=Paper= {{Paper |id=Vol-1660/caos-paper4 |storemode=property |title=The Yoneda Path to the Buddhist Monk Blend |pdfUrl=https://ceur-ws.org/Vol-1660/caos-paper4.pdf |volume=Vol-1660 |authors=Marco Schorlemmer,Roberto Confalonieri,Enric Plaza |dblpUrl=https://dblp.org/rec/conf/fois/SchorlemmerCP16 }} ==The Yoneda Path to the Buddhist Monk Blend== https://ceur-ws.org/Vol-1660/caos-paper4.pdf

The Yoneda Path to
the Buddhist Monk Blend
Marco SCHORLEMMER1 Roberto CONFALONIERI and Enric PLAZA
Artificial Intelligence Research Institute, IIIA-CSIC
Bellaterra (Barcelona), Catalonia, Spain

Abstract. Mazzola et al. propose a metaphor to describe the process by means
of which an open question is solved in a creative way, likening it to the manner
by which, due to the Yoneda Lemma of category theory, the internal structure of
any object is completely characterised by a diagram of certain selected objects and
morphisms. Building upon this metaphor, we explore how the creativity underlying
conceptual blending could be characterised by diagrams of image schemas. We
further illustrate this by providing a formalisation of the Buddhist Monk riddle
and of the blend that solves it, together with a computational realisation that views
conceptual blending as an amalgam-based process of generalisation and colimit
computation.
Keywords. Conceptual blending, image schemas, Yoneda Lemma, Buddhist Monk
riddle.

1. Introduction

The theory of conceptual blending (aka conceptual integration) has been proposed as
a fundamental cognitive operation underlying much of everyday thought and language
[7,8]. It models the way by which several mental spaces—small conceptual packets con-
structed as we think and talk for purposes of local understanding and action—are blended
into a novel space by combining the particular elements and their relations pertaining to
the initial spaces into a new whole, which eventually is more than the sum of spaces.
Conceptual blending has been thoroughly used as an analytic tool for understanding the
origin of ideas and concepts a posteriori [27], but it has also been proposed as a basis for
computational models of creativity, using the theory to guide the design and implemen-
tation of algorithms for generating novel ideas and concepts [28,24,11,12,6].
According to Fauconnier and Turner’s model [7,8], a conceptual blend is determined
by a network of relationships between mental spaces that serve as input to the blend
and generic spaces that encode the common structure underlying the input spaces. The
prototypical network consists of two input spaces and one generic space, modelling the
cross-space mapping between shared structure in input spaces, but more complex blends
require more complex networks (see, e.g., the megablends reported in Chapter 8 of [8]).
Goguen has been the first to suggest a mathematical framework for conceptual
blending, by applying techniques from category theory. He proposed to model blending
1 Corresponding Author: Marco Schorlemmer, IIIA-CSIC, Campus UAB, 08193 Bellaterra (Barcelona),

Catalonia, Spain; E-mail: marco@iiia.csic.es.
as a certain kind of generalised colimit in preordered categories, with the generalised
pushout of two inputs and one generic space as the prototypical case of blending [9].
Bou et al. have built upon Goguen’s intuitions and proposed a uniform model, also based
on category theory [3], that includes the closely related notion of amalgam initially pro-
posed for reasoning about cases in Case-Based Reasoning [23]. This latter approach has
also been given a computational realisation based on Answer-Set Programming [6].
Still, an important issue of conceptual blending, if it is to be used as a generative
technique for concept invention and computational creativity, is that of determining the
mental spaces to be blended and the generic spaces that encode their common structure,
as well as the relationships between those spaces. Generic techniques such as structure
mapping and anti-unification have been proposed [12], and several examples of blending
have been re-created using these techniques [4,5]. In this paper, we explore an alternative
approach, one that is grounded on an embodied understanding of cognition, and we shall
pay attention to the image schemas that underlie the concept invention process. To this
aim, we take inspiration from a metaphor for the creative process proposed by Mazzola
et al. [20], and which draws from the insights of the Yoneda Lemma in category theory.

2. On the Yoneda-Based Creative Process

In order to better grasp the metaphor which sees the creative process in the light of the
Yoneda Lemma, we will need to first give a very brief overview of this lemma and its
implications, before moving on to present the metaphor. For this, some basic knowledge
of category theory is assumed. But it is not necessary to fully comprehend the meaning
of the Yoneda Lemma for the purpose of understanding this paper.

2.1. The Yoneda Lemma

The Yoneda Lemma is a technically simple result but with deep implications in category
theory. It implies that, for any category C (i.e., a collection of mathematical objects such
as sets, graphs, groups, etc., and its structure-preserving mappings), a C -object C can be
completely characterised by how it relates via structure-preserving mappings —called
morphisms— to all other objects in the category; That is, an object can be understood
either by understanding all the C -morphisms from C to any other C -object X, or by
understanding all the C -morphisms from any other C -object X to C. In addition, C -
morphisms from C to D can be completely characterised by focusing on how all mor-
phisms from and to C are transformed into morphisms from and to D.
A major task in category theory is to identify a subcategory A of C , such that, with
A -objects only, we have enough to characterise all C -objects. Of particular interest are
subcategories A for which every C -object C is (structurally isomorphic to) a colimit of
a diagram of A -objects and A -morphisms. (The colimit is the C -object that includes
all structure —and not more— present in the A -objects of the diagram, in a way that
it respects the relationships between A -objects as given by the A -morphisms of the
diagram.) Such a subcategory is called dense and the A -diagrams whose colimits are
(structurally isomorphic to) C -objects are said to be canonical. Canonical diagrams ex-
ternalise, as it were, the internal structure of an object, in the form of a diagram that only
uses objects of the dense subcategory. Let’s look at an example:
Source Domain Target Domain
C ATEGORY T HEORY C REATIVITY

To understand the object A → Exhibiting the open question
Category C → Identifying the semiotic context
A → Finding the question’s critical sign or
concept in the semiotic context
The uncontrolled behaviour of Hom(−, A) → Identifying the concept’s walls
Finding a dense subcategory A → Opening the walls
Find the canonical colimit C for A → Displaying its new perspectives
To understand A via the isomorphism C ∼=A → Evaluating the extended walls
Figure 1. The Yoneda-Based Creative Process Metaphor

Take the category Grf of (directed) graphs and graph homomorphisms. Let GV be the graph
consisting of a single vertex and no edges, and let GE be the graph consisting of two vertices
and a single edge from one vertex to the other.
For any graph G, the set Hom(GV , G) of graph homomorphism from GV to G is isomorphic
to the set of G’s vertices, and the set Hom(GE , G) of graph homomorphism from GE to G is
isomorphic to the set of G’s edges.
For the category Grf, the subcategory A consisting of the two graphs GV and GE and graph
homomorphisms s : GV → GE (which sends the unique vertex in GV to the vertex at the source
of the unique edge in GE ) and t : GV → GE (which sends the unique vertex in GV to the vertex
at the target of the unique edge in GE ) is dense, and each graph can be characterised as a
diagram with objects and morphisms from A .
Let, for instance, G be the graph

a / b / c
f g

The canonical diagram whose colimit is isomorphic to G has the following shape:

7 GE h 7 GE h
s t s t
GV GV GV

2.2. A Metaphor of Creativity

Mazzola et al. propose to take the insights offered by the Yoneda Lemma as a metaphor
for the process by which an open question may be solved in a creative way [20]. They
conjecture that, in the same manner in which a canonical diagram describes an object
by making its internal structure explicit, it may well be that by externalising the inner
workings of our open question we may gain new perspectives that yield a creative answer.
These new perspectives are like the morphisms of the colimit of a canonical diagram,
which project the constitutive elements of the diagram in a structure-preserving way onto
the object under study.
Each of the metaphorical mappings that constitute this metaphor of the creative pro-
cess, and which we illustrate in Figure 1 following the notational convention of [18], can
also be seen as concrete steps of the process that the metaphor seeks to conceptualise [1].
3. Image Schemas

In the last decades, researchers in cognitive science have advocated for the central role
that the embodied mind plays in thought and language. Their main claim is that concepts
are grounded on our bodily experience with the environment. We structure concepts and
reason with them according to basic skeletal patterns that recur in our sensory and motor
experience, called image schemas [15,17].
Quoting Johnson, “an image schema is a recurring dynamic pattern of our percep-
tual interaction and motor programs that gives coherence and structure to our experi-
ence.” [15]. Hampe provides the following characterisations: Image schemas are directly
meaningful (“experiential”/“embodied”), pre-conceptual structures, which arise from or
are grounded in human recurrent bodily movements through space, perceptual interac-
tions, and ways of manipulating objects; are highly schematic gestalts that capture the
structural contours of sensory-motor experience, integrating information from multiple
modalities; exist as continuous and analogue patterns beneath conscious awareness, prior
to and independently of other concepts; and are both internally structured and highly
flexible [13].

3.1. The PATH Image Schema

An example of image schema is PATH. This image schema —sometimes also referred
to as the S OURCE -PATH -G OAL schema— is presented in slightly different ways in the
literature, but in its simplest form it consists of: a source or starting point, a goal or
end-point, and a path or sequence of contiguous locations connecting the source with
the goal. As such, PATH is a specialisation of another image schema, called the L INK
schema, which consists of two entities and a link between them. Here, we have a source
and a goal location that are linked by a path.
Since the PATH schema arises from our bodily experience of moving about in space,
it often includes the notion of a trajector moving from source to goal, and the associ-
ated notion of this trajector “being on the path”. Furthermore, because our experience of
moving about in space is tightly linked with our perception of time, often the temporal
dimension is also included into the schema, so as to take into account that the trajector
starts at a certain time at the source position and ends at a later time at the goal position,
being on the path at any intermediate time instance.
The internal logic and built-in inferences of this schema allow us to state, for in-
stance, that if somebody has travelled from A to B and from B to C, then he or she has
travelled from A to C. We shall see a formalisation of this schema in Section 4.

3.2. Conceptual Blends of Image Schemas

Lakoff and Johnson identify several of these image schemas and show how they ground
the meaning we give to abstract concepts and situations on our bodily experience through
conceptual metaphors that project the structure of image schemas upon the new domains
of thought we create [18,19]. Following this view of how conceptualisations might work,
it is reasonable to conjecture that the internal structure of an abstract concept can even-
tually be described by mainly focusing on the image schemas upon which the concept
is grounded, and on the conceptual metaphors that project the structure of these image
schemas onto the concept under consideration. Situating this conjecture in the context of
our previous discussion on the creative process proposed by Mazzola et al., we shall ex-
plore the idea that conceptual blends can be metaphorically understood as colimits over
canonical diagrams of image schemas.

4. The Buddhist Monk Riddle

We conjecture that Mazzola et al.’s metaphor of the creative process might be useful as
a guide to find out which diagrams are relevant as a basis for conceptual blending. To
explore this conjecture we shall apply the steps of this metaphor to exhibit and solve the
riddle of the Buddhist Monk, that was firstly discussed by Koestler to illustrate creativity
[16]. It goes as follows:
One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. The narrow
path, no more than a foot or two wide, spiralled around the mountain to a glittering temple
at the summit. The monk ascended the path at varying rates of speed, stopping many times
along the way to rest and to eat the dried fruit he carried with him. He reached the temple
shortly before sunset. After several days of fasting and meditation, he began his journey back
along the same path, starting at sunrise and again walking at variable speeds with many pauses
along the way. His average speed descending was, of course, greater than his average climbing
speed. Prove that there is a single spot along the path the monk will occupy on both trips at
precisely the same time of day. [16, p. 183–184]
Koestler further states how he experienced his mental process of solving the riddle by
superimposing the image of the ascending and descending monk as two figures that must
meet at some point at some time [16, p. 184].
The Buddhist Monk riddle was later picked up by Fauconnier and Turner to describe
their theory of conceptual blending and its constitutive elements [8]. The superimposi-
tion that Koestler mentions can be seen as an example of conceptual blending, where the
mental image of the monk walking up the mountain is overlaid with the image of the
monk walking down the same mountain on the same path. The composition of those im-
ages is then completed and elaborated with our experience that two objects that approach
each other on the same path will necessarily meet at some point.

4.1. The Creative Process to Solve the Riddle

Our objective is to illustrate a step-by-step process by which to solve the riddle, by look-
ing it at as a creative process—as modelled by Mazzola et al.—using their metaphor
based on the Yoneda Lemma, together with the view of conceptual blends as amalgams,
i.e., as a process of diagram generalisation and colimit computation. Let us, hence, fol-
low the steps suggested by Mazzola et al. as they apply to the riddle of the Buddhist
Monk.
Exhibiting the open question: Our open question is, obviously, the riddle. Metaphor-
ically speaking, this would be the object A that we seek to understand, and that we
will need to formalise it in some way. We will use the Common Algebraic Specifi-
cation Language (CASL) [2], a general-purpose specification language based on first-
order logic, which is supported by the tool set HETS [22]. This will allow us to rep-
resent conceptual integration networks as diagrams of CASL theories and to compute
blends by colimit computation over these diagrams. Let us take, for instance, the B UD -
DHIST M ONK S IGNATURE specification for expressing the Buddhist Monk riddle as
spec B UDDHIST M ONK S IGNATURE =
then sorts Journey, Spot, MountainPath, Person,
Time, Day, TimeOfDay
ops ascent, descent : Journey;
foot, summit : Spot;
narrow path : MountainPath;
∃ s : Spot; t1 , t2 : Time
monk : Person;
• on(s, narrow path)
startAscent, endAscent : Time;
∧ occupies(ascent, monk, t1 ) = s
startDescent, endDescent : Time;
∧ occupies(descent, monk, t2 ) = s
day1, day2 : Day;
∧ timeOfDay(t1 ) = timeOfDay(t2 )
sunrise, sunset : TimeOfDay;
occupies : Journey × Person × Time
→ Spot
timeOfDay : Time → TimeOfDay
preds on : Spot × MountainPath
end

Figure 2. Signature and open question of the Buddhist Monk riddle.

shown in Figure 2 on the left. One way to formalise the open question of our riddle, given
this signature, is shown in Figure 2 on the right.
Identifying the semiotic context: Since we use CASL to express the open question, and
also the input and generic spaces relevant to solve the riddle, the category C we would
be dealing with is the category of CASL theories and CASL theory morphisms [21].
Finding the question’s critical sign or concept in the semiotic context. Identifying the
concept’s walls: We maintain that the critical sign or concept to solve the riddle is
the relationship between time intervals and paths, because the open question is about
positions on a path with respect to time instances as this path is traversed; so image
schemas for these concepts are going to play a relevant role in order to solve the riddle.
Following our metaphor, since we cannot work with the entire set Hom(−, A), we will
need to focus solely on the relevant image schemas.
Opening the walls: Our claim is that the image schemas could play the role, in this
metaphor, of the dense subcategory A as described in Section 2.1. We are not claiming
that image schemas form a dense subcategory for the category C of CASL theories
and CASL theory morphisms. This would be unfounded. But, still, we can draw some
insights from The Yoneda-Based Creative Process Metaphor, and we conjecture that, if
concepts—concrete and abstract—are eventually grounded on image schemas, then there
might exist a sufficiently interesting collection of CASL specifications which should be
fully described in terms of image schemas, in the same way categorical objects are fully
described by objects from a dense subcategory.
For the particular example of the Buddhist Monk riddle, Figure 3 shows a specifi-
cation of the image schemas that we found relevant. We made the following modelling
decision. For each image schema, there is a generic specification of the general structure
and logic of the schema (O BJECT, L INK and PATH on the left-hand side of Figure 3), and
also an additional specification extending the generic one that introduces constants for
the particular constitutive elements of the schema (AN O BJECT, A L INK and A PATH on
the right-hand side of Figure 3). This way we are able to have, in our diagrams, several
distinct occurrences of an image schema, whose constitutive elements are kept separate,
spec O BJECT = spec AN O BJECT = O BJECT
then sort Object then ops o : Object
end end

spec L INK = spec A L INK = L INK
then sorts Link, Entity then ops e1 , e2 : Entity
pred linked : Entity × Link l : Link
∀ l : Link • ∃!2 e : Entity • linked(e,l) • e1 6= e2
end • linked(e1 ,l)
• linked(e2 ,l)
end

spec PATH = L INK with Entity 7→ Location spec A PATH = PATH
and O BJECT with Object 7→ Trajector then ops p : Path
and I NSTANT s,g : Location
then sorts Path r : Link
ops source, goal : Path → Location t : Trajector
route : Path → Link b,e : Instant
trajector : Path → Trajector • source(p) = s
start, finish : Path → Instant • goal(p) = g
position : Path × Trajector × Instant → Location • route(p) = r
preds on : Location × Link • trajector(p) = t
∀p : Path • start(p) = b
• source(p) 6= goal(p) • finish(p) = e
• linked(source(p),route(p)) end
• linked(goal(p),route(p))
• position(p, trajector(p), start(p)) = source(p)
• position(p, trajector(p), finish(p)) = goal(p)
• ∀x : Instant
• (start(p) < x ∧ x < finish(p))
⇒ on(position(p,trajector(p),x),route(p))
end

Figure 3. CASL specifications of image schemas. Quantification ∃!2 in L INK stands for there are exactly two.

but which share the general structure and logic of the schema they are occurrences of.
Additionally, we will need some rudimentary theory of time and time intervals as shown
in Figure 4.
Displaying its new perspectives: Our objective is to describe the riddle, not as a stand-
alone CASL theory, but by means of a diagram of image schemas, externalising, as it
were, its internal structure. This externalisation is captured by a diagram—the canonical
diagram, metaphorically speaking. Figure 5 shows the image-schematic structure of the
riddle, with morphisms depicted in solid lines and specified in Figure 6. By computing
the colimit of this diagram we should obtain a complete description of the riddle. In par-
ticular, we are interested in the CASL theory that is isomorphic to the theory at the apex
of the colimit and that uses the signature B UDDHIST M ONK S IGNATURE of the open
question. This colimit is shown in Figure 5 with the morphisms depicted in dashed lines
and specified in Figure 7. These morphisms constitute the Yoneda perspectives of the
Buddhist Monk riddle. We can see the entire diagram as the conceptual integration net-
work that represents a conceptual blend of several input spaces linked by several generic
spaces in a bipartite graph.
spec I NSTANT = spec T IME =
then sorts Instant then sorts Time, Day, TimeOfDay
preds < : Instant × Instant preds < : TimeOfDay × TimeOfDay
[irrefl., trans., total] ops day : Time → Day
end timeOfDay : Time → TimeOfDay
end

spec AN I NTERVAL = I NSTANT spec A TOD I NTERVAL = T IME
then ops beginning, ending : Instant then ops beginTime, endTime : Time
• beginning < ending beginDay, endDay : Day
end beginTOD, endTOD : TimeOfDay
• beginTOD < endTOD
• day(beginTime) = beginDay
• timeOfDay(beginTime) = beginTOD
• day(endTime) = endDay
• timeOfDay(endTime) = endTOD
end

Figure 4. CASL specifications of a rudimentary theory of time instants and intervals

B UDDHIST M ONK
7 < c g

Ascent Interval Descent Interval

Ascent Descent

A TOD I NTERVAL
O R _ k
A PATH
3 A= PJ ATH A TOD I NTERVAL
O
g f2
f1 g

A L INK AN O BJECT

h h

AN I NTERVAL
k k

Figure 5. “Canonical” diagram (solid arrows) and colimit (dashed arrows) of the Buddhist Monk riddle

Evaluating the extended walls: The colimit theory computed given the diagram dis-
cussed so far does not help us to answer our open question. It does, however, make the
image-schematic structure of the problem explicit for us to explore a way to solve the
riddle. Here is where our intuition deviates slightly from Mazzola et al.’s, since our idea
is to realise this exploration using the technique of conceptual blending as amalgam com-
putation: to explore several generalisations of input and generic spaces (i.e., generalisa-
tion of the canonical diagram) until the colimit computation generates a blend that allows
us to answer the open question.
view f1 : A L INK to A PATH = view f2 : A L INK to A PATH =
l 7→ r, e1 7→ s, e2 7→ g l 7→ r, e1 7→ g, e2 7→ s

view g : AN O BJECT to A PATH = view h : AN I NTERVAL to A PATH =
o 7→ t beginning 7→ b, ending 7→ e

view k : AN I NTERVAL to A TOD I NTERVAL =
Instant 7→ TimeOfDay,
beginning 7→ beginTOD, ending 7→ endTOD

Figure 6. CASL signature morphisms of the “canonical” diagram

view Ascent : A PATH to B UDDHIST M ONK =
Path 7→ Journey, Link 7→ MountainPath, Trajector 7→ Person,
p 7→ ascent, s 7→ foot, g 7→ summit, r 7→ narrow path, t 7→ monk,
b 7→ sunrise, e 7→ sunset

view Ascent Interval : A TOD I NTERVAL to B UDDHIST M ONK =
beginTime 7→ startAscent, endTime 7→ endAscent,
beginDay 7→ day1, endDay 7→ day1, beginTOD 7→ sunrise, endTOD 7→ sunset

view Descent : A PATH to B UDDHIST M ONK =
Path 7→ Journey, Link 7→ MountainPath, Trajector 7→ Person,
p 7→ descent, s 7→ summit, g 7→ foot, r 7→ narrow path, t 7→ monk,
b 7→ sunrise, e 7→ sunset

view Descent Interval : A TOD I NTERVAL to B UDDHIST M ONK =
beginTime 7→ startDescent, endTime 7→ endDescent,
beginDay 7→ day2, endDay 7→ day2, beginTOD 7→ sunrise, endTOD 7→ sunset

Figure 7. Morphisms into the apex of the colimit

5. Solving the Riddle

In order to solve the riddle by “evaluating the extended walls” we will follow a general
process that consists of the repeated application of (i) diagram generalisation, (ii) colimit
computation, (iii) image-schematic completion and (iv) reasoning at a distance, until we
reach a solution. Steps (i) and (ii) constitute the amalgam-based technique for comput-
ing conceptual blends described in [6], and we will not go further into the details in this
paper. Step (iii) is how we imagine that blend completion could be realised. For the Bud-
dhist Monk riddle in particular, we think it is cognitively more plausible that comple-
tion will draw from our sensory-motor experience that we have coded in form of image
schemas than by including an abstract specification of the Intermediate Value Theorem,
as done by Goguen in [10]. The final step (iv) is the one that allows us to answer the
riddle in its original formulation, namely for one monk ascending and descending on
different days, using the blend that involves two monks journeying on the same day. Let
us see these steps in more detail for our example.
B UDDHIST M ONK0
< _

Ascent0 Descent0

A PATH
R _ 4 A PATH
N
f1 f2

A L INK

h
h

AN I NTERVAL

Figure 8. Generalised diagram (solid arrows) and colimit (dashed arrows) of the Buddhist Monk riddle

5.1. Diagram Generalisation

Two generalisations will be required to generate the right blend to be able to solve the
Buddhist Monk riddle. The first generalisation forgets the calendrical time of the journey
and focuses only on times of day as start and finishing times of the ascent and descent
journeys, thus generalising the two occurrences of A TOD I NTERVAL in our diagram
to AN I NTERVAL, along morphism k (see Figure 6). This amounts to remove these two
occurrences and those of morphism k. The second generalisation is to forget that the
trajector—the monk— of both journeys, the ascent and the descent, are the same entity,
by removing the AN O BJECT image schema and the occurrences of morphism g that
were responsible for the identification of trajectors in the apex of the colimit.

5.2. Colimit Computation

After these two generalisations our new colimit diagram looks as shown in Figure 8.
Again, we are interested in the CASL theory that is isomorphic to the theory at the
apex of the colimit and that uses the signature B UDDHIST M ONK S IGNATURE of the
open question. (To be precise, our signature will need two separate symbols for monks
standing for two separate trajectors.)
We claim that the colimit specification over this generalised diagram allows us to
solve the riddle, not immediately, but by completing it in the appropriate way. In [10]
it was conjectured that such completion could be realised by an appropriate “meeting
space” that specifies a suitable version of the Intermediate Value Theorem, and which is
drawn into the colimit (see also [26]). We think that such meeting space, as suggested
by Goguen, is too abstract a specification to play such direct role in solving the riddle.
Instead, we conjecture that the generated colimit is completed with image-schematic
structure of an extension of the PATH schema, namely the one that has two trajectors.
spec T WO T RAJECTORS PATH = PATH
then ∀p1 , p2 : Path; t1 , t2 : Time; ∀x, y : Location
• t1 6= t2 ∧
position(p1, trajector(p1), t1 ) = x ∧ position(p1, trajector(p1), t2 ) = y ∧
position(p2, trajector(p2), t1 ) = y ∧ position(p2, trajector(p2), t2 ) = x
⇒ ∃ s : Location; t : Time
• t1 < t ∧ t < t2 ∧
position(p1, trajector(p1), t) = s ∧ position(p2, trajector(p2), t) = s
end

spec A T WO T RAJECTORS PATH = T WO T RAJECTORS PATH
then ops p1 , p2 : Path
s1 , s2 , g1 , g2 : Location
r : Link
t1 , t2 : Tra jector
b, e : Time
• source(p1 ) = s1 • source(p2 ) = s2
• goal(p1 ) = g1 • goal(p2 ) = g2
• route(p1 ) = r • route(p2 ) = r
• tra jector(p1 ) = t1 • tra jector(p2 ) = t2
• start(p1 ) = b • start(p2 ) = b
• f inish(p1 ) = e • f inish(p2 ) = e
end

Figure 9. Extension of the PATH image schema with an additional trajector moving at the same time interval
on the same route.

5.3. Image-Schematic Completion

Following Hedblom, Kutz and Neuhaus [14], our repository of images schemas (playing
the role of dense subcategory in The Yoneda-Based Creative Process Metaphor) could in-
clude several variants of a same core image schema organised in a graph of theories, such
as that suggested for PATH. The blending completion process would consist of detecting
image-schema variants that could be partially mapped into the newly created space at
the apex of the colimit diagram of Figure 8. In this case A T WO T RAJECTORS PATH
(see Figure 9) can be partially mapped onto B UDDHIST M ONK’. The specification of
this image schema includes an additional axiom specifying the inherent logic of this ex-
tension; and a particular occurrence of this schema (A T WO T RAJECTORS PATH) con-
sists of two source-path-goal structures (with their respective source, target, and trajec-
tor) that share the route and the time interval in which trajectors move. By bringing into
B UDDHIST M ONK’ the remaining structure and logic of A T WO T RAJECTORS PATH
we will be able to deduce the open question from the resulting specification; not directly,
but by moving the reasoning along the morphism induced by the generalisation of the
“canonical” diagram, as we will see next.

5.4. Reasoning at a Distance

A diagram generalisation as the one described for the Buddhist Monk riddle is formally
captured by a generalisation morphisms between diagrams. So, if diagram D is gener-
alised to D 0 , there will be a morphism (a monomorphism to be precise) from D 0 to D.
This generalisation morphism induces a morphism f from the apex of the colimit of D’
to the apex of the colimit of D. This morphism allows us to use the theory at the apex
of the colimit of the generalised diagram to reason with the entities of the apex of the
colimit of the original diagram. This is done as follows.
Let D be a diagram of CASL theories and let D’ a generalised diagram. Let B and
B0 be the CASL theories at the apexes of the colimits of D and D 0 , respectively, and let
f : B0 → B be the morphisms induced by the diagram generalisation. Let ϕ be a sentence
in the signature of B. We will say that B ` f ϕ when there exists a sentence ϕ 0 , such that
ϕ = f (ϕ 0 ) and B0 ` ϕ 0 .
For our Buddhist Monk riddle, let D be the base diagram of Figure 5, and D 0 that
of Figure 8. The morphisms f : B UDDHIST M ONK0 → B UDDHIST M ONK will be the
identity on all signature elements, except for the following cases:

f (monk1 ) = monk f (t1 ) = timeO f Day(t1 )
f (monk2 ) = monk f (t2 ) = timeO f Day(t2 )

Let ϕ be the open question as formalised in the step Exhibiting the open question
given above. In order to see if this open question holds, we will need to check if there
exists a ϕ 0 in the signature of B UDDHIST M ONK0 that holds for B UDDHIST M ONK0 .
Indeed such sentence exists, and it is:

ϕ 0 = ∃ s : Spot; t1 , t2 : Time
• on(s, narrow path)
∧ occupies(ascent, monk1 , t1 ) = s ∧ occupies(descent, monk2 , t2 ) = s ∧ t1 = t2

6. Implementation

The HEterogeneous ToolSet (HETS) [22] is a reasoning engine that supports the col-
imit computation of CASL theories and is capable of translating CASL to various in-
put languages understood by theorem provers. The HETS GUI allows us to graph-
ically represent the entire theory modelling the riddle (see Figure 8). The diagram
shows that after computing the colimit B UDDHIST M ONK’, this is completed with the
A T WO T RAJECTORS PATH instantiated schema and further elaborated by bringing in
the open question, which is then proven to follow from B UDDHIST M ONK’.
In the current implementation, blend completion is realised as the union (‘and’ in
CASL) of the A T WO T RAJECTORS PATH and B UDDHIST M ONK’ theories. It is wor-
thy noticing that blend completion can also be seen as a process of conceptual blend-
ing, by which a generated blend is further blended with the image-schematic structure
it is completed with. In Figure 8, for instance, A T WO T RAJECTORS PATH could be
blended with B UDDHIST M ONK’ to obtain B UDDHIST M ONK C OMPLETION theory.
This will amount to create another base diagram, whose inputs are B UDDHIST M ONK’
and A T WO T RAJECTORS PATH. This is left as future work. The current implementation
is available at https://ontohub.org/yoneda-path/monk-gen.casl.
Figure 10. Generalised diagram and proof of the open question displayed in the HETS GUI.

7. Conclusion

We claimed that Mazzola et al.’s metaphor for the creative process can be useful to make
explicit the external structure of the concept or idea we want to creatively explore. This
metaphor likens the creative process to the task of finding a canonical diagram that ex-
ternalises the structure of a categorical object. Besides Mazzola et al.’s model, there are
other proposals for modeling cerativity in general, and concept invention in particular,
such as [25], and it is worthwile to consider these as well. In this paper, however, we have
explored the idea of applying Mazzola et al.’s metaphor to solve the Buddhist Monk rid-
dle, by making explicit the image-schematic structure underlying the conceptual blends
required to solve the riddle. Moreover, we have modelled the blend completion process
in terms of incorporating additional relevant image-schematic structure that was initially
not present in the problem specification, but which became relevant once the concep-
tual blends were realised. In our approach, the blending process itself is modelled as a
amalgam-based process of generalisation and colimit computation.
As future work, we intend to further explore our approach in other domains, vali-
dating the hypothesis that a relevant collection of image schemas should be sufficient to
model diagrams such that, via generalisation and colimit computation, yield the expected
blends. Moreover, we surmise that for complex situations we will have not a blend but
a web of blends, e.g., situations where one (or both) input mental spaces are recursively
blended. Such a web of blends is called Hyper-Blending Web in [27]. We intend to ex-
plore the span of the hypothesis that the input concepts in such a web of blends are image
schemas and their specialisations, while the blend concepts are created by generalisation
and colimit computation of image schemas and previous blends in the web.
Acknowledgements. This work is supported by project COINVENT, which acknowl-
edges the financial support of the Future and Emerging Technologies (FET) programme
within the Seventh Framework Programme for Research of the European Commission,
under FET-Open Grant number: 611553.
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