A closer look at creativity as search
Graeme Ritchie
Computing Science
University of Aberdeen
Aberdeen AB24 3UE
g.ritchie:abdn.ac.uk
Abstract
Several papers by Wiggins (building on ideas by Boden)
have outlined a view of creative concept generation
as a very general search process, but that formalisation
has not been developed much in the past few
years. Also, there are some aspects where clarification
or spelling out of details would be useful. We present a
re-formulation of the central ideas in Wiggins’s framework,
with slightly more rigorous statements of the definitions
and a number of minor extensions. We also explain
how this framework relates to some hitherto completely
separate proposals by Ritchie.
Introduction
In recent years, there have been various formalisations of aspects
of the computational creative process ((Pease, Winterstein,
and Colton 2001), (Colton, Pease, and Ritchie 2001),
(Garc´ıa et al. 2006), (Thornton 2007), (Colton, Charnley,
and Pease 2011)). Hence there is a consensus among at least
some established researchers that it is methodologically beneficial
to have fully precise, detailed and formal accounts of
any mechanisms being considered as ‘creative’.
A prominent example is Wiggins’s creative systems
framework (CSF), presented in a number of papers (particularly
Wiggins(2006a; 2006b)). That framework emphasises
the notion of search as the central mechanism for simulating
creativity, and outlines how a metalevel search could represent
some phenomena sometimes discussed as ‘transformational’
creativity. Although these ideas are very helpful in
clarifying the nature of creative computation, the published
versions of the CSF are at best a preliminary sketch: some
details are unspecified, some natural extensions are undeveloped,
and there are some formal errors or infelicities. The
current paper starts from the central ideas of the CSF, but redefines
the formal mechanisms in a way which leaves fewer
gaps, aspires to have fewer formal inconsistencies, and includes
the description of more aspects of computationally
creative processes. The central motivation for this is that,
if we subscribe to a belief in the benefits of formal models
(as noted above), then these models should not be left undeveloped,
but should continue to be maintained, repaired, and
extended as necessary.
It is important to realise that the underlying intuitive ideas
– creation as the exploration of a ‘conceptual space’, and
possible ‘transformation’ of that space – have been set out
in numerous articles by Boden, with many illustrative examples
from human creativity. Wiggins’s contribution was
to take those informal, broadbrush ideas and outline a formal
framework which both captured the core notions and
made sense computationally. The reader is referred to publications
by Boden, Wiggins, and many others for more about
the intuitive motivation; our aim here is to refine and extend
Wiggins’s proposals.
A summary of Wiggins’s CSF
Although this paper is centrally concerned with formalisation,
we start with a very brief informal overview of the ideas
in the existing version of the CSF.
The framework posits a universal set of all concepts, a
term which covers both abstract ideas (e.g. a mathematical
theorem, a design for a better political system) and concrete
artefacts (e.g. a painting, a poem). Within this wide-ranging
set, there are particular types of idea/artefact (e.g. stories,
paintings, poems), and what counts as a recognisable example
of a story/painting/poem/etc. may be highly dependent
on socio-cultural norms. For many such creative genres, it
is not realistic for there to be a firm definition of acceptability,
as the specific concept may be vague in the sense of (van
Deemter 2010). That is, the extent to which a text is or is not
a well-formed story (or other artistic category) is a matter of
degree, rather than an all-or-nothing decision. Hence, within
the CSF, the criterion for acceptability/recognisability is represented
as a rating between 0 and 1; in effect, the set of examples
of a genre is treated as a fuzzy set. As well as whether
something falls within the definition of some artistic genre,
there is the separate question of whether it is a good (high
quality) instance (e.g. a profound poem, a beautiful painting).
This is similarly a ‘vague’ notion, and again is represented
within the CSF as a score between 0 and 1. This
means that an artefact can be an acceptable instance (e.g.
recognisably a story) without being of high quality (e.g. it
may be a poor story); hence the need for two different ratings
(mappings from concepts to values).
The inspiration for the CSF was the work of Boden(1998;
2003), in which creativity was described as occurring within
a conceptual space, which could be explored, or – in more
radical creativity – transformed into a new space. This
view has some resemblance to the traditional ideas of search
International Conference on Computational Creativity 2012 41
within AI (Nilsson 1971), and Wiggins set out both to clarify
the relationship between conventional AI search and creative
computation, and to provide a formal framework for
describing the latter. The idea is that a creative system starts
from some set of concepts, and by a series of steps creates
further concepts one after another, thus ‘exploring the
space’. Although the term ‘creative’ has connotations of
‘construction’, the CSF, following the practice in formalising
AI search, regards ‘new’ concepts as not so much ‘constructed’
as ‘reached’. That is, notionally all the possible
concepts are elements of some universal set, but the creative
system computes a route through that set to particular
concepts, and those which have been thus reached represent
‘discoveries’ or ‘inventions’.
The exploration of the space of concepts (the search
method) is modelled by an operator which acts on a sequence
of concepts (a list of the concepts the system has
already processed), and yields as its result a new (presumably
longer) list of concepts, which can then be processed in
turn as the next cycle of search. Sequences are used because
the search method, like an agenda-based AI search system,
has to maintain some record of where it has reached within
the space of available concepts, and what to work on next.
Wiggins points out that by separating the acceptability
rating from the search method, we can describe the situation
where different composers, each with a personal way of
finding new artefacts (different search procedures) are working
within the same style of music (a shared notion of acceptability).
A creative agent should be able to recognise
something as being a recognisable artefact, or a high quality
artefact, without necessarily having a search method that
would allow the agent to reach (create) that artefact.
More concretely, this is an intuitively plausible account of
certain potentially creative programs. The MCGONAGALL
poetry generator (Manurung, Ritchie, and Thompson 2012)
uses an explicit search mechanism (a genetic algorithm) to
find suitable candidate texts. Each text must at least be syntactically
well-formed according to the system’s linguistic
grammar; this could be regarded as the acceptability mapping.
During the search, items are scored for rhythmic suitability
and proximity to an initial semantic message; this
would be the quality mapping. At each stage, the system
keeps an ordered list of the current candidates, from which
each cycle in the search starts.
A small formal detail is that the CSF search operator takes
as arguments the two fuzzy criteria (for acceptability and
quality), and from there computes a way of going from an
existing concept-sequence to a new concept-sequence. This
means that the search method can be sensitive to these two
criteria if necessary, or that we can describe two systems as
having the same search strategy relative to their own different
definitions of validity and quality.
For Wiggins, the mappings (the two fuzzy sets and the
search mechanism) are to be represented as expressions in
some symbolic language, translatable to actual mappings.
Hence, in the CSF, an exploratory creative system consists
of the following seven components:
(i) the universal set of concepts
(ii) the language for expressing the relevant mappings
(iii) a symbolic representation of the acceptability mapping
(iv) a symbolic representation of the quality mappings
(v) a symbolic representation of the search mechanism
(vi) an interpreter for expressions like (iii) and (iv)
(vii) an interpreter for expressions like (v)
That constitutes the object level of the creative system,
which searches through concepts in the domain (e.g.
melodies). Wiggins also proposes that there can be a metalevel,
which searches through possible object level systems
to find an interestingly different ‘conceptual space’,
thus modelling Boden’s idea of a ‘transforming’ the space.
The metalevel in CSF is structured in the same way as the
object level (i.e. the seven parts as set out above), except
that its set for exploration (i.e. its universal set) contains expressions
in the symbolic language used at the object level.
In this way, the metalevel searches through expressions describing
object-level systems, assessing these descriptions
for acceptability and for quality (using the metalevel’s criteria
for these two measures).
Relative to the published accounts of the CSF, the revisions
or extensions made here are:
• The symbolic language for expressing the various mappings
is given a much less central role.
• The way in which the metalevel defines the object level is
explicitly stated. In particular, the notion of ‘transformation’
of an (object-level) space is defined.
• Some minor inconsistencies in definitions are removed.
• We outline how search methods within the CSF can be
compared at the metalevel.
• The CSF is related to a proposal for formal assessment of
creative systems (Ritchie 2001; 2007).
The object level
The structure of an object level system
Wiggins posits the existence of one universal set, U, the set
of all concepts, but then defines a creative system as a tuple,
one component of which is the universal set. If the set
is truly universal across all systems, it should not need to
be mentioned as a defining component of a specific system.
On the other hand, it would be useful to be able to allow
different creative systems to consider only specific subsets
of this hugely general set. The compromise here is to accept
the existence of the wholly universal set, but for the
definition of each creative system to specify a subset of this
universe; this could, in principle, be a non-proper subset.
We will use P (mnemonic for ‘possibilities’) for these subsets
in our definitions, below. The idea is that U is universal
enough to contain concepts for every type of artefact that
might ever be conceived: it includes poems, stories, sculptures,
jokes, paintings, theorems, architectural plans, designs
for food mixers, etc. On the other hand P represents the
whole range of concepts within some narrower sphere, such
as two-dimensional arrays of coloured pixels (which could
act as a ‘universal’ set for the creation of visual art), or finite
sequences of words and punctuation (a possible ‘universal’
set for various textual artistic forms).
International Conference on Computational Creativity 2012 42
Notation: For any sets A, B, BA denotes the set of mappings
from A to B. In particular, for any set X, [0, 1]X
denotes the set of mappings from X to values between 0
and 1 inclusive. Since a fuzzy set is defined by a mapping
from possible elements to values betwen 0 and 1, our fuzzy
sets of ‘acceptable’ elements and of ‘valuable’ elements will
be stated in this way; that is, as members of [0, 1]P . We
also take tuples(X) to denote the set of finite tuples (of any
length) of elements of the set X.
Definition 1: An exploratory creative system comprises:
(i) a subset P of U (possible concepts within this type or
genre)
(ii) N 2 [0, 1]P , the acceptability mapping (mnemonically,
this describes norms)
(iii) V 2 [0, 1]P , the value mapping (mnemonically, this
describes value)
(iv) a mapping Q (the exploration scheme) from [0, 1]P ⇥
[0, 1]P to the set of mappings from tuples(P) to
tuples(P) (mnemonically, this describes a quest for
creations – ‘s’ for ‘search’ is used elsewhere).
The four components in our definition are direct counterparts
of those in Wiggins’s CSF, but we have chosen different
symbols for the components of a system, to avoid confusion;
the relationships to Wiggins’s notation are: N ⇠= [[R]],
V ⇠= [[E]], Q(N , V) ⇠= hhR, T , Eii. The intuitive meanings
of the components are the same: P is the set of possible
concepts (e.g. arrays of pixels, sequences of words), N de-
fines the fuzzy set of acceptable instances of whatever domain/genre
is being explored, V indicates how ‘good’ an instance
is, and Q defines how to explore the space.
The component Q, the search method, may need some
explanation. Directly following Wiggins’s proposals, Q is
applied to a particular N and V, and from that produces a
mapping which takes sequences of concepts into sequences
of concepts; hence, N and V could in principle influence
Q, or could be ignored. It might seem odd to describe Q as
taking these two parameters, when the only possible values
for the parameters seem to be fixed as N and V – why not
just ‘compile in’ these two values, as they are specified in
the same 4-tuple package as Q? At present, this level of parameterisation
has no real advantage, but it leaves open the
possibility, as the framework is elaborated, of considering a
‘transformed’ version of an exploratory creative system in
which Q is unchanged, but one or both of N , V are altered,
with automatic consequences for the operation of Q.
As in the original Wiggins formulation, Q(N , V) maps
from sequences of concepts to sequences of concepts, providing
an agenda-like exploration of the set of possibilities,
with the sequence representing the current search state
As noted earlier, the CSF includes a symbolic language in
which mappings are expressed as rules, which are then interpreted
into mappings. Here, we abstract away from the use
of a language, and define a creative system using mappings.
The advantage of this is that it states the essential relations
within a creative system without regard for representational
issues. In a later section, we show how the symbolic language
can be incorporated, directly reflecting the Wiggins
approach.
Characterising the conceptual space
As noted above, the basic definition of an exploratory creative
system contains a fuzzy set, N , of concepts, which
are – intuitively – those concepts which conform to the current
norms of the domain. In Wiggins’s formulation, this
fuzzy set is not regarded as modelling Boden’s ‘conceptual
space’. Instead, Wiggins stipulates that conceptual spaces
are ordinary subsets (not fuzzy) of the universal set U, and
that the conceptual space C (of a given creative system) consists
of all concepts mapped by his [[R]] (our counterpart is
N ) to values greater than or equal to 0.5. Similarly, Wiggins
defines the valued set as those concepts which his [[E]]
(our equivalent is V) maps to values greater than or equal to
0.5. That is, although the CSF allows both these sets to have
graded membership (0 to 1), Wiggins immediately simpli-
fies them to non-graded sets by imposing a threshold.
Within our formulation, the counterpart to Wiggins’s nonfuzzy
definitions would be as follows:
Definition 2: Given an exploratory creative system S =
(P, N , V, Q), we define, for any ↵ 2 [0, 1] and X ✓ P:
(i) N↵(X) = {c 2 X | N (c) > ↵} (the set of concepts
which reach the threshold ↵ in their ‘normality’).
(ii) V↵(X) = {c 2 X | V(c) > ↵} (the set of concepts
which reach the threshold ↵ in their ‘quality’).
(iii) the fuzzy conceptual space of S is N (this just confirms
the status of N as outlined earlier).
(iv) the conceptual space of S is N0.5(P) (this is for backwards
compatibility with Wiggins’s 0.5 threshold).
(v) the fuzzy valued set of S is V (this just confirms the
status of V as outlined earlier)..
(vi) the valued set of S is V0.5(P) (this is for backwards
compatibility with Wiggins’s 0.5 threshold).
Searching
In the CSF, the searching process begins from an initial set
of concepts. This is to allow for the situation where the creative
system starts from some given concept set, representing
the status quo. It is also useful later, when considering
metalevel search. In Wiggins’s definitions, exploration always
starts from the single totally unspecified concept, >,
representing a situation in which the system has no known
concepts already. Here, we generalise this slightly.
If the sequences on which Q operates are to be like an
agenda in conventional AI search, then those sequences
should contain only items which have been produced from
previous steps of the search (i.e. applications of Q). This
means that the initial agenda has to include every item (concept)
which could ever participate in discoveries but is not
produced by an application of Q.
Wiggins defines ‘reachable’ concepts using indefinitely
many applications of the search operator. There is a minor
slip in his definition, in that repeated applications of
hhR, T , Eii (which corresponds to our Q) will compute sequences
(tuples) of concepts, not individual concepts. This
is easily remedied (and we can add an intermediate version,
for limited search). First we need a minor definition of all
the items appearing within a set of tuples:
International Conference on Computational Creativity 2012 43
Definition 3: Given any set of tuples A, we define
elements(A) ⌘
{x | 9hy1,...,yni 2 A, 9i 1  i  n : x = yi}
The Wiggins formalisation defines all the concepts which
can be reached with any amount of search, i.e. without limit.
Although this case is of theoretical interest, in practice any
system will search only to some finite depth, and so we also
define the notion of ‘reachable in a fixed number of steps’,
relying on the fact that a single application of the search
mapping Q corresponds to one step in the search process.
Definition 4: Given an exploratory creative system
(P, N , V, Q), and a set B ⇢ P of concepts (B is the starting
set of concepts for the search):
(i) the set reachable from B in m steps is
S
m
n=0
elements(Q(N , V)n(B))
(i.e. any number of repeated applications of Q, up to
m; this describes search up to some depth.)
(ii) the set reachable from B is
S
1
n=0
elements(Q(N , V)n(B))
(i.e. any number of repeated applications of Q; this allows
any depth of search.)
(iii) the set of reachable concepts is the set reachable from
{>}. (This matches Wiggins’s notion, where all search
starts from a single unspecified concept).
(iv) the set of concepts reachable in m steps is the set reachable
from {>} in m steps. (This is a bounded search
variant of Wiggins’s ‘start from nothing’ definition.)
In considering the behaviour of a creative system, it is
important to know which of its final output (i.e. creations)
were provided to it initially and which were computed by
the system itself. We can define these thus:
Definition 5: Given an exploratory creative system
(P, N , V, Q), a subset B of P, and a set of concepts K
reachable from B, the discoveries in K is the set of concepts
in K ) B.
Wiggins defines the set of valued concepts as being those
concepts reachable from the undefined concept, >, which
exceed a particular threshold value (0.5) when his ‘value’
mapping (our V) is applied. That definition can be restated
in the terminology here.
Definition 6: Given an exploratory creative system
(P, N , V, Q), where RC is the set of reachable concepts,
and a value ↵ 2 [0, 1]:
(i) the ↵-valued set of reachable concepts is V↵(RC).
(ii) the valued set of reachable concepts is the 0.5-valued
set of reachable concepts (i.e. V0.5(RC)); this mirrors
Wiggins’s definition.
The metalevel
Structure of the metalevel
For the metalevel, the first matter to be clarified concerns the
set of items used for exploration. In the Wiggins papers, this
is the set of possible expressions in a symbolic language L.
An expression in L is defined earlier as defining a rule-set
representing either R (acceptability rules), E (value rules)
or T (search rules), with different interpreters applying depending
on which of these is intended. If the metalevel is
considering single L-expressions, how does one such expression
represent an entire object level system, which contains
all three of R, E, and T ? Within the language-based
formalisation, a possible response would be to say that L
must contain notation which allows one expression to represent
three rule sets. If following this path, Wiggins’s definitions
of the language interpreters would also have to be
patched. As we are separating out the language aspect, we
have a different solution.
Here, the object level space is defined by the mappings N ,
V, Q, so it seems reasonable to have the metalevel searching
through triples (N,V,Q) where N, V , Q, are possible
values for N , V, Q respectively (and hence are elements of
the appropriate sets such as [0, 1]P ). The exploration set at
the metalevel will be the set of such triples; for brevity here,
we will call this set of triples ECS(P) (as it is the set of
possible exploratory creative systems for P).
Given a subset P of U, a metalevel creative system for P
is an exploratory creative system made up of:
(i) ECS(P) (i.e. this is the metalevel’s set to explore)
(ii) an element N meta of [0, 1]ECS(P)
; this rates potential
object-level systems as to how ‘normal’ they are, thus
providing a (fuzzy) set of ‘acceptable’ triples (N,V,Q).
(iii) an element Vmeta of [0, 1]ECS(P)
; this rates potential
object-level systems as to their ‘quality’, thus providing
a (fuzzy) set of ‘valuable’ triples (N,V,Q).
(iv) a mapping Qmeta from [0, 1]ECS(P) ⇥ [0, 1]ECS(P)
to the set of mappings from tuples(ECS(P)) to
tuples(ECS(P)); this is structured like the search device
Q in an object-level creative system, but operates
on elements of ECS(P) instead of P.
That is, the structure at the metalevel is exactly parallel to
the structure at the object level, as in the Wiggins version.
A metalevel has information about what an object level creative
system should look like (N meta) and what would count
as a ‘good’ object level system (Vmeta). It also contains a way
of searching through potential object level systems (Qmeta).
Characterising an object level system
Given a definition of the components of a metalevel, it is
essential to then define exactly how the parts of the metalevel
characterise an object level system. This is not discussed in
detail by Wiggins, but he indicates that the metalevel is to
operate (in terms of search, etc.) exactly as an object level
creative system.
An object level creative system will ascribe various characteristics
to a set of concepts. Each concept will be: rated
(by N ) as to how acceptable it is as an member of the conceptual
space in question, rated (by V) as to its value/quality,
and defined (by Q) as either reachable or not. As noted earlier,
Wiggins proposes that the ratings by (his equivalents of)
N and V are turned into non-fuzzy sets using a threshold.
However, even then the object level does not characterise a
International Conference on Computational Creativity 2012 44
single object, or a unique set of systems: it defines three independent
sets, via N , V and Q. Since the metalevel has exactly
the same structure as an object level system, the items
which it explores (for Wiggins, expressions in a symbolic
language L) are presumably similarly allocated to 3 sets:
the recognisable, the valued and the reachable (and for Wiggins,
reachability is always relative to >, not some specified
starting set of items). Hence, the metalevel is assigning (potential)
object level systems to these three categories. What
the metalevel does not do is characterise a single object level
system, or even a unique set of systems. This means that we
do not, from the published papers, have a definition of how
one object level system is a transformation of another, or
how a computation at the metalevel will yield a new object
level system – all that the metalevel provides is this tripartite
classification. We will remedy this by defining how a
metalevel can define (or transform) a specific object level
system.
In the next few definitions, we assume two exploratory
creative systems Sobj = (P, N , V, Q) and
S0
obj = (P, N 0
, V0
, Q0
), and a metalevel system Smeta =
(ECS(P), N meta, Vmeta, Qmeta) for P. Where the relation
‘6=’ is used here, this allows for the two items in question to
have elements in common.
Definition 7:
(i) S0
obj is a revision of Sobj using Smeta if S0
obj is in the set
reachable from {Sobj} within Smeta, and S0
obj 6= Sobj .
(ii) for any ↵ 2 [0, 1], S0
obj is ↵-valued with respect to
Smeta if Vmeta((N 0
, V0
, Q0
)) $ ↵.
(iii) S0
obj is a transformation of Sobj using Smeta if S0
obj is a
revision of Sobj using Smeta and also N 6= N 0
.
Here we have taken a ‘transformation’ to be a revision in
which the definition of the conceptual space (acceptable set)
changes, as indicated by the condition ‘N =6 N 0
’. As this
could be true even if N and N 0 differ only on one element,
proponents of transformation as a form of radical change
might wish to enhance this definition.
Relationship to the original CSF
To clarify the amendments we have made to the formalisation,
we can compare it with the original version in the
papers by Wiggins. As mentioned earlier, the original CSF
includes a symbolic language in which the components (the
counterparts of our N , V, Q) are expressed. We can add this
to our framework by defining a symbolic-language version
of an exploratory creative system, with appropriate links to
the definitions given above. In order to mimic Wiggins’s
definitions, we first have to clarify certain aspects which are
unclear in the published papers. Sometimes the mapping
N (or what corresponds to this in Wiggins’s framework) is
represented as a single expression in a symbolic language,
and sometimes it is said to be a set of expressions. Either
of these accounts could be made to work, if applied consistently.
Here, we have opted for the single expression version,
with the observation that the symbolic language could contain
connective symbols such as ‘conjunction’, ‘disjunction’
or other logical operators, thereby getting the effect of a set
of rules in one syntactic expression.
Wiggins’s version does not separate clearly the definition
of the language from the specific language expressions used
in a particular creative system. We have tried to draw this
distinction in the next two definitions.
Definition 8: Given a set of concepts P, a creative systems
language for P is a tuple (A,LR,LT , [[.]],hh.ii) where:
(i) A is a set of symbols, the alphabet.
(ii) LR and LT are languages over A (only 2 are needed
because the language LR can be used for both the ‘acceptability’
rules and the ‘value’ rules, since these both
describe fuzzy sets of concepts).
(iii) [[.]] is a mapping from LR to [0, 1]P (this is the interpreter
which takes an expression in the symbolic language
and returns a mapping; that mapping is then a
fuzzy set of concepts).
(iv) hh.ii is a mapping from LR ⇥ LT ⇥ LR to
tuples(P)tuples(P) (this is the interpreter which turns
symbolic expressions specifying a search method into
an actual mapping to carry out the search).
The above definition (which is closely modelled on Wiggins’s
proposals) provides the symbolic mechanisms, separately
from any particular creative system which might use
these representations. The next two definitions state how
these mechanisms can be used to define a specific system.
Definition 9: Given a set of concepts P and a creative systems
language (A,LR,LT , [[.]],hh.ii) for P, then a symbolically
represented exploratory creative system for P consists
of a tuple (WR, WE , WT ) where:
(i) WR 2 LR; the norms or acceptability rules.
(ii) WE 2 LR; the rules assigning value to items.
(iii) WT 2 LT ; rules which define the search method.
Definition 10: Given a set of concepts P, a creative systems
language (A,LR,LT , [[.]],hh.ii), and a symbolically represented
exploratory creative system SE = (WR, WE , WT ),
then the exploratory creative system associated with SE is
the tuple S = (P, N , V, Q) where
(i) N = [[WR]]; i.e. the meaning of this rule expression is
the normality mapping.
(ii) V = [[WE ]]; i.e. the meaning of this rule expression is
the value mapping.
(iii) Q(N , V) = hhWR, WT , WE ii; i.e. the meanings of
these rule expressions give the search mapping.
(This directly mirrors the arrangement of Wiggins’s CSF.)
Using his formalisation, (Wiggins 2006a) provides a number
of definitions of specific behaviours that a creative system
could display, in terms of what concepts are valued,
which can be reached, etc. These terms can all be defined
within the formalisation given here, as follows, assuming
an exploratory creative system S = (P, N , V, Q), and using
some of the terminology already defined above:
Hopeless uninspiration: The valued set of concepts is
empty. That is, there are no concepts anywhere within
the universal set that meet the ‘quality’ threshold.
International Conference on Computational Creativity 2012 45
Conceptual uninspiration: V0.5(N0.5(P)) = ;. That is,
there are no concepts within the acceptable (‘normal’) set
of concepts that meet the ‘quality’ threshold.
Generative uninspiration: The valued set of reachable
concepts is empty. That is, there are no concepts which
the search mechanism can reach which meet the quality
threshold.
Aberration: Where B consists of exactly those elements in
the reachable set which are not in N0.5(P), aberration
occurs if B =6 ;. That is, aberration is when the search
mechanism goes outside the ‘normal’ set of concepts.
Perfect aberration is where V0.5(B) = B (i.e. all the nonnormal
concepts meet the quality threshold); productive
aberration is where V0.5(B) 6= ; and V0.5(B) 6= B (i.e.
just some of the non-normal concepts meet the quality
threshold); pointless aberration is where V0.5(B) = ;
(i.e. no non-normal concepts meet the quality threshold).
Evaluating search methods
Ventura’s analysis
Ventura(2011) gives an analysis of the limitations of uninformed
search strategies in a creative context. His definitions
and results are general enough that they should be applicable
to the framework here, although there is one small
formal point that needs to be stipulated first. Ventura (implicitly)
makes the following assumption:
One concept per step: Each formal ‘step’ in the search
process corresponds to the generation of exactly one
concept/artefact.
That is, Ventura’s analysis does not allow for intermediate
computational steps behind the scenes which do not directly
correspond to the generation of an artefact. The Wiggins
definitions (and our reformulations) do not demand this restriction,
but it is a plausible constraint, and could be formalised
thus:
Given an exploratory creative system (P, N , V, Q), Q
is a one-concept-per-step scheme if, whenever z0 =
Q(N , V)(z), 9c0 2 P such that:
(i) c0 is an element of z0
;
(ii) c0 is not an element of z;
(iii) for every element c of z0 where c0 6= c, c is an
element of z.
This perspective could be taken even further, by revising
our definition of an exploratory creative system to include
a set OP of operators, which are mappings from tuples of
concepts to concepts; that is, each operator is a member of
PPk
for some integer k. Then we would stipulate that each
step in a search meets the constraint that it corresponds to
the invocation of one operator:
If two concept-sequences hc1,...,cni,hd1,...,dmi
are such that Q(N , V)(hc1,...,cni) = hd1,...,dmi,
then:
there must be an operator p 2 OP, and concepts
he1,...,eki (where k  n) such that for every 1 
i  k, ei = cj for some j, and p(e1,...,ek) = dr for
some dr 2 {d1,...,dm}, and dr 2/ {c1,...,cn}, and
8di 2 {d1,...,dm}, either i = r or di 2 {c1,...,cn}.
Ventura’s analysis provides one possible formalisation of
the intuitive notion of a search strategy being ‘better’ (or
‘best’). It posits a set of target elements (concepts, in our terminology)
and considers the probability of the search reaching
one of these elements. In a footnote, Ventura also offers
a definition where the desirability of elements is a function
to [0, 1] (cf. our V), and computes the probability of reaching
an element with a maximal value.
It is arguable that in the area of creative systems, there is
less emphasis on finding specific target items (or even finding
a maximal-scoring item) and more on generally reaching
acceptable or highly valued concepts (a direction in which
Ventura’s footnote moves). Our formalisation allows an alternative
perspective on the assessment or comparison of
search methods; see the next subsection.
Comparing search
Our formulation of the CSF allows the comparison of two
search methods according to how the concepts they reach are
rated by the related mappings N and V, taking into account
the depth of search involved. As we will want to apply certain
definitions to various mappings (including N and V),
we will start with a general schematic definition in which
the mapping F can be anything of the appropriate type (so
F is not mnemonic for anything, being just a placeholder
for now). Also, AGG will stand for either AV G (arithmetic
mean) or MAX (maximum) of a function applied to a set.
Definition 11: Suppose there are two exploratory creative
systems (P, N , V, Q1) and (P, N , V, Q2), with Si(k, k0
) =
the set of concepts reachable in no fewer than k and no more
than k0 steps in these two systems (i = 1, 2). Also, F 2
[0, 1]P (i.e. a rating of concepts, of some sort). Then
(i) Q1 has higher AGG F-values up to k0 steps than Q2
if AGG(F, S1(0, k0
)) > AGG(F, S2(0, k0
)).
(ii) Q1 has higher AGG F-values beyond k steps than Q2
if AGG(F, S1(k, k0
)) > AGG(F, S2(k, k0
)) for any
k0 > k.
This compares two variants of a system in which only the
search method Q is different. The above definitions will be
applied, below, to specific values for F.
Definition 12: Given a two exploratory creative systems
(P, N , V, Q1) and (P, N , V, Q2):
(i) Q1 is higher valued on average up to k steps than Q2
if Q1 has higher AVG V-values up to k steps than Q2.
(ii) Q1 is more normal on average up to k steps than Q2 if
Q1 has higher AVG N -values up to k steps than Q2.
(iii) Q1 achieves higher value up to k steps than Q2 if Q1
has higher MAX V-values up to k steps than Q2.
(iv) Q1 achieves greater conformity up to k steps than Q2
if Q1 has higher MAX N -values up to k steps than Q2.
For each of the 4 subparts of the above definition, we can
frame a corresponding definition which says that there exists
some depth k0 after which one of the search methods Qi
gives a higher value than the other; e.g.:
International Conference on Computational Creativity 2012 46
Q1 is higher valued eventually than Q2 if there is some
integer k0 > 0 such that Q1 has higher AVG V-values
beyond k0 steps than Q2.
Similar substitutions can be made in the other definitions.
In this way, we have several ways of describing one search
method Q1 as being ‘better’ than another, Q2. Next, we
consider how comparisons of search methods can be more
detailed.
Descriptive criteria
Ritchie(2001; 2007) defines a set of formal criteria which
can be used to describe aspects of a potentially creative system’s
behaviour. Central to these formal criteria are two rating
schemes for assigning values in [0, 1] to elements of the
set of basic items (i.e. the set of possible artefacts). One rating
scheme (typ) represents typicality, indicating the extent
to which an item lies within the norm for this type of artefact.
The other rating (val) is for value, and indicates the
‘quality’ of an item. Ritchie’s typ and val, Wiggins’s R and
E and our N and V all appear to capture the same intuitive
notions: that we can rate possible creations as to their membership
of a concept set, and in terms of the quality of such
creations.
There are some differences of nuance between Ritchie’s
constructs and those in the CSF, to which we will return
later, but for the moment let us consider how the central
ideas in some of Ritchie’s criteria could be used within the
CSF as stated here.
The first eight of Ritchie’s criteria are stated in terms of
a result set, R, which is the set of artefacts produced by the
computer program, and the two ratings typ and val. There is
not space here to reproduce them all, but Criterion 7 illustrates
the general idea:
ratio(V!,1(R) \ T0,"(R), T0,"(R)) > ✓,
for suitable ", #, ✓.
where V!,1(R) is the set of elements of set R which are rated
above # by val, T0,"(R) is the set of elements of R which
are rated below " by typ, and ratio computes the ratio of
the sizes of two sets. That is, this computes the proportion
of the untypical items which are of good quality.
Criteria like this could be applied to a creative system
(P, N , V, Q), using N to define Ti,j and V to define Vi,j .
There are various ways in which the result set R could be
defined in terms of reachable concepts: all reachable concepts?
concepts reachable from a starting set B? concepts
reachable after some number of steps k? All of these are
plausible models of a ‘result set’. Hence there would be a
few families of very similar formula, parameterised according
to starting set or number of steps.
Ritchie(2007) emphasises that these criteria are not all
measures of creative success, but can be used to ‘profile’
a (potentially) creative program by describing its behaviour
in more detail. In the same way, they could give a more
detailed picture of a creative system, in the CSF sense.
Ritchie also postulates an inspiring set, I, which are
the existing artefacts upon which the design of the creative
program was based. The remaining criteria (9 - 18 in
Ritchie(2007)) make comparisons of different sorts between
I and the result set R. There is no exact counterpart within
the CSF, as there is no ‘design’ stage in the formalisation.
However, the formal structure of Ritchie’s criteria which involve
I could be coerced into service within the CSF, by
replacing I with some initial set of concepts B, from which
search starts. That is, where Ritchie has a criterion such as:
ratio(V!,1(R " I) \ T↵,1(R " I)),(R " I)) > ✓, for
suitable ↵, #, ✓
(informally, ‘a high proportion of the novel results are both
highly valued and very typical of the genre’) we would have:
ratio(V!,1(R"B)\T↵,1(R"B)),(R"B)) > ✓, for
suitable ↵, #, ✓.
where R is the set of concepts reachable from B. (Again,
there is a possible variant where a number of steps k is stipulated.)
Although this seems to indicate that we can simply port
the Ritchie criteria into the CSF, there is one further issue to
consider: there is a difference in the overall perspective of
the two formal accounts. There are various viewpoints one
could assume in devising a formal model in this area. For
example, it would be possible to have an abstract declarative
formalisation of the nature of the creative task, without including
any details of how this task might be executed. Or
a model might be proposed as describing (at some suitably
abstract level) how a creative system operates. Ritchie’s criteria
arguably take a third viewpoint, in which one treats the
program as a conveyor of input/output data, and attempts,
from an external viewpoint, to say more precisely how it
has performed. The typ and val mappings are certainly not
proposed as components of the program or system. Instead,
these are measures which might be applied by, for example,
having humans judge the program’s output.
In Wiggins’s CSF, the formal definitions of R and E (the
symbolic counterparts of our N , V) would be compatible either
with a characterisation of the abstract nature of creativity,
or with a model of a creative system. However, the terminology
used, and the inclusion of the T (‘agenda’) mapping
(our Q) determine that this is a model (at a very abstract
level) of the working of a creative system. Hence, the whole
intent of the CSF is radically different from that of Ritchie’s
definitions, even though there is a clear parallel between the
intuitive meanings of N and typ, V and val.
This means that what we have sketched above is not the
direction application of Ritchie’s criteria within the framework,
but the definition of counterparts within the CSF,
where the Ti,j and Vi,j mappings are defined using internal
components (N , V) of the system, not external judgements.
However, this means that if we are scrutinising the
behaviour of a creative system, we can apply the conditions
outlined above (i.e. the counterparts of Ritchie’s criteria) in
distinct ways:
Internal: How is the system performing, in its own terms?
For this, we use N and V to define Ti,j and Vi,j .
External: How is the system performing, in terms of independent
measures such as human judgements? For this,
we use the independent measures to define Ti,j and Vi,j .
International Conference on Computational Creativity 2012 47
In both of these, we retain the notions of B (initial set) and R
(reachable concepts) discussed above. Given that the set of
reachable concepts is in effect the ‘result set’, if the inspiring
set I is known, then using I instead of B, with an External
perspective, is effectively the scenario in the Ritchie papers.
These various adaptations of the criteria to the CSF can be
viewed alongside the definitions in our section Comparing
search above, and provide a slightly finer grained and more
detailed vocabulary for comparing search strategies.
Guiding the metalevel
We have already shown how the metalevel of a creative
system can start from an existing object level system and
search for a variant system, using a metalevel value function
Vmeta. What should the content of Vmeta be? Since
the metalevel has access to the object level mappings N
and V, it would be possible for Vmeta to be defined using
composite criteria such as those we have outlined above,
the counterparts of Ritchie’s criteria. That is, the metalevel
search could be guided by how candidate object-level systems
performed according to these criteria. For this, the
distinction between internally-parameterised and externallyparameterised
versions of the criteria is important. Whereas
the externally-parameterised version (using human judgements
or other measures) is exactly appropriate for profiling
or assessing the success of the object-level system, the
internally-parameterised version (using N and V) are the
only ones that make sense within the creative (metalevel)
system itself.
This glosses over the significant question of whether
a real creative program would be implemented with the
meta/object strata of the CSF, or whether the formal framework
is only a way of describing, at some fairly abstract
level, what creative systems do (or could do). It is possible
that the actual use of structured criteria which compare ratings
of initial sets and of reachable concepts would not be
realistically applicable in implemented systems.
Summing up
We have presented a reformalisation of Wiggins’s CSF,
which:
• makes the use of a symbolic language an optional extra
• indicates how search strategies can be formally compared
within the CSF
• clarifies the metalevel, defining some metalevel constructs
in more detail and making explicit some formal comparisons.
• shows how Ritchie’s criteria can be adapted, in a number
of ways, to the CSF formalisation, thereby clarifying the
relationship between these frameworks
In this way, we have extended the development of formal
descriptive frameworks for creative systems.
Acknowledgments
The author is very grateful to Geraint Wiggins for detailed
discussions of this material, and for comments on an draft of
this paper.
<references_biblio/>
References
Boden, M. A. 1998. Creativity and Artificial Intelligence.
Artificial Intelligence 103:347–356.
Boden, M. A. 2003. The Creative Mind. London: Routledge,
2nd edition. First edition 1990.
Colton, S.; Charnley, J.; and Pease, A. 2011. Computational
creativity theory: The FACE and IDEA descriptive models.
In Ventura et al. (2011), 90–95.
Colton, S.; Pease, A.; and Ritchie, G. 2001. The effect
of input knowledge on creativity. In Weber, R., and von
Wangenheim, C. G., eds., Case-Based Reasoning: Papers
from the Workshop Programme at ICCBR 01.
Garc´ıa, R.; Gervas, P.; Herv ´ as, R.; and P ´ erez y P ´ erez, R. ´
2006. A framework for the E-R computational creativity
model. In 5th Mexican International Conference on Artifi-
cial Intelligence (MICAI-06), volume 4293 of LNAI, 70–80.
Springer Verlag.
Manurung, R.; Ritchie, G.; and Thompson, H. 2012. Using
genetic algorithms to create meaningful poetic text. Journal
of Experimental & Theoretical Artificial Intelligence
24(1):43–64.
Nilsson, N. J. 1971. Problem-solving methods in artificial
intelligence. New York: McGraw-Hill.
Pease, A.; Winterstein, D.; and Colton, S. 2001. Evaluating
machine creativity. In Weber, R., and von Wangenheim,
C. G., eds., Case-Based Reasoning: Papers from the Workshop
Programme at ICCBR 01, 129–137.
Ritchie, G. 2001. Assessing creativity. In Proceedings of the
AISB Symposium on Artificial Intelligence and Creativity in
Arts and Science, 3–11.
Ritchie, G. 2007. Some empirical criteria for attributing
creativity to a computer program. Minds and Machines
17(1):67–99.
Thornton, C. J. 2007. How thinking inside the box can
become thinking outside the box. In Proceedings of the 4th
International Joint Workshop on Computational Creativity.
van Deemter, K. 2010. Not Exactly: in praise of vagueness.
Oxford, UK: Oxford University Press.
Ventura, D.; Gervas, P.; D. Fox Harrell; Maher, M. L.; Pease, ´
A.; and Wiggins, G., eds. 2011. Proceedings of the 2nd
International Conference on Computational Creativity.
Ventura, D. 2011. No free lunch in the search for creativity.
In Ventura et al. (2011), 108–110.
Wiggins, G. 2006a. A preliminary framework for description,
analysis and comparison of creative systems.
Knowledge-Based Systems 19:449–458.
Wiggins, G. A. 2006b. Searching for computational creativity.
New Generation Computing 24(3):209–222.
International Conference on Computational Creativity 2012 48