Steps towards a Design Theory for Virtual Worlds
Steps towards a Design Theory
for Virtual Worlds
Joseph A. Goguen
Department of Computer Science and Engineering
University of California at San Diego
Abstract
Virtual worlds, construed in a broad enough sense to include textbased
systems, as well as video games, new media, and user interfaces of all kinds,
are increasingly important in scientific research, entertainment,
communication, and art. However, we lack scientific theories that can
adequately support the design of such virtual worlds, even in simple cases.
Semiotics would seem a natural source for such theories, but this field lacks
the precision needed for engineering applications, and also fails to addresses
interaction and social issues, both of which are crucial for applications to
communication and collaboration. This paper suggests an approach called
algebraic semiotics to help solve these and related problems, by providing
precise applicationoriented basic concepts such as sign, representation, and
representation quality, and a calculus of representation that includes
blending. This paper also includes some theory for narrative and metaphor,
and case studies on information visualization, proof presentation, humor, and
user interaction.
1 Introduction: Motivation, Difficulties and Approaches
The term "virtual world" is used in many ways, but perhaps virtual worlds
can be broadly characterized as the class of media experiences that provide
some sense of immersion and closure. By immersion, which is sometimes
also called virtuality, we mean a sense of being engaged with
nonphysically present entities through material mediation in the immediate
real world, but not with the other aspects of the immediate real world, and by
closure we mean that the virtual world gives an appearance of relative
completeness, although it may of course be changing. A lecture, a
conversation, a movie, a magazine, a formal paper, a video game, a user
interface, can all be virtual worlds in this sense. A major factor in
creating immersion and closure is the coherence of the world; of
course, there are many other factors, relating for example to the situation,
background, and interests of participants, but this paper is focused on ways
to achieve coherent representations.
Given the enormous cultural and economic importance of current media for
communication, entertainment, and art, as well as the promise of new media,
there would be many uses for scientific theories that could provide guidance
for difficult tasks, such as the following:
 indent 1.7ex
 designing new media (e.g., virtual reality environments with haptics);
 creating new metaphors (e.g., beyond the desktop for PCs);
 making new hardware (such as wireless applicances) more usable;
 designing new genres (such as interactive poems); and
 supporting nonstandard users (e.g., with disabilities).
Because virtual worlds are user interfaces in some broad sense, and because
user interface design is a welldeveloped area of computer science (which is
also known as humancomputer interaction, or HCI, or sometimes CHI), this
would seem another good place to look for appropriate theory. But most HCI
results are either very precise but also highly specialized and therefore not
very useful (e.g., Fitt's law), or else they are very general but of uncertain
reliability and generality (e.g., protocol analysis, questionnaires, case
studies, usability studies).
Another plausible place to seek a theory of virtual world design would be
semiotics, a subject founded by Charles Sanders Peirce [Peirce, 1965] and
Ferdinand de Saussure [Saussure, 1976] in the late nineteenth century.
Peirce was an American logician concerned with problems of meaning and
reference, who concluded that these are relational rather than denotational,
and who also made an influential distinction among modes of reference, as
symbolic, indexical, or iconic. Saussure, a Swiss linguist, wanted to
understand how features of language relate to meanings, and he emphasized
binary features and denotational meaning. More recent thinkers like the
French literary theorist Roland Barthes [Barthes, 1968] combined and extended
these theories, creating a powerful language for cultural and media studies,
which in various versions has been called semiotics, semiology, structuralism,
and finally poststructuralism. Unfortunately, this tradition:
 indent 1.7ex
 does not have mathematical precision needed to integrate well with
engineering processes;
 does not consider representing signs in one system by signs in another,
as is needed for the study and design of interfaces;
 has not addressed dynamic signs, which are necessary for the study and
design of interaction;
 has not much considered social issues, such as arise in shared worlds;
 tends to ignore the situated, embodied aspects of sign use;
 tends towards a Platonistic view of signs, as actual existing
abstract entities; and
 often considers only single (complex) signs (e.g., a novel or a film),
rather than systems of signs;
Therefore semiotics needs to address some significant problems before it can
meet all our needs. This paper sketches how algebraic semiotics attempts to
bridge this gap. The theory originated in an attempt to understand data from
an early experimental study of multimedia learning [Goguen and Linde, 1984], and
was later elaborated for applications to user interface design; more complete
expositions appear in [Goguen, 1999a,Goguen and Harrell, 2003] and [Goguen, 2003], though the theory
is still evolving. Here we focus more on motivation and applications.
There are at least two perspectives that one might take towards the study of
signs and representations: pragmatic and theoretical. The first
is the perspective of a designer, who has a job to get done, often within
constraints that include cost, time, and stylistic guidelines; we may also
call this an engineering perspective, and it will generally involve
negotiating tradeoffs among various values and constraints. The second is
the perspective of a scientist who seeks to understand principles of design,
and is thus engaged in a process of constructing and testing theories. From
the second perspective, it makes sense to describe semiotic theories in a
detailed formal way, and to test hypotheses by doing calculations and
experiments with users. But from the pragmatic perspective, it makes sense to
formalize only where this adds value to the design process, e.g., in
especially tricky cases, and even then, only to formalize to the minimum
extent that will get the job done. Our experience is that one can often get
considerable benefit from applying principles of algebraic semiotics, such as
identifying and preserving key features of the source theory, without doing a
great deal of formalization.
From either the pragmatic or theoretical perspective, one should seek to model
semiotic theories as simply as possible, since this will simplify later tasks,
whether they are engineering design or scientific theorizing and
experimentation (not forgetting that the conceptual simplicity of a theory
does not necessarily correspond to the simplicity of its expression in any
particular language). However, from a pragmatic perspective, good
representations need not be the simplest possible, for reasons that include
engineering tradeoffs, the difficulty (and inherent ambiguity) of measuring
simplicity, and social and cultural factors, e.g., relating to esthetics.
Similar considerations apply, though to a notably lesser extent, to the
simplicity of semiotic theories, since creating such theories is itself a
design task, subject to various tradeoffs. It may be reassuring to be
reminded that in general there is no unique best representation.
Sections 2 and 3 develop some basic theory of
algebraic semiotics. Two main concepts are semiotic theory and semiotic
morphism, which generalize the conceptual spaces and conceptual mappings of
Fauconnier and Turner [Fauconnier and Turner, 1998,Fauconnier and Turner, 2002], by taking account of
structure and dynamics. Some measures of quality and design principles are
given, including a tradeoff between form and content. Although similar
principles can be found in many places, none seem to be either as precise or
as general as those described here. This section also discusses metaphor and
blending in natural language, and gives some basics of a calculus of
representation. Section 4 describes a number of case studies,
including information visualization, proof presentations, humor, and user
interaction; a discussion of narrative is included here. Some conclusions,
future research directions, and social implications for virtual worlds are
discussed in Section 5.
Before beginning, it may help to be clear about the philosophical orientation
of this work, because it is very common in Western culture for mathematical
formalisms to claim and be given a status beyond what is warranted. For
example, Euclid wrote, "The laws of nature are but the mathematical thoughts
of God." Similarly, the "situations" in the situation semantics of Barwise
and Perry, which resemble conceptual spaces (but are more sophisticated 
perhaps too sophisticated) are considered to be actually existing,
ideal Platonic entities [Barwise and Perry, 1983]. Somewhat less grandly, one
might consider that conceptual spaces are somehow directly instantiated in the
brain. However, the point of view of this paper is that such formalisms are
constructed in the course of some task, with the heuristic purpose of
facilitating consideration of certain issues in that task, which might be
scientific study or engineering design. Under this nonPlatonist view, all
theories are situated social entities, mathematical theories no less than
others; of course, this by no means implies that they cannot be useful.
* Acknowledgements
I thank Drs Kirstie Bellman and Christopher Landauer of Aerospace Corp for
inviting me to the 2001 Virtual Worlds and Simulation Conference, and for
their valuable comments. I also thank the students in my UCSD classes CSE 171
and 271 for their feedback on topics in this paper.
2 Algebraic Semiotics
The basic notions of algebraic semantics are sign (or semiotic) system,
semiotic morphism, and representation quality; these are discussed in the
following subsections.
2.1 Signs and Sign Systems
The definition below of sign system incorporates the insight of Saussure that
individual signs should not be studied in isolation, but rather as elements of
systems of related signs [Saussure, 1976], of Peirce that signs may have
parts, subparts, etc., that play different roles [Peirce, 1965], and of Goguen
that sign parts have different saliencies, depending on the roles that they
play [Goguen, 1999a].
The structure of a sign system can be described by an algebraic theory, since
they are in particular abstract data types, and it is well known that these
can be defined algebraically [Goguen and Malcolm, 1996]. In addition, signs become what they
are by virtue of attributes that differ from those of other signs, as shown
for example by vowel systems (how the space of possible vowel sounds is
divided into specific vowels for a given dialect of a given language), as well
as by traffic signs, alphabets, numerals, etc. However, these attributes need
not be binary, as was supposed to Saussure and his followers in the French
structuralist movement, such as LeviStrauss and early Barthes. Also, the
same sign in a different system can have a different meaning, as illustrated
by the way similar characters in different alphabets can take different
meanings, e.g., in the Roman and Cyrillic alphabets, the token "P" denotes
different sounds.
We formalize^{1} sign
system as many sorted loose algebraic theories with data, plus two additional
items that are specifically semiotic:
Definition 1
A sign system, or semiotic system or semiotic theory,
consists of:
 indent 1.7ex
 a signature, which declares sorts, subsorts and operations
(including constructors and selectors);
 a subsignature of data sorts^{2} and data functions;
 axioms (e.g. equations) as constraints;
 a level ordering on sorts, including a maximum element called
top; and
 a priority ordering on constructors at the same level.
The nondata sorts classify signs and their parts, just as in grammar, the
"parts of speech" classify sentences and their parts. There are two kinds
of operation: constructors build new signs from old signs as parts, while
selectors pull out parts from compound signs. Data sorts classify a special
kind of sign that provides values serving as attributes of signs. Axioms act
as constraints on what count as allowable signs for this system. Levels
indicate the whole/part hierarchy of a sign, with the top sort being the level
of the whole; priorities indicate the relative significance of subsigns at a
given level; social issues play a dominant role in determining these. The
above definition follows [Goguen, 1999a], where the special treatment of data sorts
follows [Goguen and Malcolm, 2000]. The first four items constitute what is called an
algebraic theory when all axioms are equations (e.g., [Goguen and Malcolm, 1996,Goguen et al., 1978]); it
can be proved that this special case is sufficient for our needs.
The approach of Definition 1 differs from the more traditional
setbased approaches of Gentner [Gentner, 1983], Carroll [Carroll, 1982],
etc., in that it is axiomatic, i.e., it does not present signs
as particular models, but rather, a particular theory expressed in a formal
language describes a space of possible signs, which are models of the
theory, in the sense of that term in logic, providing concrete
interpretations for the things in the theory: sorts are interpreted as
sets; constant symbols are interpreted as elements; constructors are
interpreted as functions, etc.; i.e., the theory is a language for
talking about such models. This approach allows both multiple models
and open structure, both of which are important for applications. The
first point means, for example, that a semiotic space of books should allow
anything having the structure of a book as a model; it also means that
designers and implementers have the freedom to optimize implementations so
long as they respect the constraints of the given axioms. The second point
says that structure can be extended and combined without violating the
specifications, which is not necessarily the case for models. For example, we
might want to extend a basic sign system for books with some further structure
for a certain series of books from a particular publisher. In addition, it is
also more natural to treat levels and priorities in axiomatic theories than in
setbased models.
For an example of axioms, in formalizing indices of books, we might well want
to impose an axiom requiring that indexed items must be phrases of 1, 2 or 3
words, but not more. In a sign system for books, the top level might be
occupied by the sort for books, the next level by author, title, publisher,
and the third level by the first and last names of author, and by the name and
location of the publisher; see Figure 1. Here last name has
priority over first name, and publisher name has priority over publisher
location. This is similar to the nesting structure used in XML documents.
Figure
Figure 1: Levels for the Book Semiotic Theory
The following are some further informal examples of sign systems: dates;
times; bibliographies (in one or more fixed format); tables of contents (e.g.,
for books, again in fixed formats); newspapers (e.g., the New York
Times Arts Section); and a fixed website, such as the CNN homepage (in some
particular instance of its gradually evolving format). Note that each of
these has a large space of possible instances, but a single fixed structure.
There is a basic duality between theories and models. We have already
discussed one aspect: A semiotic theory determines the class of models that
satisfy it, which we call its semiotic space^{3}. The other aspect is more subtle: a class of models has
a unique (up to equivalence) most restrictive theory whose models include
it^{4}. This duality
helps to justify our occasional use of the term "space" when we really mean
"theory"; this is mainly done for consistency of terminology when discussing
conceptual blending theory.
Gilles Fauconnier introduced mental spaces for studying meaning in
natural language from a cognitive point of view [Fauconnier, 1985]. The
abstract mathematical structure of a mental space is a set of atomic elements
together with a set of relation instances among those elements [Goguen, 1999a], and
as such is a very special case of a sign system. Any such representation
necessarily omits the qualitative, experiential aspects of what is represented
(these aspects are often called "qualia"), since formal representations
cannot capture meaning in any human sense. Moreover, mental spaces are not
powerful enough for designing virtual worlds or other applications where
structure and dynamics are important; obvious examples include wikis,
web sites, and music.
The conceptual spaces of Fauconnier and Turner [Fauconnier and Turner, 1998,Fauconnier and Turner, 2002]
are mental spaces, and hence share their limitations. For example, conceptual
space theory can help us understand concepts about music, but semiotic
spaces and structural blending are needed for an adequate treatment of the
structure of music, e.g., how a melody can be combined with a sequence
of chords. Conceptual spaces are good for talking about concepts about (e.g.,
how we talk about) things, but are awkward for talking about the structure of
things. It is also interesting to notice that greater cultural variation can
be found in conceptual blending than in structural blending, because the
former deals with concepts about something, whereas they latter deals with the
structure of its instances and/or its representations. Mathematically, a
conceptual space is a single model, consisting of items and assertions that
certain relations hold of certain of those items; it is not a theory or a
class of models.
Our suggested methods for determining semiotic spaces are grounded in ideas
from sociology, especially ethnomethodology, but this paper is not the right
place to discuss such issues; see [Goguen, 1997,Goguen, 1994], beyond noting that
semiosis, which is the creation of meaning, is always situated and embodied,
and in particular always has a social context. Immersion arises in part
through embodiment (even if only metaphorical embodiment, e.g. in textbased
virtual worlds).
2.2 Representations
Mappings between structures became increasingly important in twentieth century
mathematics and its applications; examples include linear transformations (and
their representations as matrices), continuous maps of spaces, differentiable
and analytic functions, group homomorphisms, and much more. Mappings between
sign systems are only now appearing in semiotics, as uniform representations
for signs in a source space by signs in a target space. Since we formalize
sign systems as algebraic theories with additional structure, we should
formalize semiotic morphisms as theory morphisms; however, these must be
partial, because in general, not all of the sorts, constructors, etc. are preserved in real world examples. For example, the semiotic morphism that
produces an outline from a book, omits the sorts and constructors for
paragraphs, sentences, etc., while preserving those for chapters, sections,
etc. In addition to the formal structure of algebraic theories, semiotic
morphisms should also (partially) preserve the priorities and levels of the
source space. The extent to which a morphism preserves the various features
of semiotic theories is important in determining its quality, as the case
studies to follow will show.
The design of virtual worlds and more generally, of user interfaces, is the
art of creating representations, for example, representing the functionality
of an operating system using icons, menus, buttons, etc., or using haptics and
virtual reality. The basic insight is that a representation is a mapping
M \colon S_{1}®S_{2} of sign systems that preserves as much as is reasonable.
The following formalizes this insight:
Definition 2
A semiotic morphism M \colon S_{1}®S_{2} from a semiotic system S_{1} to
another S_{2} consists of the following partial mappings:
 indent 1.7ex
 from sorts of S_{2} to sorts of S_{2}, so as to preserve the subsort
relations,
 from operations of S_{2} to operations of S_{2}, so as to preserve
their source and target sorts,
 from levels of S_{1} sorts to levels of S_{2}, so as to preserve the
ordering relation, and
 from priorities of S_{1} constructors to priorities of S_{2}
constructors, so as to preserve their ordering relations,
so as to strictly preserve all data elements and their functions.
It is not always possible or even desirable for a semiotic morphism to
preserve everything. For example, sometimes we just want to summarize
some dataset, e.g., the table of contents of a book, in which case much of the
structure and information are intentionally deleted. Another important
observation is that not all representations are equally desirable. For
example, one way to parse the sentence "Time flies like an arrow" in the
following "bracket" (or "bracketwithsubscript") notation, which is
widely used in linguistics:
[[time]_{N}[[flies]_{V}[[like]_{P}[[an]_{ Det}[arrow]_{N}]_{NP}]_{PP}]_{VP}]_{S}
However, this notation is not very satisfactory for humans, who would find it
easier to discern the syntactic structure by examining a parse tree, or using
the algebraic "constructor" notation given later. Some criteria for judging
the quality of representations are discussed in Section 2.4 below.
The duality between theories and models means that there is an inherent
ambiguity about the direction of a semiotic morphism. For example, if B is
is a sign system for books and T is one for tables of contents, then books
(which are models of B) are mapped to their tables of contents, which are
models of T, but this map on models is determined by, and is dual to, the
theory inclusion T®B, which expresses the fact that the structure of
tables of contents is a substructure of that of books. In informal
discussions we will often take the direction to be that of the models, which
is perhaps more intuitive; however, in formal discussions, it is much better
to use the direction of the underlying theory morphism, which is opposite to
that of the models.
There are at least three "modes" in which one might consider
representations: analytic, synthetic, and conceptual. In the analytic
mode, we are given one or more sign from the representation (i.e., the target)
space, and we seek to reconstruct both the source space and the
representation. In the synthetic mode, we are given the source space
and seek to construct a good representation for the signs in that space, using
some given technology (such as command line, or standard GUI widgets, or
virtual reality) for the target space. In the conceptual mode, we
seek to analyze the metaphorical structure of the representation, in the style
of cognitive linguistics [Turner, 1997,Fauconnier and Turner, 2002]; for example, how is Windows XP
like a desktop, or how is a scrollbar like a scroll? A treatment in this mode
will involve conceptual spaces, in the sense of cognitive linguistics (see
Section 3.1 below). In each mode, particular aspects of the cultures
involved can be very significant.
2.3 Simple Examples
This subsection gives rather informal descriptions of some simple examples of
semiotic theories and semiotic morphisms, to illustrate the concepts, rather
than to demonstrate their applicability to virtual world design, since these
examples could only be considered virtual worlds in a trivial sense. The
following sign systems are considered:
 indent 1.7ex
 Lists of (potential) words with punctuation, denoted S_{W}.
 Parse trees for sentences of a formal grammar G, denoted S_{T}.
 A printed page format, denoted S_{P}.
Then the following are some interesting morphisms:
 indent 1.7ex
 Let P \colon S_{W} ®S_{T} give parse trees for lists from S_{W} that are
Gsentences.
 Let H \colon S_{T} ®S_{P} give bracket representations of parse trees.
 Let F \colon S_{W} ®S_{P} give bracket representations of lists from
S_{W} that are Gsentences.
The morphism F is "composed" from P and H, by first doing P and then
doing H; we denote this by P;H, where ";" denotes the composition
operation. Composing morphisms corresponds to composing representations,
which is the essence of iterative design, an important technique for any
complex design task. By Definition 2, a semiotic morphism M
has four component mappings, for sorts, operations, levels, and priorities;
let us denote these M_{1}, M_{2}, M_{3} and M_{4}, respectively. Then the
composition M;M¢ of morphisms M and M¢ can be defined by the
formula (M;M¢)_{i} = M_{i};M¢_{i} for i=1,2,3,4.
The sign system S_{W} for punctuated lists of words can be described roughly
as follows: Its sorts are char, alpha, punc,
puncword, alphaword, word, and list, where the sorts
alpha and punc are subsorts of char, the sorts alpha,
puncword and alphaword are subsorts of word, and the sort
word is a subsort of list. These subsort relationships are shown in
Figure 2. The sorts char, alpha, and punc
are respectively for character, alphabetic character, and punctuation
character; the sorts puncword, alphaword, and word are
respectively for words consisting of alphabetic characters with a final
punctuation sign, words with all alphabetic characters, and the union of these
two.
Figure
Figure 2: Sorts and Subsorts for S_{T}
The sort list is level 1, the "top" sort of this system; word is
level 2; alphaword and puncword are level 3; and char,
alpha and punc are level 4. The punctuation characters are comma and
period (of course we could add more). The following defines concatenation
constructors for constructing a list of alphabetic characters as a
alphaword, a list alphaword followed by a punctuation as a
puncword, and a list of words as a list; a functional notation is
used:
_ _ : alpha alphaword > alphaword
_ _ : alphaword punc > puncword
_ _ : word list > list
Here the two underbars give the syntactic form of the function, which is to
concatenate its two arguments, which respectively have the two types given
between the colon and the arrow; the result type then comes after the arrow.
These three operations also satisfy associativity equations, such as the
following:
(" W L L¢) (W L) L¢ = W (L L¢) . 

The context free grammar grammar G given below allows some lists of
punctuated words to be recognized as legal sentences of that grammar; such
sentences can be parsed, which means dividing them into phrases, which
are sublists, each with its "part of speech" (or syntactic category)
explicitly given. The grammar G will become the signature of the sign
system S_{T}. The nonterminals of G are S, NP, VP,
N, Det, V, PP and P, which stand for sentence, noun
phrase, verb phrase, noun, determiner, verb, prepositional phrase, and
preposition, respectively. Then the rules of a simple example G might be
the following:
S > NP VP
NP > N
NP > Det N
VP > V
VP > V PP
PP > P NP
The nonterminals of this grammar (i.e., the "parts of speech" S,
NP, VP, etc.) are the sorts of the sign system S_{T}, and the rules of
the grammar will become its constructors. For example, the first rule says
that a sentence can be constructed from an NP and a VP. There
should also be some constants of the various sorts, such as
N > time
N > arrow
V > flies
Det > an
Det > the
P > like
Then, for example, a parse tree for the sentence "time flies like an arrow"
is shown in Figure 3.
Figure
Figure 3: Parse Tree for S_{T}
By the way, if we add the productions below to the grammar G, then the
sentence gets another parse, a fact which the reader might enjoy checking.
NP > N N
V > like
There is a systematic way to convert context free rules into constructor
operations in the signature of a sign system; for the above grammar G, it is
as follows, written in a functional notation:
sen : NP VP > S
nnp : N > NP
np : N Det > N
vvp : V > VP
vp : V PP > VP
pp : P NP > PP
time : > N
flies : > V
.....
In this context, it is more elegant to regard N as a subsort of
NP, and V as a subsort of VP, rather than to have monadic
operations N > NP and V > VP. This sign system gives what
computer scientists call abstract syntax for sentences; it gives an
abstract algebraic representation for syntactic structure, in which the
operations above generate a free algebra of terms that describe
parses. For example, the term that represents the syntax of our example
sentence is:
sen(time, vp(flies, pp(like, np(an, arrow)))) .
Equations can be used in this algebraic setting to express constraints on
sentences, for example, that the number of the subject and of the verb agree
(i.e., both are singular or else both are plural). Each of the concrete ways
to realize abstract syntax (trees, terms, bracket notation, and lists) can be
considered to give a model of the sign system S_{T}, providing a set of signs
for each sort, and operations on those sets which build new signs from old
ones.
The sign system S_{P} should have sorts for lines and pages, and could also
have different fonts and subscripts, in order to display the bracket notation
to display parses. We omit the details, which are not very different from
those above, except for an equation to limit the length of lines, e.g., to 80
characters, such as the following:
(" L \colon line) length(L) £ 80 . 

The morphism P \colon S_{W} ®S_{T} is very partial, since it is defined on a list
l iff l can be parsed using G; thus the subset of lists on which
it is defined is the set of sentences generated by G, which is usually
denoted L(G). If fr(t) denotes the frontier, or list of leaf nodes,
of a parse tree t, then fr(P(l)) = l for all l Î L(G), which
is a strong preservation property, although it only holds on a small subset of
lists of words. The morphism P also preserves all of the sort hierarchy in
Figure 2.
We do not describe the morphism H \colon S_{T} ®S_{P} in as much detail as we did
the morphism P \colon S_{W} ®S_{T}. The morphism H is essentially a pretty
printer for parse trees; it could use any of the representations we have been
discussing, and it could even just print the frontier of the parse tree,
although this would preserve much less structure (see the next subsection for
more discussion of structure preservation). As already mentioned, the
morphism F \colon S_{W} ®S_{P} is the composition P;H; it pretty prints the
parse trees of those lists of words that can be parsed by G.
2.4 Quality of Representation
It is easy to define sort preserving, constructor preserving, level
preserving, content preserving (where content refers to the values of
selector operations, such as size and color), etc. But this is not as useful
as one might hope, because in practice, these are often not preserved.
Instead, we define the comparative notions of more sort preserving,
more level preserving, more constructor preserving, more content preserving,
etc. [Goguen, 1999a]. These notions define orderings on morphisms, which can be
logically combined to get the right one for a given application [Goguen, 1999a].
This is important because given morphisms M,M¢, one may preserve more
levels, while the other preserves more content, and similarly for the other
concepts. Empirical work has validated the following general principles:
 indent 1.7ex
 It is more important to preserve structure than content (this is called
Principle F/C).
 It is more important to preserve level than priority.
 Structure and content at lower levels should be sacrificed in favor of
those at higher levels.
 Lower level constructors should be sacrificed in favor of higher level
constructors.
The first principle is perhaps the most important, and at first might seem
counterintuitive, many special cases can be found in the design literature,
e.g., [Tufte, 1983]. It asserts that when a tradeoff is necessary, form
should be weighted more heavily than content; in general, the right balance
between form and content can only be determined after knowing how a
representation will actually be used. Also, we are fortunate that it is
easier to describe structure than content.
These principles do not explain everything, for example, they do not explain
why the tree representation of phrase structure is better than the bracket
representation, since these two representations have exactly the same
structure and content, but display them differently. In fact, the advantage
of the tree representation arises from human visuocognitive capabilities,
which prefer a more explicit diagrammatic representation of phrases and
subphrase relations as nodes and edges, over a linear symbolic representation
that requires counting brackets. Preservation of form and content can
respectively be formalized as preservation of constructors and selectors, in
the sense of abstract data type theory
[Goguen et al., 1978,Goguen and Malcolm, 1996].
3 Fragments of a Calculus of Representation
Section 2.3 defined the composition of semiotic morphisms and
showed that it can be important for applications. It is easy to prove that
this definition of composition obeys the following identity and associative
laws, in which A \colon R®S, B \colon S®T and C \colon T®U,



A ; (B ; C) = (A ; B) ; C 




where 1_{S} denotes the identity morphism on S. These three laws are
perhaps the most fundamental for a calculus of representation, since they
imply that semiotic theories and their morphisms form what is called a
"category" in the relatively new branch of mathematics called category
theory [Mac Lane, 1998]. The basic ingredients of a category are
objects, morphisms, and a composition operation that satisfies the above three
laws, and that is defined on two morphisms if and only if they have matching
source and target. Perhaps surprisingly, many important mathematical concepts
can be defined abstractly in the language of category theory, without
reference to how objects are represented, using only morphisms and
composition; moreover, many general laws can be proved about such concepts,
and these automatically apply to every category.
Three of the simplest categorical concepts are isomorphism, sum and product.
A morphism A \colon R ®S is an isomorphism if and only if there is
another morphism B \colon S ®R such that A ;B = 1_{R} and B ;A = 1_{S}, in
which case B is called the inverse of A and denoted A^{1}; it can
be proved that the inverse of a morphism is unique if it exists. The
following laws can also be proved, assuming that A \colon R®S and B \colon S®
T are both isomorphisms (and no longer assuming that B is the inverse of
A).



(A ; B)^{1} = B^{1} ; A^{1} 




Because sign systems and their morphisms form a category, these three laws
apply to representations. In Section 3.2 below, we discuss sums of
semiotic morphisms as a special case of blends of semiotic morphisms (blends
are discussed in the next subsection), and we also give some laws for blends
and sums in Section 3.2. The standard mathematical reference for
category theory is by Sanders Mac Lane [Mac Lane, 1998], but [Pierce, 1990]
is one computer science oriented introduction, among several others.
3.1 Metaphor and Blending
Research in cognitive linguistics by George Lakoff and others under the banner
of "conceptual metaphor theory" (abbreviated "CMT") has greatly deepened
our understanding of metaphor [Lakoff and Johnson, 1980,Lakoff, 1987], showing that
many metaphors come in families, called image schemas, that share a
common pattern. One example is better is up, as in "I'm feeling up
today," or "He's moving up into management," or "His goals are higher than
that." Some image schemas, including this one, are grounded in the human
body^{5} and are called basic
image schemas; they tend to yield the most persuasive metaphors, as well as
to enhance the sense of immersion in virtual worlds.
Fauconnier and Turner [Fauconnier and Turner, 1998,Fauconnier and Turner, 2002] study blending, or
conceptual integration, claiming it is a basic human cognitive
operation, invisible and effortless, but nonetheless fundamental and
pervasive, appearing in the construction and understanding of metaphors, as
well as in many other cognitive phenomena, including grammar and reasoning.
Many simple examples are blends of two words, such as "houseboat," "jazz
piano," "roadkill," "artificial life," "computer virus," "blend
space," and "conceptual space." To explain such phenomena, blending theory
(abbreviated "BT") posits that concepts come in clusters, called
conceptual spaces, which consist of certain items and certain relations that
hold among them. Such spaces are relatively small constructs, selected on the
fly from larger domains, to meet an immediate need, such as understanding a
particular phrase or sentence^{6}. The abstract mathematical structure
of a conceptual space consists of a set of atomic elements together with a set
of relation instances among those elements [Goguen, 1999a]. Conceptual
mappings are partial functions from the item and relation instances of one
conceptual space to those of another, and conceptual integration
networks are networks of conceptual spaces and mappings that are to be
blended together.
Picture Omitted
Figure 4: A Blend Diagram
We now describe our generalization of blending from conceptual spaces to
semiotic theories. A simple example where this generality is needed is in the
integration of a window with its scrollbar, which is structural, not
conceptual, although conceptual aspects of this blend could also be studied;
this example is discussed in considerable detail in [Goguen, 2003]. To
indicate this added generality, we will use the terms structural
blending or structural integration for the blending of semiotic
systems, which in general involves nontrivial constructors; but for
consistency with BT, we use the phrase "semiotic space" instead of
"semiotic theory" in this discussion. The simplest form of blend is as
shown in Figure 4, where I_{1} and I_{2} are called the
input spaces, B the blend space, and G the generic space,
which contains conceptual structure that is shared by the two input
spaces^{7}. Let us call I_{1}, I_{2}, and G together with the two
morphisms G®I_{1} and G®I_{2} an input diagram. Then a
blendoid over a given input diagram is a space B together with morphisms
I_{1} ®B, I_{2}®B, and G®B, called injections, such that the
diagram of Figure 4 weakly commutes, in the sense that
both compositions G®I_{1} ®B and G®I_{2}®B are weakly equal
to the morphism G®B, in the sense that each element in G gets mapped to
the same element in B under them, provided that both morphisms are defined
on it^{8}. The special
case where all four spaces are conceptual spaces gives conceptual blends.
This diagram is "upside down" from that used by Fauconnier and Turner, in
that our arrows go up, with the generic G on the bottom, and the blend B
on the top. Our convention is consistent with duality mentioned earlier, as
well as with the way that such diagrams are usually drawn in mathematics, and
with the image schema more is up (since B is "more"). Also,
Fauconnier and Turner do not include the map G®B. By definition, the
maps G®I_{1} and G®I_{2} are total, not partial, and if the input spaces
were minimal, then the maps I_{1}®B, I_{2}®B, G®B would also be
total.
Usually an input diagram has many blendoids, only a few of which are
interesting. Weak commutativity of the blend diagram, which is included in
the definition, is a good first step, but still leaves too many possibilities.
Therefore additional principles are needed for identifying the most
interesting possibilities, so that we can define a blend to be a
blendoid that is optimal with respect to these principles. Fauconnier
and Turner suggest a number of "optimality principles" that serve this
purpose (see Chapter 16 of [Fauconnier and Turner, 2002]), but they are too vague to be easily
formalized. A tentative and difficult but precise mathematical approach is
given in Appendix B of [Goguen and Malcolm, 1996], based on a modification of the category
theoretic notion of "pushout" [Mac Lane, 1998]; this modification takes
advantage of an ordering relation on morphisms, along the lines discussed in
Section 2.4. The intuition is that nothing can be added to or
subtracted from such an optimal blendoid without violating consistency or
simplicity in some way. However, there can still be more than one blend in
this sense, as an example discussed below will make very clear. It should
also be noted that this notion of blend easily generalizes to any number of
semiotic spaces, and even to arbitrary diagrams of semiotic spaces and
morphisms, for which there are many significant applications. Thus, the
emphasis on double scope blending in [Fauconnier and Turner, 2002] seems somewhat out of place
in algebraic semiotics, because its major applications typically involve
multiple "scopes" arising from multiple spaces and morphisms among them.
It has perhaps not been sufficiently emphasized in the BT literature that
blending does not always give a unique result. For example, the following are
four different blends of conceptual spaces for "house" and "boat":
 indent 1.7ex
 houseboat;
 boathouse;
 amphibious RV; and
 boat for moving houses.
The last may be a bit surprising, but I once saw such a boat in Oban,
Scotland, transporting prefabricated homes to a nearby island. There are also
some other, even less obvious blends [Goguen and Harrell, 2004].
In the UCSD Meaning and Computation Lab, Fox Harrell and I have been
experimenting with a blending algorithm, which has generated novel metaphors,
which in turn were used in generating poems [Goguen and Harrell, 2004], with some success
before a live audience. The algorithm uses dynamic programming to generate
blends in approximate order of optimality, and if requested, can generate all
possible blends, including even very bad ones. One surprise was that there
were so many blends, for example, 48 for the (small) house and boat spaces.
The CMT view of metaphor associates aspects of one domain to another, and
describes this association using a mapping, of which the target domain
concerns what the metaphor is "about." On the other hand, BT views
metaphors as "crossspace mappings" that arise from blending conceptual
spaces extracted from the domains involved. For example, the metaphor "my
love is a rose" arises from blending conceptual spaces for "my love" and
"rose," such that the identification of the two items "love" and "rose"
in the blend space gives rise to a correspondence between certain items in the
rose space and the target love space. Such metaphoric blends are
asymmetric, in that as much as possible of the target space is imported into
the blend space, whereas only key aspects from the source space, associated
with elements that have been identified with elements of the target space, are
imported, e.g., sweet smell and attractive color; moreover, names from the top
space take precedence over those in the source space, so that relations in the
source space become "attributed" to items in the target space. Our approach
differs from orthodox BT not only in that we allow many more kinds of
structure in our spaces, but also in that we do not first construct a minimal
image in the blend space and then "project" that material back to the target
space, but instead, we construct the entire picture in the blend space. Thus
it is not the case for us that, in forming the blend, elements are
preferentially omitted from the target space, only to be restored upon
projection, as described in [Grady et al., 1999]. Since CMT has been
mainly concerned with families of metaphors having a shared pattern, and BT
has been more concerned with how novel metaphors can be understood, the two
theories are compatible, and can both play a role in understanding complex
language. This and related issues are discussed with many interesting details
and examples in [Grady et al., 1999].
Algebraic semiotics also goes beyond conceptual spaces in allowing entities
that have dynamic states; this is necessary for applications to the dynamic
entities that appear in user interface design and virtual worlds. Actually,
two kinds of dynamics are involved in blending: the process of blending
itself, and entities with internal states. Whereas cognitive linguistics has
so far focussed mainly on the former, algebraic semiotics is more concerned
with the latter. Another difference from BT is that relations like causality
are represented as ordinary relations rather than being given a special
ad hoc status.
The conceptual spaces, mappings and blending of cognitive linguistics seem
well adapted for treating many aspects of literature, as in [Turner, 1997],
as well as some recent trends in art, including (the very aptly named)
conceptual art movement, and with the conceptual aspects of works in many
other styles, which are often designed to provoke conceptual conflicts or to
force unusual conceptual blends. One important application is the combination
of music with lyrics, as skillfully studied using crossdomain mappings by
Lawrence Zbikowski [Zbikowski, 2002]. Unfortunately, the framework of
conceptual blending seems too restricted for studying blending within
music, e.g., harmony, polyphony, polyrhythm, etc., because musical structure
is inherently hierarchical, and hence cannot be adequately described using
only atomic elements and relation instances among them. Understanding how a
particular melody, chord sequence, and rhythm can work together requires
attention to the component notes, phrases, chords and beats, as well as to
their subcomponents. However, it seems that the added generality of semiotic
spaces and semiotic morphisms is adequate for such purposes.
3.2 Some Laws
This subsection gives some further fragments of a calculus of representation;
see [Goguen, 1999a] for more detail. Here a,b,c are semiotic morphisms, and
à denotes some choice of a blend that is maximal with respect to some
optimality criterion:
In the following, A,B,C may be either semiotic morphisms or just semiotic
systems. Sums, denoted +, are the special case of blend where the base
theory is 1, which is the theory having exactly one constant, its top
element, and nothing else.
It should be noted that products of models correspond to sums of theories,
that is, a model of a sum of theories is a product of models of the summand
theories, and vice versa, or even more formally, there is an isomorphism
of categories of models,
Mod(A+B) @ Mod(A) ×Mod(B) , 

where A and B are semiotic theories; see [Goguen, 1999a] for details.
4 Case Studies
This section surveys some case studies applying algebraic semiotics. Noting
that we have already discussed blending and metaphor, the following additional
case studies are considered:
 indent 1.7ex
 information visualization;
 proof presentation;
 humor; and
 user interaction.
The first category in this list actually contains three small case studies,
and the second can also be considered a special case of it; proof
visualization is our most extensive case study, part of a large project to
produce a webbased system to support theorem proving. The study of humor is
somewhat of a digression, but it is hoped that the reader will find it
amusing. Proof navigation is used to illustrate how interaction is treated in
algebraic semiotics, although many details are left out, because the formal
theory of dynamic signs is technically rather complex.
4.1 Information Visualization
Visualizing complex data can help to discover, verify and predict patterns,
and to quickly locate specific information; but it can be difficult to
construct the appropriate visualizations for these purposes. Because
visualizations are representations, our theory applies to them, and in
particular, our quality measures apply. The following subsections analyze
three real visualization systems as semiotic morphisms, and on that basis,
suggest some improvements. We found it convenient to use algebraic semiotics
in a semiformal style, letting the ideas and results guide the redesign, and
introducing formal details only to the degree that they actually help with
decisions. Many aspects of these discussions follow [Goguen and Harrell, 2003].
Figure
Figure 5: A Code Browser
4.1.1 Code Visualization
A visualization tool for code developed at ATT Bell Labs, and discussed in
[Shneiderman, 1997], displays the large grain structure of code by omitting
details; see Figure 5. This is an excellent illustration of Principle
F/C (Section 2.4): commands are indistinguishable lines, but files
and procedures are easily distinguished, and the age of code is highlighted
with color (though it shows up as shades of gray in this figure), presumably
because code age is so important for software maintenance, which accounts for
most of the cost of large software systems. Moreover, code at the command
level can be viewed in a separate window, which is activated by "zooming in"
from the main overview window. However, software engineers often need to find
other specific features of code, such as:
 indent 1.7ex
 occurrences of particular variables,
 certain uses procedure calls, and
 certain uses of pointers.
Or consider what would be needed to work on the Y2K problem. To support this
kind of flexibility, the system should allow users to select and highlight a
variety of features to be displayed with color, not just code age; indeed,
each feature listed above could be highlighted with a different color, because
these features are binary (i.e., they either occur or do not occur, at any
given point in the program), rather than, like age, being measured on a
(nearly) continuous scale.
Figure
Figure 6: FilmFinder Display
4.1.2 A Film Visualizer
Figure 6 shows FilmFinder, a system to help consumers find films,
designed in Ben Shneiderman's group at the University of Maryland, as
described in [Shneiderman, 1998]. The vertical axis indicates popularity,
the horizontal axis indicates the release date, and the color^{9} indicates the
genre; the area on the right side of the display is for controlling the
system. This complex sign is the image under an appropriate semiotic morphism
of a sign in a space of information about films. From this, we infer that the
designer of the system thought users would consider the popularity, date, and
genre to be the most important attributes of films.
Instead of thinking of it as a consumer product, it is interesting to think of
this system as a scientific tool for displaying data about the movie industry.
Using it in this way, we can see that the density of films increases rapidly
in the most recent years displayed, except perhaps for those genres that are
the least popular; and we can also easily see some other facts, such as that
there has always been a higher percentage of drama, and that there are
increasing percentages of action and horror.
However, this representation is less useful than it could be for this purpose.
The problem is again that too much content and not enough structure have been
preserved. For example, it would be better to aggregate all films having
approximately the same attributes of interest into one blob, and then display
the number of films in a blob using a distinct visual attribute, such as size
or brightness. Successive blobs of the same kind could then be connected by
lines having the same color as the blobs. Users could click on a blob to see
what's in it, preferably displayed graphically in a new popup window. These
revisions would facilitate hypothesis formation, and would also make the tool
more useful for consumers, especially when (as in the most recent years that
are not represented in the figure) there are many more films.
Figure
Figure 7: SpotFire Version of FilmFinder
4.1.3 A Later Version
Figure 7 shows a later version (SpotFire from ivee Development in
Sweden) of the FilmFinder tool in Figure 6; the main improvement is
that users have more control over what is displayed and how it is displayed.
This particular display has length and date as its axes, and again uses color
for genre, although the genre color coding scheme is not explicitly shown;
prize winning films are highlighted by having a larger size. It is
interesting to observe a clustering at around 90 minutes length. But once
again, the display is difficult to use because there are too many dots, even
though this display cuts off at 1990! If the user is seeking a particular
film or class of films, she will want to narrow the search focus by imposing
additional constraints, but from this single display, it is difficult to know
how easily that could be done. We are presumably supposed to assume that the
(possibly imaginary) user who created this display considered these particular
attributes the most interesting at a certain point during a sequence of
displays constituting a search; but in fact, they do not seem especially
useful for any particular purpose.
We can also infer what the designer of this version thought would be most
important, by examining the controls on the right of the display; we can hope
that these were determined by polling an adequate pool of typical users. But
the key issue is how convenient these controls are for scenarios that typical
users find particularly important; most likely, those typical users are
looking for a good video to rent, rather than analyzing trends in the movie
industry, and so the controls should reflect the key actions involved in those
searches, rather than just the most important general attributes of films. It
would take some experimental work to determine these most relevant search
attributes, but we can still criticize the design of the control console,
because of its exclusive focus on simple attributes instead of structure. And
we can also criticize the fine grain control that it gives users over length
and year, and suggest instead that soft constraints would be more appropriate;
it also seems doubtful that length is a highly significant attribute for
search. Moreover, we can criticize the design philosophy, advocating instead
a more social approach that relates the profile of one user to the profiles of
other users to select films that similar users have found interesting (there
are numerous variations on this, such as listing films that a user's friends
have liked; Amazon has exploits similar strategies very successfully).
Finally, we can note that the design ideas proposed to improve the previous
version of this system also apply to the new version.
4.2 Proof Representation and Understanding
It is well known (perhaps too well known to many unhappy students)
that understanding mathematical proofs can be very difficult. But why
is it difficult? And how can this situation be improved? The UCSD
Tatami project [Goguen et al., 2000] aims to make proofs more interesting and even
enjoyable to read, by viewing them as representations of their underlying
mathematics; this lets us apply algebraic semiotics, including the theory of
representation quality.
The Kumo system generates websites, based on userprovided proof sketches in a
language called Duck. The pages are in XML, displayed using XSL style sheets,
and can be viewed over the web using any browser. The complex signs that
users actually see are called proofwebs, consisting of English phrases
and sentences, mathematical signs, navigation buttons, formal input and output
for a mechanical theorem prover, etc. [Goguen et al., 2000,Goguen, 1999b].
Our view of what constitutes the underlying mathematics to be displayed is
unusual: we consider it to include not just the tree structure of proofs,
decorated with formal sentences and rules, as is common among computer
scientists and logicians, but also:
 indent 1.7ex
 a dramatic structure, following Aristotle (see below);
 a narrative structure (following ideas of Labov [Labov, 1972] and
Linde [Linde, 1981], as briefly described below);
 hyperlinks to related material, including tutorials for proof rules
used, input and output to a formal theorem prover (if available), and
motivation and explanation for proof strategies and steps; and
 image schemas (in the sense of Lakoff [Lakoff and Johnson, 1980,Lakoff, 1987]).
As with any virtual world, image schemas can make the language more direct and
powerful, and hence easier to follow.
Aristotle said, "Drama is conflict" [Aristotle, 1997], which suggests
providing conflict to add drama to proofs. Finding a nontrivial proof
usually requires exploring many dead ends, errors and misconceptions, some of
which may be very subtle. Therefore the process of proving can be full of
disappointed hopes, unexpected triumphs, repeated failures, and even fear and
interpersonal conflict. All this is typically left out when proofs are
written up, leaving only the map of a path that has been cleared through the
jungle. But proofs can be made much more interesting and understandable if
some of the conflicts that motivated their difficult steps are integrated into
their structure; proof obstacles are exactly what is needed for drama. Of
course, this must be done with care, and it should not be overdone, just as in
a good novel or movie. Our Kumo theorem proving system [Goguen et al., 2000] used
these ideas to structure the websites that it generates to display proofs.
Aristotle also gave other useful suggestions, including unity of time and
place, and having a beginning, middle, and end to a drama [Aristotle, 1997].
William Labov [Labov, 1972] showed that oral narratives of personal
experience have a precise internal structure, which includes the following:
 an optional orientation section, which provides basic orientation
information, such as the time and place of the story, and perhaps some major
characters;
 a sequence of narrative clauses which describe the events of the
story;
 the narrative presupposition, which by default assumes that the
ordering of the narrative clauses corresponds to the temporal ordering of the
events that they describe;
 evaluative material interleaved with the narrative clauses, which
"evaluates" the events, in the sense of relating them to socially shared
values; and finally
 an optional closing section, which may contains a "moral" or
summary for the story.
The above follows [Linde, 1981,Linde, 1993], which describe developments
subsequent to the classic treatment of [Labov, 1972]. Although this
empirical research used oral narratives of personal experience as data, its
results apply much more broadly (though in general less precisely), since the
class of narratives is the core around which many discourse types are built.
Figure
Figure 8: A Typical Tatami Homepage and Proof Page
To aid our discussion of proofs as representations, we introduce terminology
for the source and target semiotic spaces: let us call their elements
abstract proofwebs and proof displays, respectively, and perhaps also
use the term display proofweb for target signs. In addition, the term
unit refers to a block of information of the same kind in a proof
display. The display proofwebs generated by Kumo adhere to the following
style guidelines, called the tatami conventions [Goguen et al., 2000]. The first
eight are justified mainly by narratology:
 Homepages are provided for every major proof part; homepages
introduce and motivate the problem to be solved and the approach taken to the
solution for that part, and correspond to the orientation sections of Labov's
narrative structure; they may contain graphics, applets, and of course text.
Homepages appear in the same window as their tatami pages (see next the item)
because they are part of the same narrative flow. See Figure 8.
 Tatami pages, also called proof pages, are the basic
constituents of display proofwebs; they are XML pages containing one or more
proof unit, with its inference rule applications, interleaved with one or
more explanation units. This interleaving follows the interleaving of
narrative and evaluative material in Labov's theory. Limiting the number of
nonautomatic proof steps on tatami pages to approximately 7 is justified by
classic work of Miller on limitations of human cognitive capacity
[Miller, 1956]; this limitation also makes it feasible to place both proof
and explanation units on the same proof page. See Figure 8.
 The explanation units of tatami pages are proversupplied
informal discussions of proof concepts, strategies, obstacles, etc. They
correspond to the evaluative material in Labov's theory, and motivate
important proof steps by relating them to values shared in the appropriate
community of provers.
 Tatami pages can be browsed in an order designed by the prover to be
helpful and interesting to the reader; if possible, they should tell a story
about how obstacles were overcome (or still remain). This narrative order
again comes from Labov's theory, while including obstacles comes from work of
Campbell [Campbell, 1973] and others on "heroic" narratives.
 Major proof parts, including lemmas, have their own subwebsites, each
with the same structure as the main proof, including homepage and explanation
units. These appear in a separate dedicated persistent popup window. Having
separate hyperlinked websites for major proof parts is similar to the way
that flashbacks and other temporal dislocations occur in stories. It is
helpful to have them in a separate window in order to clarify their relation
to the main sequence of proof steps.
 Tatami pages also have associated formal proof scores, which appear in
another separate popup window when summoned from a tatami page. The separate
window is convenient because users typically want to look at the formal proof
and its motivation at the same time as the proof score. Users can also
request proof score execution, and the result is displayed in the same window
as the score, so that one can easily alternate between them. (The proof
score is sent to an OBJ server and the result is returned for display.) This
hiding of routine details is similar to human proofs, which use it to
highlight the main ideas [Livingston, 1987].
 Major proof parts can have an optional closing page, to sum up
important results and lessons, again following Labov's theory. They appear
in the same window as proof pages, again because they are part of the same
narrative flow.
 A menu of open subgoals appears on each homepage, and error messages
are placed on appropriate pages. Open subgoals are important to provers when
they read a proof, since proving new results is a major value within this
community.
Now we give further style guidelines, with justifications based on algebraic
semiotics:
 Windows: The main contents of a display proofweb are its proof
steps, informal explanations, tutorials, and mechanical proof scores. These
four are also the main contents of abstract proofwebs, and their preservation
has much to do with the quality of their representation. These four basic
sorts of the abstract data type for proofwebs are reflected in our choice of
windows for displaying them. Because tatami pages are the main constituent
of proofwebs, theirs is the master window, and because explanation pages are
so closely linked, they share that window; each unit is enclosed is its own
"box." Tutorial and machine proof score pages each have a separate window.
All this preserves the hierarchical structure and priorities of the
underlying mathematics.
 Backgrounds: Each major sort of unit has its own background color:
proof units have light beige, explanations have light yellow, tutorials have
yellow marble, and proof scores have light purple. Although the choice of
colors is somewhat arbitrary, and is easily changed by editing the XSL style
file, their distinctness reflects the importance of distinguishing these four
units.
 Navigation: Similar considerations hold for navigation. Each
page has a title, supplied by the user in the Duck script (or a simple default
if no title is supplied). Buttons are used to move to other pages of the same
sort, and to open widows that display information of other sorts. Each
persistent window has somewhat different layout and navigation buttons,
reflecting its different typical uses. For example, the master tatami window
has buttons to step through the narrative ordering of tatami pages, both
forward and backward, and a button to return to the homepage.
 Mathematical Formulae: gif files are used for mathematical
symbols, in a distinctive blue color, because mathematical signs come from a
domain that is quite distinct from that of natural language.
Some additional applications of semiotic morphisms to the user interface
design of the Tatami system are described in [Goguen, 1999b], in a more precise
style than here, although they are based on an older version of the system.
For example, [Goguen, 1999b] shows that certain early designs for the status
window were incorrect because the corresponding semiotic morphisms failed to
preserve certain key constructors.
We have studied a corpus of over 50 "humorous oxymorons" (phrases like
"military intelligence," "good grief," and "almost exactly").
Dictionaries say an "oxymoron" is a phrase having contradictory (or
incongruous) components. But this is not what happens in a humorous oxymoron:
instead, there are two distinct meanings, one of which is conventional, and
the other of which has some contradictory components; i.e., there are two
different blends, one of which has conflicts. When we are told that something
is an oxymoron, we seek out that second, conflictual blend, and we feel
pleasure when we find it.
We also studied over 40 newspaper cartoons, and found that about 75 percent
have a similar pattern, but instead of two blends existing simultaneously, the
reader is first led to form one blend, and then led by new information to form
a completely different blend, usually in partial conflict with the first; that
is, there is a kind of dynamic reblending.
Thus in each case, it is not just the existence of more than one blend, but
rather the process of reblending that produces the humorous effect,
and I conjecture that reblending in fact characterizes humor. This is
relevant to HCI and the design of virtual worlds, because humor is sometimes
used in computer system interfaces, often very badly. For example, the
paperclip in MicroSoft Office creates a poor impression in part because the
sensation of reblending loses loses its effect if it is repeated many times,
and eventually becomes "stale" or even unpleasant. See [Goguen, 2004a] for
additional details. These observations, which go back to about 1999, seem to
have potential for fascinating new application areas.
4.4 Interaction
Classical semiotics is concerned with static signs; it does not allow for
signs that change in response to user input, or that move on their own. This
section sketches how algebraic semiotics handles dynamics, by extending its
foundation from classical algebra to hidden algebra. As a simple example,
consider the problem of designing that part of the Kumo interface that
supports browsing proofs. Kumo provides buttons to traverse in the proof
author's chosen narrative order, labeled with iconic triangles to indicate
forward and backward motion, as well as buttons to return to the homepage, to
view the specification, etc. (see Figure 8). Common practice
would suggest constructing an automaton with a state for each proof tree node,
and a transition label for each traversal button. But this does not allow for
the fact that different proofs have different structures, and thus different
automata, nor does it account for the different displays that are produced in
each state, nor for the variety of possible implementations of transition
lookup, e.g., using lists, arrays, or hash tables. An automaton can describe
how a single proof instance can be navigated, but it cannot describe the
general method which generates proof navigation support for any given proof,
nor the way that this method is implemented, nor the quality of the resulting
interface.
In fact, despite the formal character of the model itself, the construction
and use of transition diagrams (or the corresponding automata) in user
interface design is intuitive, and does not provide an adequate basis for a
rigorous mathematical analysis of possible designs. In order to address the
display, implementation and quality questions raised above, the automaton
model must be supplemented in various ad hoc ways, whereas hidden
algebra can handle all of these within a single unified framework. Another
example of dynamics in Kumo that would be difficult to handle with traditional
user interface modeling techniques is the facility to execute the proof script
for a proof part by downloading it to a BOBJ proof server and then viewing the
result on a local browser as it executes.
This is not the place for details (see [Goguen, 2003] for that), but we can
say that hidden algebra provides a precise way to handle both the display and
implementation aspects of examples like that described above, and the
corresponding extension of semiotic morphisms gives a precise basis for
comparing the quality of interface designs realizing the desired dynamics,
without bias towards any particular implementation. The dynamics of a window
with a scrollbar is discussed in considerable detail in [Goguen, 2003].
5 Summary, Future Research, and Social Implications
This paper has presented theory and case studies to support the claim that
algebraic semiotics is a promising foundation for virtual world design, in
both theory and practice. The case studies on information visualization,
proof presentation, metaphor, humor, and interaction are encouraging, and
suggest that design problems can be successfully confronted directly, without
unreliable ad hoc methods and assumptions, such as analyses based on
prior systems that are only remotely related, or expensive, timeconsuming
methods of experimental psychology and usability testing. These studies also
confirm our views that taking account of key social and cognitive factors is
crucial for success, and that formal methods can play a very helpful role, if
applied pragmatically rather than dogmatically. However, much more work is
still needed, such as:
 indent 1.7ex
 Combining Gibsonian affordances [Gibson, 1977] with algebraic semiotics,
to provide a sociocognitive dimension for the interaction formalism
discussed in Section 4.4.
 Studying immersion in virtual worlds, e.g., how closure and embodiment
relate to representational coherence, image schemas, affordances, choice of
media, etc.
 More work on social foundations and the processes of semiosis.
 More work on narrative structure, including flashbacks and flashforwards.
 More work on how to choose quality orderings on representations that are
appropriate to their actual use.
 More case studies, done more thoroughly.
Only the second of these is specific to virtuality, though all are related.
We hope that others may find some benefit to the algebraic semiotic approach,
and will contribute to its further development.
I close this article with some words of warning, along lines perhaps most
closely associated with Jean Baudrillard [Baudrillard, 1994], who wrote:
Simulation is no longer that of a territory, a referential being, or a
substance. It is the generation by models of a real without origin or
reality. ... By crossing into a space... no longer that of the real, nor
that of truth, the era of simulation is inaugurated by a liquidation of all
referentials  worse: with their artificial resurrection in the systems of
signs, a material more malleable than meaning, in that it lends itself to all
systems of equivalences, to all binary oppositions, to all combinatory
algebra. It is no longer a question of imitation, nor duplication, not even
parody. It is a question of substituting the signs of the real for the real,
that is to say of an operation of deterring every real process via its
operational double, a programmatic, metastable, perfectly descriptive machine
that offers all the signs of the real and shortcircuits all its vicissitudes.
Never again will the real have the chance to produce itself  such is the
vital function of the model in a system of death ...
If we translate this out of the stylistic conventions of recent French
intellectualism, the danger is that the virtual can replace the real in our
affections, so that we lose touch with our communities, our values, even the
very living quality of our lives. Baudrillard claims that exactly such
alienation is already characteristic of the contemporary world, and that it is
growing like a cancer. He does not offer any solution to this dilemma, but I
would like to suggest that compassion [Goguen, 2004b] is one way out of an
enervating absorption in virtuality. A sympathetic feeling for the suffering
of others, and action on their behalf, can generate positive emotionality and
reengagement with real experience. And, contrary to Baudrillard, it seems
quite possible that technology, including virtual world technology, can assist
with such projects.
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Contents
1 Introduction: Motivation, Difficulties and Approaches
2 Algebraic Semiotics
2.1 Signs and Sign Systems
2.2 Representations
2.3 Simple Examples
2.4 Quality of Representation
3 Fragments of a Calculus of Representation
3.1 Metaphor and Blending
3.2 Some Laws
4 Case Studies
4.1 Information Visualization
4.1.1 Code Visualization
4.1.2 A Film Visualizer
4.1.3 A Later Version
4.2 Proof Representation and Understanding
4.3 Humor
4.4 Interaction
5 Summary, Future Research, and Social Implications
Footnotes:
^{1}Due to the nature of this paper, sign systems are not fully
formalized, and in particular, signatures are treated rather informally,
because they are sufficiently complex that a formal definition would distract
from the flow of ideas; see [Goguen and Malcolm, 1996,Goguen et al., 1978] for the formal definition of
signature, and see [Goguen, 1999a] for the formal definition of sign system.
^{2}These are for fixed data types
like integers, booleans and colors, which are always interpreted in a
standard way.
^{3}This use of the word
"space" conflicts with cognitive linguistics, where conceptual spaces, which
are discused below.
^{4}This duality is a Galois connection between algebraic theories and
their models; it does not involve the levels or priorities.
^{5}The source up is grounded in our experience of gravity, and the
schema itself is grounded in everyday experiences, such as that when there is
more beer in a glass, or more peanuts in a pile, the level goes up, and that
this is a state we often prefer; therefore the image schema more is up,
discussed in [Lakoff, 1987], is even more basic.
^{6}However, we do not assume that they are
necessarily the minimal such spaces needed to understand a given blend,
since that can only be determined after the blend has been understood.
Moreover, different blends may ignore different elements of the input spaces,
and it may also be necessary to recruit additional information from other
spaces in order to understand a blend.
^{7}The term "base space" is used in [Goguen, 1999a], because it is
considered to better describe how this theory is used in applications to user
interface design.
^{8}Strict commutativity, usually called just commutativity, means
that the compositions are strictly equal, i.e., one morphism is defined on an
element if and only if the other is, and then they are equal.
^{9}But as
before, gray tones appear in our rendition of the display.
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