# A Mathematical Framework for Interpreting Playing Environments as Media for Information Flow

Authors: Tatsuo Motoyoshi; Takashi Hattori; Hiroshi Kawakami; Takayuki Shiose; Osamu Katai
Journal: Advances in Human-Computer Interaction
 (2008)

---

## Abstract

This paper proposes a novel strategy of designing play equipments.
The strategy introduces two loose constraints as a guideline for
designers. The first constraint is “describing unit of play action chain_〈O,S,V〉 based on Barthes_ semiology, and the second is the infomorphism
between designer, play equipment, and players based on channel
theory. We provide detailed explanation of the strategy through an example
of a designing process of playing environment where the players
usage of the play equipment cannot be foreseen.


---

## Body

## 1. Introduction

When  playing computer games, users sometimes tend to
add their own original rules to make games more exciting; thus the games (or
more generally speaking the play equipment) do not have a unique “right way”
to be played but have the potential to be played in various ways. This
potential is usually interpreted as a problem to be avoided in the field of
conventional system design because it may lead to unexpected danger. In the
field of designing play equipment, however, the potential is an important
element for producing entertainment.It is needless to say that designers of conventional
systems, to, often assume users' various ways of using a system and handle them
one by one. In designing play equipment in which the events caused by the equipment
are prepared by the designers (e.g., computer games), a variety of decision
branches are considered an essential element for making the play equipment
“entertaining”; the knowledge regarding adequate size of the decision
branches and how to create them is organized as gamenics [1].On the other hand, users of such play equipment are
not always satisfied with the given decision branches, and sometimes create a
new and unpredicted way of playing. In this paper, we focus on this point and
target outdoor play equipment that is often seen at playgrounds as equipment
for which it is difficult to assume players' intentions.We expect to gain broad knowledge that also covers
“event-foreseeable play equipment” through analysis of such play equipment,
for example, factors that cause entertainment or guiding principles of how to
design “enjoyable-to-use tools”. Unlike gamenics, this knowledge is not
organized at present and is a challenging task to tackle.It is impossible to design these types of targets by
combining rigidly pre-constructed building blocks; “correspondence between
playing actions and pleasure” and “correspondence between play equipment's
characteristics and pleasure” are both context-dependent. Furthermore, we must
not assume that the context can be fully clarified because we aim to construct
a framework that can tolerate unpredictable ways of playing. This paper
therefore first focuses on “correspondence between playing actions and
pleasure” in Section2 and proposes “describing
units of the play action chain”; this unit introduces play equipment's
components and its characteristics as a context that (not fully but to some
extent) regulates the correspondence. This unit functions as a first “loose”
constraint. Though play actions and pleasure cannot be uniquely correlated,
allowing spontaneous description of the correspondence will lead to a denial of
any systematic operation. The adoption of the new unit is namely an adoption of
an intermediate (loose) level of constraint; we fix only the description format
and vary the implication of the description itself, case by case.  Next in Section 3, by using the format introduced in Section
2, we adopt  a new strategy for designing targets whose influence cannot be
foreseen. The main idea of the strategy is to individually describe the part
that can be locally decided and describe what cannot be locally decided as a correspondence
between local description. That is to say, give a correspondence between the 3
factors: assumption of the designer, assumption about the user, and an event
that can be generally assumed to happen in a physical environment. Here, we
give a second loose constraint. Though bijection, the most rigid
correspondence, is too powerful, allowing spontaneous correspondence is
meaningless. On loosening the powerful bijection constraint, we can adopt
various strategies such as surjection and injection, but in this paper we adopt
“infomorphism.” Infomorphism is essentially a projection that is consolidated
to formulate qualitative information flow, and formation of an infomorphism is
assumed as formation of information flow.


## 2. Analysis of Perception-Action Chain Structure

In a playing
environment, pleasures prompt new playing actions that in turn generate new
pleasures. These perceptions of pleasures and playing actions are coupled
dynamically. In other words, the playing environment involves many “perception-action
chains” in terms of ecological psychology [2]. Vygotsky advocated in his psychological studies
that the primitive unit of linguistic thought is not a single word or a
syllable but “word meaning,” which has both properties of language and
thought [3]. These
properties are coupled dynamically, and variations of these properties generate
new “word meanings.” Interpreting this notion of linguistic thought to
playing environment, a unit of playing actions should have both properties of
actions and environment, and should be able to describe relations between play
equipment and players in order to grasp the generations of pleasures in the
playing environment as shown in Figure 1. Furthermore, variations of playing actions or environment
perceptions generating new pleasures correspond with the characteristic of the
unit of linguistic thought. We describe analysis of the perception-action chain
for outdoor play equipment in the next section, using one of four types of
pleasure actions: “imitative action,” “resting action,” “vertiginous
action,” “challenging action,” which are observed in using outdoor play
equipment [4].Figure 1
A unit
of linguistic thought and playing action.


### 2.1. Perception-Action Chain Analysis
Using Therblig

To describe
processes of generation of pleasures from playing actions, we separate playing
action using the “therblig” as a unit that is beyond an analyst's
subjectivity. Therblig analysis [5] is the notation evolved from the observation of human
manual operation and is made up of 17 types of basic motions called
“therbligs.” For example, describing two different actions “climbing” and
“swinging” observed on the rope of outdoor play equipment using therbligs is
performed as shown in Table 1.Table 1ActionsTherbligClimbingTransport empty→ grasp → transport loaded → useSwingingTransport empty→ grasp → transport loaded → useThis analysis gives the same result. This means that
using the therblig cannot derive various pleasures from different playing
actions.Therblig has much less basic motions for analysis for
playing actions, so we cannot describe all of the playing actions using
therblig. Additionally, therblig does not have properties of environment perception.
Generally each playing action does not correspond to a unique “pleasure,” so
we cannot derive pleasures generated in the relations between play equipment
and playing environment even though we can assign one of the therbligs to each
action. Furthermore, this method cannot allow us to describe generation of new
pleasures from linked actions. Thus, the therblig, which is one of the
reductionism units, is not suited for analysis of playing action. The “PTS”
has more basic motions [6], but it is not suited for them in the same way.


### 2.2. A New Unit〈O,S,V〉 Based on Barthes' Semiology

As noted in
Section2, reductionism units such as the therblig or PTS are not suited for
analysis of playing action. Additionally, a unit of playing actions should have
both properties of actions and environment and represents the “pleasure of a
combination of actions” that is not just a integration of its constituent
actions. In this section, we propose a new unit of playing action based on
Barthes' semiology, “the fashion system,” which has a syntax structure that
is suited for representing linked playing actions.The Fashion System
Barthes
proposed “a unit for describing fashion [7],” which is defined as an association of three
elements: object  (O),
support  (S),
and variant  (V) as follows:
S:
the
garment's technical existence. Support of signification (e.g., the collar),V:
the garment's
signifying existence. The point of the matrix from which signification emerges
(e.g., open/closed),O:
the material
element of the garment. Object of the signification. (e.g., the cardigan).
For example,
analysis of “a sporty cardigan” is as follows:(1)S(collar)+V(open)+O(cardigan)≡sporty.A New Unit〈O,S,V〉
We propose “a
unit of playing-actions〈O,S,V〉” based on the fashion system in Barthes'
semiology [7] as
follows:
S:
a technical
factor that determines a position of a part of play equipment (e.g., dropped
down vertically),V:
a playing
action that is carried out in a part of play equipment (e.g., climbing),O:
a device to
be a part of play equipment (e.g., a rope).
For example, describing two different actions
“climbing” and “swinging” observed on the rope of outdoor play equipment is
as follows, using one of four types of pleasure actions: “imitative action”,
“resting action”, “vertiginous action”, “challenging action”, which are
observed in using outdoor play equipment [4]: (2)S1(droppeddownvertically)+V1(climbing)+O1(rope)≡pleasureofChallengingAction,S1(droppeddownvertically)+V2(swinging)+O1(rope)≡pleasureofVertiginousAction.
These units correspond to various pleasures. In
particular, we can represent new actions and new pleasures involved in the same
device. The substance of pleasures depends on choices ofV (actions) in the same parts; also choices of V depend on S and O.
Players recognize V and S as characteristic of O.
These operations correspond to that of variant V in “the fashion system,” in that V has the right of determination of fashion mode
and becomes characteristic of O with S.


#### 2.2.1. Binding of Units

In “the
fashion system”, multiple units link together and generate a new unit and
describe fashion mode as follows:(3)〈whitebraidandwhitebuttons〉〈S1V1O1〉〈S2V2O2〉〈S0V0S0′〉S.S0 represents 〈O1,S1,V1〉.
Similarly, S0′ represents 〈O2,S2,V2〉.
Each meaning of units is transmitted to the final unit. We can apply this
representation to the playing action as follows: (4)droppeddownverticalclimbingrope〈S1V1O1〉droptheropedownswingingbarS2V2O2.〈O1,S1,V1〉,
which is represented by S2,
generates “challenging action” pleasure. S2 makes a new unit with O2 and V2 and generates a new pleasure of “vertiginous
action.” This means that we can grasp the new pleasure on a larger scale when
playing actions that are linked together, which is not described in the method
using therbligs. We can adopt this method in analyzing the playing action in
connected multiple parts of play equipment.


## 2.1. Perception-Action Chain Analysis
Using Therblig

To describe
processes of generation of pleasures from playing actions, we separate playing
action using the “therblig” as a unit that is beyond an analyst's
subjectivity. Therblig analysis [5] is the notation evolved from the observation of human
manual operation and is made up of 17 types of basic motions called
“therbligs.” For example, describing two different actions “climbing” and
“swinging” observed on the rope of outdoor play equipment using therbligs is
performed as shown in Table 1.Table 1ActionsTherbligClimbingTransport empty→ grasp → transport loaded → useSwingingTransport empty→ grasp → transport loaded → useThis analysis gives the same result. This means that
using the therblig cannot derive various pleasures from different playing
actions.Therblig has much less basic motions for analysis for
playing actions, so we cannot describe all of the playing actions using
therblig. Additionally, therblig does not have properties of environment perception.
Generally each playing action does not correspond to a unique “pleasure,” so
we cannot derive pleasures generated in the relations between play equipment
and playing environment even though we can assign one of the therbligs to each
action. Furthermore, this method cannot allow us to describe generation of new
pleasures from linked actions. Thus, the therblig, which is one of the
reductionism units, is not suited for analysis of playing action. The “PTS”
has more basic motions [6], but it is not suited for them in the same way.


## 2.2. A New Unit〈O,S,V〉 Based on Barthes' Semiology

As noted in
Section2, reductionism units such as the therblig or PTS are not suited for
analysis of playing action. Additionally, a unit of playing actions should have
both properties of actions and environment and represents the “pleasure of a
combination of actions” that is not just a integration of its constituent
actions. In this section, we propose a new unit of playing action based on
Barthes' semiology, “the fashion system,” which has a syntax structure that
is suited for representing linked playing actions.The Fashion System
Barthes
proposed “a unit for describing fashion [7],” which is defined as an association of three
elements: object  (O),
support  (S),
and variant  (V) as follows:
S:
the
garment's technical existence. Support of signification (e.g., the collar),V:
the garment's
signifying existence. The point of the matrix from which signification emerges
(e.g., open/closed),O:
the material
element of the garment. Object of the signification. (e.g., the cardigan).
For example,
analysis of “a sporty cardigan” is as follows:(1)S(collar)+V(open)+O(cardigan)≡sporty.A New Unit〈O,S,V〉
We propose “a
unit of playing-actions〈O,S,V〉” based on the fashion system in Barthes'
semiology [7] as
follows:
S:
a technical
factor that determines a position of a part of play equipment (e.g., dropped
down vertically),V:
a playing
action that is carried out in a part of play equipment (e.g., climbing),O:
a device to
be a part of play equipment (e.g., a rope).
For example, describing two different actions
“climbing” and “swinging” observed on the rope of outdoor play equipment is
as follows, using one of four types of pleasure actions: “imitative action”,
“resting action”, “vertiginous action”, “challenging action”, which are
observed in using outdoor play equipment [4]: (2)S1(droppeddownvertically)+V1(climbing)+O1(rope)≡pleasureofChallengingAction,S1(droppeddownvertically)+V2(swinging)+O1(rope)≡pleasureofVertiginousAction.
These units correspond to various pleasures. In
particular, we can represent new actions and new pleasures involved in the same
device. The substance of pleasures depends on choices ofV (actions) in the same parts; also choices of V depend on S and O.
Players recognize V and S as characteristic of O.
These operations correspond to that of variant V in “the fashion system,” in that V has the right of determination of fashion mode
and becomes characteristic of O with S.


### 2.2.1. Binding of Units

In “the
fashion system”, multiple units link together and generate a new unit and
describe fashion mode as follows:(3)〈whitebraidandwhitebuttons〉〈S1V1O1〉〈S2V2O2〉〈S0V0S0′〉S.S0 represents 〈O1,S1,V1〉.
Similarly, S0′ represents 〈O2,S2,V2〉.
Each meaning of units is transmitted to the final unit. We can apply this
representation to the playing action as follows: (4)droppeddownverticalclimbingrope〈S1V1O1〉droptheropedownswingingbarS2V2O2.〈O1,S1,V1〉,
which is represented by S2,
generates “challenging action” pleasure. S2 makes a new unit with O2 and V2 and generates a new pleasure of “vertiginous
action.” This means that we can grasp the new pleasure on a larger scale when
playing actions that are linked together, which is not described in the method
using therbligs. We can adopt this method in analyzing the playing action in
connected multiple parts of play equipment.


## 2.2.1. Binding of Units

In “the
fashion system”, multiple units link together and generate a new unit and
describe fashion mode as follows:(3)〈whitebraidandwhitebuttons〉〈S1V1O1〉〈S2V2O2〉〈S0V0S0′〉S.S0 represents 〈O1,S1,V1〉.
Similarly, S0′ represents 〈O2,S2,V2〉.
Each meaning of units is transmitted to the final unit. We can apply this
representation to the playing action as follows: (4)droppeddownverticalclimbingrope〈S1V1O1〉droptheropedownswingingbarS2V2O2.〈O1,S1,V1〉,
which is represented by S2,
generates “challenging action” pleasure. S2 makes a new unit with O2 and V2 and generates a new pleasure of “vertiginous
action.” This means that we can grasp the new pleasure on a larger scale when
playing actions that are linked together, which is not described in the method
using therbligs. We can adopt this method in analyzing the playing action in
connected multiple parts of play equipment.


## 3. Information Channels in Playing Environment

We discussed
the syntax structure of units of playing actions. In a playing environment,
there are information flows involving generation processes of new playing
actions that are observed on play equipment based on players' interpretations.
We propose a mathematical framework for qualitative information flows of
designing processes of the playing environment.In the conventional way of designing play equipment, a
designer recollects some devices (e.g., ladder) associated with a specific
action (e.g., climb) and integrates the device as a part of play equipment. But
in such a case, we cannot capture the emergence of new actions in the playing
environment. We cannot adopt the strategy of deriving an action correlated to a
combination of devices by consolidating actions correlated to its constituent
devices. In this chapter, we interpret that individual descriptions of the part
are not devices but a correspondence between three factors: assumption of the
designer about “character of play equipment and pleasures,” assumption of
the user about “character of play equipment and actions afforded from," and
equipment and actions that can be generally assumed to happen in a physical environment.
Then, we adopt a strategy that can be locally decided on and describe what
cannot be locally decided on as a correspondence between local descriptions.Bijection, the most rigid correspondence, is too
powerful, and allowing spontaneous correspondence is meaningless. On loosening
the powerful bijection constraint, we can adopt various strategies such as
surjection and injection, but in this paper we adopt “infomorphism” in
channel theory [8, 9]. Infomorphism is essentially
a projection that is consolidated to formulate qualitative information flow,
and formation of an infomorphism is assumed as formation of information flow.
So, we interpret the playing environment as a new medium that mediates play
equipment and players, by focusing on interactions between artifacts and users,
and define a model for representing designers—play equipment—players as
information channels based on channel theory [8, 9]. Channel theory is the
qualitative theory of information advocated by Barwise and Seligman.Channel theory defines a classification(D) such as 〈typ(D),tok(D),⊨D〉.
Given a set of tokens: tok(D) and a set of types: typ(D),a⊨Dα means that the type of a(∈tok(D)) is α(∈typ(D)).
A unit of playing action 〈O,S,V〉 has both properties of action 〈V〉 and environment 〈O,S〉.
It is appropriate to separate both properties to typ and tok in the classification of Channel theory. The
separation of both properties can be done in an intensional or extensional way.
In this paper, we adopt an extensional separation, in which recitation of
properties of environments defines properties of actions that are attributes of
playing environments. For playing environment, we define three classifications
as follows.Classification of Designers Assumption:Dds
tok(Dds):  a set of
“pleasures Ti”.
typ(Dds): a set of
“features of playing environment Ei.”
Ti⊨DdsEi: Pleasure Ti is observed in environment Ei.Classification of Equipment:Dpl
tok(Dpl): a set of
“playing actions Vi.”
typ(Dpl): a set of
“parts of play equipment Oi.”
Vi⊨DplOi: a part
of play equipment Oi enables an action Vi.Classification of Users:Dch
tok(Dch): a set of
“playing actions of players Vi′.”
typ(Dch): a set of
“combinations of parts and their feature SOi(=Si+Oi).”
Vi′⊨DchSOi: SOi enables an action Vi′.In the
classification of users(Dch), types SOi are defined by combining Si in terms of units of playing actions 〈O,S,V〉 with parts of play equipment Oi of Dpl. Si affects Oi in its ability to bring about some new play
actions Vi.The three classifications enable us to interpret
designing play equipment as an establishment of aninfomorphism in terms of channel theory.


### 3.1. Conventional Design of Play Equipment

Aninfomorphismf:A⇄B from A to B is defined as a contravariant pair of
functions f=〈f∧,f∨〉 in channel theory. The function f∨ is a map from tok(B) to tok(A), and f∧ is a map from typ(A) to typ(B).
It satisfies the following property: (5)f∨(b)⊨Aα⟺b⊨Bf∧(α), for each token b∈tok(B) and each type α∈typ(A).
Figure 2 shows an infomorphismf:A⇄B from A to B.Figure 2
An infomorphism from classificationA to B.In the case of playing environment, the establishment
ofinfomorphism from Dds to Dpl can be seen as an accomplishment of designing
play equipment in a conventional way. Namely, “a pleasure Ti in an environment Ei(=f∨(Oi))” in the mental world of a designer is
implemented in the physical world as the action Vi(=f∧(Ti)) by using equipment Oi.
Figure 3 illustrates an image of establishment of an informorphism. This method
describes only actions corresponding to parts of play equipment based on the
designer's logic, and no actions generated by players' intentions. So,
designers cannot foresee the emergence of new playing actions and pleasures.Figure 3
Conventional design process modeled by an
infomorphism.


### 3.2. Novel Design of Play Equipment

Channel theory
defines achannel as a set of infomorphisms sharing the same codomain.
In the case of a playing environment, establishment of an information channel between Dds and Dch via Dpl can be interpreted as an accomplishment of a
novel design from the viewpoint of interpreting play environment as a medium of
designers, play equipment, and children. Figure 4 illustrates an image of
establishment of a channel.Figure 4
Novel design process
modeled by a single channel.


### 3.2.1. A Concrete Example of Using Chu Map

ClassificationDds: first, a
designer sets features of playing environments Ei and pleasures Ti in classification Dds as follows. In this classification,
correspondence relations of Ei and Ti are trivial, like T1⊨DdsE2, T2⊨DdsE1, T2⊨DdsE4, T3⊨DdsE3,
and T3⊨DdsE4.typ(Dds)
–
E1: environment
that contains pleasure T2.–
E2: environment
that contains pleasure T1.–
E3: environment
that contains pleasure T3.–
E4: environment
that contains pleasures T2 and T3.tok(Dds)
–
T1: pleasure
of resting action.–
T2: pleasure
of challenging action.–
T3: pleasure
of vertiginous action.ClassificationDpl: second, a
designer sets parts of play equipment Oi and playing actions Vi in classification Dpl as follows. To establish an information channel between Dds and Dch via Dpl, designers must set an informorphism Dds to Dpl.
In this case, a designer sets infomorphisms f∧(E1)=O1, f∧(E2)=O2, f∧(E3)=O3, f∧(E4)=O4,
and f∨(V1)=T2, f∨(V2)=T1, f∨(V3)=T3.typ(Dpl)
–
O1: a
ladder.–
O2: a
horizontal plate.–
O3: a
slope.–
O4: a
bar.tok(Dpl)
–
V1: climb
vertically.–
V2: walk.–
V3: slide
down.ClassificationDch: finally, a
designer creates units of playing actions 〈O,S,V〉,
and sets SOi,
which are defined by combining Si in terms of 〈O,S,V〉 with parts of play equipment Oi and Vi′,
which contain new playing actions emerged in the play environment in
classification Dch as follows. In this operation, designers must
set Si and Vi′ to establish an information channel between Dds and Dch via Dpl.
Figure 5 shows a slide derived from the design model using channel theory.
Figure 6 shows an example of a channel represented by Chu maps [10].Figure 5
A slide derived from the design model using channel theory.Figure 6
A description of a “playing structure
design” by Chu maps.typ(Dch)
–
SO1: a
ladder that is put to SO2 vertically.–
SO2: a
covered horizontal plate.–
SO3: a
steep slope that is put to SO2.–
SO4: a
bar dropped down from SO2.tok(Dch)
–
V1′: climb
vertically.–
V2′: walk.–
V3′: slide
down.–
V4′: walk
into.–
V5′: walk
outside.–
V6′: jump.–
V7′: scale
up.


### 3.3. New Playing Actions Described in a
Model of Information Channel

In the novel
design, there are several alternatives for not only segmentingOi but also Si and Vi.
Therefore, designers can foresee the emergence of new playing actions and
pleasures.At Boundary Between Multiple Parts
Designers have
to select multiple elements when they select units at the boundary between
multiple parts of play equipment. They become loose constraints that guide
design processes. For example, settingS2:
“covered” generates new playing actions, V4′:
“walk in," and V5′:
“walk out”:
(i)
a playing actionV4′ at the boundary between SO1, SO3, SO4, and SO2: (6)SO1SO2SO3SO4V4′1011,(ii)
a playing actionV5′ at the boundary between SO2, SO1, SO3,
and SO4: (7)SO1SO2SO3SO4V5′0100.At One Part
In the novel
design of a play equipment, settingSi makes arrangement Oi.
Therefore, designers can set more playing actions in the same device, but
selecting elements at one part of the play equipment depends on the designer's
ability largely. For example, setting SO1 and SO4 generates new playing actions V6′:
“jump.” And setting SO3 generates V7′:
“scale up”:
(i)
a playing actionV6′ in SO1 and SO4: (8)SO1SO2SO3SO4V6′1001,(ii)
a playing actionV7′ in SO3: (9)SO1SO2SO3SO4V7′0010.


#### 3.3.1. Constraint in Classification

Channel theory
defines a constraint such asΓ⊢AΔ.
Given a classification A,
a token a∈tok(A) satisfies a sequent 〈Γ,Δ〉 of typ(A) provided that if a is of every type in Γ,
then it is of some type Δ.
A constraint is a sequent 〈Γ,Δ〉 for which, Γ⊢AΔ.
The following are part of constraints in classifications Dds, Dpl, Dch:(1)
constraint ofDds(10)E1⊢E4,E3⊢E4,(2)
constraint ofDpl(11)O1⊢O4,O3⊢O4,(3)
constraint ofDch(12)SO1⊢SO4. Constraints E1⊢E4, E3⊢E4 in Dds are directly translated in Dpl, but not in Dch.
This means that a designer's inference is described in Dds and Dpl,
because of Dpl being a step in which designers correspond
parts of play equipment to playing actions, but which is not translated in Dch.
In other words, generating V7′ in tok(Dch) disturbs establishment of SO3⊢SO4.


## 3.1. Conventional Design of Play Equipment

Aninfomorphismf:A⇄B from A to B is defined as a contravariant pair of
functions f=〈f∧,f∨〉 in channel theory. The function f∨ is a map from tok(B) to tok(A), and f∧ is a map from typ(A) to typ(B).
It satisfies the following property: (5)f∨(b)⊨Aα⟺b⊨Bf∧(α), for each token b∈tok(B) and each type α∈typ(A).
Figure 2 shows an infomorphismf:A⇄B from A to B.Figure 2
An infomorphism from classificationA to B.In the case of playing environment, the establishment
ofinfomorphism from Dds to Dpl can be seen as an accomplishment of designing
play equipment in a conventional way. Namely, “a pleasure Ti in an environment Ei(=f∨(Oi))” in the mental world of a designer is
implemented in the physical world as the action Vi(=f∧(Ti)) by using equipment Oi.
Figure 3 illustrates an image of establishment of an informorphism. This method
describes only actions corresponding to parts of play equipment based on the
designer's logic, and no actions generated by players' intentions. So,
designers cannot foresee the emergence of new playing actions and pleasures.Figure 3
Conventional design process modeled by an
infomorphism.


## 3.2. Novel Design of Play Equipment

Channel theory
defines achannel as a set of infomorphisms sharing the same codomain.
In the case of a playing environment, establishment of an information channel between Dds and Dch via Dpl can be interpreted as an accomplishment of a
novel design from the viewpoint of interpreting play environment as a medium of
designers, play equipment, and children. Figure 4 illustrates an image of
establishment of a channel.Figure 4
Novel design process
modeled by a single channel.


## 3.2.1. A Concrete Example of Using Chu Map

ClassificationDds: first, a
designer sets features of playing environments Ei and pleasures Ti in classification Dds as follows. In this classification,
correspondence relations of Ei and Ti are trivial, like T1⊨DdsE2, T2⊨DdsE1, T2⊨DdsE4, T3⊨DdsE3,
and T3⊨DdsE4.typ(Dds)
–
E1: environment
that contains pleasure T2.–
E2: environment
that contains pleasure T1.–
E3: environment
that contains pleasure T3.–
E4: environment
that contains pleasures T2 and T3.tok(Dds)
–
T1: pleasure
of resting action.–
T2: pleasure
of challenging action.–
T3: pleasure
of vertiginous action.ClassificationDpl: second, a
designer sets parts of play equipment Oi and playing actions Vi in classification Dpl as follows. To establish an information channel between Dds and Dch via Dpl, designers must set an informorphism Dds to Dpl.
In this case, a designer sets infomorphisms f∧(E1)=O1, f∧(E2)=O2, f∧(E3)=O3, f∧(E4)=O4,
and f∨(V1)=T2, f∨(V2)=T1, f∨(V3)=T3.typ(Dpl)
–
O1: a
ladder.–
O2: a
horizontal plate.–
O3: a
slope.–
O4: a
bar.tok(Dpl)
–
V1: climb
vertically.–
V2: walk.–
V3: slide
down.ClassificationDch: finally, a
designer creates units of playing actions 〈O,S,V〉,
and sets SOi,
which are defined by combining Si in terms of 〈O,S,V〉 with parts of play equipment Oi and Vi′,
which contain new playing actions emerged in the play environment in
classification Dch as follows. In this operation, designers must
set Si and Vi′ to establish an information channel between Dds and Dch via Dpl.
Figure 5 shows a slide derived from the design model using channel theory.
Figure 6 shows an example of a channel represented by Chu maps [10].Figure 5
A slide derived from the design model using channel theory.Figure 6
A description of a “playing structure
design” by Chu maps.typ(Dch)
–
SO1: a
ladder that is put to SO2 vertically.–
SO2: a
covered horizontal plate.–
SO3: a
steep slope that is put to SO2.–
SO4: a
bar dropped down from SO2.tok(Dch)
–
V1′: climb
vertically.–
V2′: walk.–
V3′: slide
down.–
V4′: walk
into.–
V5′: walk
outside.–
V6′: jump.–
V7′: scale
up.


## 3.3. New Playing Actions Described in a
Model of Information Channel

In the novel
design, there are several alternatives for not only segmentingOi but also Si and Vi.
Therefore, designers can foresee the emergence of new playing actions and
pleasures.At Boundary Between Multiple Parts
Designers have
to select multiple elements when they select units at the boundary between
multiple parts of play equipment. They become loose constraints that guide
design processes. For example, settingS2:
“covered” generates new playing actions, V4′:
“walk in," and V5′:
“walk out”:
(i)
a playing actionV4′ at the boundary between SO1, SO3, SO4, and SO2: (6)SO1SO2SO3SO4V4′1011,(ii)
a playing actionV5′ at the boundary between SO2, SO1, SO3,
and SO4: (7)SO1SO2SO3SO4V5′0100.At One Part
In the novel
design of a play equipment, settingSi makes arrangement Oi.
Therefore, designers can set more playing actions in the same device, but
selecting elements at one part of the play equipment depends on the designer's
ability largely. For example, setting SO1 and SO4 generates new playing actions V6′:
“jump.” And setting SO3 generates V7′:
“scale up”:
(i)
a playing actionV6′ in SO1 and SO4: (8)SO1SO2SO3SO4V6′1001,(ii)
a playing actionV7′ in SO3: (9)SO1SO2SO3SO4V7′0010.


### 3.3.1. Constraint in Classification

Channel theory
defines a constraint such asΓ⊢AΔ.
Given a classification A,
a token a∈tok(A) satisfies a sequent 〈Γ,Δ〉 of typ(A) provided that if a is of every type in Γ,
then it is of some type Δ.
A constraint is a sequent 〈Γ,Δ〉 for which, Γ⊢AΔ.
The following are part of constraints in classifications Dds, Dpl, Dch:(1)
constraint ofDds(10)E1⊢E4,E3⊢E4,(2)
constraint ofDpl(11)O1⊢O4,O3⊢O4,(3)
constraint ofDch(12)SO1⊢SO4. Constraints E1⊢E4, E3⊢E4 in Dds are directly translated in Dpl, but not in Dch.
This means that a designer's inference is described in Dds and Dpl,
because of Dpl being a step in which designers correspond
parts of play equipment to playing actions, but which is not translated in Dch.
In other words, generating V7′ in tok(Dch) disturbs establishment of SO3⊢SO4.


## 3.3.1. Constraint in Classification

Channel theory
defines a constraint such asΓ⊢AΔ.
Given a classification A,
a token a∈tok(A) satisfies a sequent 〈Γ,Δ〉 of typ(A) provided that if a is of every type in Γ,
then it is of some type Δ.
A constraint is a sequent 〈Γ,Δ〉 for which, Γ⊢AΔ.
The following are part of constraints in classifications Dds, Dpl, Dch:(1)
constraint ofDds(10)E1⊢E4,E3⊢E4,(2)
constraint ofDpl(11)O1⊢O4,O3⊢O4,(3)
constraint ofDch(12)SO1⊢SO4. Constraints E1⊢E4, E3⊢E4 in Dds are directly translated in Dpl, but not in Dch.
This means that a designer's inference is described in Dds and Dpl,
because of Dpl being a step in which designers correspond
parts of play equipment to playing actions, but which is not translated in Dch.
In other words, generating V7′ in tok(Dch) disturbs establishment of SO3⊢SO4.


## 4. Discussion

In Channel
theory, there are several ways of setting ofinfomorphism between same classifications.
For example, we can set different infomorphism between 3 classifications Dds, Dpl, Dch shown in Figure 6. We can conclude that this
case corresponds to players interpreting playing environments in different ways
from designers' intentions, even though players seem to adopt the ways
satisfying designers' intentions. We are now exploring the setting of
classification in depth to describe this representation.In this work, we discussed linked playing actions
using the syntax structure of〈O,S,V〉 binding them. Therefore, we are investigating
a way of representing the linked playing actions by operation of classification
in channel theory.Furthermore, we discussed an environment where only a
single play equipment and a single player exist. But a real playing environment
can be seen as a medium for mediating plural equipment and players. Therefore,
we are now investigating a way to develop our framework for multiactors.


## 5. Conclusions

We focused on a
playing environment and analyzed the perception-action chains in order to argue
playing environment on a theoretical ground. We showed that reductionism units
such as therbligs are not suitable for analyzing perception-action chains.
Then, we proposed representational units of playing actions〈O,S,V〉,
which have both properties of actions and environment.There are high degrees of freedom for segmenting
playing environments.〈O,S,V〉 can be seen as one of the loose constraints for
the segmentation. There are several alternatives for not only segmenting Oi but also selecting Si and Vi.
These constraints provide a guide for design. They are not excessively strict,
therefore designers' ability can be utilized properly. But as long as designers
follow the format 〈O,S,V〉, they are forced to pay attention to boundaries
of 〈O,S,V〉,
which enable designers to foresee the emergence of new playing actions.Furthermore,〈O,S,V〉 enables designers to interpret the playing
environment as an information channel in terms of channel theory, which is also an appropriate loose constraint for
designing a play environment. The establishment of an information channel between a designer and
a player via play equipment supports in a mathematical way
a good environment from the viewpoint of
information flow.



---

*Source: 258516-2008-10-12.xml*