The dataset was manually segmented into episodes. Each of the 48 episodes
contains a varying number of instances of seven distinct episode types,
labeled as A, B1, B2, C1, C2, D, and E. Collectively, the dataset comprises
356 instances of these episode types.

Utilizing domain knowledge, we can define a library $L$ to characterize
relations between attributes. One straightforward approach to constructing
this library is by directly specifying its parsing graph $\mathcal{G}(L)$. In
this regard, we designate the following signs: _Move_ , _Objects_ , _Path_ ,
_Node_ , _Episode_ , _Rotate_ , _Stalled_ , and _Class_. Additionally, we
specify components such as _pushing_ , _direction_ , $\text{stat}_{1}$,
_proximity_ , _source_ , _target_ , _start_ , _end_ , direction,
$\text{stat}_{2}$, and _type_ , with their constraints delineated in the graph
below.

Figure 8.2: Parsing Graph and an example query.

Assigning a model to this structure involves fixing an interpretation for each
sign and component. The semiotic system defined by this structure is
consistent with the available data if there exists a word in the associated
graphic language that contains the provided dataset. However, the expressive
limitations of the chosen language render this notion of consistency too
restrictive for practical purposes. To address this, we introduce the concept
of $\lambda$-consistency with the data for a semiotic: a specification
expressed in the graphic language is $\lambda$-consistent with the data if
there exists an interpretation that provides a "good approximation" to the
given data.

For the parsing graph depicted in Figure 8.2, the signs’ interpretation
domains come equipped with a similarity relation. The limit of the diagram in
Figure 8.2, where we identify the diagram sources as
$\\{\text{Move},\text{Object},\text{Rotate},\text{Stalled}\\}$, the target as
$\\{\text{Class}\\}$, and auxiliary signs as
$\\{\text{Path},\text{Node},\text{Episode}\\}$, serves as a "good"
approximation to the dataset. This limit can be represented as an
$\Omega$-set:

$\alpha:\text{Move}\times\text{Object}\times\text{Rotate}\times\text{Stalled}\times\text{Class}$

The discrepancies between $\alpha$ and the actual data should be viewed as
information that is not semantically valid within the defined semiotic
framework. Such limits provide a perspective of the data as described by the
semiotic. However, the information encapsulated in this generated limit is not
suitable for solving the proposed pattern detection problem using machine
learning algorithms. This is because it fails to encode the temporal relations
between states in the time series generated by the robot’s perceptual system.

We utilized limits of admissible configurations to extract potentially
valuable information from the universe modeled by the semiotic. The existence
of patterns associated with the robot stalling in the first three states of
each episode should be discernible in the limit for diagram $D(a)$ in Figure
8.3, using the appropriate machine learning tools.

Figure 8.3: Queries $D(a)$ and $D(b)$.

However, for episode classification, the last robot states appear to be more
relevant. Information patterns associated with the last three states of a
robot are evident in the limit for diagram $D(b)$ from Figure 8.3.

To utilize relational information available for three consecutive states in
episode class prediction, we need to modify the current library. A challenge
arises: the existing library structure lacks the expressive power required to
describe this type of query comprehensively. Enhancing the library’s
expressive capacity involves adding an additional component: associating each
automaton state with its respective episode. The query can then be defined by
diagram $D(c)$ in Figure 8.3.

Library updates can also be implemented to constrain their interpretations. In
Figure 8.4, we enriched the sign system by imposing interpretive restrictions
through the addition of two equalizers. These were employed to mandate that in
a path, the source and the target must differ. This constraint can be codified
by restricting the equalizer between the _source_ and _target_ components to
an initial relation. If the consideration is also limited to episodes with
more than one state, the limit of the diagram defined by the _start_ and _end_
components must be the initial relation. The specification below is further
enriched with new components $\text{Lim}\;D(a)$, $\text{Lim}\;D(b)$,
$\text{Lim}\;D(c)$, and $\text{Lim}\;D(d)$ – interpreted as the limits of the
presented queries $D(a)$, $D(b)$, $D(c)$, and $D(d)$, respectively – added as
signs interpreted as the source of these new components. These new signs
introduce an additional layer to the sign ontology, being more general than
the initially defined signs.

Figure 8.4: Enriched sign system.

Note that the limit of diagram $D(c)$, presented in Figure 8.3, describes
available information about signs interpreted as three consecutive states,
while the limit of $D(d)$ characterizes three consecutive states of episodes
where the robot stalls.

Figure 8.5: $D(c)$ querying three consecutive states and $D(d)$ querying three
consecutive states of episodes where the robot stalls.

In the example provided, we utilized limits as a method to query the structure
of a semiotic system. In the following chapters, we will formalize some of the
concepts introduced in this example, particularly elucidating what we mean by
an "approximation" to the given data.

######  Example 16 (Neural Networks [19])

A neural network is comprised of a network of simple processing elements,
known as neurons, which can collectively exhibit complex global behavior. This
behavior is determined by the connections between the processing elements and
their associated parameters. Formally, a neural network can be represented as
a function $f$, defined as a composition of other functions $g_{i}(x)$, which
can themselves be further defined as compositions of additional functions.
These dependencies within the network can be conveniently represented using a
network structure, with arrows denoting the dependencies between variables.
One of the most intriguing aspects of neural networks is their capacity to be
parameterized for learning specific tasks. Given a particular task and a class
of functions $F$ defined using a network structure and a set of neurons,
learning entails utilizing a set of observations to identify $f^{*}\in F$ that
optimally solves the task.

In this context, an artificial neural network can be viewed as an admissible
diagram defined within a semiotic framework, which codifies every potential
parametrization of each processing element. More precisely, the associated
sign system describes the possible neural network structures, while a model
for it represents a set of parameterizations that characterize the behavior of
each processing element.

There are three major learning paradigms in neural networks, each
corresponding to a distinct learning task [2]. These paradigms are supervised
learning, unsupervised learning, and reinforcement learning. They can be
regarded as different approaches to exploring the space defined by models of
admissible configurations in order to identify the optimal solution—i.e., the
model that best fits the data.

Given a diagram $D$ in a Neural Network Semiotic, the multi-morphism
$\text{Lim}\;D$ describes the functional behavior of the network $D$ when
applied to its input space. In this context, a network $D$ has learned a
concept described in a dataset if a portion of its limit $\text{Lim}\;D$
serves as a good approximation to the dataset.

Additional examples can be drawn from the realm of generic programming
applications. For a comprehensive exploration of this topic, one can refer to
[16].

## Chapter 9 Logics

We can conceptualize a semiotic as a formal mechanism to specify a Universe of
Discourse for a given problem. We are particularly interested in the
specification of universes where entities can be characterized by propositions
in monoidal logics. To achieve this, we introduce the following definition:

###### Definition 17

A semiotic system $(S,M)$ is termed a _logic semiotic system_ if it is
characterized by the following components:

$S=(L:|L|\rightarrow(Chains\downarrow\Sigma^{+}),\mathcal{E},\mathcal{U},co\mathcal{U}),$

The system satisfies the following conditions:

  1. 1.

There exists a sign in $S$, denoted as the support $\Omega$, which is
interpreted as an ML-algebra. This algebra has operator interpretations
corresponding to the components labeled with the signs
$\vee,\wedge,\otimes,\Rightarrow,\bot$, and $\top$.

  2. 2.

For every string $w=s_{0}s_{1}\ldots s_{n}$ consisting of signs in $S$, there
exists a label $=_{w}$. This label is interpreted by $M$ as the similarity
measure $\bigotimes_{i}^{n}[\cdot=\cdot]_{i}$, where $[\cdot=\cdot]_{i}$
represents the similarity on the $\Omega$-set $M(s_{i})$, as defined in 8.

  3. 3.

For every sign $s$ in $S$ and every natural number $n$, there exists a
component in $S$ labeled by $\lhd^{n}_{s}$. This component is interpreted by
$M$ as a diagonal relation

$\lhd^{n}_{M(s)}:M(s)\rightharpoonup\prod^{n}_{i=1}M(s),$

given by

$\lhd^{n}_{M(s)}(a,a_{1},a_{2},\ldots,a_{n})=\bigotimes_{i}^{n}[a=a_{i}].$

  4. 4.

Analogously, for every sign $s$ in $S$ and every natural number $n$, there
exists a component labeled by $\rhd^{n}_{s}$. This component is interpreted by
$M$ as a codiagonal relation

$\rhd^{n}_{M(s)}:\prod^{n}_{i=1}M(s)\rightharpoonup M(s),$

and is given by

$\rhd^{n}_{M(s)}(a_{1},a_{2},\ldots,a_{n},a)=\bigotimes_{i}^{n}[a_{i}=a].$

  5. 5.

We assume the existence of a disjoint decomposition for the set of signs
$\Sigma$, which is given by $\Sigma_{aux}$ and $\Sigma_{pri}$. Here, signs in
$\Sigma_{aux}$ are termed _auxiliary_. For every pair
$(s,u)\in\Sigma_{pri}\times\Sigma_{aux}$, there exist components
$r(s,u):su^{+}$ and $r(u,s):us^{+}$ in $L$, referred to as _rename
components_. These components are interpreted by $M$ as the identity in
$M(s)$, i.e.,

$M(r(s,u))=id_{M(s)}\text{ and }M(r(u,s))=id_{M(s)}.$

In a semiotic logic, the signs $\triangleleft$ and $\triangleright$ play a
crucial role in connecting similar component inputs and outputs, respectively.
These signs serve as essential tools for establishing relationships between
the components, ensuring a coherent flow of information within the system.
Rename components, on the other hand, serve as a mechanism to codify the
connections or links between the inputs and outputs of various components
represented in the diagram.

Equations in this context can be elegantly represented using commutative
diagrams. As illustrated in Figure 9.1, we specify the property

$e_{s}+x=x+e_{s}=x$

which denotes the existence of an identity element $e_{s}$. This property is
depicted using a commutative diagram to visually capture the relationships and
operations involved.

A model $M$ for an additive library Ł${}_{A}(S,C)$ defines an additive
operator with identity if the diagram’s limit yields a total multi-morphism.
This ensures that the system possesses an additive structure where the
identity element interacts with the other components in a consistent and
coherent manner.

---
$\textstyle{\top}$$\textstyle{e_{s}}$$\textstyle{\lhd}$$\textstyle{+}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{=}$$\textstyle{\rhd}$$\textstyle{+}$$\textstyle{\lhd}$$\textstyle{=}$
Figure 9.1: Diagrams codifying $e_{s}+x=x+e_{s}=x$.

The diagram illustrated in Figure 9.1 can be represented in a string-based
notation by the chain of signs:

$\lhd^{3}_{s}r(s,x)r(s,x)r(s,x)e_{s}\lhd^{2}_{s}r(s,y)r(x,s)+r(s,z)r(x,s)r(y,s)+\lhd^{2}_{s}r(s,w)r(s,w)r(z,s)r(w,s)=r(w,s)r(x,s)=\top\rhd^{3}_{\Omega}.$

This notation demonstrates that any diagram that defines a word in a graphical
language can be codified using string-based notation with the aid of rename
components.

###### Example 17

Consider $Set(\Omega)$ defined using the ML-algebra
$\Omega=([0,1],\otimes,\Rightarrow,\vee,\wedge,0,1)$, where
$([0,1],\otimes,\Rightarrow)$ serves as a model for product logic, and
$([0,1],\vee,\wedge,0,1)$ represents the standard complete lattice defined in
$[0,1]$ by the "less than" relation. The $\Omega$-set, defined in
$A=\\{0,1,2\\}$ and characterized by the similarity relation provided in the
table below:

$[\\_\;=\;\\_]$ | 0 | 1 | 2
---|---|---|---
0 | 1.0 | 0.5 | 0.0
1 | 0.5 | 1 | 0.5
2 | 0.0 | 0.5 | 1.0

and the additive relational operators depicted in the following tables:

$\\_\;+\;0$ | 0 | 1 | 2
---|---|---|---
0 | 1.0 | 0.5 | 0.0
1 | 0.5 | 1 | 0.5
2 | 0.0 | 0.5 | 1.0
$\\_\;+\;1$ | 0 | 1 | 2
---|---|---|---
0 | 0.5 | 1.0 | 0.5
1 | 0.0 | 0.5 | 1.0
2 | 1.0 | 0.5 | 0.5
$\\_\;+\;2$ | 0 | 1 | 2
---|---|---|---
0 | 0.0 | 0.5 | 1.0
1 | 1.0 | 0.0 | 0.5
2 | 0.5 | 1.0 | 0.0

provide a model for an additive library. The identity element $e_{s}=0$ since
the diagram in Figure 9.1 yields the following limit:

A | $\Omega$
---|---
0 | 1.0=[0]
1 | 1.0=[1]
2 | 1.0=[2]

.

In a logic semiotic system $(S,M)$, the language $Lang(L)$ is referred to as a
_logic language_. Logic semiotics possesses sufficient expressive power to
differentiate between diagrams that define relations and those that define
entities. To achieve this, we classify words as follows:

###### Definition 18

(Graphic relations) A diagram $D\in Lang(S)$, for a logic semiotic $(S,M)$, is
termed a _relation_ when its output $o(D)$ is interpreted by $M$ as the set of
truth values $\Omega$ on $Set(\Omega)$.

###### Definition 19

A relation $D$ is designated as an _equation_ if the diagram $D$ can be
decomposed as

$D=I\otimes D_{0}\otimes D_{1}\otimes`=`,$

where $I=\lhd^{n_{1}}_{s_{1}}\ldots\lhd^{n_{m}}_{s_{m}}$ is defined through
realizations of diagonals. The diagrams $D_{0}$ and $D_{1}$ are diagrams that
satisfy $o(D_{0})=o(D_{1})$, and ’=’ is a diagram defined using the unique
component, interpreted as a similarity relation, and satisfying
$i(`=`)=o(D_{0})\cdot o(D_{1})$. In this case, we simplify the notation by
denoting the diagram $D$ as $D_{0}=D_{1}$. It is noteworthy that the diagram
$I$ codifies the dependencies between interpretations of signs used as inputs
for the diagram $D_{0}\otimes D_{1}\otimes`=`$, linking together signs with
similar values.

In the given definition, the diagram
$I=\lhd^{n_{1}}_{s_{1}}\otimes\ldots\otimes\lhd^{n_{m}}_{s_{m}}$ encapsulates,
within a graphic logic, the dependence relations defined in string-based logic
by repeatedly utilizing bounded variables in a proposition.

Figure 9.2: Sketch for a multi-morphism of type $D=I\otimes D_{0}\otimes
D_{1}\otimes`=`$.

The relation $D\in Lang(S)$ is termed _true_ by $M$ if the limit
$M(D\otimes\top\otimes\rhd^{2}_{\Omega})$ is a total multi-morphism. In this
context, an equation $D$ is considered universal if the interpretation of
$D.\top.\rhd^{2}_{\Omega}$ by $M$ is a total multi-morphism.

Given a logic semiotic system $(S,M)$, let $Lang_{R}(S,M)$ be the subcategory
of $Lang(S)$ consisting of diagrams defining relations. Utilizing the
operation of diagram gluing, we define the following operators for pairs of
relations $D_{0},D_{1}\in Lang_{R}(S,M)$:

  1. 1.

$D_{0}\otimes D_{1}$ corresponds to the diagram $I\otimes D_{0}\otimes
D_{1}\otimes`\otimes`$,

  2. 2.

$D_{0}\Rightarrow D_{1}$ corresponds to the diagram $I\otimes D_{0}\otimes
D_{1}\otimes`\Rightarrow`$,

  3. 3.

$D_{0}\wedge D_{1}$ corresponds to the diagram $I\otimes D_{0}\otimes
D_{1}\otimes`\wedge`$,

  4. 4.

$D_{0}\vee D_{1}$ corresponds to the diagram $I\otimes D_{0}\otimes
D_{1}\otimes`\vee`$,

where $I$ is defined through realizations of diagonals, linking together
inputs that have the same meaning according to $M$. This framework enables the
definition of new relations from pairs of simpler ones, representing the
lifting of the $\Omega$ structure to diagrams in $Lang_{R}(S,M)$.

In the subsequent chapters, we present some illustrative examples of logic
semiotics that are pivotal for characterizing the expressive power of
languages utilized by certain machine learning algorithms:

######  Example 18 (Binary semiotic $S_{B}(S,C)$)

A binary semiotic is a logic semiotic where sets $S$ and $C$ define a binary
library Ł${}_{B}(S,C)$ (as presented in example 9). We refer to this type of
semiotic as a dataset semiotic or table semiotic, as instantiations of
relations in $Lang_{R}(S_{B}(S,C))$ can be used to encode datasets or tables.

Binary semiotics find applications in machine learning algorithms designed to
generate decision rules, such as Apriori, as described in [19].

######  Example 19 (Linear semiotic $S_{L}(S,C)$)

A linear semiotic $S_{L}(S,C)$ serves as an extension of a binary semiotic. It
is characterized by a linear library Ł${}_{L}(S,C)$ (as illustrated in example
10). Within this framework, $\geq:ssl^{+}$ is interpreted as a partial order
in the interpretation of a sign $s$, denoted as $M(s)$, for all symbols $s\in
S$. This type of relation is encapsulated in the model of a linear semiotic
$S_{L}(S,C)=(L,\mathcal{E},\mathcal{U},co\mathcal{U})$ if it encompasses the
diagrams depicted in fig. 9.3. These diagrams codify propositions represented
in string-based first-order logic as follows:

$\forall x:x\geq x,$ $\forall x,y:(x\geq y\wedge y\geq x)\Rightarrow(x=y),$
$\forall x,y,z:(x\geq y\wedge y\geq z)\Rightarrow(x\geq z),$

and these propositions are "preserved" by its models.

$\textstyle{\top}$$\textstyle{\rhd}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{\geq}$$\textstyle{\top}$$\textstyle{\geq}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{\geq}$$\textstyle{\wedge}$$\textstyle{\rhd}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{\Rightarrow}$$\textstyle{=}$$\textstyle{\top}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{\geq}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{\geq}$$\textstyle{\wedge}$$\textstyle{\rhd}$$\textstyle{\geq}$$\textstyle{\Rightarrow}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$
Figure 9.3: Diagrams codifying $\forall x:x\geq x,\,\,\,\,\forall x,y:(x\geq
y\wedge y\geq x)\Rightarrow(x=y)$ and $\forall x,y,z:(x\geq y\wedge y\geq
z)\Rightarrow(x\geq z)$.

Let’s consider an example on $Set(\Omega)$ where $\Omega$ has the structure of
an ML-algebra. Here, $([0,1],\otimes,\Rightarrow)$ serves as a model for
product logic, and $([0,1],\vee,\wedge,0,1)$ represents the typical complete
lattice defined in $[0,1]$.

For the $\Omega$-set defined with the support $A=\\{0,1,2,3,4\\}$ and the
similarity relation:

$[\\_\;=\;\\_]$ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
0 | 1.0 | 0.5 | .25 | 0.0 | 0.0
1 | 0.5 | 1.0 | 0.5 | .25 | 0.0
2 | .25 | 0.5 | 1.0 | 0.5 | .25
3 | 0.0 | .25 | 0.5 | 1.0 | 0.5
4 | 0.0 | 0.0 | .25 | 0.5 | 1.0

and the relational operator defined from $A$ to $A$ as:

$[\\_\;\geq\;\\_]$ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
0 | 1.0 | 0.5 | .25 | 0.0 | 0.0
1 | 1.0 | 1.0 | 0.5 | .25 | 0.0
2 | 1.0 | 1.0 | 1.0 | 0.5 | .25
3 | 1.0 | 1.0 | 1.0 | 1.0 | 0.5
4 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0

When these relations are employed for sign interpretation in the three
diagrams depicted in fig. 9.3, they yield the following limits, respectively:

$\small\begin{tabular}[]{c|c}$A$&$\Omega$\\\ \hline\cr 0&1\\\ 1&1\\\ 2&1\\\
3&1\\\ 4&1\\\ \end{tabular}\;\;\begin{tabular}[]{c|c}$A\times A$&$\Omega$\\\
\hline\cr(0,0)&1\\\ (1,0)&1\\\ $\vdots$&$\vdots$\\\ (3,4)&1\\\ (4,4)&1\\\
\end{tabular}\;\;\text{ and }\begin{tabular}[]{c|c}$A\times A\times
A$&$\Omega$\\\ \hline\cr(0,0,0)&1\\\ (1,0,0)&1\\\ $\vdots$&$\vdots$\\\
(3,4,4)&1\\\ (4,4,4)&1\\\ \end{tabular}.$

This confirms the commutativity of each diagram.

We refer to this type of semiotics as grid semiotics. They often appear in
associated machine learning algorithms used to generate decision rules, such
as the C4.5Rules by J.R. Quinlan, as described in [19].

######  Example 20 (Additive semiotic $S_{A}(S,C)$)

An _additive semiotic_ $S_{A}(S,C)$ is a linear semiotic $S_{L}(S,C)$ such
that it possesses an additive library Ł${}_{A}(S,C)$ (as illustrated in
example 11). In this context, $(M(s),M(+:sss^{+}))$ forms a monoid for every
symbol $s\in S$. Furthermore, the interpretation for $e_{s}:s^{+}$ corresponds
to the monoid identity, denoted by $e_{s}\in C$. The properties of a monoid
can be effectively encoded within the semiotic structure if the model
transforms each of the diagrams depicted in fig. 9.4 into a total multi-
morphism for every sign $s$ in the semiotic.

$\textstyle{\top}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{+}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{+}$$\textstyle{=}$$\textstyle{\rhd}$$\textstyle{\top}$$\textstyle{e_{s}}$$\textstyle{+}$$\textstyle{=}$$\textstyle{\rhd}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$
---
$\textstyle{\top}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{+}$$\textstyle{+}$$\textstyle{=}$$\textstyle{\rhd}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{+}$$\textstyle{+}$$\textstyle{:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$
Figure 9.4: Diagrams codifying $\forall x,y:x+y=y+x,\forall x:x+e_{s}=x$ and
$\forall x,y,z:(x+y)+z=x+(y+z)$.

######  Example 21 (Multiplicative Semiotic $S_{M}(S,C)$)

An _multiplicative semiotic_ $S_{M}(S,C)$ is an additive semiotic $S_{L}(S,C)$
defined by a multiplicative library Ł${}_{M}(S,C)$ (as presented in example
12). In this context, $(M(s),\times)$ forms a commutative semigroup for every
symbol $s\in S$.

Within this semiotic framework, we can encode rules established through
regression, akin to the rules generated by machine learning algorithms such as
M5 by J.R. Quinlan, as described in [22].

The definition of a Domain of Discourse structure may impose constraints on
the interpretation of signs. In a binary semiotic, we might impose sign
interpretation rules defined by Horn clauses of the form:

$(x_{1}=c_{1}\wedge x_{2}=c_{2}\wedge x_{3}=c_{3})\Rightarrow y=c_{4}.$

The expressive power of a linear semiotic enables the encoding of rules like:

$(x_{1}\leq c_{1}\wedge x_{2}\leq c_{2}\wedge c_{3}\leq x_{1})\Rightarrow
y\leq c_{4},$

While in a multiplicative semiotic, sign interpretation can be confined to
semiotics defined by models that satisfy regression rules such as:

$(x_{1}\leq c_{1}\wedge x_{2}\leq(c_{3}\times x_{1}+c_{2})\wedge
x_{3}=c_{3})\Rightarrow y\leq c_{4}\times x_{1}+c_{5}\times x_{2}+c_{6}\times
x_{3}+c_{7}.$

This type of regression rules can be represented through diagrams, similar to
the one depicted in fig. 9.5, where frames highlight the evident subdiagrams.

---
$\textstyle{\top}$$\textstyle{x_{1}:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{{}_{x_{1}\leq
c_{1}}}$$\textstyle{\rhd}$$\textstyle{x_{2}:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{{}_{c_{3}x_{1}+c_{2}\leq
x_{2}}}$$\textstyle{\wedge}$$\textstyle{x_{3}:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{{}_{x_{3}=c_{3}}}$$\textstyle{\wedge}$$\textstyle{\Rightarrow}$$\textstyle{y:s\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{}_{y=c_{4}\times
x_{1}+c_{5}\times x_{2}+c_{6}\times x_{3}+c_{7}}}$ Figure 9.5: Diagram
codifying a regression rule.

Models generated by various Data Mining and Machine Learning tools can be
encoded within these sign systems. This implies that rules extracted from
models produced by learning algorithms can reflect the underlying data
structure. Such structures can enhance the richness of the sign system used to
specify the information system. This enhancement allows for the definition of
constraints on its models that are compatible with the existing data or
perspectives on reality. Essentially, this suggests that the most accurate
description we can have for an information system is the best generalization
derived from the stored data.

## Chapter 10 Lagrangian syntactic operators

The expressive power of our specification languages can be increased using
_Lagrangian syntactic operators_ or _sign operators_. This operator are
defined at the level of sign systems signs or/and components, and must be
preserved via sign systems models, allowing the generation of new signs or
components based on others signs or components.

An example of this operators, with evident applicability, are the differential
operators. For that we define:

######  Definition 20 (Differential semiotic)

A logic semiotic system $(S,M)$ is termed a _differential semiotic system_ if
it is a multiplicative semiotic that contains a sign $R$, which is interpreted
as the support for a ring $(M(R),\times,+,0,1)$. Additionally, the system
includes labels $\partial_{w}f$ in $S$, corresponding to components
$f:i(f)\rightarrow R$ in $S$ with an output of $R$, and where $w$ is a word
composed of its input symbols, subject to the following conditions:

  1. 1.

$w\leq i(f)$,

  2. 2.

$\partial_{w}f:d(f)\rightarrow R$,

  3. 3.

For the component label definitions presented below, utilizing
${}^{\prime}\times^{\prime}$ and ${}^{\prime}+^{\prime}$ in infix notation,
the following conditions must be met:

    1. (a)

If $f\equiv_{l}d_{0}\times d_{1}$, then $M(\partial_{w}(d_{0}\times
d_{1}))=M(\partial_{w}(d_{0})\times d_{1}+d_{0}\times\partial_{w}(d_{1}))$,

    2. (b)

If $f\equiv_{l}d_{0}+d_{1}$, then
$M(\partial_{w}(d_{0}+d_{1}))=M(\partial_{w}(d_{0})+\partial_{w}(d_{1}))$, and

    3. (c)

If $ww^{\prime}\leq i(f)$, then
$M(\partial_{ww^{\prime}}f)=M(\partial_{w}(\partial_{w^{\prime}}f))=M(\partial_{w^{\prime}}(\partial_{w}f))$.

The component label operator $\partial$ facilitates the characterization of
multi-morphisms that are otherwise impossible to represent within the
associated logic semiotic. For instance, in a differential semiotic system, we
can interpret a component $f:xy\rightarrow R$ as a multi-morphism that adheres
to the conservative law when the following diagram is interpreted by the model
as a total multi-morphism.

$\textstyle{\top}$$\textstyle{\partial_{x}f}$$\textstyle{\rhd}$$\textstyle{:xy\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\lhd}$$\textstyle{=}$$\textstyle{\partial_{y}f}$
Figure 10.1: Conservative law in a diferencial semiotic.

Let’s consider a naive application:

######  Example 22 (Modeling traffic)

In the Lighthill-Whitham-Richards (LWR) model, as presented in [31] and [23],
the traffic state is represented from a macroscopic viewpoint by the function
$\rho(x,t)$, which denotes the density of vehicles at position $x$ and time
$t$. The dynamics of the traffic are encapsulated by a conservation law
expressed as

$\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v)}{\partial x}=0,$

where $v=v(x,t)$ represents the velocity of cars at $(x,t)$. The fundamental
assumption of the LWR model posits that drivers adjust their speed
instantaneously based on the surrounding density:

$v(x,t)=V(\rho(x,t)),$

where the function $f(\rho)=\rho V(\rho)$ serves as the "flow rate," denoting
the number of vehicles per unit time. Consequently, we have

$\frac{\partial\rho}{\partial t}+\frac{\partial f(\rho)}{\partial x}=0.$

The model is formulated for a single unidirectional road, defining a
differential semiotic system with the foundational library depicted in Figure
10.2.

$\textstyle{\lambda}$$\textstyle{0}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\leq}$$\textstyle{0}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\rho}$$\textstyle{[0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{f}$
Figure 10.2: Library for a single unidirectional road model.

The associated sign system is characterized by the total diagrams

$\partial_{t}\rho+\partial_{x}f(\rho)=0\quad\text{and}\quad
0\leq\rho\leq\lambda,$

where the latter condition defines the maximum density of the road. A model
for this sign system can be conceptualized as an admissible distribution of
cars on the road.

The initial library lacks the descriptive power required to specify a road
network comprehensively. In Figure 10.4, we introduce an extension to the
initial library that enables the graphical modeling of a single network.

$\scriptstyle{X_{1}}$$\scriptstyle{X_{3}}$$\scriptstyle{X_{2}}$ Figure 10.3: A
road network defined by the junction of two incoming roads, $X_{1}$ and
$X_{2}$, and one outgoing road, $X_{3}$, all with a single direction.

This network is characterized by the junction of two incoming roads, $X_{1}$
and $X_{2}$, and one outgoing road, $X_{3}$, all with a single direction. The
semiotic for this problem is defined using the extended library depicted in
Figure 10.4.

$\textstyle{\lambda_{1}}$$\textstyle{\lambda_{2}}$$\textstyle{\lambda_{3}}$$\textstyle{0}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\leq}$$\textstyle{0}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\rho_{1}}$$\textstyle{[0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{c}$$\textstyle{\rho_{2}}$$\textstyle{\rho_{3}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 10.4: Extended library for the presented road network.

In this extended library, we have also included three "flow rate" components:

$f_{1}:X_{1}\times R\times[0,1]\rightharpoonup R,\quad f_{2}:X_{2}\times
R\times[0,1]\rightharpoonup R,\quad\text{and}\quad f_{3}:X_{3}\times
R\times[0,1]\rightharpoonup R,$

one for each road. Additionally, we introduce initial condition constants
$\rho_{1,0},\rho_{2,0},\rho_{3,0}:\bot\rightharpoonup R$. The restrictions of
the network traffic model are described by the following conservation laws and
flow restrictions for each road:

  1. 1.

$\partial_{t}\rho_{1}+\partial_{x}f_{1}(\rho_{1})=0$ and
$0\leq\rho_{1}\leq\lambda_{1}$,

  2. 2.

$\partial_{t}\rho_{2}+\partial_{x}f_{2}(\rho_{2})=0$ and
$0\leq\rho_{2}\leq\lambda_{2}$, and

  3. 3.

$\partial_{t}\rho_{3}+\partial_{x}f_{3}(\rho_{3})=0$ and
$0\leq\rho_{3}\leq\lambda_{3}$.

To complete the model description, we define the mechanisms that occur at the
junction. A fundamental condition is the conservation of flows:

$f_{1}(\rho_{1}(0,t))+f_{2}(\rho_{2}(0,t))=f_{3}(\rho_{3}(0,t)),\quad\forall
t.$

One elementary problem we can study, which also forms the basis for a specific
model, is the Riemann problem. For a Riemann problem at a junction, we
consider a constant density on all three roads as the initial condition:

  1. 1.

$\rho_{1}(x,0)=\rho_{1,0}$,

  2. 2.

$\rho_{2}(x,0)=\rho_{2,0}$, and

  3. 3.

$\rho_{3}(x,0)=\rho_{3,0}$.

Models derived from this sign system may be inherently unrealistic. For
example, constants such as $\rho_{1}=\lambda_{1}$, $\rho_{2}=\lambda_{2}$, and
$\rho_{3}=0$ can be associated with a conceivable semiotic representation.
However, this model is clearly counterintuitive in most scenarios (except, of
course, in the presence of a red light at the entrance of the third road). A
natural approach to constructing a realistic model for a specific road
junction involves incorporating rules that describe driver behavior at the
junction. We can enrich the sign system by incorporating additional
information obtained from models generated by machine learning algorithms for
the data. This integration necessitates combining the defined semiotic with a
semiotic associated with the language used to describe machine learning
models, a challenge we address in the following chapter.

## Chapter 11 Concept description

A _concept description_ , using attributes $(A_{i})$, in the logic semiotic
system $(S,M)$ is represented by an $\Omega$-map:

$d:\prod_{i}A_{i}\rightarrow\Omega,$

where the family $(A_{i})$ forms a sequence of $\Omega$-sets. A concept
description is deemed to be defined in a semiotic $(S,M)$ if the sequence
$(A_{i})$ is constructed using interpretations of signs within the language
$Lang(S)$. For brevity, we denote $d\in M(S)$ when we aim to select a concept
description within the semiotic $(S,M)$. It’s important to note that $d$
defines a relation in a monoidal logic, which may not necessarily be
interpretable by $M$ as a relation in $Lang(S)$. Intuitively, a concept
description can be visualized as the state of knowledge about a concept at a
given moment. A database specified by the sign system $S$ can encapsulate the
concept $d$ if there exists a model $M\in Mod(S)$ and a diagram $D\in Lang(S)$
such that

$M(D)=d.$

In this scenario, we assert that the query $D$ on the information system
defined by semiotic $(S,M)$ yields the answer $d$.

Given two concept descriptions

$d_{0}:\prod_{i}A_{i}\rightarrow\Omega\quad\text{and}\quad
d_{1}:\prod_{i}A_{i}\rightarrow\Omega,$

we denote $d_{0}\leq d_{1}$ if $d_{0}(\bar{x})\leq d_{1}(\bar{x})$ for every
$\bar{x}$ in $\Pi A_{i}$. It is noteworthy that every $\Omega$-set $A$ defines
a concept description via the extent map $[\cdot]:A\rightarrow\Omega$. In this
context, we regard every $\Omega$-set as a concept description. Within
$Set(\Omega)$, every $\Omega$-set with support $\Pi A_{i}$ is associated with
a complete lattice of concept descriptions, having a bottom element $\bot:\Pi
A_{i}\rightarrow\Omega$ and a top element $\top:\Pi A_{i}\rightarrow\Omega$.
Notably, the limit $M(D)$ serves as a concept description for every $D\in
Lang(S)$.

While some concept descriptions can be codified semantically, others cannot.
Given a pair of concept descriptions

$d_{0}:\Pi A_{i}\rightarrow\Omega\quad\text{and}\quad d_{1}:\Pi
A_{i}\rightarrow\Omega,$

we define

$\Gamma(d_{0},d_{1})=d_{0}\Leftrightarrow d_{1}.$

The biimplication $\Gamma$ is a $\otimes$-_similarity relation_ in $\Pi A_{i}$
due to the following properties:

  1. 1.

$\Gamma(d_{0},d_{0})=\top$ (reflexivity),

  2. 2.

$\Gamma(d_{0},d_{1})=\Gamma(d_{1},d_{0})$ (symmetry), and

  3. 3.

$\Gamma(d_{0},d_{1})\otimes\Gamma(d_{1},d_{2})=\Gamma(d_{0},d_{2})$
(transitivity) (as per Proposition 2).

When $\Omega=\\{\bot,\top\\}$ represents a two-valued logic, $\Gamma$ clearly
emerges as an equivalence relation on $\Pi A_{i}$.

###### Definition 21

Given a semiotic $(S,M)$ and a concept $d$, a diagram $D$ serves as a
$\lambda$-codification or a $\lambda$-description for $d$ if

$\Gamma(d,M(D))\geq\lambda.$

In this context, we also refer to $M(D)$ as an approximation to the concept
$d$. Consequently, the relation $D$ acts as a hypothesis that describes the
concept encapsulated by $d$, chosen from the language $Lang(S)$.

This definition can be extended to concept descriptions with differing support
sets. Given concept descriptions

$d_{I}:\Pi_{i\in I}A_{i}\rightarrow\Omega\quad\text{and}\quad d_{J}:\Pi_{j\in
J}A_{j}\rightarrow\Omega,$

such that a projection map $\pi:\Pi_{i\in I}A_{i}\rightarrow\Pi_{j\in J}A_{j}$
exists, we denote $d_{J}\preceq d_{I}$. We term $d_{J}$ a $\lambda$-projection
of $d_{I}$ if

$\Gamma(\pi\otimes d_{I},d_{J})\geq\lambda.$

For a given concept description $d$ and a hypothesis $D$ in $Lang(S)$, the
quality of $D$ as a descriptor for $d$ is quantified by:

$[d=D]=\bigwedge_{\bar{x}}\Gamma(d,D)(\bar{x}),$

where

  1. 1.

$\Gamma(d,D)(\bar{x})=(\pi\otimes M(D)\Leftrightarrow d)$ if $d\preceq M(D)$,

  2. 2.

$\Gamma(d,D)(\bar{x})=(M(D)\Leftrightarrow\pi\otimes d)$ if $M(D)\preceq d$.

We view a model as the fuzzy responses to a query within a semiotic, which we
formally define as:

###### Definition 22

A concept $d$ serves as a $\lambda$-model for $D$ in $Lang(S)$ if either
$d\preceq M(D)$ or $M(D)\preceq d$. Additionally, considering the diagram
depicted in Fig. 11.1, we define a $\lambda$-model as a pullback where

$\Gamma_{\lambda}(d,D)=\Pi A_{i},$

given that $[\lambda,\top]$ forms a chain within the lattice $\Omega$.
Consequently, we denote this relationship as

$d\models_{\lambda}\forall D.$

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\Gamma_{\lambda}(d,D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lrcorner}$$\scriptstyle{\subset}$$\textstyle{[\lambda,\top]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{\Pi
A_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma(d,D)}$$\textstyle{\Omega}$
Figure 11.1: Diagram evaluation.

In the aforementioned pullback diagram, if

$\Gamma_{\lambda}(d,D)\subseteq\Pi A_{i}\text{ and
}\Gamma_{\lambda}(d,D)\neq\emptyset,$

we express this relationship as

$d\models_{\lambda}\exists D,$

or, with more precision,

$d\models_{\lambda}\forall_{B}D,$

where $B=\Gamma_{\lambda}(d,D)$. This notation is further employed as
$d\models_{\lambda}\forall_{C}D$ when $C\subseteq\Gamma_{\lambda}(d,D)$.

When $d\models_{\top}\forall D$, we simplify this to $d\models\forall D$,
indicating that $d$ can be regarded as the answer to the query $D$ within the
information system defined by $(S,M)$. Similarly, if $d\models_{\top}\exists
D$, we denote it as $d\models\exists D$, suggesting that a part of $d$ is
$\lambda$-consistent with the interpretation of $D$ in the semiotic $(S,M)$.

It’s important to note that $\forall_{B}D$ might not always be interpretable
as a formula within the language associated with the employed semiotic, as the
language might lack the necessary expressive power to define $B$. However, if

$B=\Gamma_{\lambda}(d,D)=\Gamma_{\top}(\chi_{B},D^{\prime}),$

meaning if the domain $B$ can be explicitly defined using the diagram
$D^{\prime}$ in the language, we express this relationship as

$d\models_{\lambda}\forall_{D^{\prime}}D.$

Additionally, in this scenario, for every description $d$, we have

$d\models_{\lambda}\forall_{D^{\prime}}D\Leftrightarrow
d\models_{\lambda}\forall D^{\prime}\Rightarrow D.$

Given $d_{0}\models_{\lambda_{0}}\forall_{B_{0}}D_{0}$ and
$d_{1}\models_{\lambda_{1}}\forall_{B_{1}}D_{1}$, we derive:

$d_{0}\otimes d_{1}\models_{\lambda_{0}\otimes\lambda_{1}}\forall_{B_{0}\cap
B_{1}}D_{0}\otimes D_{1},$ $d_{0}\vee
d_{1}\models_{\lambda_{0}\vee\lambda_{1}}\forall_{B_{0}\cap B_{1}}D_{0}\vee
D_{1},$ $d_{0}\Rightarrow
d_{1}\models_{\lambda_{0}\Rightarrow\lambda_{1}}\forall_{B_{0}\cap
B_{1}}D_{0}\Rightarrow D_{1}.$

Given the proposed definition, it follows that every diagram possesses a
$\lambda$-model:

###### Proposition 17

In a logical semiotic $(S,M)$, if $D$ is a relation defined in the language
$Lang(S)$, then

$MD\models\forall D.$

From the introduced notion of similarity, characterized by biimplication, we
can identify as $\lambda$-models for $D$ the concepts that are
$\lambda$-similar to its interpretation $MD$:

###### Proposition 18

If $\Gamma(d,MD)\geq\lambda$, then $d\models_{\lambda}\forall D$.

Naturally, we utilize the definition of similarity to formalize our
understanding of concepts that are consistent with relations.

######  Definition 23 (Consistence)

Given a semiotic $(S,M)$, a relation $D$ from $Lang(S)$ is said to be
_consistent with_ $d\in M(S)$ if $d\models\forall D$. It is
$\lambda$-_consistent with_ $d$ when $d\models_{\lambda}\forall D$. Moreover,
the relation $D$ is consistent with a part of $d$ if $d\models\exists D$, and
it is $\lambda$-_consistent with a part of_ $d$ when
$d\models_{\lambda}\exists D$.

The set of hypotheses consistent with $d$ is denoted by $Hy_{(S,M)}(d)$. For
every $\lambda\in\Omega$, the set of hypotheses $\lambda$-consistent with $d$
is represented as $\lambda$-$Hy_{(S,M)}(d)$. For a chain of truth values in
$\Omega$ given by

$\top\geq\lambda_{0}\geq\lambda_{1}\geq\ldots\geq\lambda_{n},$

we establish the following inclusion relationships:

$Hy_{(S,M)}(d)\subseteq\lambda_{0}\text{-}Hy_{(S,M)}(d)\subseteq\lambda_{1}\text{-}Hy_{(S,M)}(d)\subseteq\ldots\subseteq\lambda_{n}\text{-}Hy_{(S,M)}(d).$

######  Example 23 (Description Consistent with a Dataset)

Consider a binary semiotic $(S,M)$ with signs $A,B,C,D$. Let

$d:M(A)\times M(B)\times M(C)\times M(D)\rightarrow\Omega$

be a finite crisp concept description. This means that for every entity
$\bar{x}$, the value of $d(\bar{x})$ is either $\top$ or $\bot$, and the
number of entities $\bar{x}$ for which $d(\bar{x})=\top$ is finite. In this
context, there exists a word $D$ in the language associated with the semiotic
that is consistent with $d$, known as the dataset used to describe $d$.

To be more specific, suppose the signs $A,B,C,D$ share the same
interpretation, such that $M(A)=M(B)=M(C)=M(D)=[0,1]$. Let’s also assume the
following:

  1. 1.

$d(1.0,0.5,0.2,0.2)=\top$,

  2. 2.

$d(1.0,1.0,0.2,0.2)=\top$, and

  3. 3.

$d(1.0,1.0,0.0,0.2)=\top$.

These are the only tuples where $d$ evaluates to $\top$. The relation $d$ is
consistent with the diagram

(A=1.0⊗B=0.5⊗C=0.2⊗D=0.2)⊗(A=1.0⊗B=1.0⊗C=0.2⊗D=0.2)⊗(A=1.0⊗B=1.0⊗C=0.0⊗D=0.2)

or, equivalently, $d$ serves as the answer to the query defined by the
diagram. This can be represented using table notation as shown in Figure 11.2.

A | B | C | D
---|---|---|---
1.0 | 0.5 | 0.2 | 0.2
1.0 | 1.0 | 0.2 | 0.2
1.0 | 1.0 | 0.0 | 0.2
Figure 11.2: Dataset.

## Chapter 12 Fuzzy computability

When the interpretation of a diagram is consistent with a multi-morphism, we
consider the multi-morphism to be computable within the semiotic. Formally:

######  Definition 24 (Computability)

Given a semiotic $(S,M)$, a multi-morphism $f:A\rightharpoonup B$ is said to
be computable in $(S,M)$ if there exists a diagram $D$ in $Lang(S)$
satisfying:

  1. 1.

$A$ serves as the input, with $A=i(D)$, and $B$ as the output, with $B=o(D)$,
and

  2. 2.

$D$ codifies $f$, meaning that $f\models\forall D$.

The multi-morphism $f$ is said to be $\lambda$-computable in $(S,M)$ if
$A=i(D)$, $B=o(D)$, and $f\models_{\lambda}\forall D$. These definitions are
quite restrictive, so we relax them by considering a diagram $D$ as a
specification to compute part of $f$ if $d\models\exists D$. When the domain
of the computable part of $f$ can be described by a diagram $D^{\prime}$, we
write

$f\models\forall_{D^{\prime}}D.$

When $f\models\forall D$, with $A=i(D)$ and $B=o(D)$, we refer to the diagram
$D$ as a _program_ or _specification_ in the language $Lang(S)$. Its image
under $M$ serves as an implementation for the multi-morphism
$f:A\rightharpoonup B$.

In this context, every interpretation of words from $Lang(S)$ is computable in
the semiotic $(S,M)$. Furthermore, since words in $Lang(S)$ are generated from
atomic components, we can state the following proposition:

###### Proposition 19

If $f$ and $g$ are computable multi-morphisms in the semiotic $(S,M)$, then
$f\otimes g$ is also computable in $(S,M)$.

Moreover, since $Lang(S)$ is defined by finite diagrams, every finite diagram
$D$ in $Set(\Omega)$, comprising arrows of computable multi-morphisms, will
have a limit that is a computable multi-morphism.

The set of interpretations of words from $Lang(S)$ and computable multi-
morphisms together define a category, denoted by $Hy_{(S,M)}$. In this
category, we write $f:d_{1}\rightarrow d_{2}$ if $f$ is a computable multi-
morphism and $d_{1}$ and $d_{2}$ are consistent descriptions in the semiotic,
satisfying $d_{1}\otimes f=d_{2}$. It’s worth noting that if $D$ is consistent
with $d_{1}$ and $D_{f}$ is the specification for $f$, then the diagram
$D\otimes D_{f}$ is consistent with $d_{1}\otimes f$.

More generally, if $(d\Leftrightarrow MD)\geq\lambda$ and $(f\Leftrightarrow
MD_{f})=\top$, then $(d\otimes f\Leftrightarrow MD\otimes f)\geq\lambda$,
i.e., $(d\otimes f\Leftrightarrow MD\otimes MD_{f})\geq\lambda$. Formally:

###### Proposition 20

Let $d$ be a description $\lambda$-consistent with $D$, and let $f$ be a
computable multi-morphism specified by $D_{f}$. Then, $d\otimes f$ is a
description $\lambda$-consistent with the diagram $D\otimes D_{f}$.

In this context, a computable multi-morphism is recognized as a pre-processing
tool in the data mining community. This leads to the definition of
$\lambda$-$Hy_{(S,M)}$, the category of concepts that are $\lambda$-consistent
and computable multi-morphisms in the semiotic $(S,M)$. Naturally, both the
limit and the colimit, in the conventional sense, of finite diagrams in
$\lambda$-$Hy_{(S,M)}$ define computable relations. We refer to this type of
finite diagrams as _mining schemas_. Given a mining schema $D$ in semiotic
$(S,M)$ and a $\lambda$-consistent concept $d$, the limit $Lim\;D$ defines a
computable multi-morphism, and $d\otimes Lim\;D$ is a $\lambda$-consistent
concept, interpreted as the output of schema $D$ when applied to concept $d$.

As is customary, we extend the notion of computability by defining:

######  Definition 25 (Turing computable)

A concept $d$ is called _Turing computable_ in the semiotic $(S,M)$ if there
exists a diagram $D$, possibly infinite but enumerable, such that

$Lim\;D=d.$

Computability is typically associated with state-based systems. In the context
of a semiotic, the interpretation of a state must be time-dependent. Given the
static definition of sign interpretation presented earlier, we can only
capture this dynamic behavior by employing an ontological hierarchy. We
envision the possible interpretation of a sign as a class of structures that
could serve as potential instantiations for it during system execution. To
achieve this, we utilize a syntactic operator that links together signs within
the same class, representing different perspectives of the same entity. It is
imperative that the class of related signs, connected by the syntactic
operator, share the same generalization sign within the sign ontology. The
presence of such a syntactic operator in a semiotic is what we referred to as
a syntactic operator in chapter 10.

######  Definition 26 (Temporal semiotics)

A _temporal semiotic_ is defined as a semiotic $(S,M)$, characterized by a
library $L:|L|\rightarrow(Chains\downarrow\Sigma^{+})$ and augmented with a
syntactic operator

$t:\Sigma^{+}\rightarrow\Sigma^{+}$

that adheres to the following conditions:

  1. 1.

It maintains the polarization of signs, specifically, $t(s^{+})=t(s)^{+}$ for
every $s\in\Sigma$;

  2. 2.

It upholds concatenation, meaning $t(w_{0}.w_{1})=t(w_{0}).t(w_{1})$ holds
true for every pair of words $w_{0},w_{1}$;

  3. 3.

It ensures the functionality of components; for any function $f:w\rightarrow
w^{\prime}$, there exists a corresponding component

$t(f):t(w)\rightarrow t(w^{\prime}).$

We further require the existence of a component

$t(r):i(t(w))\rightarrow o(t(w))$

for every component $r:i(w)\rightarrow o(w)$. Additionally, we introduce an
ontological hierarchy for signs that remains time-invariant when relating
time-dependent signs. That is, if $s_{1}=t(s_{0})$, then there exists a sign
$s$ such that $s_{1}\leq s$, $s_{0}\leq s$, and $s=t(s)$. Consequently, every
sequence of time-dependent signs

$s_{0},t(s_{0}),t(t(s_{0})),t(t(t(s_{0}))),\ldots$

is generalized by the same sign $s$ in the ontology. We refer to $s$ as a
_time-invariant sign_.

In a temporal semiotic $(S,M)$, if $r:i(w)\rightarrow o(s(w))$ is a component
within the semiotic, then its interpretation $M(r):M(i(w))\rightharpoonup
M(o(s(w)))$ is termed a _coalgebra_. A sign $s\in\Sigma$ is deemed _time-
invariant_ in the semiotic if $M(s)=M(t(s))$.

A _temporal logic semiotic_ is characterized as a semiotic that is both a
logic semiotic and a temporal semiotic.

######  Example 24 (Fuzzy Turing machine)

A fuzzy Turing machine, whose tape is defined using signs from $F$, can be
conceptualized as a word within the language associated with a temporal logic
semiotic $(S,M)$. The interpretation of this word can be viewed as its
execution. The structure of the machine can be represented within a sign
system $S$ with a library $L$, where the signs correspond to a set of machine
states, denoted as $Q$, and the components represent Turing machine
instructions labeled within a set $I$.

Each instruction in $I$ takes a conditional form: it dictates an action based
on the distribution of symbols currently being scanned on the tape.
Specifically, there are three categories of actions that can be executed:

  1. 1.

Print: Modify the symbol distribution in the square currently being scanned;

  2. 2.

Move one square to the right;

  3. 3.

Move one square to the left.

Thus, depending on the active instruction and the symbol distribution being
scanned, the machine or its operator will execute one of these actions.

An instruction defines a link between two states and is encoded as component
labels with the following structure:

  1. 1.

$q_{0}[f]q_{1}$: If in state $q_{0}$ the scanned distribution is modified
using the interpretation of component $f$, the machine transitions to state
$q_{1}$;

  2. 2.

$q_{0}[d_{0}:L]_{\lambda}q_{1}$: If in state $q_{0}$ the machine reads a
distribution $d$ and $d\otimes M(d_{0})\geq\lambda$, then the machine moves
left and transitions to state $q_{1}$;

  3. 3.

$q_{0}[d_{0}:R]_{\lambda}q_{1}$: If in state $q_{0}$ the machine reads a
distribution $d$ and $d\otimes M(d_{0})\geq\lambda$, then the machine moves
right and transitions to state $q_{1}$.

An instruction is executed if its condition is satisfied.

In this context, a diagram in $Lang(L)$, defined using time-invariant signs,
serves as a _Turing machine specification_ , with states in $Q$ and tape
symbols from $F$. Every refinement of a Turing machine specification in
$Lang(L)$, defined using only time-variant signs, is referred to as a _flow
chart_ , and it encodes a Turing machine’s execution. To ensure the correct
interpretation of an instruction, for each state $q_{i}\in Q$ in the sign
system, we have signs

$q_{i}^{(r)},q_{i}^{(m)},q_{i}^{(l)},q_{i}^{(hr)},q_{i}^{(tr)},q_{i}^{(hl)},q_{i}^{(tl)}$

where $q_{i}^{(r)}$ is interpreted as the tape’s right half, $q_{i}^{(m)}$
denotes the reading square, and $q_{i}^{(l)}$ represents the tape’s left half.
Additionally, for each tape half, we select the right half head
$q_{i}^{(hr)}$, the right tail head $q_{i}^{(tr)}$, the left half head
$q_{i}^{(hl)}$, and the left tail head $q_{i}^{(tl)}$. The structure of the
sign system is defined in such a way that the relationship between these signs
and $q_{i}$ is preserved if a model $M$ satisfies:

  1. 1.

$M(q_{i})=M(q_{i}^{(r)})\otimes_{I}M(q_{i}^{(m)})\otimes_{I}M(q_{i}^{(l)})$;

  2. 2.

$M(q_{i}^{(r)})=M(q_{i}^{(hr)})\otimes_{I}M(q_{i}^{(tr)})$;

  3. 3.

$M(q_{i}^{(l)})=M(q_{i}^{(hl)})\otimes_{I}M(q_{i}^{(tl)})$;

This interpretation for signs reflects the relations between I-projections
(see Example 2) as expressed in the following diagram:

$\textstyle{M(q_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(q_{i}^{(r)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(q_{i}^{(m)})}$$\textstyle{M(q_{i}^{(l)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M(q_{i}^{(hr)})}$$\textstyle{M(q_{i}^{(tr)})}$$\textstyle{M(q_{i}^{(hl)})}$$\textstyle{M(q_{i}^{(tl)})}$
Figure 12.1: Sign interpretation structure.

And models of each instruction must satisfy the following conditions:

  1. 1.

For the print instruction $q_{0}[f]q_{1}$, we should have:

${}_{M(q_{i}[f]t(q_{j}))^{\circ}\otimes M(q_{i})\otimes
M(q_{i}[f]t(q_{j}))=M(t(q_{j}))}\Rightarrow\left\\{\begin{array}[]{l}{}_{M(t(q_{j})^{(r)})=M(q_{i}^{(r)})}\\\
{}_{M(t(q_{j})^{(m)})=M(f)^{\circ}\otimes M(q_{i}^{(m)})\otimes M(f)}\\\
{}_{M(t(q_{j})^{(l)})=M(q_{i}^{(l)})}\\\ \end{array}\right.$

  2. 2.

For instructions of the type "Move one square to the left"
$q_{0}[d_{0}:L]_{\lambda}q_{1}$, we must have:

${}_{\left\\{\begin{array}[]{l}{}_{M(q_{i}^{m})\otimes M(d_{0})\geq\lambda}\\\
{}_{M(q_{i}[d_{0}:L]_{\lambda}t(q_{j}))^{\circ}\otimes M(q_{i})\otimes
M(q_{i}[d_{0}:L]_{\lambda}t(q_{j}))=M(t(q_{j}))}\\\
\end{array}\right.\Rightarrow\left\\{\begin{array}[]{l}{}_{M(t(q_{j})^{(r)})=M(q_{j}^{(r)})\otimes_{I}M(q_{i}^{(m)})}\\\
{}_{M(t(q_{j})^{(m)})=M(q_{i}^{(hl)})}\\\
{}_{M(t(q_{j})^{(l)})=M(q_{i}^{(tl)})}\\\ \end{array}\right.}$

  3. 3.

For instructions of the type "Move one square to the right"
$q_{0}[d_{0}:R]_{\lambda}q_{1}$, we must have:

${}_{\left\\{\begin{array}[]{l}{}_{M(q_{i}^{m})\otimes M(d_{0})\geq\lambda}\\\
{}_{M(q_{i}[d_{0}:R]_{\lambda}t(q_{j}))^{\circ}\otimes M(q_{i})\otimes
M(q_{i}[d_{0}:R]_{\lambda}t(q_{j}))=M(t(q_{j}))}\\\
\end{array}\right.\Rightarrow\left\\{\begin{array}[]{l}{}_{M(t(q_{j})^{(r)})=M(q_{i}^{(hr)})}\\\
{}_{M(t(q_{j})^{(m)})=M(q_{i}^{(tr)})}\\\
{}_{M(t(q_{j})^{(l)})=M(q_{i}^{(m)})\otimes_{I}M(q_{i}^{(tl)})}\\\
\end{array}\right.}$

Thus, a model $M$ assigns to each state a _fuzzy tape_ with signs in $F$,
which can be seen as an infinite chain of indexed products (see example 2):

$t=\underbrace{\overbrace{\cdots\otimes_{I}d_{5}}^{t^{(hr)}}\otimes_{I}\overbrace{d_{3}}^{t^{(tr)}}}_{t^{(r)}}\otimes_{I}\underbrace{d_{1}}_{t^{(m)}}\otimes_{I}\underbrace{\overbrace{d_{2}}^{t^{(hl)}}\otimes_{I}\overbrace{d_{4}\otimes_{I}\cdots}^{t^{(tl)}}}_{t^{(l)}}$

where we fix a component $t^{(m)}=d_{1}$, and such that each $d_{i}$ is a
concept description $d_{i}:F\times I\rightarrow\Omega$. Moreover, the model
$M$ associates with each possible instruction (component) a relation between
fuzzy tapes $t_{0}$ and $t_{1}$, satisfying the described properties.

A fuzzy Turing machine begins its execution in an initial state and is a
parallel device; at any given moment, it can be in more than one state. It
finishes its execution when it is stalled in a state or set of states.

## Chapter 13 Consequence relation

In a semiotic $(S,M)$, we define for every relation $D$ in $Lang(S)$ the set
of its $\lambda$-answers as:

$ans_{\lambda}(D)=\\{g\in M(S):g\models_{\lambda}\forall_{D^{\prime}}D\\}$

This set can be interpreted as the collection of concepts that are
$\lambda$-consistent with $D$, defined on the domain specified by
$D^{\prime}\in Lang(S)$.

###### Example 25

The examples presented in this chapter utilize a grid semiotic, which has the
expressive power to encode structures on a grid, employing a three truth-
values logic $\Omega=\\{\bot,\frac{1}{2},\top\\}$.

Let $D$ be the diagram that defines a relation between pairs of entities in a
grid, as illustrated in Fig. 13.1. In this figure, white points $\bar{x}$
correspond to $M(D)(\bar{x})=\bot$, gray points represent
$M(D)(\bar{x})=\frac{1}{2}$, and darker points indicate $M(D)(\bar{x})=\top$.

Figure 13.1: Relation defined by interpreting $D$ and finite relations
$g_{1}$. A point $\bar{x}$ marked with an X represents $g_{1}(\bar{x})=\top$,
while a point marked with 0 corresponds to $g_{1}(\bar{x})=\bot$.

The interior of the box, denoted as $D_{1}$ in the figure, can be viewed as
the set of points described by this diagram. The relation $g_{1}$ depicted can
be considered as an example that satisfies

$g_{1}\models_{\frac{1}{2}}\forall D,\quad g_{1}\models\forall_{D_{1}}D,$

which can be expressed concisely as

$g_{1}\in ans_{\frac{1}{2}}(D).$

Note that given a relation $D$, the set $ans_{\lambda}(D)$ contains at least
one element, namely $M(D)\in ans_{\lambda}(D)$. Naturally, we have:

###### Theorem 1

If $D$ is a relation in $Lang(S)$ and $\lambda_{0}\leq\lambda_{1}$, then

$ans_{\lambda_{1}}(D)\subseteq ans_{\lambda_{0}}(D).$

If $g\in ans_{\lambda}(D)$, and $g\models_{\lambda}\forall_{D^{\prime}}D$, we
denote this relation by writing $g_{D^{\prime}}\in ans_{\lambda}(D)$.

Let $D$ be a relation defined in a semiotic by

$f\leq_{D}g,$

this implies that

$\text{if }M(D)(\bar{x})=\top,\text{ then }f(\bar{x})\leq g(\bar{x}).$

We utilize this relation and the operator $ans_{\lambda}$ to introduce two
modal operators, $\diamond_{\lambda}g$ and $\Box_{\lambda}g$, which represent
the weak and strong images, respectively, of the description $g\in M(S)$ along
the relation $\models_{\lambda}$:

$\diamond_{\lambda}g=\\{D\in Lang_{R}(S):(\exists f_{D^{\prime}}\in
ans_{\lambda}(D))(g\leq_{D^{\prime}}f)\\}$ $\Box_{\lambda}g=\\{D\in
Lang_{R}(S):(\forall f_{D^{\prime}}\in
ans_{\lambda}(D))(f\leq_{D^{\prime}}g)\\}$

Here, $\diamond_{\lambda}g$ and $\Box_{\lambda}g$ can be interpreted,
respectively, as the set of models that are $\lambda$-consistent with parts of
$g$ and the set of models that are $\lambda$-consistent with $g$ in the
language $Lang(S)$.

###### Example 26

For a grid semiotic with three truth-values, we present in Fig. 13.2 two
possible diagrams: $D_{1}\in\diamond_{\frac{1}{2}}g_{1}$ and
$D_{2}\in\Box_{\frac{1}{2}}g_{2}$.

Figure 13.2: Examples of $D_{1}\in\diamond_{\frac{1}{2}}g_{1}$ and
$D_{2}\in\Box_{\frac{1}{2}}g_{2}$.

We have the following theorem:

###### Theorem 2

Given relations $D_{0}$ and $D_{1}$ in $Lang_{R}(S)$ and a description $g$:

  1. 1.

If $\lambda_{0}\leq\lambda_{1}$, then
$\Box_{\lambda_{1}}g\subseteq\Box_{\lambda_{0}}g$.

  2. 2.

If $\lambda_{0}\leq\lambda_{1}$, then
$\diamond_{\lambda_{1}}g\subseteq\diamond_{\lambda_{0}}g$.

  3. 3.

If $D_{0},D_{1}\in\Box_{\lambda}g$, then $D_{0}\vee D_{1}\in\Box_{\lambda}g$.

  4. 4.

If $D_{0},D_{1}\in\diamond_{\lambda}g$, then $D_{0}\wedge
D_{1}\in\diamond_{\lambda}g$.

In the other direction, we can extend $ans_{\lambda}$ to a set of relations
$U$ in $Lang(S)$ as follows:

$ans_{\lambda}(U)=\bigvee\\{g\in M(S):(\exists D\in\Box_{\lambda}g)(D\in
U)\\}$

This provides the greatest description $\lambda$-consistent with a model from
$U$. Additionally, let

$mod_{\lambda}(U)=\bigwedge\\{g\in M(S):(\forall D\in\Box_{\lambda}g)(D\in
U)\\}$

be a description that is $\lambda$-consistent with every model in $U$.

###### Theorem 3

Let $U$ and $V$ be sets of relations in $S$. Then:

  1. 1.

If $\lambda_{0}\leq\lambda_{1}$, then $ans_{\lambda_{0}}(U)\geq
ans_{\lambda_{1}}(U)$.

  2. 2.

$ans_{\lambda}(U\cup V)=ans_{\lambda}(U)\vee ans_{\lambda}(V)$.

  3. 3.

$mod_{\lambda}(U\cup V)=mod_{\lambda}(U)\wedge mod_{\lambda}(V)$.

Moreover, if $U\subseteq V$, we have:

  1. 1.

$ans_{\lambda}(U)\leq ans_{\lambda}(V)$.

  2. 2.

$mod_{\lambda}(V)\leq mod_{\lambda}(U)$.

The $\lambda$-interior of a concept $g$ within the semiotic system $(S,M)$ is
defined as the largest portion of $g$ that is $\lambda$-consistent with a
model defined in the associated language. Mathematically, it is represented
as:

$int_{\lambda}(g)=\bigvee\\{h\in M(S):(\exists D\in\Box_{\lambda}h)(\forall
f_{D^{\prime}}\in ans_{\lambda}(D))(f\leq_{D^{\prime}}g)\\},$

This can be understood as the maximal fragment of $g$ that has a
$\lambda$-consistent model within the language of the semiotic. The operator
$int_{\lambda}$ is considered an _interior operator_ because:

  1. 1.

$int_{\lambda}(g)\leq g$,

  2. 2.

if $g\leq f$, then $int_{\lambda}(g)\leq int_{\lambda}(f)$, and

  3. 3.

$int_{\lambda}(g)=int_{\lambda}(int_{\lambda}(g))$.

Furthermore, given $\lambda_{0}\leq\lambda_{1}$, we have
$int_{\lambda_{1}}(g)\leq int_{\lambda_{0}}(g)$. A concept description $g$ is
termed $\lambda$-_open_ in $(S,M)$ if:

$int_{\lambda}(g)=g.$

Both $ans_{\lambda}(U)$ and $mod_{\lambda}(U)$ for any set of relations $U$
serve as examples of $\lambda$-open concepts due to the following proposition:

###### Proposition 21

In a semiotic, for any set of relations $U$ and for any $\lambda\in\Omega$:

  1. 1.

$int_{\lambda}(ans_{\lambda}(U))=ans_{\lambda}(U)$, and

  2. 2.

$int_{\lambda}(mod_{\lambda}(U))=mod_{\lambda}(U)$.

To delve into the details:

###### Theorem 4

In a semiotic system, for any set of relations $U$:

$ans_{\top}(U)\models\bigvee U\quad\text{and}\quad
mod_{\top}(U)\models\bigwedge U.$

The closure of a concept $g$ within the semiotic system $(S,M)$ is defined as
the minimal cover of $g$ represented in the language $L(S)$. Mathematically,
it is expressed as:

$cl_{\lambda}(g)=\bigwedge\\{h\in M(S):(\forall D\in\Box_{\lambda}h)(\exists
f_{D^{\prime}}\in ans_{\lambda}(D))(g\leq_{D^{\prime}}f)\\},$

This can be interpreted as the most concise cover that includes $g$,
represented in the language associated with the semiotic system. The operator
$cl_{\lambda}$ is termed a _closure operator_ because:

  1. 1.

$g\leq cl_{\lambda}(g)$,

  2. 2.

if $g\leq f$, then $cl_{\lambda}(g)\leq cl_{\lambda}(f)$, and

  3. 3.

$cl_{\lambda}(cl_{\lambda}(g))=cl_{\lambda}(g)$.

Furthermore, for $\lambda_{0}\leq\lambda_{1}$, we have
$cl_{\lambda_{1}}(g)\leq cl_{\lambda_{0}}(g)$. It is also evident that:

###### Proposition 22

For a given semiotic system $(S,M)$, for every $g\in M(S)$:

$int_{\lambda}(g)\leq g\leq cl_{\lambda}(g).$

A concept description $g$ is termed $\lambda$-_close_ in the semiotic system
$(S,M)$ if it satisfies:

$cl_{\lambda}(g)=g.$

Both the descriptions $ans_{\lambda}(U)$ and $mod_{\lambda}(U)$ also qualify
as $\lambda$-closed concepts. This concept can be generalized to encompass
every $\lambda$-open description:

###### Proposition 23

For a given semiotic system $(S,M)$, every $g\in M(S)$ is $\lambda$-closed if
and only if it is $\lambda$-open.

In this context, when a description is either $\lambda$_-open_ or
$\lambda$_-close_ , we refer to it as a description that is
$\lambda$-representable within the semiotic system. This leads us to the
following proposition:

###### Proposition 24

Let $g$ be a description within the semiotic system $(S,M)$. There exists a
relation $D$ such that $g\models_{\lambda}D$ if and only if $g$ is
$\lambda$-open or $\lambda$-close.

Due to the symmetry between the left and right sides of $d\models D$, the
definitions yield:

$int_{\lambda}=ans_{\lambda}\Box_{\lambda}\quad\text{and}\quad
cl_{\lambda}=mod_{\lambda}\diamond_{\lambda}$

These definitions also possess symmetric counterparts, which can be derived by
replacing each operator with its symmetric equivalent:

$\mathcal{A}_{\lambda}=\Box_{\lambda}ans_{\lambda}\quad\text{and}\quad\mathcal{C}_{\lambda}=\diamond_{\lambda}mod_{\lambda}.$

By this symmetry, it is immediate to conclude that $\mathcal{C}_{\lambda}$ is
an interior operator and $\mathcal{A}_{\lambda}$ is a closure operator.

Elaborating on the definition of $\mathcal{A}_{\lambda}$, for every set of
relations $U$:

$\mathcal{A}_{\lambda}(U)=\\{D\in Lang(S):(\forall f_{D^{\prime}}\in
ans_{\lambda}(D))(f\leq_{D^{\prime}}ans_{\lambda}(U))\\},$

This means that all $\lambda$-answers for $D$ are $\lambda$-codified using
relations from $U$. This leads us to:

###### Theorem 5

For every pair of relations $U$ and $V$ in the semiotic system $(S,M)$, the
following properties hold:

  1. 1.

$\mathcal{A}_{\lambda}(U\cup
V)\supseteq\mathcal{A}_{\lambda}(U)\cup\mathcal{A}_{\lambda}(V)$,

  2. 2.

if $U\subseteq V$, then
$\mathcal{A}_{\lambda}(U)\subseteq\mathcal{A}_{\lambda}(V)$,

  3. 3.

if $\lambda_{0}\leq\lambda_{1}$, then
$\mathcal{A}_{\lambda_{1}}(U)\subseteq\mathcal{A}_{\lambda_{0}}(U)$,

  4. 4.

if $D\in\mathcal{A}_{\lambda}(U)$, then $\bigvee ans_{\lambda}(D)\leq
ans_{\lambda}(U)$, and

  5. 5.

if $D\in\mathcal{A}_{\lambda}(U)$, then $ans_{\lambda}(U)\models_{\lambda}D$.

Expanding on the definition of the operator $\mathcal{C}_{\lambda}$, we have:

$\mathcal{C}_{\lambda}(U)=\\{D\in Lang(S):(\exists f_{D^{\prime}}\in
ans_{\lambda}(D))(mod_{\lambda}(U)\leq_{D^{\prime}}f)\\},$

Where $D\in\mathcal{C}_{\lambda}(U)$ implies that $D$ has a $\lambda$-answer,
and every $\lambda$-codification for it is in $U$. Based on this, we can
establish:

###### Theorem 6

For every pair of relations $U$ and $V$ in the semiotic system $(S,M)$:

$\mathcal{C}_{\lambda}(U\cup
V)\subseteq\mathcal{C}_{\lambda}(U)\cap\mathcal{C}_{\lambda}(V)$

and when $U\subseteq V$, we have
$\mathcal{C}_{\lambda}(U)\subseteq\mathcal{C}_{\lambda}(V)$.

Let’s define:

$U\vdash_{\lambda}D\text{ iff }D\in\mathcal{A}_{\lambda}(U),$

Given that $\mathcal{A}_{\lambda}$ is a closure operator, the following
properties hold:

###### Theorem 7

In a semiotic system $(S,M)$, for every $\lambda$, we have:

  1. 1.

if $D\in U$, then $U\vdash_{\lambda}D$ (Inclusion),

  2. 2.

if $U\vdash_{\lambda}D$, then $U\cup V\vdash_{\lambda}D$ (Monotony), and

  3. 3.

if $V\vdash_{\lambda}D$ and $U\cup\\{D\\}\vdash_{\lambda}D^{\prime}$, then
$U\cup V\vdash_{\lambda}D^{\prime}$ (Cut).

This implies that $(Lang_{R}(S),\vdash_{\lambda})$ serves as an _inference
system_ [18] for every $\lambda$.

###### Example 27

Consider the interpretations, depicted in Fig. 13.3, for three diagrams
$D_{0}$, $D_{1}$, and $D_{2}$ within the grid semiotic employing three-valued
logic:

Figure 13.3: Interpretations for diagrams $D_{0}$, $D_{1}$, $D_{2}$, and
$D_{3}$.

The diagram $D_{3}$, with its represented interpretation, can be viewed as the
result of applying inference to the set of diagrams $\\{D_{0},D_{1},D_{2}\\}$.
Symbolically, we express this as:

$D_{0},D_{1},D_{2}\vdash D_{3}.$

Given that $\mathcal{A}_{\lambda_{0}}\leq\mathcal{A}_{\lambda_{1}}$ whenever
$\lambda_{0}\geq\lambda_{1}$, we can represent this relationship as:

$U\vdash_{\lambda_{1}}D_{0}\lx@proof@logical@and
U\vdash_{\lambda_{0}}D_{0}\lambda_{0}\geq\lambda_{1}$

Utilizing the definition of $\lambda$-consistency, we can state the following
theorem:

###### Theorem 8

Let $g\models_{\lambda_{0}}\forall_{D^{\prime}_{0}}D_{0}$ and
$f\models_{\lambda_{1}}\forall_{D^{\prime}_{1}}D_{1}$:

  1. 1.

$g\wedge f\models_{\lambda_{0}\wedge\lambda_{1}}\forall_{D^{\prime}_{0}\otimes
D^{\prime}_{1}}(D_{0}\wedge D_{1})$,

  2. 2.

$g\vee f\models_{\lambda_{0}\vee\lambda_{1}}\forall_{D^{\prime}_{0}\otimes
D^{\prime}_{1}}(D_{0}\vee D_{1})$,

  3. 3.

$g\otimes
f\models_{\lambda_{0}\otimes\lambda_{1}}\forall_{D^{\prime}_{0}\otimes
D^{\prime}_{1}}(D_{0}\otimes D_{1})$, and

  4. 4.

$g\Rightarrow
f\models_{\lambda_{0}\Rightarrow\lambda_{1}}\forall_{D^{\prime}_{0}\otimes
D^{\prime}_{1}}(D_{0}\Rightarrow D_{1})$.

Applying these properties to the definition of $\lambda$-answer, we obtain:

###### Theorem 9

Within a semiotic system, we have:

  1. 1.

For every diagram $D\in Lang_{R}(S)$:

    1. (a)

$ans_{\lambda_{0}\vee\lambda_{1}}(D)=ans_{\lambda_{0}}(D)\vee
ans_{\lambda_{1}}(D)$,

    2. (b)

$ans_{\lambda_{0}\wedge\lambda_{1}}(D)=ans_{\lambda_{0}}(D)\wedge
ans_{\lambda_{1}}(D)$, and

    3. (c)

$ans_{\lambda_{0}\otimes\lambda_{1}}(D)=ans_{\lambda_{0}}(D)\otimes
ans_{\lambda_{1}}(D)$;

  2. 2.

For every concept description $g\in M(S)$:

    1. (a)

$\Box_{\lambda_{0}\vee\lambda_{1}}(g)=\Box_{\lambda_{0}}(g)\vee\Box_{\lambda_{1}}(g)$,

    2. (b)

$\Box_{\lambda_{0}\wedge\lambda_{1}}(g)=\Box_{\lambda_{0}}(g)\wedge\Box_{\lambda_{1}}(g)$,
and

    3. (c)

$\Box_{\lambda_{0}\otimes\lambda_{1}}(g)=\Box_{\lambda_{0}}(g)\otimes\Box_{\lambda_{1}}(g)$;

  3. 3.

For every set of diagrams $U\subset Lang_{R}(S)$:

    1. (a)

$ans_{\lambda_{0}\vee\lambda_{1}}(U)=ans_{\lambda_{0}}(U)\vee
ans_{\lambda_{1}}(U)$,

    2. (b)

$ans_{\lambda_{0}\wedge\lambda_{1}}(U)=ans_{\lambda_{0}}(U)\wedge
ans_{\lambda_{1}}(U)$, and

    3. (c)

$ans_{\lambda_{0}\otimes\lambda_{1}}(U)=ans_{\lambda_{0}}(U)\otimes
ans_{\lambda_{1}}(U)$.

This provides support for the introduction rules’ definition:

$U\vdash_{\lambda_{0}\vee\lambda_{1}}(D_{0}\vee D_{1}),\quad
U\vdash_{\lambda_{0}\wedge\lambda_{1}}(D_{0}\wedge D_{1}),\quad
U\vdash_{\lambda_{0}\otimes\lambda_{1}}(D_{0}\otimes
D_{1})U\vdash_{\lambda_{0}}D_{0}\quad\text{and}\quad
U\vdash_{\lambda_{1}}D_{1}$

The assertion that if $D_{0}\wedge D_{1}\in\mathcal{A}_{\lambda}(U)$ then both
$D_{0}\in\mathcal{A}_{\lambda}(U)$ and $D_{1}\in\mathcal{A}_{\lambda}(U)$ can
be captured by the elimination rule:

$U\vdash_{\lambda}D_{0},\quad
U\vdash_{\lambda}D_{1}U\vdash_{\lambda}(D_{0}\wedge D_{1})$

In a divisible logic setting, we observe:

$U\vdash_{\lambda_{0}\otimes\lambda_{1}}(D_{0}\wedge
D_{1})U\vdash_{\lambda_{0}}D_{0}\quad\text{and}\quad
U\vdash_{\lambda_{1}}(D_{0}\Rightarrow D_{1})$

This is because, if $g\models_{\lambda_{0}}\exists D_{0}$ and
$g\models_{\lambda_{1}}\exists(D_{0}\Rightarrow D_{1})$, then
$g\models_{\lambda_{0}\otimes\lambda_{1}}\exists D_{1}$. Specifically, if
$D_{0}\in\mathcal{A}_{\lambda_{0}}(U)$ and $D_{0}\Rightarrow
D_{1}\in\mathcal{A}_{\lambda_{1}}(U)$, it follows that $D_{0}\wedge
D_{1}\in\mathcal{A}_{\lambda_{0}\otimes\lambda_{1}}(U)$. This implies that for
every $f\in ans_{\lambda_{0}}(D_{0})$, we have $f\leq ans_{\lambda_{0}}(U)$,
and for every $h\in ans_{\lambda_{0}}(D_{1})$, $f\Rightarrow h\in
ans_{\lambda_{1}}(D_{0}\Rightarrow D_{1})$. Consequently,
$f\otimes(f\Rightarrow h)\in ans_{\lambda_{0}\otimes\lambda_{1}}(U)$. It’s
worth noting that in a divisible ML-algebra, $f\otimes(f\Rightarrow h)\leq
f\wedge h$. Hence, $f\wedge h\in ans_{\lambda_{0}\otimes\lambda_{1}}(U)$.

A diagram $D$ encapsulates all the information present in a concept $d$,
employing the syntax associated with the semiotic $(S,M)$. This occurs if, for
every diagram $D_{1}$ such that $d\models_{\lambda}\forall D_{1}$, we have
$M(D\vee D_{1})=M(D)$. Such diagrams are termed _total_ and can be defined as:

$D=\bigvee_{d\models_{\lambda}\forall D_{i}}D_{i}.$

In the category $Lang(S)$, whose objects are diagrams encoding relations, and
where a diagram $D$ serves as a morphism from relation $D_{0}$ to relation
$D_{1}$ if $D_{0}\otimes D=D_{1}$, we conceptualize composition as the
operation of diagram gluing. The consequence operator $\vdash_{\lambda}$ can
be viewed as a functor:

$\vdash_{\lambda}:Lang(S)\rightarrow Lang(S).$

A diagram $D$ is denoted as a _theory_ in the $\lambda$-semiotic $(S,M)$ if it
stands as a fixed-point for the consequence operator:

$\vdash_{\lambda}(D)=D.$

The semiotic model $M$ can be interpreted as a functor:

$M:Lang(S)\rightarrow\lambda\text{-}Hy_{(S,M)},$

within the category of $\lambda$-consistent concepts and computable multi-
morphisms. Conversely, a functor in the opposite direction can be formulated
using the consistency operator:

$\models_{\lambda}:\lambda\text{-}Hy_{(S,M)}\rightarrow Lang(S),$

which assigns to each $\lambda$-consistent description a total diagram along
with its codification in the semiotic.

Given $M(D)\models_{\lambda}\forall D$, it follows that:

$M\circ\models_{\lambda}=\text{id},$

and, by the definition of the consequence relation:

$\models_{\lambda}\circ M=\vdash_{\lambda}.$

If $D$ represents a $\lambda$-theory in the semiotic, then $M(D)$ serves as
the $\lambda$-consistent model associated with this theory. .

## Chapter 14 Integration

Our objective is to construct an integrated semiotic foundation derived from
multiple distinct semiotics. This necessity may arise, for instance, when
knowledge bases are independently acquired through interactions with various
domain experts. Similarly, this challenge can emerge when separate knowledge
bases are produced by distinct learning algorithms. The ultimate aim of
integration is to create a unified system that leverages all available
knowledge, ensuring high performance, i.e., a substantial degree of
consistency with the data.

It is crucial to differentiate between two forms of integration: semiotic
integration and model integration within a semiotic framework. Semiotic
integration seeks to define a semiotic that amalgamates the syntax and
semantics of a given set of semiotics. Conversely, model integration within a
semiotic refers to the enhancement of concept descriptions by integrating
models derived from diverse data sources or differing perspectives on the same
data. The integration of models is governed by an integration schema that
delineates the relationships between various models within the same semiotic
framework. In semiotic integration, we amalgamate different logics within a
single semiotic, which are associated with distinct languages employed by
domain experts or structural specification languages. In both contexts,
knowledge integration, combined with inference, can significantly facilitate
the knowledge acquisition process.

We impose a critical constraint on semiotic integration: Given a family of
semiotics $(S_{i},M_{i})_{I}$, its integration is defined if and only if equal
signs and components bearing identical labels across different semiotics share
the same interpretation. The only potential exception to this rule pertains to
the interpretations of the sign $\Omega$, which is associated with the logic
of the semiotics and its operators and may vary.

The integration of semiotics $(S_{i},M_{i})_{I}$ is denoted by
$(\bigcup_{I}S_{i},\bigcup_{I}M_{i})$ and is represented by the sign system:

$\bigcup_{I}S_{i}=(\bigcup_{I}L_{i},\bigcup_{I}\mathcal{E}_{i},\bigcup_{I}\mathcal{U}_{i},\bigcup_{I}co\mathcal{U}_{i}),$

where each semiotic $S_{i}$ is defined by the structure
$(L_{i},\mathcal{E}_{i},\mathcal{U}_{i},co\mathcal{U}_{i})$ for $i\in I$.
Here, $\bigcup_{I}L_{i}$ represents the library defined as the union of
libraries $(L_{i}:|L_{i}|\rightarrow(Chains\downarrow\Sigma_{i}^{+}))_{I}$
associated with each semiotic. This library is given by:

$\bigcup_{I}L_{i}:\bigcup_{I}|L_{i}|\rightarrow(Chains\downarrow\bigcup_{I}\Sigma_{i}^{+}),$

where the signs consist of the union of ontologies
$\bigcup_{I}\Sigma_{i}^{+}$, defined by the signs present in each library, and
the component labels comprise the union of labels existing in both libraries.
It’s worth noting that the integration of libraries must maintain the
functionalities of components. Hence, the union of libraries is only defined
if components in different libraries, with identical labels, exhibit the same
functionalities. The graphical language associated with $\bigcup_{I}L_{i}$ is
denoted as $Lang(\bigcup_{I}L_{i})$, and we have
$\bigcup_{I}\mathcal{E}_{i}\subset Lang(\bigcup_{I}L_{i})$.

Recalling the description for the language associated with $\bigcup_{I}L_{i}$,
given two graphs $G_{0}$ and $G_{1}$, we define $G_{0}\bigcup G_{1}$ as the
graph having vertices from both $G_{0}$ and $G_{1}$ and arrows from both
$G_{0}$ and $G_{1}$. If each library $L_{i}$ is associated with multi-graphs
$\mathcal{G}(L_{i})$, then:

$\mathcal{G}(\bigcup_{I}L_{i})=\bigcup_{I}\mathcal{G}(L_{i}).$

If we have models $(M_{i}:\mathcal{G}(L_{i}^{\ast})\rightarrow
Set(\Omega))_{I}$ for different libraries, the homomorphism $\bigcup_{I}M_{i}$
serves as a model for the sign system $\bigcup_{I}S_{i}$:

$\bigcup_{I}M_{i}:\mathcal{G}((\bigcup_{I}L_{i})^{\ast})\rightarrow
Set(\Omega)$

constructed using the union of models
$(M_{i}:\mathcal{G}(L_{i}^{\ast})\rightarrow Set(\Omega))_{I}$, assigning to
nodes $\bigcup_{I}M_{i}(v)=M_{i}(v)$ if $v\in\mathcal{G}(L_{i}^{\ast})$ and
$v\neq\Omega$ for some $i\in I$, and $\bigcup_{I}M_{i}(v)=M_{i}(v)$ for multi-
arrows $f:w\rightarrow w^{\prime}$ in $\mathcal{G}(L_{i}^{\ast})$ where
$w^{\prime}\neq\Omega$ for some $i\in I$. This implies that logical signs and
multi-arrows not representing relations are interpreted as if they were in
their respective libraries. This definition is meaningful only when identical
signs and labels are interpreted consistently across different libraries.

For the family of logical signs $(\Omega_{i})_{I}$ associated with the family
of logical semiotics $(S_{i})_{I}$, we define the sign $\prod_{I}\Omega_{i}$
interpreted by $\bigcup_{I}M_{i}$ as the ML-algebra product
$\prod_{I}M_{i}(\Omega_{i})$. The interpretation of the sign
$\prod_{I}\Omega_{i}$ yields an ML-algebra, and for every relation
$r:w\rightarrow\Omega_{i}$ present in each semiotic $S_{i}$, its
interpretation by $\bigcup_{I}M_{i}$ is given by:

$\bigcup_{I}M_{i}(r:w\rightarrow\Omega_{i})=M_{i}(r)\otimes\pi^{\top}_{\Omega_{i}}$

where $\pi^{\top}_{\Omega_{i}}:\Omega_{i}\rightarrow\prod_{I}\Omega_{i}$ is
the mapping defined as:

$\pi^{\top}_{\Omega_{i}}(\alpha)=(\top,\ldots,\top,\alpha,\top,\ldots,\top)$

with $\alpha$ appearing in the component of order $i$ and differing from
$\top$. Formally,

###### Proposition 25

If

$S_{0}=(L_{0},\mathcal{E}_{0},\mathcal{U}_{0},co\mathcal{U}_{0}),S_{1}=(L_{1},\mathcal{E}_{1},\mathcal{U}_{1},co\mathcal{U}_{1}),\ldots,S_{n}=(L_{n},\mathcal{E}_{n},\mathcal{U}_{n},co\mathcal{U}_{n})$

are sign systems with models $M_{0},M_{1},\ldots,M_{n}$, then the mapping

$\bigcup_{I}M_{i}:\mathcal{G}((\bigcup_{I}L_{i})^{\ast})\rightarrow
Set(\Omega)$

defined as above serves as a model for the sign system

$\bigcup_{I}S_{i}=(\bigcup_{I}L_{i},\bigcup_{I}\mathcal{E}_{i},\bigcup_{I}\mathcal{U}_{i},\bigcup_{I}co\mathcal{U}_{i}).$

Given models $M_{0},M_{1},\ldots,M_{n}$ for sign systems
$S_{0},S_{1},\ldots,S_{n}$, by Definition 15, for every $j=1,\ldots,n$:

  1. 1.

If $D\in\mathcal{E}_{j}$, then $\bigcup_{I}M_{i}(D)=M_{j}(D)$ is a total
multi-morphism.

  2. 2.

If $(s,D,i(D),o(D))\in\mathcal{U}_{j}$, then $\bigcup_{I}M_{i}(s)=M_{j}(s)$
represents the $\Omega$-set defined by $Lim\;MD$.

  3. 3.

If $(s,D,i(D),o(D))\in co\mathcal{U}_{j}$, then $\bigcup_{I}M_{i}(s)=M_{j}(s)$
represents the $\Omega$-set defined by $coLim\;MD$.

Naturally, the resulting semiotic from an integration process inherits both
the syntax and semantics generated by the syntax and semantics associated with
the individual semiotics. A similar principle applies to certain syntactic
operators. The integration of a family of semiotics, where at least one is a
differential semiotic, yields a differential semiotic. The same holds true for
temporal semiotics. If $(S_{i})_{I}$ is a family where $(S_{j})_{J}$ forms a
subfamily of temporal semiotics, defined by the syntactic operator
$t_{i}:\Sigma^{+}_{i}\rightarrow\Sigma^{+}_{i}$, then the integrated semiotic
$\bigcup_{I}S_{i}$ is a temporal semiotic. Here, the syntactic operator
$t:\bigcup_{I}\Sigma^{+}_{i}\rightarrow\bigcup_{I}\Sigma^{+}_{i}$ is defined
as follows:

  1. 1.

$t(s)=t_{j}(s)$ if $s\in\Sigma^{+}_{j}$ where $j\in J$, and

  2. 2.

$t(s)=s$ if $s\notin\bigcup_{J}\Sigma^{+}_{j}$.

An integration schema is represented by a diagram

$\mathcal{J}:\mathcal{G}\rightarrow Set(\Omega),$

defined within the category of interpretations and computable multi-morphisms,
such that $\mathcal{J}(i)=MD_{i}$ for every vertex $i$ in $\mathcal{J}$. Let
$(D_{i})$ be a family of diagrams used in the definition of integration schema
$\mathcal{J}$. The concept description $\Omega(\mathcal{J})$ defined by
$\mathcal{J}$ is given by the colimit of $\mathcal{J}$:

$\Omega(\mathcal{J})=colim_{i}\;MD_{i}=colim_{i}\;Lim\;M(D_{i}),$

where the colimit is computed as defined in 11, i.e.,

${}_{(coLim\;\mathcal{J})(\ldots,\bar{x}_{i},\ldots,\bar{x}_{j},\ldots)=\ldots\otimes
Lim\;M(D_{i})(\bar{x}_{i})\otimes\ldots\otimes
Lim\;M(D_{j})(\bar{x}_{j})\otimes\ldots\otimes\bigvee_{f:MD_{i}\rightarrow
MD_{j}\in\mathcal{J}}f(\bar{x}_{i},\bar{x}_{j}).}$

## Chapter 15 Reasoning about models of concepts

The language $\lambda$-RL$(S)$ of $\lambda$-representable logic serves as a
formalism for discussing structures that are $\lambda$-representable within a
semiotic $(S,M)$. Fundamentally, it is a string-based modal logic, defined by
a generative grammar where propositional variables are interpreted as diagrams
belonging to the language associated with the sign system $S$.

$\lambda$-RL$(S)$ is constructed using relations from $Lang(S)$, modal
operators such as limit, closure, interior, and the lifting of the monoidal
logic connectives $\otimes$, $\Rightarrow$, $\wedge$, and $\vee$ to relations.

Every semiotic $(S,M)$ provides semantics for $\lambda$-RL$(S)$ through the
truth-relation

$g\models_{\lambda}\varphi,$

which is defined as follows for every formula $\varphi\in\lambda$-RL$(S)$ and
every concept description $g$ in $(S,M)$:

  1. 1.

$g\models_{\lambda}\varphi$ if and only if $\varphi$ corresponds to the
diagram $D$ and $\Gamma(g,MD)\geq\lambda$,

  2. 2.

$g\models_{\lambda}[I]\varphi$ if and only if
$int(g)\models_{\lambda}\varphi$,

  3. 3.

$g\models_{\lambda}[C]\varphi$ if and only if $cl(g)\models_{\lambda}\varphi$.

For any formulas $\varphi_{0}$ and $\varphi_{1}$ in $\lambda$-RL$(S)$, if

$g\models_{\lambda_{0}}\varphi_{0}\text{ and
}g\models_{\lambda_{1}}\varphi_{1},$

then:

  1. 1.

$g\models_{\lambda_{0}\otimes\lambda_{1}}(\varphi_{0}\otimes\varphi_{1})$,

  2. 2.

$g\models_{\lambda_{0}\Rightarrow\lambda_{1}}(\varphi_{0}\Rightarrow\varphi_{1})$,

  3. 3.

$g\models_{\lambda_{0}\wedge\lambda_{1}}(\varphi_{0}\wedge\varphi_{1})$, and

  4. 4.

$g\models_{\lambda_{0}\vee\lambda_{1}}(\varphi_{0}\vee\varphi_{1})$.

Utilizing the structural compatibility between multi-morphism composition and
diagram gluing, we have:

###### Proposition 26

Given multi-morphisms $g$ and $h$ such that

$g\models_{\lambda_{0}}\varphi_{0}\quad\text{and}\quad
h\models_{\lambda_{1}}\varphi_{1},$

we can conclude that

$g\otimes
h\models_{\lambda_{0}\otimes\lambda_{1}}(\varphi_{0}\otimes\varphi_{1}).$

By extending the ML-algebra structure to the set of concept descriptions, we
derive:

###### Proposition 27

Given concept descriptions $g$ and $h$ in $\oplus_{i}A_{i}$ such that

$g\models_{\lambda_{0}}\varphi\quad\text{and}\quad
h\models_{\lambda_{1}}\varphi,$

we have:

  1. 1.

$(g\otimes h)\models_{\lambda_{0}\otimes\lambda_{1}}\varphi$,

  2. 2.

$(g\Rightarrow h)\models_{\lambda_{0}\Rightarrow\lambda_{1}}\varphi$,

  3. 3.

$(g\wedge h)\models_{\lambda_{0}\wedge\lambda_{1}}\varphi$, and

  4. 4.

$(g\vee h)\models_{\lambda_{0}\vee\lambda_{1}}\varphi$.

Given a set of relations $U$ from $\text{Lang}_{R}(S)$ and $\varphi$ a
relation in $\lambda$-RL$(S)$, we define:

$U\models_{\lambda}\varphi\quad\text{iff}\quad\text{ans}_{\lambda}(U)\models_{\lambda}\varphi.$

Utilizing Theorem 5, we can state:

######  Theorem 10 (Soundness)

Given a set of relations $U$ from $\text{Lang}_{R}(S)$ and $\varphi$ a
relation in $\lambda$-RL$(S)$,

$\text{if }U\vdash_{\lambda}\varphi\text{ then }U\models_{\lambda}\varphi$

for every $\lambda$.

Naturally, completeness does not hold universally. If
$U\models_{\lambda}\varphi$, it does not necessarily mean that we can prove
$U\vdash_{\lambda}\varphi$ by deduction.

## Chapter 16 Conclusions and Future Work

The use of semiotics appears to be an appropriate formalism for defining both
the syntax and semantics of graphic languages. This is especially true when
these languages are based on a library of functional components interpreted as
relations evaluated in multi-valued logic. Such an approach simplifies the
integration of knowledge expressed in different languages and facilitates the
inference of new knowledge.

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