Complexity equals change

Abstract

Traditionally, models of complexity used in psychology have been based on probabilistic and algorithmic paradigms. While these models have inspired a great deal of research, they are generally opaque about the relationship between complexity and the cost of information processing. We argue that the psychological complexity is easily defined and quantified in terms of change and support this argument with a measure of complexity for binary patterns. We extend our measure to 2-D binary arrays, and show that it correlates well with a number of existing complexity and randomness measures, both subjective and objective. We suggest that measuring change represents an intuitively and mathematically transparent way of defining and quantifying psychological complexity which provides the missing link between subjective and objective approaches to complexity.

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Complexity equals change
Action editor: Gregg Oden
Aleksandar Aksentijevic
a,⇑
, Keith Gibson
b
a
Roehampton University, London, UK
b
Birkbeck College, London, UK
Received 26 July 2010; received in revised form 30 December 2010; accepted 3 January 2011
Available online 21 January 2011
Abstract
Traditionally, models of complexity used in psychology have been based on probabilistic and algorithmic paradigms. While these
models have inspired a great deal of research, they are generally opaque about the relationship between complexity and the cost of infor-
mation processing. We argue that the psychological complexity is easily defined and quantified in terms of change and support this argu-
ment with a measure of complexity for binary patterns. We extend our measure to 2-D binary arrays, and show that it correlates well
with a number of existing complexity and randomness measures, both subjective and objective. We suggest that measuring change rep-
resents an intuitively and mathematically transparent way of defining and quantifying psychological complexity which provides the miss-
ing link between subjective and objective approaches to complexity.
Ó2011 Elsevier B.V. All rights reserved.
Keywords: Complexity; Entropy; Pattern; Structure; Change
1. Introduction
We propose a measure of pattern complexity, C(s),
which connects the subjective, first-person perspective of
the human observer and the third-person perspective of
mathematics and science. Such a measure should be based
on a simple, primitive notion, which underpins perception
and cognition. Perhaps the most primitive is the notion of
change. Change is of fundamental importance for psychol-
ogy as well as science and computation. Study of changes
in sensation represents the basis of experimental psychol-
ogy and psychophysics. Any textbook on the subject
clearly demonstrates that enquiry into perception and cog-
nition must begin with change. Any account of sensory
processing stresses the importance of changes in physical
parameters of the stimulus for its encoding.
Change allows direct quantification of the relationship
between pattern elements. One of the reasons for the failure
of information theory (Shannon, 1948) to capture the
essence of psychological complexity (see Chater, 1996;
Luce, 2003) has been its focus on individual symbols and
their frequencies at the expense of a description of their
relationships (structure). In the words of Luce (2003),
“...the stimuli of psychological experiments are to some
degree structured, and so, in a fundamental way, they are
not in any sense interchangeable (p. 185).”In other words,
different levels of structural organization within a pattern
should not be treated as mutually independent statistical
events. It is here that the importance of change becomes
clear. Structural information is contained in the transition
from one symbol (or element) to another and not in the
symbols themselves (see e.g. Attneave, 1954).
An alternative to information theory was offered by
transformational approaches to pattern goodness/com-
plexity (Palmer, 1983) that begin with certain prescribed
forms of invariance assuming that these govern the percep-
tion of structure. Many influential mathematical theories
1389-0417/$ - see front matter Ó2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.cogsys.2011.01.002
⇑
Corresponding author. Address: Department of Psychology, Roe-
hampton University, Whitelands College, Holybourne Avenue, London
SW154JD, UK. Tel.: +44 208 392 5756; fax: +44 208 392 3527.
E-mail address: a.aksentijevic@roehampton.ac.uk (A. Aksentijevic).
www.elsevier.com/locate/cogsys
Available online at www.sciencedirect.com
Cognitive Systems Research 15–16 (2012) 1–16

Author's personal copy
are founded on the notion of invariance (e.g. Weyl, 1952)
and invariance has been given a special place in psycholog-
ical theorizing on complexity and goodness (e.g. Garner,
1974; Leyton, 1986a, 1986b; Palmer, 1977).
While acknowledging the importance of invariance, we
start from change because change has not been sufficiently
considered in complexity literature thus far. Change repre-
sents the inverse of invariance and as we show here, it offers
a viable and potentially useful way of quantifying complex-
ity. Another important motivation for equating psycholog-
ical complexity with change has to do with the ease with
which change links psychological, physical and computa-
tional interpretations. The traditional approaches to com-
plexity have ignored the crucial fact that information
processing involves effort and cost (Falk & Konold,
1997). Our approach avoids the dichotomy between “mes-
sage”and “process”complexity (a pattern containing little
change is easy to describe AND compress). Consequently,
we focus on the information in a pattern (message) and
quantify the effort needed to describe and/or compress it.
We then argue that our measure is meaningfully related
to the way in which the brain processes change and
complexity.
An important attempt to quantify pattern structure has
been Algorithmic Information Theory (AIT; Chaitin, 1969;
Kolmogorov, 1965; Solomonoff, 1964; see Li & Vitanyi,
1997 for an overview).The development of the AIT repre-
sents an attempt to reconcile structural complexity with
the probabilistic nature of information theory. Algorithmic
complexity represents the length of the shortest algorithm
in any programming language, which computes a particu-
lar binary string. A string of length x is incompressible if
the shortest program that can produce it is at least x bits
long.
Algorithmic complexity provides a more intimate link
between the observer and the observed by introducing
structure into computational complexity. Specifically, this
approach makes it possible, at least theoretically, to com-
pute the complexity of individual, albeit infinite, strings.
Many pattern-coding languages have been proposed based
on algorithmic compression, which are widely used in psy-
chology for describing structure. (e.g. Leeuwenberg, 1969;
Restle, 1970; Simon & Kotovsky, 1963; Vitz & Todd,
1969). These models involve different kinds of algorithmic
notation aimed at providing compact description of pat-
terns (Simon, 1972). While many of these approaches have
faced criticisms (see Simon, 1972), others have advanced
complexity research by incorporating perceptually and psy-
chologically relevant forms of invariance into their coding
scheme (e.g. van der Helm, 2000).
Theoretically, the notion of algorithmic complexity is
useful because it describes the relationship between an
algorithm and its output. However, it is difficult if not
impossible to apply this idea to human observers. For
sequences of a certain level of complexity, it is not possible
to know if a better (more efficient) algorithm does not exist
(Chaitin, 2001). In addition, algorithmic compression is
complex and often irreversible. Any simple pattern can
be encoded in a complex way and at the same time, any
encoding can hide almost infinitely many meanings. Fur-
thermore, simple algorithms can produce highly complex
outputs (e.g. Wolfram, 2001, p. 27). Consequently, algo-
rithms can shed little light on the human response to
complexity.
Understanding involves effort and cost and a meaning-
ful measure of psychological complexity should reflect this.
We propose that any perception, cognition or action
involves change, and is accompanied by an irreversible
expenditure (conversion) of energy. Change equals increase
in entropy, and this in turn equals cost. Any action, how-
ever trivial, must incur cost and this cost is reflected in
an increase in (physical or computational) entropy. Regis-
tering change always costs more than registering no
change. This means that in its interaction with the environ-
ment, the agent converts a certain amount of available
energy, irrespective of its scale. In the context of what fol-
lows, we are assuming that all computing (human or other-
wise) is thermodynamically irreversible. This formulation
brings together physical, computational and psychological
meanings of entropy.
In order to apply the concept of information cost, we
describe a measure of complexity of a binary string based
on the amount of change present in the string. We have
restricted ourselves to discussing binary patterns because
we believe that binary representation offers the most trans-
parent way of encoding and describing change. In the
words of Vitz (1968), the simplicity of binary representa-
tion “presumably exposes the process of perceptual organi-
zation more clearly than other patterns (p. 275).”In
addition, the complexity of structures encoded by larger
alphabets cannot be judged or interpreted without intro-
ducing an appropriate metric which might differ from con-
text to context. This requires additional assumptions,
which might have little to do with the structure of the
pattern.
Our account of complexity has parallels with the Gestalt
approach to perception. The notion of interdependence of
different levels of structure represents one of the corner-
stones of Gestalt psychology, whose impact on complexity
research in psychology cannot be overestimated (e.g.
Hochberg & McAlister, 1953; see also van der Helm,
van Lier, & Leeuwenberg, 1992). The first serious attempt
to define and explain pattern goodness was offered by
Gestalt psychologists in the 1920s and 1930s. They pro-
posed that human observers organized sensory/perceptual
and cognitive information according to a number of simple
rules. Fundamentally, an individual perceptual scene is
organized in such a way as to minimize the expenditure
(or rather conversion) of energy. This is what the Gestal-
tists named the “Law of Pra
¨gnanz”or “Minimum Princi-
ple”(e.g., Koffka, 1935). Patterns are considered “good”
if they are compact, symmetrical, repetitive, or predictable.
Such patterns contain little change and have low entropy.
They are predictable, resistant to disruption and easy to
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process, assimilate and memorize. Part of the aesthetic
appeal of periodicity and symmetry might lie in the fact
that they appear to defy entropy (increase in disorder). In
the words of Garner (1970),“good patterns have few alter-
natives”. By contrast, “poor”patterns – those that contain
a great deal of change – are complex, asymmetrical and
defy easy description. They contain more information than
simple patterns and are consequently unpredictable and
more changeable when subjected to spatial transforma-
tions. The observer searches for informational “oases”pro-
vided by symmetry, similarity and regularity in order to
minimize the expenditure (conversion) of energy associated
with understanding.
2. The measure
In order to quantify our intuition, we propose an index
C(s) of structural complexity, based on the amount of
change present in one- or two-dimensional binary arrays.
In this section we describe its mathematical properties
and describe a form of generalized structure invariant that
emerges from the properties of the model.
2.1. Informal description and motivation
Let Sbe a binary string of length LP2. We could scan
Stwo symbols at a time, and register a change every time
we encountered a substring “01”or “10”of S. The crudest
measure of the complexity of Swould then be the number
of changes registered. However this will not do for our pur-
poses because the maximum complexity would then be
ascribed to the string “0101 ... 01”, which is clearly very
simple for a human observer. We therefore conduct a scan
jsymbols at a time, j=2toL, and ask which substrings of
length jof Sshould register a change. Suppose we encoun-
ter the substring x=“010”of S. Then xwill register a
change if there is in some sense a change in passing from
the first two symbols of xto the last two, that is, in passing
from “01”to “10”. These two strings are not equal, but are
essentially the same. They both display a change, and any-
way differ only in the encoding of their symbols. Thus x
should register no change. On the other hand, a substring
y=“001”should register a change, because the first and
last two symbols of y are the strings “00”and “01”, and
these strings are essentially different. It is not true that they
differ only in the encoding of their symbols, and one dis-
plays a change while the other does not. If we look again
at the string “0101 ... 01”we will find that no substring T
of length j> 2 will register a change, because the strings
formed from the first and last j1 symbols of Twill differ
only in the encoding of their symbols.
We proceed therefore to define a change function on bin-
ary strings that determines when a substring of S should
register a change, and then use it to construct the change
profile P =(p
2
,p
3
, ..., , p
L
)ofS, where p
j
is the number
of substrings of length jof Sthat register a change. We
then obtain the complexity C of Sas a suitably weighted
average of the coordinates of P. Our choice of change func-
tion and complexity are as follows. A substring of Sof
length 2 registers a change if its two symbols are not the
same. A substring of Sof length j> 2 registers a change
if the strings formed from its first and last j1 symbols
have the same change profile. Noting that there are
Lj+ 1 substrings of Sof length jwe define Cto be the
sum of the quantities p
j
/(L+j1), j=2to L. The string
“0101 ... 01”has a complexity value of 1 under this defini-
tion, and only strings of all ones or all zeros, with complex-
ity value 0, have a smaller complexity value.
It is not clear a priori that our choice of change function
is a good one. It succeeds because it provides a strong con-
nection between change and symmetry, even though sym-
metry is not mentioned in the definition, and it is this
connection that we now explore. A number of lemmas
and theorems are needed, the proofs of which are not
required for a basic understanding of this paper. They
are given in Appendix A.
2.2. Symmetric equivalence
We define two operators, r(reverse), and c(comple-
ment) on binary strings, and use them to define the notions
of symmetric equivalence and palindromicity. In the fol-
lowing definition the complement of 0 is 1 and the comple-
ment of 1 is 0.
Definition 1. Let S=(s
1
,s
2
,... ,s
L
) be a binary string, and
let t
i
be the complement of s
i
,i=1to L.
rS =(s
L
,... ,s
2
,s
1
) = the reverse of S.
cS =(t
1
,t
2
,... ,t
L
) = the complement of S.
Note that rand ccommute, i.e. rcS =crS.
Definition 2. Let Sand Tbe two binary strings of the same
length.
Sis symmetrically equivalent to T, written ST,ifTis
one of S,rS,cS,rcS.
Definition 3. A binary string Sis a symmetric palindrome if
Sis one of rS,rcS.
If the length Lof Sis odd we cannot have S=rcS,so
if Sis a symmetric palindrome then we must have S=rS,
i.e. Sis a palindrome in the classical sense. However if L
is even we can have S=rcS, and then Sis not a palin-
drome in the classical sense. For example, “0101”is a
symmetric palindrome, but not a classical palindrome.
Note that all strings of length 2 are symmetric palin-
dromes. For the rest of this paper the words equivalence
and palindrome will refer to symmetric equivalence and
symmetric palindrome.
It is easy to see that is an equivalence relation, and
that if a binary string Sis a palindrome then its equivalence
class has two members, Sand cS, both of which are palin-
dromes, while if Sis not a palindrome its equivalence class
has four members, S,rS,cS,rcS, none of which are
palindromes.
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The following lemma relates palindromicity of a string
of length Lto symmetry between its first and last L1
symbols, paving the way for the change function of the
next section.
Lemma 1. Let S =(s
1
,s
2
,... ,s
L
)be a binary string, L >1,
and let U =(s
1
,s
2
,... ,s
L1
), V =(s
2
,s
3
,... ,s
L
).
Then S is a palindrome if and only if U V.
2.3. The change function
Definition 4. Achange function on binary strings is any
map from binary strings of length >1 to {0, 1}.
We denote a change function by [], so for a binary string
Sof length >1, [S] is either 0 or 1. We will refer to [S] as the
change in S.
Definition 5. Let Sbe the binary string (s
1
,s
2
,... ,s
L
),
L> 1, and let [] be a change function.
Let X
ij
be the change in the substring of length jof S
starting at s
i
,j=2to L,i=1 toLj+1.
Let p
j
be the number of substrings of length jof Swhose
change is 1, j=2to L. Thus
Xij ¼½si;siþ1;...;siþj1;j¼2toL;
i¼1toLjþ1ð1aÞ
pj¼X
Ljþ1
i¼1
Xij;j¼2toLð1bÞ
The change matrix of Sis the matrix Xwhose entry in
row iand column jis X
ij
.
The change profile of S is the array P=(p
2
,p
3
,... ,p
L
).
Note that p
L
=[S].
Our task is to define a suitable change function.
Although we are not going to refer to symmetry in the def-
inition, there are nevertheless two symmetry properties
that any reasonable change function can be expected to
have. The first is that equivalent strings have the same
change profile. The second is that if S,U, and Vare as
in Lemma 1, then [S]=0ifUV, since there is then really
no change in passing from the first to the last L1 charac-
ters of S. It is tempting therefore to define [S]tobe0if
UV, and 1 otherwise, which in view of Lemma 1 makes
[S] = 0 if and only if Sis a palindrome. This would cer-
tainly guarantee equivalent strings had the same change,
and as we shall see later, that would be enough to guaran-
tee they had the same change profile. However as we shall
also see later, such a definition does not pick up periodicity
in a string. We therefore make the following recursive
definition:
Definition 6. Let S,U,Vbe as in Lemma 1.
Define [s
1
s
2
] to be 0 if and only if s
1
=s
2
.
If L> 2 then define [S] to be 0 if and only if Uand V
have the same change profile.
The first thing we do is remove the recursion from this
definition, and show how to compute the change function
efficiently. The next result is central to our theory, and in
view of its importance we state it first for a string of length
5, so as to make the general statement easier to grasp.
We have
[s
1
s
2
s
3
s
4
s
5
] = 0 if and only if [s
1
s
2
]=[s
4
s
5
],
[s
1
s
2
s
3
]=[s
3
s
4
s
5
], [s
1
s
2
s
3
s
4
]=[s
2
s
3
s
4
s
5
].
Theorem 1. Let the binary string S = (s
1
,s
2
,... ,s
L
), L>2.
Then [S] = 0 if and only if
½s1;s2;...;sj¼½sLjþ1;sLjþ2;...;sL;j¼2toL1ð2Þ
Numerical example 1:
As an example we use theorem 1 to show [1 1 0 1 1] = 0.
We have to check that [1 1] = [1 1], [1 1 0] = [0 1 1],
[1 1 0 1] = [1 0 1 1].
The first is vacuously true, and applying theorem 1
shows [1 1 0] = [0 1 1] = 1.
To check the third, we have by theorem 1 to check
[1 1] = [1 1], [1 1 0] = [0 1 1], both now proven.
Theorem 1 allows us to calculate the change function for
all substrings of a string. The following algorithm com-
putes the change matrix, and hence the change profile, of
a binary string Sof length L, at a cost of L(L
2
1)/6 bin-
ary comparisons and space for L(L1)/2 binary values. In
particular it calculates [S].
Algorithm 1.
Let Sand X
ij
be as in Definition 5.
Initialize all the X
ij
to 0.
For i=1to L1
If s
i
is not equal to s
i+1
then set X
i2
=1.
For j=3to L,i=1 toLj+1,k=2 toj1
If X
ik
–X
rk
, where ris i+jk, then set X
ij
=1, and
continue with the next value of i.
Numerical example 2:
As an example we use algorithm 1 to calculate the
change matrix Xand change profile Pof the string
S=“11011”. Recall X
ij
= [substring of length jstarting at
position i].
Substrings of length 2:
X
12
= [11] = 0, X
22
= [10] = 1, X
32
= [01] = 1,
X
42
= [11] = 0
Column 2 of X
X12 X22 X32 X42
0110
Substrings of length 3:
X
13
= [110]: X
12
–X
22
Set X
13
=1
X
23
= [101]: X
22
=X
32
Set X
23
=0
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X
33
= [011]: X
32
–X
42
Set X
33
=1
Column 3 of X
X13 X23 X33
101
Substrings of length 4
X
14
= [1101]: X
12
–X
32
. No need to check whether
X
13
=X
23
. Set X
14
=1
X
24
= [1011]: X
22
–X
42
. No need to check whether
X
23
=X
33
. Set X
24
=1
Column 4 of X
X14 X24
11
Substrings of length 5
X
15
= [11011]: X
12
=X
42
,X
13
=X
33
,X
14
=X
24
Set X
15
=0
Column 5 of X
X15
0
X¼0110
101
11
0
P= 2 2 2 0 obtained by summing the entries in each col-
umn of X.
The next two results show that the change profile is
invariant under equivalence, the first of the two symmetry
properties we expect our change function to have. Lemma
2states that if equivalent strings always have the same
change then they always have the same change profile,
and Theorem 2 states that equivalent strings do always
have the same change.
Lemma 2. Let S, T be binary strings of length L P2, and
let [] be any change function.
If S T implies [S] = [T] then S T implies that S and
T have the same change profile.
Theorem 2. Let S, T be binary strings of length L P2. If
ST then [S] = [T].
Corollary (using Lemma 2): If S T then S and T have
the same change profile.
The converse is not true for strings of length P5. The
strings “01110”and “00100”have the same change profile
but are not symmetrically equivalent. This is encouraging,
because these two strings could be regarded as being per-
ceptually equivalent in respect of simplicity, so that change
profile is capturing something beyond what symmetric
equivalence can capture.
We note that the change matrix is not invariant under
equivalence. If ST, it will not normally be the case that
Sand Thave the same change matrix. Their change matri-
ces will have the same set of entries in each column, but the
entries will be in different places.
2.4. Generalized palindromes
The next lemma relates change to palindromicity, and
together with Lemma 1 is sufficient to guarantee the second
symmetry property expected of our change function.
Lemma 3. Let S be a binary string of length L > 2. If S is a
palindrome then [S] = 0.
Note that L> 2 is needed since “01”is a palindrome but
we have defined [01] to be 1.
Corollary 1 (using Lemma 1). If the first and last L1
symbols of Sform equivalent strings, then [S]=0.
The converse is not true for LP9. Consider the string
S=“111011000”of length 9. Then [S] = 0, but Sis not a
palindrome. However if 2 < L< 9, then [S] = 0 does
imply Sis a palindrome. In view of this we make the
following definition.
Definition 7. Let Sbe a binary string of length L>2. We
call Sageneralized palindrome if [S]=0.
Thus a palindrome is a generalized palindrome, but for
LP9 there are generalized palindromes that are not palin-
dromes, and we shall see that these strings play an impor-
tant part in the ability of the change function to pick up
periodicity.
Theorem 1 highlights a kind of palindromic structure in
generalized palindromes. Reading the values of square
brackets from the left is the same as reading them from
the right.
2.5. Complexity
Definition 8. Let Sbe a binary string of length LP2, with
change profile P=(p
2
,p
3
,... ,p
L
).
The complexity C of Sis defined to be
C¼X
L
j¼2
pjwj;where wj¼1=ðLjþ1Þð3Þ
The change profile Pof Sis an array of integers that
forms a representation of the amount of change in S,and
a natural way to derive a single number Cfrom Pis to take
a weighted average of its coordinates. We can use the
weights to regulate the contribution of different lengths of
substrings of Sto the overall complexity of S, and this
allows us to model different levels of structure in our com-
plexity measure.
Psychological complexity research has shown that the
importance of different levels of structure in judging com-
plexity depends on a number of factors. Here we refer to
the hypothesis put forward by Chipman (1977) and
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supported by Ichikawa (1985), according to which subjec-
tive appreciation of pattern complexity is underpinned by
two processes: quantitative and structural. Ichikawa sug-
gests that quantitative processes rely on enumeration of
distinct elements in a pattern, which might include the
number of dots, runs or clusters. This can be contrasted
to structural processes, which evaluate periodicity and
symmetry within the pattern. He proposed a model of the
processing of structure consisting of two stages: primary
and higher cognitive. With short stimulus presentation
times, quantitative factors dominate the judgment and
the structural aspects are “discovered”only given sufficient
time. We model the quantitative aspect with weights that
favour short substrings of S, and the structural aspect with
weights that favour all lengths equally.
Adjusting weights to favour the contribution of short
substrings can be achieved by taking all the w
j
in (3) to
be 1, but that provides only L(L1)/2 different complexity
values over 2
L
strings. The low number of distinct values
could be avoided by viewing Pas a mixed radix represen-
tation of C,withp
j
being a digit in radix Lj+2,p
2
being
the most significant digit. To understand mixed radix
representations, think of time as a triple (days, hours, min-
utes). The number of minutes represented is minutes +
60 h + 60.24 days. The weights for our “mixed radix”com-
plexity are given by w
L
=1, w
Lj
=(j+1)w
Lj+1,
j=1to
L2, which makes w
Lj
=(j+ 1)! = (j+1)j(j1) ...
3.2.1. There would then be as many different complexity
values as there are different change profiles, and the rank-
ing of complexities would be intuitively good. Moreover
comparison of complexity values would be easy, carried
out by comparing profile entries from the left. The contri-
bution to the complexity from long substrings would be
negligible however. We will see later that both “profile
sum”complexity (w
j
= 1) and mixed radix complexity can
be useful in modelling the “quantitative”mode of complex-
ity perception.
The actual choice of w
j
=1/(Lj+ 1) in (3) derives
from the fact that there are Lj+ 1 substrings of S of
length j, so we have defined Cto be the sum over jP2
of the proportion of substrings of length jof S whose
change is 1. This choice gives equal weight to all substring
lengths, and models the “structural”mode of perception.
Furthermore it provides a sufficient number of complexity
values. Indeed for medium length strings it makes complex-
ity a very efficient representation of profile. As an example,
for L= 16 there are 11,889 different complexity values and
13,420 different profiles, and the ratio of these two numbers
is 0.886. For L625 this ratio is P0.725.
Since each p
j
in (3) is in the range 0 to Lj+ 1, we see
that the complexity Cof a binary string Sof length Lis in
the range 0 to L1. We therefore define the normalized
complexity N of Sto be C/(L1), so Nis in the range 0
to 1. Where necessary we refer to Cas the unnormalized
complexity of S. The upper bounds for Cand Nare not
attained, but we can expect that as Lgets large both the
mean and maximum values of Napproach 1, and the stan-
dard deviation of Napproaches 0, so that the normalized
complexity of almost all long strings is near to the maxi-
mum value of 1.
For strings of length up to 32 we have determined the
distribution of complexity values by exhaustive search,
and for strings of length up to 128 we have estimated it
from a sample of one million strings chosen uniformly at
random. The distribution of unnormalized complexity for
strings of length 24 is shown in Fig. 1. A feature of the dis-
tribution is that the values are tightly bunched at the right
end of the distribution curve, which could be seen as
reflecting the decreasing ability of our measure to discrim-
inate between increasingly complex structures that is also
characteristic of human observers.
Generalized palindromes tend to have low complexity.
The reason is that if Sis a binary string of length Lwith
unnormalized complexity C, then [S] contributes 1 to Cif
Sis not a generalized palindrome, and contributes 0 if S
is a generalized palindrome. If LP5 a difference of 1 in
unnormalized complexity translates to more than 1.2 stan-
dard deviations, and for LP21 it translates to more than
2 standard deviations.
2.6. Periodicity
One of the most striking features of our complexity mea-
sure and change function is their ability to pick up period-
icity in a string. This is of considerable interest since our
definition of complexity is not based on structure, and in
particular does not attempt to encode any specific form
of structure. Rather we have taken the view that change
can quantify complexity, and based our definitions on
change accordingly. We have already seen that generalized
palindromes have low complexity. That our complexity
measure is related to both palindromic and periodic struc-
ture is a partial vindication of our approach, and is one of
the most important features of our model. Coupled with
the results in the final part of this paper, which show that
20.6
20.2
19.8
19.4
19.0
Frequency (million)
3
2
1
Unnormalized Complexity
0
Fig. 1. The right-hand tail of the population frequency distribution of
C(s) for binary strings of length 24. Mean unnormalized complexity is
20.14 with a standard deviation of 0.43. The mode is 20.37. Maximum
complexity equals 20.85, and mean normalized complexity is 0.88. Inset: A
plot of the entire distribution from C(s) = 0 to 20.8.
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our measure performs well alongside other measures in the
literature, it strongly suggests that one of the primary activ-
ities performed by the brain when it assimilates structure in
a pattern is to process the elements of change present in the
pattern.
First, a periodic string with period Tand period length t
is a string Sthat takes the form TTT..., where Tis a string
of length t, and tis minimal with this property, i.e. Sis not
of the form RRR..., where Ris a string of length r<t.An
ultimately periodic string is one that is periodic from some
point onwards. We give results for periodic strings, but
similar results can be observed for ultimately periodic
strings.
Let Sbe a periodic binary string with period length t,
and let S
n
be the string formed from the first ncharacters
of S. Numerical calculations indicate that the percentile
complexity of S
n
quite rapidly approaches zero as ngets
large. It is usually zero for n>t
2
/2. This behaviour could
perhaps have been expected. A partial explanation is that
our calculations also indicated that a periodic string S
tends to have an above average number of large substrings
that are generalized palindromes, and these depress the
complexity of Sfor much the same reasons as generalized
palindromes tend to have low complexity. As an example,
let Sbe the periodic string with period length 9 and period
“101101110”. Then the percentile complexity of S
n
is 0 for
nP33.
Even more striking, and unexpected, is the following,
which while almost certainly true, must remain a conjecture
until we find a proof. It says that the change function is
itself an indicator of structure, which would provide fur-
ther evidence that processing change is fundamental to
the detection of structure.
Conjecture 1. Let Sbe a periodic binary string with period
Tand period length tP2.
Let S
n
be the string formed from the first ncharacters of
S, let u
n
=[S
n
], n> 1, and u
1
=0.
Then the string U=(u
1
,u
2
,u
3
, ...) takes the form IXX
..., where Xhas just one zero entry.
If Tis of the form U V, where V is the complement of U,
then Iand Xare of length t/2, otherwise they are of
length t.
Shown below are the string Swith period length 9 we
considered earlier, and the associated string U.
S 101101110 101101110 101101110 101101110...
U 010110111 111110111 111110111 111110111...
It is important to note that if we had defined [S]tobe0
if Sis a palindrome, periodic strings would not have low
complexity, and the sequence Uwould not detect periodic-
ity in periodic strings. In the example just given, we would
find with this definition of [S] that [S
n
] is 1 for every nP7,
and that the percentile complexity of S
n
is more than 90%
for nP9.
2.7. Instability and non-intuitive low complexity
A small change to a string can result in a large percentile
change in its complexity, with the consequence that some
strings that do not appear at first sight to be intuitively sim-
ple can have low complexity. Two examples of strings dif-
fering in only one position are
“101100 010001011101110010”(C= 18.97, p= 2%)
“101100 110001011101110010”(C= 20.49, p= 86%)
“101 1011101011011101011011”(C= 20.04, p= 2%)
“101 0011101011011101011011”(C= 21.33, p= 66%)
Here Cdenotes unnormalized complexity and pdenotes
percentile. In the first example, the string with low com-
plexity is a palindrome of length 24, and in the second
example it consists of the first 25 symbols of a periodic
string of period length 9. A brief subjective view of these
strings will probably not see these structures, and will con-
sequently assess the strings as intuitively complex. Our
complexity measure is acting as an agent with sufficient
resources to pick up structure which is not immediately
obvious to a human observer on a brief glance, and so gives
these strings low complexity.
Although it is not surprising that a one bit change to a
string with low complexity can destroy internal structure to
the extent that our measure can no longer detect the struc-
ture, it is not true that a one bit change always turns low
complexity to high complexity. If we change the bit in posi-
tion 7 of the first string in the second example from 1 to 0,
we obtain the string “1011010101011011101011011”, which
still has C= 20.04 and p= 2%. This string is not a palin-
drome, nor even a generalized palindrome, and it is not
the start of a periodic string. Its first 24 characters do form
a generalized palindrome, and this has helped to depress its
complexity. We are unable to specify exactly when a string
will get low complexity, but we believe that low complexity
will generally indicate some kind of internal structure. The
existence of strings whose internal structure is not immedi-
ately obvious does however mean there will be an imperfect
correlation between our measure and brief subjective
assessment of intuitive complexity.
We have found that strings which are intuitively simple
do get low complexity, and that the complexity ranking of
strings with low complexity is intuitively correct, though
there will always be room for argument. Thus the string
“010101010101”has complexity 1.0 and percentile com-
plexity 0%, while the string “001100110011”has complex-
ity 5.45 and percentile complexity 1%. Intuitively these
strings are little different in complexity, and indeed at the
percentile level our measure barely distinguishes them.
2.8. Array complexity
It is possible to extend our theory of binary strings to
binary arrays, though the results and their proofs are con-
siderably more intricate, and the best change function may
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not be binary. In this paper we have taken an easier road,
and defined the complexity of a binary array in terms of the
complexities of its rows, columns, and diagonals, showing
in the final part of the paper that this rather crude
approach works surprisingly well. We experimented with
several ways of doing it, and describe the one that worked
best.
Let Abe an m by n binary array. We consider four sets
of strings, the rows of A, the columns of A, the main diag-
onals of A(left to right slanting), and the back diagonals of
A(right to left slanting), and calculate the average unnor-
malized complexity of each of these four sets of strings.
Diagonals of length 1 are included and taken to have zero
complexity. We define the normalized complexity Nof Ato
be the sum of these four averages, scaled so that N<1.We
found it convenient to work with U=(L1)N, where Lis
the average length of all the rows, columns, and diagonals
of A, and call Uthe unnormalized complexity of A, because
Nand Ureduce to normalized and unnormalized string
complexity when mor nis 1. The details of the calculation
of Nand Uare given in Appendix B.
Fig. 2 shows the sample distribution of unnormalized
complexity for 24 24 arrays, obtained from a sample
of 100,000 arrays chosen uniformly at random. The figure
is a bit misleading because it hides a long left hand tail of
outliers which do not show up in a sample of any feasible
size, and there is in fact even tighter bunching of values
at the right hand end of the distribution of array com-
plexity than there is for string complexity. As with
strings, most sufficiently large arrays have nearly the same
complexity, so that array complexity does not distinguish
well between arrays of average (perceptually high) com-
plexity. The outliers with low complexity correspond to
images possessing some structure, and array complexity
does provide good discriminability for such images. This
means that for our purposes the distribution of array
complexity is of limited use, but that the ranking of intu-
itively simple arrays provided by array complexity is
useful.
3. Comparisons with existing measures
We have examined the relationship between C(s)and
several subjective and objective complexity measures
reported in the psychological literature. The purpose of this
overview is to demonstrate how well C(s) correlates with
different measures of complexity based on apparently
diverse principles. Given the assumed ordinal level of our
measure, all reported correlations involving C(s) are non-
parametric.
3.1. String complexity: simultaneous presentation
A well-known study of the effects of pattern complexity
on recall was carried out by Glanzer and Clark (1962) who
presented participants with the exhaustive set of binary
patterns of length 8. The stimuli were arrays of symbols
(diamond, circle, square, spade, diagonal cross, club, heart
and triangle) which were patterned in all possible combina-
tions of black and white. Two complement subsets of 128
patterns were presented for 500 ms to two different groups
with the dependent variable being the accuracy of repro-
duction. Specifically, participants were asked to reproduce
the pattern by writing Bfor “black”and Wfor “white”in
the blank arrays. The authors reported that patterns which
appeared simple were also reproduced more correctly.
They measured the complexity of the patterns using differ-
ent methods including the number of runs, Attneave’s
(1959) redundancy measure and an ad hoc measure based
on the Gestalt principles. They obtained a significant corre-
lation between the number of runs and subjects’ accuracy
scores (r=.771) but rejected the measure because it (a)
had no theoretical basis and (b) there was a curvilinear
relationship between the number of runs and accuracy
scores. This was caused by the fact that subjects processed
patterns with few runs (e.g. BWWWWWWW) and those
with many runs but regular alternations (BWBWBWBW)
with comparable ease. Note that similar curvilinear rela-
tionships are obtained in distinct but related contexts such
as subjective randomness research (e.g. Falk & Konold,
1997). Our measure addresses this issue by “removing”
periodic strings from the right-hand tail of the distribution
and placing them among strings which have few
alternations.
Not satisfied by Attneave’s redundancy measure (low
correlation) or ad hoc Gestalt-inspired measures (no
theoretical rationale), the authors used the length of the
subjects’ verbal descriptions of the patterns (Mean Verbal-
ization Length; MVL) as a measure of stimulus complexity.
Perhaps, not surprisingly, they obtained a very high corre-
lation between MVL and accuracy scores (.826).
Although the result seems impressive, there is a serious
objection to using MVL as a measure (as opposed to a cor-
relate) of complexity. Given the complexity of the underly-
ing psychological processes, one can justifiably view the
ease or difficulty with which subjects verbally describe a
pattern as a consequence rather than as a predictor of
Unnormalized Complexity
13.09
13.0112.93
12.85
Frequency (thousand)
12
8
4
0
12.77
Fig. 2. Sample distribution of C(s) for 24 24 arrays (N= 100,000).
Mean unnormalized complexity is 12.97 with standard deviation of 0.04.
Average length of rows, columns and diagonals is 16.22. Mean normalized
complexity is 0.85. The left tail of the distribution is omitted.
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stimulus complexity (see Vitz, 1968 for a related criticism).
In a sense, MVL must be highly correlated with other
performance indicators because it can be seen as one of
them. More seriously, MVL tells us nothing about the
qualities and structural properties of the pattern which
are associated with subjective perception of complexity.
A similarly “agnostic”position on pattern complexity
was adopted by Lordahl (1970), who proposed a
40-parameter model of sequential prediction for binary
patterns.
In order to examine the relationship between our mea-
sure and subjects’ performance, we collapsed the two sub-
sets of 128 complementary patterns and calculated a mean
of mean accuracy scores for each pattern. We obtained a
highly significant correlation between C(s) and accuracy
scores (r=.349, p< .001). Following Glanzer and
Clark’s example however, we had to conclude that our
measure accounted for about 10% of the variance in
response accuracy, hardly a satisfactory result.
The explanation for the low correlation was found in the
distinction between the quantitative and structural pro-
cesses described in Section 2.4. Out of all studies analyzed
here, Glanzer and Clark’s study is the only one to have
included short presentation time (500 ms). The dominance
of the quantitative factors can be illustrated by the fact that
a very simple pattern (0 10101010) was recalled very
poorly (mean accuracy score of around .350 out of the
maximum of 1).
After considering the high task demands (fast stimulus
presentation and possible distraction caused by the symbol
outlines), we correlated the profile sums with mean accu-
racy scores and after removing three outliers (including
01010101) obtained a correlation of .695 (p< .001),
accounting for almost 50% of the variance. Restricting
the profiles to the first three levels increased the correlation
to .744. We then used the mixed radix representation of
the pattern profile, which gives even more weight to
changes at the lowest level of structure. Interestingly, the
correlation (.830) was higher than that between the num-
ber of runs and accuracy (.771) and slightly higher than
the correlation between MVL and accuracy. This evidence
allows us to interpret the quantitative/structural distinction
in terms of levels of change. Quantitative factors are related
to low-level change (e.g. number of runs and alternations
and second-order entropy) whereas structural processes
require the system to ascend the hierarchy of change to
higher levels. As we show below, in its current form our
measure appears suited to situations in which observers
have sufficient time to consider the structural properties
of a pattern.
An influential study using black and white 1-D patterns
of length 7 was carried out by Alexander and Carey (1968),
who investigated the effect of pattern complexity on the
performance of different tasks. Participants were presented
with 35 patterns – linear arrangements of three black and
four white squares differing in complexity – and were asked
to complete four tasks (search, reconstruction, memoriza-
tion and verbal description). The data for the four experi-
ments were ranked with respect to the patterns and
correlations showed remarkable agreement. The authors
pointed out that simple concepts such as overall symmetry
or number of blocks could not account for the agreement.
They proposed the concept of “subsymmetry”, that is, sym-
metry of the parts of the pattern. Without offering a theo-
retical rationale, they suggested that patterns possessing
more symmetry at all levels would be perceived as simpler.
The measure was highly correlated with the average rank-
ing of the patterns (r= .808, p< .001). Although ad hoc,
subsymmetry is related to our measure because patterns
containing symmetries at all levels also contain less change
at all levels. Instead of counting instances of symmetry at
every level, we count instances of (recursively-defined)
change. As expected, the average number of subsymmetries
was significantly correlated with C(s)(r= .672, p< .001).
The correlation between pattern goodness rank averaged
across the experiments and the corresponding values of
C(s) was highly significant (r= .694, p< .001). Interest-
ingly, the number of runs was weakly correlated with the
number of subsymmetries and not correlated with subjec-
tive pattern ranks. This result should be contrasted to that
reported by Glanzer and Clark (1962), whose subjects
showed clear reliance on low-level quantitative informa-
tion. The difference can be explained by the fact that in
Alexander and Carey’s study, subjects had sufficient time
to inspect the patterns and consider higher-level structural
information.
A context closely related to complexity is subjective ran-
domness. Although in psychology the two have not been
explicitly linked, algorithmic information theory considers
random those patterns that possess high Kolmogorov com-
plexity. Consequently, we could predict that our measure
would correlate significantly with subjective randomness
judgments. An influential study on the relationship
between complexity and subjective randomness was carried
out by Falk and Konold (1997), in which objective, infor-
mation-theoretic predictors of complexity (e.g. second-
order entropy and probability of alternation) were
compared with subjective ones (apparent randomness,
copying difficulty and memorization time) using a set of
40 binary strings of length 21. The strings were selected
so that although most were perceptually different, four sets
of ten strings were matched for number of runs and sec-
ond-order entropy. Our measure was highly correlated
with all three performance variables, namely, apparent ran-
domness (r= .720), copying difficulty (r= .796) and mem-
orization time (r= .865; all p< .001). In addition,
distributions of apparent randomness and copying diffi-
culty were negatively skewed. In contrast to Glanzer and
Clark’s study, low-level change did not seem to be impor-
tant to Falk and Konold’s subjects, since the number of
runs correlated modestly only with apparent randomness
(r= .364, p= .021) but not with other measures. Again,
this can be interpreted in terms of the tasks employed by
Falk and Konold. Copying and memorization tasks give
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subjects time to consider higher levels of structure and our
measure in its current form appears to be particularly well
suited to such tasks.
In another study of subjective randomness, Griffiths and
Tenenbaum (2003) asked subjects to rank 128 binary pat-
terns of length 8 according to how random they perceived
them to be. The authors also provided a multi-parameter
model of subjective randomness which corresponded clo-
sely to the subjective scores. The predictions of the model
correlated highly with the subjective rankings (r= .806,
p< .001). We observed highly significant correlations
between our measure and the two variables (subjective
rankings and the model; r= .663, p< .001 and r= .695,
p< .001 respectively). Controlling for complementary and
mirror symmetrical strings increased the correlations some-
what (.715 and .713 respectively).
From the above examples, we can conclude that C(s)
correlates highly with disparate measures of complexity
and randomness, both subjective and objective. Further-
more, these results support the view, originally put forward
by Kolmogorov, that subjective randomness equals high
complexity.
3.2. String complexity: sequential presentation
C(s) does not take into account sequential effects which
have been demonstrated with serial presentation (e.g.
Feldman & Hanna, 1966; Restle, 1970) and as such it does
not appear suited to quantifying complexity in this context.
Yet, if complexity is viewed even partly as a property of a
pattern, the relationship between change and processing
difficulty should hold irrespective of the mode of presenta-
tion. Two objective (statistical) models of binary pattern
complexity, called H(k-span) and H(run-span), were pro-
posed by Vitz (1968) as elaborations of an earlier model
(Vitz & Todd, 1967). The former is based on the transi-
tional uncertainties between pattern elements of different
length and equals log
2
of the length of a string, whereas
the latter is log
2
of the total number of runs in the pattern.
Vitz correlated these two measures with subjective judged
complexity (JC) and mean verbalization length (MVL)
for 26 binary patterns of varying length, obtained over
two experiments. In order to increase sample size, three
measures [H(k-span), H(run-span) and judged complexity]
were collapsed across experiments and MVL was excluded
because it had been used with only eight patterns in Exper-
iment 1. This did not inflate the result, because the correla-
tions were highly significant within individual experiments.
Again, correlation coefficients between C(s) and the three
measures were highly significant (C(s)/H(k-span) = .838,
p< .001; C(s)/H(run-span) = .786, p< .001; C(s)/JC =
.861, p< .001). Although C(s) was very highly correlated
with the participants’ performance, so were Vitz’s mea-
sures. Given that those measures were based on quantita-
tive aspects of information (string length and number of
runs), there appeared to be no reason for claiming that
C(s) was more psychologically valid.
Vitz (1968), Vitz and Todd (1967) and Leeuwenberg
(1969) reported high correlations between their measures
and subjective data as evidence of the validity of their
approaches. However, high correlations can be obtained
between variables even when one of them possesses few dis-
tinct values. Thus, correlation on its own does not guaran-
tee that a measure is a realistic index of psychological
complexity. Humans are reasonably (if not infinitely), dis-
criminating with regard to complexity and a subjective
complexity scale can assume a number of distinct values.
Given the importance of subjective judgment, in addition
to correlation, what is needed is some way of knowing
how sensitive a measure is, that is, how close it is to the sen-
sitivity of the human observer. Vitz (1968) was aware of
this problem and stated that “... probably... coding into
runs is too simple a description of the coding process (p.
280).”
One possibility is to compute the ratio of distinct subjec-
tive responses based on the number of distinct stimuli and
compare this with the ratios obtained from different mea-
sures. Clearly, a measure should be capable of assuming
a number of distinct values, which is sufficiently large to
approximate the complexity of the underlying psychologi-
cal process. The sensitivity of C(s) (.92) was virtually indis-
tinguishable from subjective judgment (.96) and much
higher than the sensitivity of Vitz and Todd’s measures
(.23 and .17 for k-code and run-code measures respec-
tively). With the number of subjective judgments used as
the baseline, Vitz’s measures appear highly insensitive
and significantly different from subjective judgment
(p= .001 and p< .001 respectively; Binomial test). By con-
trast, there was no statistical difference between the number
of subjective judgments and C(s)(p= 1). This suggests that
C(s) corresponds better to the subjective complexity judg-
ments than do Vitz’s probabilistic measures.
Vitz and Todd (1969) extended their approach by devel-
oping a measure of complexity called Hcode. Based on the
hierarchical coding of (binary or trinary) pattern elements
and the application of Garner’s (1962) multivariate uncer-
tainty analysis, the measure was developed specifically to
account for sequential pattern learning. The authors
obtained very high correlations between their measure
and judged complexity of the 20 binary patterns 1–8 sym-
bols long (.941, p< .001). C(s) was significantly correlated
with both Hcode and judged complexity scores (.605 and
.581, respectively, both p= .005).We reiterate that unlike
Hcode, which assumes sequential pattern processing, our
model makes no such assumptions.
Psotka (1975) proposed a measure of sequential pattern
structure called “syntely”, based on expectancies created by
runs and alterations within a pattern. He investigated 35
binary patterns of length 8 comparing six measures of pat-
tern structure: an information-theoretic complexity mea-
sure by Vitz and Todd (Hcode; 1969; discussed in Simon,
1972), judged complexity, measured and judged symmetry,
and measured and judged syntely. As can be seen from
Table 1,C(s) showed no correlation with Vitz and Todd’s
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measure or with judged syntely. In contrast, it correlated
significantly with judged complexity, measured symmetry,
judged symmetry and measured syntely. Interestingly, mea-
sured syntely correlated only with judged syntely, indicat-
ing that these measures were not adequate models of
subjective complexity. Furthermore, Vitz and Todd’s mea-
sure showed no correlation with judged symmetry. Table 1
shows correlations between the measures reported by Pso-
tka and C(s). In order to enable a comprehensive compar-
ison, the table also includes the following complexity
measures: number of runs, Leeuwenberg’s (1969) original
SIT code and van der Helm’s (2000) SIT code. It should
be noted that out of eight measures (seven objective and
one subjective) C(s) exhibited the highest correlation both
with subjective complexity and subjective symmetry scores
(highlighted). This is particularly surprising given that our
measure is based on a simple theoretical premise and has
not been adapted to the requirements of sequential
presentation.
Fig. 3 illustrates the relationship between four different
measures and subjective complexity judgments. It can be
seen that our measure (Cs) not only shows the highest cor-
relation with subjective judgments but also provides more
distinct values. By contrast, runs, which take into account
only low-level structure correlate poorly. Hcode, which was
specifically designed for sequential patterns performs
slightly better but shows a pronounced clustering at the
upper end of response distribution. Van der Helm’s SIT
code shows an even higher correlation (.55) but it possesses
only five different values over a range of 35 stimuli.
Although C(s) appeared be non-linearly related to the
responses, a quadratic fit was only slightly better than a lin-
ear one (R
2
= .542 and .527 respectively).
Our claim that C(s) performed better than other mea-
sures was confirmed in terms of sensitivity. As can be seen
from Table 2,C(s) is the closer than other objective models
in terms of the number of distinct values to the subjective
complexity judgment. There was no difference in sensitivity
between C(s) and subjective complexity judgments
(p= .560 on a Binomial test). By contrast, Vitz’s probabi-
listic model, for example, provides significantly fewer dis-
tinct values (p= .001).
Finally, we conducted a visual inspection of complexity
distributions for different measures. Traditionally, invari-
ance has been viewed as the opposite of change. This is
implicitly represented in the entropy/redundancy distinc-
tion, which views the former as the inverse of the latter
(entropy = 1–redundancy; Attneave, 1954). Yet, psycho-
logical research has indicated that change is more difficult
to process than the absence of change and that symmetrical
information-theoretic probability distributions which treat
change and its absence as equiprobable do not agree with
distributions of scores on a number of complexity and ran-
domness-related tasks, which are negatively skewed (Falk
& Konold, 1997, p. 306). Briefly, observers consider as
most random or complex those patterns which contain a
large amount of change as long as the change is not
regular.
An examination of distributions of different complexity
measures in Fig. 4 shows that C(s) behaves reasonably well
in modelling the distribution of subjective judgments. By
contrast, Hcode contains too few distinct values to repre-
sent a plausible model of subjective complexity. Note the
negative skew of the distribution of subjective complexity
judgments, which agrees with our theoretical distribution.
From this, it appears that, at least for short binary pat-
terns, our measure successfully approximates the shape of
the subjective complexity scaling distribution.
A point worth mentioning is that the reason our mea-
sure produced identical values for different strings is that
in most cases the strings in question were structurally
equivalent (or “logically equivalent”according to Vitz) a
point not generally raised by other authors. According to
our definition, mirror symmetrical (001 = 100) and comple-
mentary (001 = 110) strings possess the same complexity.
In other words, mirror symmetrical and complementary
strings contain an equal amount of change or information.
Clearly, subjective judgments will not agree with this all of
the time. To illustrate, in Psotka’s study, strings 01101010
and 01010110 have very different subjective complexity
Table 1
Rank order correlations between C(s) and other complexity measures on the data provided by Psotka (1975).
Measure JC MS JS Msyn Jsyn C(s) Runs SIT(L) SIT(vdH)
MC 468
**
.560
***
.056 .006 .101 .247 .673
**
.084 .419
**
JC – .605
***
.601
***
.273 .186 .668
***
.338
***
.307 .550
***
MS – – .356
*
.185 .117 .389
*
.431
**
.339
*
.642
***
JS – – – .246 .062 .800
***
.120 .422
*
.345
*
Msyn – – – – .677
***
.388
*
.100 .433
**
.221
Jsyn – – – – – .263 .063 .243 .193
C(s) – – – – – – .052 .290 .442
**
Runs – – – – – – – .250 .451
**
SIT(L) – – – – – – – – .385
*
Note. MC = measured complexity; JC = judged complexity; MS = measured symmetry; JS = judged symmetry; Msyn = measured syntely; Jsyn = judged
syntely; C(s) = our measure; Runs = number of runs; SIT(L) = SIT code (Leeuwenberg, 1969); SIT(vdH) = SIT code (van der Helm, 2000).
*
p< .05.
**
p< .01.
***
p< .001.
A. Aksentijevic, K. Gibson / Cognitive Systems Research 15–16 (2012) 1–16 11

Author's personal copy
values (525 and 400 respectively) despite the fact that the
latter is a mirror-symmetrical version of the former. The
latter string could have been judged less complex because
it began with a sequence of 01 pairs – an important consid-
eration in the context of sequential presentation. Neverthe-
less, the high correlation between our measure and
subjective complexity indicates that local variations in
complexity are less important than the overall structure
of the pattern.
The supramodal nature of perceptual organization (e.g.
Aksentijevic, Elliott, & Barber, 2001) suggests that the
effectiveness of C(s) in quantifying complexity/information
might extend to auditory pattern perception. In a famous
study, Royer and Garner (1966) investigated the ability
of participants to synchronize their tapping responses with
19 repeated binary auditory sequences of length 8. The
sequences were played continuously and the subjects were
required to tap in synchrony with the perceived start of
the pattern. As expected, the more complex patterns were
more difficult to organize and were associated with higher
response uncertainty, longer response delays and higher
error rate. Response uncertainty (in bits), median delay
and mean number of errors for each pattern were signifi-
cantly correlated with C(s)(r= .638, p= .003; .595,
p= .007; and .580, p= .009, respectively). While the corre-
lations are not very high, they are comparable to the ones
reported above. The number of runs in individual patterns
was not correlated with any dependent variables, suggest-
ing that the subjects relied substantially on higher-order
structural information.
Another well-known study of sequential pattern pro-
cessing, this time using visual presentation, was carried
out by Garner and Gottwald (1967). Subjects were pre-
sented with two binary patterns of length 5. The patterns
were labeled “simple”(RRRLL) and “complex”(LLRLR)
and presented to subjects via two lights (left or right). The
patterns were presented from each possible starting point
Subjective complexity
23456
100
600
200
300
400
500
Runs
78
r = .34
2345
SIT code (van der Helm)
r = .55
100
600
200
300
400
500
6912 15 18
H(code)
21
r = .47
123456
C(s)
r = .67
Fig. 3. Correlations between four objective measures of complexity and subjective responses for sequentially presented patterns of length 8 (Psotka, 1975).
See text for details.
Table 2
Sensitivity of measures employed by Psotka (1975) and C(s).
Measure Distinct values Sensitivity ratio
MC 7 .20
JC 26 .74
MS 8 .23
JS 34 .97
Msyn 17 .48
Jsyn 11 .31
C(s) 21 .60
Runs 7 .20
SIT (L) 4 .11
SIT (vdH) 4 .11
123456
100 300 500
Judged complexity
6101418
Hcode
0
5
10
15
20
Frequency
C(s)
Fig. 4. Frequency distributions of the complexity values for patterns
studied by Psotka (1975). Hcode corresponds to the measure proposed by
Vitz and Todd (1969) and structural complexity refers to our measure.
12 A. Aksentijevic, K. Gibson / Cognitive Systems Research 15–16 (2012) 1–16

Author's personal copy
giving 10 different patterns. Subjects were asked either to
predict the location of the next light from the beginning
of the sequence (immediate responding) or when they were
confident about the form of the sequence (delayed respond-
ing). The authors examined a number of dependent vari-
ables including the mean number of trials needed to
reach criterion and number of errors for each starting
point. C(s) correlated significantly with the former variable
(r= .749, p= .013) and there was a marginally significant
correlation with the latter (r= .625, p= .054). Although
the number of data points was small (10), the significant
correlation indicates the importance of change in sequen-
tial pattern prediction. In contrast to the Royer and Garner
study, the number of runs was significantly correlated with
both variables.
3.3. Array complexity
Although C(s) is proposed primarily as an index of
string complexity, preliminary investigations suggest that
it could be useful in determining the complexity of binary
arrays. In a large and comprehensive study, Chipman
(1977) investigated the importance of different structural
properties on the judgment of the complexity of black
and white patterns (6 6 matrices containing 12 black
squares). Judging the complexity of 2-D arrays is a highly
complex process which might involve a large number of
different quantitative and structural factors. To illustrate,
Chipman examined the following: The number of turns,
(perimeter)
2
/area, horizontal and vertical symmetry, diago-
nal symmetry, opposition symmetry and number of repeti-
tions. We examined the complexity of 45 patterns used in
Experiment 1 (of 7) because the patterns were assigned
judged complexity values, making them accessible for anal-
ysis. The correlation between C(s) and judged complexity
was highly significant (r= .754, p< .001). Interestingly,
the sensitivity of C(s) (.98) exceeded that of the subjective
judgment (.87).
As discussed earlier, symmetry represents one of the pri-
mary determinants of pattern goodness. Howe (1980) con-
ducted a large-scale study on the effects of partial symmetry
(symmetry of parts of an object) on different tasks such as
exposure duration, masking, immediate memory and
reproduction. Participants were presented with 60 dot pat-
terns containing varying degrees of symmetry. The stimu-
lus set was constructed by gradually reducing the amount
of symmetry from 12 randomly chosen “good”patterns.
Subjects’ performance was clearly governed by the amount
of symmetry. To illustrate, subjective judgments of good-
ness highly correlated with the change in degree of partial
symmetry and the performance on exposure, masking,
memory and reproduction tasks was inversely proportional
to the complexity (absence of symmetry) in the stimuli. To
test the validity of C(s) in the context of the relationship
between goodness and symmetry, each of Howe’s patterns
was assigned a structural complexity value and these values
were correlated with mean subjective goodness ratings
given by the subjects. The correlation between subjective
judgment and C(s) was highly significant (r= .673,
p< .001) confirming that C(s) in its current form represents
a good predictor of subjective perception of goodness. The
sensitivity ratios for the subjective judgments and our mea-
sure were .91 and .65, respectively.
Related to this, Yodogawa (1982) proposed an objective
measure of symmetry for 2-D binary patterns, based on the
two-dimensional discrete Walsh transform. It should be
noted that Yodogawa’s measure is very successful in pre-
dicting judged goodness. He compared his measure “sym-
metropy”with goodness judgments using the patterns
previously employed by Howe (1980). The rank order cor-
relation between symmetropy and C(s) for ten binary pat-
terns presented in the paper was .790 (p= .007).
4. Discussion
The fact that our approach is based on a simple premise
(amount of change), together with excellent correspon-
dence with empirical data, suggests that it could be more
powerful than other comparable models (see Table 3).
One of the interesting properties of the model is its ability
to detect periodicities in binary strings, which appear inac-
cessible to a measure defined in terms of symmetry.
Although partly motivated by physical and computational
accounts of entropy, our measure successfully models the
well-documented negative skew of subjective complexity
and randomness distributions. In addition, our measure
appears to be more sensitive (i.e. closer to subjective perfor-
mance in terms of distinct complexity values) than other
measures, both probabilistic and algorithmic.
Our approach elucidates the relationship between infor-
mational entropy on the one hand and McKay’s (1950)
metric/structural distinction on the other. Informational
entropy is a metric and frequentist concept defined by the
size of the source or frequencies of occurrence of different
outcomes. This is why information theory is not well suited
to quantifying structure, which refers to relationships
between elements/symbols/objects. Our measure integrates
the quantitative and structural aspects of information by
quantifying the occurrence of the simplest and most gen-
eral of relationships – same or different.
As the complexity of a pattern increases, the more effort
is needed to encode, assimilate or compress the pattern and
this is reflected in our measure. As expected, simple pat-
terns are few and represent statistical aberrations of high
order. An overwhelming proportion of possible structural
arrangements is far too complex (contains too much
change) for the observer to relate to immediately. In order
to achieve this, the observer has to consider structural rela-
tions at all levels of the pattern. Our approach sheds light
on the well-documented relationship between the amount
of information present in a pattern and the amount of time
needed to assimilate it. A brief exposure to a stimulus com-
pels the perceptual/cognitive system to rely on surface,
quantitative information. This information is represented,
A. Aksentijevic, K. Gibson / Cognitive Systems Research 15–16 (2012) 1–16 13

Author's personal copy
for example, by the number of uniform patches, turns or
angles (Ichikawa, 1985). The more time the observers are
given to study the pattern, the more their judgment is influ-
enced by the relations between the elements and different
levels of structure. A broad analogy could be drawn here
with evidence that visual perception follows a similar
course.
In agreement with Falk and Konold (1997), we propose
that psychological complexity reflects effort (or cost)
required in order to convert available into useful informa-
tion. In a sense, C(s) measures the distance between the two
domains. For simple, orderly patterns, this distance is small
and this is reflected in our measure. Patterns are easily dis-
criminated and little effort is required to assimilate them.
The increase in the distance between the two domains
caused by the limitation of the human observer is reflected
in the fact that complex patterns are increasingly more dif-
ficult to discriminate. When complexity reaches a certain
level, different patterns become indistinguishable. This
observation can be linked to the current debate on the nat-
ure of randomness with some authors equating random-
ness with simplicity (e.g. Adami & Cerf, 2000;
Gell-Mann, 1995). We suggest that the perceived simplicity
of random patterns is due to the fundamental limitation of
the human observer who has to abandon trying to assimi-
late highly complex contexts and is compelled to treat them
as undifferentiated noise.
To summarize, the notion of cost allows us to relate psy-
chological understanding of complexity to the physical and
computational contexts. The fact that C(s) correlates reli-
ably with a wide range of disparate measures of complexity
suggests that change represents the conceptual core of com-
plexity. Increase in change implies an increase in entropy.
Symmetry and periodicity denote transformational invari-
ance, that is, absence of change under transformation.
Sequences (visual and auditory) and arrays that contain
Table 3
Summary of comparisons of C(s) with different complexity measures.
Study N(patterns) Modality
(auditory/
visual)
Presentation
(simultaneous/
sequential)
Dimensions Measure Subjective/
objective
Correlation
with C(s)
Glanzer and Clark (1962) 128 V Sim 1 8 Reproduction accuracy O .83
***
Alexander and Carey (1968) 35 V Sim 1 7 Perceived goodness S .69
***
N subsymmetries O .67
***
Falk and Konold (1997) 40 V Sim 1 21 Apparent randomness S 72
***
Copying difficulty S .80
***
Memorization time S .86
***
Griffiths and Tenenbaum (2003) 128 V Sim 1 8 Perceived randomness S 71
***
G & T model O 71
***
Vitz (1968) 26 V Seq 1 1–1 8 H(k-span) O .84
***
H(run-span) O .79
***
Judged complexity S .86
***
Vitz and Todd (1969) 20 V Seq 1 1–1 8 H(code) O .58
**
Judged complexity S .61
**
Psotka(1975) 35 V Seq 1 8 H(code) O .25
Judged complexity S .67
***
Measured symmetry O .39
*
Judged symmetry S .80
***
Measured syntely O .39
*
Judged syntely S .26
Van der Helm (2000) 35 V N/A 1 8 SIT code O .44
**
Garner and Gottwald(1967) 10 V Seq 1 5 Trials to criterion S .75
*
Number of errors S .65

Royer and Garner (1966) 19 A Seq 1 8 Response uncertainty S .64
**
Response delay S .59
**
Error rate S .58
**
Chipman (1977) 45 V Sim 6 6 Judged complexity S .75
***
Howe (1980) 60 V Sim 5 5 Perceived goodness S .67
***
Yodogawa (1982) 10 V Sim 5 5 Symmetropy O .79
***
***
p< .001.
**
p< .01.
*
p< .05;

p< .1.
14 A. Aksentijevic, K. Gibson / Cognitive Systems Research 15–16 (2012) 1–16

Author's personal copy
little change are easier to compress, perceive, analyze or
describe. This is true of computers as well as human
observers.
Acknowledgments
The authors wish to thank Slobodan Markovic for his
help in surveying psychological complexity literature. We
also thank Ruma Falk and Cliff Konold for providing
the complete data set from their 1997 study and for their
encouraging suggestions as well as Mike Oaksford for his
editorial advice. Finally, we extend special thanks to Peter
van der Helm for his exhaustive and helpful comments as
well as for his help with data coding.
Appendix A. Proofs of the lemmas and theorems in Sections
2.2 and 2.3
Lemma 1. Let S =(s
1
,s
2
,... ,s
L
)be a binary string, L >1,
and let U =(s
1
,s
2
,... ,s
L1
), V =(s
2
,s
3
,... ,s
L
).
Then Sis a palindrome if and only if UV.
Proof. Suppose Sis a palindrome. Then S=rS, where ris
one of r,rc. It is straightforward to verify that in both cases
U=rVso that UV.
Conversely, suppose UV. Then V is one of U, cU, rU,
rcU. It is straightforward to verify the following statements.
If V =Uthen Sis one of “000 ...”,“111 ...”.IfV=cU
then Sis one of “0101 ...”,“1010 ...”.IfV=rU then
S=rS.IfV=rcU then S=rcS with Leven. In all cases
Sis a palindrome. h
Theorem 1. Let the binary string S =(s
1
,s
2
,... ,,s
L
), L>2.
Then [S]=0 if and only if [s
1
,s
2
,... ,s
j
]=[s
Lj+1
,
s
Lj+2
,... ,s
L
], j=2to L 1.
Proof. Let (s
1
,s
2
,...,,s
L1),
(s
2
,s
3
,... ,s
L)
have change
profiles P =(p
2
,p
3
,... ,p
L1
), Q =(q
2
,q
3
,... ,q
L1
)
respectively. By Definition 6, [S] = 0 if and only if P =Q.
From Eq. (1b), we have that for j=2toL1, h
pj¼½s1;s2;...;sjþ½s2;s3;...;sjþ1þ 
þ½sLj;sLjþ1;...;sL1
qj¼½s2;s3;...;sjþ1þþ½sLj;sLjþ1;...;sL1
þ½sLjþ1;sLjþ2;...;sL
Thus P=Qif and only if [s
1
,s
2
,... ,s
j
]=[s
Lj+1
,
s
Lj+2
,... ,s
L
], j=2to L1, proving the theorem.
Lemma 2. Let S,T be binary strings of length L P2, and let
[] be any change function.
If STimplies [S]=[T] then STimplies that Sand
Thave the same change profile.
Proof. Write S=(s
1
,s
2
,... ,s
L
), T=(t
1
,t
2
,... ,t
L
).
Suppose ST. Then S=rT, where ris one of e,r,c,
rc. Here edenotes the identity operator. Consider the
following one to one correspondence ubetween the
substrings of Sand T.
Let X
ij
be the substring of Sof length jthat starts at s
i
,
j=2to L,i=1 toLj+1.
If ris eor cthen umaps X
ij
to the substring of Tof
length jthat starts at t
i
.
If ris ror rc then umaps X
ij
to the substring of Tof
length jthat finishes at t
Li+1
.
It is straightforward to verify that in all cases uX
ij
X
ij
.
Now suppose that for all string lengths P2, UV
implies [U]=[V].
Then we have that [uX]=[X] for all substrings Xof Sof
length P2.
By the definition of change profile, this means S and T
have the same change profile. h
Theorem 2. Let S, T be binary strings of length L P2. If
ST then [S]=[T].
Proof. The proof is by induction on L. We first observe
that the theorem is true for L= 2. We then make the
“induction hypothesis”that the theorem is true for string
lengths j=2toL1, L > 2, and use the induction hypoth-
esis to show that the theorem is true for string length j =L.
For L= 2 just note that [00] = [11] = 0, and [01] =
[10] = 1. Suppose therefore L> 2 and that the theorem
holds for string lengths j=2toL1. Let U
j
,V
j
be formed
from the first and last jsymbols of S, and W
j
,Z
j
be formed
from the first and last jsymbols of T,j=2 to L1.
Suppose ST. Then S=rT, where ris one of e,r,c,rc.
Here edenotes the identity operator. It is straightforward
in each case to verify that U
j
=rW
j
and V
j
=rZ
j
,j=2to
L1. In other words U
j
W
j
and V
j
Z
j
for each j=2to
L1. By the induction hypothesis this means that
[U
j
]=[W
j
] and [V
j
]=[Z
j
] for each j=2 to L1. By
Theorem 1 this makes [S]=[T]. h
Lemma 3. Let S be a binary string of length L >2.
If S is a palindrome then [S]=0.
Proof (using theorems 1 and 2). Let U
j
,V
j
be the strings
formed from the first and last jsymbols of S,j=2 to L.
Suppose Sis a palindrome. Then S=rS, where ris one
of r,rc. It is straightforward to verify that in both cases
U
j
=rV
j
for each j, i.e. U
j
V
j
, so that [U
j
]=[V
j
] from
Theorem 2. The result now follows from Theorem 1.h
Appendix B. The calculation of array complexity
Let Abe an mby nbinary array, and let R,C,M,Bbe
the sums of the unnormalized complexities of the rows,
columns, main diagonals, and back diagonals of A. To cal-
culate the normalized and unnormalized complexities N
and Uof A, compute
A. Aksentijevic, K. Gibson / Cognitive Systems Research 15–16 (2012) 1–16 15

Author's personal copy
d¼mþn1
S¼R=mþC=nþM=d:þB=d
X¼d1þ2ðm1Þðn1Þ=d
L¼4mn=ð3dþ1Þ
N¼S=X
U¼ðL1ÞN
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