Representing Actions with an Assumption-Based Truth Maintenance

From: AAAI-86 Proceedings. Copyright ©1986, AAAI (www.aaai.org). All rights reserved.

Representing Actions with an Assumption-Based Truth Maintenance Paul

H. Morris

and Robert

IntelliCorp El Camino Real

1975

Mountain

View,

ABSTRACT

A.

the

94040 following

The

Assumption-based

modeling

Truth

Maintenance

System,

introduced

by

is a powerful new tool for organizing a search through a However, the ATMS is oriented towards alternatives. problem solving, and provides no special mechanisms for We describe an approach to actions or state changes.

the ATMS to the task of representing the effects of actions. context extends traditional tree-structured approach The mechanisms to allow context merges. It also takes advantage of the underlying ATMS to detect inconsistent contexts and to maintain Some results are presented concerning possible derived results. questionable merges in treatment of approaches to the

sections,

we give a functional overview of the We then describe the underlying representation in terms of the ATMS. Special attention is given to the situation where a world has multiple parents. This is followed by a discussion of non-monotonic reasoning about actions in a more KEEworlds

de Kleer, space of inferential

Nado

West

California In

System

facility.

general TMS setting, suggested by the worlds with some remarks about related systems.

mechanism.

We close

applying

Finally,

circumstances.

the

analysis

of actions

in terms

of a truth

2. Worlds The

basic

world

together

ordered

the need for a more elaborate maintenance system suggests treatment of contradiction in such systems than exists at present.

Assumption-Based

efficient

Truth Maintenance System (ATMS), new tool for organizing an [2], is a powerful By explicitly through a space of alternatives.

by de Kleer search

recording the dependence choices, a truth maintenance across

different

of

branches

the system

of the

reasoning steps is able to share

search

on individual partial results

In effect, knowledge gleaned in one context is automatically transfered to other contexts where it is relevant. The ATMS permits simultaneous reasoning about multiple, possibly conflicting contexts, avoiding the cost of context switching. The

ATMS

purely

as

presently

inferential.

constraint temporal

This

satisfaction changes

constituted

problems.

or actions

de Kleer [5] points out, world, and this cannot

views

is an appropriate

problem

stance

solving

for a broad

for

modeling

actions

its ancestors

in the graph

Each

successor

represents

conflicting,

thus

action

be regarded networks.

as representing

Each

partially

require

‘I... problem be modeled

This

KEEworldsTM

of

the

by

application somewhat operators

the

action.

of an operator,

not

Since an action corresponds to an operator itself, this assumption

less restrictive than that of STRIPS to have fixed add and delete lists.

[8], which

and

underlying contexts,

has

been

KEETM

ATMS and to

-on(c,d)

implemented

(Knowledge

This

research

was

(DARPA)

reported the official

here

are

position

supported

in part

under

contract

those

of the

or policy

+on(c,

to allow maintain in

Engineering

Environment

by

the

No. F30602 authors

of DARPA

and

Defense

Advanced

85 C 0065.

The

trademarks

should

or the U.S.

not

Research views

be construed

government.

and

w3

W2

Engineering

are

tab)

the

IntelliCorp.

Agency

requires

Wl

Figure Figure

Knowledge

an is

of

system.*

KEE

action

a.s

class

* KEEworlds,

The world multiple,

ordered

solvers [may] act, changing the in a pure ATMS in which there

approach

facility

EnvironmentTM)

graph

associated

However, problems involving some additional mechanism. h

advantage of an and QA4 [l]), taking context merges, to detect inconsistent results.

in the

a hypothetical extension of the world’s to include a new subsequent action. may

as A

a partially

of a world

is no way to prevent the inheritance of a fact into a daughter context.” In this paper we explore one approach to using the ATMS to support the modeling of actions. The basic idea is to extend a traditional tree-structured context mechanism (as in CONNIVER

derived

is a directed

network resembles a procedural net of NOAH [9], or NONLIN [lo], where the actions are fully specified. We assume that the effects of a fully specified action can be represented by additions and deletions of base facts, so each world has a set of additions and deletions associated with it that represent the actual primitive changes determined

space.

provided

of actions.

as a whole

possibly

1. Introduction The

with

network

then represents action network graph

introduced

structure

acyclic graph of worlds. Each world may be regarded representing an individual, fully specified action or state change.

Projects conclusions

as representing

of

2-l

The additions and addition table. Those block These

2-l:

A Worlds

Graph

from

shows

an example

worlds

graph

at

Wl produce at W2 represent at W3 correspond

c to the table. are

W2 and recorded

assumed

Blocks from

World the

Notice

that

preconditions

are not

to have

been

tested

parent

W3 were constructed. in the worlds.

blocks

world.

the initial state, while the deletion the effect of moving block a to the to an alternative action of moving

Only

in the

the

AUTOMATED

effects

of the

represented. world

before

actions

REASONING

are

/ 13

To simplify graph

the

discussion

is a tree,

parent

and

we will

as in figure

2-1,

assume i.e.,

for the moment

each

world

has

a branch of the tree corresponds to a linear Later, we will consider the consequences

actions.

that

at

most

the

double

one

world

sequence of of multiple

parents. Observe

that

we

may

associate

each

world

with

the

state

that

results from applying the changes encoded by the world and all of its ancestors (in figure 2-l the states are depicted inside the worlds). Hence, a world plays a double role, representing both a state change

and

a world

role:

assumption may

action.

also

be thought

The

world

the state,

and

the

world

given

assumption

corresponds

of as the

environment,

actually

choice

the world’s

by the

or decision

all of its ancestors.

action

environment.

encoded

that

hand,

of the set of world

ATMS itself to compute the accomplished by having a special world. This node may be thought that

action

on the other

consists

and

a world

and

to the

The world,

led to the

corresponds

assumptions

It is convenient

to from

to use the

world environment. This is world node associated with each of as representing the statement

occurs.

and a state. The facts in the state will in general be augmented with deductions using general knowledge of the domain. Thus, the facts that are true at a world fall into the following three categories: 1. facts

inherited

2. direct

additions

3. deductions In

from

keeping

at this

from with

ancestor world

facts

the

worlds

in 1 and 2

view

that

additions

actual changes, they are only recorded that is, an addition only occurs where hold. and a deletion where it did hold. The

inherited

facts

follow

and

deletions

where they the fact did

a principle

of

represent

are effective, not previously

inertia

(essentially

STRIPS assumption [ll]): a fact that is added at a world to be true in succeeding worlds, up until (but not including) where

it is deleted.

The

deduced

representing is

the

continues a world

marked

as

reasoning useful

facts

may

include

a contradiction. in

The

inconsistent. such

worlds

to do meta-level

the

A world

system

(however,

reasoning

distinguished

where it

about

fact

FALSE

node’

FALSE,

can be deduced

generally

avoids

is possible

and

inconsistent

Figure

further

sometimes

The world

worlds).

Before the

in ATMS

discussing

how

underlying

ATMS,

the worlds we

graph

give

a

mechanisms that are used, primarily reader is urged to consult de Kleer the ATMS.

is implemented

brief

sketch

in terms

of

the

of

nodes

ATMS

to establish terminology. [3, 4, 51 for a full description

The of

The

basic

elements

used as an propositional nodes,

of the ATMS

in the ATMS

By

nodes.

An

tracing

be justified back

ultimate

justification

support

for

a

it may also have multiple environments. The set of Computing environments for a node is called its label. of nodes is one of the major activities of the ATMS.

primary

transaction This causes

justification. recomputed. contradiction, justifications. called nogoods Environments

the ATMS labels of

There is a special which is similar The environments and that

of nogoods,

are not

that the

constitute are

discovered

used for further

is adding nodes to

a be

element called FALSE, denoting may have and to a node, that would be in its label are

minimal

are removed

supports affected

inconsistent

to be inconsistent, from

the

labels

environments. i.e., of nodes

that

are

so that

world

has two ATMS

1-i / SCIENCE

world

node

a single

Node

justification

of the

environment,

adding a fact a justification

for the

parent,

and

Thus,

of the form

A,

is the

world

described.

F at a world can now in terms of the world

possibility

is included.

of later

a nondeletion

deletion,

the justification

be accomplished node. However,

by to

assumption

has the form

NwAA,w - F where

A,,

is the

world (to subsequent A,

allow world A A,,

is supplied Wl.

-

This

the

assumption. each

separate

A distinct addition If

F

where

Awl

resulting

scheme

justifications

is the from

for addition

supplied

world

and

deleted

at

a

at

a

assumption

by the

for

of this

form

is shown

system

to represent

the

facts,

which

a fact

deletion

deletions, and justifications for world installed by the user to represent

primitive

is

justifications

additions and be justifications as the presence determination,

nondeletion

of

FALSE ATMS,

call nogoods

deletion nogoods. figure 3-2. from

for

independent deletion). WI, the justification

to the

We will

Apart

nondeletion

is required

in

nodes, there will deductions from

These deductions need be performed only once of the justifications in the ATMS allows the efficient via label propagation, of which derived facts hold in

worlds.

reasoning. Derivations

Each

for World

a single

of the given world. This is depicted graphically in It is not difficult to see that this results in all world

assumption

it

they

the

the

of other

of a node as a set of assumptions. Such a set is called an Since a node may have multiple for the node.

supersets

determine

through

in terms

structure,

The

to

may

derivation environment derivations, (minimal) the labels

is possible

and

to a decision or choice, and is Nodes correspond to descriptor.

Justification

is given NW

is the

having

Directly supplying

corresponds

elementary context facts or data, which

or assumptions.

are assumptions

N,

assumption figure 3-1.

allow assumption

3-l:

N,,

N,&AW+ where

3. Worlds

node,

entities

associated

with

it, reflecting

its

determine

of FALSE inconsistent

arising worlds,

from

user

representing

justifications dead

ends

are used in the

to

search.

non-deletion assumption

-on(c,d) +on(c,tab)

assumptqon Figure

3-2:

Addition

The nogoods determined nondeletion assumptions However, only the wish these to take further

in section

a feedback

loop

5).

Thus,

that

the

by the ATMS in addition to

latter represent all the “blame”

To

in

test

with

the

feedback only

are,

ensures

left

contain

that

nondeletion

assumptions

is necessarily

one

nondeletion

then

checked

unique

of

applied,

assumption). as true

environment

When

as are consistent

in the world.

at

extended,

the

of the fact

W2

includes

with

changes

examples

environment

example,

the

world

also

contains

in figure

on

in an ambiguous of

such

merges.

the

the

order

merge. In

in

cases,

branches

in

which

In figure

both

two

arises

4-2,

the

all

that

the

they

are

we show

state

at

W5

Thus, on(c,d) is true assumption for the the the

Figure depends

definition

now

consider

the

multiple

parents:

we

perform

merges

allows

more call

components, As recombined. Thus,

are the

to the table.

a simple

union

which before,

combined.

state

to

the

be

where a world The ability

decomposed

into

has to

nearly and the

In

is a

figure

in

4-1,

the

to W4 will

to stress

facts

a merge.

can be worked on separately the changes represented by

corresponding

We wish of

situation

a world

a problem

independent

worlds

complex

such

the

that

a merge

parent

+P

-P

-P

+P

for

4. Merges

as

depend

along

a difficulty

from

the nondeletion

There

moved

changes

changes

2-1,

assumption

world have

W4

both

is not

worlds,

blocks

the same

but

rather

4-2:

Ambiguous

on the order

of the preceding

are basically

two

is to forbid

merge.

the

of the

If so,

addition of on(a,b) at Wl cannot be consistently added to environment at W2 because it shares a deletion nogood with world assumption for W2. Hence on(a,b) fails to be true at W2.

ancestor

4-1,

a union

generally,

may

resulting

from

it (the

environment.

For

assumption for the addition of on(c,d) at Wl. at W2. Note, however, that the nondeletion

later

More

since each nogood contains at most The extended world environment is

to see if it is a superset

is regarded

world

We

of figure

A Merge

of

procedure

ones

effect two

minimality

nogoods

example

independent.

a fact holds in a world, we can compare each The in the node label with the world environment. is done as follows (in principle; the actual algorithm is The world environment is extended but more efficient).

extension

Wl.

the

nogoods with the are subsets of the

Deletion

In the are

incorporates

resulting

whether

equivalent, with as many

the fact

reduced nogoods

accordance

removed. the process; are

system

4-l:

assumptions.

environment comparison

the

worlds

are

course, exempt from this that the deletion nogoods nondeletion

the multiple

so,

produces the state the ancestor worlds.

choices in the search, and we for dead ends (we discuss this

in the ATMS These removed.

latter

Figure

Justifications may, however, contain the world assumptions.

installs

nondeletion assumptions and original ones, requirement,

and Deletion

the merge of the

merge

ways

possible to adopt an intermediate and further specifying others.

changes.

of dealing

in ambiguous so that

Merges

cases.

the ambiguity position, We

now

with

this

The

other

difficulty.

is removed. forbidding

consider

One

is to refine

the

It is also some

examples

merges of each

approach. We have already deletions at worlds from of

actions fully

required

introduced be effective

in ancestor

specified

worlds.

actions,

to be effective

the requirement that with respect to the However,

the additions even with respect

from

additions and state resulting

a strict

standpoint

and deletions to possible states

AUTOMATED

REASONING

could be resulting

/

15

from

actions

merge

in sibling

linearization then

subgraph

in which

prove

Theorem

Thus,

next

result

or deletion

is ineffective.

1: A merge

forbid

possesses

a any

One can

that

is not

assists

in the identification that

of such

additions

2: A graph of worlds an addition is ineffective

at

two

that

neither result

worlds

in which

is an ancestor holds

4-2 would

It is of interest

the

forbidden

merges

effective

with

addition

occurs

With

this

approach,

and a

the

above

restriction

deletions

that

merge

constraints

is the necessity

would

to be a high

that

led

of quickly

produce

The

a

the

world,

operation. as

to

schemes

determining

a consistent

frequency

allow the merge to be computed environments. The other constraint of unambiguous cases where there the merge.

resembles

that

be

whether

since

schemes

simple

that

is

described

union

of

ATMS

is the existence of a large core is only one reasonable value for

has some

for the merge.

ancestor

in every addition that every valid

to show

Thus,

one might

appeal,

but

arises

subgraph

but

does not

as follows.

Let

of a proposed

and deletion linearization define

any such In figure

W5 in the left example,

intuitive

implementation,

of the

value (if there is forbidden otherwise).

required

that

merge

being effective. gives the same

the merge

to be this

It is result

common

linearization; the merge could 4-2, this would lead to P holding

not

be at

in the right.

5. Actions

is

definiteness, and for contrast, this will be cast in terms of a Doylestyle truth maintenance system [S]. The general approach we follow is to use a form of nonmonotonic inference to reason about the

If

excessive:

one with

it

forbids

does not respect to

sufficient

condition

unambiguous

guarantees

type

an efficient

are mandatory are, in effect, the separate branches of the because each branch deletes a

effective

weaker

merge

to admit

valid if it results

nets [lOI . . where actions that violate each not be unordered relative to each other.

above restriction unambiguous.

somewhat

necessary)

a potential

possible

the merges

of the other.

additions

overriding One

US call a linearization

such

be disallowed. that

the are

two here.

appear

of the other.

Indeed, additions and deletions that preconditions. From this perspective, networks of figure 4-2 are in conflict

In a sense, merges that

are

described

A further

admits a linearization if and only if there are

the

for deletions.

for conflict-free procedural others’ preconditions may

precondition

by

are always

Theorem

A similar

There

worlds).

in which least

forbidden

is unambiguous.

to ancestor

actions,

might

expected

we are assuming

in figure

one merge

an addition

criterion

(remember respect

worlds.

of the proposed

the following.

above The

or cousin

if the ancestor

many

require that non-ancestor (but

still

merges

effects

of actions.

contradiction

not

as

can be obtained,

and NonMonotonicity

It is instructive to consider how actions might be represented more general TMS setting, as suggested by the worlds system.

truth

However,

the

is somewhat

behavior

different

maintenance

systems.

will

a

we require

from

the

in a For

in response

to

approach

in

system,

as

standard

follows.

Theorem

3: A sufficient

unambiguous

is

that

contain two worlds, other of which adds the other. This

criterion

We now and adopt

for

ancestor

a merge

subgraph

We

to be

may

the examples

of figure

Besides

which

then

fact,

and

the absence

ignorance, the

not

original

merge,

fact

when

Notice

the

holds,

that

merge

irrespective

of the

order

in

Otherwise, we are ignorant of the from the merge simply denotes such

of the fact

falsity.

the fact

the

are performed.

this

definition

is unambiguous.

is consistent With

the

dual

to the pessimistic

true

in the

merge

consistent

with

optimistic

merge,

at W5 in both examples of figure 4-2. A merge is the optimistic merge where a fact is

if it is true the

in some

original

for

P is present

the

linearization.

Again,

unambiguous

at W5 in both

case.

this

With

the

examples.

deletions

is unchanged.

The

different

merge,

the

descendant effective

with

merge

tends

defensible One

16

/

deletions additions. respect to

be

on semantic

might

imagine

SCIENCE

are

effective

For

the

to easier

with

implement

are obtained

deletions

case,

additions

affect,

the to

the

only

i.e..

pessimistic all except

deletions (the

efficiently,

are

optimistic

although

less

grounds). a wide

variety

of possible

For

merge

algorithms.

state

about

facts

context b”.

of

an

of

the

to a particular in the

records Note

that

of past

in time.

“present”

past

actions

there

may

such like

as A3:

be several

same description; we giving them unique the identifiers is not so far - the relative

can

action

occurrence

point

be

actions

represented

by

to present

facts.

u was not we justify

moved P5 as

example, A3 A I% + block

a is on block

(05) Here,

-

“(D5)”

it a nonmonotonic

justification

of the form

F5 indicates

that

a subsequent

b.

condition of the form “block In order to allow deletion, A3.”

off block b after an assumption by giving

statement

of merge For respect

optimistic

ancestor to

kinds

additions the are entered.

current

of individual between the

effects The positive justifications linking the

is

We now discuss the effect of merges on the ATMS representation. When a world has multiple parents, the justification for the world node must be expanded to include each of the parent world nodes among the justifiers. The justification scheme for additions and by different selections of which which justifications for FALSE

assertions

P5 is a preservation

P is absent

or

of a situation

actions with the occurrences by The numbering of identifiers such as A3. Thus intended to imply temporal order. timing of past actions has not been represented.

with

pessimistic

context,

evolution

containing

occurrences distinguish

4-2.

other approach to removing criteria for defining the

be assured

the actions

the

a is on block b,” the a was placed on block

“block “block

the ambiguity, merge. In the pessimistic merge, an individual fact belongs to the merge if it survives in every linearization of the actions. The rationale is that we may

regard

describing

not

one of which deletes a fact and the it, such that neither is an ancestor of

also prohibits follow the additional

condition

the

that

“some

D5 is an OUT-justifier,

action

action,

say

where

D5 is the

after A3 moves block u off block 6”. the block off, we supply A4, moves

If a

justification A4-+05 causing the OUT-justifier to come IN, thereby undercutting t,he derivation of “block a is on block b.” Note that the information about

the

these A

difficulty

solving the

relative

timing

of actions

is now implicitly

represented

by

justifications. with

process

search

space.

this

generates We

representation contradictions

do not

wish

arises that

the

when

represent

preservation

the dead

problem ends

assumptions

in to

be implicated in these; rather, we wish the assumptions representing choices of actions to be the ones considered for revision. Choosing a as culprit during backtracking would preservation assumption

amount

to

postulating

the

deletes

one of the facts

make

the

maintenance the

TMS, has

ensure

that

have

separation

between by de Kleer,

only

informed the

like

directly,

Incidentally, identification

but

a

then

exist Some

handles

an

unknown

contradiction. problem

that

it about.

TMS

something

refuted

actions

of

to the

suggested the

solver

existence

leading

this

solving

from

are

the

“sheltered”

that

One

not indirectly

in response

of

problem

is required

to

to a contradiction.

the need for a more discriminating is not confined to the difficulty

system

suggests

a

schemes

The

approach

KEEworlds

possibility is to which could be

assumption,

TMS

maintenance

shortcoming

handle

efficient

tool

KEEworlds existing

described

facility for

is to

consequence defeat

his

assumption standard

break

in

of waking

a window.

However,

the occupants,

purpose.

Let

if they

us suppose

the

this

may

are home,

burglar

have

which

makes

the

“contradiction” of waking the occupants, might elect to revise the assumption that the occupants are home, even though that is not subject to the burglar’s control, instead of the real culprit, breaking the window. The system would in effect regard the undesired of waking

the

occupants

However, it is only when there occupants being absent that this This example of “wishful thinking” systems

in general

need

as evidence

for

their

refined

treatment

outlined

here could

PI

frame-based

existing

implemented

the

exploration during either

to

as

provide

multiple

multiple

representation

truth

of

system

provides

the and The

situations.

worlds

system,

part a useful

with

an

a graphical

of worlds and allows rule-based forward or backward chaining.

discussion of the K.EEworlds of its use, may be found in [8].

Bobrow,

facility,

G. and B. Raphael.

New Programming

Languages

for Artificial

Intelligence

Research. Surveys

Computer

PI

PI

absence.

is independent evidence for the possibility is worth considering. suggests that truth maintenance

a more

way

References

default

that the occupants are home. The difficulty is that a truth maintenance system, in attempting to resolve the

consequence

the

appears

about

integrates

An application-oriented together with an example

would

the

been

and

reasoning

facility

browser for manual generation of worlds

process of culprit with preservation

has

of I(EE

assumptions. As another example, consider a situation where a burglar is planning to break into a house late at night. To accomplish his purpose, he must choose some method of entry. One method

in

contradictions.

truth

of view the

mechanism

correctly.

that if we

and

point

those

new

action However,

141

de Kleer, J. Choices Without In Proceedings,

1974.

Backtracking. Austin,

AAAI-84.

Texas,

de Kleer, J. An Assumption-Based

Truth

Maintenance

Artificial

28(l),

1986.

28(l),

198G.

de Kleer,

Intelligence

1984.

System.

J.

Extending Artificial

of contradiction

6(3):153-174,

the ATMS. IntelEigence

handling.

i51 Although

the approach

ATMS more directly for modeling actions, for a user to have to input the justifications and deletions by hand. provides a framework that an action

modeling

be adapted

to using

The worlds facility described earlier presents a more convenient interface to

PI

which graphs

items with

are recorded in data the same apparent

different results for a merge. apparently have no notion mentioned McDermott’s justifiers.

by McDermott approach However,

Boolean

equations,

pools. external

This means that structure may

Another difference of contradiction

in the is that

this

requires

rather

than

paper). One justifications that

the

labels

simple

PI

two have

is that data pools (at least none is

be computed

PI

of

illI ViewpointsTM appears quite

described comparisons

here.* since

the underlying

facility similar

of Inference in behavior

However, it little information

mechanisms

is

changes.

terms

of

System.

A realization

an An

underlying examination

* Viewpoints

and

difficult has been

to made

ARTTM facility

make detailed available about

of ART.

LVe have described an approach mechanism that represents a partially state

Corporation’s to the worlds

to constructing ordered network

of the mechanism Assumption of a similar

ART are trademarks

of Inference

Based

has been Truth

representation

a context of actions or described

1986.

System.

Intelligence

12(3),

1979.

Dependencies:

Nardi, B. and A. Paulson. Multiple Worlds With Truth ECAI-86.

A Synthesis.

Publishing

Sacerdoti,

and

Maintenance

Brighton,

Nilsson, N. J. Principles of Artificial

England,

Machine

In AI Applications. 1986.

Intelligence.

Company,

Palo

Alto,

Ca.,

1980.

E.D.

A Structure Elsevier

the ATMS. The system

Maintenance

McDermott, D. Contexts and Data

Tioga

ilO1

by solving procedure

28(l),

J.

In Proc.

attractive aspect of may have OUT-

propagation

the ATMS.

IEEE Transactions on Pattern Analysis Intelligence 5(3):237-246, May, 1983.

of

data pool order in

with

Intelligence

Artificial

Remarks pools

Doyle, A Truth

system.

The worlds considered here resemble the data h?cDermott [7]. H owever, the result of a merge in the approach is determined by the arbitrary chronological

J. Solving

Artificial

it would be cumbersome representing additions

PI 6. Closing

de Kleer, Problem

the

for

Plans

North-Holland,

Tate, A. Generating

Project

Tn I.1C,-U-77,

and Behavior. 1977.

Networks.

pages

888-893.

Cambridge,

Massachusetts,

1977.

[12]

lj’aldinger, Achieving In Elcock, pages

R. J. Several Goals Simultaneously. E. and Michie, D. (editor), Muchi?ze Intelligence 94-136.

Ellis

Hoi-wood,

Chichester,

8,

1977.

in

Maintenance in a classical

Corporation.

AUTOMATED

REASONING

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