bank, the missionaries

will be eaten.

How shall they cross?

We call this original version of the problem MCP0.

Saul Amarel proposed [Ama71]: Let a state (mcb) be given by the num-

bers of missionaries, cannibals and boats on the initial bank of the river. The

initial situation is represented by 331 and the goal situation by 000.

Most AI texts that mention the problem accept this formulation and give

us the solution:

331 → 310 → 321 → 300 → 311 → 110 → 221 → 020 → 031 → 010 → 021 → 000.3Drosophilas are the fruit flies that have been used by geneticists to study inheritance

since 1910. Their short generation times, large chromosomes and the ability to keep 1,000

of them in a bottle make them valuable, even though the Drosophilas of today are no

better than those of 1910. The utility of suitable Drosophilas for scientific research in AI

needs to be emphasized, because of a recent fad for demanding that all research promise

a practical payoff on a three year schedule. They aren’t getting their payoffs and are

learning much less than a more scientific approach would get them.

331 → 310 → 321 → 300 → 311 → 110 → 221 → 020 → 031 → 010 → 021 → 000.The state space of the Amarel repreentation has 32 elements some of

which are forbidden and two of which are unreachable. It is an elementary

student exercise to write a program to search the space and get the above

sequence of states, and people are always solving it without a computer or

without even a pencil. Saul Amarel [Ama71] points out that this represen-

tation has fewer states than a representation with named missionaries and

cannibals.

What more does this problem offer AI?

If one indeed begins with the Amarel representation, the problem is indeed

trivial. However, suppose we want a program that begins, as people do,

with a natural language presentation of the problem.

It is still trivial if

the program need only solve the missionaries and cannibals problem. The

programmer can then cheat as much as he likes by making his program

exactly suited to the MCP. The extreme of cheating is to make a program

that merely prints

Readers will rightly complain that this cheats, but it isn’t clear what does

and doesn’t count as cheating when a method for solving a single problem is

asked for.

The way to disallow cheating is to demand a program that can solve any

problem in a suitable set of problems. To illustrate this we consider a large

set of elaborations of MCP. It won’t be trivial to make a program that can

solve all of them unless the human sets up each of them as a state space

search analogous to the original MCP. We demand that the program use

background common sense knowledge like that about rivers and boats that

is used by a human solver.

We skip the part about going from an English statement of the problem

to a logical statement for two reasons. First, we don’t have anything new

to say about parsing English or about the semantics of English. Second, we

don’t yet have the logical target language that the parsing program should

aim at. Progress toward establishing this language is the goal of the paper.

The problem is then to make a program that will solve any of the problems

using logically expressed background knowledge. The background knowledge

should be described in a general way, not specifically oriented to MCP and

related problems.

This much was already proposed in [McC59]. What is new in the present

paper is spelling out the idea of elaboration tolerance that was distantly

implicit in the 1959 paper. We require a formulation of MCP that readily

tolerates elaborations of the problem and allows them to be described by

sentences added to the statement of the problem rather than by surgery

on the problem. We can call these additive elaborations. English language

formulations allow this, but the Amarel-type formulations do not. AI requires

a logical language that allows elaboration tolerant formulations.

We begin a few examples of English language elaboration tolerance. After

discussing situation calculus formalisms, there will be a lot more.

• The boat is a rowboat. (Or the boat is a motorboat). This elaboration

by itself should not affect the reasoning. By default, a tool is usable.

Later elaborations make use of specific properties of rowboats.

• There are four missionaries and four cannibals. The problem is now

unsolvable.

• There is an oar on each bank. One person can cross in the boat with

just one oar, but two oars are needed if the boat is to carry two people.

• One of the missionaries is Jesus Christ. Four can cross. Here we are

using cultural literacy. However, a human will not have had to have

read Mark 6: 48–49 to have heard of Jesus walking on water.

• Three missionaries with a lone cannibal can convert him into a mis-

A later section discusses the formal problems of these and other elabora-

sionary.

tions.

3 Nonmonotonic reasoning

Elaboration tolerance clearly requires nonmonotonic reasoning. For exam-

ple, elaborating MCP0 with a requirement for oars adds preconditions to the

action of going somewhere in the boat. If oars are not mentioned, nonmono-

tonic reasoning prevents such additional preconditions.

However, it is still not clear how to formulate the nonmonotonic reasoning

so as to obtain tolerance of a wide class of elaborations, such as those of

section 7. We propose to use some variant of circumscription, but this still

leaves open what is to be circumscribed.

[McC80] discusses several nonmonotonic aspects of the human under-

standing of MCP0. They all have a Gricean [Gri89] character. They all

concern the non-existence of features of the problem that should have been

mentioned, were they supposed to exist. What can be inferred from such

contexts includes the Gricean implicatures. Very likely, the formal theory of

contexts [McC93] can be used, but that is beyond the scope of this article.

Here are a few nonmonotonic inferences that come up. Each of them

seems to present its own formal problems.

• If there were a bridge, it should have been mentioned. A puzzle problem

like MCP is given in a context.

• The river can’t be forded, and there isn’t an extra boat.

• There isn’t a requirement for a permit or visa to cross the river.

• There is nothing wrong with the boat.

In general, when a tool is

mentioned, it is supposed to be usable in the normal way.

• The group of missionaries and cannibals is minimized. Mentioning

that one of the missionaries is Jesus Christ will include him in the

number otherwise inferrable rather than adding him as an additional

missionary. Of course, assertions about him in the general database

should have no effect unless he is mentioned in the problem.

• If you keep transporting cannibals to the island (in the variant with an

island) they will eventually all be at the island.

These kinds of nonmonotonic reasoning were anticipated in [McC80] and

have been accomodated in situation calculus based nonmonotonic formalisms,

although the Yale shooting problem and others have forced some of the ax-

iomatizations into unintuitive forms.

The elaborations discussed in this article mainly require the same kinds

of nonmonotonic reasoning.

4 A Typology of Elaborations

There are many kinds of elaborations a person can tolerate, and they pose

different problems to different logical formalizations. Here are some kinds of

elaborations.

irrelevant actors, actions and objects Sentences establishing the exis-

tence of such entities should not vitiate the reasoning leading to a

solution.

adding preconditions, actions, effects of actions and objects The ex-

ample of the oars adds a precondition to rowing and adds the action

of picking up the oars. Several situation calculus and event calculus

formalisms allow this—assuming sentences are added before the non-

monotonic reasoning is done. Tolerating added preconditions is a cri-

terion for good solutions of the qualification problem, and tolerating

adding effects relates similarly to the ramification problem.

changing a parameter This is needed when the numbers of missionaries

and cannibals are changed from 3 to 4. In English, this is accomplished

by an added sentence. Doing it that way in logic requires a suitable

belief revision method as part of the basic logical formalism. At present

we must use minor brain surgery to replace certain occurrences of the

number 3.

making a property situation dependent Whether x is a missionary is

not situation dependent in MCP0, but we can elaborate to a missionary

becoming a cannibal. It is tempting to say that all properties should be

situation dependent from the beginning, and such a formalism would

admit this elaboration easily.

I think this might lead to an infinite

regress, but I can’t formulate the problem yet.

specialization In one situation calculus formalization we have the action

M ove(b1, b2).

If there are guaranteed to be exactly two places, we

can replace this action by M ove(b), regarding this as M ove(b, Opp(b)),

where Opp(b) designates the opposite bank and satisfies Opp(Opp(b)) =

b. We regard this kind of specialization as an easy kind of elaboration.

generalization Some of our elaborations can be composed of an general-

ization of the language—replacing a function by a function of more

arguments, e.g. making whether a person is a cannibal or missionary

situation dependent or replacing going from a bank b to the opposite

bank Opp(b) by going from b1 to b2. Many elaborations consist of a

generalization followed by the addition of sentences, e.g. adding pre-

conditions or effects to an action.

unabbreviation This is a particular case of generalization. Suppose we

write (∀a ∈ Actions)Abbreviates[a, Does(person, a)]. We mean to use

it in elaborating Result(a, s) to Result(Does(person, a), s) in sentences

where it is somehow clear which person is referred to. The square brack-

ets mean that this is a metalinguistic statement, but I don’t presently

understand precisely how unabbreviation is to work.

going into detail An event like the action of crossing the river is made up

of subactions. However, the relation between an event and its subevents

is not often like the relation between a program and its subroutines,

because asserting that the action occurs does not imply specific subac-

tions. Rowing is a detail of crossing a river when rowboats are used,

but rowing is not a part of the general notion of crossing a river.. Bail-

ing if necessary is another detail. Getting oars or a bailing can are

associated details. There is more about this apparently controversial

point in [McC95].

m. . . s and c. . . s as actors MCP and almost all of the elaborations we

have considered take a god-like view of the actions, e.g. we send a

cannibal to get an oar. We can also elaborate in the direction of

supposing that the actions of cannibals and missionaries are some-

times determined by the situation.

In this case, it may be conve-

nient to use a predicate Occurs(event, s) and let one possible event

be Does(C1, Enter-Boat). The situation calculus treatment has to be

altered and looks more like event calculus.

simple parallel actions If one of the missionaries is Jesus Christ, we can

transport 4 missionaries and 4 cannibals. We get 3 cannibals on the far

bank and one on the initial bank. Then two ordinary missionaries and

Jesus cross, the ordinaries in the boat and Jesus walking on water. The

rest of the solution is essentially the same as in MCP0. A missionary

and a cannibal row back and now the remaining two missionaries cross.

We then send a cannibal to ferry the remaining two cannibals. We

haven’t tackled the problem of being able to say “essentially the same”

in logic.

The formalization must permit Jesus to cross in parallel with the other

missionaries so that the missionaries are never outnumbered. This isn’t

the same as having Jesus cross as a separate action.

full parallelism This is what permits requiring that the boat be bailed.

events other than actions The simple Result(a, s) doesn’t allow for events

other than actions. To handle them we have used [McC95] a predicate

Occurs(e, s) asserting that the event e occurs in the situation s. Then

Result(e, s) can be used.

comparing different situations This works ok in situation calculus, but

some other formalisms don’t allow it or make it awkward. Thus we

can have s <better Result(e, s) to say that the situation is better after

event e occurs. We may also want Result(a1, s) <better Result(a2, s),

comparing the result of doing a1 with the result of doing a2.

splitting an entity Sometimes an entity, e.g. a node in a graphy, an edge,

or a concept needs to be split into two entities of the same type so that

separate properties can be assigned to each subentity. Thus we may

split cannibals into strong and weak cannibals.

continuous time and discrete time If Achilles runs enough faster than

the tortoise, there is a time when Achilles catches up. We use the

fluent F uture(π, s) to assert that the situation s will be followed in the

future by a situation satisfying π. We do not require formalizing real

numbers to express the Achilles catching up sentence

F uture((λs)(V alue(Distance-covered-by(Achilles), s)

= V alue(Distance-covered-by(T ortoise), s)), S0).

5 Formalizing the Amarel Representation

We use logic and set theory to formalize MCP0 and call the formalization

MCP0a.

In this formalization we are not concerned with elaboration tol-

erance. My opinion is that set theory needs to be used freely in logical

AI in order to get enough expressiveness. The designers of problem-solving

programs will just have to face up to the difficulties this gives for them.

States = Z4 × Z4 × Z2

(∀state)(Ok(state) ≡

Ok1(P 1(state), P 2(state)) ∧ Ok1(3 − P 1(state), 3 − P 2(state)))

Here Z2 = {0, 1} and Z4 = {0, 1, 2, 3} are standard set-theory names

for the first two and the first four natural numbers respectively, and P 1,

P 2 and P 3 are the projections on the components of the cartesian product

Z4 × Z4 × Z2.

Note that having used 3 − P 1(state) for the number of missionaries on

the other bank put information into posing the problem that is really part

of solving it, i.e. it uses a law of conservation of missionaries.

(∀m c)(Ok1(m, c) ≡ m ∈ Z4 ∧ c ∈ Z4 ∧ (m = 0 ∨ m ≥ c))

M oves = {(1, 0), (2, 0), (0, 1), (0, 2), (1, 1)}

(∀move state)

(Result(move, state) = M kstate( P 1(state) − (2P 3(state) − 1)P 1(move),P 2(state) − (2P 3(state) − 1)P 2(move),1 − P 3(state))),

where M kstate(m, c, b) is the element of States with the three given compo-

nents.

(∀s1 s2)(Step(s1, s2) ≡ (∃move)(s2 = Result(move, s1) ∧ Ok(s2)))

Attainable1 = T ransitive-closure(Step)

Attainable(s) ≡ s = (3, 3, 1) ∨ Attainable1((3, 3, 1), s)

Notice that all the above sentences are definitions, so there is no question

of the existence of the required sets, functions, constants and relations. The

existence of the transitive closure of a relation defined on a set is a theorem

of set theory. No fact about the real world is assumed, i.e. nothing about

rivers, people, boats or even about actions.

From these we can prove

attainable((0, 0, 0)).

The applicability of MCP0a to MCP must be done by postulating a cor-

respondence between states of the missionaries and cannibals problem and

states of MCP0a and then showing that actions in MCP have suitable corre-

sponding actions in MCP0a. We will postpone this until we have a suitable

elaboration tolerant formalization of MCP.

MCP0a has very little elaboration tolerance, in a large measure because

of fixing the state space in the axioms. Situation calculus will be more

elaboration tolerant, because the notation doesn’t fix the set of all situations.

6 Situation Calculus Representations

The term situation calculus is used for a variety of formalisms treating situa-

tions as objects, considering fluents that take values in situations, and events

(including actions) that generate new situations from old.

At present I do not know how to write a situation calculus formalization

that tolerates all (or even most) of the elaborations of section 7. Nevertheless,

I think it is useful to give some formulas that accomplish some elaborations

and discuss some issues of elaboration tolerance that these formulas present.

6.1 Simple situation calculus

We begin with some axioms in a formalism like that of [McC86] using a Result

function, a single abnormality predicate and aspects. This suffers from the

Yale shooting problem if we simply minimize Ab. However, as long as the

problem requires only projection, i.e. predicting the future from the present

without allowing premises about the future, chronological minimization of

Ab [Sho88] avoids the Yale shooting problem. It is certainly a limitation on

elaboration tolerance to not allow premises about the future.

Here are some axioms and associated issues of elaboration tolerance.

The basic operation of moving some people from one bank to the other

is conveniently described without distinguishing between missionaries and

cannibals.

¬Ab(Aspect1(group, b1, b2, s)) →

V alue(Inhabitants(b1), Result(Cross(group, b1, b2), s)) =

V alue(Inhabitants(b1), s) \ group

(1)

V alue(Inhabitants(b2), Result(Cross(group, b1, b2), s)) =

V alue(Inhabitants(b2), s) ∪ group,

where \ denotes the difference of sets.

The fact that (1) can’t be used to infer the result of moving a group if

some member of the group is not at b1 is expressed by

¬(group ⊂ V alue(Inhabitants(b1), s)) → Ab(Aspect1(group, b1, b2, s)).

We extend the notion of an individual being at a bank to that of a group

being at a bank.

Holds(At(group, b), s) ≡ (∀x ∈ group)Holds(At(x, b), s).

(2)

¬Ab(Aspect2(group, b1, b2, s)) ∧ Crossable(group, b1, b2, s)

→ ¬Ab(Aspect1(group, b1, b2, s))

relates two abnormalities.

(3)

Crossable(group, b1, b2, s) → 0 < Card(group) < 3

tells us that the boat can’t cross alone and can’t hold more than two.

Card(u) denotes the cardinality of the set u.

We can sneak in Jesus by replaceing (3) by

(4)

Crossable(group, b1, b2, s) → 0 < Card(group \ {Jesus}) < 3,

but this is not in the spirit of elaboration tolerance, because it isn’t an added

sentence but is accomplished by a precise modification of an existing sentence

(3) and depends on knowing the form of (3). It’s education by brain surgery.

It’s bad if the cannibals outnumber the missionaries.

Holds(Bad(bank), s)

0 < Card({x|x ∈ M issionaries ∧ Holds(At(x, bank), s)})

< Card({x|x ∈ Cannibals ∧ Holds(At(x, bank), s)})

(6)

Holds(Bad, s) ≡ (∃bank)Holds(Bad(bank), s).

Many unique names axioms will be required. We won’t list them in this

6.2 Not so simple situation calculus

The notion of Bad in the previous subsection avoids any actual notion of the

missionaries being eaten. More generally, it avoids any notion that in certain

situations, certain events other than actions will occur. We can put part of

this back.

We would like to handle the requirement for oars and the ability of Jesus

Christ to walk on water in a uniform way, so that we could have either, both

or neither of these elaborations.

(5)

and

version.

To say that the missionaries will be eaten if the cannibals outnumber

them can be done with the formalism of [McC95].

Holds(Bad(bank), s) →

(7)

(∀x)(x ∈ M issionaries

∧Holds(At(x, bank), s) → Occurs(Eaten(x), s)).

As sketched in [McC95], the consequences of the occurrence of an event

may be described by a predicate F uture(f, s), asserting that in some situ-

ation in the future of the situation s, the fluent f will hold. We can write

this

(8)

F uture(f, s) → (∃s0)(s <time s0 ∧ Holds(f, s0)),

and treat the specific case by

(9)

occurs(Eaten(x), s) → F (Dead-soon(x), s)

To say that something will be true in the future of a situation is more

general than using Result, because there is no commitment to a specific next

situation as the result of the event. Indeed an event can have consequences

at many different times in the future. The Result(e, s) formalism is very

convenient when applicable, and is compatible with the formalism of Occurs

and F . We have

(10)

¬Ab(Aspect2(e, s))∧Occurs(e, s) → F uture((λs0)(s0 = Result(e, s)), s),

where something has to be done to replace the lambda-expression (λs0)(s0 =

Result(e, s)) by a syntactically proper fluent expression. One way of doing

that is to regard Equal(Result(e, s)) as a fluent and write

(11)

¬Ab(Aspect2(e, s)) ∧ Occurs(e, s) → F uture(Equal(Result(e, s))).

We may get yet more mileage from the Result formalism. Suppose

Result(e, s) is taken to be a situation after all the events consequential

to e have taken place. We then have one or more consequences of the

form P ast(f, Result(e, s)), and these permit us to refer to the consequences

of e that are distributed in time. The advantage is that we can the use

Result(e, s) as a base situation for further events.

6.3 Actions by Persons and Joint Actions of Groups

When there is more than one actor acting, we can consider three levels of

complexity. The simplest level is when the actors act jointly to achive the

goal. The second level is when one actor (or more than one) does something

to motivate the others, e.g. one person pays another to do something. This

generalizes to a hierarchy of influence. The hard level is when the actors

have competing motivations and must negotiate or fight. This is the subject

of game theory, and we won’t pursue it in this article.

As MCP was originally formulated, the missionaries and cannibals are

moved like pieces on a chessboard. Let’s consider elaborations in which the

actions of individual missionaries and cannibals are considered. One eventual

goal might be to allow a formalization in which a cannibal has to be persuaded

to row another cannibal across the river and bring the boat back. However,

our discussion starts with simpler phenomena.

We now consider an action by a person as a particular kind of event.

What we have written Result(a, s) we now write Result(Does(person, a), s).

If there is only one person, nothing is gained by the expansion.

Consider a proposition Can-Achieve(person, goal, s), meaning that the

person person can achieve the goal goal starting from the situation s. For the

time being we shall not say what goals are, because our present considerations

are independent of that decision. The simplest case is that there is a sequence

of actions {a1, . . . , an} such that

Result(Does(person, an), Result(. . . Result(Does(person, a1), s) . . .))

satisfies goal.

Now let’s consider achievement by a group. We will say Can-Achieve(group, goal, s)provided there is a sequence of events {Does(person1, a1), . . . , Does(personn, an)},where each personi is in group, and the personis are not assumed to be dis-

tinct, and such that

Result(Does(personn, an), Result(. . . Result(Does(person1, a1), s) . . .))

satisfies goal.

We can now introduce a simple notion of a person leading a group, written

leads(person, group) or more generally leads(person, group, s). We want the

axioms

leads(person, group) ∧ Can-Achieve(group, goal, s) → Can-Achieve(person, s)Thus a leader of a group can achieve whatever the group can achieve.

Note that person need not be a member of group for this definition to work.

We could give the same definition for leads(person, group, s), but maybe

it would be better to make a definition that requires that person maintain

his leadership of group in the succeeding situations.

Leads(person, group) is too strong a statement in general, because the

members of a group only accept leadership in some activities.

7 Formalizing some elaborations

  1. The boat is a rowboat. (Or the boat is a motorboat). By itself this is a trivial elaboration. Adding it should not affect the reasoning. By

    default, a tool, i.e. the boat, is usable. Further elaborations might use

    specific properties of rowboats.

  2. The missionaries and cannibals have hats, all different—another trivial elaboration. These hats may be exchanged among the missionaries and

    cannibals. In all the elaborations mentioned below, exchanging hats is

    an action irrelevant to crossing the river. There are two demands on

    the reasoner. Epistemologically, whatever reasoning that establishes a

    plan for crossing the river without the hats should be valid with the

    hats. This includes any nonmonotonic reasoning.

    Heuristically, the problem may not be trivial. Why should it be obvious

    that exchanging hats is of no use? Certainly we can make elaborations

    in which it is of use, e.g. we can assert that if the smallest missionary

    wears the hat belonging to the largest missionary, the largest cannibal

    won’t eat him even if they go together.

    However, it should be possible to tell a problem solver: Look for a

    solution that has no hat change actions. After that, the reasoner should

    find the solution as easily as it would if hats were never mentioned.

  3. There are four missionaries and four cannibals. The problem is now unsolvable. In ordinary logic, adding sentences that there are four of

    each produces a contradiction. Belief revision systems ought to make

    the correct change. It seems to me that people take a metalinguistic

    stance, just saying “Change the numbers of missionaries and cannibals

    to four”, thus regarding the original statement of the problem as an

    object. Actually what is regarded as an object is the sense of the

    original statement, since people ordinarily don’t remember the words

    used.

    Proofs of impossibility take the following form. Choose a predicate

    formula φ(s) on situations. Show φ(S0) and (∀s)(φ(s) → ¬Goal(s)).

    Also show

    (∀s a)(φ(s) → Bad(Result(a, s)) ∨ φ(Result(a, s))).

    Thus you can’t get out of the situations satisfying φ, and the goal

    isn’t included. The simplest φ(s) is a disjunction of specific locations

    of the missionaries and cannibals in the reachable situations, but this

    disjunction is long, and it is very likely possible to do better.

    We can regard the argument that four can’t cross as a kind of elabo-

    ration. A formalism that doesn’t permit expressing the best argument

    is then deficient in elaboration tolerance.

  4. The boat can carry three. Four can cross but not five. If the boat can carry four an arbitrary number can cross. [2003 Sept: This is mistaken.

    Joohyung Lee showed that if the boat holds three, five can cross.]

  5. There is an oar on each bank. One person can cross in the boat with just one oar, but two oars are needed if the boat is to carry two people.

    We can send a cannibal to get the oar and then we are reduced to the

    original problem. 4

    A formalism using preconditions can accept this elaboration as just

    adding a precondition for rowing, the action of putting an oar in the

    boat and adding facts about the locations of the oars in S0.

    The oar-on-each-bank elaboration can be expressed by conjoining to

    (12),

    Card(group) > Card({x|Oar(x) ∧ Holds(In(x, Boat), s)})

    → Ab(Aspect1(group, b1, b2, s)),

    4It was not mentioned before that the boat was a rowboat. Once oars are mentioned,

    it is a Gricean implicature that the boat is a rowboat. The philosopher Paul Grice [Gri89]

    studied what can be inferred from statements under the assumption that the person posing

    the problem is not trying to be misleading. That the boat is a rowboat follows, because

    the speaker should have said so if it wasn’t.

    but this looks a bit ad hoc.

    In particular, it wouldn’t tolerate the

    further elaboration of making the boat hold three if that elaboration

    were expressed as the single sentence

    Crossable(group, b1, b2, s) → 0 < Card(group) < 4

    In order to admit the reasoning that getting the oar reduces the problem

    to MCP0, we will need a notion of one problem reducing to another—or

    one theory reducing to another.

  6. Only one missionary and one cannibal can row. The problem is still solvable. Before this elaboration, we did not need to distinguish among

    the missionaries or among the cannibals. An elaboration tolerant lan-

    guage must permit this as an addition. We use

    (12)

    ¬(∃x)(x ∈ group ∧ Rower(x)) → Ab(Aspect1(group, b1, b2, s)).

    and

    (13)

    (∃!x ∈ Cannibals)Rower(x) ∧ (∃!x ∈ M issionaries)Rower(x)

  7. The missionaries can’t row. This makes the problem impossible, since any solution requires two missionaries in the boat at some time. The

    formalism must admit the statement and proof of this lemma.

    For this we need (12) and (∀x ∈ M issionaries)¬Rower(x).

  8. The biggest cannibal cannot fit in the boat with another person. The problem is solvable. However, if the biggest missionary cannot fit in

    the boat with another person the problem becomes unsolvable. We

    can imagine having to elaborated in the direction of saying what sets

    of people can fit in the boat. The elaborations are BigC ∈ Cannibals

    and

    (14)

    Crossable(group) ∧ BigC ∈ group → group = {BigC}.

    Note that the defining property of the biggest cannibal is unnecessary

    to make the elaboration work.

    I assume we’d pay for this shortcut,

    were further elaboration necessary.

    The corresponding elaboration about the biggest missionary is formal-

    ized in the same way; only the conclusion is different.

  9. If the biggest cannibal is isolated with the smallest missionary, the lat- ter will be eaten. A solution to the basic problem can be specialized to

    avoid this contingency. We have the Gricean implicature that the can-

    nibals aren’t all the same size, and need to have language for referring

    to an individual as the biggest cannibal and not just language to refer

    to him by name. We have

    (15)

    group = {BigC, SmallM } → ¬Crossable(group, b1, b2, s),

    and

    (16)

    Inhabitants(bank, s) = {BigC, SmallM } → Holds(Bad(bank), s).10. One of the missionaries is Jesus Christ. Four can cross. Here we are

    using cultural literacy. However, a human will not have had to have

    read Mark 6: 48–49 to have heard of Jesus walking on water. The

    formalism of Section 6 permits this elaboration just by adjoining the

    sentence

    (17)

    Crossable(group, b1, b2, s) → Crossable(group∪{Jesus}, b1, b2, s).However, this elaboration says nothing about walking on water and

    therefore seems to be a cheat.

    11. Three missionaries alone with a cannibal can convert him into a mis-

    sionary. The problem for elaboration tolerance is to change a predicate

    that doesn’t depend on situation or time to one that does. Note that

    a sorted logical language with missionaries and cannibals as distinct

    sorts would freeze the intolerance into the language itself.

    12. The probability is 1/10 that a cannibal alone in a boat will steal it.

    We can ask what is the probability that a given plan will succeed, say

    the Amarel plan. The formalism of [McC79a] treats propositions as

    objects. Using that formalism P r(p) = 1/10 can be expressed for any

    proposition p. I see at least two problems. The language of propositions

    as objects needs to be rich enough to express notions like the probability

    of a cannibal stealing the boat on an occasion—or of being a thief who

    always steals boats if alone. The second problem is that we need to be

    able to assert independence or joint distributions without letting the

    entire formalism be taken over by its probabilistic aspects. In MCP0,

    cannibals have to be alone in the boat several times. We can write a

    formula that states that probabilities are independent by default.

    We now need to infer that the probability of successfully completing

    the task is 0.9.

    13. There is a bridge. This makes it obvious to a person that any number

    can cross provided two people can cross at once. It should also be an

    obvious inductive argument in the sense of McAllester [McA]. This

    is a straightforward elaboration in situation calculus formalisms, since

    adding the bridge is accomplished just by adding sentences. There is

    no need to get rid of the boat unless this is part of the elaboration

    wanted.

    14. The boat leaks and must be bailed concurrently with rowing. Elabo-

    ration tolerance requires that treating a concurrent action be a small

    change in the statement of the problem, and this will show the limita-

    tions of some versions of situation calculus.

    15. The boat may suffer damage and have to be taken back to the left

    bank for repair. This may happen at any time. This requires that

    the formalism permit splitting the event of crossing the river into two

    parts.

    16. There is an island. Then any number can cross, but showing it requires

    inductive arguments. Though inductive, these arguments should be

    obvious. Defining the three stages—moving the cannibals to the island,

    moving the missionaries to the opposite bank and then moving the

    cannibals to the opposite bank—is an easy three step problem, provided

    moving the sets of missionaries and cannibals can be regarded as tasks.

    Whether the elaboration is easy depends on the original representation.

    There may be a nonmonotonic rule that if you keep getting closer to

    a goal and there is no inferrable obstacle you will achieve the goal.

    Zeno’s “paradox” of Achilles and the tortoise involves noting that this

    rule doesn’t always hold, i.e. is nonmonotonic. Such a rule would make

    the above induction easy and maybe obvious.

    17. There are four cannibals and four missionaries, but if the strongest of

    the missionaries rows fast enough, the cannibals won’t have gotten so

    hungry that they will eat the missionaries. This could be made precise

    in various ways, but the information is usable even in vague form.5

    18. There are four missionaries and four cannibals, but the cannibals are

    not hungry initially, and the missionaries have a limited amount of

    cannibal food. They can tell if a cannibal is hungrier than he was

    and can avoid trouble by giving the food to the cannibal who has got

    hungrier. This requires comparing a situation and a successor situation.

    19. There are two sets of missionaries and cannibals too far apart along

    the river to interact. The two problem should be solvable separately

    without considering interleaving actions at the two sites.

    If the two

    problems are different elaborations, the work required and the length

    of the proof should be the sum of the lengths for the separate problems

    plus a small constant.

    The theory of two sets of missionaries should be a conservative exten-

    sion of each of the subtheories. We have called this property conjunc-

    tivity.

    There are N sites along the river with identical conditions. The rea-

    soning should be able to do one site, or a generalized site, and, with a

    constant amount of additional reasoning, say that all N crossings are

    the same.

    20. After rowing twice, a person becomes too tired to row any more. [Added

    2003 April 1].

    8 Remarks and Acknowledgements

  10. The English language elaborations don’t refer to an original English text. If someone has read about the problem and understands it, he

    usually won’t be able to quote the text he read. Moreover, if he tells

    the problem to someone else more than once, he is unlikely to use the

    same words each time. We conclude from this that that a person’s

    understanding of MCP is represented in the brain in some other way

    5“Pull, pull, my good boys”, said Starbuck.—Moby Dick, XLVIII

    than as an English text. For the purposes of this paper we don’t need

    to speculate about how it is represented, since the formal elaboration

    tolerance applies to logical formulations.

  11. Some commonly adopted conventions in theories of actions interfere with elaboration tolerance. An example is identifying situations or

    events with intervals of time. You can get away with it sometimes, but

    eventually you will be sorry. For example, you may want to say that a

    good move is one that leads to a better situation with

    Good(a, s) ≡ s <good Result(a, s).

  12. Elaboration tolerance and belief revision have much in common, but we are looking at the problem from the opposite direction from researchers

    in belief revision. Belief revision studies have mainly concerned the

    effect of adding or removing a given sentence, whereas our treatment

    of elaboration tolerance concerns what you must add or change to get

    the effect you want. Moreover, the effect of an elaboration can involve

    changing the first order language and not just replacing one expression

    in the language by another.

  13. Elaboration tolerance is rather straightforward when the theory to be changed has the structure of a cartesian product, and the elaboration

    can be describes as giving some components of the product new val-

    ues. [McC79b] discusses theories with cartesian product structures in

    connection with counterfactuals, and [McC62] discusses the semantics

    of assignment, i.e. the semantics of changing components of a state.

  14. Murray Shanahan [Sha97] considers many issues of elaboration toler- ance in his discussions of action formalisms. In particular, his solutions

    for the frame problem are considerably elaboration tolerant. I quali-

    fied the above, because I consider elaboration tolerance an open ended

    problem.

  15. I suspect that elaboration tolerance requires a proper treatment of hy- pothetical causality and this involves counterfactual conditional sen-

    tences. Counterfactuals will be treated in a shortly forthcoming paper

    by Tom Costello and John McCarthy. For example, we need a non-

    trivial interpretation of “If another car had come over the hill while

    you were passing, there would have been a head-on collision” that is

    compatible with the fact that no car came. By non-trivial interpreta-

    tion, I mean one that could have as a consequence that a person should

    change his driving habits, whereas no such conclusion can be reached

    from sentences of the form p → q when p is false.

  16. We can distinguish between a formalism admitting a particular elabora- tion and the consequences of the elaboration being entirely determined.

    For example, the Jesus Christ elaboration could be given alternate in-

    terpretations and not just the one about his ability to walk on water.

    Another example (suggested by Tom Costello) has the original story say

    that the capacity of the boat is one less than the number of missionaries.

    Then changing the number of missionaries and cannibals to 4 leaves the

    problem still solvable, even though the set of logical consequences of

    the sentences of the two formalisms is the same. This tells us that

    if we translate the English to logic and take all logical consequences,

    information that determines the effects of elaborations can be lost.

    This paper has benefitted from discussions with Eyal Amir, Tom Costello,

    Aarati Parmar and Josephina Sierra. The present version is somewhat im-

    proved from the version presented at Common Sense-98 in January 1998. It

    may be further improved without warning.

    The on-line version is http://www-formal.stanford.edu/jmc/elaboration.html.References

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