A PR OPOSED NUTATION FO R THE ADV I CE TA~ E R

by Er Ik Sandewall

"

I NDEX ==== = 0 , Int rod uct i on

1, Rep r esentat i on of facts In s i de a situa ti on . Repre sentation of kno wled ge,

2, Samp le en cycloPaedia for facts inside a situation, 2A , Dedu ctio n schemata for " Tha, Any , Some 28 , Gene r al-purPose axioms 2C , Speclal-purnose axioms

3, Rep r esentation of transitions between situations, Causality, The frame problem ,

4, Sample en c yc l oPaedia for transitions between situat ion s, 4A, Rules for uPdating situations 4B , Special-purPoSe axioms

5, ExamP l es of PrOblem - solving

6, Assorted problemS; 6A , The problem of 'Un l ess' 68 , The pro blem of undetermined Re sult 6C , The Closure problem

7, Imp l ementat i on

.. " "

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0, INTRODUCTION ================

This paper presents a set of relations, functions, and specIal operators for use In an advice taker, I,e, a deduction prOgram which reasons about common~sense situations, solves common~sense problems (such as the 'hoW do I get to the airport' problem), etc, We are not particularlY concerned about finding a notation whloh also Is sufficient fOr physical laws (such as the law of failing bodies) or other Information usually expressed In formulas,

The work here Is In several respects sImIlar to work towards a representatIon of thA 'meaning' of natural"language sentences, except that we shal I be a lot concerned wIth deduction rules Which wi I I enable the system to perform non~tr'v'al deduotlons (this Is a problem on Which lInguists do not seem to spend too much effort), and We sha II be I ess concerned about f' nd I ng a notat i on that W II I be adequate f~r re ... express I ng a.1 I (Or near I y a II) natura~~language sentences,

We shal I be ooncerned both wIth faots that hold "nslde' situations and faots that relate Situations to each other. 'The monkey Is In the corner' Is an example of the 'facts Inside sItuatIons'; 'If the monkey gOes out In the rain, he will be wet' ,S a fact that relates between situatIons, 'Eternal' facts (e,g, 'rock Is Inanimate' wi I I be treated among the facts 'InsIde; situations.

We shal I use different approaches for facts 'Inside' and 'between' situations, The facts 'inside' situations wi I' be handled' In a many-sorted first-Order oalculus, where only some smal I gadgets (like a oounterpar t of the Iota operator) have been added, 'facts 'between' situations are handled by some extraloglcal operatIons Which enable the sYstem to move between sItuations,

The notation for facts "Inside' situations Is Intended for c6n­ventlonal Questlon"answerlng, Since the notation Is (almost) a flrst"order calculus, It can east Iy be handled by present .. day theorem-proving prOgrams. (It should be noted that D. Luckham et ai, at Stanford are presently developIng programs for theorem­proving In many-sorted logIc), The notatIon for transformations 'between' situatIons 1s Instead oriented towards 'problem-solving',

We claim that the notation proposed In thIs paper has the fol lowing advantages over the notation proposed In the earl fer advlce"taker paper (McCarthy & Hayes, 1969): (a) It covers a wider spectrum of semantic (= natural~language-type)

concepts; (b) It oan be used on existing theorem·provers; (c) It contains sOlutions or at least, partial remedies to some

classical advice taker problems, ViZ, 1, the frame problem; 2. the opacIty problem for funotlons/relatlons whIch express

knowledge, belief, etc,

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3. the problem of how to represent ~knowftwhat'(as dls~lngu,shed from 'know-that' or \know~whether'). (ThIs Is the" problem where (McCarthy & Hayes suggest the use of functIons suoh as 'ldea~of-telePhoneNnumber'). "

We shal I give some cOmments on ho~ an ImplementatIon of the system proposed here could be performed, but we shal I not be able to report any existing implementation,

1. REPRESENTATION Or FACTS INSIDE A SITUATION. REPRESENTATION OF KNOWLEDGE -_ ... _--------------------

The purpose of this seotlon Is to develop a flrst"order oalculus for describing a person's knowledge of simple, speolflo facts 1n the real world, By 'specIfIc facts', we hint that we wish to represent

'Peter knows that JOhn Is ta III but not necessar IIY

'Peter knows that If x Is tal ter than y, and y Is taller than z, then x Is taller than z'.

In our calculus, A notation for representing the simple f act s t h ems e I ve s (e, 9 " \ J 0 h n 1st a I I ') I san e c e s sa r y pre r e qui sit e ,

An obvious requ'rement is that the developed notatIon shal I generalize easilY to other concepts that are similar to 'know', e.g, 'belleve', 'remember', 'tell', etC, .

Let us first Introduce the notation for the 'simple facts',

Sorts

We use flrstsorder predicate calculus with at least three sorts: 1. ObJeots (for physIcal objects, persons, etc,) 2, Properties (for Counterparts of nouns (except proper names),

adJectives, and some verbs) 3, Events (for hypothetical or real events In the world, e,g,

"that IJk Is Peteris tel-nr" "that the monkey Is under the bananas" "that the monkey Jumps to the ce II I n9"

More sorts will be introduced later,

The relation IS

IS Is a relation on Cobjects * propertles~, and assigns a property to an obJect,

Examples: John IS bOy peter IS father IJk IS tel-nr

Modlfloatlon functions: OF, AT, TO".,

These are functionS (properties ~ objects ~ propertlesJ, They are used more or less fntultlvelY, followIng the pattern of" natural language (Eng II sh) t

Examples: IJk lJk John peter

IS tel-nr OF John IS ((te I .. nr OF John) AT

IS walking TO school IS father OF John

home) --...... -.-----------AS barefoot

Expressions Involving several modifIcation functions are parsed with left association.

Notational conventions

----~-----------------Before we proceed, we had better specify clearly the notational conventIons that were used In the previous paragraphs, and that wIll be used I n the seQue I.

Binary functions and relatIons are usually written Infixed, and with capital letters throughout: TO, wERE, IS, F"unctlons have higher priority than relations,

Monary functIons and relations (and operators, see below) are written prefixed, and with an InItial capital letter: When, Move, lstrue, The, Any. Monary functions have hIgher priorIty than binary ones, The arguments are not necessar Ily enc losed by parentheses,

Parentheses are used freely to clarify or modify the order of application of fUnctions or relations,

Constants and variables for objects and properties are written with smal I letters throughout, Variables are written wIth only one letter.

Some binary functiOns wi I I be wrItten In prefix or In 'blflx' form (the algol constrUction IF a THEN ~ Is an examPle of a blflx), The blflx notation wI I I be Introduced and defined In section 2,

Verb fUnctions: Give, Move, etc,

----------~---~---------------~-~-

Intransitive verbs can usually best b~ represented as properties, I I ke I n the examp I e "peter Is wa I k I n9 to schoo I barefoot" t tor transitive verbs, one maY feel that the dIrect obJect Is an Indispensable part of the constructIon, It Is then better to let It be an argument Of the verb, than the argument of a modt~ flcatlon functIon, Transitive verbs therefore show UP as functlons (objects ~ propert1esJ.

Example: peter IS GivIng (could be parsed like 'x ~

fldo TO mary f(y) + z')

A s I mil ar dev I ce can be used I f more than one part of speech Is believed to be Indispensable for a verb,

The function WERE -~----~---~------

One more preliminary definition remains before we proceed to the notation for knowledge, The function wERE, (objects ~ properties ~ eventsJ Is used to construct an event fro~ an object and a property, Thus

321~578 WERE tel-nr OF John Is a functional term, whIch stands for an event (and therefore an Individual), WERE Is not a relation: Its use wi II Immediately be "bvlous,

Representation of 'know' ----~-~------~--~-~-----

We use the following functions:

Knowing [proPerties ~ propertlesJ Knowing-whether (events ~ propertlesJ Knowlngpthat (events ~ propertles~

We say that a person knows a property, Iff he can determine fOr each object t (given by Its name (assumed to be unique), rather than by a description), whether t has the property, This knowledge could cOnceivably be Implemented e,g. by main­taInIng a lIst of all objects that have the property (or of those that do not have it), A person Is said to know whether an event Iff he knOWS whether It Is true. He Is said to know that the event, Iff he knows whether It, AND It Is true,

None of these functiOns shall be used for 'be aCQuainted with', as used In e,g, 'I know John', ThIs would even be a violation of types, If we later Introduce a function ACQualnted"wlth, It Is to take objects, not properties, as arguments,

Examples: peter IS Knowing (tel-nr OF John) peter IS Knowlng""whether ( I J k WERE tel",nr OF John) peter IS Knowlng .. that C I Jk WERE te!-nr OF John) dJck IS Knowlng .. that (peter WERE Knowing

(tel"'nr OF john) )

In order to relate Knowing-whether and Knowlng ... that In a fOrmal way, we need

The property 'truel

-~----------------~-

'true' Is the property an event has when It does Indeed 6ccur In the real world~ Thus

x WERE y IS true - x IS y

We can then define KnowIng-that In terms of Knowlng-whetherl x IS Knowing-that y -

[y IS true A x IS Knowlngwwhether yJ

Notice that we have tacltlY extended (objects u events) * Properties], and similar Iy.

IS to be a relation WERE has been extended

Ax loms and othe r comments about 'know I w I I I appear I n sect I on 2C. KNOW In the next chapter~

Property funot'ons: Qfprop, Atprop"., ----~-------~---------~-----~-~--~~-----

In order to handle e.g, 'Peter knows when John goes to school', we have for each mOdification function OF an associated . property funotlon Of prop Cevents ~ propertles~, satisfying

x IS (y OF z) _ z IS Of prop (x WERE y)

Example: John IS GIvIng fldo TO mary Is equivalent to

mary IS ToProp (john WERE giving ffdo) In natural language, the latter phrase would be 'marY fs the one John gives fldo to'~

Example: peter IS In natural language:

KnowIng Toprop (john WERE Giving peter knows whom John gives fido to,

If we wish to express 'it Is fldo that john gives to mary' In a simi lar fashIon. We should let 'give' be a property Instead of a verb function, and Indicate the given obJect with a modification functIon.

fldo)

The function 'The', Predicate 'Unique', and operators 'AnY', 'Some' -~------~---~------------------------------------------------------

In those examples where we translate simple natural-language facts Into our notation, we can gain much convenience by using the functions or oPerators The, Any, and Some.

The function The (Properties ~ objects] assumes that the argument Is a property which Is satisfied by exactly one obJect, and has this object as value.

The predfcate UniQue is defined on properties, and true for those properties which are assumed by exactly one object. Thus ~UnlQue pI Is a necessary requirement for using 'The P'. In fact, we shall later Introduce a deductfon rule whIch (roughly speaking) enables us to

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Any and Some are used for those oases where Is not guaranteed to satisfy the unIqueness

The expression 'AnY p' (where p Is a property) Is used as a free variable ranging over al I x such that x IS p.

The expression \Some p' 's eQuivalent to writing a new constant symbol (generated In a gensym~llke manner) pn, and stating somewhere that pn IS P.

There Is an obvious algorithm for rewriting expressions that Involve Any or Some Into clean predicate calculus,

'The' Is obviously similar, although not Identical, to Church's Iota operator,

The function Whatls --------------------Next, we need some way of handling situations where a perSOn knows (or believes", i) something about an object Which he knows by Its descrlpt On only, We Introduce the function 'Whatls' for this purpose, If p Is a property, then 'Whatls p' Is taken to mean 'the object (whatever It Is) that has property P', Or more crudely, 'the Idea of an object with property p'~ The function ~hatls eliminates the need for constructions such as \Ideawof-telephoneftnumber' Which are used In (Mccarthy & Hayes, 1969),

Exarr;ole 1: 'John's telephone number Is next to johanna's', vIz', 'peter believes that JOhn's telephone number Is next to Johanna's' can be represented as:

The (tel-nr OF (tell!9nr Of

JOhn) IS Johanna)

The

peter IS Believing [Whatls (tel~nr OF John) wERE Next-to Whatls (tel~nr OF Johanna)J

Notice that peter maY hold this belief without knowing John's or johanna's telephone numbers, Therefore, we should not wrIte 'The' Instead of 'Whatls' In the second expression,

ExaffiPle 2: ConSider the two eXpressions

and

peter IS Knowjng"whether (Whatls tel-nr OF John W~RE tel-nr OF dick)

peter IS Knowing-whether (The (tel~nr OF John) WERE tel-nr OF dick)

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If John's actual telephone number Is 321, then the first of the above sentences SaYS that Peter would be able to ansWer correctly the Question

'Do John and Dick have the same telephone number?' whereas according to the second sentence, he would be able to answer the questlon

'Is 321 the telePhone number of Dlok?', In vague words, If \The' Is used, then the desorlptlon Is 'evaluated' during the conversation between you and me, whereas the 'Whatjsi function performs a kind of QuotIng,

Questtons of transParencY ----~--~.---------~------

In every use of modal operators, such as \Knowlng~whether', the questJon arIses of what kinds of deductIons shal I be permitted among the arguments Of the operator, Shal I we permit sUbstitutlon of eQuals for equals? More precisely, shal I we permIt substitution of (a) what we know are equals, (b) what he Is known to know are equals, (c) what Is universally known to be eQuals, for eQuals? Shal I we assume that the agent who 'knows' also oarrles on logIcal deductions soundly and completelY (at least as completely as we do)? If not, how can we descrfbe what amount of logical reasonlng he can perform? Etc, etc.

The notation whloh has been gJven In thIs section Impllcttly specifies our apprOach to these problems, Let us now specify this approacn eXPlicitly, I * S 0 mer e fer e n c est 0 Phi los 0 phi c a I ~I 0 r k s h 0 u I d bel n s e r ted her e .. I

We do have transparencY with respect to equality (represented by the Infix =), However' many things Which could have been expressed using eqUal ity (e.g, John = father(peter» are Instead expressed In our notatJon using properties and the relation IS, which of course Is handled with opacity, This Is what keeps us from getting In trOUble with examples such as the 'Walter Scott paradox'.

The reader may ask whethAr we ever use eQua II ty at a II. we do, but only to IdentIfy 'analytically equal' concepts, An example: If Q and R are modification functlons,we have

p R m Q n = P Q n R m This equality can be USed to Identify 'peter knows the telephone number of John at home' and 'peter knows the telephone number at home of john', Equalities of a similar trivial nature can be oonstructed for the simple functions 'AND', 'OR', etc. Which we shall use for combining events Into larger events Clf m knows that (e and f), then he knows that (f and e».

With these conventions, every equal tty that we set UP carries with It a statement that the agents who 'know' also know about this equality, But clearly, the amount of deduotlon

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which oan be performed with this strong eQuallt~ will be rather limited.

However, there exists a more generous device for enabl In; deduct 1 on 'I n' the arguments of 'Know I ng' and 'Know I ng.-whether'. Let us first Introduce

The relations SUB and EXCLUDES

-----------------~~---~----------If p Is a more specIalized property than '1, we write , p SUB '1', 1 t f 0 I lOw 5 . (p SUB QJ ~ [(V x) x IS p ~ x IS Q~

Examples: boy SUB man tel-nr OF John SUB tel-nr

If p and Q are mutuallY exclusive properties, we write 'p EXCLUDES Q'. I t f 0 I lows - [p EXCLUDES QJ ~ [(V x) ~ (x IS P A X IS Q)~

We shal I only use these relations In those cases where the sub-ness or excludes-ness Is 'analytic' or 'Inherent In the concepts themselves' or (using less abstract terms) can be assumed to be known about b~ a II agents who 'know' at a II, (This Is wh~ we have ~ rather than - In the above axioms), Thus we shal I have the axiom

~ IS Knowlng.that (n WERE p) A p SUB Q ~

m IS Knowing-that (n WERE a) which of course Can be abbreviated Into

p SUB Q ~ Knowjngthat(n WERE p) SUB Knowlng~that(n WERE Q)

If a relation of the "P SUB Q' typo Is considered to be (for practIcal purPoSeS) non-anal~tlc, I,e. If we do not trust everybody with knowing about It, then we write Instead 'Any p IS a', Similarly, a non-anal~tlc 'p EXCLUDES '1' oan be phrased as 'AnY p IS Not '1', where Not Is a function whioh constructs the negation of a property (cf, sectIon 2C,COMP), Expressions about knowledge are clearly opaQue wIth respect to such relations,

It ~ay of course be hard to determine whIch relatJons between propertIes shal I be knIghted as'analytlc', by being expressed with =, SUB, or EXCLuDES. A relatIon such as 'All whales are mammals' should prObably not be analytIc. An eQuality such as 'The sum of 25 and 67 Is 92' probably should, although some people may be unable to perform the addItIon. If we declare lt anal~tlc, we Immediately forfeit the ability to say 'm believes that the sum of 25 and 67 is 82',

The approach to transparency which has been outlIned above Is certainlY not Ideal, but we thInk It wi I I be sufficient for handl In9 at least s~me simple common-sense deductIons,

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Use of higher-order logic

----------~-----~~"------

The notat Ion wh I ch has been given In th I s chapter can eas I l'i be re-wrltten In higher-order logic, Taking that step upwards would actually have some advantages: The relation between modlflcatlon functions and corresponding property functions (Of vs, Of prop) can be ffiade more expljcjt, and Inversion of verb funotlons (for expressIng 'he knows what Peter gaVe to MarY') becomes easy to do. By and large, however, It would Seem that the many-sorted flrst"order notation and the higher-order notation Is equivalent, With present technology In the field of automat'jc theorem-proving, we should then Prefer the first-order notation.

Conclusion

In this section, we haVe Introduced a notation for expressing knowledge, belief, etc. The notatJon Is essentiallY based on the trlok of writIng simPle facts on the form 'm IS p', where one could conceIVablY have written 'pCm)' Instead, We have worked out a few different types of 'Kno~' ('Know~whether' etc.) to some detal I,

The Idea of writing 'm IS p' had the further advantage of enabling modification Of properties Ccf. 'peter IS father' vIz, 'peter IS father OF John'), This advantage wi I I be further exploited In the next Section,

A third advantage, whiCh will appear In a later section, Is that the same Idea gives us a trick for handling the frame problem,

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2. SAMPLE ENCYCLOPAEDIA FOR FACTS INSIDE A SITUATION =================:=:=:=:=============================

When a Question Or a problem Is given to an advice taker or another similar system, we clearly ,.,Ish (In the long run) that the problem statement shal I consist only of very speolflc statements {'In a roOm 15 a monkey and a box'), More general sta tements (' I f a monkey I sat a box, he can c I I mb It') sha I I not need to be part of the problem statement, but shal I be kno~n to the advice taker beforehand.

We should ask, therefore, what general axioms are necessar~ for such a system, and equally Important, how we can select functions and relationS so that the amount of knowledge that has to be stored aWaY Is held reasonably finite, The purpose of this section Is to 1 I IUstrate how this problem can be tackled, for some classes of semantic Information (e,9, knowledge, . spatial location, comparison of adJectives),

In general, when the advice taker (or whatever) starts to work with a Question Or a aujzz, It will need the followIng background machinery:

A, Deduction schemata for =, The, Any, and Some 8, General-purpose axIoms (for IS, WERE, modlfloatlon

functions, etc,) C, Axioms for the not~so-general functions and relations

such as Know-that, At, Move, etc, ('encyclopaedic axioms')

In this section, we shal I give tentative versions of A and B, and a sample of C. No clalm of completeness Is made, We be! leve, but have not proved, and do not Intend to try to prove, that the given axIoms are cOnsiste~t,

NotatIon "IP-II!'I'-U-~

k,m,n p, a, r d,e,f x,y I , J a,b,c Q,R Rprop I,h s,t,u v,w s(x)

objects propertfes events objects, Properties, or events objects or events properties Or events modificatIon functions property function corresp. to R locations time-segments Integers expression Where x Is one occurrence of a sub-eXpression

Section 2C 15 divided Into Sub"5ec t tons, which have Indlv,ldual, more or 1$55 mnem6nfc names (COMP, KNOW, eto.~,

In some subsections Of 2C (e,g, CADJ, we use b I f I x fUnctIons, A function Is blf1xed Ii It Is Introduced In the form

More " THAN ., In suoh a case, We reallY mean to have one binary functl6n MORETHAN of two arguments, and We write

More tal I THAN peter when we mean

MORETHAN (ta I I, pete r) An Inflxwto~preflx translator whfch also takes care of blf)xes fa descrIbed In appendIx A~

2A. DeductIon schemata for =, The, An~, Some ----------~-------~---~------------------------~~

C 1 )

( 2 )

x = y,

m IS p,

SCx) Sty)

SCm)

(3) Let [S(The p), CJ be a clausel C being empty or a new (posslbl~ unit) Clause, We may then add the clause

rUnIQuecp), CJ to our set of Clauses

.( 4 ) m IS p, S(An~ p) 1- SCm)

( 5 ) S(Some p) 1-' (3v) v IS p " s(v)

Remark: In (5), only ONE occurrence of 'Some p' In S can be substituted fOr at a tIme. 'v' can be selected as any variable which does not occur In S or P.

Some fUnctions (propertIes" properties] wi' I be extended to functions [events" eVentsJ In a general way. We use the sohema

J WERE rn p = Fn (J WERE p) Similarly, for a binary function FN,

J WERE (p FN q) = (J WERE p) FN (J WERE Q) or, If only the fjrst argument Is to be extended,

J WERE (p FN q) = (j WERE p) FN 0 Finally. for a blnar~ relatIon R,

[p R oJ = C(~x) (x WERE p) R (x WERE q)J

28. General ... purPose ~xjoms --~-~-----~---"---"-~-~-~-~

( 1 ) WERE p Op true - I OP P (where OP = IS, BEGINS, ENDS)

(2) e WERE true = e

( 3 ) Cp SUB QJ :l [(V I ) I IS p ::». IS Q~

( 4 ) p R m SUB p

(5 ) p R m Q n = p Q n R m

( 6 ) m IS p R n - n IS Rprop (m WERE p)

( 7 ) ( p EXCLUDES q] :;) ( ( V I ) ... ( , IS p 1\ IS q)J

( 8 ) C(VJ)(VJ) i IS p " J IS p :;) = JJ - Unique p

1 '1J

2C, SpecIal-purPoSe axioms

The axioms under headlng (28) were all concerned with the system functions and relations, such as WERE, BEGINS, etc, In. ... thls sectIon (2 C), We shall give examples of axioms con" eerned with oonstants, verb functions, etc, The axioms are sorted UP under dIfferent head'ng~, acoordlng to topic, The laYout has been seleoted so as to make the axioms readable from LISP!

(AXIOMS COMP) fOr cOmPounding properties and events

(COMMENT) We use the fun c t Ion sAN D ,OR, and i~ 0 t to b u I I d com p 0 sit e events from simpler events Inside a situation, and THEN to bu I I d compos I te events tha t cut th r Qugh s' tua t Ions, THEN has lower pr lor I ty than AND, OR, and Not; and a II of these have lower priority than functions Introduced earlier. The priorities between AND, OR, and Not are as usual.

Example: monkey1 IS Plannln9to (monkey! WERE Moving box1 TO Under bananas1 THEN mOnkey1 WERE C! Imbl n9 boxl THEN monkey1 WERE ReaChlng.for bananasl)

We extend these functlons to apply to property arguments, We obVIouSlY have

x WERE y THEN x WERE z = x WERE (y THEN Z) and similarly for AND, OR, and Not, This follows from the rules In section 2A,

Example: monkey1 IS Planning

THEN Climbing boX1 (Moving boxl TO Under bananas1

THEN Reaching-for bananas1)

THEN Is taken to mean 'immediatelY after' rather than 'then, at some later time'.

The set of axioms given here Is obviously Incomplete,

(COMMENT)

(COMP 1)

(CaMP 2)

(COMP 3a)

(COMP 3b)

(COMMENT)

(COMP 5a)

(COMP 5b)

(COMP 5c)

axioms that relate AND, OR, THEN, and Not to each other

conventional rules for AND, OR, Not

assocjatlvlty for THEN

(a OR b) THEN c = (a THEN c) OR ( b THEN c)

c THEN (a OR b) = ( c THEN a) OR ( c THEN b)

axioms that relate AND, OR, Not, and THEN to the system relations

( e AND f ) IS true -e I S true A f Is true

( e OR f ) IS t.r ue -e IS true v f IS true

(Not e ) IS true - ... ( e IS true)

~. .~.

l I ~

1

(COMMENT) THEN has to be hand I ad in ohapt~r 4, with· the tj me"d 9p9ndentfacts.

(AXIOMS KNOW) for knowledge and transmission of knowledge

(COMMENT) we use the following functions: KnowIng [proPerties ~ properties] Knowing-whether [events ~ properties] Knowlng~that [events ~ propertles~

Due to heavy editing, the axioms In this section are not numbered concecutjvely.

(KNOW 6) m IS KnowIng p ('Ik) m IS Knowlng .. whether (k

(KNOW 11) m IS Knowing pAm IS Knowing Q ~ m IS Knowing-whether (What,s p IS Q)

(KNOW 5) m IS KnoWIng-that e e IS true A m IS Knowing-whether

(KNOW 3) p SUB Q ~ Knowing-that (m WERE p) SUB Knowing-that (m WERE q)

(KNOW 8) Knowing-whether (p AND q) SUB (Knowing-whether p AND Knowing-whether Q)

(KNOW 9) Knowing-whether (p OR q) SUB (Knowlng"whether p AND Knowing-whether q)

(KNOW 10) Knowing-whether (Not p) = Knowing-whether p

( KNOl~ 7) Knowing (p AND q) SUB (Knowing P AND Knowing

(KNOW 12) Knowing (0 OR Q) SUB (Knowing p AND Knowing

(KNOW 13) Knowing (Not p) = Knowing 0

e

g)

Q)

(AXIOMS ET~U) for the connectives ET and ~U.

(COMMENT) The function ET Is used to construct composite objects from stmple obJects, fOr use e,g, In constructions such as 'Peter and Mary are marrIed',

In Engl Ish (I Ike in several other European languages) there Is a number of eQuivalent formulatIons suoh as

Peter and Mary are married Peter Is married to Mary Mary Is marrIed to peter

Peter and Mary are QUarreling Peter Is Quarrelling with Mary

Peter and Mary meet In the cIty Peter meets Mary In the city -

, . , , , ,

We shal I make universal use of the connective ~U prepositions used In natural English (to, with, _, Would write e.g,

peter IS married tU mary

peter peter

IS (meeting ET mary IS

IN The city) ~u mary meeting IN The city

for the varlous ,.,), Thus we

Moreover, we use a special (lexicographic) predicate tuable to mark those properties (married, meeting, .,,) whIch can be treated In this way.

We easily obtain the fOllowing axioms:

(ETi!U 1) m ET n = n ET m

(ET;!U 2) (m ET n) ET k = m ET ( n ET k)

(ETi!U 3) m ET m = m

(ETtU 4 ) ~uable P :J

m WERE p ~U n = m ET n WERE p

(AXIOMS LOC) spatial location

(COMMENT) We shall In principle follow the phi losophy In Schank - Tesler ~ Weber, page 78.

We Introduce a new sOrt, LOCATION, and the fol lowing functionsl

Loc [locations .. propertles~

Inside [objects .. locatlonsJ Outside -,,-Near -,,-Farfrom -"-Atlnslde .. "-At -I, "" Upon -"-Under -,,-Above .. 1,_ Below -,,-Beside -"-Between [objects .. 10catlonsJ

A location Is thought ot as [1] having an EXTENSION In space, and [2J optionally, having an IMPERATIVE SURFACE, An object 'S .thoUght of as hav I n9 an OUTSIDE, an INSIDE, and an EDGE. We write 'm IS LOc I' Iff [lJ the object m 15 contained In the extension Of I, and [2~ the edge of m has some segment In common with the Imperative surface of I, If I has one,

The fol lowing functfons generate locations with an ImperatIve surface: Atlnslde, At, Upon, Below, The other functions do not, The meaning of al I functions should be rather obvious.

The function 'Between; IS supposed to take argument of the for 'm ET n' or 'm1 ET m2 ET m3' •

We use a relation SUBL on [locations ~ 10cationsJ to describe location-Inclusion,

(LOC 1 ) [ I SUBL hJ .- [Loc SUB Loc hJ (LOC 2) Near m SUBL Outside m (LOC 3) Farfrom m SUBL Outside m (LaC 4) Atlnsl de m SUBL Inside m (LOC 5) At m SUBL Near m (LOC 6) Upon m SUBL At m (LOC 7) Above m SUBL Near m (LOC 8) Upon m SUBL Above m (LaC 9) Under m SUB'L At m (LaC 10) BeloW m SUBL Near m (LOC 11) Under m SUBL Below m

m IS LOC R n - n IS LOC R m (WHERE R = Near. Farfrom, At, Beside) m IS LOC Upon n - n IS Loc Under m m IS LOC Above n _ n IS Loc Below m

(LOC 12)

(LaC 13) (LOC 14) (LOC 15) m IS LOC R n ~ Between m ET n SUBP R n

(LaC 16)

(COMMENT) (LOC 17)

(WHERE R = Beside, Above, Under, Near) m IS LOC Above n ~

Between(m ET n) SUBP Under m axioms 15 and 16 are Incorrect, m IS LOC R n ~

~ k IS Loc Between m ET n (WHERE R = At, Inside)

(COMMENT) We should also have a relation EXCLUDEL on [locations" locatlonsJ to say that two locations are mutuallY exclusive, Axioms:

(LaC (LoC (LOC (LOC (LOC CLOC

31) 32) 33) 34) 35) 36)

h -EXCLUDEL

I EXCLUDEL Inside m Near m Under m Under m Above m

EXCLUDEL EXCLUDEL EXCLUDEL EXCLUDEL

Loc I outside

Farfrom m Above m Beside m t3estde m

EXCLUDES m

(COMMENT) Deduction USyng Loc axioms certainly needs to be supported by a natural model~

Loc h

(AXIOMS CADJ) comparison of adJectives

(COMMENT) we use the fOllowing functions

More , . THAN • • [properties * objects As AS - " -•• • • Less ~ . THAN • • - It -

.. propertlesJ

Themost t • AMONG . , (properties -It properties .. propert t es~. Theleast AMONG - " -, t ••

The meanIng of these functions should be clear.

To explain that two properties are each other's opposites, we I ntroduce a oonstant " oppos I te', wh I ch I s to be a 'i!uab Ie' (cf. section ETi!U) property on properties, It is to be used I Ike In the examples

peter IS ta I I ta II IS oPPosite 2!U short

We obviously also have

long IS opposite i!U short

and therefore we dO NOT have

ta II = The opposite i!U short

whereas we do have

short = The opPoslte 2U tal I

The general ET~U mechanisms enable us to say

short short

IS oPPosite ET tall Is

2U ta II opposite

Some examples of the new functions:

I S t~ 0 ret a I I T HAN J 0 h n IS Less ta II THAN peter IS As tall AS dick

peter john john peter IS Themost tal I AMONG (brother

Now some axioms:

(CADJ 1)

(CADJ 2)

(CADJ 3)

m

...

m

IS j

m

IS k

More P m IS

IS More

Less p IS More

THAN I< 1\

More p THAN

p THAN m

THAN k -p THAN m

OF dick)

k IS More p n

THAN n

'. I

(CADJ 4)

(CADJ 5)

(CADJ 6)

(CADJ 7)

(CADJ 8)

(CADJ 9)

(CADJ 10)

(CADJ 11)

(CADJ 12)

(CADJ 13)

(CADJ 14)

(CADJ 15)

(CADJ 16)

(COMMENT)

m IS More P THAN K h k IS As p AS n ~ m IS More p THAN n

m IS More p THAN K A m IS As p AS n ~ n IS More P THAN k

m IS AS P n IS As

AS n AS n

n IS AS P AS n

m IS Themost p AMONG Q A n 15 Q m = n v m IS More p THAN n

m IS Theleast m = n v m

('in) n IS q

p IS

AMONG Less

Q

P

A

THAN

m = n em IS

v m IS Themost p

More p THAN AMONG q~

ceVn) n IS q m = n v m IS Less p

em 15 Theleast p AMONG

p ET Q em IS

IS opPosIte More p THAN n

m I S AS ... m IS

p AS n :;) l"iore p THAN

2uable opposite

n

THAN Q~.

m

p EXCLUDES Any opposite ~U p

n n

nJ

n~.

IS

IS Q

Less a THAN n~_

m IS More p THAN n h p ET Q IS Opposite A UnjQue(opposlte ~u p) ~

m IS Less Q THAN m

O~ second thought, It Is probably better to change the notation and re~wrlte

m IS Themost p AMONG Q as

m = The Most P AMONG Q This makes It possible for several m In a to be 'Most p AMONG q', by writing

mj IS Most p AMONG Q

for each of these mi.

'. -,

(AXIOMS MADJ) measureS on adjectives

(COMMENT) we use the functions

Very Rather Slightly

[properties ~ properties] _ t, _

- " -We also assume that eyerY obJect has exactly one of the following properties:

Very p Rather p Slightly p Not p

for eyery 'basic property' P. We do not have p ET Q IS oPPosite ~ Not p = 0

because perhaps the properties p and Q are not at al I applicable to an object, Then It has tne property Not p, but not the proPerty 0, (For example, a stone would be 'Not happY; using these concepts),

Some axioms are:

(MADJ 1) (MADJ 2) (MADJ 3) (MADJ 4 ) (MADJ 5) (MADJ 6)

(MADJ 7)

(MADJ 8 )

(MADJ 9 a-c)

Very p EXCLUDES Rather p Rather p EXCLUDES Slightly p Very p EXCLUDES Slightly p Very p SUB p Rather p SUB P Slightly p SUB p

m IS verY pAn IS Rather p ~ m IS More p THAN n

n IS Rather p f\ k IS Slightly p n IS More p THAN k

f\ m IS Op p m IS As p AS n n IS Op p

(WHERE Op = VerY, Rather, S II ght I y)

(COMMENT) for several reasons, Including the dlffloultles with Very Rather Very Slightly p, we do not wfsh to haye an axiom such as

m IS p :;) m IS Very p v m IS Rather P v m IS Sllgntly p

- ,

(AXIOMS BIIM) references backwards In time

(COMMENT) the functions here are to be used for saying that something has happened, and PossiblY to describe when It happened, The underlying philosophy Is vaguelY similar to the one used for spatial location, We Introduce a new type, time~segment, Every event which takes place, has a duration of a certain time-segment, and we need functions for going back and forth between events and time-segments, We start with

Tlmeof (events ~ propertles~

IN

which IS supposed to generate properties that are taken by time-segments, Thus 'The Tlmeof e' Is the tlme~segment durIng which e took place, If It has only happened onoe.

DURING

(propertles * tIme-segments ~ propertlesJ

Cproperties * tlme"segments ~ propertles~

m IS p IN t means that m HAD the property p at some time (not necessarilY that he has It noW), and that the dUration of 'm WERE p' Is a subset of t, WIth DURING Instead of IN, we Intend that the duration of e was a superset Instead,

Examples:

peter IS swImming IN march23

peter IS sWimming DURING break2

The (teacher The Tlmeof

OF Peter) IS (peter WERE

Watching swimming

peter DURING IN yesterday)

However, since 'Tlmeof e' can be an amblgous description, we need a way to say 'the second time e happened' or 'the latest (3rd latest) time e happened', Functions:

TH (IntegerS * propertIes ~ propertlesJ

TH-LATEST Clntegers * properties .. propertlesJ

(Oh, we have Just estab II shed I ntegers as another sort, We II, f thad to happen sooner or later). The meaning of TH and TH-LAST is rather obvious, especiallY after we have given these examples:

Exarr.ples:

peter IS swImming DURING The 2 TH~LATEST Tlmeof (The class OF Petar WERE resting IN yesterday)

There are at least two reasons why the 'The' should not be bul It Into the functions TH and TH-LATEST: (1) perhaps the second argument of TH (, •• ) did not occur, or did not occur a sufficient number of times, Then our function would not have any meaningful

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In order to formulate the axioms convenIently, we need two auxt Ilary relations:

SUBT [time-segments * tlme-segmentsJ.

PREC [time-segments * tlme~segments~

for 'sub-segrnent' and 'precedes' (with zero over lap, but not nee e s sa r I y do vet a I lin 9 ) '. Now the a x 10m s •

(STIM 1)

(BTIM 2)

(BTIM 3)

(BTIM 4)

(BTIM 5)

(BTIM 6)

(BTIM 7)

(BTIM 11)

(COMMENT)

(BTIM 12)

(BTIM 13)

(BTIM 14)

(BTIM 15)

s SUBT t t SUBT u s SUBT u

s SUBT s

s SUBT t t SUBT s s = t

sPREe t t PREC u s PREC u

.. 5 PREC s

s PREC u

u PREC t

Tlmeof (n WERE p) SUB Tlmeof (n WERE Q)

actuallY, that axiom Is not correct

S PREC.< t u SUBT t

sPREe t u SUBT s

p SUB Q

s SUBT t

s SUBT t

s IS v TH [ v < w

p

(p DURING t) SUB (p DURING s)

(p IN s) SUB (p IN t)

" 5

tIS w PREC t]

TH p

s IS v TH-LATEST pAt IS w TH-~ATEST p C v < w _ t PREC s J

(COMMENT) one may feel that more axioms are needed

(COMMENT) Some of the above axioms mIx up 'analytic' and 'empiric' Information and are therefore Incorrect, This 15 the same problem as In section LOC.

REPRESENTATION OF TRANSITIONS BETWEEN SITUATIONS, CAUSALITY, THE FRAME PROBLEM, ~-------~-~---~-­-----------------

In order to do problem-solving In this system, we need some way of representing transitions between situations, There are many possible ways of doing that, e,g, (a) tack on a situation variable as a third argument of

the relation IS (b) extend the apParatus for references backwards In time

(BTIM) which WaS gIven at the end of the previous section, so that It also handles references forwards In time, causality, and condItional futures (which depend on the actlon(s) of the thinking agent (= the advice taker),

Pursuing any of these approaches, we would revisIt the cla s slcal advice taker problems, notably the frame problem, (For a description of the frame problem, see (McCarthy & Hayes, 1969», It seems to us that no satisfactorY solution to the frame problem has yet been given. In this section, we shall propose a new (we th'nk) method for handlln9 transformations between situations, This method Is not similar to the approaches (a) or (b) above, It has the Important advantage of offering a remedy for the frame problem.

We estab II sh the fO I I owj ng convent Ions:

1, For~ulas refer to One particular situation ('now'), If one wishes to refer to other situations (I,e. to what has happened, or what will happen), he has to use special operators (see below),

2. Properties are of two types: permanent or tranSient, If an object has started having a permanent property, It Is assumed to retain this property untl I the property Is eXPlicitlY 'turned off', using the ENDS relation (see below). A transient property, on the other hand, Is something which vanishes In the next time-step unless eXpllclty renewed.

3. Each property constant and each verb function Is assigned Its property type, Modification functions preserve this type. Permanenoy type carries over to events In the obvious way.

Deduotlons In the system must obviously handle time In a special way, The simPlest mode of deductive process (corresponding to the British museum algorithm) can be characterized as a simUlation forward In time, where conventional deductiOn IS performed Inside each tlmewstep. If we are deduolng/slmulatlng a world where no active actions are taken, we do as fo II ows: a. start with a set S0 of statements C= ground unit claUses) about

the Initial situation, and a set S of general facts about the world. CA

fact such as 'the monkey Is In the corner' would be In S0, whereas a fact such as 'If m goes from p to r, then m wI I I henceforth be at r' would be In S),

b, Run the British muSeum algorithm to generate S0* = the set of al I statements which are derivable from 50 u S but not from S0 alone. S0~ wi I I normally contain some 'unit clauses where the relation Is BEGINS or ENDS, See below,

c. Generate Sl from S0* as foilowsl el. remove a I I statements that ass I gn trans lent

propertIeS to objects, c2. update statements that assign permanent properties

to obJects, using the BEGINS- and ENDS-expressions In S0*.

03, remove al I statements that Involve any other relations than IS,

d. Generate S1* from Sl u S as In step b, and continue around the cYcle.

In problem environments where active actIons can be taken, each Sn* may contain sOme CANBE-expresslons on the form

x CANBE y In such cases, we add one more step under c, above:

c4. add optionallY to Sn+1 some expresslon(s) x IS Y

where Sn* cOntained the expreSsion )( CAN8E y.

Step 04 Involves a choice, so the simulation may go off In several dIfferent directions, and branch UP I Ike a tree. This Is then the problem~solvlng tree.

In practice, one obviOUslY does not generate the whole sets Sn* at the first shot, Instead, one searc~es the problem­solving tree (= the various possible futures), us'ng very Incomplete deduction In the step from Sn to Sn*. When a tentative solutIon to the given problem has been found, this solution Is analyzed mOre carefully,

The oprators BEGINS and ENDS

---~--------------------------~

BEGINS and ENDS are operations on [(objects u events) ~ propertles~.

From the point of view of deduction InSide a situation, they are perfectlY normal relations. Only In situation updating are they used In a special waY,

If 'x BEGINS pi Is derivable inside a sltuatJon, then thIs I s taken to mean tha t 'x IS p' sha I I be a sta tement In the succeeding sltuatlon, It does not Imply that 'x IS p' should be In the same situation as 'x BEGINs p', rurthermore, If ,~ x IS p' 'is a statement In the present situation, -,t shall be removed In the next situation,

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BEGINS overr,des ENDS, if both are found In the same situation,

Example of use: x IS candle

Another example: x IS LOC Beside y

x BEGINS Loc z

)( IS lit x BEGINS burning

x IS Moving y TO z y BEGINS Loc z

Other examples are in the monkey-banana thriller (below),

The operator CAN8E

------------------~ Is an operator on Similar to BEGINS and ENDS, CAN8E

(obJeots u events) * PropertlesJ situation, but whiCh gets special sItuations, Rule: It 'x CANBE then we are allowed to put 'x IS

which acts I Ike any relation treatment In the updating of

pI Is present In one sn*, pI In Sn+1.

We make It a convention that only one CANBE expression can be used for a certa1n x In any situation updating, If we want to say that \x IS p' and 'x IS q' can be permitted together, we use the AND function (below) to write

x CANBE p AND Q

Example: The monkey-banana story ~-~-----~---------~--------~-----

The following Is a partial description of the traditional monkey~banana situation, We give It as an example of the uSe of BEGINS, CANBE, etc., but the reader Is warned that some of the functiOns used here (Under, Beside, etc,) are pre-historic and incomPatible with the functions that were Introduced in section 2C,LOC.

1 , monkeyl CANBE going TO boxl

2 • monkeyl IS going TO P ~ monkeyl BEGINS Beside

3. monkeyl IS Beside x 1\ x IS movable " Y :::» monk eYl CANSE Moving x TO y

4 • monkeyl IS MOving x TO y :::»

monkeyl BEGINS At y 1\ x BEGINS At y

IS

Inside a

p

place

5, x IS At Y A Z IS At Y :) x IS BesIde z

6. monkey1 IS Beside x 1\ x IS o I I mbab I e :)

monk ey1 CANBE ClimbIng x

7 • monkey1 IS Climbing x :) monkeyl BEGINS Upon x

8 • monkey1 IS UPon boxl 1\ boxl IS Under banan as1 monkey1 CANBE Reaohlng bananasl

9, x IS Reaohlng y ::l x BEGINS Having y

10. x IS At Y 1\ x BEGINS At z ::l x END~ At

Formulas 1 - 9 contaJn the desorlptlon of the problem enylron­mente Some further, Yery general rules are needed (e,9, to specify the translenoy ~f the properties (nvolved) before one oan proye that the monkey can reach the bananas,

y

SAMPL~ .EN~YCLOPAED 1 A OF. T .1. ME ... OEPENDENT ~ ACTS ======================~=====================

We'll have to write an lntroductlon tothts chapter.

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-~~~~~I~~ f~r ·~Pd~tjng.5Ituatlons ~_~_.e_~_~~ ___ m_"_"_"~~_~---______ .

Pend'lng,

(AXIOMS TELL) fOr transmission of knowledge

(KNOW 1) m IS KnowIng p 1\ m CANBE Talking TO n :J rn CANBE Te I II ng n p

(COMMENT) 'Telling' Is a verb function of two arguments

(KNOW 2) m IS Telling n p :J n BEGINS KnowIng p

(KNOW 3) m IS Knowing-whether e " m CANBE Talking TO n :J m CANBE Te III ng n e

(KNOW 4) m IS TellIng n e :J

n BEGINS Knowlng .. whether e

(AX I O·MS PLAN) for kn~wiedge of and use of plans

(COMMENT) This sectl~n Is Quite tentatIve and will certainly be revised

(PLAN 1) m IS Knowing Howto p :J

m CANBE Accomplishing p

(PL.AN 2) m IS KnowIng Howto p 1\ m IS Accompllshtng p ~ m IS Executing Howto p

(COMMENT) or should it be , . , Some Howto p ?

(PLAN 3) m IS EXecutIng (p WHEN e) 1\ e IS true ~ m lS TrYing p

(PLAN 4 ) m IS Trying, p 1\ m CANBE p m BEGINS p

(PL,AN 5) m IS Exeoutlng (p THEN Q) m IS EXecuting p

(PLAN 6) m IS EXecuting Cp THEN Q) 1\ m IS p !:) m IS Executing q

(AXIOMS MOV) movements

(MOV 1) m IS Lac I 1\ m BEGINS Loc r m ENDS -Lac I

(MOV 2) m IS Golng.,to I ~

m BEGINS Loc

(MOV 3) m IS Moving n 1\, n IS Loc n ENDS Loo I

(MOV 4) m IS Movlng n TO I ~

m BEGINS Lao I 1\ n BEGINS Loc

(COMMENT) 'm IS goIng TO n' Is I I legal and must b~ re~wrl~ten 'm IS going TO Near n' vIz. 'm IS Going TO Inside n', Notice the difference between 'she went to her mother' (Near) ~nd 'she went to Lond6n; (Inside). The same dlstlnotlon must 'be ~ade wIth 'MovIng' and other simi lar verbs,

5 , EXAMPLES OF PROBLEM_SOLVING === ======= ==== ===:= :=: =========

Write up the monkey - banana. and hopefully something mo r e ,

I

I

~\ ,

7. ,I MPLEMENTA T I ON ~--~-~----~-----­-----------------

T h 'J s c hap t e r w I I I 5 0 0 n b e w r Itt en, I t w I I I dis c U 5 S so me pro b I ems ~onnected, wJth ImplementIng ari AT,

Thomas }. "/alsofl Rescarch Cenler P. O. Box 218

Yorh-Iown Heights, New York 10598

International Business Machines Corporation IVC 5·3000 (Code 914)

/

Prafessor John McCorthy Computotion Center Stonford University Stonford, Col ifornia

Dear John:

February 11, 1966

A progrom in the Iverson language for the 'a -~ heuristic' is enclosed (Exhibit 1). I have used your functions ISTER and VAL and SUC (for successors). The voriable Q is a simple kind of list: It con be added to, and selected from, like an ordinary vectar in the longuage. Each element of the list has its own structure, according to how it was specified before being added to the list. Thus , in the present instance (line 4), Q is composed of elements whose structure is given by whatever the function SUC returns. In line 4, new list elements are tacked onto the head end; and in line 6, the head is lopped off. M is a two-column matrix which holds, in its top row, the current values of a and ~. Maxseek is a variable that corresponds to choosing v+ or v- as the starting point.

Yop may be interested in the way in which I arrived at this program. As a first step, I made a one-one transl iteration of your functional statements, with the exception that I did not use the A-notation for specifying, as a parameter, the functions to be used in SUP and INF, but rather wrote the functions right into their definitions. (This is something I bel ieve can be done in general; i. e., the range of po;sible functions which might be specified as parameters in call ing a function can always be included explic itl y in the called function, ond the proper one chosen at time of use by means of a simpler parameter.) The programs are shown in Exhibit 2.

As the next step, I recognized that si nce VP and VM are not in themselves recursive, and each calls respectively only SUP or INF, the system of programs could be reduced from four to two. Up to this time, each program was assumed to have access to its own variables only. I found it useful now to think of the two remaining programs as operating as "system" programs, as defined in our S/ 360 description . This imposes the condition that variabl es with the same name are the same variable, so that direct

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Professor John McCarthy -2- February 11, 1966

interaction among the programs is possible; and this, in turn, helped me to recognize and organize the necessary push-down stores. After some cleaning up, this resulted in Exhibit 3.

It can now be seen that the two programs never actually lido" anything simultaneously, since one is always dwell ing on its line 2 while the other is going through its more active portions. This, and the obvious symmetry between them, suggested that they could easily be combined by using a single bit to indicate the state corresponding to each, and resulted in Exhibit 4.

A straightforward logi cal analysis can then show that the last I ine is unnecessary, and, after that, placing the test on P ~ before the specification of M? can some­times save useless work. The result is then Exhibit 1.

By and large, the foregoing process was a formal one, in the sense that each successive step was taken by examining the form of the programs, without necessarily understand­ing what they were supposed to do. While the manipulations have not been formal ized, one device that proved very effective, particularly In going from Exhibit 3 to Exhibit 4, was the elimination of crossing branch lines wherever possible. This has long been a standard device in transforming Iverson programs.

ADF:nh Enclosures (4)

Aih~ Adin D. Falkoff

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COMMUNICATIONS OF THE ACM A Publication of the Association for Computing Machinery

1133 AVE~UE OF THE AMERICAS NEW YORK, NEW YORK 10036 212 265-6300

M. STUART LYNN, Editor-in-Chief MYRTLE R. KElliNGTON, Executive Editor

Augus t 27, 1970

Professor J. McCarthy Computer Science Department Stanford University Stanford, California 94305

Dear Professor McCarthy:

AEPl Y TO: rBM Scien tific Cente r 6900 Fann in Stroet Hous ton , Texas 77025

Could I prevail upon your ti me to referee the enclosed paper entitled, "A Strategy for Developing Man-Machine Systems, with a Detai led Example," whi ch was wri tten by Dr . L. I. Press.

Because of the possibility of rapid publication in the Communi ­cations, it is currently the practice of ACM editors to request reports from referees wi thi n four weeks and it woul d be greatly appreciated if you could meet with this deadline. If your schedule will not permit it, I should appreci ate the return of the manuscri pt.

Than k you in advance for your kind assistance.

MS L:ch

~

COMMUNICATIONS OF THE ACM Numerical Mathematics Section

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COMMUNICATIONS OF THE ACM: Numerical Mathematics Section

Article:

Author:

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Referee's Report

Date Sent to Referee:

(Hc:p rldditional blank sheets if necessary.)

4 ....

A Strate a . ;. : ~i .1 1 i r .... ,. ~ ~ ~ ') PI ~ for Developlng Nan->:achine S stems'~ 'wiTh· 'a :":HeT·a:iled: Exam Ie .

I

Laurence I. Press and Miles S. Ro~ers. o

Key "Tords

Man-machine, heuristic, pattern recognition, concept acquisition, data

analysis, data display.

ABSTRACT

Man-machine systems may be characterized by the relative dominance of

the two par~ners. For instance, the machine may suggest actions to be

taken or may merely evaluate functions when directed to do so. This paper

suggests a strategy for developing systems in which the machine is an

active partner: The extension of current artificial intelligence or heu­

ristic programs (machine alone) to provide for man-machine interaction.

The developm~~t of a system for the analysis of multivariate data using

the above approach is shown.

Several of the heuristic developed for that system are also described.

These heuristics are concerne~ with the problem of finding a function

which partitions a multivariate spac~ into disjoint regions such that

all the data points falling in a region are accurately fit by a simple

model associated with that region. ~stems for concept acquisition,

pattern recognition, cluster seeking, decision table analysis and statis­

tical analysis ~re related in that they attempt to solve this problem.

Ackno"Tledgements:

The authors wish to acknowledge Hillia";l Mdfni~ .... ~ey of UCLA for his guidance

and encouragement. Resetl.rch reported herein ",as conducted under Contract

Nurr~er DAHC15-67-C-0277 with the Advanced Research Projects Agency, De- I partment of Defense. I

! I I

I

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Introduction

There ,vas a tj.mc '",hen intelligent behavior was the exclusive province of

man. Hhen a decision had to be made, a problem solved, or a theory induced

from observation of phenoIJena man was called upon. However, intelligent man

was intrigued by the task of. building machines which could also solve problems

and perform other tasks requiring intelligence. Early efforts, such as chess

playing machines, were few and far between, and it was not until recently with

the develop:t.ent of computer technology that. we hav.e seen much interest in

. intelligent mCl:chines. Hachines--properly programmed computers--now perform

tasks which twenty-rive years ago \V'ould have been for man only and \-lould have

gcne~ally been accepted as requiring intelligence.

Intelligent behavior is no longer the exclusive province of man. Arti-

ficial intelligence research has produced an alternative for intelligent I

behavior. However,. a man alone or. a machine alone are not our only alternatives--

they are extreme positions. Between these extremes we may use ~he man-machine

team. This notion of man-machin'e, intelligence is not novel. Han has long

collaborated with his slide rule, adding machine, telescope, etc. More recent;y o

Licklidder (1960), arid others have.·suggested r:1~n-computer symbiosis.

It is not sufficient, t9. categorize all.ma,n-machine systems together; this

mid-range encompasses varying degrees of.man-machine.dominance •. In the case

of 'man-slide rule problem solving,' man is clearly the dominant partner. He

cerely selects one of the several operations that the machine is capable of

e..xecuting, supplies necessary p~ramcter values) and directs the machine to

perforc the operation 'and report. the results •. The slide rule never USt~ggests"

that a certain operation r.:izht be useful or "tells" the man whClt to do next.

This same dominance of man over his machine ?artner is pervasive in nearly

all current man-comput~r systems. But this is not our only option, we can

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build systecs where a man augments artificial intelligence as well as systems

where th~ co~puter augments man's intelligence.

One strategy for building such systems is to modify current artificial

intelligence and heuristic (machine alone) programs by allowing interaction

with a human operator. This strategy 'vas used in developing IDEA--a system

for Inductive Data Exploration and Analysis. In this proj ect our goal vlas to

provide a useful system for multivari?te data analysis by extending the Concept

Learning Systern eCLS) of Hunt, Harin and Stone (1966).

After a brief description of CLS, \.;re v1i11 shot., the general modifica tions

necessary to !:lake it interactive and suitable for' data analysis. Next we: \vil1

turn to the general problem of partitioning a multivariate space into disjoint

regions (\-:hich is central not only to CLS and IDEA but to several other important

types of sys te~s) and describe t' .. :o of the' 'IDEA heuristics for this proble-it. He

will conclude with a sUillmary of our attempts to evaluate IDEA and suggestions

for future work.

Concept Learning System eCLS)

CLS is a oodel of concept formation. A concept is for~med by learning a

r.ule for discriminating between objects that. are exemplars of the conc(:pt and

tho~e that are nonexemplars. The rule is in the form of'tests of the values

assumed by the attributes of the objects. For instance, the class of objects

being considered mi~h t have ,three at l:ri~utes: color, numher of lc[;s, and

.10se leng th. The concept If elephant" would be defined by the rule: if COLOR =

grey and LEGS = 4 and ~OSE > :3 fect the object is an exemplar of the concept;

othenvisC! it is a nonexemplar. In the context of CLS this rule would he

represented by an equiv.:llent dcci;:;:ion tree as shmID in Figure 1.

·. 3 -

Insert Figure 1 about here.

In order to learn a concept, CLS nlust be presented with the descriptions,

(attributes, anc va lues) of n objects, each of ",hich is identified as an

exemplar or nonexemplar. The CLS heur-istic develops the tree one node at

a time, starting with the top. All n objects are considered in choosing

the partitioning rule at the first node.

Once a partitioning rule is chosen the objects are routed to the

appropriate point in the evolving tree. The same selection procedure is

then applied to each of the subsets of objects producing smaller subsets,

which are in turn partitioned, etc. Tne process continues until all endpoints

at"e er.1pty or contai.n only exer:~:Jlars 0::' nonexemplars.

In grm-ling the tree of Figu:-, 1, CLS selects the attribute COl-OR to

partition all n objects. Those ,,;rith value "not grey" are sorte.d to an

endpoint which is terminated since all of ~hesc objects are nonexemplars. o

For the subset of the objects v7hich are "grey , II the attribute LEGS' is .

chosen by CLS and the objects valued J~ot. 4" are sorted to an endpoint·

\olhich is terminated. CLS then operates on the subset which 1's "grey"

and "4;: and decides to partition those objects using the attribute NOSE.

Tne tree growing process is complete at this point since all endpoints

contain only exemplars 0::' nonemcxplars. Figure 2 presents" flow charc

of the CLS procedure~ . I

Insert Figure 2 about h(:re.

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If ~c think of the attributes in a concept learning problem as

predictor variables and nonc:-:0.i:1plnr" as a binnry criterion . I

variable, CIS may be thought of as tryin~ :0 explain the criterion variable

in terrr~ of ~he pred~ctor variabl~s. Tne criterion variable is not

explained by sllmi"i:ting the effects of. the pl'~dictor variables as in linear

statistical techniques, but by the joint occurence of various predictor

values, i.e. by the region of the data space. in which a data point occurs.

Hunt, Narin, and Stone (1966) recogni~e the isomorphism of concept

acquisition a~d d~ta a~alysis and dcsc~ibe applicatiou3 of.CLS to so~e

live data. Figure 3 depicts an, ID&\ rcpr~sentation for soma hypothetical

multivariate data. Note the similarity in form to Figure 1 •.

Insert Figure 3 .about here.

"

'llle goal of the IDEA project was,to extend the data analysis capability

of CLS by introducing several major. modif.icaotions:

1. Tne use of analytic techn:i.ques· approI?riate to the leve 1 .of .

I

measurement in the data •. CIS. tr~.ats all. data as nominal,

but IDEA can be applied to data ~f varying levels o~:

measurement through appropriate~y ch~~en heuristic~ from

·its library. Two of 'these heuristics are described in detail

* belm" •

* A unique heuristic is requirc:d for each combinat'ion of' predictor-crit·-;rion . levels of measurement, e.g. nominal-normal, nominal-interval, etc. \.Je have imple~cn:cd h~uristics for four such pairs. It should also be n~ted that.£!l.'i. characteristic of a vari'-1ble, not just scale 1!";vc1, may be m~de explicit and a heuristic v:hich assumes that ch~ractcristic may be pro.~rammed, e.g. a heuristic ~ay assume that a predictor variable has a monotonic relationship with the criterion variable.

- 5 -

2. Generalization of the decision tree. Depending upon the

version~ CLS either creates a unique branch for every value

of the variable assigned to u node or it splits one value frem

all of the re~t. Instc~d of restricting the form of the tree,

IDEA .employs heuristics. \vhich al1m" the relationships in the

data to determine the groupings of values at a node.

3. Provision for interaction bet\leen the inves tiga tor and the

analysis p~o~ram. CLS operates in the batch processing mode.

Any batch proc~ss{nB program is restric~cd to operating on

information which can be explicitly represented in a computer

and for which processing routines can be devised. For instance,

CLS is told only the nUr.1ber of poss'ible valUeS for' each variable.

Tne inves tigator;) hm.]cver;) has· knoY71ecge of. the theoretical

frame'tvork of his discipline, knoHlecl!;)e of the reliability of

various 'measures, tentative. hypotheses or hunches, pattern

recognition skills, etc. 'vhieh cannot be. made explicit (for

technical or economic reasons)., Fj.9gl:lre 4 presents a. flow chart.

of IDEA shO\oling the points at. "V7hich user. interactions. may occur •

. '

Insert Fi~urc 4· about here.

Interaction

The box numbering of Fieu!."c· l~ is keyed to that of Figure 2. The same

~rabic numeral is used to clesignntc analogous functions, and the alphebetic . "

I

,

•

.. v -

labels refer to additions made in IDEA. The follmoling notes refer to

these figures.

1. At ~ tcps a and b, th~ user u1ay dire ct the ana lys is to other than

the largest subset. No~~ that he ~~ecifies S as a node rather

than a previously un?~rtitioned subset; S ,·,ill contain all

observations that h3VC been sorted b(;! 1m·} tha t node, i. e. the

analysis will have been I1backed up" to a previous point.

2. At step c termination tests that are a' function of S alone may be

applied. For instance, we may desire to test the size of:S ageinst

a mini~um t.hreshold) or the observations in S may already be well'

predicted.

3. At step 3 we apply ap~ropriate heuristics and choose the best

partitioning rule for each predictor.

4. Late termination tests at step 4 are those which are a function of: .t. '4~

Sand pA where P o

is the best partitioning'rule found.

5. At step i the user may call tor other displays than the. one

presented. at step h. He ~y evaluate other partitions, compute

statistics, display data distributions, etc. for. any subset of the

data base (usually for S).

6. Ey step 6 the analyst has either specified his own partitioning rule,

select~d one of th~ rcco~T.endcd rules, or decided to terminate

partitioning of S. At step ~ his decision is added to the tree

anJ the observsti0ns in S are routed accordingly.

·1

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P.?.rtit5.oni.n~ 1I:~t1risti.cs

We have seen the addition of user interaction at several points in

the tree gro\o1ing process. Let us nOH turn to a discussion of the heuristics

used to select partitioning rules.

In concept acquisition, pattern recognition, discriminant analysis,

cluster seeking, decision t.::i)le analysis, an~l statistical analysis we are

given a set of points (~ sample) in a multivariate space. We then find a

function which partitions the space into disjoint regions. Our goal is to

have the observations falling within a rcgio~ accurately fit a si~?le ~odel

a~soci.2..ted with the region •.. Hith the exception of cluster seeking, in ""hich

no distinction is ffiadc between a criterion variable and predictor variables,

all bf the analyses listed in the opening sentence. may. be characterized as

. seeking a function \·lhich produces r~gions in "ihich the observations are

accurately fit by the bomogE;.neity rr.odel: y. ="('j. + t;, which states that in· 1.. 1. "

.region i the criterion variable equals ~i. plus a random error.

Since each of these problelP.s·may be thought of as seeking regions in

which the observations are accurately.fit by Eo hOl:",ogencity model, any program

for concept formation, pattern recognition, discriminant analysis, or decision

table analysis, as well as for some unconventional data.analysis .procedures,

im?licitly embodies a routine for. the solution of the. problem which we have

defined. All such routines may be characterized by the strategy used in

seeking partitions. Several analytic and adaptive strategies have been

employed.

If we make assumptions about the population [rom "-'hich our observations

arc drawn and the levc Is of meaSUrCl"l1Cnt of the variab les, and then choose

certain criteria, an analytic solution to optimize that criteria may be kno~n.

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Several of the pattc1:"11 rcco::::ni~crs dcscribCli by SC}'('stycn (1962) are of this

type, as are discrimin.;:.r.t analysis tC!ch71ique~.

Another strategy (see Figure 5) is to consider the observations o'nc

at a time. in order to group them into clusters in \Olhich the observations

with a given value of the criteria variable, i.e. pattern type, are "close"

using a given ~etric. 1nese clusters are used to define the regions into

'~7hich the space is partitioned. Sebestyen r s (1962) "adaptive sample set II

l!'.achinc and HcQuccn I s (1966) r:k - llic~ns II techniques nre of this type ~

Except for ouj;" conc1ud ing remarks cOl~cE:rning future \vork, \".,here \ole wi 11

mention other strategies, the remainder of ' this paper assumes a sequential

parti tiol~ing s tra tegy, such as that used in' CLS or IDEA.

Insert Figure 5 about here.

The' central part of a sequential partitioning program is the ~outine

(step 3 in Figure 4) that selects the partittoning rule to .be used for a

subset. Since the number of possible partitioning rules for'a subset is very

large, a heuris tic routine (\o!hich dccs not guarantee an opt imal solution)

* selects the rule to be used. Sequential partitioning heuristics are used

for modeling co~cept acquisition by Hunt, ~t al (1966), and for the

analysis of survey data by Sonquis t ar:.cl Horgan (1964). T'ncse are described

be lot., , along \vith two of the heuristics' in the IDEA library.

* If the an~lysis is part of an interactive system, it is desirable to iimit comput~tio:1 to Lho point \·!here a partition may be sclecte:d \olithin a few seconds. In a b.:ltcil processing r.:oac, th8 amount of computation which is lIexc~ssivclJ is ~n cccnor;:ic questio,n. I

\ I

.~e lated Hcuris tics

Several different CLS heuris tics have been experimented \'1i th and the

one described here was found most powcr£ul~ TIl~ only partition considered

for each of the predictors is the one \'lhich produces a sub-region for each

value tha't the predictor may assume (r~ 11 variables are assumed to be nominal).

F9r example, if a variable has five possible values, only the five-way

split on that variable is cV3luatcd. Therefore, the CT~ heuristic must only

* select a predictor in order to specify a split. Note that this heuristic I

co~ld not produce a tI'ee such [is that in Figure 1. Ins tead . the top node.

would be,

assuming that COLOR had four v~lues.

Tne criteria for'predictor s~lcction' is to select the predicto~ that

would minilnize the number of ~isclassificatio~s if no further splitting were o

to occur in the region. Specifically,

*

~

.n-

K

V. 1.

= {A.1 . ~

= {k!'

n .. , 1.Jt<

n * ij

H. 1.

= the set of predic~ors not used in defining the current region;

:: the set of values which the criterion variable may assum~;

= the number of possible values for predictor i;

= the number of observations in the current region with value

j for predictor i and criterion value k;

:: m3xir.1Um of over 1-:;

.Vi :: J. n:i.j~·". :......J

j=l

Stcrli.p.g ct :11 (19G(,~, I:['·scrib:· an ir:formation theoretic criteria for predictor selection in a different context.

\ , I I

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The heuris ti.c cri tcria is H.; CLS dot(:~·:-::: n('.~ 1 .

~ T ~ , .

1. of A and

selects the predictor for \oJhich ·i.t is r.1~:d.:d.:~c;1.

Sonquist and Morgan (1964) have' developed a related proeram while

working directly on the problem of the ~n~1ysis of survey d~ta. Like CLS,

they utilize a sequ~ntial partitioning strateby and display their results as

a decision tree. TIley consider only binary partitions (partitions into two

subsets) and evaluate them using the criteria of m&ximizing the "between

groups sum of squarc:s ll of the criterion '.,.~-:;'l.ble for the: tHO groups

that the criterion vari3blc is measured on an interval scale.

More precisely, a partition into two ~roup~ of size nl~ and n2

.would be

evaluated using the criteria

·BSS

\Olhere

y ; n 1 Yl + n2 Y2-

nl

+ n2

Furthermore, they sho,," that in using thi's criteria it. is. not·.necessary to

consid~r all the possible binary partitions of a predictor. If. the predictor

values are ordered by the mean of the criterion variable of all observations

having that value, the partition which maximizes BSS given' this ordering will.

maxlmize BSS for all possibie binary partitions. Therefore, it is only

·necessary co evaluate V. - 1 partition for each predictor. 1.

IDEA Hour i ~!.ics

In IDEA the critC'ria used to evaluate a partitioning rule depends u?on

the level of l1;('aSllr(!;;::.::nt of the: critc·rion variahle. In the case that it is

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nominal, partitions arc ranked by the significance level of the corresponding

chi-square.. In the case of an i.nterval critel-ion variable, a good parti.tion

is one that produces a large reduction in unexplained variance, nor~ali~ed

for the number of subsets produced. These cri teria ,·.'ere chosen because of

several properties.

They favor a partition into re\o1 subsets over a partition into many,

even though the latter may yield a more precise estimate. While this property

is pleasing T,yhen ,V'e are interested in data 'a'nalysis, \·:rhere results must be

interpreted and cornmunicated, it rr~ay be inapp:ropria~e in a purely predic:tive

setting such as p.:lttcrn recoznition. They allo~~ for the di.scovery of any

Single-valued functional relationship, again favorine the relatively sinplc.

Tnis property is mandatory in the complex structure seeking processes of data

analysis and is also require~ in many estimation and pattern recognition

problems. The latter is true whenever the problem is in fact difficult,

i.e. ,.;then assumptions such as linear. separability of classes or .knowledt:e

of class frequency distribut{?ns cannot be made. Finally, these criteria

tend to favor variables with strong main' effects •. This is beneficial in all_

but one case. When no interacting variable has a significant-main effect,

the presence of a significant interaction will.not·be.discovered. Although

more conplex heuristics could be desig~ed to ferret out such occurrences,

the investigator will often be able to'inf~r this state by observing the

structure of the tree produced. Tnerefore, we su?stitute human pattern

recognition skills for more complex heuristics •.

Using th~se criteria, \V'e have dc:vc:l.op~d several heuristics. ~]e will

de~cribe, first, the heuristic used ~lcn bo~h the predictor and the criterion

va!."i~b10 nrC' measured on n no:ninal scale and) then, the one used ,,,hen they

arc me~surcd on an interval sc~le.

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No:ninal Ex.;:.::-.~)l£. This heuristic l.S i.!s(!d \.:1:c::-;, the predictor variable and the

criterion varic.blc arc both r.:c.:lsurt::d 0:1 a nO::1in~l scale. The routine seeks

the par-t';tic';·Ll.· n? rul~ \.,1,.1.' cl~ ':"1:"I.a·.··.1.· ",", .. ,.,,_. ','l~''-' -L' ~"l~. S1' rrn1.' ::1' c~nce of c"n1.' so" .... r·e J.. • _ - • • ~'-... "'- ..,l J.. a. •• - ..... '-1. • The

rows in a predictor-criterion contingency t~ble are cOffibined in order to

increase the signific~nce level of chi-square. Note that this is nontrivial

since degrees, of freedo::i ch.::.nec as \Ole}l as the value of chi-square ylherever

rows are co~bined.

The change in chi-square from combining, t,vo rOHS (x and y) is:

2 n I [~ AX = - G~ ')

(

+ Ry) 0':"-2 0 0 C. IR R yi.

xi yi i=l

~ \ x x

Wheie G is the grand frequency count for this table, n is the number of

. th columns, C, the marginal total for the i coluwn, R, the marginal total for

~ 1.

h . th dO' b d" " th d ,th 1 tel rm'j, an " tne 0 serve rrequency 1n 1 rO't·} anr J co umns. ~J

2 Since ~X becomes small as the marginal tot'als of. the rO'\'7S to' be cc:nbined

become large, or as the entries i~j the rOHS bccbme proportional (0 . =k 0 ,), X~ . yl. .

the first step is to sort t:he rows oE the contingency table. The modQ,l

column is the .major field sorted; \'lithin this the rO\"s are sorted on the

relative frequency of the mcd£tl value to the ro\oJ total. This is illustrated

in Figure 6.

Insert Figure 6 about here.

The he:uristic considers comhining Oilly rO\l1s \vhich arc adjacent after the 2

sorting i:.; co.;ipletC'u. Th~ t\-..'o best rOvlS :'0 combine are found by COl.lputillg ~X

- 13 -

directly. The significance level.Inay n(,',·] be co:nputed for ollly that combination

of ro\~'s, thereby saving considerab12 C01::put~ti.on. If sienificance is increased,

the rm.;s are combined and the ?rocf~h:rc is repca ted; othcr~'lis e, the heur fs tic

has chosen a rule for this predi·ctor. The heuristic is then used for ~ll the

other riominal predictors, and the one with maximum significance is chosen.

Note that the procedure docs not eU2rantce finding the optimal (maximum

significance level) partitioning rule. Figure 7 shows the flow chart for this

·procedure.

Insert Figure 7 .1bout here.

Interval exanmle. With an inccrval scale predictor variable the number of

possible p2rtitions is very large; therefore, an immediate decision was made

to coasider only partitions \"hich maintained the order of ' the observations on

the predictor variable. This restriction \'leakens the heuristic in that it

undervalues non-monotonic relationships; however, a quadratic (for instance)

shm·}s up as a chree-y}ay partition, and since ~veral alternative partitions

are displayed aiter the heuristic search, the user is able to i.dent:ify the

relationship. Therefor, it was felt that the task of identifying good, non-

monotonic partitions was better and more economically accomplished by the

user than by the computer routines. ?rior to implementing this heuristic,

several altcrn.1tivcs \Olere considC!Tcd and r(:~, . .'cted. One was an algorithm to

search exhaustively for ar. optin,al solution. It was hoped that.a lemma by

"1: Fisher (1958) \vould make tills co;nputa~ion(llly feasible, but such was not the

* If A: denotes a p~:::-ti ti.on of set A into ... ·.·.'0 disjoint subsets, Al and A2

) if P ... 1 " ~. 1 .. (. TT S S ) f \. G b t d' -... Gc:-:otcs a cast sqU<1l"CS Pc.1r'tl tu:m i.lin. • .... , 0 • lnto l su sc S an It P~ .:: dC:!1ot:s, a lc.:::t sc!lt:lrcs p:::' , ,or. of :12 into C

2 sunsets; tnC!n, of trw cl:lSS

01 SUbP2:"tl:· :~1:1;-: :I:: ."': : A? Ci~iplI)Yl,~g G1 SllDsets over Al and G2

subset3 over A2 a least Sr,L:.~tres suhp.:lrti'lion is P "k: P? of.;. Therefore, once a least squares partitio~i. over set A, is founJ, this work need not be repeated \"hen testing for various partitions Lover A

2•

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case. Hmvever, Fisher' s lem~1a is used in searching for the estimate of the

* optim31 p~rtilion (r ) in step three below. n

Dissatisf.1ction \·Jith the above approach and \yith scvc·ral other appr(}a~hes

led us to develop an estimate of the optimal partition by computing a fWlction,

the within groups' sum of squares (HSS) , of a random sa~ple of the

* obscrv~tions. This approach, as given' in the six steps \.Jhich follow, is

analogous to forming an estimate of the height of the tallest person in ~l

population by observing the height of the tallest person in a sample.

1. Select a ra.udom Sa1TI1)lc of size N (N being a specified pararaete:-) trom the: tlata blocks sorted to the current nOdE!.

If less than N obscrvatious exist, usc all of them.

2. Order thes~ according to the val~c of the predictor variable.

3. Find the minimum WSS for 2-,3-, ••• , M-way partitions of the criterj.on varia~12 c·r being a sp2ci£i~d parcuncter), '-lhile respecting the ordering of step 2. These ~vill define partition:; of the predictor variable for the sample (cal] these u-way partitions p * and call the associated WSS, WSS *). . n n

4. Find p.'" ~·]hich is the partition that minimizes

* \oJSS

N n

n

(~.]here N is the normalizing f.actor discussed belo\.J), to accoun i :

for the Ualue of n. Q

5. As three cane! id~tes f or the final choi ce cons ider p.':, ?o': ,yi::h all p~rtition points shi~ted dmvn to the first unique predictor value, and ?":: with all partition points shifted up_

6. Select the one that minimizes .WSS (this tine computed for criterion variaqle for all of tne obs~rvations at the node).

Tois .procedure yields an estim~te of the partition 'vhieh minimizes the \oriS of the dependent variable for all observations at the node.

* This .:lpproacii Has sug~;estcd 1)y the samp1ins scheme err.ploJ'cd by Herbert Solomon [or dcfinin~ clustcr~ o[ correl~cLJ varinble~. This sampling scheme wns described in an informal seminar at the Systenl Development Corporation at the time we wcra seeking to formulate an interval heuri.itic.

\ ;

.' . - 15 -

Statements about the pow~r ~nd efficiency of this technique require

a knm'lledgc of the frequency distribution of t~c HSS of all possible

parti.tions., Until an ~n.:11yt:ical or e:r.p:i.;~ica1 descri.pti.on of the

distribution . , 1S m.:1G(!, CO:;~Pl1l"J.tio:l~J c;~pacity (systr.m response time:) 1,o/i11 be

used to determine the sample size (~).

A similar situation arise~ when we attempt to arrive at normalizing

factors for ccmparin~ p~rtitions into different numbers of subsets. One

approach would be to estimate the mc&n n-\"ay HSS ( ~'lSSn ) for the populati')l1

of partitions. He would compute the mean 'HSS for a random sample.

Evoking the centrnl limit theorcr.l, \·]e knmv thnt the. distribu.tion of samp '_e

HSS means is normal around the popu1ntion HSS mean. Therefore, a sampling

technique to obtai.n an arbitrarily precis(! estimate of '~'!SS as a normali::ing n

factor could be implemented, but at considerable cost in terms of computation.

TIlis procedure would possibly yiel~:dif~erent results in terms of select~ng

pk than the' current one of using the mea.n \-iSS of all partitions explicit-_y

considerecl for final selection. Since the latter approach deals with a,

screened (biased) sample, it produces a ImoJer normalizing factor. o

Cone 1 us i 0.1.2

Empir'ical evicieGce for IDEA's utility 1,olaS obtained by a study of thl!

application of IDEA to data collected in three empirical studies (Press,

1967). TIle IDEA results were contrasted with tllOSC of a discriminant an.llysis

til. lht' !,,:1;1It.' ddl':'l in order Lu pruvide a pui.nt of referencL' 1(11: our conclu,li.od~;.

Using D.ccuracy of classification as a criterion, it was fou!1d that: 1) -:OEA,

without user interaction, pcrform~d batter than discrininant analysis; 2:, by

interacting during IDEA analysis, the tlsC'r was always able to im?rove th,!

,iccur:'cy of cl.:lssificati0n; .:;r.~:' J) Hhen the resu1 rs of the IDE.t1. analysis \-Jere

· '. - 16 -

used to guide the step1,·.'i.sc discri:nin~:~t an.:d.ysis of the sarr.e data, the

discriminant analysis made fewer errors tilan the unguided version, but more

errors than IDEA. I

Finally and p2rhaps most important, it was found t~at

the opportunity that IDEA provide.::; for a user to explore the finc-grai.n

structure of his duta has v.J.lu~ble ,mcill[.ry results such as pointing out

interesting new directions for furthc~ c~pirical or theoretical study.

In addi tion, IDE.:'\. has been used in the analysis of data havi41g h'li 1 t- in

structure, with various amounts of random e~ror. Routines to gcnerat(~

artificial data bases were used to produce complex data structures wi:h

various Ifnoise" or Ifr.:easurer.:ent cr:1.'or" levels (10 percent to 60. percent)

superi~?oscd on the underlying structure. For a description of these artificial

data analyse~ see Press, Rogers and Shure (1969).

Future work on the problem of partitioning multivariate spaces mny

diverge fro~ what has bcien done in several ways. We might retain the

sequential partitioning strategy, but.seek regions which fit well with

other models than the homogeneity model. Another alternative is to drop

the sequential partitioning strategy in favor of other analytic ~r ad&?tive ~

strategies. An exa~ple would be to devise heuristics which start with a

maximally complex tree and a~tempt to simplify it while minimizing loss of

fit. Further research might also b~ aimed at an ~naly~ic understandi~~ of

the adaptive and sequential partitioning ~ethods described in this paper.

Regardless of the n.:lturc of the future re:search it \-,1ould seem that the

problem of partitioning a multivariate space is of sufficiently general

interest to support ~tlrther investLgatiqn.

AlternativE.:s also exist for our strntegy of developing "machine-rr:;:n"

systems hy extending m:1chine-only progr[!ms. For instE'lncc Hormann (196E:) is

working ~ t: snc t O\']:lrd the salT!<? cnd by pro'J i.eli ng the human par- tner \oli.::h a

\ I

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17

languugc in \·,hich he can cqnstruct pO'.·:~!.'fllJ and incrcflsin21y 'complex commands

for his machine pa::.-tner. Others Hill certai.nly design such systems. Our

experience \-lith IDEA has been encouraging and :i.ndicates that further efforts

in extending artificial intelliience systems is justified.

" ----.-.

"".

' .. - 18 -

REFER!~~C!::S

H1.tnt, E." HoRrin, J., and Ston(2) p. ,]:;:::"i?~rJ...r::.~~il~LJnr-L~~-:j.2.!l, Ne't· .. Yor}:, London: Academic Press, 1966. I

Fisher, H., "On Grouping for ?·la:d.!ilum HO:-.logcnci ty, II J. Amer. S ta t·. As s ,:~. , .2l (28 l .) , 1958, pp. 789-98.'

Horme.nn, Aiko, IJ1tc;.act}'v~_I?c:cision Nakin~ b'Ll1an.:·Nach~.ne Teams, WNSI Colloquism on Hathe:natics in the B~havioral Sciences, UCL..A.., November, 1968 (unpublished).

HacQueen, J., SOr:i~l'!ethods :01" Cl?ss!fl~~tion a!1d_Anal,,;~§J.? of Hltltiva:riat~ Observations, Working Paper 96, Western Xanagcrnent Science Institute, UCLA, Harch) 1966. '

Press, L. I., IDEA: A Tec1!Jlf:.9uc for __ ):ng.}!S:.~ .. ~~ __ !2::}..tf::..E}~')loration and ALalvsis, unpublished doctoral disser~ation) University of California, Los Angeles J 1967.

Press, L •. I.) Rogers, N. S .. ;) and Shure) G. H., "An In ter ac t ive Technic. ue for the Analysis of }!111tivariate Data," Behavi.orc:l Science (In rress).

Scbestyen, G., Decision ~akin~ Processes iq Pattern Recognition, New York: Macmillan, 1962.

Sonquist;) J., and Horgan, J .. :> The D.§tectiQ.n of Interaction Efft¥cts, Nonograph !,f35 , SRC, Univc?Tsity o£·Hic.higan, Ann Arbor, i964.

o Sterling) T." Glosser, N., Haberman: S., and Pollack, S. J IIRobot Da te Screening:

A Solution to Multivariate Type Problems in the Biological and S0cial Sciences," Communications of the ACH, 1966, pp. 375-38~.

I

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- 19 -

r;?l.l.T 1 o nO"lCXemp ar

o ex.:!rnplar

not

Figure 1. Dacision tree repL'esentation for the concept of "ne lephant ll 4

..

-",

all sub·~ets .. terminated:

CD

- 20 -

STAnT

,l o ~

hoose largcst~ nOl1- terr.1i He:l subset S

1 '0' yes ------------~> )

t apply a heuri~tiC . to choose a

'partitioning rule for S

1

entE.r nod~L I

in ::=J __ J J sort Observatio~

~ ®

Figure 2. Flow chart of the CLS pr~ccdures.

·1 make S a '=:-l )I~ _t_er_mi_n_al_l.:J-~®

, I

I ! I I

I I

I !

I !

I r I

\ I I ,

•

under 24

Hale

r~ 112 ,14J

Attend less than once per \-leek

Over 40

years

..J

B

- 21 -

'Female

6-12 years

UndcrQ

27

Attend more than once per \-leek

Under. £ years or. over )2 years

3,28

\ 27

\

years and.over

117;~ This illustration depicts a sample of 16L~ questionnaires including response infor~ation on age, sex, education level, frequency of reli$ious atte~dance, and participation in the 1965 revolt in the Watts area'of Los Angeles. Participation in the revolt is the criterion variable. Each of the seven terminal subsets (leaves) is represented by a box containing a freque~cy distribution of the criterion variable for the subset. Nominal varia~les such as sex are ~scd along with ordinal or inte~val scale variables; ~

variable may occur at different brn~ch points (nodes); and, the parti:ioning rule at a node may be specified by any grouping of the values of the ~ariablc.

Tne tree is developed from the top elm..'n. The entire sClmple is first :;pU.t, by frequency of rcll~ious att(:ncl.· ) into suh£ets of 139 and 25. As the subset of 25 cont:.:.1in!.~ 2!~ parti cipant5, it is not spli t furtht::r; the o·.:her subset of 1J9 questiOn!1:1 i.l .::~~ ::.s next split by sex. This process rcp0·~ts for 311 subsL!'1 L!\. ... n t 5 ubsc t s unt i 1 c.:i.ch i.:; tcn:ii.;·::: ted in a 1 C! ::f •

Figure 3. A decision tree 5unrn~ry of the struct~re of a set of hyp)thctic~l dat.J.

• 0

\ \

. ..

a..

- 22 -

'~I 0) ,II r--'--'---'

,'iI005(' 1':1'[',c·::t,1 non-tcrminal subset S

b

user specify ~ ____ Y_e_s _____ ~next terminal S

~~----------________________ -J

c.

d.

I apPl~ ea:ly ·1 term~na'C ~on I tests to S

Yes ~------~

e f I 1 Imal~c S a I

'~-----~; terminal 1eaJ no

No E------------------~ ®

\!/

a pp ly hl2ur i s t i.c's to choose a

partitioning rule for S to ~ach predictor

apply late I termination tests 9

using best rule' J found

"------:---' . ~.

Yes

h. f\ dT s pr71y -;;u-t11iila j;y

of best partitio~ing rules found for each predictor

i.

\V -----------~ exit on

~_l_· l_l_t c_. _r_ti_c_t_J ~~1I1!11.1 nd

g 6)

No ~lake S a I 7\ terminal ~eafj

'l/

®

® )[ ur,c!':lte tree-=t->r-sort.-' --. -·~i

L-_____ ---' L obSC';"V.:ltlons I

t . "lork- \vi th ~4@

j '. tree display j

I .1

\ \

.-" -

I cs taL 1ish a nC\., -I

cluster Hith

centroid of W

No I ----r

I

I

.- 23 -

START

i I

I I

is W within & of an

established centroid

for observations of

the sarna criterion

value

Yes

'..--1 _-::. j.~, --add ill to neares t

cluster

reco~pute the I centroid of· t~e I

cluster

L_---J

no more 1-------

1. define regions

in terms of (·xis ting

centroids

\!/. end'.

Figure 5. Flov] chart of a grouping strategy. These m.::LY use various metrics for es[ablishing the cistance of an observation to a cluster centroid. ~ is typically a parameter. The regions in the last step reay either be de£ine~ by a threshold distance from each centroid or by the closest centroid.

\ ,

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-40 ~

.(' 't~

- 2~;. -Criterion

0 1 2 }~odal

V':ilue

p' 0 17 6 9 0 r e 1 12 19 3 1 d i 2. 22 lr 8 0 c ;

t 3~ 14 2 0

4'- 10' 23 r. 15 1

Figure 6a _w Contingency TabJe Prior to Reordering. The column on the right indicates the crit~rion value.of me.x.imum frequency

IS01-:( Tablei ,1/

Find optiG~l pair of rOHS to combine

I Record!

1_ rCG~lt I ~n 11St

J/ f

Criterion

0 1 2 110rJ a 1 Value

P 2 22 4 9 " r ! 0

0 17 6 9 e d i 1 12 19 3 c } . 1

t 4 10 23 15 0

r 3 1 6 14 2

Figure 6b -- Sorted Contingency Table. Adjacent rows are more nearly proportional.

! combine] ~ ________________ ~ them

'/'_1 ___ _

yes

repeat for next preciictor

Figure 7~ ---Flow diagram for nominal data heuristic.

----~--

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