In Proceedings of the Ninth National Conference on Artificial Intelligence (AAAI-91)

Cambridge, MA: AAAI/MIT Press, 1991.

Negation and Pro of by Contradiction in



Logic AccessLimited

J M Crawford B J Kuip ers

Department of Computer Sciences

The University of Texas At Austin

Austin Texas

jccsutexasedu

kuip erscsutexasedu

Abstract of a tractable and So cratically Completeness inference

metho d for a language in which complete inference is

AccessLimited ALL is a language for

complete Co ok The heart known to b e NP

knowledge representation which formalizes the ac

of the extension is the addition of classical negation

cess limitations inherent in a network structured

ie not negation by failure to ALL The addition of

Where a deductive metho d such knowledgebase

negation is imp ortant b ecause it allows one to express

as resolution would retrieve all assertions that sat

negative and disjunctive information F urther the ad

isfy a given pattern an accesslimite d logic re

dition of negation allows us to augment inference by

trieves all assertions reachable by following an

dus ponens the only inference metho d required in mo

available access path

a deductive with inference by reductio ad ab

In this pap er we extend previous work to include

surdum the ability to make assumptions and reason

negation disjunction and the ability to make as

by contradiction

sumptions and reason by contradiction We show

So cratic Completeness guarantees that for any fact

remains So cratically that the extended ALL

neg

which is a logical consequence of the knowledgebase

Complete thus guaranteeing that for any fact

ther e exists a series of preliminary queries and assump

which is a logical consequence of the knowledge

tions after which a query of the fact will succeed Since

base there exists a series of preliminary queries

complete inference in rst order logic without existen

and assumptions after which a query of the fact

tial quantication is known to b e NP complete the

will succeed and computationally tractable We

ating such series must b e NP com problem of gener

show further that the key factor determining the

plete Ho wever we will show that the key factor de

computational dicult y of nding such a series of

termining the diculty of nding such a series is the

preliminary queries and assumptions is the depth

en any depth of assumption nesting in the series Giv

of assumption nesting We thus demonstrate the

knowledgebase and any fact if there exists a series

existence of a family of increasingly p owerful in

of preliminary queries and assumptions after which a

ference metho ds parameterized by the depth of

query of the fact will succeed and the series only nests

assumption nesting ranging from incomplete and

assumptions to depth n then the series can b e found

tractable to complete and intractable

in time prop ortional to the size of the knowledgebase

raised to the nth p ower This result is particularly

signicant since we exp ect commonsense reasoning to

tro duction In

require large knowledgebases but relatively shallow

P ast work in ALL Crawford Kuip ers

assumption nesting

has shown So cratic Completeness and computational

It has b een apparent at least since the work of

tractability for a language with expressive p ower equiv

that a knowledge representation language Wo o ds

alent to that of a deductive database In this pa

sucient to supp ort the building of large knowledge

p er we extend the expressive p ower of ALL to that

bases must have a clear semantics Without a clear

of rst order logic without existential quantication

semantics one can never b e sure exactly what a given

and thus for the rst time demonstrate the existence

expression represents which deductions should follow



This work has taken place in the Qualitative Reason

from it or how it compares to an expression in a dif

ing Group at the Articial Intelligence Lab oratory The

ferent knowledge representation language Exp erience

ersity of Texas at Austin Research of the Qualita Univ

with formally sp ecied knowledge representation sys

tive Reasoning Group is supp orted in part by the Texas

tems has revealed a tradeo b etween the expressive

under grant no Advanced Research Program

p ower of knowledge representation systems and their

NSF grants IRI and IRI and by NASA

If for exam grant NAG computational complexity Levesque

2

describ ed op erationally ple a knowledge representation language is as expres In ALL deductions which

sive as rstorder predicate calculus then the prob follow simply by rule chaining are obvious and can

lem of deciding what an agent could logically deduce b e made by a single op eration The deductive p ower of

from its knowledge is unsolvable Bo ALL on a single op eration is thus roughly equivalent to olos Jerey

3

that of a deductive database There are several p ossible solutions to this problem As will b e seen b elow

the more complex deductions are those which require restrict the expressive p ower of the representation

reasoning by contradiction or equivalently reasoning language describ e the reasoning ability of the sys

by cases tem with an op erational semantics rather than a mo del

Our approach in ALL b egins with the well known give up completeness but theoretic semantics or

mapping b etween atomic prop ositions in predicate cal guarantee So cratic Completeness

culus and slots in frames the atomic prop osition that

Unfortunately there are problems with each of these

the ob ject a stands in relation r to the ob ject b can b e

solutions If one is interested in expressing the full

written logically as r a b or expressed in frames by

range of human commonsense knowledge then re

including ob ject b in the r slot of ob ject a Hayes

stricting the expressive p ower of the representation lan

guage is not an option An op erational semantics may

a

suce to describ e the reasoning ability of the system

r

but it do es not dene the semantics of statements in

r a b

values f b g

the representation language

can attempt to give a mo del theoretic semantics One

for the representation language and an op erational se

ALL allows forwardchaining and backwardchaining

mantics to reasoning However reasoning will then b e

inference rules Antecedents of ALL rules must al

incomplete with resp ect to the semantics of the rep

ways dene access paths through the network of frames

1

resentation language That is there will b e cases

Access paths are sequences of predicates which can

in which according to the mo del theoretic semantics

b e evaluated by retrieving values from known slots of

h j f but the system cannot derive f from T h T

known frames For example in ALL one could write

Thus the meaning ascrib ed to statements by the mo del

the backwardchaining rule

theoretic semantics will not always match their mean

aunt J ohn y par entJ ohn x sister x y

ing to the system

The third solution So cratic Completeness do es

But one cannot write the logically equivalent rule

guarantee that the mo del theoretic semantics describ es

auntJ ohn y sister x y par entJ ohn x

the p otential reasoning ability of the system and thus

together with soundness guarantees that the mean

since evaluation of the antecedent would require a

ing ascrib ed to statements by the mo del theoretic se

global search of the knowledgebase for pairs of frames

mantics is the same as that ascrib ed to statements by

4

in a sister relation The use of access paths allows

the system However So cratic Completeness is a weak

ALL op erations to b e p erformed in time prop ortional

prop erty and can b e satised by any system of infer

to the size of the accessible p ortion of the knowledge

ence with a complete set of pro of rules

base Crawford Kuip ers

In ALL we use a mo del theoretic semantics to de

In order to add negation to ALL we b egin by adding

scrib e the p otential reasoning ability of the system

r If we for each relation r an additional relation

by guaranteeing So cratic Completeness but uses an

made no other changes this would give us a language

op erational semantics to describ e the b ehavior of the

with the expressive p ower to represent negation but

system on individual op erations In a sense it is not

we would have lost So cratic Completeness In order

surprising that complete inference with resp ect to a

to maintain So cratic Completeness we have to con

mo del theoretic semantics is intractable the mo del

nect r with r We do this by adding the notion of an

theoretic semantics was not intended to describ e which

inconsistent knowledgebase and the ability to make

inferences are easy to make or are useful in a given con

2

A mo del theoretic semantics may well not b e appropri

text The mo del theoretic semantics describ es which

ate since obviousness often dep ends as much on the form

inferences are p ermissible and is thus an appropriate

of knowledge as its meaning eg p is obvious from q and

description of the p otential long term reasoning ability

p but is not so obvious from q and :p !:q q !

of the system We exp ect a single op eration to p ermit

3

ALL allows b oth backward and forward chaining rules

only deductions which are ob vious The b ehavior

If one uses only backward chaining rules or only forward

of the system on single op erations can thus b est b e

chaining rules then the deductive p ower of ALL on a single

alent to that of a deductive database op eration is equiv

1 4

One could achieve completeness by giving up tractabil The restriction to access paths limits the syntax of

ity however if a system is intractable then in practice this ALL but is not a fundamental limit on its expressive p ower

means that it do es not return on some inputs or since one could always add a new constant and make it the returns

and thus is rst argument to every predicate This would amount to only after some unacceptable p erio d of time

ely incomplete eectiv making the entire knowledgebase a single frame

b oth of which op erate on a knowledgebase by applying

Example

all accessible inference rules and b oth of which return

pairs

If Mary waters the owers then the owers blo om

h set of substitutions new knowledgebase i

If the owers blo om then Mary is happy

We use sub and k b as accessors on the rst and second Mary waters the owers

comp onents resp ectively

Q Is Mary happy

In ALL we use the function neg ate to return

neg

the negation of a relation or prop osition in the obvi

Example

ous way neg ater t t r t t and

1 n 1 n

5

neg ater t t r t t A knowledge

1 n 1 n

If Mary waters the owers and the owers blo om

base is finc onsistent i there exists a fact f such that

then Mary is happy

6

f K and neg atef K

If Mary do es not water the owers then they do not

When an assumption is made in ALL the state

neg

blo om

of the knowledgebase is saved so that if the assump

Mary is not happy

tion leads to a contradiction it can b e restored Thus

in general ALL op erations op erate on stacks of

neg

Q Do the owers blo om

knowledgebases which we refer to as know ledgebase

structures Formally a knowledgebase structure or

To a human the solution to rst example is obvious

is a stack of pairs kbs for short K

while the second requires some thought To a reso

lution theoremprover b oth are twostep pro ofs In

ha K i ha K i

n n 1 1

ALL the rst question is answered immediately

neg

such that for i n K is a knowledgebase and

i

while the second requires a nested pro of by contradic

7

or a fact allowed in K We access the top a is nil

i i

tion It is our goal to build systems that reason e

element of such a stack using the function head and

ciently on problems p eople nd easy even at the cost

rest of the stack using r est heig htK denotes the

of requiring help on problems p eople nd dicult

numb er of pairs in the structure For any pair ha K i

we access the rst and second comp onents using the

Figure Two Examples

functions assump and k b resp ectively We use top k b

and top assump to access the top knowledgebase and

assumption in a structure top k bK k bheadK

assumpK assumpheadK We denote and top

assumptions and reason by reductio ad absurdum Fig

the structure formed by adding the pair ha K i to the

ure shows an example of a conclusion which can b e

top of the the structure K by pushha K i K

drawn using rules alone and a conclusion which re

To dierentiate them from ALL op erations we sub

quires reasoning by reductio ad absurdum

script ALL op erations by neg Thus q uer y q is

neg

The rst section b elow presents the formal develop

an ALL op eration and op erates on a knowledgebase

ment of ALL including sketches of the pro ofs of So

neg

while q uer y q is an ALL op eration which op er

neg neg

cratic Completeness and p olynomial time complexity

ates on a kbs The same holds for variables ranging

We next turn to the problem of generating the prelim

over op erations if O q uer y q then O is the cor

neg

inary queries and assumptions and shows that the key

resp onds ALL op eration q uer y q

neg neg

factor determining the complexity of generating such

Three typ es of op erations are supp orted in ALL

neg

series is the depth of assumption nesting Finally we

is a non queries assertions and assumptions If

overview related work and discuss current and future

empty path then q uer y is a query If f is a fact

neg

work

then asser t f is an assertion Finally if a is a fact

neg

then assume a is an assumption If an op eration

neg

F ormal Development

O is allowed in a kbs K then O K is an ALL

neg neg neg

This section formally develops ALL with nega

formula

tion A more detailed development can b e found in

5

Crawford

The distinction b etween : and neg ate is imp ortant :

is simply a character which can o ccur in names of relations

while neg ate is a function : is a part of ALL while neg ate

Syn tax

is a part of the mathematical language used to dene ALL

ALL relies on the denitions of queries and asser

6

neg

Since knowledgebases may not b e deductively closed

tions in ALL The syntax and semantics of ALL with

it may not have a knowledgebase may b e inconsistent ie

out negation are overviewed in Crawford Kuip ers

a mo del but not b e finconsistent

7

b and discussed in detail in Crawford For

A fact is allowed in a knowledgebase i it contains only

purp oses of the discussion b elow it suces to know

constants and relations which are allowed in the knowledge

uer y and asser t that ALL denes the op erations q base

For a knowledgebase K PC K simply returns the Knowledge Theory

conjunction of the facts and rules in K with all vari

The knowledge theory denes the values of ALL

neg

ables in the rules universally quantied Note that

formulas by dening the eect of op erations ALL

neg

we overload PC in that it maps knowledgebase struc

op erations map a kbs to a set of substitutions and a

tures knowledgebases and facts to predicate calculus

new kbs We denote these returned values with pairs h

A kbs K is consistent i there exists a mo del M such

set of substitutions knowledgebase structure i We

that M j PC K

use sub and k bs as accessors on the rst and second

A fact is provable in ALL i there exists a series

neg

comp onents resp ectivel y

of queries and assumptions deriving it For a series

Performing an ALL op eration involves p erform

neg

we use the notation K to denote of op erations

ing it as an ALL op eration on the top knowledgebase

the result pro duced by p erforming the op erations in

in the structure and then checking for and dealing

on K in turn eg unless of length zero K

with contradictions Consider a kbs K and a query

k bsheadr estK

or assertion O allowed in K If k bO top k bK

neg

Def Given a kbs K ha K i and a ground path

ie the knowledgebase pro duced when O is p er

f f al lowed in K

formed as an ALL op eration on the top knowledge

1 n

base in the structure is fconsistent then it just re

K f f

1 n ALL

places the old top knowledgebase in K If how

ever k bO top k bK is finconsistent then the most

there exists a series of queries and assumptions i

recent assumption is retracted ie the stack is

al lowed in K such that

p opp ed and the negation of the most recent assump

K ha K i i i n f K

tion is asserted as a fact unless the stack has only

i

one element in which case there are no assumptions to

The key lemmas for are the pro of rules Modus

8

ALL

Figure drop and the knowledgebase is inconsistent

Ponens and Reductio Ad Absurdum In stating these

shows the formal denitions of ALL op erations

f to denote the ad lemmas we use the shorthand K

dition of f to the top knowledgebase in K

Completeness So cratic

Lemma Reductio Ad Absurdum

As discussed in the intro duction reasoning in ALL

neg

For any wel lformed kbs K and any fact a al lowed in

is not complete but it is So cratically Complete for any

K if there exists a fact f al lowed in K such that

fact which is a logical consequence of a knowledgebase

K a f K a neg atef then

ALL ALL

there exists a series of preliminary queries and assump

K neg atea

ALL

tions after which a query of the fact will succeed To

of Sketch Intuitively the series of op erations Pro

prove this we rst dene the semantics of ALL by a

deriving neg atea is one which rst assumes a and

mapping to predicate calculus and then dene prov

then derives f and f one can show such a series

ability in ALL and show it satises the pro of rules

neg

9

exists since K a f and K a neg atef

Mo dus Ponens and Reductio Ad Absurdum From these

ALL ALL

This leads to a contradiction causing the assumption

pro of rules we show So cratic Completeness

a to b e retracted and neg atea asserted

Mapping ALL to predicate calculus is straightfor

neg

ward each knowledgebase in the structure is b elieved

Lemma Modus Ponens For any wel lformed kbs

under its assumption and all assumptions under it

d or ifneeded rule top k bK such K any ifadde

For any kbs K such that K ha K i ha K i

n n 1 1

b b and any ground substitution that Ant

1 n

V

such that v ar s domain if i i n

C K i i n P

onseq K b then K C

ALL i ALL

j j i PC a PC K

j i

Pro of Sketch i i n K b

ALL i

implies that for all i there is some such that b

i i

8

The assumption management techniques used in this

top k b K Intuitively one can app end these

i i

formalism have b een chosen to make the formal develop

together and pro duce some such that Ant

ecient techniques ment as transparent as p ossible More

top k bK formally a bit more work is required

are used in the implementation of ALL The ma jor dier

but one can show that such a exists Finally if

ence b etween the formalism and the implementation is that

a knowledgebase contains the antecedent of a rule

onological backtracking while the im the formalism uses chr

then we know from the study of ALL without negation

plementation uses dep endency directed backtracking In

Crawford that there must always exist a series of

chronological backtracking when a contradiction is found

the most recent assumption is retracted In dep endency

queries deriving the consequent of the rule

directed backtracking the dep endencies of facts in the

9

knowledgebase on assumptions are explicitly maintained Actually such a series may not exist as some other acci

often as lab els on the facts Stallman Sussman dental contradiction may o ccur b efore f and neg atef are

When a contradiction is found an assumption which the derived Such contradictions are not a problem however

fact dep ends on is retracted as they only cause neg atea to b e derived more quickly

For any query or assertion O allowed in a kbs K

neg

If k bO top k bK is fconsistent or heig htK then

k bK pushhtop assumpK k bO top k bK i r estK i O K hsubO top

neg

else an inconsistenc y has b een found and there is an assumption to drop

O K h k bsasser t neg atetop assumpK r estK i

neg neg

For any fact a allowed in a kbs K assuming a requires adding a new pair to the stack and then asserting a

k bK i K assumeaK asser t apushha top

neg

Figure Formal denitions of ALL op erations

We also formalize the idea of limiting the depth of Proving So cratic Completeness is now only a matter

assumption nesting Consider a kbs K and series of of using lemmas and to show that PC K j PC f

op erations allowed in K O O implies K f This can b e done using a standard Let K K

1 m ALL 0

Henkin style pro of Hunter We use the shorthand and K O K

1 i i i

max heig htK M ax i i m heig htK

i

Theorem So cratic Completeness for ALL

neg

If K is a wel lformed kbs of height one and f is a fact

For any kbs K any fact f al lowed in K and Def

any n al lowed in K then PC K j PC f K f

ALL

n

K f

ALL

Time Complexity

i K f by some series of queries and assump

ALL

We show in Crawford Kuip ers Crawford

tions such that max heig htK n

that ALL op erations can b e p erformed in time p oly

Finally we dene parameters which measure the size nomial in the size of the accessible p ortion of the

of a kbs Consider a kbs K Let l en b e the maxi knowledgebase ALL op erations require only a lin

neg

mum length of any rule in K mv ar s b e the maximum ear numb er of ALL op erations and thus must also b e

numb er of variables in any rule in K and ma b e the computable in time p olynomial in the size of the ac

maximum arity of any relation allowed in K Further cessible p ortion of the knowledgebase

let r b e the numb er of rules in K f b e the numb er

of frames eg constants in K and s b e the numb er

the Preliminary Op erations Generating

of slots eg relations in K One imp ortant measure

Consider a kbs K and a fact f logically entailed by K

of the size of K is the b ound on the numb er of facts

So cratic Completeness guarantees that there exists a

which could b e added to K by any op eration or series

series of op erations deriving f but says nothing ab out

10

of op erations

the diculty of nding such a series Intuitively such a

ma

m s c

series makes assumptions and then p erforms queries to

uncover contradictions In this section we show that

The other imp ortant measure is the b ound on the time

the depth of assumptions nesting in the series eg

complexity of any single ALL op eration Past work

the maximum height reached by the kbs as the series

has shown that the time complexity of any ALL op

of op erations is p erformed determines the complexity

eration is b ounded by a p olynomial function of the

of generating the series

size of the accessible p ortion of the knowledgebase

In order to state this result formally we need some

The size of the accessible p ortion of the knowledge

additional notation

base varies from one op eration to the next but it

clearly cannot exceed the b ound on the p ossible size

Def For a kbs K let be the series of queries

K

of the knowledgebase From the previously derived

of ground instances of consequents of the backward

b ound on the time complexity of ALL op erations

chaining rules in K

Crawford Kuip ers Crawford one can show

is imp ortant b ecause one can show from lemmas

that for a kbs of height n or less the time complex K

used in proving the So cratic Completeness of ALL

ity of any ALL op eration p erformed on K or on any

without negation Crawford that iterating a

K

10

sucient numb er of times is sucient to derive any fact

Such a b ound exists since ALL op erations never create

new frames or new slots which can b e derived without making assumptions

A comparison b etween ALL and is illuminat

Function P r ov eK f n

ing ALL implements classical negation ie nega

Until K unchanged do

tion as in predicate calculus while Prolog implements

K K

K

negation as failure ie the negation of a goal succeeds

Unless n

i the goal fails In ALL the negation of a fact is im

For each fact a allowed in K do

plied by a knowledgebase i the fact is false in all

K P r ov e assumeaK a n

mo dels of the knowledgebase ALLs implementation

if heig htK heig htK then K K

of negation allows one to express disjunctions eg

o d

John is the banker or the lawyer which cannot b e

o d

expressed directly in Prolog Negation as failure is pro

vided in Algernon the lisp implementation of ALL by

a syntactically distinct form

Figure Algorithm for determining whether

n

A complete natural deduction system or resolution

K f

ALL

based theorem prover with a b ound on pro of length

is trivially So cratically Complete Such systems dier

from ALL in several imp ortant resp ects While nat

ural deduction like inference rules do app ear in the

kbs pro duced by a series of ALL op erations on K is

metatheory of ALL single op erations in ALL do not

b ounded by

corresp ond to a b ounded numb er of applications of

5 2+6 mv ar s+ma

o n r l en m

these rules In a single ALL op eration rules may for

ward and backward chain to a depth b ounded only

Thus we have

by the size of the accessible p ortion of the knowledge

Theorem For any kbs K any fact f al lowed in K

base Thus any deduction which follows simply by rule

and any n there exists an algorithm with worst

chaining will succeed in a single ALL op eration This

2n

case time complexity of order o m for determining

gives a considerably simpler picture of the p ower of a

n

whether K f

ALL

single op eration than is p ossible in a system in which

tractability is guaranteed by a b ound on pro of length

Pro of Sketch The algorithm is shown in gure

Further when preliminary op erations are required the

It p erforms a search for a series of op erations deriving

diculty of nding a series of preliminary op erations

f A more ecient algorithm could certainly b e found

is related to the depth of assumption nesting which is

but our fo cus here is on illustrating the dep endence

a considerably more intuitive metric that pro of length

of the time complexity of inference on the depth of

Recen tly there has b een growing interest in dif

assumption nesting

ferent asp ects of tractable inference One interest

The analysis of the time complexity of P r ov e is

ing result is that the tractability of inference rules

straightforward one need simply notice that b oth

is dep endent on the syntax of the language used

lo ops can b e executed at most m times The key cor

McAllester et al McAllester p

rectness prop erty for P r ov e is that

Shastri Ajjanagadde show that using a highly

n

K f f P r ov eK f n

ALL

parallel implementation certain queries can b e an

swered in time prop ortional to the length of the

can b e seen by observ The reverse implication

shortest derivation of the query Term subsumption

ing that P r ov e only p erforms ALL op erations and

languages overviewed in PatelSchneider et al

the depth of assumption nesting never exceeds n

also supp ort a particular kind of tractable inference

The forward implication is more complex As

n

the computation of subsumption relations b etween de

f This implies that there exists some sume K

ALL

scriptions ALL provides a tractable inference metho d

such that f K One can show that every

which applies to a network structured knowledgebase

query and pro of by contradiction p erformed by is

and which is p owerful enough to guarantee So cratic

also p erformed by P r ov e or is unnecessary and so

Completeness

f P r ov eK f n

Current and Future Work

Related Work

ALL cannot currently represent mixed existential and Work on vivid reasoning has similar goals to ALL A

universal quantication When mixed quantication is vivid knowledgebase trades accuracy for sp eed

allowed inference can result in the creation of a p oten Etherington et al by constructing a database

tially unb ounded numb er new frames The key require of ground facts from statements in a more expressive

ment for tractable inference with full quantication is language ALL represents all the knowledge that has

thus careful control over the creation of new frames b een asserted though some of it may not b e accessible

We exp ect that commonsense reasoning requires the at a given time while a vivid knowledgebase is an

creation of a relatively small numb ers of new frames approximation of the asserted knowledge thus So cratic

while the cleverness required for more complex reason Completeness do es not hold for vivid reasoning

and tractable reasoning preliminary rep ort IJCAI ing often involves knowing which sets of new frames to pp

construct

Algernon our Lisp implementation of ALL is an

Fine Kit Reasoning With Arbitrary Objects Aris

totelian So ciety Series Volume Basil Blackwell Oxford

eective knowledgerepresentation language which has

b een used to implement several substantial knowledge

Grub er T Pang D and Rice J Ontolingua A Lan

based

guage to Support Shared Ontologies Technical Rep ort

Stanford Knowledge Systems Lab Palo Alto California

systems including QPC a qualitative mo del builder

for QSIM Crawford Farquhar Kuip ers Alger

Hayes Patrick J The logic of frames In Frame

non has b een an integral part of our work on ALL Understanding ed D Metzing Conceptions and Text

Walter de Gruyter and Co Berlin pp Reprinted

since our research metho dology involves an interplay

in Brachman Levesque pp

b etween theory and exp erimentation for example the

Hunter G Metalogic An Introduction to the

imp ortance of the depth of assumption nesting was rst

Metatheory of Standard First Order Logic University of

observed in solving logic puzzles using Algernon The

California Press Berkeley CA

natural language group at MCC has recently b egun us

Levesque H J Knowledge representation and rea

ing Algernon and has develop ed functions which trans

soning In Ann Rev Comput Sci Annual Re

late from an abstract knowledgebase interface based

views Inc Palo Alto California

on Ontolingua Grub er down to Algernon queries

McAllester D Givan R and Fatima T Taxo

and assertions Barnett et al Algernon is some

nomic syntax for rst order inference In Proceedings of the

what more p owerful than the currently formalized ver

First International Conference on Principles of Know l

sion of ALL and supp orts full quantication and cer

edge Representation and Reasoning Morgan Kaufmann

tain typ es of default reasoning

Los Altos California pp

PatelSchneider PF et at Term subsumption

References

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Barnett J Rich E and Wroblewski D A func

11

tional interface to a for use by a natu

An Optimally Ecient Limited Inference System

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