,rians,e aaa

Perception,
1992,volume21, pages517-544
Serialpatterncomplexity:irregularityand hierarchy
PeterA van der Helm,RobJ van Lier,EmanuelL J Leeuwenberg
(NlCl),Psychology
Nilmegenlnstitutefor Cognition
and Information
Laboratory,
University
of
Nilmegen,
POBox9104,6500HENijmegen,
The Netherlands
Received
8 March1991,in revisedform 21 November1991
.{bstract. In perception research, various models have been designed for the encoding of, for
example. visual patterns, in order to predict the human interpretation of such patterns. Each of
these encoding models provides a few coding rules to obtain codes for a pattern, each code
expressingregularity and hierarchy in that pattern. Some of these models employ the minimum
principle which states that the human interpretation of a pattern is reflected by the simplest
code for that pattern. ie the simplest code according to a given complexitv metric. In rhis paper
a new complexitv metric is proposed. This metric is based on a formal analysis of the concept
of regularitv. Some conclusions of this analysis are sketched. The new metric does not depend
on artifacts of the coding rules. [t accounts for the amounts of irregularity and hierarchy as
representedin a code of a pattern. such that these two amounts can be added to determine the
complexity of a code. An experiment is discussed that shows that the new metric performs
significantly better than the metrics used previously. In particular, the new metric predicts more
local pattern organizations than the old metrics. This implies that various local pattern
organizationsdo not falsif-vthe minimum prirrciple anymore.
I Introduction
Regulariryis a rather intuitive concept that seemsto delv formal description. This
aspectbecomesrelevantwhen regularityexplicitly plays a crucial role, as it does in
many formal modelsof perception(Simonand Kotovsky 1963: Vitz 1966; Vitz and
Todd 1969: Leeuwenberg1969, L97i: Garner 1970; Restle1970, 1979; Jonesand
Z a m o s t n v 1 9 7 5 : D e u t s c ha n d F e r o e 1 9 8 1 ; P a l m e r 1 9 8 3 : L e y t o n 1 9 8 6 a . 1 9 8 6 b ) .
These modelsare designedto explain the phenomenonthat although,in principle, a
pattern can be interpretedin many wavs,usuallyone interpretationis preferred(see
figure I for an examplein visualshapeperception).This preferenceis assumedto be
guided by the regularity that is present in a pattern. For instance,in figure 1, the
usually preferred interpretation is assumedtcj be induced by the repetition, ie the
aaa
,rians,e
A
zigzag
Figure l. In visual shape perception. a major problem is how to predict the preferred interpretation of a pattern which. in principle, can be interpreted in many ways. Here. two possible
interpretations of a line drawing are visualized. Usually, the triangle interpretation is preferred,
not the ziezaginterpretation.
ct6
P A van der Helm, R J van Lier, E L J Leeuwenberg
regularoccurrenceof the triangle part. This assumptionis in line with the minimum
principle(Hochbergand McAlister 1953)which,historically.fits in with the tradition
of GestaltpsychologyiWertheimer1912; Kohler 1920; Koffka 1935). The minimum
principle states that the preferred interpretationof a pattern is reflectedby the
simplestdescriptionof that pattern. Now, for each model mentionedabove,it hoids
that patterns are describedin terms of a formal 'language'designedto express
reeularityin patterns. Yet, the modelsshow great variety in the formal definition and
in the justificationof the complexitvmetric that is used to decide which of the
possibledescriptionsis the simplestone. The fact that, for manyof thesemetrics,the
predictionsare rathersimilariSimon L972)is. in our view. not verv surprisingsince.
for ail metrics. it holds more or less that a pattern descriptionis simpler if it
expressesa larger amount of regularitvin the pattern. However,in each model.
regularityis incorporatedand discussedonly in terms of examplesshowingonly a few
kinds of regularity like repetitionsand symmetries).These few kinds of reqularity
may be relevantin perception,but representonly a choiceout of manvpossiblekinds
of reqularity.This choicemay be plausibleand mav even be empiricallvsupported.
yet it is not the samefor all models. That is, as long as regularitvas such is nor
specified, one can hardly assign psychologicalsignificanceto a specific choice
concerningpatterndescriptions
and complexitymetrics(Simon 1972). In the presenr
paper.we will proposea solution to this problem.which may be introducedbriet-lvas
follows.
In the paper of van der Helm and Leeuwenberg(1991), the intuitiveconceptof
reqularity has been formalized within the framework provided by the structural
i1969. 1971.).This model is an encodingmodel,
informationmodel of Leeuwenberg
focusingmainlyon visualshapeperception.In this model.a patterncan be described.
in severalways. by means of coding rules expressingregularitv in the pattern. In
agreementwith the minimum principle and given a complexity metric, the simplest
pattern description.or simplestcode. is assumedto reflect the preferred interpretation of the pattern. So. the structural information model consistsof the minimum
principle plus an operationalization
ol the minimum principle,and the operationalization consistsof coding rules plus a complexity metric. In the present paper.
we rvill not compare Leeuwenberg'smodel with other models [for a review
on these topics and for referencesto related literature, see van der Helm and
{1991)]. Instead.we will focusprimarilyon the choiceof a psychologiLeeuwenberg
cally significantcomplexitymetric, on the basisof an analysisof the intuitive concepr
of regularity. To that end, first we will go into the structural information model
in some detail. Second. we will sketch the results of an analysis-of regularity.
The actualanalysisis presentedin van der Helm and Leeuwenberg
{1991). Third. rve
will propose a new metric of complexity. This metric stems from the analysisof
regularity. Finally. we will discussan experimentthat was designedto test how well
preferred interpretationscan be predicted by the new complexity metric. In the
experiment, subjects had to indicate their preference for one out of three
of patternedsequences
segmentations
of. for example,blackand whitedots.
2 The structural informationmodel
The encoding of visual patterns,as performed in the structural information model
(Leeuwenberg1969. I97l), proceedsas follows. In order to describeregularityin a
pattern. the pattern is first representedby a symbol series. For instance,in figure 3.
the contour of the pattern consistsof subsequentanglesand line segments,each of
which is labelled with a symbol. Angles or line segmentsof equal size are labelled
rvith an identicalsymbol. These symbolswill be called pattern symbols,indicating
that a symbol refers only to a pattern part and not to its meaningin. for example.the
and hierarchy
Serialpatterncomplexity:inegularity
519
Roman alphabet. Now. tracing the contour in a clockwisedirection would yield the
symbol series'kalckalckalc',representingthe pattern. Note that, in theory, a pattern
is not consideredto be mapped onto a symbol series. On the contrary, the symbol
serieshas to be such that the pattern can be reconstructedby substitutingthe acrual
sizesof anglesand line segmentsfor the patternsymbols(cf Leyton 1986a, 1986b).
This substitutionis called the semantic mapping from the symbol series onto the
pattern. The semanticmappingas such may alreadygive rise to severalquestions,for
instancewith respectto the number of possiblesymbol seriesby which a pattern can
be represented.In the presentpaper,however.we will not deal with thesequestions.
In particular,in the experimentto be discussedin this paper, we chose stimuli for
which the semanticmappinedoesnot raiseproblems.
The actual encodingconsistsof describingregularityin terms of identical symbols
in the symbol serieswhich. becauseof the semantic mapping, correspondsto the
regularity in the pattern. In the structural information model, only three classes
of regularitiesplay an essentialrole, namely iterations, symmetries,and so-called
alternations.Each of thesethree classesis describedby meansof one coding rule.
As we will argue in the next section.thesethree coding rules are indeed the proper
ones to use accordingto the formalizationof regularityas elaboratedin the paper of
van der Helm and Leeuwenberg(1991). Next. we will proceed by giving the
definitions of these coding rules, and the way in which these coding rules can be
appliedto symbolseries.
First. the iteration rule. which can be applied to expressthat a series contains
identicalsymbols.is definedasfollows:
successive
k k k . . . k k- . V x r k ; .
The expressionon the right hand side is called an l-form. in which N equalsthe
number of symbols'k' in the seriesat the Ieft-handside (N ) 2), while (k) is called
'aaaaa'can be
the l-argument. For instance.accordingto the iteration rule. the series
encodedas5 x (a).
Second.the symmetryrule. which can be applied to expressthat a seriescontains
pairsof identicalsymbols.nestedarounda so-calledpivot. is definedasfollows:
k , k , . . . k ^ p k ,. . . k = k ,- S [ ( k),( k = .) . . ( k , )(, p ) ].
The expressionon the right-handside is called an S-form, in which (p) is the pivot,
a n d t h e s e r i e s( k , ) ( k r ) . . . i k " ) i s c a l l e dt h e S - a r g u m e nat n d c o n s i s t so f e l e m e n t (sk i ) ,
I
Figure2. Tracingthe contour of the pattern (the arrow indicatesthe startingpoint and direcangles
the subsequent
This symbolseriesrepresents
tion) yieldsthe symbolseries'kalckalckalc'.
and line segmentsin the contour so that. by substitutingthe actualvaluesfor the symbols,the
pattern can be reconstructed.According to Leeuwenberg'sstructural information model, interpretationsof the patterncanbe obtainedby encodingthe symbolseries.
520
P A van der Helm,R J van Lier,E L J Leeuwenberg
w h e r ei : 1 , 2 , . . . . n 1 n ) l ) . F o r i n s t a n c et,h e s e r i e s ' k a p m p a k ' c abne e n c o d e da s
S t ( k ) ( a )( m
p )),1 .
Third, there is the alternationrule which can be applied to expressthat a series
containssuccessive
subseriesthat either all begin or all end identically.Both cases
aregivenin the followingdefinitions:
k x ,k x . . . .k x , - 1 ( t c t l ' ( ( x , ) ( x. :. ).( x , ), ) .
and
x , k r . k . . .x nk - / ( x ,) ( x :) . . .( x , ), ) 1 ( t 1 .; ;
The expressionon the right-handside is called an A-form, in which the series(x,)(x,,r
... i.x") is called the A-argumentand consistsof elements(xi), where i : 1,, ?, ....n
( n > 2 ) . F o r i n s t a n c er.h e s e r i e s ' a r a s a r ' c a b
n e e n c o d e da s ( ( a ) )t ( r i ( s ) ( t ) ) a, n d t h e
reversalof thatseries.ie'tasara',
canbe encodedas((t)(st(r))r
((a)).
In applying the coding rules to a symbol series,one should take notice of rhe
following two aspects.First, in the definition of the coding rules. rhe symbols are
consideredto be variablesstanding for arbitrary subseries(with identical symbols
standingtor identicalsubseries).
This impliesthat the codingrulescan be appliednot
just to expressthe identity of singlesymbolsbut, in general,to expressthe identity of
subseries
in a series.For instance:
a b a b a b- 3 x i a b r .
and
b a d p q v w p q b a -d S t ( b a d ) ( p q()v, w r l.
Note that the parenthesesin an l-form, S-form, or A-form (which we shall call an
ISA-iormt yield an unambiguousnotation. Any subseriesbetween parenthesesin
an ISA-form is called a chunk. It should be noted that the pivot in an S-form is a
chunk which, as a limiting caseand without distorting the symmetricalstrucrure,may
be'emptv'. This impliesthat. for instance.the series'abppab'can be encodedinto an
S-formdenotedby S[(abt(p)].
Second.the subseriesinside a chunk in an ISA-form can be encodediust like anv
symbolseries,for instance:
b a p a b a p a* 2 x i b a p a )- 2 x ( U S [ ( a(tp, ) ] ),
and
a a b p p a a *b s [ ( a a b ) ( R-) ] S t ( 2x ( a ) b ) ( p ).l
ln such cases,the lSA-forms are said to be hierarchicallynested. Similarly,but less
triviallv, a hierarchicalnestingof ISA-formsmay result in the following. Whereasan
I-argumentconsistsof only one chunk, an S-argumentor A-argumentis, in general,a
seriesconsistingof severalchunks. Such a chunk seriescan be encodedthe sameway
as a symbolseries. Consider,for instance,the following encoding:
a b a b b a b a- S [ ( a ) ( b ) ( a ) ( b*) ] S [ 2x ( ( a ) ( b ) ).]
In this example,the S-argument(a)(b)(a)(b)constirutesa chunk series'xyxy' in which
the chunk 'x' : (a) and the chunk 'y' : (b). Just like a symbol series,the chunk
series 'xyxy' can be encoded by applying the coding rules. This yields, for example,
the code 2 x (xy) which, by substituting(a) for 'x' and (b) for 'y', yields the code
2x((a)(b)), as given in the example. Similarly, the argument of an A-form can be
encoded. Whereasthe symbol seriesis said to representthe lowest hierarchicallevel.
an l-argument.S-argument.or A-argumentis said to representa higher hierarchical
Serralpatterncomplexity:irregularity
and hierarchy
521
level, and the encoding of an S-argumentor an A-argument leads to still higher
hierarchicallevels.
Now that we have consideredthe way in which the coding rules can be used to
encode symbol seriesinto codes,we can go into the meaningof the codes in more
detail. First of all, by expressingidentitiesin a series,a code provides a description
of regularity in that series. However, this is not the ultimate meaningof a code. The
ultimate meaningof a code is constitutedby the fact that a code providesa meansof
obtaininga classificationand an organizationof the series,which togetherreflect an
interpretationof the pattern that is representedby the series.This may be illustrated
asfollows.
just the identity
First. for the four-symbolseries'aaba',the code 2 x (a)baexpresses
of the first and secondsymbols.which correspondsto all the identity in a four-symbol
series like 'ppqr'. Thus. 'ppqr' can be seen as a representativeof the class of
symbol seriesto which 'aaba'belongsaccordingto the code 2 x (a)ba(cf Collard and
Buffart 1983). In general.such a classrepresentative
can be found as follows. First,
replace all pattern symbols in the code by arbitrary but different symbols; then,
decodethecode. For instance:
a a b a- l x a j b a- ? x i p ) q r - p p q r .
Note that. accordinqto another code, the svmbol series 'aaba' belonss to another
class.For instance:
a a b a - a S [ i a i ,b ) ] - p S [ ( q )(,r ) ] - p q r q .
According to this classification.the secondand fourth symbols are taken as being
identical. Whether 'aaba' is encodedand classifiedin the first way or in the second
wav is determined by some complexiry metric. That is. in agreementwith the
minimum principle,the humanlv preferred interpretationof a pattern is assumedto
be ret-lected
by the simplestcode for the pattern. We will discusscomplexirymetrics
in a later section.
Second.a code not onlv prescribesa classification.but also an organizationof a
pattern. For the series'ababab',the code 3 x (ab) expressesthat 'ababab'is similar
to 'yyy' rvhere 'y' = 'ab'. So. the code can be said to induce the organization
(ab)(ab)(abrinthe series.ie a partitioningof the seriesinto chunks. Suchorganizing
in terms of chunkswill be calleda chunkingicf Geissleret al 1978). In general,the
chunking induced by an ISA-form can be found by decoding the ISA-form without
removingthe parentheses
in the ISA-form.eg:
(vw)] - (bad)(pqXvw)(pq)(bld),
badpqvwpqbad- S[(bad)(pq)),
and
h k g h k p q- i ( h k ) ) r ( ( e ) ( p q*) ) ( h k ) ( e ) ( h k ) ( p q. )
As illustratedin figure 3. the chunking of a symbol seriesreflectsan organizationin
the pattern that is representedby the symbol series. Note that, perceprually,'ababab'
does indeed seemto consistof the three repetitionsof (ab) in line with the chunking
( a b ) ( a b ) ( a ba)s i n d u c e db y t h e c o d e3 x ( a b ) ,b u t t h a t ' a b a b a b p q q p ' s e e m
t osc o n s i s ot f
the parts 'ababab'and 'pqqp' which does not imply a chunking in the above sense.
Yet the latter organizationis relevanttoo (Leeuwenbergand van der Helm 199i)and
will be called clustering.One aspectof clusteringis that eachISA-form is considered
to induce a clustercontainingall the chunks neededto constructthat ISA-form. For
instance,for isub)series'ababab',
the I-form 3x(ab) groupsthe threechunks(ab)into
o n ec l u s t e (r ( a b ) ( a b ) ( a b ) ) .
522
P A van der Helm,R J van Lier,E L J Leeuwenberg
Another aspect of clusteringis related to the intrinsic characterof a regulariry
structure, and has consequences
for S-forms and A-forms only. (I-forms are too
simple to show such an extra aspect.)First, we will discussthe S-forms. We stated
that an S-form indicatesthat a symbol series contains pairs of identical subseries
nested around a pivot. For instance,the series 'abcpbca'can be encoded into the
(p)j, inducing the chunking (a)(bc)(p)(bc)(a).Now, wirh respect
S-form S[(a)(bc),
to that chunking, the S-form can also be said to induce a tri-partitioning into
the S-argument(a)(bc),the pivot-chunk(p),and rhe reversedS-argument(bc)(a).
Therefore, the S-form will be said to induce the grouping of the chunks in the
S-argumentinto one cluster((at(bcl),as well as rhe groupingof the chunks in the
reversedS-argumentinto one cluster((bc)(a)).This clusteringcan be indicatedin the
chunk seriesby ((a)(bc))(p)((bc)(at).
We also statedthat an A-form expresses
that a
symbol series contains successivesubserieswhich either all begin or all end identically. Let us focus on the 'all-beein-identically'-case.
(The 'all-end-identically'-case
is completeiyanalogous.)For instance,the series'abpqabrsabt'can be encodedby the
t )o. w ,
A - f o r m ( ( a b ) ) r ( ( p q t ( r s ) ( tw) )h, i c h i n d u c e st h e c h u n k i n g( a b ) ( p q t ( a b ) ( r s ) ( a b ) (N
'successive
in that chunking,eachof the above-mentioned
subseries'is chunked into a
pair of chunks called an A-pair. So. each A-pair constitutesa unit that is characteristic for the regularitydescribedby the A-form. Moreover,as we will seelater on. the
A-pairs are essential for understandingthe hierarchical character of the A-ruie.
Therefore, the A-form above will be said to induce a clustering by grouping
each A-pair into a ciuster, which can be indicated in rhe chunk series by
( ( a b , i p q t ) ( ( a b t ( r s i ) ( ( a b r tS
t )o) .. i n s u m m a r y b
, e s i d e sa c h u n k i n g ,a n I S A - f o r m a l s o
inducesa clusteringbasedon that chunking:each ISA-form inducesone cluster consistingof all chunks in that chunking;an S-torm inducestwo further clusters,namely
the S-argumentand the reversedS-argument;and an A-form inducesfurther clusters
by groupingeachA-pair into a clusrer.
The perceptualorganizationof a pattern.as induced by a code of the pattern, will
be said to consistof the combinationof the chunking and the clusreringinduced by
that code. Furthermore.the dominant segmentationin a perceptualorganizationof a
pattern is said to be the segmentationthat consistsof at least two. maximally sized,
segmentsichunksor clusters)presentin that perceptualorganization.As an illustration. we reconsiderthe pattern in figure l. The patternin tigure l can be represented
by a symbol seriesin severalways: in this example.we will focus on the two most
relevant representations(see figure-l). Rememberthat a symbol series representsa
pattern.if that patterncan be reconstructedfrom the svmbol seriesby -substitutingthe
/
/
(a)
\
ib)
F i g u r e 3 . I f t h e p a t t e r n i n i a ) i s r e p r e s e n t e db y r h e s y m b o l s e r i e s ' k a l c k a l c k a l c then,
'
according
t o t h e s t r u c t u r a l i n f o r m a t i o n m o d e l . r h i s s v m b o l s e r i e s c a n b e e n c o d e d i n t o 3 x (kalc). This
c o d e i n d u c e s . i n t h e s y m b o l s e r i e s . t h e o r g a n i z a t i o n i k a l c r ( k a l c ) ( k a l c )w h i c h . in the pattern.
correspondsto the organizationas shown in ib i.
523
Serialpanem complexity:inegularityand hierarchy
actual sizes of angles and line segments for the pattern symbols. One way to
'kakakakkakakakkakakak'
represent the pattern, is by means of the symbol series
(see figure 4a). For all of the complexity metrics to be discussed later on, the
simplestcode for this symbol seriesis 3 x (k3 x (ak)). This code yields the chunking
(kakakak)(kakakak)(kakakak)
as the dominant segnrentation,correspondingto the
triangle interpretation of the pattern. Another way to represent the pattern is by
'kakbkakbkakbcl'(seefigure 4b). The simplestcode of this
meansof rhe symbolseries
symbol seriesis 3 x (((k))l((a)(b)))cl.This code yields the clustering(kakbkakbkakb)cl
as the dominant segmentation.corresponding to the zigzag interpretation of the
pattern. Now, the code that reflects the triangle interpretation is simpler than the
code that reflects the zigzaginterpretation and, indeed, the triangle interpretation is
usuallypreferred.
---+
I
kakbkakbkakbcl
3xi(k)) /(a)(br)rcl
kakakakkakakakkakakak
,3x {lqj x rnli r
rkakb kakb kakb tc I
k a k a k a k ; k, a k a k a kr;k a k a k a k
AAA
ia)
ibl
Figure ,1. The partern in figure I can be representedin two ways by means of the symbol series
s h o w n i n ' a r a n d r b ) . F o r e a c h s y m b o l s e r i e s .t h e s i m p l e s t c o d e i s s h o w n w i t h t h e i n d u c e d
dominanr segmenration in rhe symbol series and, correspondingly, in the pattern. The code
in (a) is simpler rhan the code in (b) and, indeed, reflects the usually preferred triangle
interpretation.
3 The concept of regularity
In the paper of van der Helm and Leeuwenberg(1991), a formalization of the
intuitive concept of regularity has been given within the framework provided by the
structural information model. In this section,we will summarizethis formalization in
a nonformal way, ie we will, by means of examples,give a gist of the model in order
to pur the new complexity metric in perspective. The formalization is twofold: first,
the intrinsic character of regularity is specified by the formal notion of holographic
regularity, and second.the way in which casesof regularity can be related hierarchically is specifiedby the formal notion of transparenthierarchy. In this section, these
rwo formal notions will be discussedsuccessively,after which several psychologically
relevantimplicationswill be discussed.
3.1 Holographic regulariw
As we have seen. the structural information model represents a visual pattern by
means of a symbol seriesin which the symbols refer to pattern parts (the semantic
mapping).Therefore,an interpretationof the pattern is consideredto be reflectedin
the classificationand organizationof the symbol serieswhich. becauseof the semantic
524
P A van der Helm,R J van Liec E L J Leeuwenberg
mapping,correspondsto a classificationand an organizationof the pattern. By using
coding rules.the symbol seriesis classifiedand organizedon the basisof a hierarchical descriptionof regularityin the symbol series. Note that the symbol seriesonly
representsinformation about the order and the identity of pattern parts. So, in this
context.the formalizationof regularitycan only be basedon arrangements
of identical
symbolsin a symbolseries.Now, the formalizationproceedsasfollows.
First. any arrangementof identicalsymbolsin a seriesis formally called a caseof
regularity,no matter whetheror not it reflectssomethingintuitively regular. That is,
we start with all theoreticallypossiblecases,and eg a repetition of a specificsomething a specificnumber of times (suchas nine times the svmbol 'p') is just one of
thesecases.So, the arrangementof identicalsymbolsin a seriesis formally called a
'aaaa'and 'abccba',but also for
caseof regularity,not only for serieslike
serieslike
'kpfzpkf' and 'kyppzfkf'. Now. for example.in the series 'vy', the arrangementof
identical symbols is formally denoted by the expression(1) : (2), which simply
indicatesthat the first svmbol is identicalto the secondsymbol. The expression
( I i = I I ) is called an identity. Note that the sameidentitv denotesthe arrangementof
identicalchunksin. for example,the chunkseriesiabr(ab).A set of identitiesis called
an identitystructure.if the set meetssomeformal requirements.One of the requirementsis that the identitiesin the set are ordered. For instance.the caseof reeularity
reflected by the arrangementof identical symbols in the series 'aaaa',is formally
d e n o t e db y t h e i d e n t i t vs t r u c t u r ei ( l ' = t 2 ) , t l ) : 3 ) , r 3 ) : t - l ) l w h i c hc o n s i s t so f
threeidentitiesin the givenorder.
Second.all different casesof resularity are categorizedinto kinds of regularity.
which may be illustratedas follows. For the series'abab'.the arrangement
of identical
:
symbolscan be denotedby, amongothers.the identitystructuret(12) (34)f which
indicatesthat the two subseries'ab' are identical. In this identity structure,each of
the two subseries'ab' is taken as one unit, just like when the series'abab' would be
chunked into the chunk series , abr/abr. In other words. the identity structure
that 'abab' is the sameas 'vv' under the substitutionof 'ab' for 'y'. Now.
e.xpresses
o b s e r v et h a t , i n t h e s e r i e s ' y y ' o r a b r ( 3 $ )t,h e i d e n t i t ys t r u c t u r et ( 1 ) : ( 2 ) l a c t u a l l y
the sameas the identitvstructure1(12) = i3a)) in the symbolseries'abab'.
expresses
namel.""
the identitv of the two subseries'ab'. Theretbre,althoughthesetwo identity
structuresdescribedifferent casesof regularity,they are said to describethe same
kind of regularityinamelv,in words.a repetitionof two timesan arbitrarysomething/.
Note that, formally. two times an arbitrary somethingis a different kind of regularity
from that of three times an arbitrary something.In summary,a repetitio-nof a specific
somethinga specific number of times is a case of regularity,and a repetition of an
arbitrary somethinga specific number of times is a kind of regularitv. As we will
argue next, repetition in general (an arbitrary something an arbitrary number of
timesr.is a caseof holographicregularitv.
Above, we saw that an identity structure reflects a property of a series, namelv
identity of elements in that series. Now. holographic regularity specifies a possible
property of identity strucnrres. Metaphorically, the notion of holographic regularity
may be illustrated by a jigsaw puzzle that is to represent a landscapeconsistingof a
green lawn and a sky with clouds. Each green piece in the disorderedset of pieces
can be classifiedas a lawn-piece,since such a piece shows the holographicproperty
of havingthe samecolor as the entire lawn. In our view, such a holographicproperty
appliesquite well to the intuitive conceptof regularity. For instance,a repetition of
somethingis a repetitionno matter whetherthe number of times that the somethingis
repeatedis large (analogousto the jigsaw lawn) or small (analogousto the jigsaw
piece).The formal notionof holographicregularityhasbeenelaboratedasfollows.
Serialpattemcomplexity:inegularityand hierarchy
525
An identity stnrcture can be seen as a chain of identities becauseit is an ordered
set. This implies that one can define a substructureof an identity structure as an
ordered subsetof successive
identitiesin the identity structureor, in other words,as a
subchain.For instance,the earlier-mentioned
identity structure{(l) : (Z), (Z) : (3),
i 3 ) : ( - l i f h a s f i v e s u b s t r u c t u r ensa, m e l y{ ( 1 ) : ( 2 ) , ( 2 ): ( 3 ) 1 ,l Q ) : ( 3 ) , ( 3 ): ( 4 ) } ,
i ( 1 ) : ( 2 ) i , { ( 2 ) = ( 3 ) } ,a n d { ( 3 ) : ( 4 ) } ( s e ea l s o f i g u r e s5 a n d 7 ) . A n a l o g o u st o t h e
jigsaw puzzle and expressedsimply, an identity structure is said to describea holographic kind of regularityif its substructuresall describethe samekind of regularity
i seealsofigure5 ).
Observe the recursive character of the notion of holographic regularit-v;if an
identity structuredescribesa holographickind of regularity,then any of its substructures describesa holographickind of regularity too, since all substrucruresof the
identity structure{including the substructuresof a substructure)describethe same
kind of regularity. This recursivecharacteris essentialfor elaboratingthe formal
implicationsof the notion of holographicregularity. Without going into details,these
formal implicationscan be summarizedas follows [for the details,see van der Helm
( t 99I )j.
and Leeurvenberg
lnstead of consideringall possible identity structures an infinite number), it is
more suitable and suffices to consider only all identity strucrures consisting of
precisely three identities (also an infinite number). One can prove, first, that the
infinite number of different cases of regularity, as described by these identity
structures consisting of three identities, can be categorized into precisely 648
different kinds of regularityand, second,that only 20 of these648 kinds of regularity
are holographic.Then, becauseof the recursivecharacterof holographicregularity,
one can prove two further things. First, for each of those 20 holographickinds of
regularity,a representative
identity strucrure(consisting,as before,of three identities)
can be generalizeduniquely into an identity structure consistingof an arbitrary
number of identities(analogousto simply increasingthe number of times in a repetition). Each of these20 generalizedidentiw structuresis called a caseof holographic
regularity. Second.one can prove that thesel0 casesconstiruteall possiblecasesof
holographicregularity. Figure 6 shows some of these. Finally, ignoring syntactical
variations in the definitions of specific coding rules lsince such variations are
irrelevantwith respectto the meaningof coding rules),one can prove that eachof the
rstPtsr
kpfkfp
l ( l i : r . t )t.2 ) : t 6 ) . 3 ) - { 5 ) }
i ( lr = r 7 )i.2 ) = t 6 )r, 3 ): t 5 ) i
( l ; : i 7 ) , r ? i= ( 6 i l l ( 2 ) = ( 6 ) t, 3 ) :
.st.ts.
rs...sr
ra)
i ( l ) = r 4 i . r l t: I 6 ) ii ( 2 )= r 6 i , t 3 )= { 5 , r i
kp.k.p
.pf.fp
(b)
Figure5. Holographicregularity.In (a),the arrangement
of identicalsymbolsin the series'rstptsr'
is denotedby an identity structureconsistingof three identities.Below this identity stnrcture
are shownits two substructures
consistingof two identitieseach. Each of thesesubstructures
is
visualizedby substituting,in 'rstptsr'.a dot for thosesymbolsto which the substructuredoes
not apply. In (b) exactlythe sameprocedurehas been followed for the series'kpfkfp'. In (a),
both substructures
reflectthe samearrangement
of identicalsymbols.That is, the two symbols
'r' and the two symbols's', in the visualizationof the first
substructure,
are arrangedin the same
way as the two symbols 's' and the two symbols 't' in the visualizationof the second
substructure.[n other words.both substructures
describethe samekind of regulanty,which is
preciselythe reasonthat the identitystructureshownat the top is said to describea holographic
kind of regularity.In contrast,in (b), the two substructuresclearly do not reflect the same
arrangement.
ie do not describethe samekind of regularity,so that the identity structureshown
at the top doesnot describea holographickind of regularitv.
P A van der Helm,R J van Lier,E L J Leeuwenberg
czo
20 casesof holographicregularitycan be describedby preciselyfour different coding
rules. So, the final resultis that only preciselyeightyso-calledholographiccodingrules
exist. Among theseeighty coding rules are the lSA-rulesemployedin the structural
information model. Figure 7 shows, for the iteration rule, a schemethat typically
holdsfor any holographiccodingrule.
Now, if holographicregularityis acceptedas being relevantand if eighty holographic coding rules exist, one might wonder why the structural information model
employs only the lSA-rules. Well, the answer lies in the fact that, above, only the
intrinsic characterof regularity has been dealt with. So far. nothing has been said
about the way casesof regularitycan be relatedhierarchically.The latter aspectis, in
the formalization, specified by the formal notion of transparenthierarchy. This
notion is, even more than the notion of holographicregularity,relevantwith respect
next.
to the complexitymetricto be proposed,and will be discussed
nnnnn
a
t
l
l
]
i
l
a
r]nnr]r]
a
a b
b c
c d
d e
e
ffi
a
b a
c b
d c
e d
e
Figure 6. Five characteristic visualizations of holographic regularity. In the five prototvpical
s y m b o l s e r i e s .e a c h i d e n t i t y r e l a t i o n s h i pb e t w e e nt w o s v m b o l si s v i s u a i i z e db y a n a r c . F o r e a c h
series. the set of arcs shows a regular ordering. The holographic propertv of this ordering is
reflected by the fact that the first or the last arc in each set can be removed without distorting
t h e r e g u l a ro r d e r i n g i n t h e s e to f a r c s .
i ( l ) = { 2 ) . r l )= i 3 t i
3xta)a
1 ( l )= l r i
lxrataa
3l x 1a)a
i ( 2 ): i 3 ) .i 3 ) : 1 { ) l
a3x{a)
(3t = r-t)l
aalx(a)
'aaaa'
can be encoded into the l-form
Figure 7. The holographic iteration rule. The series
(chain)
consisting of three identities. For
shown at the top. expressing the identity stmcture
each substructure qsubchain)of this identity stmcture. an I-form exists such that it expresses
that substructure. This indicatesthat the iteration rule is a holographic coding rule.
3.2 Trarsparenthierarchy
Metaphorically,transparenthierarchymay be illustratedby the hierarchicalstructure
of an industrial organization.In such an organization,the position of the manager
may rely on that structure,but the managercan also be approachedindependentlyof
the other employees.The analoguein terms of symbol series may be illustrated as
follows.
Serialpattemcomplexity:inegularityand hierarchy
527
ln encodingmodels like the structural information model, the coding rules yield
hierarchicaldescriptionsof regularity in a symbol series. For instan".l u, we saw
before'the series'ababbaba'can
be encodedinto the S-form St(a)(b)(aXb)].
In this
(a)(b)(a)(b)is said to representa higherhierarchicallevel
S-form,the S-argument
and
can be encodedinto the I-form 2x((a)(b)). The latter code can be nestedin the
S-form,yieldingthe hierarchicalcode S[2x((a)(b))].Note that the S-form describes
regularity in the symbol series,but that the I-form describesregularity in the S-argument at a higher hierarchicallevel. So. one might wonder what the meaningof ihe
I-form is with respectto the descriptionof regularityin the symbol series.ababbaba'.
Now. observethar the l-form 2x((a)(b)) in the S-argument(a)(b)(a)(b)describes
a
kind of regularity (a repetition of two times something)that is also describedby
the
I-form 2 x (ab) in the subseries'abab' of the symbol series'ababbaba'.This example
illustrates a generalcharacteristicof the S-ruIe, namely that any kind of regularity
in
the argumentof an S-form correspondsunambiguouslyto the samekind of regularity
in the symbol series. So. in an almost visual sense,any S-form is .transparent,,
ie regularity in the symbol series can be 'seen through' the S-form. And indeed,
becauseof this generalcharacteristic,the S-rule is called a transparentcoding rule.
For the exampleabove.this impliesthat the hierarchicalcode SiZx((a)(Ut)lJan be
seenas indicatingthat the S-formS[(a)(b)(a)(b)j
and the I-form 2x(ab) can be related
hierarchically.Inversely,this implies rhat. insteadof the l-form,2x((a)(bl), in the
S-argument.one could just as well considerthe I-form, 2 x (ab), in the symbol series.
Thus. analogousto the managerin the metaphor, the higher-levelI-iorm can be
consideredindependentlyof the lower-level S-form. Moreover, note that. in the
exampleabove.the I-form 2x(ab) inducesthe chunking(ab)(ab)in the subseries
'abab' of
the symbol series'ababbaba'.This chunking can be superimposedon the
chunkingia)(b)(ar(b)(b)(a)(b)(a)
inducedby the S-form in the symbolr.ri"r. ro yield
the hierarchical
chunking((a)(b))((a)(b))(b)(a)(b)(a)
in the symbolseries(seefigure8).
Such a hierarchicalchunkingcan be assignedunambiguouslyto any hierarchicalcode
obtainedby meansof transparentcoding rules and is, therefore,called a rransparenr
hierarchy. So' a hierarchicalcode obtained by means of transparentcoding rules
indicateshow differentcasesof regularityin a symbol series.un b. relatedhierarchically, and the resulting hierarchical code induces a hierarchical chunkine in the
symbolseries.
Si(ai(bt(arrbil
level
A D
a b
b a
J
J
b a
1 a) ( bI
ta)(b)
rb)(a)
lbt(a)
(ar(b))
rb)(a)
(bl(a)
level
S fl x i i a r r b r r l
level 3 : r ( a ) t b t l
Figure8. The transParent
svmmetryrule. The S-form S[(a)(b)(aXb)]
inducesa chunkingin the
ievel I symbol series, representedby the level2 series. The l-form, 2 x {(a)(b)), in the
S-argument
corresponds
unambiguously
to the I-form.2x(ab), in the levelI series.inducingthe
chunking{ab)(ab)which can be superimposed
on the level2 series.and leadinsto rhe hierarchical chunkingrepresented
in level3.
From the example just given, one may get the impression that, for coding rules,
transparency is only a plausible requirement and may even be rather trivial. This may
be the case for the S-rule but, for coding rules in general, transparency is far from
trivial. This may be illustrated by means of the so-called M-rule which is one of the
eighty holographic coding rules mentioned earlier, and is defined as follows:
k , x , k , k . x . k , . . k n x n k " - M [ ( k , ) ( k : ) . . . ( k , ) , ( x , ) ( x .: . ( x " ) ].
528
P A van der Helm, R J van Ller, E L J Leeuwenberg
The M-rule can be applied to expressthat a seriescontainssuccessive
subseries
which each begin and end identically. Now, consider the symbol series 'arabsb
ayabzb' which. by means of the M-rule, can be encoded into the M-form
M [ ( a ) ( b ) ( a ) ( b () r, ) ( s ) ( V l Q 1 ]T. h e f i r s t a r g u m e n o
t f t h i s M - f o r m , ( a ) ( b ) ( a ) ( b )c,a n b e
x((aribi).
encoded into the l-form 2
However, this I-form describes,in the first
M-argument,a kind of regularityta repetition of somethingtwice, ie two successive
which does not occur in the symbol seriesitself. Therefore,the
identicalsubseries)
M-rule is not a transparentcoding rule. In fact, one finds in this way that only nine
of the eighty holographiccoding rules are transparentcoding rules.amongwhich are
the ISA-rulesemployedin the structuralinformationmodel.
Since the transparencyof the ISA-rules is important with respect to the new
complexitymetric. we will now go into the transparencyof the l-rule and the A-rule
in some detail. The l-rule is transparent'by default', since the l-argument of an
I-form consistsof only one chunk so that. at the higher hierarchicallevel, no
regularity can be described. The transparencyof the A-rule, however,is not that
trivial. We saw that the A-rule can be applied to express that a series contains
successive
subseriesrvhicheitherall beginor all end identically,as follows:
k x , k x , . . . k x n- : ' ( k t()( x , ) ( x : ) . . . . x " i )
x , k x , k . . . X n k- . ( x , , ( . \.:.). ( x ,) , ( k , ) .
Clearlv.the two casesare verv similar.and we will discussonly the transparency
for
'k.xkykykz'
the first case. Supposethe symbol series
is encoded into the A-form
1 ( k 1,)i 1 " r ( y t ( y r ( z tT) .h e n t h e s u b s e r i e sy ) ( y ) i n t h e A - a r s u m e nct a n b e e n c o d e di n t o
t h e I - t o r m 2 x t ( y ) , v i e l d i n et h e h i e r a r c h i c aclo d e { ( k ; ) ( ( x ) 2x ( ( v i ) i z ; ) . N o w . o b s e r v e
t h a t t h e l - f o r m . 2 x ( i y l ) . i n t h e A - a r s u m e ndt o e sn o t c o r r e s p o n dt o a n l - f o r m , l x i y l .
in the symbol series'kxkykykz'. Yet the regularitvdescribedby the I-form as in the
A-argumentdoes correspondunambiguouslyto the same kind of regularitv in the
symbol series.That is. the l-form in the A-argumentcorrespondsunambiguouslv
to
(kv1.
x
the I-form. 2
in the symbol series. Note that. in the latter I-form. the
I-argument(ky) correspondsto an A-pair cluster as discussedbefore. Similarlv,in
general(seedefinitionabove),for an A-form ((k)) ((x,)...(x,,)), any regularityin the
a r g u m e n(tx , ) ( x : ) . . . , x , , ,c o r r e s p o n dusn a m b i g u o u stl o
y t h e s a m ek i n d o f r e g u l a r i r iyn
t h e s e r i e si k x , ) ( k x . , . . . l k x , , ) , c o n s i s t i n go f A - p a i r c l u s t e r s .C l e a r l v .a n y k i n d o f
regularityin this clusterseriescorresponds
unambiguously
to the samekind of regularity in the symbol series 'kx, kx. ... kxu', thus showing that the ,trule is indeed
a transparent coding rule. Furthermore. in the example above. the A-form
( ( k ) )( ( . x ) ( y ) ( v ) ( zi )n)d u c e st h e c h u n k i n g( k ) ( x ) ( k ) ( V ) ( k ) ( y ) ( k ) (i zn) t h e s y m b o ls e r i e s .
w h i l e t h e l - f o r m 2 x 1 k y ) i n d u c e st h e c h u n k i n g( k v ) ( k V )i n t h e s u b s e d e s ' k y k y ' o ft h e
symbol series. Clearly. the latter chunking can be superimposedon the former
levell:
I
' ( k r \/ { x ) l v l l x ) )
|
k
x
k
v
k x
\t/
level:
( k ) ) i S [ { ( x )()l.y ) ) l )
'ktix)
,ki(vr
tkt{x)
t1k:rvr)
t(kttxt)
I
|
\y
l e v e l- 1 : , , k l L x ) )
F i g u r e9 . T h e t r a n s p a r e n tA l t e r n a t i o n r u l e . T h e A - f o r m ( ( k ) ) ( ( x ) ( V ) ( x ) )i n d u c e s a c h u n k i n e i n
t h e l e v e l I s e r i e s . r e p r e s e n t e db y t h e l e v e l 2 s e r i e s . T h e S - f o r m S [ ( ( x r )i,i y r ) j i n t h e s e c o n d
argument of the A-form corresponds unambiguously to the S-form S[(kxt,rkyl] in the level I
s e r i e s . i n d u c i n g t h e c h u n k i n g ( k x ) ( k v r ( k x ) w h i c h c a n b e s u p e r i m p o s e do n t h e l e v e l 2 s e r i e s .
l e a d i n et o t h e h i e r a r c h i c a cl h u n k i n gr e p r e s e n t e da t l e v e l 3 .
Serralpanerncomplexity:inegularityand hierarchy
529
chunking, yielding the hierarchicalchunking (k)(x)((k)(y))((t<)(V))(k)(z)
(see also
figure 9). So, in order to understandthe hierarchicalcharacterof the A-ruIe, one
should replaceeachchunk in the A-argumentby the relatedA-pair, in order to obtain
the chunkingat the higherhierarchicallevel.
3.3 Implications of formal regulairy
We have given an overviewof the formalizationof the intuitive conceptof regularity,
specifiedby the formal notions of holographicregularity and transparenthierarchy.
Holographicregularityappliesto the intrinsic characterof regularity,and reflectsthe
fact that, for example,an arbitrary repetition belongs to the set of all repetitions.
Transparenthierarchyappliesto the way different casesof regularitycan be related
hierarchically,and implies that the hierarchical character of codes is not just a
syntacticalartifact of the coding language,but a psychologicallymeaningfulaspectof
the descriptionof regularity.The result of the formalizationis that only eightycoding
rules are such that they describeholographicregularity,and that only nine of these
eighty coding rules are such that they describe transparenthierarchy too. Among
these nine transparentholographiccoding rules are the ISA-rules employed in the
structural information model. Actually. the other transparentholographic coding
rules are superfluousin the sensethat they describeonly identitiesthat can also be
describedbv the lSA-rules.whereasthe lSA-rulescan describeorher identitiesas well.
This impliesthat, accordingto the formalization,a set of appropiatecoding rules may
consistof just the lSA-rules.This result as such is not very surprisingsince the kinds
of regularity describedby the ISA-rules are widely acceptedas being relevant in
perception and have been emphasizedby various scientists(eg Wertheimer 1923;
Koffka 19351Palmer 1977). The way this result has been obtained is what matters
here, becausethe lSA-rulesnow have a unique formal statuswhich is psychologically
relevant. The psychologicalplausibilityis supportedby severalfurther implicationsof
the formalization,aswe will arguenext.
If holographiccoding rules are used to extractpattern information(regulariqvtfrom
a symbol series that representsa pattern, then the extraction is very easy,ie the
pattern informationto be extractedis verv accessible.This may be illustratedby the
holographicI-ruIe, for which l-forms can be constructedin a simple stepwisefashion,
starting with any single identity and with each step adding just one identity, eg as
follows(seealsofigure 7):
a a a a- Z x ( a ) a a* 3 x ( a ) a - 4 x ( a )
So, for the i-forms, the construction proceeds from one I=form towards another
I-form, each step enlargingthe expressedidentity structureby one identity. Clearly,
this is possiblebecauseof the holographicproperty that any iteration belongsto the
collectionof all iterations. Thus, complex combinationsof identitiesdo not have to
be matched.sincethe constructioninvolves'atomic'stepsof one identityat a time.
Maybe even more than the notion of holographic regularity, the notion of transparent hierarchy results in the accessibility of pattern information as contained in a
symbol series. As we saw before, the transparencyof coding rules implies that
regularityat a higher hierarchicallevel in a code of the seriescorrespondsunambiguously to the samekind of regularityin the seriesitself. Sincethe classificationand the
organization of the series is based on the described regularity, the transparency
implies that higher cognitivelevels will deal with (or have accessto) basic partern
information itself. That is, the information that is consideredto be passedon to
higher cognitivelevelsis not. as with an artifact of the employedmodel, basedon just
regularity at a higher hierarchical level in a code, but on regularity in the pattern.
P A van der Helm, R J van Lier, E L J Leeuwenbero
Moreover, the notion of transparenthierarchy yields the possibility of a largely
parallel encoding process, as follows. As we saw before, the hierarchical code
S [ 2 x 1 ( a ) ( b ) )cla n b e s e e na s i n d i c a t i n gt h a t t h e S - f o r mS [ ( a t ( b ) ( a ) ( ba) n
] d the I-form
2 x (abt can be relatedhierarchically.
This impliesthat the I-form and the S-form can
be constructedindependently
of eachother,and ther-rest€dfor hierarchicalcompatibility. This shows that, in general,all single ISA-forms in a svmbol series can be
constructedin parallel after which they can be tested. in parallel. in pairs for
hierarchicalcompatibilty.Thus, on the one hand. the encodingmav result in a
hierarchvconsistingof sequentiallyordered hierarchicallevels.but. on the other
hand. this hierarchydoes not have to be establishedin a strictly sequentialway. This
illustratesthat, in our view, the notion of hierarch-tshould not be basedon some
processmodel as such.as in. for example,the so-calledhierarchicalsequentialsearch
(196-t),or as in other so-calledtop-down models
model of Simon and Feigenbaum
that involvereasoningor unconscious
inferences(von Helmholtz1909/1962:Neisser
1966: Gresorv 197?: Rock 1983). Nor should hierarchvbe seen,as suggestedin
Buffart 11987),as a property of a single pattern representationin which several
differenthierarchicallevelscan be distinguished
by decomposinsthat represenrarion
i s e ea l s ov a n d e r H e l m 1 1 9 8 8 ) 1O. n t h e c o n t r a r yh. i e r a r c h vs h o u l dr a t h e rb e s e e na s a
relation between severaldifferent representations
of the same pattern. That is.
severaldifferent [SA-forms.obtained in parallel.may be related hierarchicallyin
order to composea hierarchyconsistingof severaldifferenthierarchicallevels.In the
Iatter sense.the notion of hierarchvagreeswith a bottom-upextractionof gradually
more structuredinformation.That is. startingfrom the singleidentitiesin a 'raw'
pattern registration.resularityis describedfirst and then different kinds of regularity
are relatedhierarchicallv.
resultingin a classification
and an organizationwhich can
be'embedded'
in storedknowledgestructures.
Another implication of the formalization is related to the problem that the
minimumprincipleseemsto requirean unrealisticsearchfor simplestcodessince.for
an arbitrarysymbolseries.the numberof possiblecodesis combinatoriallyexplosive
(cf Hatfieldand Epstein1985). This problemdoesnot dependon the exactcomplexitv metric that is used. For instance,if a seriescan be encodedentirely into one
S-form ior A-form) then. in general.the seriescan be encodedentirely into an
exponentialnumber of S-forms(or A-forms). Becauseof the notionsof holographic
regularitvand transparenthierarchy.however.theseS-formsior A-forms)can all be
stored and encodedsimultaneouslv.
as if it were just one S-forn (or A-form). Thus,
the searchfor the simplestS-form (or A-form) becomesrealistic.since it does not
involve an explosiveamount of storagespaceor processingtime. For detailed information on this problemand its solution.we refer to van der Helm and Leeuwenberg
( 1 9 8 6 .I 9 9 1) a n dv a nd e r H e l m( 1 9 8 8) .
The discussionin this subsectionshows that the formalizationof regularitynot
only provides a psychologicalbasis for the choice of appropriatecoding rules, but
also has further implicationsthat are psychologicallyrelevant. The implicarionthat is
most relevantin the presentpaper is the fact that the formalizationpavesthe way for
a newcomplexitvmetric.This implicationwill be discussed
next.
4 The new complexit-vmetric
In this section,we will discussand evaluatecomplexity metrics on the basis of the
analysisin the previoussection.First. in order to clarify the need for a new metric.
we consider three metrics that have been used more or less frequently in earlier
researchon the structuralinformationmodel. Then we introducethe new metric.
Serialpanerncomplexity:inegularityand hierarchy
4.1 I",o-load
In many studies concerning the structural information model, the complexity of a
code has been measuredby meansof the /o,o-load(Leeuwenbergl97L). This load
was meant to reflect the amount of memory spaceneededto representa code, ie the
preferredpatterninterpretationis assumedto be reflectedby the code that requiresa
minimum of storagespace. Therefore,an assumptionhas been made with respectto
the relation betweenthe syntacticalcomponentsin a code and the memory space
needed to representthe code. as follows. First, the encodingof a series yields a
reduction of the seriesinto a code such that not all of the pattern symbols in the
series,but only those in the code, need to be representedin memory. Second,the
meansof establishing
the reductionhaveto be takeninto accountas well. Thesemeans
consist of the ISA-forms, and are taken into account as follows. For decodirg u
stored I-form like 5 x (ab) into the series'ababababab',
the numericvalue '5' has to be
known and is, therefore,assumedto be representedin just as much memory spaceas
each of the two pattern symbols,so that this l-form requiresthree units of memory
space. In decoding a stored S-form like S[(a)(b)(c)]into the series 'abccba',the
S-argumenthas to be reversedin order to produce the secondhalf of the series;this
reversaloperationis assumedto be representedin, again,just as much memory space
as each of the three pattern symbols, so that this S-form requires four units of
memory sPace. For decoding a stored A-form like ((at)i((r)(s)(t))into the series
'arasat'.the
number of times that the part 'a' has to be repeateddoes not have to be
stored since this number is implicitly given by the number of elements in the
A-argument(r)(s)(t); therefore,the storageof this A-form requiresonly four units of
memory space.ie only for the four pattern symbols. So, for an arbitrary code, one
has to count the pattern symbols,the I-forms, and the S-forms in the code to determine the 1o,o-load
of the code,ie the oomplexityof the code in termsof storagespace.
The strucrural information model has gained empirical support by using rhe /o,oIoad as a complexitymetric. However,the /o,o-loadgives rise to severalconceprual
problems. First. the patternsymbols,the numericvalue in an I-form, and the reversal
operation for an S-form, are given equal value but are in fact incomparableentities,
or at least very different entities. For example,why not assign the value of the
numeric value and of the reversaloperation as being equal to two or more pattern
symbols? Second, the reversal operation for an S-form is counted, but not the
iteration operationfor an I-form, nor the alternationoperationof an A-form. These
conceptualproblemsindicatethat the /o,6-loadis basedon a dgbious,and in our view
unacceptable,assumptionconcerningthe relation betweensynracticalcomponentsin
a code and the memory spaceneededto representthe code. That is, in our view [see
also Hatfield and Epstein (1985)] this assumptiondependstoo much on artifactsof
the encoding language.Moreover, we think that the semanticalimplication of a code
(ie an interpretation)shouldnot be judged on the basisof the requiredmemory space
but, psychologicallymore meaningfully,on the basis of the semanticalcontent of the
code (ie the descriptionof regularity).The conceptualproblemswith respectro the
.1,,,u-load
have been reuognized in research concerning the structural information
model; the followingcomplexitymetricwasdevisedto overcometheseproblems.
1.? P-load
In order to determine the P-load of a code, one simply counts the number of pattern
symbols in that code (cf Collard and Buffart 1983). For instance,for the code
3 x (ab), the value of the P-load, is P = 2, becauseit contains only two pattern
symbols,'a' and 'b'. This metric implies a judgementof the resultingclassification,as
follows. For instance, as we saw in the second section. if the series 'aaaaaa' is
encoded into the I-form 3 x (aa), for which p : 2, then the series is considered
3Jl
P A van der Helm, R J van Lier, E L J Leeuwenberg
to be classifiedinto the class of seriesthat can be representedby, for example,the
series'yzyzyz'. Occasionally,such a class representativehas been called an abstract
code (cf Collard and Buffart 1983). Now note that, in the exampleabove,the l-form
3x(aa) describedregularityin the symbol series'aaaaaa'byexpressingthe identity
'yzyzyz'. Therefore, the residual
of the elements contained in the abstract code
nonidentityof elementsin the abstractcode can be regardedas the irregularityin the
symbol series(at least,accordingto the I-form). Now, the P-load of a code equalsthe
number of different elementsin the correspondingabstract code and, therefore,
implies a judgementof the resultingclassificationby measuringirregularitv:the more
differentelements.the more irregularity,the more complex.
So. the P-load appliesto the output of the encodingmodel. ie it does not depend
on artifacts of the encoding language.Therefore, conceptually,the P-load seems
better than the In,o-load.Yet the P-load has not been used frequently becauseit
yields worse predictionsthan does the /.,u-load. A possibleexplanationfor this leads,
asfollows.to a third metric.
1.3 I ^-load
not for the
On the one hand.the P-loadaccountsonly for the resultingclassification.
and the organizationare part of
resultinqorganization.Sinceboth the classification
the output. one could sav that the P-load is incompleteso that the P-load 51illgives
rise to a conceptualproblem. On the other hand. the {,,u-loadcan be seen as
accounringfor nor only the resultingclassificationbut also, although only to some
extenr. for the resulting organization.as follows. First. note that the In,u-loadof a
code equals the P-load of the code plus the number of l-forms and S-forms in
in the
can be seenas accountingfor the classification
the code. That is. the d,,u-load
same wav as does the P-load. Now supposethat, insteadof only the l-forms and
S-forms.the A-forms in a code are also counted. Let us call this adaptedmetric the
I*-load. Now. recall that the argumentof an ISA-form representsa hieher hierarchical leveli in the code as rvellas.as we saw before.in the resultingorganizationl.That
is. each ISA-form in a code yields a transition to a higher hierarchicallevel. Thus,
counting all [SA-formsin a code correspondsto counting all hierarchicallevels.
In this wav the 1.-load can be seenas taking into accountthe resultingorganization.
So. similarly, the {,,0-load can be said to account for the resulting organization
ithough only to some extent since the A-forms are not counted),which might explain
why it yields better predictionsthan does the P-load. For the samereason.we expect
(and hypothesizein the experiment to be discussed)that the /"-lo4d. which also
countsthe A-forms.shouldyield betterpredictionsthan the /o,u-load.
However.this wav of accountingfor the resultingorganizationgivesrise. as before,
to a conceptualproblem. Namely,hierarchicallevelsand pattern symbolsare valued
equallybut are, in fact. incomparableentities. That is, whereasthe number of pattern
symbolsrefers directly to something(namely,irregularity)that is presentin a symbol
series,the number of hierarchicallevelsrefers only to the presenceof those levelsas
such. ie not to somethingthat is presentat those levels. The new complexitymetric
doesnot showsucha conceptualproblem.aswe will seenext.
1.1 In *-load
To a large extent,the new complexitymetric. the 1n.*-load,is basedon the conceptof
transparenthierarchy,as discussedin section3.2. tn the presentstudy,we have seen
that if a symbol seriesis encodedby meansof the lSA-rules,then the resultingcode
unambiguouslyinduces a transparenthierarchy, ie a hierarchical chunking, in the
symbol series(seefigures8 and 9). Now. imposingthe samehierarchicalchunkingon
the correspondingabstractcode (as defined when discussingthe P-load), yields an
orsanized class reDresentativewhich reflects both the resulting classificationand
Serialpattemcomplexity:inegularityand hierarchy
tr?a
the resulting organizationof the symbol series. For instance,if the symbol series
'abcabcab'is encoded into the
(ab)], then the classificationis
S-form S[(ab)(c),
'xztpqtxz',
representedby the abstractcode
whereasthe organizationis represented
by the chunk series(ab)(c)(ab)(c)(ab).
Imposingthis samechunkingon the abstract
code vields the expression(xz)(t)(pq)(t)(xz).Such an expressionwill be said to
representan abstractchunking. Note that, in general,suchan abstractchunkingexists
only for codesobtainedby meansof transparentcodingrules.
Now. in the new complexitymetric, the /n.*-load,all the different elementsover all
the hierarchicallevelsin such an abstractchunkingare counted. For instance,for the
S-form above, /n.* : 7, since the abstract chunking contains the following seven
d i f f e r e ne
t l e m e n t s' x: ' . ' z ' . ' ( x z ) '',t ' , ' p ' , ' q ' a n d ' ( p q ) ' . N o t e t h a t t h e l o w e r - l e v eel l e m e n t
't' is not differentfrom the higher-level
element'(t)'since,in this case,the parentheses
merely indicatea chunk that equalsone symbol. That is. in addition to rhe different
singlesymbols.onlv the different chunksconsistingof at leasttwo symbolsor chunks
are counted. This may be illustratedby two further examples.In figure8. we saw
t h a t t h e e n c o d i n go f t h e s y m b o ls e r i e s ' a b a b b a b a ' i n t ho e c o d eS [ 2 x ( ( a ) ( b r )y] i e l d sa
t r a n s p a r e nhti e r a r c h w
y h i c hi s r e p r e s e n t ebdy t h e e x p r e s s i oi n
(ar(b))((at(bi)(bt(a)(b)(at.
Sincethe code characterizes
all the identity of elementsin the symbolseries,the latter
expressionalso representsthe abstractchunkinq. This abstractchunkingcontains
t h r e e d i f f e r e n te l e m e n t sn. a m e l y , ' a ' . ' b ' .a n d ' ( ( a t ( b ) ) 's. o t h a t t h e c o d e h a s a n 1 n " * load value of /n.* = 3. Furthermore,in figure 9. we saw that the encodingof the
((y))])yieldsa transparenr
symbolseries'kxkykx' into the code ((k)) (S[((xi),
hierarchy
w h i c hi s r e p r e s e n t ebdy t h e e x p r e s s i oin( k i ( x t ) t ( k i ( y t ) ( ( k i ( x tA
) .g a i n .t h e c o d ec h a r a c terizesall the identity of elementsin the symbol series,so that the latter expression
also representsthe abstractchunking.This abstractchunkinqcontainsfive different
e l e m e n t sn.a m e l y , ' k ' . ' x ' , ' ( ( k i ( x ) ) ' . ' a
y 'n. d ' ( ( k ) ( y ) )s' .o t h a t t h e c o d eh a sa n . / n . * - l o a d
valueof 1n.* : 5.
Now that we have introducedthe 1n"*-load.we first of all want to emphasizethat
this complexitvmetricdoes not dependon artifactsof the codinglanquaqebecauseit
is basedsolely on the output. ie on the classificationand orqanization.Therefore, it
does not show the conceptual problems connected with the 1o,o-load.A better
account is now given of the hierarchicalstructure by counting not just hierarchical
levels but the different elementsat those levels. Counting the different elementsar
some level corresponds,as we saw when discussingthe P-load, to measuringthe
irregularityat that level. In other words, the 1n"*-loadcan be-saidto accountfor the
hierarchicalstructureby quantifyingits contributionto patterncomplexityin terms of
the irregularity at higher levels. In this way, the /n"*-load can be said to measure
patterncomplexityaddingirregularityand hierarchy.
In section5, we will discussan experimentthat has been designedto test the new
complexitymetric by consideringthe dominant segmentarion
in the perceptualorganization as predictedon the basisof the simplestcode (as discussedbefore). We argueC
that one has to take notice of not only the chunkingbut also the clusteringas part of
the perceptualorganizationof a pattern. That is, for instance,the series'abab' is
chunked into (ab)(ab) in order to describethe iteration regularity by meansof the
I-form 2 x (ab), which implies that the entire series 'abab' is clustered inro one
regularitystructurewhich is representedby the cluster(abab). Note that clusteringis
not taken into accountin the new complexitymetric. Whereaschunks are neededto
construct lSA-forms, clusters are 'only' a consequenceof ISA-forms. Therefore,
clusteringmay be relevantin experimentalsettingsbut is not relevantwith respectto
the complexityof ISA-forms.
P A van der Helm,R J van Lier,E L J Leeuwenberg
534
5 Experiment
The following experimenthas been designedto test the structuralinformation model
by using the new metric of complexity,ie the /n"*-load as discussedin the previous
section. This test comprises a comparison, with respect to preferred pattern
betweenthe new metric and the other metrics discussedin section4,
segmentations,
ie the P-load,the 1.,0-load,and the 1o-load.Each pattern,as used in the experiment,
is a patternedsequencein which the elementsare drawn from one of three different
setsof graphicsymbois(seefiguresl0 and l1). For eachpattern.subjectswere asked
to indicatethe preferredpattern segmentation.In each trial, subjectswere forced to
was
choosebetweentwo given pattern segmentations.Each of the two segmentations
a
code
the dominant segmenrationin the perceptual organization as induced by
obtained by means of the ISA-rules. So, given one of the complexity metrics. one
could check in each trial which of the two codeswas the simpler one and whetheror
one can investigate
not thar code indeed induces the preferred segmentation.Thus,
performs
best.
whichof the complexitymetrics
Patternedsequenceswere chosenbecausea straightforwardsemanticmappingcan
be assumedbetweena iserial)pattern code and the pattern. That is, still existing
unclaritieswith respectto the semanticmapping,eg for (nonserial)two-dimensional
parterns.or iserial)auditorvpatterns,are excluded.The specificstimuliwere selected
such that the variousmetricsmostly yield different predictions(exceptfor the P-load
'baseline').
which.aswill be clear.is usedasa sortof
oo
set I
set I
x l lo
set i
Figure 10. The three sers of graphic symbols used in the experiment. To construct a stimulus.
graphic symbols from one se! are drawn and composed into a patterned sequence. The graphic
iymbols within a set are verv distinctive phenomenally (short versus long; white versus black:
crossed versus parallel versus circular), so that they can be considered to be basic pattern
eiements.
ooo oo
ooooo
oooo
lal
(b)
Figurell. A target (a) and responsealternatives(b) used in the experiment.The pattern
r.qu.n"" at the lift was constructedby assigninggraphicsymbolsfrom set 2 in figure l0 to the
'ababb'. In the segmentations
(shownin b) a large spacebetween
symbolsin the svmbol series
two graphicsymbolsindicatesa border betweentwo segments.Subjectswere askedto indicate
theypreferaspartitioningof the targetinto coherentparts.
whichof the two segmentations
Serialpattemcomplexity:inegularityand hierarchy
535
5.1 Hypothesis
In line with the minimum p-rinciplewe predict that the simpler the code of a pattern,
the greaterthe preferencefor the segmentationinduced by that code. Now, for some
complexity metric M, let Gv be the goodnessof M, ie, the amount of preferred
pattern segmentations
that M predictscorrectly. Then, on the basisof the theoretical
analysisin the previoussections,we hypothesizethat:
Gp-t.u,t <
Gq,,u-tr"c <
Gl.-to"a (
G1"..-to"o.
A brief explanationof this is as follows: in the previous section,we arguedthat the
complexitv of a pattern code is constituted by the irregularity and the hierarchy
representedin that code. Schematically,the four metrics count the following code
components:
P-load -pattern svmbols.
/,,r-load-pattern svmbolsplus l-forms and S-forms,
1r-load -pattern symbolsplus lSA-forms.
/n"*-load-patternsymbolsplus chunks.
Bv countins the pattern symbols,the four metrics all account in the same way for
irreqularitv.so that differencesbetweenthe metrics have to be sought in the way of
accountinqfor hierarchy. According to the theoreticalanalysisand the hvpothesis,
the successive
metricsaccountfor hierarchyin a better way: the P-load does not take
hierarchvinto account:the /",0-loadcounts someof the higher hierarchicallevelsin a
code: the /n-load counts all higher hierarchicallevels in a code; and the /nu*-load
counts hierarchvin terms of irregularity at higher hierarchicallevels. Note that the
main issuewill be to contrastthe 1n.*-loadwith the {,,0-load,the most frequentlyused
metric in earlier empirical researchon the structural information model. The less
lrequentlv used P-load and /*-load are consideredmainly to obtain a more detailed
viewof the adequateness
of the theoreticalanalvsisin the previoussections.
5.2 .Vlethod
5.2.1 .Subjects.
Thirty-one undergraduates
receivedcoursecredit to participatein the
experiment.
5.2.2.VIaterials.To construct the patterns for the experiment, ie the patterned
sequencesof graphic symbols,three different sets of graphic symbolswere used (see
figure l0). For each pattern, graphic symbols were drawn from one set of such
symbolsto constructa sequence(seefigure 11a). In each set,-lhegraphicsymbolsare
'semanticallyindependent'.ie are very
distinct phenomenally(set 1: short versuslong;
set 2: white versusblack: set 3: cross versusparallel versuscircular). This ensures
that subsequentgraphic symbolsin a pattern will not be grouped togetherbecauseof
the'visualdistance'(cfTverskyand Gati 1982),suchas for a regularincreaseof greyvalue. or becausethey consritute,for example,a complementarypair of parentheses.
This, togetherwith the symmetryin each graphic symbol, implies that the symbolscan
be consideredas 'primitives', ie, as basic pattern elementswhich are not sensitiveto
biases thirt are irrelevant with respect to the goal of this experiment. This ensures
that a straightforwardsemanticmapping can be assumedbetweensuch a patterned
sequence and a symbol series as used in the structural information model.
For instance, the pattern in figure 11a can be represented by the symbol series
'ababb'. This series can be encoded by the ISA-rules, yield
to
the code S[(a),(b)]
2x(b) or the code 2x(ab)b. The former code is the simplestcode accordingto the
^fn"*-load.
implying that laba)(bb) is the dominant segmentation,whereasthe latter
code is the simplestcode accordingto the /o,o-load,and implies that (abab)(b)is the
dominant segmentation.These two segmentationsof the symbol series can be
P A van der Helm,R J van Lier,E L J Leeuwenberg
mapped straightforwardlyonto two segmentationsof the patterned sequenceof
graphic symbols(seefigure 11b in which the large spacebetweentwo of the graphic
symbolsstandsfor the border betweenthe two segments).Thus, each stimuluscan be
constructed from a patterned sequenceof graphic symbols (the target) and two
(responsealternatives)out of which subjectscan choose the
relevant segmentations
preferred segmentation.These preferencescan be checked to determine which
complexitymetriccorrectlypredictthem.
The total stimulusset consistedof 140 stimuli. derived from forty different symbol
also below). Among theseforty series.
serieslike the above-mentioned'ababb'(see
twenty-four contained two different symbols,and sixteen contained three different
symbols. The lengthof the seriesvaried from five up to eighteensymbols. For each
were considered. For example,for
of the forty series.three different segmentations
'ababb'the two sesmentations
(ab)(abb),as
mentionedabove.plus the segmentation
induced bv the code 1(a))((b)(2x(b))) were considered.Out of these three segwere each used in a stimulus.
mentations.the three differentpairs of segmentations
T h i s y i e l d sl 0 x 3 : 1 2 0 ' s y m b o l i c ' s t i m u li ie. s t i m u l i n t e r m so f s y m b o l su s e di n t h e
structuralinformationmodel and not yet in terms of graphic symbolsused in the
experiment. Furthermore.from each of the 120 symbolic stimuli. one additional
symbolicstimuluswas derived by reversingthe order of the symbolsboth in the
eg for the series'ababb',the reversalof
symbol seriesand in the two segmentations.
(ab)(abb)is lbba)(ba).Note
the seriesis'bbaba'.and the reversalof the segmentation
( ( 2 x ( b ) ) ( b ) ) r ( ( a )a)n, d
t h a t t h e l a t t e rs e s m e n t a t i oi sn i n d u c e db y t h e ' r e v e r s e d ' A - f o r m
that any code can be reversed in a similar way. Thus. one gets a total of
ll0 x 2 : 2.10svmbolicstimuli.
During the experiment.eachof the 340 symbolicstimuli was assignedrandomly to
one of the three graphic symbol sets. (Clearlv, symbolic stimuli containing
three different symbolscan only be assignedto set 3 in figure 10.) Then, each of the
different symbols in the symbolic stimulus was assignedrandomly to one of the
graphicsymbolsin that set. Thus. one obtainsthe actual stimulusset consistingof
l40 stimuli. In order to cancel out any residual bias with respect to the graphic
symbols.this random transformationof the svmbolic stimuli into real stimuli was
pertormed for each subject individually. The transformationwas performed by a
computer program that was developed to run the experiment and to present the
stimulion a monitor.
For selectingthe forty symbol seriesfrom which the 240 stimuli werg derived.and
for each of the forty series,three criteria were
for selectingthe three segmentations
used:
(i) For each symbol series,the three codesthat induce the three segmentations
should
have the same value tor P but different complexitiesaccordingto each of the other
three metricssuch that. for,eachof thesethree metrics,the simplestcode is included.
That is, the P-load was used as a baselinebecausethe P-load takes irregularity into
account in the same way as do the other metrics but does not accountfor hierarchy,
whereas the other three metrics precisely differ in how they take into account
hierarchy. In this way the experimentalresults can be related to the differences
betweenthe metrics.
(ii) The forty symbol series should be balanced with respect to the length of a
subseriesthat is coveredby a specificISA-ruIe. For instance,the series'ababb'shows
an iteration of the subseries'ab' and a partly overlappingiteration of subseries'b'.
The preferencefor one of these iterations may depend on how prominent such an
iteration is in a series. Therefore,also included were seriessuch as 'ababababb'and
'ababbbb'.
in whichoneof the iterationsis moreredundantlypresent.
Serialpanerncomplexity:irregularity
and hierarchy
q?7
(iii) For each of the fortv symbol series,each ISA-rule should be used in at least one
of the threecodes,so that the resultsindeedapplyto the entireencodingmodel.
These criteria are hard to meet simultaneously;thus they are met to a large extent
for the entire stimulusset but not for everv stimulus. This holds in particularfor the
requirementfor differentcomplexitiesfor eachof the three codes,as given in the first
criterion. That is, in the set of 240 stimuli two subsetscan be distinguishedfor each
metric. One subsetcontainsthe stimuli in which the two segmentations
are induced
by equally complexcodesso that the metric predicts ambiguity,ie no preferencefor
one of the two segmentations.The other subsetcontainsthe stimuli for which the
metric predictsnonambiquity,ie a preferencefor one of the two segmentations
since
the two codesdiffer with respectto complexity. In the analysisof the results,these
two subsetsare consideredseparately.In figure i2. the sizesof the two subsetsare
shownfor eachof the metrics,exceptfor the P-load. As mentionedabove,the P-load
is used as the 'baseline'.and predicts ambiguityfor almost all stimuli. Furthermore,
the P-load was alreadvknown to be inadequate,so that the experimentalresultsare
not verv interestingwith respectto the P-load iunless,of course,the resultsshowed
that most of the stimuli are indeed ambiguousbut, as wilt be clear, this is not the
casel. Therefore.the P-load is not onlv omitted in figure12 but also in the analysis
of the results. To concludethis subsectionon stimulus selection,some examplesof
the symbol series,as used in the experiment.are given in figure 13. Each symbol
seriesis encodedin three differentways. Each pair out of the three segmentations
for
one seriesrepresents
the responsealternativesfor the stimulus.
t00
@ ambiguityset
!
nonambiguitv
set
I
't)
/ro
/^
Metric
1n"_
Figure 12. For each complexitymetric, the sizesof the two subsets(expressedas a percenrage
of the total 240 stimuli) that contain the stimuli for which that metric predictsnonambiguiiy
(ie preferencefor one of the mo responsealternatives)and ambiguiiy (ie no prefereicej,
respectively.[n the analysisof the results,thesesubsetsare dealtwith si:paratelv.
Code
Segmentation
P
1.,,t
1\
/n"*
:x{abrb
S[1ar.ib)]2x(b)
,(al)((br(2x(b)))
rababt(br
{abarrbbr
tab)tabb)
3
3
3
I
5
I
-l
.t
5
5
3
-t
aabaabb :xilx{a)b)b
S I 2x ( ( a ) )(, b l l 2x ( b r
. ( ? x ( a ) ) )(i( b ) ( 2x ( b ) ) )
(aabaab)(b)
( a a b a a(ib b r
l a a b )( a a b b )
3
3
3
5
6
5
5
6
6
-t
abcabcc :x(abc)c
S [ ( a b ri.c t ] Zx i c t
. ( a b t )( ( c t ( 2x ( c l ) )
rabcabc)(c)
( a b c a b )i c c )
( a b c )( a b c c )
{
-l
-l
5
6
5
5
6
6
5
5
6
ababb
3
5
Figure13. Three examplesof symbolseriesusedin the experiment.E a c h p a i r o u t o f t h e t h r e e
segmentations
for one series representsthe responsealternatives.The values for the four
metricsaregiven.
s38
P A van der Helm,R J van Lier,E L J Leeuwenberg
5.3 Procedure
All subjectswere tested individually. As the total duration of the experiment was
about 1.5 h, there was a break halfuay through the experiment. All subjectswere
given the same instructions,projectedonto a monitor. Each stimuluswas presented
as follows. First, only the target (a patterned sequenceof graphic symbols) was
presentedin the middle of the left-hand side of the screen. After 7 s, the response
alternatives(two segmentations
of the target)were presentedin addition,at the righthand side of the screen,one in the upper half and one in the lower half of the screen
(as in figure l1). The task was "to decide how you would partition the pattern into
coherentparts" (during the 7 s), and then "to select,from the two partitionings.the
partitioning that resemblesyour own partitioning most closely". Pilot investigations
showed that subjects had a clear preference within a period of 7 s. Subjects
respondedby pressingone of two buttons on the table in front of them, the buttons
correspondingin position to the positionsof the responsealternativeson the screen
(top versusbottom).
First. l0 trials were presentedso as to get acquaintedwith the task.then the actual
experiment began by the 2-10 stimuli being presented in a random order. As
mentionedbefore. the random transformationof the 240 symbolic stimuli into real
stimuli was performed for each subjectindividuallv. The position of each response
alternative(top or bottom) was randomizedtoo. In the responsealternatives,the
Iarger spacesbetweenthe segmentswere made such that the visual angle for both
rvasalwaysequal{asin figure I I ), independent
responsealternatives
of the numberof
segments.The computerregisterednot only each response.but also each response
time. this being the time betweenthe onsetof the responsealternatives
and pressing
the button.
5.-l Resrrlts
Each of the thirtv-one subjectspert'ormedall l-10 trials, so that the total number of
triais was 3l x ?-10 : 7-l-10trials. As mentionedbefore,the P-load is not considered
and. for each of the other three metrics.the nonambiguityset and the ambiguitv set
are consideredseparately(see figurel2). Since the subjectsperformeda forcedchoice task. the metrics cannot be tested for correcr predictions for stimuli in the
respectiveambiguitvsets. Only if. experimentally,stimuli in an ambiguityset appear
to be significantlynonambiguous.
can the respectivemetric be said ro have predicted
falsely. The latter caseswill be dealt with together with the false predictions for
stimuli in the nonambiguitysets. First. the correct and falsepredictionsfor stimuli in
only the nonambiguitysetsare considered.
Figure 1.4shows.for each of the three nonambiguitysets.a histogramof the raw
data. In each histogram, the bars represent disjunct stimulus subsets,together
constitutingthe entire nonambiguityset. The heightof a bar representsthe size of the
subset.and the position on the horizontalaxis representsthe number of subjectsfor
which the respectivemetric correctlv predicted the responsesto the stimuli in that
subset. For instance,the 1n.*-histogram
appliesto a nonambiguityset containing 186
of the 240 stimuli. From that histogram,one can read, for example,that, for one
subsetof 15 of the 186 stimuli, the /n"*-load correctly predicted the responsesof
twentv of the thirty-one subjectsand that, for another (disjunct)subsetof 15 of the
186 stimuli, the 1n"*-loadgave the correct predictionsfor twenty-four of the thirtyone subjects. So, from figure 14. one readily seesthat the /o-load roughly performs
better than the {,,u-loadsince.in the 1.-histogram.larger subsetstend to be relatedto
a larger number of subjects.ie more of the responseswere predictedcorrectly. In the
sameway, one readily seesthat the /n.*-loadroughly performsbetter than the /o-load.
Serial pattern complexity: inegulanty and hierarchy
So, at first glance, figure 1-l confirms the hypothesis. For a second glance, more
detailedstatisticsare presentednext.
First, for each metric. the predictionswere compared to choice at chance level
(p : 0.5). The resultsof the t-test performedwith the data (pooled acrosssubjects)
in table 1, show that the /",.-loadscoresare highly significantlyfalse(!)(r:o = 7.785,
p < 0.00i), that the /^-load scores are just significantlycorrect (t:o : 2.088,
p < 0.05), and that the (..-load scoresare highly significantlycorrect (/:o : 7.736,
p < 0.001). Furthermore,within subjects.and again comparedto choice at chance
level, the 1n,o-load
scored significantlyi,p 10.05) more correct predictionsfor only
one subject. the 1o-load scored significantlymore correct predictions for fifteen
subjects,and the 1n"*-loadscoredsignificantlymore for twenty-foursubjects.
l0
,{
)
3l
Number of subiects
I
't)
il
Number of subjects
Figure 14. Histograms of the raw data for each metric on its nbnambiguity set. [n each
histogram the bars represent disjunct stimulus subsets. together constituting the entire nonambiguity set. For each subset the histogram shows the size of the stimulus subset and the
number of subjects for which the respective metric correctlv predicted the responses to the
stimuli in that subset. In each histogram. the eleven leftmost horizontal positions show the
subsets for which the respective metric prediction was significantly false. and the eleven
rightmost horizontal positions show the subsetsfor which the respective metric prediction was
significantlycorrect.
T a b l e l . T h e m e a n s( + S D i a n d p e r c e n t a g e so f r e s p o n s e sa l l p r e d i c t e d c o r r e c t l y b y e a c h m e t r i c
for stimuli in the respectivenonambiguiryset.
Metric
Size of set
Responsespredicted correctly
ot
,o
meant SD
{,r,r
/A
/n"*
150
r36
r86
5 6 . 0t 1 3 . 6
7 1 . 6 +t 7 . 6
l't I I !
L -4.1
!
1a
--.
t
I
37 . 3
54.9
66.9
P A van der Helm,R J van Lier,E L J Leeuwenberg
540
Second.the metrics were comparedby consideringthe respectiveproportions of
significantly(a :0.05) false and correct predictions(pooledacrosssubjects);these
resultsare presentedin figure 15. All differencesbetweenthe proportions are highly
significant, both for the significantly false predictions (/" compared with Ionl
p < 0.001; 1n.*comparedwith In: z = 5.253,p < 0.001)as well as for
I : .1.56.1,
the significantlycorrect predictions(1o comparedwith 1o,o:i : 5.071, p < 0.001;
/n.* comparedwith I^: z:3.682, p < 0.005). Note that this analysismay seemto
betweenthe three nonambiguitysets. However,for
be tricky becauseof dependencies
the given stimulusset. it is clear a priori that a more complex analysisto accountfor
those dependencieswould yield an even strongereffect. The same argumentapplies
to the next test.
Third. in figure 16, the proportions of significantlyfalse predictions for stimuli
in the respectivenonambiguitysets are plotted again, but now together with the
proportions of significantlyfalse predictions for stimuli in the respectiveambiguity
betweenthe latterproportionslie for just the ambiguitysets)are
sets.The differences
not significanr,bur the differencesbetweenthe summedproportionsiambiguityset
: 3.776pius nonambiguityset) are highlv significantr1^ compared with l.ral :
=
5 . 5 1 1 .p < 0 . 0 0 1 ) .
p < 0 . 0 0 5 ; / n " *c o m p a r e dw i t h ^ 1 . ::
Fourth and finally, each metric was testedfor a possibledifferentiationin response
time betweenresponsespredictedcorrectlyand responsespredictedfalsely(which can
Q
f
z
r''o
"i;"
significantlv correct
significantlyfalse
/"'"
Figure15. The proportionsof significantlyfalse and significantlycorrect predictionsfor the
stimuli in the respectivenonambiguityset for each metric. All differencesbetweenproportions
aresignificant.
:<
I
S
z
1',u
1.
Metric
ambiguityset
set
nonambiguity
1n"*
Figure 16. The proportions of significantly false predictions for the entire stimulus set. ie for
both the ambiguity set and the nonambieuity set. for each metric. For just the ambiguitv sets.
the differenc.j b"t*een rhe proportions are not significant. but for the entire stimulus set the
differencesare sienificant.
and hierarchy
Serialpanerncomplexity:irregularity
541
be done only for the nonambiguity sets). Figure 17 shows, for each metric, the
averageresponsetimes over all responsespredictedcorrectly and over all responses
'false' response times tend to be
shorter than
predicted falsely; for the /o,o-load,
'correct' responsetimes, whereasthe other metrics yield an opposite tendency. A
MANOVA was performed to check whether these tendenciesmight be influencedby
the number of different elementsin a target (two or three symbols)or by the total
number of elementsin a target (ie target'length' which varied from five to eighteen
symbols).The factor Length was set to four levels,and the factor Symbolswas set to
two levels. Then, for the /n.*-load only, the factor Prediction(two levels: false or
correct) yields a significant differentiation in response times (uexOve over
4 x 2 x 7 : 16 cells;five subjectswere rejectedbecauseof emptycells):F,.r, : 5.88,
p < 0.05). The factor, Prediction,does not interact with the factor Length nor with
the interactionterm Length x Symbols.
_5.00
1
false
correct
/ot,l
faise
corTecl
ialse
correct
I
Figure 17. The average response times over all responses predicted falselv and over all
responsespredicredcorrectly l only for the respectivenonambiguitvsets) for each metric.
5.5 Discussion
The experimentalresultsconfirm the hypothesis.That is. the resultsnot only confirm
but
that the In.*-load is significantlybetter than both the /^-load and the 1o,o-load,
also that the /o-load is already significantlybetter than the /o,o-load.The P-load
scored 58-2/" significantlyfalse predictionsfor the stimuli in its ambiguityset which,
as indicatedbefore,almostequalsthe entire stimulusset. This implies that the P-load
scoredworst of all (seealso figure 16), and supportsthe earliermentionedreasonsfor
excluding the P-load from the analysis. For the three metrics consideredin the
analysis,the orderingin the goodnessof the metricsis significantwith respectto both
the number of significantlycorrect predictionsfor the respectivelynonambiguitysets.
as well as the number of significantlyfalse predictionsfor the entire stimulusset (see
figures15 and 16).
The superiority of the In.*-load is supported further by its differentiation in
response times between falsely and correctly predicted responsesfor stimuli in its
nonambiguityset (see figure 17). In general.we assumethat such a differentiation
takes place for a good predictor. That is, if subjectshave more doubt about their
preference,then an increase in responsetime results. So, inversely, if such an
increaseoccurs consistentlyin casesof false predictions,then the predictionsmay
'compete' with the actual
have been false but are apparently still good enough to
responses.In the presentexperiment,the MANOVA on the responsetimes showeda
significantincreasein responsetime on ralselypredicted responsesfor the 1n"*-load
only. So, adoptingthe assumptionabove,the responsetime resultssupport the /n.*load onlv.
g ,A a
P A van der Helm,R J van Lier,E L J Leeuwenberg
The goodnessof the (.*-load may seem to be underminedsomewhatby the fact
'onlv' 57.5"/"significantlvcorrect predictionsfor stimuli in its
that this metric scored
nonambiguityset (seefigure 15). However,as mentionedbefore, the selectedstimuli
had to be rather critical with respectto the metrics. For an arbitrary stimulusset. the
predictionsof the metricsare not very different.and the metricsscoremuch better,as
shown in earlier experimentalwork on the structural information model. Moreover,
the main issue in this experimentwas to compare the In.*-load with the 1o,o-load
which scored the five times smallerpercentageof LI.3Yosignificantlycorrect predictions(seefigure 15 ).
One further remarkin connectionwith the stimulusselectionis that, as mentioned
before.one of the criteria for selectinga target was that all the lSA-rulesshould be
used to generatethe three possibleresponsealternatives.so that the results indeed
apply to rhe entire encodingmodel. Now, for all stimuli (target plus two response
is basedmainly on the I-rule.
in which one of the responsealternatives
alternatives)
is about two times as
In.*-load
by
the
predictedcorrectly
the number of responses
high as the number of responsespredictedfalsely. E.ractlythe samehoids for the
'treats'the tSA-rulesin an equivalent
S-rule.and also for the A-rule, So. the 1n.*-load
in thecorrectness
of thismetric.
rvav.whichgivesfurtherconfidence
Finally,the confirmationof the hypothesisimpliesstrongsupportfor the theoretical
analysisof regularityand hierarchy.as elaboratedin van der Helm and Leeuwenberg
(1991). As arguedin the previoussections,that analysisleadsdirectlyto the ordering
of the metrics accordingto goodness,that is now confirmed experimentally.So. the
I".*-load is not only a good metric. but it also has a firm theoreticalbasis which is
supportedby the experimentalresults.
6 Summar.vand conclusion
In this paper. we introduced a new metric to measure the complexity of serial
parterns.Such a complexitymetric is neededin pattern encodingmodels.such as
Leeuwenberg'si969. 1971) structural information model. Such models employ
several coding rules to describe possible structures of a pattern, and adopt the
minimum principle by assumingthat the simplest code of a pattern retlects the
are
of that pattern. In brief. patterninterpretations
humanlypreferredinrerpretation
(parts
plus
between
relations
pattern
segmentation
pattern
parts.
ie
a
given in terms of
parts)represenrsrhe pattern structureaccordingto an interpretation.and the simplest
code of a pattern is a code that describesthe largest amount of regularity in that
partern. In such encodingmodels.the employedcomplexitymetricsqre generallyjust
'good guesses',formulated in terms of the encodingsyntax and lacking an intrinsic
merelyin an intuitivesense.
whereasregularityis discussed
psychologicaljustificarion.
The new complexitymetric introducedin this paper,however.is basedon a strictly
formal analysisof regularityand hierarchyin patterns,as elaboratedin the paper of
van der Helm and Leeuwenberg(199i). This formal analysisresultedin the notions
of holographic regularity and of transparent hierarchy. The notion of transparent
hierarchy applies to the conditions under which different but (partly) overlapping
regulariry structures can be combined hierarchically. The notion of holographic
regularityrepresentsa formalizationof the intuitive notion of regularity,and resultsin
a restricrednumber of basickinds of regularity.Assumingthat holographicregularity
and transparenthierarchy are psychologicallyrelevant notions, only three coding
rules are needed to describe pattern structures, namely the iteration rule, the
symmetryrule. and the alternationrule, as used in Leeuwenberg'sstructuralinformation model. The kinds of regularity described by these three coding rules were
alreadywidely accepted(intuitively)as being psychologicallyrelevantbut now aPpear
to have a unique formal statustoo. That is, in a strictly formal way, preciselythese
Serialpanem complexity:inegularityand hierarchy
543
kinds of regularityresult from the notions of holographicregularityand transparent
hierarchy. This implies that thesetwo notions can be seenas the underlyingpsychologicalbasisof the descriptionof patternstructures.
The plausibilityof the formal analysisis supported further by two facts. First, it
allows for a largely parallel encoding process. This supports the analysissince the
fascinatingspeed with which the human perceptual system processespatterns is
generallythoughtto result from parallelprocessing.Second,it allows for the elimination of combinatorialexplosions.That is. it enablesthe selectionof a simplestcode
out of the exponentialnumberof possiblecodes,by takinginto accountall codesbut
without generatingevery code separately.This also impliessupport of the analyses
since it shows that the minimum principle is realistic in the sensethat it does not
requirean unrealisticsearchfor simplestcodes.
All in all, the formal analysisconstitutesa firm basisfor further research.In the
present paper, we elaboratedthe fact that the formal analysisenablesa detailed
investigationinto the wav in which pattern complexity is quantifiedby meansof the
resultedin proposinga
complexitymetricsusedin earlierresearch.This investigation
new complexitymetric which. accordingto the formal analysis,should be superior to
the metrics used betore. That is. the new metric quantifiescomplexity by taking
into account the irregularityin a code in the same way as in the other metrics,but
it accountsfor the hierarchvin a code in a better wav. In particular,this improved
accountof hierarchyis relevantwith respectto so-calledlocal-effectcases,ie casesin
which (rviththe other metncsran a priori patternpartitioninghas to be assumedin
order for the simplestcode to reflect the preferred interpretation.This requirement
contradicts the minimum principle. With the new metric, simplest codes tend to
representless hierarchicallyorganizedpattern stnrcturesand. therefore,tend to look
that many local
like codesobtainedby encodingpatternpartsseparately.This suggests
effects may 'disappear'with the new metric, since it is not necessaryto assumethe
properpatternsegmentation
it followsdirectlyfrom thesimplestcode.
a priori because
The experimentdiscussedin this paper shows that the new compledry metric is
indeed sienificantlybetter than the metrics used before. Moreover, and equally
important. the experiment significantlysupports the ordering of the metrics with
respect to goodness.as was hypothesizedon the basis of the formal analysis.This
implies that we may concludethat the new metric is not merely a 'better guess',but a
plausible choice based on a formal analysis which is supported by experimental
results.
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