Generalize and Blend:
Concept Blending Based on Generalization, Analogy, and Amalgams
Tarek R. Besold
Institute of Cognitive Science
University of Osnabruck ¨
D-49069 Osnabruck, Germany ¨
tarek.besold@uni-osnabrueck.de
Enric Plaza
IIIA, Artificial Intelligence Research Institute
CSIC, Spanish Council for Scientific Research
Campus U.A.B., 08193 Bellaterra, Catalonia (Spain)
enric@iiia.csic.es
Abstract
Concept blending, a cognitive process which allows for
the combination of certain elements (and their relations)
from originally distinct conceptual spaces into a new
unified space combining these previously separate elements
and allowing the performance of reasoning and
inference over the combination, is taken as a key element
of creative thought and combinatorial creativity.
In this paper, we provide an intermediate report on work
towards the development of a computational-level and
algorithmic-level account of concept blending, presenting
the theoretical background together with the main
model characteristics, as well as two case studies.
Creativity and Concept Blending
The term “combinatorial creativity” (Boden 2003) refers to
creativity which arises from a combinatorial process joining
familiar ideas (in the form of, for instance, concepts,
theories, or artworks) in an unfamiliar way, thereby producing
novel ideas. But although the overall idea of combining
preexisting ideas into new ones seems fairly intuitive and
straightforward, computationally modeling this form of creativity
turns out to be surprisingly complicated: When looking
at it from a more formal perspective at the current stage
neither can a precise algorithmic characterization be given,
nor are at least the details of a possible computational-level
theory describing the process(es) at work well understood.
Still, in recent years a proposal by (Fauconnier and Turner
1998) called concept blending (or conceptual integration)
has influenced and reinvigorated studies trying to unravel
the general cognitive principles operating during creative
thought. In their theory, concept blending constitutes a cognitive
process which allows for the combination of certain
elements (and their relations) from originally distinct conceptual
spaces into a new unified space combining these previously
separate elements and allowing the performance of
reasoning and inference over the combination.
Unfortunately, a proper computational modeling of concept
blending as cognitive capacity again is lacking. Neither
do (Fauconnier and Turner 1998) provide a fully worked out
and formalized theory themselves, nor does their informal
account capture key properties and functionalities as, for example,
the retrieval of input spaces, the selection and transfer
of elements from the input into the blend space, or the
further combination of possibly mutually contradictory elements
in the blend.
These shortcomings notwithstanding, several researchers
in AI and computational cognitive modeling have used
the provided conceptual descriptions as a starting point
for proposing possible refinements and implementations:
(Goguen and Harrell 2010) propose a concept blendingbased
approach to the analysis of the style of multimedia
content in terms of blending principles and also provide
an experimental implementation, (Pereira 2007) tries
to develop a computationally plausible model of several
hypothesized sub-parts of concept blending, (Thagard and
Stewart 2011) exemplify how creative thinking could arise
from using convolution to combine neural patterns into ones
which are potentially novel and useful, and (Veale and
O’Donoghue 2000) present their computational model of
conceptual integration and propose several extensions to the
(at that time prevailing) view on concept blending.
Since 2013, another attempt at developing a computationally
feasible, cognitively-inspired formal model of concept
creation, grounded on a sound mathematical theory of concepts
and implemented in a generic, creative computational
system is undertaken by a European research consortium
in the so called Concept Invention Theory (COINVENT)
project (Schorlemmer et al. 2014)1. One of the main goals
of the COINVENT research program is the development
of a computational-level and algorithmic-level account of
concept blending based on insights from psychology, AI,
and cognitive modeling, the heart of which are made up
by results from cognitive systems studies on computational
analogy-making and knowledge transfer and combination
(i.e., the computation of so called amalgams) from casebased
reasoning. In the following we present an analogyinspired
perspective on the COINVENT core model for concept
blending and show how the respective mechanisms and
systems interact.
Two Mechanisms at the Heart of COINVENT:
Generalization-Based Analogy and Amalgams
As analogy seems to play a crucial role in human cognition
(Gentner and Smith 2013), researchers on the computa-
1
Also see http://www.coinvent-project.eu for details
on the consortium and the project.
Proceedings of the Sixth International Conference on Computational Creativity June 2015 150
tional side of cognitive science and in AI also very quickly
got interested in the topic and have been creating computational
models of analogy-making since the advent of computer
systems, among others giving rise to (Winston 1980)’s
work on analogy and learning, (Hofstadter and Mitchell
1994)’s Copycat system, or (Falkenhainer, Forbus, and Gentner
1989)’s well-known Structure-Mapping Engine (SME).
Generally speaking there are (at least) two families of
computational analogy models: one family is based on a
(generalization-free) direct mapping approach, the other one
relies on a two-step procedure with a generalization stage
followed by a subsequent mapping stage. While the former
type of analogy engine is, among others, exemplified
in the SME and its immediate pairwise mapping of domain
elements between elements of source and target of the potential
analogy, followed by the accumulation of individual
mappings into more complex structures, the latter category
is represented by the Heuristic-Driven Theory Projection
(HDTP) framework (Schmidt et al. 2014). As COINVENT,
for principled conceptual reasons (see the section on
the idea(s) behind concept blending in COINVENT below),
relies on the generalization-based view on analogy-making,
we shortly introduce this model category in the following
subsection.
In a conceptually related, but mostly independently conducted
line of work researchers in case-based reasoning
(CBR) have been trying to develop problem solving methodologies
based on the principle that similar problems tend to
have similar solutions. CBR tries to solve problems by retrieving
one or several cases relevant for the issue at hand
from a case-base with already solved previous problems
(cases), and then reusing the past case(s) to also solve the
new task (Aamodt and Plaza 1994). While the retrieval
stage has received significant attention over the last two
decades, the transfer and combination of knowledge from
the retrieved case to the current problem has been studied
in an domain-specific way, with (Ontanon and Plaza 2012) ´
being a recent attempt at also gaining insights on this phase
of the CBR cycle by suggesting the framework of amalgams
(Ontanon and Plaza 2010) as a formal model for reuse of ´
multiple cases. The second subsection gives an overview of
amalgams as used in COINVENT.
Generalization-Based Models of Analogy
Generalization-based models of analogy-making share a
close conceptual connection to models of inductive generalization
(Smaling 2003). Similar to these, the basic principle
is the recognition of a common core between source
and target of the potential analogy, which is then used for
guiding the formation process of the analogy and the subsequent
content transfer and reasoning steps. Fig. 1 gives a
schematic overview: The common conceptual elements between
source S and target T correspond to a shared generalization
G (subsuming both, S and T), which also induces
mappings between the respective domain elements,
establishing an analogical relation. These mappings, governed
by the generalization, then also subsequently define
how (previously unmatched) knowledge from the source domain
can be transferred to and integrated into the target doGeneralization
(G)
% KK KK KK KK KK
ysssssssss
SOURCE (S) analogical
relation
TARGET (T)
Figure 1: A schematic overview of a generalization-based
approach to analogy.
main, namely by converting elements from S into their corresponding
counterparts within T.
The precise nature of the subsumption relation between
generalization and source or target domain, respectively,
is defined by the specific analogy model, possibly ranging
from semantic subsumption in a suitable ontology, through
taxonomic subsumption based on names and labels, logical
subsumption in a model-theoretic sense, to purely syntactic
subsumption in a formal language.
One example for a generalization-based computational
analogy-model (and the system used in COINVENT) is
the already aforementioned HDTP (Schmidt et al. 2014).
The framework has been conceived as a mathematically
sound theoretical model and implemented engine for computational
analogy-making, on a syntax basis computing
analogical relations and inferences for domains which are
presented in (when allowing for re-representation possibly
different) many-sorted first-order logic (FOL) languages.
Source and target domains are handed over to the system
in terms of finite axiomatizations and HDTP tries to compute
a generalization between both domains. This is done by
aligning pairs of formulae from the two domains by means
of restricted higher-order anti-unification (Schwering et al.
2009): Given two terms, one from each domain, HDTP
computes an anti-instance in which distinct subterms have
been replaced by variables so that the anti-instance can be
seen as a meaningful generalization of the input terms. As
already indicated by the name, the class of admissible substitution
operations is limited. On each expression, only renamings,
fixations, argument insertions, and permutations
may be performed. By this process, HDTP tries to find
the least general generalization of the input terms, which
(due to the higher-order nature of the anti-unification) is not
unique. In order to solve this problem, current implementations
of HDTP rank possible generalizations according to a
complexity measure on the chain of substitutions — the respective
values of which are taken as heuristic costs — and
returns the least expensive solution as the preferred one.
HDTP extends the notion of generalization from terms to
formulae by basically treating formulae in clause form and
terms alike. Finally, as analogies rarely rely exclusively on
one isolated pair of formulae from source and target domain,
but usually encompass sets of formulae (possibly completely
covering one or even both input domains), a process iteratively
selecting pairs of formulae for generalization has been
included. The selection of formulae is again based on a
heuristic component. Mappings in which substitutions can
be reused get assigned a lower cost than isolated substituProceedings
of the Sixth International Conference on Computational Creativity June 2015 151
tions, leading to a preference for coherent over incoherent
mappings.
Due to the use of many-sorted FOL as an expressive representation
language, and the purely syntax-based generalization
approach underlying HDTP, over the last years the
framework has shown remarkable generalizability and generality.
Having originally been conceived and applied for
modelling the Rutherford analogy and poetic metaphors, as
well as for providing an alternate account of (Falkenhainer,
Forbus, and Gentner 1989)’s heat-flow analogy in (Schwering
et al. 2009), without major changes to the model HDTP
has by now been applied to different tasks from different domains,
such as modeling a potential inductive analogy-based
process for establishing the fundamental concepts of arithmetics
(Guhe et al. 2010), or studies applying the framework
to modeling analogy use in education and teaching situations
(Besold 2014).
Combining Conceptual Theories Using Amalgams
The notion of amalgams was developed in the context of
CBR (Ontanon and Plaza 2010), where new problems are ´
solved based on previously solved problems (or cases, residing
on a case base). Solving a new problem often requires
more than one case from the case base, so their content has to
be combined in some way to solve the new problem. The notion
of an amalgam of two cases (two descriptions of problems
and their solutions) is a proposal to formalize the ways
in which cases can be combined to produce a new, coherent
case.
Formally, the notion of amalgams can be defined in any
representation language L for which a subsumption relation
v between the formulae (or descriptions) of L can be de-
fined. We say that a description I1 subsumes another description
I2 (I1 v I2) when I1 is more general (or equal)
than I2. Additionally, we assume that L contains the infi-
mum element ? (or ‘any’), and the supremum element >
(or ‘none’) with respect to the subsumption order.
Next, for any two descriptions I1 and I2 in L we can
define their unification, (I1 t I2), which is the most general
specialization of two given descriptions, and their antiunification,
(I1 u I2), defined as the least general generalization
of two descriptions, representing the most specific
description that subsumes both. Intuitively, a unifier is a
description that has all the information in both the original
descriptions; if joining this information leads to inconsistency,
this is equivalent to saying that I1 tI2 = > (i.e., they
have no common specialization except ‘none’). The antiunification
I1uI2 contains all that is common to both I1 and
I2; when they have nothing in common, then I1 u I2 = ?.
Depending on L anti-unification and unification might be
unique or not.
The notion of an amalgam can be conceived of as a generalization
of the notion of unification: as ‘partial unification’
(Ontanon and Plaza 2010). Unification means that what is ´
true for I1 or I2 is also true for I1tI2; e.g., if I1 describes ‘a
red vehicle’ and I2 describes ‘a German minivan’ then their
unification yields a common specialization like ‘a red German
minivan.’ Two descriptions may contain information
that produces an inconsistency when unified; for instance
I1 I2
¯I2 ¯I1
G = I1 u I2
A = ¯I1 t ¯I2
v
v
v
v v v
v v
Figure 2: A diagram of an amalgam A from inputs I1 and I2
where A = ¯I1 t ¯I2.
v
v v v
v v
A = S0 t T
S0
S
T
G = S u T
Figure 3: A diagram that transfers content from source S to
a target T via an asymmetric amalgam A.
‘a red French sedan’ and ‘a blue German minivan’ have
no common specialization except >. An amalgam of two
descriptions is a new description that contains parts from
these two descriptions. For instance, an amalgam of ‘a red
French sedan’ and ‘a blue German minivan’ is ‘a red German
sedan’; clearly there are always multiple possibilities
for amalgams, like ‘a blue French minivan’.
For the purposes of this paper we can define an amalgam
of two input descriptions as follows:
Definition 1 (Amalgam) A description A 2 L is an amalgam
of two inputs I1 and I2 (with anti-unification G =
I1 u I2) if there exist two generalizations ¯I1 and ¯I2 such
that (1) G v ¯I1 v I1, (2) G v ¯I2 v I2, and (3) A = ¯I1 t ¯I2
When ¯I1 and ¯I2 have no common specialization then trivially
A = >, since their only unifier is “none”. For our
purpose we will be only interested in non-trivial amalgams.
This definition is illustrated in Fig. 2, where the antiunification
of the inputs is indicated as G, and the amalgam
A is the unification of two concrete generalizations ¯I1 and ¯I2 of the inputs. Equality here should be understood as vequivalence:
X ⌘ Y iff X v Y and Y v X. Conventionally,
we call the space of amalgams of I1 and I2 the set of
all amalgams A that satisfy Definition 1.
Usually we are interested only in maximal amalgams of
two input descriptions, i.e., those amalgams that contain
maximal parts of their inputs that can be unified into a new
coherent description. Formally, an amalgam A of inputs I1
and I2 is maximal if there is no other non-trivial amalgam
A0 of inputs I1 and I2 such that A @ A0
. The reason why
we are interested in maximal amalgams is very simple: a
non-maximal amalgam A¯ @ A preserves less compatible
information from the inputs than the maximal amalgam A.
Conversely, any non-maximal amalgam A¯ can be obtained
by generalizing a maximal amalgam A, since A¯ @ A.
There is a special case of particular interest that is called
Proceedings of the Sixth International Conference on Computational Creativity June 2015 152
an asymmetric amalgam, in which the two inputs play different
roles. The inputs are called source and target, and while
the source is allowed to be generalized, the target is not.
Definition 2 (Asymmetric Amalgam) An asymmetric
amalgam A 2 L of two inputs S (source) and T (target)
satisfies that A = S0 t T for some generalization of the
source S0 v S.
As shown in Fig. 3, the content of target T is transferred
completely into the asymmetric amalgam, while the source
S is generalized. The result is a form of partial unification
that preserves all information in T while relaxing S by generalization
and then unifying one of those generalizations S0
with T itself. As before, we will usually be interested in
maximal amalgams: in this case, a maximal amalgam corresponds
to transferring maximal content from S to T while
keeping the resulting amalgam A consistent. For these reasons
asymmetric amalgams can be seen as models of a form
of analogical inference, transferring information from the
source to the target by creating a new amalgam that enriches
the latter with the content of S0 (Ontanon and Plaza 2012). ´
Analogy-Based Concept Blending in
COINVENT
Taking the concept of generalization-based analogies (and
HDTP as suitable framework for the computation of the latter)
together with the notion of asymmetric amalgams, we
now can introduce the core idea(s) behind concept blending
as performed in COINVENT in the next subsection, subsequently
also showing the feasibility of the approach in two
examples. The general suitability of the approach is demonstrated
revisiting the “sign forest” metaphor from (Kutz et
al. 2012), an implementation using HDTP is exemplified
(re-)constructing the concept of a foldable toothbrush.
The Core Model: An Analogy-Inspired View
One of the early formal accounts on concept blending, which
is especially influential to the approach applied in COINVENT,
is the classical work by Goguen using notions from
algebraic specification and category theory (Goguen 2006).
This version of concept blending can be described by the diagram
in Fig. 4, where each node stands for a representation
an agent has of some concept or conceptual domain. We will
call these representations “conceptual spaces” and in some
cases abuse terminology by using the word “concept” to really
refer to its representation by the agent. The arrows stand
for morphisms, that is, functions that preserve at least part of
the internal structure of the related conceptual spaces. The
idea is that, given two conceptual spaces I1 and I2 as input,
we look for a generalization G and then construct a blend
space B in such a way as to preserve as many as possible
structural alignments between I1 and I2 established by the
generalization. This may involve taking the functions to B
to be partial, in that not all the structure from I1 and I2
might be mapped to B. In any case, as the blend respects (to
the largest possible extent) the relationship between I1 and
I2, the diagram will commute.
Concept invention by concept blending can then be
phrased as the following task: given two representations of
G
~~~~~~
✏✏
%@@@@@@@
I1
%@@@@@@@ I2
~~~~~~
B
Figure 4: A conceptual overview of (Goguen 2006)’s account
of conceptual blending.
two domain theories I1 and I2, we need first, to compute a
generalized theory G of I1 and I2 (which codes the commonalities
between I1 and I2) and second, to compute the
blend theory B in a structure preserving way such that new
properties hold in B. Ideally, these new properties in B are
considered to be (moderately) interesting properties. In what
follows, for reasons of simplicity and without loss of generality
we assume that the additional properties are just provided
by one of the two domains, i.e., we align the situation
with a standard setting in computational analogy-making by
renaming I1 and I2. The domain providing the additional
properties for the concept blend will be called source S, the
domain providing the conceptual basis and receiving the additional
features will be called target T.
In COINVENT’s account, the reasoning process is then
triggered by the computation of the generalization G
(generic space), where for concept invention we will only
need the mapping mechanism and replace the transfer phase
by a new blending algorithm. The mapping is achieved via
the usual generalization process between S and T, in which
a generalized theory is created that reflects common aspects
of both spaces. The generalized theory can be projected
back into the original spaces by specializations !S and !T ,
respectively. As S and T might contain elements which
are not reflected in the shared generalization, it holds that
!S(G) ✓ S and !T (G) ✓ T. While in analogy making the
analogical relations are used in the transfer phase to translate
additional uncovered knowledge from the source to the target
space, blending combines additional facts (i.e., elements
from S \ SC or T \ TC ) from one or both spaces. Therefore
the process of blending can build on the generalization and
specializations provided by the analogy engine, but has to
include a new mechanism for transfer and concept combination.
Here, amalgams naturally come into play: The set
of specializations can be inverted and applied to generalize
the original source theory S into a more general version S0
(forming a superset of the shared generalization G, also including
previously uncovered knowledge from the source)
which then can be combined into an asymmetric amalgam
with the target theory T, forming the (possibly underspeci-
fied) proto-blend T0 of both. In a final step, T0 is then completed
into the blended theory and output of the process TB
by applying corresponding specialization steps stored from
the generalization process between S and T (see also Fig. 5).
If we now take the domains to be represented in the form
of finite axiomatizations as processed by HDTP, in an imProceedings
of the Sixth International Conference on Computational Creativity June 2015 153
G
!T

? ? ? ? !S
%
#_ #_ #_ #_
G = S0 u T
v
ysssssssss v
%JJJJJJJJJJ
Tc
v
 ~ ~ ~ Sc
v
%? ? ? ? S0
v %KKKKKKKKKKK !S
v 
? ? ? ?
T
v yttttttttttt
T analogical
relation S T0
!T v
✏✏
✏O✏O✏O
TB
Figure 5: A general overview of COINVENT’s account of concept blending using generalization-based analogy and asymmetric
amalgams: The shared generalization G from S and T is computed with !S(G) = Sc. The relation !S is subsequently re-used
in the generalization of S into S0
, which is then combined in an asymmetric amalgam with T into the proto-blend T0 = S0 t T
and finally, by application of !T , completed into the blended output theory TB. (Here v indicates subsumption between theories
in the direction of the respective arrows.)
plementation of the general model we can use the analogyengine
for computing the generalizations and deriving the
corresponding substitutions. In the generalization step between
S and T, as usual pairs of formulas from the source
and target spaces are anti-unified for deriving the generalized
theory G, and the specializations !S and !T become
substitutions which are computed during anti-unification.
Example 1: The Sign Forest
We now want to revisit the example of the blend sign forest
discussed in (Kutz et al. 2012), providing an interpretation
of the concept from a metaphor-centered perspective
and showing how the general COINVENT model can serve
for reconstructing the blending process. In what follows we
consider sign forest equivalent to the expression “a forest of
signs”, that shows more clearly its metaphorical nature.
The original sign forest blend was defined in the context
of blending ontologies, which means that the involved inputs
for blending were ontological descriptions of trees, forests,
and (traffic) signs. This approach views a concept such as
tree defined as an ontological specification of the concept of
tree: a specification that is ideally so general as to cover all
kinds of trees; the same can be said about forest, and (traffic)
signs. As such, certain properties and relations are selected
to form these specifications that are useful for an ontology
framework. However, our approach follows the notion that
concepts in human cognition can often be viewed, in cognitive
science, as bundles of their most typical properties
(albeit typicality may certainly be context-dependent). This
view is also taken in examples by (Fauconnier and Turner
1998) that are used to show how conceptual blending works:
a boathouse has typical properties of boat and house —but
not other properties that may appear in an ontological specification
of boat and house.
Thus, in this approach, the concept of tree is typically
formed by a plant having roots, a trunk and a crown (even if
there may be plants categorized as trees that do not have a
trunk, this is ignored as it does not belong to the bundle of
properties that are typical); this view is depicted as I2 in the
bottom right of Fig. 6, where other properties are included,
like plants being not mobile and the roots fixing the (typical)
tree to the ground. Finally, a forest is commonsensically de-
fined as a group of trees. The second concept, (traffic) sign,
may come in many forms (as we know from own experience),
but the first that comes to mind is the most typical one:
the signpost. The signpost is typically fixed on the ground
near a road, and has a post supporting a surface panel depicting
some traffic related information (labeled I1 in the lower
left corner of Fig. 6). The cognitive advantage of a signpost
is that it has a recognizable physical structure, while “traffic
sign” is so generic as to be a merely functional-based concept:
any kind of surface panel depicting some traffic-related
information is a traffic sign.
The generic space G of conceptual blending corresponds
to the anti-unification shown as G = I1 u I2 in Fig. 6;
G depicts common structure between a signpost and a tree:
a stem-like object, fixed to the ground, and supporting another
object on top. As discussed later, this common structure
is the basis for a metaphor like “a forest of signs” to
make sense — in contradistinction to a metaphor that does
not make sense such as “a forest of chairs”, even when a
typical chair is made of wood.
Now, the construction of the blended metaphor for
sign forest can be interpreted easily in the combined
generalization-based analogy and amalgam framework: the
input spaces can be generalized in different ways (although
always satisfying what they already have in common,
namely G). Different generalizations would yield different
amalgams, but the one we are considering here can be
seen as generalizing I2 into ¯I2, as shown in Fig. 6. Now
this generalization ¯I2 can directly be unified with I1, since ¯I1 is identical to I1; this unification yields the amalgam
A = ¯I1 t ¯I2 that, as shown in Fig. 6, represents a “forest
of signposts”. Moreover, since I1 ⌘ ¯I1, this model is an
asymmetric amalgam, as evidenced by the fact we generalize
the source (Forest) until it unifies with the target (Signpost),
while the latter remains fixed (i.e., is not generalized).
In order to support our perspective that a metaphor
(viewed as an analogy and amalgam combination in natural
language) is based on some (strong enough) common
Proceedings of the Sixth International Conference on Computational Creativity June 2015 154
Tree
Crown
Trunk
Root
has
has
has
above
above
Ground
attached
Signpost
Panel
Post
Post
holder
has
has
has
above
above
Ground
attached
Information displays Plant False mobile
Living
Being
Physical
Object
fixes-into
fixes-into
Forest
group-of
Physical
Object
Physical
Object
Vertical
Stick
Physical
Object
has
has
has
above
above
Ground
attached
fixes-into
Postsign
Panel
Post
Post
holder
has
has
has
above
above
Ground
attached
Information displays
fixes-into
Forest
group-of
Physical
Object
Physical
Object
Vertical
Stick
Physical
Object
has
has
has
above
above
Ground
attached
fixes-into
Forest
Signpost
Panel
Post
Post
holder
has
has
has
above
above
Ground
attached
Information displays
fixes-into
GENERIC SPACE
FOREST OF (TYPICAL) TREES TRAFFIC (TYPICAL) SIGNPOST
GENERALIZATION
is (basically) identical
GENERALIZATION OF FOREST
BLEND “SIGN-FOREST”
I1 I2
G = I1 u I2
¯I1 ¯I2
A = ¯I1 t ¯I2
Figure 6: Blending schema for “Sign Forest” when inputs are typical concepts for “Sign” (traffic signpost) and “Forest” (forest
of typical trees); the arrows indicate subsumption (v) as in Figure 2.
structure of the typical concepts participating in the blending
process, we checked if other metaphors can be constructed,
or better yet, have already been constructed, that are based
on the same kind of generic space G. We used Google’s ngrams
database to search for existing phrases in which “forest
of X” is used metaphorically2. Most n-grams starting
with “forest of” were about places or kinds of trees, as is to
be expected; still, we found the following metaphors used
on the web: (1) forest of spears, (2) forest of masts, and (3)
forest of marble columns. These three cases have a generic
space that is very similar to G: they all represent a multitude
of vertical stem-like objects. Some differences are:
while masts and columns are fixed, spears are not fixed to
the ground, but may be used in a context where they are vertical
and immobile stems, supporting a pointed tip; masts
2
Google’s 3-grams starting with “fo” are available at: http:
//storage.googleapis.com/books/ngrams/books/
googlebooks-eng-all-3gram-20120701-fo.gz
and columns support different kinds of objects, but all three
examples have generic spaces resembling G in Fig. 6.
What about counterexamples? We did not find “forest of
chairs” of course, and there were other metaphors on forest,
but they were based on different generic spaces and different
input spaces; we found these metaphors: “forest of X”,
where X could be opinions, possibilities, desires, words, human
experience. Clearly, these metaphors were not based on
the trees being elements of a “forest”, but on the human experience
of (walking in) the forest as a place of multiplicity
of paths, options, destinations. We think they are not counterexamples,
but rather examples of blends from different
input spaces.
Example 2: The Folding Toothbrush
Having given an example for the general model in the previous
subsection, we now want to also exemplify a concrete
implementation of the approach using HDTP as analProceedings
of the Sixth International Conference on Computational Creativity June 2015 155
Figure 7: Brillo, an example of a foldable toothbrush as produced
by Metaphys.
ogy framework. As application example, we will use the
blending-driven (re-)invention of foldable toothbrushes as,
for instance, the one depicted in Fig. 7.
Currently, when using HDTP, the required subsumption
relation between theories is given by logical semantic consequence
|=, i.e., A v A0 if A0 |= A for any two theories
A and A0
. In order to make sure that this relationship
is preserved by HDTP’s syntax-based operations, the range
of admissible substitutions for restricted higher-order antiunifications
has to be further constrained to only allow for
fixations and renamings.
Foldable toothbrushes are a conceptual combination between
a typical toothbrush and a folding mechanism like
that of pocketknives. In order to reconstruct the underlying
blending process, we start with the stereotypical characterizations
of a toothbrush and a pocketknife in a many-sorted
first-order logic representation from Table 1.
Sorts:
entity, part, functionality
Entities:
toothbrush, pocketknife : entity handle, brush head, blade, hinge : part
brush, cut, fold : functionality
Predicates:
has part : entity ⇥ part, has functionality : entity ⇥ functionality
Laws of the pocketknife characterization:
(↵1) has part(pocketknife, handle) (↵2) has part(pocketknife, blade)
(↵3) has functionality(pocketknife, cut) (↵4) has part(pocketknife, hinge)
(↵5) has functionality(pocketknife, fold)
Laws of the horse characterization:
(#1) has part(toothbrush, handle) (#2) has part(toothbrush, brush head)
(#3) has functionality(toothbrush, brush)
Table 1: Example formalizations of stereotypical characterizations
for a pocketknife S and a toothbrush T.
Given these characterizations, HDTP can be used for
finding a common generalization of both, for instance (due
to the syntactic similarities and the system’s heuristics)
aligning and generalizing ↵1 with #1, ↵2 with #2, and ↵3
with #3. Subsequently, reusing the same anti-unifications
(corresponding to !S), the source theory S is generalized
into S0 as given in Table 2: $1 corresponds to ↵1/#1, $2 to
↵2/#2, $3 to ↵3/#3, and $4 and $5 are obtained by generalizing
↵4 and ↵5, respectively.
Entities:
E : entity, P : part, F : functionality
Laws:
($1) has part(E, handle) ($2) has part(E, P )
($3) has functionality(E, F )
($4⇤) has part(E, hinge) ($5⇤) has functionality(E, fold)
Table 2: Abbreviated representation of the generalized
source theory S0 based on the stereotypical characterizations
for a toothbrush and a pocketknife (axioms not obtained
from the covered subset Sc are highlighted by *).
Computing the asymmetric amalgam of S0 with the
(fixed) target theory T, we obtain the proto-blend T0 from
Table 3. As T0 still features axioms containing noninstantiated
variables, !T is applied to the theory resulting in
the (with respect to !T ) fully instantiated blend theory TB
from Table 4, describing the concept of a hinge-equipped
toothbrush that can be folded.
Entities:
E : entity
Laws:
(%1) has part(toothbrush, handle) (%2) has part(toothbrush, brush head)
(%3) has functionality(toothbrush, brush)
(%4) has part(E, hinge) (%5) has functionality(E, fold)
Table 3: Abbreviated representation of the proto-blend T0
obtained from computing the asymmetric amalgam between
S0 and T.
Laws:
(%1) has part(toothbrush, handle) (%2) has part(toothbrush, brush head)
(%3) has functionality(toothbrush, brush)
(%4) has part(toothbrush, hinge) (%5) has functionality(toothbrush, fold)
Table 4: Abbreviated representation of TB = !T (T0
).
Conclusions
We presented a perspective on the blending of concept theories
building on generalization-based analogy and the amalgam
framework: Building upon analogy models of generalization
and domain matching, asymmetric amalgams allow
to provide a sound model for the controlled computation of
the concept blend(s) of two input theories.
Clearly, this is not the only attempt at developing a computational
model of (some facet of) concept blending: (Martinez
et al. 2014) present an algorithmic approach for blending
mathematical theories, (Kutz et al. 2015) give an account
Proceedings of the Sixth International Conference on Computational Creativity June 2015 156
of ontological blending, (Li et al. 2012) describe the goaland
context-sensitive blending-based production of creative
artifacts, and (Martinez et al. 2012) consider concept blending
in a human-level AI context. Still, in combining the
generality of generalization-based analogies and the amalgam
framework, COINVENT’s approach stands out as highlevel,
cognitively-inspired perspective on concept blending.
Acknowledgements
The authors acknowledge the financial support of the Future
and Emerging Technologies within the 7th Framework Programme
for Research of the European Commission, under
FET-Open grant 611553 (COINVENT).
<references_biblio/>
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