Algebraic Model for Describing and Analyzing
Spatial Situations
Filatov I. Yu.
Faculty of Computer Engineering
Ryazan State Radio Engineering University
Ryazan, Russian Federation
rgrta@inbox.ru
Abstract—The word "algebra" refers not only to a section of
mathematics, but also to one of the specific objects studied in this
section. "Algebra" in the narrow sense of the word is not
considered here, and therefore for brevity and to avoid
misunderstandings, the word "algebra" in this paper will be used
as a generic concept to refer to a variety of algebraic structures.
To formalize the subject area using spatial relations between
objects it is possible to define algebraic model. When solving the
problem of positioning, for example, an aircraft and its
surrounding objects, using universal algebras, it is possible to
create terms that describe the surrounding environment. To solve
the problem of positioning an aircraft and its surrounding
objects, the unification theory can be used as shown in this paper.
Keywords–universal algebra, unification theory, logics of space,
situation analysis, formalization of spatial relationships, fuzzy logic,
artificial intelligence systems.
I.
INTRODUCTION
Algebraic methods of model description are widely use to
formalize a domain model. Development begins with the
introduction of appropriate designations for operations and
relationships, followed by research on their properties, when a
domain model is constructed. Knowledge of algebraic
terminology is included in the set of tools necessary for
abstract modeling that precedes practical programming of
problems in a specific subject area [1,2,3,4].
Briefly consider the basic concepts and components of
algebra. This is necessary to avoid contradictions and establish
certainty.
Using M , we will denote any set as a certain set of objects
called elements of the set M . The degree of the set M is
called its direct product on itself and denoted: M n . We will
write the function as follows: f : A  B , where, A – the
domain of the function, B – the range of values of the
function. Everywhere defined (total) function  : M n  M
called n-ary operation on M . If the  operation is binary
( : M  M  M ) , then we will write ab instead of
( a , b ) or a  b , where "" is the operation sign. This form of
recording is called infix.
M
The
set
with
operations
ni
  {1 ,...,  m },  i : M  M , where ni is an arity of the
operation i , is called an algebraic structure, a universal
algebra (UA), or simply an algebra. The set M is called the
main (carrier) set or the base; arity vector ( n1 ,..., nm ) is
called a type; operations  is called a signature. We will write
as follows:  M ;   or  M ; 1 ,...,  m  . Operations i
have finite arity, and the signature  is finite. The base is not
necessarily finite, but it is not empty[5].
If only relations are included in the signature  , then the
set M together with the set of relations is called an algebraic
model (AM).
The closure of the set X  M relative to the signature 
(denoted by [ X ] ) is the set of all elements (including the
elements X themselves) that can be obtained from X by
applying operations from  .
A subset of X  M is called closed relative to the
operation  , if the result of this operation applied to any
element of the set X , belongs to this set, i.e.:
x1 ,..., xn  X : ( x1 ,..., xn )  X .
II.
DESCRIPTION AND FORMALIZATION
OF ALGEBRAIC MODEL
To formalize the subject area using spatial relations
between objects [6], we define AM as follows. Spatial logic
relations:
R  {Rr , Rru , Ru , Rlu , Rl , Rld , Rd , Rrd , Rcl , Rvn , Rn , Rnn , Rf , Rvf , Rtf } , are
binary relations of direction and distance. The base of the
algebra M is a set of objects selected on the radar information
(RI) and electronic maps (EM) with coordinates – Obj ( x; y ) .
AM can be written as follows:
A  M ,{Rr , Rru , Ru , Rlu , Rl , Rld , Rd , Rrd , Rcl , Rvn, Rn , Rnn, Rf , Rvf , Rtf } 
Object coordinates can be set in different coordinate systems.
This does not change the description, whether it is coordinates
in the image or spatial geographical coordinates.
Let's define the semantics of relations.
Direction relations. Object Obj1 ( x1 ; y1 ) is located to the
right of object Obj 2 ( x2 ; y 2 ) if the x coordinate of object
Obj1 ( x1 ; y1 ) is greater than the x coordinate of object
Obj 2 ( x2 ; y 2 ) . This can be represented as
Obj1 ( x1 ; y1 ) Rr Obj 2 ( x2 ; y 2 ) if x1  x2 and y1  y 2 .
follows:

A similar relationship is established for other relationships:
Obj1 ( x1 ; y1 ) Rru Obj 2 ( x2 ; y 2 ) if x1  x2 and y1  y 2 ;
Similarly, after analyzing the distance relations, we can
conclude that they have properties:
Obj1 ( x1 ; y1 ) Ru Obj 2 ( x2 ; y 2 ) if x1  x2 and y1  y 2 ;
Obj1 ( x1 ; y1 ) Rlu Obj 2 ( x2 ; y 2 ) if x1  x2 and y1  y 2 ;
Obj1 ( x1 ; y1 ) Rl Obj 2 ( x2 ; y 2 ) if x1  x2 and y1  y 2 ; 


Obj1 ( x1 ; y1 ) Rld Obj 2 ( x2 ; y 2 ) if x1  x2 and y1  y 2 ;
Obj1 ( x1 ; y1 ) Rd Obj 2 ( x2 ; y 2 ) if x1  x2 and y1  y 2 ;
Obj1 ( x1 ; y1 ) Rrd Obj 2 ( x2 ; y 2 ) if x1  x2 and y1  y 2 .
Distance relations. Let's consider the semantics of distance
relations based on the assumption that objects have the same
dimensions. The distance between objects will be calculated as
the Euclidean distance between two points of space with the
specified coordinates:
D  ( x1  x2 )2  ( y1  y2 )2 .
Object Obj1 ( x1 ; y1 ) is close to object Obj 2 ( x2 ; y 2 ) if the
Euclidean distance from object Obj1 ( x1 ; y1 ) to object
Obj 2 ( x2 ; y 2 ) falls within a specified interval defined, for
example, in meters. The semantics of distance relations
between objects can be described as follows:
Obj1 ( x1 ; y1 ) Rcl Obj 2 ( x2 ; y2 ) if D  [1;10 ];
anti-reflexive, because none of the relationships for the
same object;

symmetry, because if the first object is at a certain
distance, say close, from the second object, then the
second object is at the same distance from the first, i.e.
if the ratio Obj1 ( x1 ; y1 ) Rn Obj 2 ( x2 ; y2 ) is true, then it is
true that Obj 2 ( x2 ; y2 ) RnObj1 ( x2 ; y1 ) ;
Let's define the composition of relations as follows:
R1 ( a , b )  R2 ( a , b )  R1, 2 ( a , b ) .
For example: Rru ( a , b )  Rvn ( a , b )  Rru , vn ( a , b ) ; or an
equivalent
statement
in
the
infix
entry:
( a Rru b )  ( a Rvn b )  ( a Rru , vn b ) .
It can be comment as follows. If the direction and distance
relations are set for two objects, i.e. if, for example, object a is
in front of object b and if object a is "very near" (vn) from
object b , then we can say that object a is in front of and very
near to object b . The composition of relations will have the
properties of anti-reflexivity, anti-symmetry.
( a Rru , vn b ); ( a Rl , f c );...; (b Rd , n c );...
Obj1 ( x1 ; y1 ) RnObj 2 ( x2 ; y 2 ) if D  [301;1500 ];
Obj1 ( x1 ; y1 ) Rnn Obj 2 ( x2 ; y2 ) if D  [1501;2800 ]; 

In the entered AM, we can describe the EM as follows:
Obj1 ( x1 ; y1 ) Rvn Obj 2 ( x2 ; y2 ) if D  [11;300 ];

transitivity, because if x1  x2 and x2  x3 , then
x1  x3 , and, therefore, if Obj1 ( x1 ; y1 ) Rl Obj 2 ( x2 ; y2 )
Obj 2 ( x2 ; y2 ) Rl Obj 3 ( x3 ; y3 ) ,
then
and
Obj1 ( x1 ; y1 ) Rl Obj 3 ( x3 ; y3 ) .

Obj1 ( x1 ; y1 ) R f Obj 2 ( x2 ; y2 ) if D  [ 2801;4000 ];
Obj1 ( x1 ; y1 ) Rvf Obj 2 ( x2 ; y2 ) if D  [ 4001;5000 ];
Obj1 ( x1 ; y1 ) Rtf Obj 2 ( x2 ; y2 ) if D  5000 .
Consider the properties of the entered relationships. For
simplicity, we will consider the properties of the group of
relations of direction and distance, since each relation within
the group has the same properties.
Direction relations have properties:

anti-reflexive, because none of the relationships for the
same object;

anti-symmetry, because it is not true that x1  x2 and
x1  x2 , but if Obj1 ( x1 ; y1 ) Rlu Obj 2 ( x2 ; y2 ) , then
Obj 2 ( x2 ; y 2 ) Rrd Obj1 ( x1 ; y1 ) , i.e. when the elements of
the relationship are rearranged in places, the
relationship itself changes to the opposite;

where, a , b , c are constants corresponding to EM objects
that have known spatial coordinates.
Similarly, for RI, sets of objects and relationships are
compiled, which can be written as follows:
( x Ru , nn y ); ( x Rrd , n z );...; ( y Rld , vn z );...

where, x , y , z are variables corresponding to the RI
objects whose coordinates need to be determined.
It should be noted that when compiling terms for RI, we
can use information obtained from other systems that allow us
to get some knowledge about objects on the RI. In this case, the
built term may contain not only variables, but also constants,
which will make it more clear and allow more accurate
description of the environment.
Now we introduce the universal algebra B  [ L] ;   ,
where L is a set whose elements are sets (triples of the form:
(a R b) ) obtained in AM A . [ L] –closure of the set L with
respect to the signature of the algebra. The  – signature of an
algebra consists of a single binary operation "And", let's denote
it «  ». In this case, the type of UA is set by the set {(2)} ,
i.e.   {(2)} . Otherwise, we can say that the UA base is a
set of complete situations that describe the location of objects
on a certain area of terrain according to data from the ECM or
radar, and the operation «  » - allows us to combine all the
components (point situations) of complete situations.
The semantics of the «  » operation can be described as a
composition of spatial logic situations.
The «  » operation has properties:

associativity, since it does not matter in what sequence
to combine situations, the result of their combining in
any case will be a more complete situation that
describes the spatial logic; formally, this property can
be written as follows: (a  b)  c  a  (b  c) ;

commutativity, since the essence of the resulting
description does not depend on the permutation of
places of the combined situations, i.e. a  b  b  a ;

idempotency, since the composition of two identical
descriptions does not add any additional information or
formally a  a  a .
Since the constructed UA contains one binary operation
that has the properties of associativity, commutativity, and
idempotence, we can conclude that this UA is a commutative
and idempotent semigroup.
In the entered UA B , we can create terms that describe the
EM and RI for a specific area of terrain. For an EM, the term
can be represented as follows:
t  ( a Rru ,vn b )  ( a Rl , f c )  ...  (b Rd ,n c ),
For RI we can similarly create a term of the form:
s  ( x Ru , nn y )  ( x Rrd , n z )  ...  ( y Rld , vn z ).
Each of the composed terms can be considered as a finite
set whose elements are "point" situations.
M 1  { x | x  t}; M 2  {x | x  s}.
Where x is an element of the set M of the form (a R b) .
Adding and removing elements of the M set can be
described as follows:
M  x  {y | y  M  y  x};
M  x  { y | y  M & y  x}.
Where y is an element of the set M of the form (a R b) .
When deleting and adding a finite number of elements, the
finite sets remain finite.
III. POSITIONING OF OBJECTS
BASED ON THE UNIFICATION THEORY
Formalization of this kind is necessary in our case. When
solving the problem of positioning, for example, an aircraft and
its surrounding objects, using universal algebras, it is possible
to create terms that describe the surrounding environment. If
we can combine RI and EM, i.e. determine which part of the
earth's surface corresponds to the image received on the radar
screen, then the positioning problem is solved. Taking into
account the spatial logic described earlier and its formalization,
it is possible to create terms for both EM and RI. In order to
compare them and determine their correspondence, the
unification theory is best suited [7,8,9,10].
In a general, unification means the following: "is it possible
for two descriptions x and y to make a transformations
(substitutions) that would equalize these descriptions?". It is
described in the following context. We have two logical terms,
which can consist of constants, variables, and functional
symbols. Is it possible to substitute terms instead of variables,
making the original terms identical?
For example, two terms f ( x, y ) and f ( g ( y, a), h(a)) are
given. We can say that these terms are unified by substituting
g (h(a), a) instead of x , and h(a) instead of y . As a result of
substitution, both source terms will look like this:
f ( g (h(a), a ), h(a)) .
In unification theory, variables are denoted by the letters
{x, y, z ,...} , constants by the letters {a, b, c,...} , and function
symbols by the letters { f , g , h,...} . Substitutions can be
indicated by letters of the Greek alphabet {, , ,...} or by a
set of specific substitutions enclosed in curly brackets:
{x  g (h(a), a), y  h(a)} . We can perform a composition
operation on substitutions. For example, the substitution (t ) ,
means that for the term t , we must apply the substitution 
first, and then  . The composition of substitutions is
associative, but not commutative. Terms are usually denoted by
the letters {s, t , u,...} . Terms are constants, variables, as well as
terms that stand under the sign of the functional symbol in
parentheses, separated by a comma. Two terms s and t are
unified if there is a substitution ( s )  (t ) .  – unifier for s
and t . The unifier  for terms s and t is called the most
common unifier (MCU) if there is a unifier  for any other
unifier  , such that    . In other words, MCU is the
simplest unifier. The MCU is unique, but there are many
unifiers. Two terms are unified at infinity if there is a
substitution that is an infinite unifier containing an infinite
term. For example, the terms x and f (x ) are not unified, but
they are unified at infinity. The infinite unifier for these terms
is the substitution   {x  f ( f ( f (...)))} .
IV. EXPERIMENTAL STUDY
To solve the problem of positioning the aircraft and its
surrounding objects, the unification theory can be used as
follows. As noted earlier, for RI and EM, using the results of
creating spatial logic and formalizing it with the help of
universal algebras, it is possible to construct terms that will
describe the surrounding object environment. These are the
terms to apply the unification algorithm to. The result of the
algorithm will be the answer to the question: "is it possible to
unify them?". If the result is positive, we will know which
substitutions must be made to make the original terms
identical. Successful unification will allow you to determine
the coordinates of the aircraft and its surrounding objects,
information about which is received from the radar. The
substitution vector will provide additional information about
which type of underlying surface or which object known from
the EM belongs to a particular object obtained from the RI
analysis.
The first unification algorithm in a modern setting was
developed in 1965 by G. Robinson. This algorithm is not
applicable to the problem under consideration. The author
developed a unique basic algorithm adapted to the conditions
of the problem being solved, which has no analogues in the
unification theory.
The unifier obtained as a result of the algorithm, if two
terms are successfully unified, establishes a correspondence
between the objects allocated to the RI and the EM. Since the
coordinates of objects on the EM are known, the coordinates of
the corresponding objects on the RI are known also. Based on
the calculated coordinates, we can determine the type of
underlying surface for these objects.
Information about the type of underlying surface is, in our
case, crucial for identifying objects. The reason is that,
knowing the type of underlying surface on which the object
resides, we can with some probability assign it to any class of
real world objects. Further, the classification system, just as a
human thinks, must identify an object in a compressed time
based on a given assessment of the typicality of the situation.
V. CONCLUSION
On the basis of experimental studies it can be concluded
that The formalization of the subject area is performed by
developing a unique algebraic model of spatial logic. For the
first time, the algebra of spatial situations was modeled, which
allows us to adequately and effectively describe and analyze
the environment around the aircraft based on the analysis of RI
and EM information. The type and class of the algebra are
defined, the properties of its operations and the properties of
relations of the corresponding algebraic model are investigated.
A unique basic algorithm for unifying terms for describing
spatial situations has been developed in order to determine the
coordinates of the aircraft and position the objects surrounding
it in autonomous navigation conditions. We have created an
algebra for describing the properties of an object, on the basis
of which it is possible to identify objects selected when
analyzing information from various sources. The type and class
of the algebra are defined, and the properties of its operations
are investigated. Using the proposed algorithms for unification
of terms of description of spatial situations, it is possible to
solve a wide range of problems in the conditions of large flows
of information, with the lowest cost of resources.
ACKNOWLEDGMENT
The research is carried out due to the support of the Ryazan
State Radio Engineering University and Ryazan State Device
Plant.
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