One of the main open problems in knowledge engineering is to understand the nature of organizational knowledge. By using a representation of directed graphs in terms of first-order logical structures, we defined organizational knowledge as integrated relevant information about relational structures. We provide an algorithm to measure the amount of organizational knowledge obtained via a research and exhibit empirical results about simulations of this algorithm. This preliminary analysis shows that the definition proposed is a fruitful ontological analysis of knowledge management.
In this paper we provide an logical method to quantify knowledge that can be used
in all the four views of organizational knowledge and present computational results about
them all. Quantitative indicators of knowledge can create benefits such as decreasing
operational cost, product cycle time and production time while increasing productivity,
market share, shareholder equity and patent income. They can drive decisions to invest on
employees skills, quality strategies, and define better core business processes. Moreover,
if applied to the customers, quantitative indicators can create an innovative
communication platform, where the information of the clients can be quickly collected and processed
into relevant decision indicators in specific terms such as abandoning one line of product,
on the one hand, and investing, on the other [
One way to unify this different approaches to KM is to outline a minimal ontology
of business processes, in a Quinean sense. According to Quine, as it is well known, “to
be is to be the value of a variable” [
Definition 2.1. A signature is a set of symbols S = C [ P [ R such that C = fc1; : : : ; ckg is a set of constants, P = fP1; : : : ; Pmg is a set of property symbols, R = fR1; : : : ; Rng is a set of relation symbols. A formula over S is recursively defined in the following way: 1. If ; 2 C, 2 P ,
formulas;
called propositional formulas.
A theory over S is just a set of formulas. and
are formulas, called predicative ^ , _ , ! and $ are formulas,
Definition 2.2. Given a signature S, a structure A over S is compounded of: 1. A non-empty set dom(A), called the domain of A; 2. For each constant in S, an element A in dom(A); 3. For each property symbol in S, a subset A of dom(A). 4. For each relation symbol in S, a binary relation A on dom(A).
We write A( ) = 1 and A( ) = 0 to indicate, respectively, that the formula is true, false, in the structure A. Besides, we have the usual definitions of the logical operators : , ^ , _ , ! and $ on a structure A. In particular, we say that a theory T is correct about A if A( ) = 1 for all 2 T . Definition 2.3. Given a signature S, an organizational structure AT over S is compounded of a structure A over S, a theory T over S which is correct about A and expresses facts about a business process.
Proposition 2.1. Business processes are organizational structures.
3. : N ! T is a function that maps nodes to types T ; 4. : N ! L is a function that maps nodes to labels L.
Let L = fl1; : : : ; lkg be the set of labels and T = fT1; : : : ; Tng be the set of types. Thus, we can define the organization structure A with domain dom(A) = fli : 1 i kg, subsets T1; : : : ; Tn of L and the relation E.
In what follows, we write “ ” = to mean that the symbol is a formal representation of the expression . Besides, X is the interpretation of in the structure A, where X is a set over the domain of A. The elements of the domain dom(A) of a structure
Example 2.1. Let S = C [ P [ R be the signature such that C = fi; f; o; r; s; vg, P = fE; Ag and R = fLg, in which “initial” = i, “final” = f , “order” = o, “receive goods” = r, “store goods” = s, “verify invoice” = v, “is event” = E, “is activity” = A and “is linked to” = L. In this case, we can define the organizational structure AT over S below, where T = : 1. dom(A) = fi; f ; o; r; s; vg; 2. iA = i, f A = f , oA = o, rA = r, sA = s, vA = v; 3. EA = fi; f g and AA = fo; r; s; vg; 4. LA = f(i; o); (o; r); (r; s); (r; v); (s; f ); (v; f )g.
The organizational structure AT defined above represents the bussiness process in Figure 1.
in semantic terms. Since we are going to define a notion of information about organizational structures, we will call it structural information. Roughly speaking, the structural information of an organizational structure is the set of insertions and extractions that we need to perform in order to create this structure.
Definition 3.1. Let AT be an organizational structure over S. An insertion of the symbol ! into AT is an organizational structure AiT such that AiT is an structure over S0 = S [ f!g with the following properties: 1. AiT ( ) = A( ) for all 6= ! such that 2 S; 2. If ! is a constant in S, then dom(AiT ) = dom(A) and AiT (!) 6= A(!), but if ! is a constant not in S, then dom(AiT ) = dom(A) [ fag and AiT (!) = a; 3. If ! is a property symbol, then dom(AiT ) = dom(A) [ fag and AiT (!) = A(!) [ a ; f g 4. If ! is a relation symbol, then dom(AiT ) = dom(A) [ fa1; a2g and AiT (!) =
A(!) [ f(a1; a2)g.
Example 3.1. Consider the organizational structure AT over S from example 2.1. Define the signature S0 = C0 [ P 0 [ R0 equals to S except by the fact that C0 = C [ ftg, where “transfer goods” = t. Thus, the organizational structure AiT defined below is an insertion of t into AT : 1. dom(Ai) = dom(A) [ ftg; 2. tA = t and Ai = A for 2 C; 3. EAi = EA and AAi = AA [ ftg; 4. LAi = LA f(o; r)g [ f(o; t); (t; r)g.
If is a relation symbol, we write A( )i to denote element ai of (a1; a2) 2 A( ). Definition 3.3. Let AT be an organization structure over S. An extraction of the symbol ! from AT is a database AeT such that AeT is an structure over S0 = S f!g with the following properties: 1. AeT ( ) = A( ) for all 6= ! such that 2. If ! is a constant not in S, then dom(AeT ) = dom(A), but if ! is a constant in S, dom(AeT ) = dom(A) fA(!)g in the case of A(!) being free for !, otherwise, dom(AeT ) = dom(A); 3. If ! is a property symbol not in S, then dom(AeT ) = dom(A), but if ! is a property symbol in S, then dom(AeT ) = dom(A) fA(!)g, where A(!) is an element free for !, and AeT (!) = A(!) fA(!)g; 4. If ! is a relational symbol not in S, then dom(AeT ) = dom(A), but if ! is a relational symbol in S, then dom(AeT ) = dom(A) fA(!)1; A(!)2g, where A(!)i is an element free for !, and AeT (!) = A(!) f(A(!)1; A(!)2)g.
Example 3.2. Consider the organizational structure AT over S from example 2.1. Define the signature S00 = C00 [ P 00 [ R00 equals to S except by the fact that C0 = C frg. Thus, the organizational structure AeT defined below is an extraction of r from AT : 1. dom(Ae) = dom(A) 2. Ae = A for 2 C00; 3. EAe = EA and AAe = AA 4. LAe = LA
r ; f g
r ; f g f(o; r)g [ f(o; s); (o; v)g.
Strictly speaking, the organizational structure AeT in example 3.2 is not an extraction from AT . For example, LAe = LA f(o; r)g [ f(o; s); (o; v)g, which means that LAe was made of insertions in AT as well. Since we are interested here in practical applications, we will not enter in such a subtle detail - we delegate that to a future mathematically oriented article. This point is important because it shows that to build new organizational structures from a given one is, in general, a process that use many steps. We explore this idea to define a notion of structural information.
Definition 3.4. An update U A of an organizational structure AT over S is a finite sequence U A = (AjT : 0 j n) such that A0T = AT and each AjT+1 is an insertion into or an extraction from AjT . An update U A = (AjT : 0 j n) is satisfactory for a formula if, and only if, either An( ) = 1 or An( ) = 0. In the case of a satisfactory update U A for , we write U A( ) = 1 to denote that An( ) = 1 and U A( ) = 0 to designate that An( ) = 0. A recipient over in organizational structure AT for a formula is a non-empty collection of updates U of AT satisfactory for .
examples. The sequences (AT ; AiT ) and (AT ; AeT ) are updates of AT that generates, respectively, the business processes in Figures 2 and 3.
Definition 3.5. Given a recipient U over a fixed organizational structure AT , the (structural) information of a sentence is the set
IU ( ) = fU A 2 U : U A( ) = 1g:
Besides that, for a finite set of sentences information of is the set = f 0; 1; : : : ; ng, the (structural) n IU ( ) = [ IU ( i):
i=0 Example 3.4. Consider the recipient U = f(AT ; AiT ); (AT ; A2T )g. In this case, we have the following: 1. IU (Lrs _ Lrv) = f(AT ; AiT )g; 2. IU (Lio) = U .
can now define mathematically what is organizational knowledge. The intuition behind
our formal definition is that knowledge is information plus something else [
Definition 4.1. Given an organizational structure AT over S = C [ P [ R such that C = fc1; : : : ; ckg, P = fP1; : : : ; Pmg, and R = fR1; : : : ; Rng, the organizational graph associated to AT is the multi-graph GA = (V; fElgl<n) such that: 1. V = f(a; PjA) 2 dom(A) }(dom(A)) : A(Pj(a)) = 1g for 1 j m; 2. El = f(b; d) 2 V 2 : b = (a; PjA)) 2 V; d = (c; PkA)) 2 V; A(Rl(a; c)) = 1g for 1 l n.
Example 4.1. Let AT be the organizational structure from example 2.1. The organizational graph associated to AT is graph GA = (V; E) such that: Definition 4.2. Let R+ be set of non-negative real numbers. Given an organizational graph G = (V; fEigi<n) associated to an organizational structure AT over S, an objectual relevancy is a function d : V ! R+ and a relational relevance is a function D : fEigi<n ! R+ such that d(a) [d] and
D(Ei)
[D]; for all a 2 V and i < n.
nodes and types of edges between nodes. Given that, we provide some axioms for functions that every measure of organizational knowledge must satisfy.
Definition 4.3. We write UA(G) to indicate an update U A = (AjT : 0 j n) such that A0T = A and ATn = G. In special, UA(G) denotes the set of all updates UA(G). In this way, we define that K : U (Gb) U (Gr) ! R+ is an knowledge function if, and only if: 1. K(U (Gb); U (Gr)) = K(U (Gr); U (Gb)); 2. If Gb = Gr then K(U (Gb); U (Gr)) = 0; 3. If Gb \ Gr = then K(U (Gb); U (Gr)) = 1; 4. If Gb G then K(U (Gb); U (Gr)) K(U (G); U (Gr)); 5. If Gr G then K(U (Gb); U (Gr)) K(U (Gb); U (G)).
search base. This is a consequence of the fact that insertions and extractions are dual operations and so it does not matter whether we consider the order of the structures. The second and third axioms are immediate and the forth and fifth represent the monotonicity of the structural information.
Definition 4.4. Let AT be an organizational structure and K a knowledge function over an organizational graph Gb = (Vb; fEigi<n) associated to AT , called knowledge base, and an organizational graph Gr = (Vr; fEjgj<n) associated to an organizational structure BT , called research base. Thus, the organizational knowledge of BT with respect to AT and K is the number k such that
K = minfK(UGb\Gr (Gb); UGb\Gr (Gr)) :
UGb\Gr 2 U (Gb) [ U (Gr)g:
organizational knowledge. We could provide a mathematical proof that this algorithm computes an knowledge function, but we prefer to present empirical data about its execution - in a mathematical oriented article we will give all the details. The simulations provided in this section were implemented in a program wrote in Python.
The figure Fig. 4 is a graphic K jV j, where jV j is the number of nodes of a graph G = (V; fEjgj<n), generated with a number of nodes from 1 to 100 with step of 5 nodes, 5 types of edges with 10 possible values, i.e., with n = 5 and D : fEjgj<n ! R+ with 10 possibles values. Each knowledge measure is a result of the mean of 10 trials. This graph shows that the variation in an research base with respect to nodes are irrelevant to knowledge. This is in accordance with axiom 3. As we randomly choose new organizational graphs bigger and bigger, the probability of finding completely different graphs increase, and so knowledge approaches to 1.
The figure Fig. 5 is a graphic K jEj, where jEj is the number of edges of a
graph G = (V; fEigi<n), generated with a number of nodes from 1 to 100 with step of 5
nodes, 5 types of edges with 10 possible values. Each knowledge measure is a result of
the mean of 10 trials. This graph shows that the variation in an research base with respect
to edges is relevant to knowledge. This is a sigmoid function, a special case of learning
curve [
K(x) = 1=(1 + 0:001010e 0:385636px)1=0:000098:
The square root px is just due to the factor of redundancy 2:19721208941247 generated by the fact that we have chosen the graphs randomly. This redundancy implies
a decreasing in the growing of knowledge. This is a very important result because, first, it
shows a clear connection between our definition of knowledge and the usual empirical
approaches to learning and, besides that, it is evidence that knowledge is indeed a relational
property of organizational structures, as it have been sustained, for example, [
The main focus of the quantitative measure discussed in this paper is to use dynamic
data taken from research methods about knowledge management. Our results shows that
knowledge is a relational property of organizational structures. Nonetheless, much more
should be done in order to understand the consequences of these results. At first, the
organizational knowledge management techniques comprehend aspects of how to
understand knowledge, using the right attitudes to the right environments. Once the knowledge
meaning is defined, the knowledge sharing behaviour should be identified in order to
apply quantitative measures and then driving the KM process toward a more certain path
[
Concepts, Languages, Architectures.