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\begin{document}
\author{Lech Polkowski \\
%EndAName
Institute of Mathematics\\
Warsaw University of Technology\\
Pl. Politechniki 1, 00661 Warsaw\\
e-mail: polk@mimuw.edu.pl \and Andrzej Skowron \\
%EndAName
Institute of Mathematics\\
Warsaw University\\
Banacha 2, 02097 Warsaw\\
e-mail: skowron@mimuw.edu.pl}
\title{Approximate Reasoning about Complex Objects in Distributed Systems: Rough
Mereological Formalization}
\date{}
\maketitle
\begin{abstract}
This is an extended version of the lecture delivered at the Workshop: Logic,
Algebra and Computer Science (LACS) held in the Banach International
Mathematical Center in Warsaw on December 16, 1996. We propose an approach
to approximate reasoning by systems of intelligent agents based on the
paradigm of rough mereology. In this approach, the knowledge of each agent
is formalized as an information system (a data table) from which similarity
measures on objects manipulated by this agent are inferred. These similarity
measures are based on rough mereological inclusions which formally render
degrees for one object to be a part of another. Each agent constructs in
this way its own rough mereological logic in which it is possible to express
approximative statements of the type : '' {\it an object} $x$ {\it satisfies
a predicate }$\Psi $ {\it in degree} $r$''. The agents communicate by means
of mereological functors (connectives among distinct rough mereological
logics) propagating similarity measures from simpler to more complex agents;
establishing these connectives is the main goal of negotiations among
agents. The presented model of approximate reasoning entails such models of
approximative reasoning like fuzzy controllers, neural networks etc. Our
approach may be termed analytic, in the sense that all basic constructs are
inferred from data.
\end{abstract}
{\it \ }
\section{INTRODUCTION}
Approximate reasoning concerns synthesis of a solution to a given problem in
the case when our knowledge about the context of the problem is insufficient
to bring forth an exact solution e.g. when our knowledge is incomplete or
uncertain, or the problem is formulated in an imprecise (e.g natural)
language.
Many formal models of approximate reasoning are described in the literature
e.g. Dempster-Schafer theory of evidence [35], [36], [39], bayesian
reasoning [23], [36], belief networks [23], [36], many-valued logics [11]
and fuzzy logics [11], non-monotonic logics [36] and neural network logics
[17].
We can extract from these formal models a general scheme for approximate
reasoning.
It is manifest that this scheme encompasses classical models of reasoning
adopted in mathematical logic [19]: the process of derivation of a formula
from instances of axioms can be regarded as a synthesis process of a complex
artifact from simple inventory parts and each application of a derivation
rule (e.g. {\it Modus ponens}) can be regarded as an action of an agent
applying its specialized operation to simpler objects in order to construct
a more complex object.
\subsection{A formal model of approximate reasoning}
We present a formal model of approximate reasoning about processes of
synthesis of complex objects. This approach have been developed and
presented in [15], [24], [25], [27], [28], [29], [30], [31], [32], [33]. Our
research has been stimulated by the demand for solutions of the following
groups of problems concerned with the treatment of:
\medskip\
{\bf \ 1.} Poorly defined, poorly understood or incomplete design
specifications.
{\bf \ 2.} Negotiations among interacting goals and constraints.
{\bf \ 3.} Decomposition of problems into subproblems (including the problem
of formation of a hierarchical scheme for solving the problem).
{\bf \ 4.} Adaptation problems (including redesign and reuse problems).
{\bf \ 5.} Problems of knowledge representation and reasoning about
knowledge.
\medskip\
These groups of problems are considered in [1] as the most important in the
area of automated design and manufacturing.
Design as well as manufacturing processes involve the space of
specifications and the space of structures. These spaces are present in our
approach at each local process site and they meet each other at the
inventory level where primitive (indecomposable ) specifications are
converted into primitive (inventory) constructs.
Our analysis can be applied to the following fields concerned with complex
systems:
\medskip\
{\bf \ 6.} Computer-aided design [1], [34], [44] or computer-aided
manufacturing [1], [4], [14], [44]. In this field, a complex system is
synthesized , or designed, from elementary subsystems.
{\bf 7.} Adaptive control of complex systems [13], [18], [28]. In this
field, given specification (constraint) is maintained by adaptive adjustment
of specifications for some subsystems.
{\bf 8.} Business re-engineering [2], [22] (including software reuse). In
this area, a complex system is adaptively modified according to a current
requirement.
{\bf 9.} Cooperative and distributive problem solving [7], [8], [9], [10],
14], [34], [44], [50], [51]. In this field a complex system of local agents
is organized from a set of agents in order to synthesize a solution to a
problem.
\medskip\
The accessible knowledge on the basis of which constructs in the synthesis
process are selected and classified (evaluated) is as a rule incomplete,
poorly defined, or inconsistent. In consequence, we are bound to evaluate
the basic ingredients of the synthesis process approximately only, in terms
of values of some uncertainty measures which express a degree in which a
given construct satisfies a given specification and in terms of some
functors which propagate uncertainty measures along the synthesis scheme.
The general scheme for approximate reasoning can be represented by the
following tuple
\medskip\
\[
Appr\_Reas=(Ag,Link,U,St,Dec\_Sch,O,Inv,Unc\_mes,Unc\_prop)
\]
where
(i) The symbol $Ag$ denotes the set of agent names.
(ii) The symbol $Link$ denotes a set of non-empty strings over the alphabet $%
Ag$; for $v(ag)=ag_1ag_2...ag_kag\in Link$, we say that $v(ag)$ defines an
{\it elementary synthesis scheme} with the {\it root ag} and the {\it leaf
agents} $ag_1,ag_2,...,ag_k.$ The intended meaning of $v(ag)$ is that the
agents $ag_1,ag_2,..,ag_k$ are the children of the agent $ag$ which can send
to $ag$ some simpler constructs for assembling a more complex artifact. The
relation $\leq $defined via $ag\leq ag^{\prime }$ iff $ag$ is a leaf agent
in $v(ag^{\prime })$ for some $v(ag^{\prime })$ is usually assumed to be at
least an ordering of $Ag$ into a type of an acyclic graph; we assume for
simplicity that $(Ag,$ $\leq )$ is a tree with the root $root(Ag)$ and leaf
agents in the set $Leaf(Ag)$.
(iii) The symbol $U$ denotes the set $\{U(ag):ag\in Ag\}$ of {\it universes}
of agents.
(iv) The symbol $St$ denotes the set $\{St(ag):ag\in Ag\}$ where $%
St(ag)\subset U(ag)$ is the set of {\it standard objects} at the agent $ag$.
(v) The symbol $O$ denotes the set $\{O(ag):ag\in Ag\}$ of {\it operations }%
where $O(ag)=\{o_i(ag)\}$ is the set of {\it operations at} $ag$.
(vi) The symbol $Dec\_Sch$ denotes the set of {\it decomposition schemes}; a
particular decomposition scheme $dec\_sch_j$ is a tuple
\[
(\{st(ag)_j:ag\in Ag\},\text{ }\{o_j(ag):ag\in Ag\})
\]
which satisfies the property that if $v(ag)=ag_1ag_2...ag_kag\in Link$ then
\[
o_j(ag)(st(ag_1)_j,st(ag_2)_j,..,st(ag_k)_j)=st(ag)_j\text{ for each }j\text{%
.}
\]
The intended meaning of $dec\_sch_j$ is that when any child $ag_i$ of $ag$
submits the standard construct $st(ag_i)_j$ then the agent $ag$ assembles
from
\[
st(ag_1)_j,st(ag_2)_j,..,st(ag_k)_j
\]
the standard construct $st(ag)_j$ by means of the operation $o_j(ag)$. %
\medskip\
The rule $dec\_sch_j$ establishes therefore a decomposition scheme of any
standard construct at the agent {\it root }({\it Ag}) into a set of
consecutively simpler standards at all other agents. The standard constructs
of leaf agents are {\it primitive (inventory) standards}. We can regard the
set of decomposition schemes as a skeleton about which the approximate
reasoning is organized. Any rule $dec\_sch_j$ conveys a certain knowledge
that standard constructs are synthesized from specified simpler standard
constructs by means of specified operations. This ideal knowledge is a
reference point for real synthesis processes in which we deal as a rule with
constructs which are not standard: in adaptive tasks, for instance, we
process new, unseen yet, constructs (objects, signals).
(vii) The symbol $Inv$ denotes the {\it inventory set} of primitive
constructs. We have $Inv=\cup \{U(ag):ag\in Leaf(Ag)\}.$
(viii) The symbol $Unc\_mes$ denotes the set $\{Unc\_mes(ag):ag\in Ag\}$ of
{\it uncertainty measures of agents}, where $Unc\_mes(ag)=\{\mu _j(ag)\}$
and $\mu _j(ag)$ $\subseteq U(ag)\times U(ag)\times V(ag)$ is a relation
(possibly function) which determines a distance between constructs in $U(ag)$
valued in a set $V(ag)$; usually, $V(ag)=[0,1]$, the unit interval.
(ix) The symbol $Unc\_prop$ denotes the set of {\it uncertainty propagation
rules} $\{Unc\_prop(v(ag)):v(ag)\in Link\}$; for $v(ag)=ag_1ag_2...ag_kag\in
Link$, the set $Unc\_prop(v(ag))$ consists of functions $f_j$ :$%
V(ag_1)\times V(ag_2)\times ...\times V(ag_k)\longrightarrow V(ag)$ such that
if $\mu _j(ag_i)(x_i,st(ag_i)_j)=\varepsilon _i$ for $i=1,2,..,k$
\begin{center}
then $\mu _j(ag)(o_j(x_1,x_2,..,x_k),st(ag)_j)=\varepsilon \geq
f_j(\varepsilon _1,\varepsilon _2,..,\varepsilon _k).$
\end{center}
The functions $f_j$ relate values of uncertainty measures at the children of
$ag$ and at $ag$.
This general scheme may be adapted to the particular cases.
\medskip\
As an example, we will interpret this scheme in the case of a fuzzy
controller [11]. In its version due to Mamdani [18], in its simplest form,
we have two agents: {\it input,} {\it output}, and standards of agents are
expressed in terms of linguistic labels like {\it positively small,
negative, zero} etc. Operations of the agent {\it output} express the
control rules of the controller e.g. the symbol $o(positively$ $%
small,negative)=zero$ is equivalent to the control rule of the form if $%
st(input)_i$ is $positively$ $small$ and $st(input)_j$ is {\it negative}
then $st(output)_k$ is $zero$. Uncertainty measures of agents are introduced
as fuzzy membership functions [11], [49] corresponding to fuzzy sets
representing standards i.e. linguistic labels. An input construct (signal) $%
x(input)$ is fuzzified i.e. its distances from input standards are
calculated and then the fuzzy logic rules are applied [11]. By means of
these rules uncertainty propagating functions are defined which allow for
calculating the distances of the output construct $x(output)$ from the
output standards. On the basis of these distances the construct $x(output)$
is evaluated by the defuzzification procedure.
\medskip\
Knowledge is represented in our approach by means of rough set theory [20].
This theory assumes that constructs in a given universe are perceived by
means of the available information, expressed in the form of values on these
objects of certain attributes (features), and in consequence these
constructs which bear the same information about them are perceived as
identical and form collections of indiscernible constructs . The resulting
granularity of knowledge is the effect of incompleteness of knowledge. We
are not able therefore to discuss individual constructs but their
collections consisting of pairwise indiscernible objects; in consequence, we
discuss not the membership relation but the containment relation. The
counterpart of the notion of a fuzzy membership function would be the more
general notion of a partial containment.
The formal treatment of partial containment is provided by the notion of a
rough inclusion [24], [27], [29]. Rough inclusions are construed as most
general functional objects conveying the intuitive meaning of the relation
of being a part in a degree. In particular, it turns out that the relation
of being a part in the greatest possible degree is the relation of being a
(possibly, improper) part in the sense of mereology of Stanislaw
Le\'{s}niewski [16]. We can regard therefore a rough inclusion as a tool by
means of which we extend the mereological relation of being an ingredient to
a weaker relation (of being a part to a degree) over all pairs of objects in
the universe. In this way, we create a family of similarity measures on
pairs of objects.
In mereology of Le\'{s}niewski the notions of a (possibly improper) part, of
a subset and of an element are equivalent (see [16]) and therefore we can
interpret rough inclusions as global fuzzy membership functions on the
universe of discourse which satisfy certain general requirements (stemming
from our intuition).
We take rough inclusions of agents as measures of uncertainty in their
respective universes.
{\bf Remark 1.1.} Any non-leaf agent $ag$ is able to establish a local
decomposition scheme of complex constructs in its universe into some simpler
parts by means of its rough inclusion $\mu (ag)$ and the relation $part$ (of
being a (proper) part) in the induced model of mereology of Le\'{s}niewski.
{\bf Remark 1.2.} The mereological relation of being a part is not
transitive globally over the whole synthesis scheme as distinct agents use
distinct mereological languages.
\medskip\
The process of synthesis by a scheme of agents of a complex object $x$ which
is an approximate solution to a requirement $\Phi $ consists in our approach
of the two communication stages viz. the top - down
communication/negotiation process and the bottom - up synthesis process. We
outline the two stages here.
In the process of top - down communication, a requirement $\Phi $ received
by the scheme from an external source is decomposed into approximate
specifications of the form
\[
(\Phi (ag),\varepsilon (ag))
\]
for any agent $ag$ of the scheme. The intended meaning of the approximate
specification $(\Phi (ag),\varepsilon (ag))$ is that a construct $z$ $\in
U(ag)$ satisfies $(\Phi (ag),\varepsilon (ag))$ iff there exists a standard $%
st(ag)$ with the properties that $st(ag)$ satisfies the predicate $\Phi (ag)$
and
\[
\mu (ag)(z,st(ag))\geq \varepsilon (ag).
\]
The uncertainty bounds of the form $\varepsilon (ag)$ are defined by the
agents viz. the root agent {\it root(Ag) }chooses $\varepsilon (root(Ag))$
and $\Phi (root(Ag))$ as such that according to it any construct $x$
satisfying ($\Phi (root(Ag),\varepsilon (root(Ag))$ should satisfy the
external requirement $\Phi $ in an acceptable degree. The choice of ($\Phi
(root(Ag),\varepsilon (root(Ag))$ can be based on the previous learning
process; the other agents choose their approximate specifications in
negotiations within each elementary scheme $v(ag)$ $\in Link$. The result of
the negotiations is succesful when there exists a decomposition scheme $%
dec\_sch_j$ such that for any $v(ag)\in Link$, where $v(ag)=ag_1ag_2...ag_kag
$, from the conditions
$\mu (ag_i)(x_i,st(ag_i)_j)\geq \varepsilon (ag_i)$
and
$st(ag_i)_j$ satisfies $\Phi (ag_i)$ for $i=1,2,..,k,$ it follows that
$\mu (ag)(o_j(x_1x_2,..,x_k),st(ag)_j)\geq \varepsilon (ag)$
and
st(ag)$_j$ satisfies $\Phi (ag)$.
The uncertainty bounds $\varepsilon (ag)$ are evaluated on the basis of
uncertainty propagating functions whose approximations are extracted from
information systems of agents.
The synthesis of a complex object $x$ is initiated at the leaf agents: they
select primitive constructs (objects) and calculate their distances from
their respective standards; then, the selected constructs are sent to the
parent nodes of leaf agents along with vectors of distance values. The
parent nodes synthesize complex constructs from the sent primitives and
calculate the new vectors of distances from their respective standards.
Finally, the root agent $root(Ag)$ receives from its children the constructs
from which it assembles the final construct and calculates the distances of
this construct from the root standards. On the basis of the found values,
the root agent classifies the final construct $x$ with respect to the root
standards as eventually satisfying $(\Phi (root(Ag),$ $\varepsilon
(root(Ag)).$
Our approach is analytic : all logical components ( uncertainty measures,
uncertainty functions etc.) necessary for the synthesis process are
extracted from the empirical knowledge of agents represented in their
information systems; it is also intensional in the sense that rules for
propagating uncertainty are local as they depend on a particular elementary
synthesis scheme and on a particular local standard.
We conclude this section with a concise rendering of basic notions of rough
set theory.
\subsection{Rough set theory preliminaries}
An {\it information system } is a pair ${\sf A}=\left(U,A\right)$ where $U$
is a finite set called the {\it universe of objects} and $A$ is a finite set
of {\it attributes}; any attribute $a\in A$ is a mapping on the universe $U$%
. We denote by the symbol $V_{a_{}}$ the range of the attribute $a$; the set
$V_a$ is called the {\it value set} of $a$. We let $V=\cup \{V_a:a\in A\}$.
In consequence of the above assumption some objects may become
indiscernible. For an object $x\in U,$ we define for a set $B\subseteq A$ the%
{\it \ information vector} $Inf_B(x)=\{(a,a(x)):a\in B\}$. We say that
objects $x,y\in U$ are $B$-{\it indiscernible }when $Inf_B(x)=$ $Inf_B(y)$;
the $B$-{\it indiscernibility relation} $IND(B)$ is defined as follows: $%
IND(B)=\{(x,y)\in U\times U:Inf_B(x)=Inf_B(y)\}$. The relation $IND(B)$ is
an equivalence relation and we denote by the symbol $[x]_B$ the equivalence
class of this relation which contains $x$. We will use the term {\it concept}
for subsets of the universe $U$; for a concept $X\subseteq U$, we define the
two approximations of $X$ relative to a set $B\subseteq A$:
\begin{center}
$\underline{B}X=\{x\in U:[x]_B\subseteq X\}$ and $\overline{B}X=\{x\in
U:[x]_B\cap X\neq \emptyset \}$.
\end{center}
The set $BN_B(X)$ = $\overline{B}X-\underline{B}X$ is called the $B-${\it %
boundary region }of $X.$ In the case when $BN_B(X)=\emptyset $ the concept X
is said to be $B$-{\it exact}, otherwise $X$ is $B$-{\it rough}. The
concepts $\overline{B}X$, $\underline{B}X$ and $BN_B(X)$ have clear
epistemic interpretation viz. the concept $\overline{B}X$ collects all
objects which belong certainly in X, the concept $U-\overline{B}X$ collects
all objects which certainly do not belong in X and the concept $BN_B(X)$
collects all objects which are vague with respect to $X$ i.e. have
representatives both in $X$ and in the complement of $X$. It follows that $%
BN_B(X)$ is a non-sharp boundary of $X$ in the sense of Frege (cf.[20]).
Given a concept $X$, the numerical characterization of a degree in which an
object $x$ belongs in the concept $X$ relative to the knowledge represented
by an attribute set $B\subseteq A$ is provided by the rough membership
function $\mu _{X,B}$ [21]. For $B\subseteq A$, $X\subseteq U$ and $x\in U$,
we let
\begin{center}
$\mu _{X,B}(x)=\frac{\Vert X\cap [x]_B\Vert }{\Vert [x]_B\Vert }$
\end{center}
where $\Vert Z\Vert $ denotes the cardinality of a set $Z$. In the case when
$B=A$ we use the symbol $\mu _X$ instead of the symbol $\mu _{X,A}$.
We can extend the rough membership function $\mu _X$ to the function $\mu _U$
on the power set $\exp (U)$ of $U$. To this end, we define $\mu _U:\exp
(U)\times \exp (U)\longrightarrow [0,1]$ by letting
\begin{center}
$\mu _U(X,Y)=\frac{\Vert X\cap Y\Vert }{\Vert X\Vert }$ in case $X\neq
\emptyset $ and $\mu _U(\emptyset ,Y)=1.$
\end{center}
We denote by the symbol $Stand$ the class of pairs of the form $(U,\mu _U)$
where $U$ is a finite set and $\mu _U$ is the standard rough inclusion on
the set $U$.
The reader will find in [26], [37], [38], [41], [42] a thorough discussion
of rough set-theoretic tools for decision rules generation and for synthesis
of adaptive decision systems.
\section{Mereology of St. Le\'{s}niewski}
The importance for logic of the fundamental study of relations of being a
part was already stressed by Aristotle ({\it Metaphysics}, Book IV). The
first modern mathematical system based on the notion of the relation of $%
being$ $a$ $(proper)$ $part$ was proposed by Stanislaw Le\'{s}niewski [16].
We recall here the basic notions of the mereological system of
Le\'{s}niewski; in the next section the mereological system of
Le\'{s}niewski will be extended to the system of approximate mereological
calculus called {\it rough mereology.}
We consider a finite set $U$; we assume that $U$ is non-empty. A binary
relation $part$ on the set $U$ will be called the{\it \ relation of being a }%
({\it proper}) {\it part} in the case when the following conditions are
fulfilled
\medskip\
(P1) (irreflexivity) for any $x\in U$, it is not true that $x$ $part$ $x$;
(P2) (transitivity) for any triple $x,y,z\in U$, if $x$ $part$ $y$ and $y$ $%
part$ $z$, then $x$ $part$ $z$.
\medskip\
It follows obviously from (P1) and (P2) that the following property holds
(P3) for any pair $x,y\in U$, if $x$ $part$ $y$ then it is not true that $y$
$part$ $x$.
In the case when $x$ $part$ $y$ we say that the object $x$ is a (p{\it roper}%
) {\it part} of the object $y.$ The notion of being (possibly) an improper
part is rendered by the notion of an ingredient [16]; for objects $x,y\in U$%
, we say that the object $x$ is a {\it \ }$part$-{\it ingredient} of the
object $y$ when either $x$ $part$ $y$ or $x=y$. We denote the relation of
being a $part$-ingredient by the symbol $ingr$($part$); hence we can write
(I1) for $x,y\in U$, $x$ $ingr(part)$ $y$ iff $x$ $part$ $y$ or $x=y.$
It follows immediately from the definition that the relation of being an
ingredient has the following properties:
(I2) (reflexivity) for any $x\in U$, we have $x$ $ingr(part)$ $x$;
(I3) (weak antisymmetry) for any pair $x,y\in U$, if $x$ $ingr(part)$ $y$
and $y$ $ingr(part)$ $x$ then $x=y$;
(I4) (transitivity) for any triple $x,y,z\in U$, if $x$ $ingr(part)$ $y$ and
$y$ $ingr(part)$ $z$ then
$x$ $ingr($ $part)$ $z$
i.e. the relation $ingr(part)$ is a partial order on the universe $U$.
We will call any pair $(U,$ $part),$ where $U$ is a finite set and $part$ a
binary relation on the set $U,$ which satisfies the conditions (P1) and
(P2), a {\it pre-model of mereology. }
We now recall the notions of a set of objects and of a class of objects
[16]. For a given pre-model $(U,$ $part)$ of mereology and a property $m$
which can be attributed to objects in $U$, we say that an object $x$ is an
object $m$ ($x$ $object$ $m$, for short) when the object $x$ has the
property $m$. The property $m$ is said to be non-void when there exists an
object $x\in U$ such that $x$ $object$ $m.$ Consider a non-void property $m$
of objects in a set $U$ where $(U,part)$ is a pre-model of mereology.
An object $x\in U$ is said to be a{\it \ set of objects with the property} $%
m $ when the following condition is fulfilled:
(SET$m$) for any $y\in U$, if $y$ $object$ $m$ and $y$ $ingr(part)$ $x$ then
there exist $z,t\in U$ with the properties: $z$ $ingr(part)$ $y$, $z$ $%
ingr(part)$ $t$, $t$ $ingr(part)$ $x$ and $t$ $object$ $m.$
We will use the symbol $x$ $set$ $m$ to denote the fact that an object $x$
is a set of objects with the property $m$.
Assume that $x$ $set$ $m$; if, in addition, the object $x$ satisfies the
condition
(CL$m$) for any $y\in U$, if $y$ $object$ $m$ then $y$ $ingr(part)$ $x$ then
we say that the object $x$ is {\it a class of objects with the property} $m$
and we denote this fact by the symbol $x$ $class$ $m$. We will say that a
pair ($U,part$) is a model of mereology when the pair $(U,part)$ is a
pre-model of mereology and the condition
(EUC) for any non-void property $m$ of objects in the set $U$, there exists
a unique object $x$ such that $x$ $class$ $m$ holds.
The following proposition [16] recapitulates the fundamental
metamathematical properties of mereology of Le\'{s}niewski; observe that in
mereology there is no hierarchy of objects contrary to the Cantorian naive
set theory. We denote for an object $x\in U$ by the symbol $ingr(x)$ the
property of being an ingredient of $x$ (non-void in virtue of (I2)) and for
a property $m,$ we denote by the symbol $s(m)$ the property of being a set
of objects with the property $m$.
\medskip\
Proposition 2.1
(i) $x$ $class(s(m))$ if and only if $x$ $classm$;
(ii) $x$ $class(ingr(x));$
(iii)$x$ $set(s(m))$ if and only if $x$ $setm$.
\medskip\
We recall the notions of an element and of a subset in mereology of
Le\'{s}niewski. For $x,y\in U$, we will say that
(SUB) the object $x$ is a {\it subset} of the object $y$ $(x$ $sub$ $y,$ for
short$)$ when for any $z\in U,$
if $z$ $ingr(part)$ $x$ then $z$ $ingr(part)$ $y$
and
(EL) the object $x$ is an element of the object $y$ ($x$ $el$ $y$, for
short) when there exists a non-void property $m$ such that $x$ $object$ $m$
and $y$ $class$ $m$.
The following propositionj is a direct consequence of (I4) and Proposition
2.1(ii).
\medskip\
Proposition 2.2
The following are equivalent for any pair $x,y$ of objects
(i) $x$ $ingr$ $y$;
(ii) $x$ $sub$ $y;$
(iii) $x$ $el$ $y.$
\medskip\
\section{Rough Mereology}
An approximate mereological calculus called rough mereology has been
proposed as a formal treatment of the hierarchy of relations of being a part
in a degree. We begin with an exposition of rough mereological calculus in
the form of a rough mereological logic $L_{rm}$.
We begin with the syntactical part.
\medskip\
\subsection{Syntax of $L_{rm}$}
We have the following basic ingredients of the syntactic part of the logic $%
L_{rm}.$
{\bf Variables:} $x,x_1,x_2$,..., $y,y_1,y_2$,..., $z,z_1,z_2,...$ of type
{\it set\_element} and $r,$ $r_1,r_2$,..., $s,s_1,s_2,...$ of type {\it %
lattice\_element};
{\bf Constants:} $\omega$ of type {\it lattice\_element};
{\bf Predicate symbols, function symbols:} $\leq$ of type ({\it %
lat\-tice\_element, lat\-tice\_element}) and $\mu$ of type ({\it %
set\_el\-e\-ment, set\_el\-e\-ment}, {\it lat\-tice\_el\-e\-ment)};
{\bf Auxiliary symbols:} propositional connectives: $\vee$, $\wedge$, $%
\Longrightarrow$, $\neg$, quantifier symbols: $\forall$, $\exists$ and
commas, parentheses.
{\bf Formulae:} atomic formulae are of the form $\mu (x,y,r)$, $s\leq r$ and
formulae are built from atomic formulae as in the predicate calculus.
{\bf Axioms:} the following are axioms of $L_{rm}$
(A1) $\forall x.\mu (x,x,\omega )$;
(A2) $\forall x.\forall y.\{\mu (x,y,\omega )\Longrightarrow \forall
s.\forall r.\forall z.[\mu (z,x,s)\wedge \mu (z,y,r)\Longrightarrow (s\leq
r)]\}$;
(A3) $\forall x.\forall y.\{\mu (x,y,\omega )\wedge \mu (y,x,\omega
)\Longrightarrow $
\begin{center}
$\forall s.\forall r.\forall z.[\mu (x,z,s)\wedge \mu (y,z,r)\Longrightarrow
(s\leq r)]\}$;
\end{center}
(A4) $\exists x.\forall y.\mu (x,y,\omega );$
(A5) $\forall x.\forall y.\{[\forall z.[[\exists u.\neg (\mu (z,u,\omega
))\wedge \mu (z,x,\omega )]\Longrightarrow $
\begin{center}
$\exists t.(\exists w.(\neg \mu (t,w,\omega ))\wedge \mu (t,z,\omega )\wedge
\mu (t,y,\omega )]\Longrightarrow \mu (x,y,\omega )\};$
\end{center}
and the axiom schemata (A6)$_n$ for n = 2,3,.... where
(A6)$_n$
$\forall x_1.\forall x_2....\forall x_n.\exists y.
(\alpha_n(x_1,x_2,...,x_n,y)\wedge$
\[
\beta _n(x_1,x_2,...,x_n,y)\wedge \gamma_n(x_1,x_2,...,x_n,y))
\]
where
$\qquad \alpha _n(x_1,x_2,..,x_n,y)$ : $\forall z.\{[\exists t.(\neg \mu
(z,t,\omega ))\wedge \mu (z,y,\omega )]\Longrightarrow $
\begin{center}
$\exists x_i.\exists w.[(\exists u.(\neg \mu (w,u,\omega )))\wedge \mu
(w,z,\omega )\wedge \mu (w,x_i,\omega )]\}$;
\end{center}
$\qquad \beta _n(x_1,x_2,..,x_n,y):\mu (x_1,y,\omega )\wedge \mu
(x_2,y,\omega )\wedge ...\wedge \mu (x_n,y,\omega )$;
$\qquad \gamma _n(x_1,x_2,..,x_n,y):$
\begin{center}
$\forall z.\{[\alpha _n(x_1,x_2,..,x_n,z)\wedge \beta
_n(x_1,x_2,..,x_n,z)]\Longrightarrow \mu (y,z,\omega )\}.$
\end{center}
\medskip
\subsection{Semantics of $L_{rm}$}
\medskip
We will call an {\it interpretation} of $L_{rm}$ a triple $M=(U^M,L^M,F^M)$
where $U^M$ is a finite set, $L^M$ is a (complete) lattice with the lattice
partial order $\leq ^M$and with the greatest element $\Omega ^M$ and $F^M$
is a mapping which assigns to constants and predicate symbols of $L_{rm}$
their denotations in $M$ in the following manner: $F^M(\omega )=\Omega ^M$, $%
F^M(\leq )=\leq ^M$and $F^M(\mu )=\mu ^M\subseteq U^M\times U^M\times L^M,$
where the relation $\mu ^M\subseteq U^M\times U^M\times L^M$ is a function
i.e. $\mu ^M:U^M\times U^M\longrightarrow L^M$$.$
\medskip
An $M$-value assignment $g$ is a mapping which assigns to any variable $x$
of $L_{rm}$ of type {\it set\_element} the element $g(x)\in U^M$ and to any
variable $r$ of $L_{rm}$ of type {\it lattice\_element} the element $g(r)\in
L^M.$ For an $M$-value assignment $g$, a variable $x$ of $L_{rm}$ of type
{\it set\_element} and an element $u\in U^M,$ we denote by the symbol $%
g[u/x] $ the $M$-value assignment defined by the conditions: $g[u/x](v)=g(v)$
in case $v\neq x$ and $g[u/x](x)=u$; the same convention will define $g[p/r]$
in case of a variable $r$ of type {\it lattice\_element} and $p\in L^M.$
\medskip
For a formula $\alpha $ of $L_{rm},$ we denote by the symbol $[\alpha
]^{M,g} $ the meaning of the formula $\alpha $ in the model $M$ relative to
an $M$-value assignment $g$ by the following conditions
(M1) $[\mu (x,y,r)]^{M,g}=true$ iff $\mu ^M(g(x),g(y))=p$ for some $p\geq
^Mg(r)$;
(M2) $[s\leq r]^{M,g}=true$ iff $g(s)\leq ^Mg(r)$;
(M3) [$\alpha \vee \beta ]^{M,g}=true$ iff $[\alpha ]^{M,g}=true$ or $[\beta
]^{M,g}=true$;
(M4) $[\neg \alpha ]^{M,g}=true$ iff [$\alpha ]^{M,g}=false$;
(M5) [$\exists x.\alpha ]^{M,g}=true$ iff there exists $u\in U^M$ such that $%
[\alpha ]^{M,g[u/x]}=true$;
(M6) $[\exists r.\alpha ]^{M,g}=true$ iff there exists $p\in L^M$ such that $%
[\alpha ]^{M,g[p/r]}=true.$
\medskip
It follows that the intended meaning of a formula $\mu (x,y,r)$ is that
''the object $x$ is a part of the object $y$ in degree at least $r$''.
A formula $\alpha $ is {\it true in an interpretation} $M$ iff $\alpha $ is $%
M,g$-true (i.e. $[\alpha ]^{M,g}=true)$ for any $M$-value assignment $g$.
An interpretation $M$ is a {\it model }of $L_{rm}$ iff all axioms (A1)-(A6)
are true in $M$.
\medskip
We will give the basic deduction rules for $L_{rm}$; recall that a deduction
rule in the form $\frac{\alpha ,\beta ,...}\psi $ is said to be {\it valid}
{\it in a model }$M$ iff for any $M$-value assignment $g$ if the premises $%
\alpha ,\beta ,...$ are $M,g$ - true then the conclusion $\psi $ is $M,g$%
-true. The deduction rule is {\it valid} when it is valid in any model $M$
of $L_{rm}.$ We have the following valid deduction rules
\begin{center}
(D1) $\frac{\mu (x,y,\omega ),\mu (y,z,\omega )}{\mu (x,z,\omega )}$
\end{center}
\medskip
\begin{center}
(D2) $\frac{\mu (y,z,\omega ),\neg \mu (y,x,\omega )}{\neg \mu (z,x,\omega )}%
;$
\end{center}
\medskip
\begin{center}
(D3) $\frac{\mu (x,y,\omega ),\neg \mu (z,y,\omega )}{\neg \mu (z,x,\omega )}
$.
\end{center}
\medskip
\medskip
\medskip
We show the consistency of the axiom system (A1)-(A6) by revealing a class
of models of $L_{rm}$. We denote by {\it Stand} the class consisting of
pairs $(U,\mu _U)$ where $U$ is a finite set and $\mu _U$ is the standard
rough inclusion on the set $\exp (U)$. For a pair $M=(U,\mu _U)$ , we let $%
L^M=[0,1],$ the unit interval , $\leq ^M=$the natural linear ordering on
[0,1] , $\mu ^M=\mu _U$ and $U^M=\exp (U).$Then we denote by {\it Stand\_Mod
}the class of triples $M^{*}=(U^M,L^M,F^M)$ where $M=(U,\mu _U)$ and $%
F^M(\omega )=1,$ $F^M(\leq )=\leq ^M$ and $F^M(\mu )=\mu _U.$ We have the
following statement whose proof is straightforward.
\medskip\
Proposition 3.1
Any $M^{*}=(U^M,L^M,F^M)$ in {\it Stand\_Mod }is a model of $L_{rm}.$
\medskip\
\section{Rough inclusions}
In this section we are concerned with the structure in models of $L_{rm}$
induced by rough inclusions. We show that in any model of $L_{rm}$ we have a
canonical model of mereology of Le\'{s}niewski introduced by means of the
rough inclusion of this model. One can apply the Tarski idea of fusion of
sets [46] in order to define in a model of $L_{rm}$ the structure of a
(complete) Boolean algebra which contains isomorphically the quasi-boolean
structure (without the least (zero) element) corresponding to the model of
mereology of Le\'{s}niewski. We show that the rough inclusion satisfies with
respect to boolean operations of join and meet the same formal conditions
which the rough membership function satisfies with respect to the
set-theoretic operations of union and intersection.
\medskip
We study relations of rough inclusions with many-valued logic and fuzzy
logic; in particular, we show that when the rough inclusion is regarded as a
fuzzy membership function then any fuzzy containment induced by a residual
implication [11] is again a rough inclusion and, moreover, the hierarchy of
objects set by the induced model of mereology of Le\'{s}niewski is invariant
under these fuzzy containment operators.
\medskip
We are concerned also with the problem of consistency of deduction rules of
the form
\medskip
\begin{center}
(D$_f$) $\frac{\mu (x,y,r),\mu (y,z,s)}{\mu (x,z,f(r,s))}$
\end{center}
\medskip
where $f$ is a functional symbol of type ({\it lattice\_element}, {\it %
lattice\_element}, {\it lattice\_element})$.$
We demonstrate the consistency of (A1)-(A6)+(D$_f$) by revealing a class of
models in which the deduction rule (D$_f$) is valid under an appropriate
interpretation of $f$.
\medskip
Given a model $M$ of $L_{rm},$ $M=(U^M,L^M,F^M)$, we will call the function $%
\mu ^M:U^M\times U^M\longrightarrow L^M$ the $M$-rough inclusion. We define
a relation $congr(\mu ^M)$ on the set $U^M$ by letting for $u,w\in U^M:u$ $%
congr(\mu ^M)$ $w$ iff $\mu ^M(u,w)=\Omega ^M$ = $\mu ^M(w,u)$ . The
following proposition, whose proof follows immediately by (A2) and (A3) and
is therefore omitted, establishes the basic properties of the relation $%
congr(\mu ^M)$ and demonstrates it to be a $\mu ^M$$-$congruence.
\medskip\
Proposition 4.1
\medskip
%\newtheorem{p8}{prop}
\begin{prop}
The relation $congr(\mu ^M)$ is an equivalence relation on the set $U^M$ and
we have
(i) if $u$ $congr(\mu ^M)$ $w$ then $\mu ^M$$(v,w)=\mu ^M(v,u);$
(ii) if $u$ $congr(\mu ^M)$ $w$ then $\mu ^M$$(u,v)$= $\mu ^M$$(w,v)$
for any triple $u,v,w\in U^M$.
\end{prop}
It follows from this proposition that the rough inclusion can be factored
throughout the relation $congr(\mu ^M)$ i.e. we define the quotient set $%
U_\mu ^M=U^M/congr(\mu ^M)$ and the quotient function
\[
\mu _{\sim }^M:U_\mu ^M\times U_\mu ^M\longrightarrow L^M
\]
by letting $\mu _\mu ^M(u_\mu ,w_\mu )=\mu ^M(u,w)$; clearly, the pair $%
(U_\mu ^M,\mu _{\sim }^M)$ introduces a model $M_{\sim }$ of $L_{rm}$. In
the sequel we will always work with a fixed reduced model $M_{\sim }$. We
denote by the symbol $n_\mu $ the {\it null object} i.e. the object existing
in virtue of (A4) and such that $\mu _{\sim }^M(n_\mu ,w_\mu )=\Omega ^M$
for any $w_\mu \in U_\mu ^M.$ We will write $u_\mu \neq _\mu n_\mu $ to
denote the fact that the object $u_\mu $ is not the null object. Let us
recall that the existence of a null object in a model of mereology of
Le\'{s}niewski reduces the model to a singleton, as observed in Tarski [45].
In the sequel, for simplicity of notation, we will write $\mu $ in place of $%
\mu _{\sim }^M$, $U$ in place of $U_\mu ^M$, $u$ in place of $u_\mu $ etc.
We will call the rough inclusion $\mu $ a {\it strict rough inclusion} when
it satisfies the condition $\mu (x,n)=0$ for any non-null object $x$; we
observe that any standard rough inclusion is strict.
We now show how the rough inclusion $\mu $ introduces in $U$ a model of
mereology of Le\'{s}niewski. To this end, we define a binary relation $%
part(\mu )$ on the set $U$ by letting
$u$ $part(\mu )$ $w$ iff $\mu $$(u,w)=\Omega ^M$ and it is not true that $%
\mu $$(w,u)=\Omega ^M.$
Then we have the following proposition whose straightforward proof is
omitted:
\medskip\
Proposition 4.2
The relation $part(\mu )$ satisfies the conditions (P1) and (P2) i.e. it is
a relation of being a (proper) part in the sense of Lesniewski.
\medskip\
We now define in the model $M_{\sim }$ for any collection $\Psi $ of objects
in $U$, the notions of a set of objects in $\Psi $ and of a class of objects
in $\Psi $. We will say then that $u\in U$ is a {\it set of objects in} $%
\Psi $, $u$ $set$ $\Psi $ for short, when
(S1) for any $w\neq _\mu n$ such that $w$ $ingr(part(\mu ))$ $u$ there exist
$v\neq _\mu n$ and $t\in \Psi $ such that $v$ $ingr(part(\mu )\ )$ $w$, $v$ $%
ingr(part(\mu ))$ $t$, $t$ $ingr(part(\mu )\ )$ $u$;
if in addition, we have
(S2) $t$ $ingr(part(\mu )\ )$ $u$ for any $t\in \Psi $ ;
(S3) for any $t$, if $t$ satisfies (S1) and (S2) with $\Psi $ then $u$ $%
ingr(part(\mu )\ )$ $t$
\medskip
then we say that $u$ is a {\it class of objects in} $\Psi $, $u$ $class$ $%
\Psi ,$ for short. It follows from (A6) that for any collection $\Psi $
there exists a unique object $u$ such that $u$ $class$ $\Psi $ and there
exists objects of the form $set$ $\Psi .$
We sum up the last few observations.
\medskip\
Proposition 4.3
For any model $M=(U^M,L^M,F^M)$ of $L_{rm},$ the pair ($U_{\symbol{126}%
}^M-\{n_\mu \},$ $part(\mu _{\symbol{126}}))$ is a model of mereology of
Lesniewski.
\medskip
\subsection{Rough inclusion vs. fuzzy containment}
In this section we will reveal some of the basic connections between rough
mereology and many-valued logic [11], announced above. We recall that a {\it %
t-norm $\top $} is a mapping $\top :[0,1]\times [0,1]\longrightarrow [0,1]$
which satisfies the conditions $\top (r,1)=r$ , $\top (r,s)=\top (s,r)$ , if
$r\leq s$ then $\top (r,t)\leq \top (s,t)$ and $\top (r,\top (s,t))=\top
(\top (r,s),t)$. A {\it residual implication} $\overrightarrow{\top }$
induced by a t-norm $\top $ is a mapping $\overrightarrow{\top }:[0,1]\times
[0,1]\longrightarrow [0,1]$ which satisfies the condition
\begin{center}
$\top (r,s)\leq t$ iff $r\leq \overrightarrow{\top }(s,t).$
\end{center}
Clearly, when a t-norm $\top $ is a continuous mapping then we have a unique
residual implication
\[
\overrightarrow{\top }(s,t)=\sup \{r:\top (r,s)\leq t).
\]
We consider a model $M_{\sim }$ of $L_{rm}$. As the induced model of
mereology has the property that the notions of a set and of a subset are
equivalent, we can interpret the value $\mu (u,w)$ as the value of a fuzzy
membership function $\mu _w(u)$ in the sense of fuzzy set theory [46]. The
partial containment can be expressed in this theory [11] by means of a
many-valued implication viz. for a given many-valued implication $%
I:[0,1]\times [0,1]\longrightarrow [0,1]$, the induced partial containment
function $\sigma _I(u,w)$ is defined by the formula: $\sigma _I(u,w)=\inf
\{I(\mu _u(z),\mu _w(z)):z\in U\}.$ We show that when the implication $I$ is
a residual implication $\overrightarrow{\top }$ induced by a continuous
t-norm $\top $ then the resulting function $\sigma _{\top }$ is a rough
inclusion and, moreover, the function $\sigma _{\top }$ preserves the
relation $ingr(part(\mu ))$ .
Our next proposition reads as follows.
\medskip\
Proposition 4.3
\medskip
%\newtheorem{p17}{prop}
\begin{prop}
For a continuous t-norm $\top $ and a model $M_{\sim }$ of $L_{rm}$ with the
strict rough inclusion $\mu $, the function
\begin{center}
(i) $\sigma _{\top }(u,w)=\inf \{\overrightarrow{\top }(\mu _u(z),\mu
_w(z)):z\in U\}$
\end{center}
is a rough inclusion; moreover, we have
\begin{center}
(ii) $\sigma _{\top }(u,w)=1$ iff $\mu (u,w)=1.$
\end{center}
\end{prop}
$\sigma _{\top }(u,w)=1$ if and only if $\mu (u,w)=1.$
\medskip\
We denote by $M_{\top }$ the model which is produced from a model $M_{\sim }$
with a strict rough in\-clu\-sion $\mu $ by replacing $\mu $ with $\sigma
_{\top }$. By the symbol ${\it Stand\_Mod}(\top )$ we denote the class of
models of the form $M_{\top }$ where $M$ is a standard model of $L_{rm}$.
We now prove the consistency of the deduction rule of the form (D$_f$); the
symbol Con((A1)-(A6)+(D$_f$)) denotes the consistency of (D$_f$) i.e. the
existence of the model of $L_{rm}$ in which (D$_f$) is valid under a
plausible interpretation of the function symbol $f$. We extend the syntax of
the $L_{rm}$ by adding a functional constant symbol $f$ of type
\begin{center}
({\it lattice\_element, lattice\_element, lattice\_element}).
\end{center}
we extend accordingly the domain of $F^M$. Then we have
\medskip\
Proposition 4.4
\medskip
%\newtheorem{p18}{prop}
\begin{prop}
Con((A1)-(A6) +(D$_f))$; more specifically, the deduction rule (D$_f)$ is
valid in any model $M$ in {\it Stand\_Mod}($\top )$ where $F^M(f)=\top .$
\end{prop}
\medskip
We present a general scheme for synthesis of approximate solutions to a
given requirement. We begin with introductory remarks which provide a
motivation and explain our methodological assumptions.
\section{Approximate reasoning in distributed systems: methodology}
We present a general scheme for reasoning with uncertainty by a system of
intelligent cooperating agents. We begin with an account of the structure of
an agent.
\subsection{The agent structure}
We will discuss here the structure of a single agent $ag$ in a scheme $S$ of
agents.
A {\it pre-rough inclusion} $\mu _o$ on a set $U$ is any function
\begin{center}
$\mu _o:U\times U\rightarrow [0,1]$
\end{center}
which satisfies the following conditions
(i) $\mu _o(x,x)=1;$
(ii) if $\mu _o(x,y)=1$ then $\mu _o(z,y)\geq \mu _o(z,x)$ for any $z\in U$;
(iii) $\mu _o(x,y)=\mu _o(y,x).$
We recall that a {\it t-conorm $\bot $} [11] is a function $\bot
:[0,1]\times [0,1]\rightarrow [0,1]$ such that $\bot $ is increasing
coordinate-wise, commutative, associative and $\bot (r,0)=r$. We extend the
operators $\top ,\bot $ over the empty set of arguments and over singletons
by adopting the following convention : $\top (\emptyset )=1,\bot (\emptyset
)=1,\top (r)=r=\bot (r).$ We observe that by the associativity and
commutativity of $\top $ and $\bot $, the values $\top (x_1,...,x_k)$ and $%
\bot (x_1,...,x_k)$ are defined uniquely for any finite set of arguments. We
have the following proposition, whose straightforward proof is omitted.
\medskip\
Proposition 5.1
For any {\it pre-rough inclusion} $\mu _o$ on the set $U,$ the function
\begin{center}
$\mu (A,B)=\top \{\bot \{\mu _0(a,b):b\in B\}:a\in A\}$
\end{center}
defined for any pair $A,B$ of finite subsets of $U$, is a rough inclusion on
the universe $U^{<\omega }$ of finite subsets of $U$.
\medskip\
We will work from now rather with pre - rough inclusions. Any interpretation
of $L_{rm}$ in which requirements for $\mu _0$ are satisfied is called a{\it %
\ pre - model} of $L_{rm}.$
We now define formally the ingredients of our scheme of agents.
We consider an agent {\it ag} in the scheme. We will call the {\it label of
the agent ag} the tuple
\medskip
$\qquad lab(ag)=({\sf A}(ag),M(ag),L(ag),Link(ag),O(ag),St(ag)${\sf ,}
\begin{center}
$Unc\_rel(ag),H(ag),Unc\_rule(ag),$ $Dec\_rule(ag))$
\end{center}
\medskip\
where
1. {\sf A}$(ag)=(U(ag),A(ag))$ is an information system of the agent $ag$;
2. $M(ag)=(U(ag),[0,1],F(ag))$ is a pre - model of $L_{rm}$ with a
quasi-rough inclusion $F(ag)(\mu )$ $=\mu _o(ag)$ in the universe $U(ag).$
3. $L(ag)$ is a set of unary predicates in a predicate calculus interpreted
in the set$U(ag)$;
4. $St(ag)=\{st(ag)_1,...,st(ag)_n\}$ $\subset U(ag)$ is the set of standard
objects at $ag$;
5. $Link(ag)$ is a collection of strings of the form $ag_1ag_2...ag_kag;$
the intended meaning of a string $ag_1ag_2...ag_kag$ is that $%
ag_1,ag_2,..,ag_k$ are children of $ag$ in the sense that $ag$ can assemble
complex objects (constructs) from simpler objects sent by $%
ag_1,ag_2,...,ag_k.$ In general we can assume that for some agents $ag$ we
may have more than one element in $Link(ag)$ which represents the
possibility of re-negotiating the synthesis scheme.
6. $O(ag)$ is the set of operations at $ag$; any $o\in O(ag)$ is a mapping
from the cartesian product $U(ag_1)\times U(ag_2)\times ...\times U(ag_k)$
into the universe $U(ag)$ where $ag_1ag_2...ag_k\in Link(ag);$
7. {\it Unc\_rel(ag) }is the set of uncertainty relations $unc\_rel_i$ of
type
\[
(o_i,\rho _i,ag_1,ag_2,...,ag_k,ag,\mu _o(ag_1),...,\mu _o(ag_k),\mu_o(ag))
\]
where $ag_1ag_2...ag_kag\in Link(ag)$ and $\rho _i$ is such that
\[
\rho _i((x_1,\epsilon _1),(x_2,\epsilon _2),.,(x_k,\epsilon
_k),(x,\varepsilon ))
\]
holds for $x_1\in U(ag_1),x_2\in U(ag_2),..,x_k\in U(ag_k)$ and $\varepsilon
_1,\varepsilon _2,..,\varepsilon _k\in [0,1]$ iff $\mu
_o(x_j,st(ag_j)_i=\epsilon _j$ for $j=1,2,..,k$ and $\mu
_o(x,st(ag)_i)=\epsilon $ for the collection of standards $%
st(ag_1)_i,st(ag_2)_i,..$ $.,$ $st(ag_k)_i,st(ag)_i$ such that
\[
o_i(st(ag_1)_i,st(ag_2)_i,..,st(ag_k)_i)=st(ag)_i.
\]
Uncertainty relations express the agents knowledge about relationships among
uncertainty coefficients of any agent $ag$ and uncertainty coefficients of
its children. The relational character of these dependencies expresses their
intensionality.
8. $Unc\_rule(ag)$ is the set of uncertainty rules $unc\_rule_j$ of type
\begin{center}
$(o_j,f_j,\mu _o(ag_1),\mu _o(ag_2),.,\mu _o(ag_k),\mu _o(ag))$
\end{center}
of the agent $ag$ where $ag_1ag_2...ag_kag\in Link(ag)$ and $f_j:$ $%
[0,1]^k\longrightarrow [0,1]$ is a function which has the property that
there exists a collection of standards $%
st(ag_1),st(ag_2),...,st(ag_k),st(ag) $ and
{\bf if} objects $x_1\in U(ag_1),x_2\in U(ag_2),..,x_k\in U(ag_k)$
\qquad satisfy the conditions $\mu _o(x_i,st(ag_i))\geq \varepsilon (ag_i)$
for $i=1,2,..,k$
\begin{center}
{\bf then} $\mu _o(o_j(x_1,x_2,...,x_k),st(ag))\geq f_j(\varepsilon
(ag_1),\varepsilon (ag_2),..,\varepsilon (ag_k)).$
\end{center}
Uncertainty rules provide functional operators for propagating uncertainty
measure values from the children of an agent to the agent; their application
is in negotiation processes where they inform agents about plausible
uncertainty bounds.
9. $H(ag)$ is a strategy which produces uncertainty rules from uncertainty
relations; to this end, various rigorous formulas as well as various
heuristics can be applied.
10. {\it Dec-rule}({\it ag}) is a set of decomposition rules $dec\_rule_i$
of type
\begin{center}
$(o_i,\Phi (ag_1),\Phi (ag_2),..,\Phi (ag_k),\Phi (ag))$
\end{center}
where $\Phi (ag_1)\in L(ag_1),\Phi (ag_2)\in L(ag_2),..,\Phi (ag_k)\in
L(ag_k),\Phi (ag)\in L(ag)$ and $ag_1ag_2...ag_kag\in Link(ag)$ such that
there exists a collection of standards $st(ag_1),st(ag_2),.$ $.,st(ag_k),$ $%
st(ag)$ with the properties that
\begin{center}
$o_j(st(ag_1),st(ag_2),..,st(ag_k))=st(ag),$
\end{center}
$st(ag_i)$ satisfies $\Phi (ag_i)$ for $i=1,2,..,k$ and $st(ag)$ satisfies $%
\Phi (ag).$
\medskip\
Decomposition rules are decomposition schemes in the sense they describe the
standard $st(ag)$ and the standards $st(ag_1),...,st(ag_k)$ from which the
standard $st(ag)_{}$ is assembled under $o_i.$
We may sum up the content of 1 - 10 above by saying that for any agent $ag$
the possible sets of children of this agent are specified and, relative to
each team of children, decompositions of standard objects at $ag$ into sets
of standard objects at the children, uncertainty relations as well as
uncertainty rules, which relate similarity degrees of objects at the
children to their respective standards and similarity degree of the object
built by $ag$ to the corresponding standard object at $ag$, are given.
\subsection{The approximate logic of an agent}
We present in this section a formal approach to reasoning by any agent $ag$
in the form of an approximate logic ${\cal L}_{app}(ag).$ We recall that any
agent $ag$ is endowed with a label $lab(ag)=({\sf A}%
(ag),M(ag),L(ag),Link(ag),O(ag),St(ag)${\sf ,}
\begin{center}
$Unc\_rel(ag),H(ag),Unc\_rule(ag),$ $Dec\_rule(ag)).$
\end{center}
{\it Atomic formulae} of the logic ${\cal L}_{app}(ag)$ are of the form $%
(ag;\Phi ,\varepsilon )$ where $\Phi \in L(ag)$ and $\varepsilon \in [0,1]$.
The set of formulae of the logic ${\cal L}_{app}(ag)$ is defined as the
smallest set containing all atomic formulae and closed under propositional
connectives : $\vee ,$ $\wedge ,$ $\neg .$ The approximate formula ($ag$; $%
\Phi ,\varepsilon )$ has the intended meaning of a formula $\Phi $ satisfied
in a degree $\varepsilon ;$ formally, we will say that a construct (object) $%
x$ $\in U(ag)$ {\it satisfies} the approximate formula $(ag;\Phi
,\varepsilon ),$ symbolically: $x\models (ag;\Phi ,\varepsilon ),$ iff there
exists a standard $st(ag)$ such that $st(ag)$ satisfies the formula $\Phi $
and $\mu _o(ag)(x,st(ag))\geq \varepsilon ;$ we write $x\models
_{st(ag)}(ag;\Phi ,\varepsilon )$ in order to stress that the satisfiability
is achieved with respect to the standard $st(ag)$. In particular, for a
decomposition rule $dec\_rule_i$ as in (10) above $x$ satisfies $(ag;\Phi
(ag),\varepsilon )$ whenever $\mu _o(ag)(x,st(ag))\geq \varepsilon $;
clearly, $st(ag)$ satisfies the approximate formula $(ag;\Phi (ag),1)$. For
any pair of formulae $\alpha ,\beta $ of the logic ${\cal L}_{app}(ag)$ and
any $x\in U(ag)$, we let $x\models \alpha \vee \beta $ iff either $x\models
\alpha $ or $x\models \beta $ and $x\models \neg \alpha $ iff it is not the
case that $x\models \alpha .$
\subsection{The approximate reasoning by a system of agents}
We now consider a system $S$ of agents over an inventory $INV$. We assume
that the relation $\leq$, defined by $ag^{\prime}\leq ag$ iff $%
ag_1ag_2...ag_kag \in Link(ag)$ and there exists $i\leq k$ such that $%
ag^{\prime}=ag_i$, orders $S$ into a tree; we assume that any agent $ag$ in $%
S$ has exactly $n$ standards which satisfy the composition rule in the sense
that if $ag_1ag_2...ag_kag \in Link(ag)$ and $%
ag_1^{i}ag_2^{i}...ag_{k_i}^{i}ag_i \in Link(ag_i)$ for $i=1,2,..,k$ then
for any $j=1,2,..,n$ the composition
\[
o_j(ag)\circ (o_j(ag_1),...,o_j(ag_k))
\]
produces from standards $st(ag_1^1)_j$,...., $st(ag_{k_k}^k)_j$ the standard
$st(ag)_j$. We denote by the symbol $Root(S)$ the root agent of the scheme $%
S $ and the symbol $Leaf(S)$ will denote the set of leaf (inventory) agents
of $S$.
Let us recall that our approach is motivated by the following observations.
{\bf 1.} The knowledge of an agent in a scheme for reasoning under
uncertainty is incomplete. In particular, an agent may not be able to
distinguish among certain requirements(specifications, formulas etc.) and
its understanding of requirements is approximate only.
{\bf 2.} The local decomposition knowledge of an agent may also be uncertain
and this knowledge may not be understood fully by other agents as the agents
possess incomplete fragments of the knowledge about the world.
{\bf 3.} The leaf agents having an access to the inventory of elementary
objects may be able to select objects which satisfy the requirements not
exactly but in an acceptable degree only.
{\bf 4.} Agents may be able to classify objects approximately only, in terms
of their closeness to certain model objects (standards, logical values etc.).
{\bf 5.} The general form of an inference rule under uncertainty of an agent
$ag$ whose children are $ag_1,ag_2,..,ag_k$ is of the form
(inf\_rule) {\bf if } [$x_1\models (ag_1;\Phi _1,\epsilon _1)$ $\wedge $ $%
x_2\models (ag_2;\Phi _2,\epsilon _2)$ $\wedge $...$\wedge $ $x_k\models
(ag_k;\Phi _k,\epsilon _k)]$
\begin{center}
{\bf then }$o(x_1,x_2,..,x_k)\models (ag;\Phi ,\epsilon )$
\end{center}
where $x_1$,$x_2$,...,$x_k$ are objects submitted by, respectively, $ag_1$, $%
ag_2$,...,$ag_k$ and $(ag_1;\Phi _1,\epsilon _1)$,..., $(ag_k;\Phi
_k,\epsilon _k)$ are ap\-prox\-i\-mate spec\-i\-fi\-ca\-tions (for\-mu\-las)
at agents $ag_1$ ,..., $ag_k$, $o(x_1,...,x_k)$ is the object produced by $%
ag $ from $x_1,...,x_k$ by means of an operation $o$ and $(ag;\Phi ,\epsilon
)$ is the approximate specification at $ag$.
The intended meaning of (inf\_rule) is as follows: if the agent $ag_1$ can
submit an object $x_1$ satisfying the approximate specification $(ag_1;\Phi
_1,\epsilon _1)$ and ... and the agent $ag_k$ can submit an object $x_k$
satisfying the approximate specification $(ag_k;\Phi _k,\epsilon _k)$ then $%
ag$ can apply the operation $o$ to assembly the object $x=o(x_1,...,x_k)$
which satisfies the approximate specification $(ag;\Phi ,\epsilon )$.
{\bf 6.} Problem specifications are issued by the external agent {\it cag}
(the{\it \ customer agent}) in a language understandable to some agents in
the scheme (in particular, to the root agent {\it R}). The specific form of
the language depends on the particular synthesis process.
The object $x$ synthesized by the scheme as an approximate solution to a
requirement is evaluated by the agent {\it cag} with respect to its local
knowledge. The process of learning the correct synthesis of solutions to a
given specification is concluded when the two evaluations are consistent.
{\bf 7.} Universes of objects (universes of discourse) of agents are models
of $L_{rm}$ in which certain collections of objects, called {\it standard
objects}, are distinguished. The rough inclusions of the universes induce
rough mereological distance functions in their respective domains by means
of which objects are perceived and characterized with respect to the
standards in the respective universe.
{\bf 8.} The semantics of the approximate logic of formulas of the form $%
(ag;\Phi ,\epsilon )$ of any agent $ag$ is defined in terms of standards of $%
ag$ and the rough mereological distance function in the universe of objects
of the agent $ag$.
\medskip\
We observe that for any agent $ag$ in $S$ there exists a unique $v(ag)$ in $%
Link$ such that $v(ag)$ is a subtree of $S$; we denote this $v(ag)$by the
symbol $v_S(ag).$
An {\it S-object assignment} is a mapping $g$ which assigns to any agent $ag$
in $S$ an object $g(ag)$ $\in U(ag).$ An $S$ - object assignment $g$ is {\it %
S-admissible} if for some $j$ , denoted $j_g,$ and any agent $ag$ in $S$, we
have
$o_j(g(ag_1),g(ag_2),...,g(ag_k))=g(ag)$
where $v_S(ag)=ag_1ag_2...ag_kag.$
An $S$ - {\it selector} $\Theta $ is a mapping which for some fixed $j$
denoted $j_\Theta $ assigns to any non - leaf agent $ag$ in $S$ a tuple $%
$ and to any leaf agent $%
ag$ a tuple $<\Phi _j(ag),st_j(ag),\epsilon (ag)>$ where $\Phi _j(ag)\in
L(ag),$ $f_j(ag)$ $\in Unc\_rule(ag).$ An $S$ - selector $\Theta $ defines a
$\Theta -$ {\it constructive subscheme} $S_\Theta $ with $j=j_\Theta $ if $%
(i)$ $f_j(ag)(\epsilon (ag_1),...,\epsilon (ag_k))\geq \epsilon (ag)$ for
any $v_S(ag);$ $(ii)$ $st_j(ag)$ satisfies $\Phi _j(ag)$ for any $ag$ in $S$%
; $(iii)$ for any $v_S(ag)=$ $ag_1ag_2...ag_kag$ and for any choice $%
\{x(ag_i):x(ag_i)\in U(ag_i),i=1,2,...,k\}$ such that
$\mu _0(ag_i)(x(ag_i),st_j(ag_i))\geq \epsilon (ag_i)$
we have
$\mu _0(ag)(o_j(ag)(x(ag_1),...,x(ag_k)),st_j(ag))\geq f_j(ag)(\epsilon
(ag_1),...,\epsilon (ag_k)).$
A $\Theta -$ constructive subscheme $S_\Theta $ is {\it compatible} with an
admissible $S$ - object assignment $g$ if $(i)$ $j_g=j_\Theta =j$ ; $(ii)$
for any leaf agent $ag$ in $S$, we have $\mu _0(ag)(g(ag),st_j(ag))\geq
\epsilon (ag).$
For the root agent $R(S)$ of the scheme S, we say that a formula $(R(S);\Phi
,\epsilon )$ in the logic ${\cal L}_{app}(R(S))$ is {\it consistent} with a $%
\Theta -$ constructive subscheme $S_\Theta $ if $(i)$ $\Phi
(R(s))\Longrightarrow \Phi ;$ $(ii)$ $\epsilon (R(S))\geq \epsilon .$ A
formula $(R(S);\Phi ,\epsilon )$ is satisfied by an $S$ - object assignment $%
g$ if $g(R(s))\models _{st_j(R(S))}(R(S);\Phi ,\epsilon )$ where $j=j_g;$ we
write in this case $g\models _S(R(S);\Phi ,\epsilon ).$
The following proposition which follows directly from the above definitions
expresses a sufficiency criterium for synthesis of an object $x\in U(R(S))$
which satisfies an approximate requirement $(R(S);\Phi ,\epsilon ).$
\medskip\
Proposition 5.2
If a formula $(R(S);\Phi ,\epsilon )$ is consistent with a $\Theta -$
constructive subscheme $S_\Theta $ and the $\Theta -$ constructive subscheme
$S_\Theta $ is compatible with an admissible $S$ - object assignment $g,$
then $g\models _S(R(S);\Phi ,\epsilon ).$
\medskip\
It follows from this proposition that the object $x=g(R(S))$ which satisfies
the approximate requirement $(R(S));\Phi ,\epsilon )$ can be constructed by
the scheme S in the case when a $\Theta -$ constructive subscheme $S_\Theta $
and an $S$ - object assignment $g$ can be negotiated by the agents in $S$
which are such that : the approximate requirement $(R(S);\Phi ,\epsilon )$
is consistent with $S_\Theta $ and $S_\Theta $ is compatible with the $S$ -
object assignment $g.$
\section{Approximate synthesis of a solution from data}
In this, final, section, we present some examples concerning approximate
reasoning by systems of intelligent agents along the scheme set above.The
reader will find in [24] an example of a negotiation process based on
boolean reasoning [3], [47] and in [25], [28], [31], [40] an idea of a rough
mereological controller.
\subsection{\bf Rough inclusions from information systems}
Rough inclusions can be generated from the information system ${\bf A}$; for
instance, for a given partition $P=\{A_1,\ldots ,A_k\}$ of the set $A$ of
attributes into non-empty sets $A_1,\ldots ,A_k$, and a given set $%
W=\{w_1,\ldots ,w_k\}$ of weights, $w_i\in [0,1]$ for $i=1,2,\ldots ,k$ and $%
\sum_{i=1}^kw_i=1$ we let
\[
\mu _{o,P,W}(x,y)=\sum_{i=1}^kw_i\cdot \frac{\Vert IND_i(x,y)\Vert }{\Vert
A_i\Vert }
\]
where $IND_i(x,y)=\{a\in A_i:\,a(x)=a(y)\}$. It is easy to check that $\mu
_{o,P,W}$ is a pre-rough{\it \ }inclusion. We give an example illustrating
this procedure.
\medskip\
Example 6.1{\bf \ }
Consider an information system ($H)$
\begin{tabular}{llll}
& hat & ker & pig \\
$x_1$ & 1 & 0 & 0 \\
$x_2$ & 0 & 0 & 1 \\
$x_3$ & 0 & 1 & 0 \\
$x_4$ & 1 & 1 & 0
\end{tabular}
Table 1. The information system ($H)$
The table below shows values of the initial rough inclusion $\mu
_{o,P,W}(x,y)=\sum_{i=1}^3\frac{\Vert IND(x,y)\Vert }3$ i.e. we consider the
simplest case when $k=1,$ $w_1=1.$
\begin{tabular}{lllll}
& x$_1$ & x$_2$ & x$_3$ & x$_4$ \\
x$_1$ & 1 & 0.33 & 0.33 & 0.66 \\
x$_2$ & 0.33 & 1 & 0.33 & 0.00 \\
x$_3$ & 0.33 & 0.33 & 1 & 0.66 \\
x$_4$ & 0.66 & 0.00 & 0.66 & 1
\end{tabular}
Table 2. Initial rough inclusion for ($H)$
\medskip\
Example 6. 2
In addition to information system (agent) $(H)$ from Example 1, we consider
agents $(B)$ and $(HB)$ . Together with $H$ they form the string ${\bf ag}%
=(H)(B)(HB)$ in $Link$ i.e. $(HB)$ takes objects: $x$ sent by $(H)$ and $y$
sent by $(B)$ and assembles a complex object $xy.$ The $\inf $ormation
systems of $(B)$ and $(HB)$ are shown below. The standards are taken as: $%
x_1,x_2$ at $(H)$ and $y_1,y_3$ at $(B)$ and in the table for $(HB)$ we have
only standards $x_1y_1,x_1y_3,x_2y_1$ and $x_2y_3.$ The attributes at $(HB)$
are related to those of $(H)$ and $(B)$ by formulas: $har=pis\wedge
cut\wedge kni,$ $lar=pis\wedge \neg cut\wedge \neg kni$ $\vee \neg pis\wedge
cut\wedge \neg kni$ $\vee \neg pis\wedge \neg cut\wedge kni$, $off=hat\wedge
pis$, $tar=kni\wedge pig\wedge \neg cr$ hence one can easily complete the
table for $(HB)$.
\begin{tabular}{lllll}
& $pis$ & $cut$ & $kni$ & $cr$ \\
$y_1$ & 1 & 1 & 1 & 1 \\
$y_2$ & 1 & 0 & 0 & 0 \\
$y_3$ & 0 & 0 & 1 & 0 \\
$y_4$ & 1 & 1 & 1 & 0
\end{tabular}
Table 3. The information system $(B)$
\begin{tabular}{lllll}
& $har$ & $lar$ & $off$ & $tar$ \\
$x_1y_1$ & 1 & 0 & 1 & 0 \\
$x_1y_3$ & 0 & 1 & 0 & 0 \\
$x_2y_1$ & 1 & 0 & 0 & 0 \\
$x_2y_3$ & 0 & 1 & 0 & 1
\end{tabular}
Table 4. The information system $(HB)$
\medskip\
The values of the initial rough inclusion $\mu =$ $\mu _{o,P,W}$ are
calculated for $(B)$ and $(BH)$ by the same procedure as in Example 1.
\subsection{Uncertainty functions from data}
We outline an algorithm which may be used to extract from information
systems of agents in a scheme $S$ approximations to uncertainty functions (
rough mereological connectives).
\medskip\
Example 6.3
We will determine an approximation to the mereological connective at the
standard $x_1y_1$ i.e$.$ a function $f$ such that:
for any pair $\epsilon _1,\epsilon _2,$
{\bf if } $\mu (x,x_1)\geq \epsilon _1$ and $\mu (y,y_1)\geq \epsilon _2$
{\bf then } $\mu (xy,x_1y_1)\geq f(\epsilon _1,\epsilon _2).$
The following tables show conditions which $f$ is to fulfill.
\begin{tabular}{lllll}
$x$ & $\mu (x,x_1)$ & $y$ & $\mu (y,y_1)$ & $\mu (xy,x_1y_1)$ \\
$x_1$ & 1 & $y_1$ & 1 & 1 \\
$x_1$ & 1 & $y_2$ & 0.25 & 0.5 \\
$x_1$ & 1 & $y_3$ & 0.25 & 0.25 \\
$x_1$ & 1 & $y_4$ & 0.75 & 1 \\
$x_2$ & 0.33 & $y_1$ & 1 & 0.5 \\
$x_2$ & 0.33 & $y_2$ & 0.25 & 0.25 \\
$x_2$ & 0.33 & $y_3$ & 0.25 & 0.00 \\
$x_2$ & 0.33 & $y_4$ & 0.75 & 0.5
\end{tabular}
Table 5. The conditions for $f$ (first part)
\begin{tabular}{lllll}
$x$ & $\mu (x,x_1)$ & $y$ & $\mu (y,y_1)$ & $\mu (xy,x_1y_1)$ \\
$x_3$ & 0.33 & $y_1$ & 1 & 0.75 \\
$x_3$ & 0.33 & $y_2$ & 0.25 & 0.25 \\
$x_3$ & 0.33 & $y_3$ & 0.25 & 0.25 \\
$x_3$ & 0.33 & $y_4$ & 0.75 & 0.75 \\
$x_4$ & 0.66 & $y_1$ & 1 & 1 \\
$x_4$ & 0.66 & $y_2$ & 0.25 & 0.5 \\
$x_4$ & 0.66 & $y_3$ & 0.25 & 0.25 \\
$x_4$ & 0.66 & $y_4$ & 0.75 & 1
\end{tabular}
Table 6. The conditions for $f$ (second part)
\medskip\
This full set $T_0$ of conditions can be reduced: we can find a minimal set $%
T$ of vectors of the form $(\varepsilon _1^{^{\prime }},$ $\varepsilon
_2^{^{\prime }},$ $\varepsilon )$ such that if $f$ satisfies the condition $%
f(\varepsilon _1^{^{\prime }},\varepsilon _2^{^{\prime }})=\varepsilon $ for
each $(\varepsilon _1^{^{\prime }},\varepsilon _2^{^{\prime }},\varepsilon
)\in T$ then $f$ extends by the formula: for $(\varepsilon _1,\varepsilon
_2,\varepsilon _3)\in T_0$ we let $f(\varepsilon _1,\varepsilon
_2)=\varepsilon ^{*}$ where $(\varepsilon _1^{*},\varepsilon
_2^{*},\varepsilon ^{*})\in T$ and:{\bf \ if }$\varepsilon _1\geq
\varepsilon _1^{^{\prime }},\varepsilon _2\geq \varepsilon _2^{^{\prime }}$
for some $(\varepsilon _1^{^{\prime }},$ $\varepsilon _2^{^{\prime }},$ $%
\varepsilon )\in T$ {\bf then} $\varepsilon _1^{*}\geq \varepsilon
_1^{^{\prime }},\varepsilon _2^{*}\geq \varepsilon _2^{^{\prime }}$ . This
follows by definition of $f.$
The following algorithm produces a minimal set $T$ of conditions.
\medskip\
{\bf Algorithm}
\medskip\
$Input:$ table $T_0$ of vectors ($\mu (x,x_1),\mu (y,y_1),\mu (xy,x_1y_1));$
Step 1. For each pair ($\mu (x,x_1)=\varepsilon _1,\mu (y,y_1)=\varepsilon
_2),$ find $\varepsilon (\varepsilon _1,\varepsilon _2)=\min \{\varepsilon
:\varepsilon _1^{\prime }\geq \varepsilon _1,\varepsilon _2^{\prime }\geq
\varepsilon _2,(\varepsilon _1^{\prime },\varepsilon _2^{\prime
},\varepsilon )\in T_0\}.$ Let $T_1$ be the table of vectors $(\varepsilon
_1,\varepsilon _2,\varepsilon (\varepsilon _1,\varepsilon _2))$.
Step 2. For each $\varepsilon ^{*}$ such that $(\varepsilon _1,\varepsilon
_2,\varepsilon (\varepsilon _1,\varepsilon _2)=\varepsilon ^{*})\in T_1,$
find: $row$($\varepsilon ^{*})=(\varepsilon _1^{*},\varepsilon
_2^{*},\varepsilon ^{*})$ where $(\varepsilon _1^{*},\varepsilon
_2^{*},\varepsilon ^{*})$ $\in T_1$ and if $(\varepsilon _1^{^{\prime
}},\varepsilon _2^{^{\prime }},\varepsilon ^{*})\in T_1$ then $\varepsilon
_1^{^{\prime }}\geq \varepsilon _1^{*},\varepsilon _2^{^{\prime }}\geq
\varepsilon _2^{*}.$
$Output$: table $T$ of vectors of the form $row$($\varepsilon )$ .
\medskip\
One can check that the following table shows a minimal set $T$ of vectors
for the case of Tables 5,6.
\begin{tabular}{lll}
$\varepsilon _1$ & $\varepsilon _2$ & $\varepsilon $ \\
0.66 & 0.75 & 1 \\
0.33 & 0.75 & 0.5 \\
0.66 & 0.25 & 0.25 \\
0.33 & 0.25 & 0.00
\end{tabular}
Table 7. A minimal set $T$ of vectors
One can extract from the algorithm the synthesis formula of $f$ from
conditions $T_0:$
$f(\varepsilon _1,\varepsilon _2)=\min \{\varepsilon ^{^{\prime
}}:(\varepsilon _1^{^{\prime }},\varepsilon _2^{^{\prime }},\varepsilon
^{^{\prime }})\in T_0$,$\varepsilon _1^{^{\prime }}\geq \varepsilon
_1,\varepsilon _2^{^{\prime }}\geq \varepsilon _2\}.$
\medskip\
We can now give an example of a synthesis.
\subsection{Approximate synthesis from data: an example}
We give an example of synthesis; in the synthesis process, we make use of
pre - rough inclusions and approximations to rough mereological connectives
which were found above.
\medskip\
Example 6.4
Assume that we want to apply the scheme $S$ with $R(S)=(HB)$ and the unique $%
v_S((HB))=(H)(B)(HB)$ in order to synthesize $x\in U(R(S))$ which satisfies
the approximate requirement $((HB);$ $\Phi (HB)=har\wedge off,$ $\varepsilon
(HB)=0.5).$ It follows by Table 7 that we can take $((H);\Phi (H)=hat\wedge
\neg pig,\varepsilon (H)=0.33)$ as the approximate requirement at the agent $%
(H)$ and $((B),\Phi (B)$ $=pis\wedge cut\wedge kni,\varepsilon (B)$ $=0.75)$
as the approximate requirement at the agent $(B),$ for the approximation $f$
satisfying the set $T$ of conditions. The $\Theta -$ constructive subscheme $%
S_\Theta $ is determined by selecting the standards: $x_1$ at the agent $(H)$
and $y_1$ at the agent $(B)$ and $\Phi (HB)=har\wedge off,$ $\varepsilon
(HB)=0.5$. We can see from Tables 1, 3 that any $S$- object assignment $g$
such that $g(<(H),(B)>$ equals one of the following: $(x_1,y_1),(x_1,y_4),$ $%
(x_2,y_1),(x_2,y_4),$ $(x_4,y_1),$ $(x_4,y_4)$ has the property that $%
S_\Theta $ is compatible with $g$ and , according to Proposition 5.1, $%
g\models ((HB);$ $\Phi (HB)=har\wedge off,$ $\varepsilon (HB)=0.5).$
\medskip\
\section{$\frac {}{}$}
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\end{document}