Vivid Knowledge and Tractable Reasoning

Preliminary Rep ort

David W Etherington Alex Borgida Ronald J Brachman

ATT Bell Lab oratories Department of Computer Science Henry Kautz

Mountain Avenue Rutgers University ATT Bell Lab oratories

Murray Hill NJ New Brunswick NJ Murray Hill NJ

Abstract intractability of such systems have generally fallen under

two headings limited languages eg PatelSchneider

Mundane everyday reasoning is fast Given

Borgida et al and limited inference eg

the inherent complexity of sound and com

Frisch PatelSchneider In the former what

plete reasoning with representations expressive

can b e expressed in the knowledge base KB is restricted

enough to capture what p eople seem to know

sometimes severely to guarantee that queries can b e

commonsense reasoning must require shortcuts

answered in more or less reasonable time In the lat

and assumptions Some means of simplifying

ter restrictions like avoiding chaining or fourvalued in

the retrieval of the inferential consequences of

terpretations yield limited conclusions alb eit from rela

a set of facts is obviously required Instead of

tively expressive KBs

lo oking as others have at limited inference or

We conjecture that a key to ecient problemsolving

syntactic restrictions on the representation we

lies in a notion of commonsense reasoning the kind of

explore the use of vivid forms for knowledge

reasoning that p eople engage in all the time without re

in which determining the truth of a sentence is

course to pap er and p encil reasoning by cases back

on the order of a database retrieval

tracking or particularly deep thought Commonsense

reasoning is fast if it were a problemsolvers normal

In order to base a reasoning system on

mo de of reasoning then the problemsolver would b e

vivid knowledge we consider ways to construct

fast Paradoxically studies of commonsense reasoning

a vivid KBa complete database of ground

in AI eg nonmonotonic logics have frequently led to

atomic factsgiven facts that may b e pre

mechanisms that are even less tractable than logical de

sented in a more expressive language that al

duction

lows incompleteness eg rstorder logic Be

This pap er describ es an attempt to bridge the gulf b e

sides oering an architecture for examining

tween principled theories of inference and practical infer

these problems our results show that some

forms of incomplete knowledge can still b e han ence systems We discuss some comp onents that might

combine to supp ort fast reasoning and a uniform ar

dled eciently if we extend a vivid KB in a

chitecture that incorp orates them Obviously common

natural way Most interesting is the way that

sense reasoning is inherently approximate and fallible

this approach trades accuracy for sp eed

Our architecture lets us move towards commonsense p er

formance and yet still say something substantive ab out

Intro duction

the systems relationship to ideal comp etence

People p erform quickly and comp etently in most ev

eryday situationsdespite an overwhelming barrage of Vivid Reasoning

information that nonetheless do es not unambiguously

What would b e a go o d basis on which to build a

characterize the state of the world In contrast com

fast reasoner Given the massive amounts of infor

puter problemsolversesp ecially those with clear for

mation agents are faced with we cannot even inter

mal foundationsare extremely slow in most circum

pret fast as p olynomialtimewe really need p er

stances even when presented with little information

formance sublinear in the total size of the KB for simple

Consider a problemsolver that relies on a knowledge

queries The natural candidate from Computer Science

representation KR system to answer queries ab out

is something like a relational database where query

what follows from a knowledge base Although there are

many factors that contribute to the overall p erformance

We call reasoning that do es not t this description puzzle

of the problemsolver clearly the eciency of the KR

mode reasoning after logic puzzles of the form The man who

owns the camel lives next to the orangejuice drinker system is imp ortant Recent attempts to deal with the

answeringreasoning is merely lo okup for the kinds of plicitly represent disjunction negation or any form of

simple questions that we exp ect to b e frequently asked incompleteness This makes it imp ortant to determine

of the KB the relationship b etween the answers that a complete

Analyses such as Levesques and Reiters theoremprover would return when queried given the

suggest that a crucial factor in the eciency of databases KB and the answers that would b e retrieved from the

is the assumption that the database has a complete and vivid knowledge in the VKBie b etween and in

accurate view of the world Generalizing from this we the gure This relationship can b e thought of as the de

conjecture that the prop er basis for commonsense rea gree of soundness and completeness of the VKB Vivid

soning is some vivid representation of knowledgeone reasoning will not b e very useful if is to o small a sub

that b ears a strong and direct resemblance to the world set of or b ears no understandable relationship to

it represents A vivid representation has symb ols that

stand in a onetoone corresp ondence to ob jects of inter

est in the world with connections b etween those symb ols

corresp onding to relationships of concern For example

Levesque a KB containing the sentences Dan

drank ounces of gin and Jack drank ounces of gin

would b e vivid with resp ect to the amount Jack and

Dan drank individually while one containing Jack and

Dan together p olished o ounces of gin and Dan

had one more ounce drink than Jack would not de

spite the fact that the same information follows from

b oth KBs

The notion of vivid representations is app ealing for

reasons b eyond supp orting reasoning as databasestyle

Figure Simple view of a vivid knowledge base

lo okup it corresp onds well to the kind of information

expressed in pictures thus it is reasonable to think

Because not all reasoning ts our commonsense rea

that much of the information we gain ie p ercep

soning paradigm we prop ose a hybrid system that re

tually o ccurs naturally in vivid form Also various

tains the original information to supplement as nec

psychologicallyoriented explanations of cognition sug

essary the vivid form We attempt to answer queries

gest that p eople often seem to reason directly from men

by simple retrieval directly from the vivid KB If that

tal mo dels JohnsonLaird rather than by syntac

provides inadequate answers general or sp ecialpurp ose

tic manipulation of sentential constructs

reasoning with the original KB may b e tried p erhaps

Of course not all information we obtain ab out the

dep ending on the imp ortance of the query Ultimately

world is in vivid form linguistic communication for ex

one measure of success will b e the prop ortion of reason

ample may yield disjunctive or otherwise incomplete or

ing that can b e delegated to the VKB

general input eg Jo e do esnt have his PhD yet

or Everyone in the department has an advanced de

gree Fortunately much of this information can b e

co erced into a vivid form in a principled way

System Architecture

What is needed is an appropriate architecture that would

allow an AI system to fall back on more general reason

ing eg rstorder logic when necessary but would

dep end primarily on ecient vivid reasoning The ap

proach of standard hybrid reasoners which delegate

questions to submo dules that can handle them eciently

will not suce We need a much more active approach

in which incoming information is pro cessed to augment

and maintain a vivid view of the world We have b een

investigating an architecture that exemplies this view

Figure A more general architecture

see Fig rstorder facts are vivied into a knowl

edge base of a sp ecial form the VKB This vivica

A generalization of the architecture of Fig and a

tion may lose information since the VKB cannot ex

more realistic view is illustrated in Fig Notice rst

We distingui sh b elow b etween the KBthe knowledge

given to the systemand the VKBthe systems vivid rep resentation of that knowledge

the inuence of a variety of comp onents on the vivica by database techniques eg memb ership in dened re

tion of the original facts Universal rules aect vivica lations need not b e computed

tion simply and directly see b elow However where the

Disjunctive and negative information do not t read

available knowledge is incomplete we can often do b etter

ily into the database worldview and are ma jor contrib

than simply leaving the information in nonvivid form It

utors to the complexity of logical reasoning We address

may b e p ossible for example to eliminate the ambiguity

disjunction and negation piecemeal distinguishing sev

of the given disjunction in favour of denite factsfacts

eral dierent forms and treating each dierently Ab ove

not strictly equivalent but sucient for the purp oses

all we strive to avoid reasoning by cases We hyp othe

of the system For example defaults or preferences size that commonsense reasoning achieves its eciency

can b e used to capture the contribution of previous ex

in part by not resorting to case analysis and we treat

p erience Gricean communication conventions Grice

problems that absolutely require reasoning by cases as

and linguistic context eects in forming mental

puzzlemo de problems

mo dels In other cases abstraction provides a p owerful

Perhaps the b est way to discuss the various versions

to ol In some circumstances it may even suce to make

of vivication is to consider progressively weaker restric

arbitrary choices as suggested in Levesques Computers

tions on the forms of negative and disjunctive informa

and Thought lecture Thus the information in the VKB

tion that can b e vivied and consider how each new

may b e the consensus of multiple knowledge sources as

class of facts can b e converted into vivid form

suggested by Fig

A Simple Case

It also seems useful to separate out parts of the orig

inal KB that are essentially taxonomic As we show

The simplest extension b eyond ground and universally

b elow taxonomies provide another form of disjunctive

quantied atoms is to allow disjunctions of the form

information that can b e used eciently in vivication

x Ax B x equivalently simple implications like

and retrieval

those found in inheritance hierarchies The vivication

algorithm treats these by asserting B whenever A

Constructing a Vivid Knowledge Base

is entered into the VKB The VKB is then queried as

a normal relational database with negation determined

A vivication pro cess for the simplest casethat of

by the closedworld assumption CWA Reiter

nondisjunctive p ositive p ossibly universally quanti

For example vivifying the KB fM anS ocr ates

ed sentencesis easy to imagine All that is necessary

W omanO phel ia x M anx M or tal xg re

is to take the set of instances of the universallyquantied

sults in the VKB fM anS ocr ates W omanO phel ia

formulae over the set of known individuals and store the

M or tal S ocr atesg The query M or tal S ocr ates is an

result as a collection of p ositive ground atomic pred

swered by lo okup in the VKB and returns Yes The

icates eg a relational database However we also

query M or tal O phel ia fails in the VKB so the CWA

intend to take information that would app ear suitable

sanctions the answer No

only for the KB and use it in vivication andor in con

The KBs considered so far corresp ond to denite

junction with the VKB in queryanswering

databases ie databases of clauses each containing ex

The architecture describ ed ab ove trades eort and

actly one p ositive literal Reiter shows that the

space as knowledge is added to the system in favour of

CWA is always consistent with denite databases We

rapid queryanswering Although there are fallback p o

have proved that the answers returned by closedworld

sitions that make vivication less demanding some of

querying of the VKB are identical to those returned

which are discussed b elow and in Borgida Ethering

by closedworld querying of the original knowledge base

ton it is useful to ignore the cost of vivication

under the domainclosure assumption DCA Reiter

at rst to make some of the underlying theoretical is

sues more apparent Notice however that vivication is

Because negative information is not explicitly repre

not the same as computing all consequences of the KB

sented in the vivid KB it is not necessary to consider

only ground atomic consequences are develop ed Fur

the contrap ositive forms of the disjunctive rules any rule

thermore any ground consequences that can b e obtained

not instantiated by the vivication pro cess will b e cor

rectly instantiated by the CWA during queryanswering

At the moment we assume the defaults are presented

to the system in the same declarative way as other facts

This ensures that the computational complexity of the

eventually defaults should b e created by insp ecting the VKB

ie from exp erience

To simplify the rest of our discussion of vivication we

For example when someone says Some of the chemists restrict ourselves to monadic predicates In some cases this

are b eekeep ers they typically mean to imply that some of hides only messy details In others some details remain to

them are not JohnsonLaird b e worked out Readers are welcome to make whichever as

sumption their credulity allows

Interestingly mathematicians and computer scientists

have indep endentl y studied vivid representations of partial The DCA which says that the individ ual s mentioned by

orders where transitive relationships can b e directly read the theory constitute the entire set of individ ual s is used in

o the representation viz Agrawal et al database theory to facilitate handling quantied queries

vivication pro cess do es not get out of hand In par vivication simply asserts memb ership in the subsuming

ticular it is not necessary to reason by cases since the class and discards the alternatives thus obtaining an

negative case can never b e explicitly asserted in the KB atomic fact that can b e stored in the VKB eg using

Any technology suitable for reasoning with monotonic Fig I nstr uctor J oe

semantic networks eg Thomason et al can b e

The price of this substitution dep ends on the density

used to vivify the KB Of course the system describ ed

of the available subsumption information If the sub

so far is not signicantly more useful than a monotonic

sumption hierarchy is complete no information is lost

semantic network In the following sections we discuss

anything deducible from the KB will follow from the

extensions that move in the direction of a useful com

VKB In what we exp ect to b e the more common case

monsense reasoning system

of a relatively sparse hierarchy a certain amount of pre

cision may b e lost Exactly how much will dep end on

A Slightly More Complicated Case

how natural the given disjunction is Disjunctions

The knowledge presented to a system sometimes con

that are useful for commonsense reasoning will often b e

tains b ona de alternatives and provides no means for

subsumed by predicates nearby in the hierarchy Less

deciding amongst them It is sometimes p ossible to trade

natural disjunctionsrequiring reasoning closer to puz

the given ambiguity for vagueness and thereby avoid

zle mo dewould b e subsumed only by much more gen

disjunction That is a list of alternatives concerning

eral conceptsconcepts that also subsume many other

an individual can sometimes b e replaced by a lessne

concepts not represented in the original disjunction

grained but atomic description that subsumes the al

For example think again of the hierarchy in Fig

ternatives For example if we are told only that Jo e is

The information that Jo e is a professor or a do ctor would

or we might represent the fact that Jo e is in his

b e vivied by asserting P r of essional J oe allowing the

early s

p ossibility that he is a teacher a lecturer or a lawyer

To substitute vagueness for ambiguity we assume that

Being told that he is a professor or a student would yield

the given KB provides certain subsumption information

Adul tJ oe losing among other things the fact that he

This may range from the extreme of a complete upp er

is not a visitor Learning that Jo e is a lawyer or a shark

semilattice containing a subsuming predicate for every

might give rise only to T hing J oe

subset of the set of predicates eg Fig through

Unnatural disjunctions do not slow vivication

more natural taxonomic hierarchies eg Fig to the

down appreciably but uselessly vague answers can b e

trivial case where everything is subsumed only by T hing

exp ected concerning the sub jects of these disjunctions

This coincides with our intention that the vivid reason

ing comp onent should not b e exp ected to handle puzzle

mo de problems well

We have not said exactly how the VKB should han

dle negation in this extended representation scheme In

particular since the KB is no longer denite it is in

appropriate simply to use the CWA which may in

tro duce inconsistency For example if T eacher J oe

P r of essor J oe is made vivid by representing only

I nstr uctor J oe then the CWA would justify b oth

T eacher J oe and P r of essor J oe since neither fol

lows from I nstr uctor J oe

One solution is not to make the CWA at all then fail

ure to nd a fact in the VKB would simply mean that

the VKB didnt know the fact to hold This solution

seems a bit radical however While avoiding overcom

mitment when given vague knowledge it prevents mak

ing the CWA even for things not even represented in the

KB This would make the VKB much less vivid Fortu

nately there is a less drastic solution

Figure Fragment of a subsumption hierarchy

In the simplest case amenable to substitution the We realize of course that natural is not a welldened

term In this context however we can dene a disjunction as

given information asserts that a particular individual is

natural if its elements are subsumed by a predicate nearby

a memb er of one of n classes ie has one of n prop

in the abstraction hierarchy We can justify this name by

erties without sp ecifying which eg T eacher J oe

b egging the question we assume that those resp onsible for

P r of essor J oe If the information available in the KB

building the KB will include no des for natural disjunctions

provides a class that subsumes all the mentioned classes of concepts and not for unnatural ones

Figure Complete structure for fTeacher Lecturer Professorg

For KBs of the form we are considering the appropri A Still More Complicated Case

ate form of the CWA is the Generalized CWA GCWA

Another naturalseeming form of disjunction involves al

Minker which is much like the CWA except that

ternation of the same predicate over more than one in

it avoids asserting the negation of terms involved in ir

dividual eg T eacher J oe T eacher B il l We treat

reducible disjunctions It turns out to b e straightfor

these using a technique similar to that discussed in the

ward to augment the representation mechanism used in

previous section abstracting a set of individuals to a

the VKB to allow it to distinguish unknown by virtue

typ e containing them In this case a disjunction is viv

of no information from unknown by virtue of vague

ied by intro ducing a Skolem constant a null value in

ness The CWA can then b e applied in the extended

database terminology to represent whichever individual

representation to infer the negations of terms for which

satises the predication We assert that the predicate

no information is available We have shown Borgida

holds of the null and that the null is a memb er of the

Etherington that this approach yields the same re

appropriate typ e As in the predicate case the infor

sults as the GCWA applied to the original KB assuming

mation lost by vivifying this way is prop ortional to the

a complete subsumption hierarchy A sparse hierarchy

density of the typ e hierarchy The empirical question of

of course may result in weaker statements due to the

whether there will generally b e typ es available that cover

loss of precision in the construction of the VKB

enough of the disjunctions over small sets of individuals

esp ecially sets of two that o ccur in commonsense rea

soning remains op en We can construct intuitively plau

sible arguments that there will b e no problem but we

The representation and algorithms we have develop ed

have not yet compiled data

are particularly attractive b ecause they have the prop

It is p ossible that the hierarchy will b e to o sparse

erty that their accuracy degrades gracefully as their e

in which case some combination of the ab ove technique

ciency improves and do es not degrade for unambiguous

with another we are studying may b e necessary This

information Thus the retrieval algorithms are sound

second approach maintains a list of constraints on which

and complete in cases where the hierarchy is complete

individuals may replace the null These constraints can

or where the given knowledge either is atomic or corre

b e used in a variety of ways to provide answers of vary

sp onds to concepts directly representable in the hierar

ing detail In the limit case a generalpurp ose equality

chy In exchange for the loss of representational delity

reasoner can b e applied to recover all the information

in other cases we achieve signicant p erformance im

inherent in the original problem statement At the other

provements assuming the hierarchy has O p predicates

extreme it is p ossible to manipulate a unication al

where p is the numb er of primitive predicates query

gorithm in such a way that it provides maximally op

answering is sublinear in the numb er of facts told to the

timistic or p essimistic views of the eects of the con

KB and linear in the size of the query Complete query

straints on answers to queries while p erforming only a

answering on the other hand is at least O n log n in

fraction of the work necessary to determine the correct

the size of the query and linear in the size of the KB

maximally informative answer This allows the system

We can also achieve signicant improvements in the com

to quickly provide upp er and lower b ounds Lipski

plexity of telling the KB facts the NPcomplete problem

of converting inputs to conjunctive normal form suitable

The same techniques can b e applied directly to

for vivication can b e approximated without additional

rangelimited existentiall yqua nti ed statements such as

9xT eacher S al ar y x loss of information in p olynomial time

on the answers to queries while avoiding the eort nec it was told Reasoning by cases is avoided by only rep

essary to determine the maximallyinformative answer resenting versions of the input that can b e made unam

biguous Levesque suggests that p eople sometimes

Again the approximations which are motivated by

avoid the exp ense inherent in disjunctive information by

the structure imp osed on the world by the hierarchy

simply picking one disjunct He argues that a tremen

made in vivifying the KB result in gratifying p erfor

dous amount of vague and ambiguous information is pre

mance gains like those discussed in the previous section

sented to an agent all the time and it is often sucient

while retaining soundness and completeness where p os

or even necessary to disambiguate it either arbitrarily

sible In this case the gains are even more signicant

or according to some default principles in eect coming

since the general query problem for the class of formu

to know more than was told

lae treated here is NPcomplete in the size of the query

Research on default reasoning has concentrated on

Imielinski Our algorithms can approximate an

developing default theories that are epistemically ade

swers in p olynomial time

quate but has ignored computational complexity Con

Negation

versely we are interested in using defaults to supp ort fast

vivid problem solvingincluding problems that could b e

The ma jor deciency in the system as it stands is that

solved more slowly without defaultsand are centrally

it provides no mechanism for explicitly telling the VKB

concerned with pro cessing the default rules quickly

negative information For many applications however

To see how defaults might b e used in vivication sup

this is not a particular problem since the system has

p ose a partial description of a ro om is input to an agents

the GCWA to provide implicit negation In fact this

KB Some items are precisely sp ecied eg the co or

is no less than is provided by many AI knowledge

dinates of the do or but others are missing eg the

representation systems eg Prolog Occasionally

width of the replace op ening Geometriclevel vivica

however it would b e useful to have explicit negation

tion might ll in the missing information with typical or

While we are still working out the details it app ears

random values For instance replaces are typically

to b e simple to add capabilities for representing sim

wide and the windows could b e assigned random p osi

ple ground atomic negative facts and p erhaps uniformly

tions along the exterior walls Finally supp ose an agent

negative ground disjunctive facts like the uniformly p osi

is told that there is a stack of long rewo o d outside

tive disjunctions discussed ab ove eg T eacher J oe

and is instructed to build a re in the replace She can

P r of essor J oe Such an extension would allow the

create a simple plan to carry some rewo o d indo ors and

system to distinguish b etween denitely false false

place it in the replace b ecause she has made the default

by the CWA and unknown in some cases

assumption that the replace op ening is larger than the

wo o d Lacking this b elief she would need to generate a

Life in the SpaceTime Continuum

conditional plan measure the replace if it is less than

In describing our vivication algorithm we have b een

then cut the rewo o d into suitablysized pieces

proigate with the space and to a lesser extent time

The utility of defaults do es not dep end on a coord

required to represent knowledge in an eort to achieve

inatelevel view of the world For example supp ose an

optimal p erformance for queryanswering Dep ending on

agent is driving a car when a tire go es at Her vivid

the availability of storage and the relative frequencies of

mo del of the car has a go o d inated tire and a jack in

up date and query it may b e desirable to retreat from the

the trunk This b elief together with the goal of replacing

extreme p osition of a totally vivid representation Since

the at tire generate the obvious plan op en the trunk

the architecture of our hybrid system assumes that the

remove the spare tire and jack raise the car and replace

original KB is available for the use of the problem solver

the at tire with the spare She do es not create a plan

it is not particularly dicult to b eat this retreat

that considers the p ossibility that there is no spare in the

The hierarchies that we use are exceptionfree and

trunk and conditionally sends her o in search of one

supp ort ecient computation Provided certain conven

Vivid reasoners never need to generate conditional

tions are followed during vivication and up date it

plans or reason ab out the p ossible ways the world could

is easy to eliminate some of the worst space consump

b e The vivid mo del forms the basis for a direct solution

tion In particular the transitive closure of the inheri

to the problem at hand If the solution fails b ecause the

tance hierarchy need not b e computed in advance and

default assumptions are incorrect then the vivid mo del

the distinction b etween unknown and assumed false

is revised the blatantly erroneous assumptions are re

can b e determined as required to answer queries rather

placed by new observations the KB is revivied and

than explicitly stored

problemsolving is rep eated

Knowing More or Less

See Etherington for a discussion of the p ervasive

ness of default reasoning in intelligent b ehaviour

The techniques we have describ ed for vivifying a KB re

Obviously there are situations in which the problem

sult in the system knowing less at least no more than solver might cho ose a more conservative and exp ensive

This use of defaults for vivication is closely related problems The essential idea is to use an array of tech

to the qualication problem in planning McCarthy niques to transform information ab out the world which

In many domains the list of circumstances that may b e incomplete into a vivid representation in which

would require dierent solutions to a planning problem inference approaches simple insp ection of the represen

is innite so that it is imp ossible even in principle to tation By trading representational delity for sp eed

solve the problem by reasoning by cases we are able to achieve attractive p erformance in certain

situations In any case the loss of accuracy can b e mo

We have b egun the task of integrating defaults into

tivated predicted and controlled by decisions made as

a vivid reasoner by analyzing the complexity of simple

knowledge is presented to the system

default systems Selman and Kautz rep ort our

analysis of mo del preference default rules which en

Acknowledgements

force a simple preference relation over the space of mo d

els of a theory which corresp onds to the space of vivid

We are grateful to Hector Levesque and Bart Selman for

mo dels While nding a mostpreferred mo del is in

their contributions to the work describ ed here

general NPhard one can b e found in p olynomial time

if the preference rules are acyclic Kautz and Selman

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