An ontology of organizational knowledge Anderson Beraldo de Araújo1 , Mateus Zitelli1 , Vitor Gabriel de Araújo1 1 Federal University of ABC (UFABC) Santo André, São Paulo – Brazil anderson.araujo@ufabc.edu.br,{zitellimateus,araujo.vitorgabriel}@gmail.com Abstract. One of the main open problems in knowledge engineering is to un- derstand the nature of organizational knowledge. By using a representation of directed graphs in terms of first-order logical structures, we defined organiza- tional 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. 1. Introduction According to [1], Knowledge management (KM) has produced a bunch of definitions that helps us to understand organizational knowledge, the kind of knowledge that we find in organizations. Nonetheless, there is no universal approach to the different kind of defi- nitions available. We are in need of an ontological analysis of organizational knowledge that is capable to unify the different notion of knowledge relevant to bussiness. Indeed, organizational knowledge has been thought according to four fundamen- tal types [3, 4]. The first one we can call the mental view of knowledge. According to this standpoint, knowledge is a state of mind. In the mental view to manage knowledge involves to regulate the provision of information controls and to improve individuals ca- pacity of applying such a knowledge. The second view is the objectual view of knowledge. Here knowledge is an object, something that we can store and manipulate. In the objec- tual approach manage knowledge becomes a process of stock managing, in which we could control the offers and the demands of individuals as parts of an integrated process inside a company. To take knowledge as a procedural phenomenon of information is the third approach, which we can call the procedural view of knowledge. In the procedural perspective knowledge becomes a process of applying expertise, so to manage means to manage the flows of information, such as creation process, conversion techniques, circu- lation processes and carrying out processes. The fourth perspective is the credential view of knowledge. In this approach knowledge is a credential for accessing information. In this case, KM focus on how you manage the credentials to access and what you expect to retrieve, granting the content as the result of a process. The credential view of knowledge is the standard approach that has been applied in companies nowadays [5]. KM faces knowledge as the potential of influencing actions. By doing so companies consider KM as a process of granting the right competences to the chosen individuals. The focus is to provide the specific know-how to the realization of the processes and to grant that every processes has its correspond knowledge unit correlated. 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 op- erational cost, product cycle time and production time while increasing productivity, mar- ket 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 communica- tion 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 [6, 7]. 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” [8]. In other words, ontology is the collection of entities admitted by a theory that is committed to their existence. In the present context, we call minimal ontology the ontology shared by every theory that successfully describes a processes as a organizational one. Our fundamental idea is to define organizational structures, using the general concept of first-order logical structure (Section 2). Thus, we propose a mathematical definition of information about organizational structures, based on the abstract notion of information introduced here for the first time (Section 3). The next step is to conceive organizational knowledge as justified relevant information about organizational structures (Section 4). From this approach we formulate an algorithm and simulate them (Section 5). 2. Organizational structures We begin by some usual definitions in logic - more details can be found in [9]. The first one is associated to the syntax of organizational structures. Definition 2.1. A signature is a set of symbols S = C ∪ P ∪ R such that C = {c1 , . . . , ck } is a set of constants, P = {P1 , . . . , Pm } is a set of property symbols, R = {R1 , . . . , Rn } is a set of relation symbols. A formula over S is recursively defined in the following way: 1. If τ, σ ∈ C, ρ ∈ P , δ ∈ R, then ρτ and δτ σ are formulas, called predicative formulas; 2. If φ and ψ are formulas, then ¬φ, φ ∧ ψ, φ ∨ ψ, φ → ψ and φ ↔ ψ are formulas, called propositional formulas. A theory over S is just a set of formulas. Now we recall the general notion of first-order structure. 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 τ in S, an element τ A in dom(A); 3. For each property symbol ρ in S, a subset ρA of dom(A). 4. For each relation symbol δ in S, a binary relation δ A on dom(A). We write A(φ) = 1 and A(φ) = 0 to indicate, respectively, that the formula φ is true, false, in the structure A. Besides, we have the usual definitions of the logical operators ¬φ, φ ∧ ψ, φ ∨ ψ, φ → ψ and φ ↔ ψ on a structure A. In particular, we say that a theory T is correct about A if A(φ) = 1 for all φ ∈ T . Figure 1. Bussiness process Definition 2.3. Given a signature S, an organizational structure AT over S is com- pounded of a structure A over S, a theory T over S which is correct about A and ex- presses facts about a business process. The idea inside the definition of organizational structures is that they are just log- ical structures with a fundamental theory about how the processes works. In the propo- sition 2.1, we show that business processes are indeed special cases of organizational structures. Proposition 2.1. Business processes are organizational structures. Proof. According to [10], a business processes is a tuple (N, E, κ, λ), in which: 1. N is the set of nodes; 2. E ⊆ N × N is the set of edges; 3. κ : N → T is a function that maps nodes to types T ; 4. λ : N → L is a function that maps nodes to labels L. Let L = {l1 , . . . , lk } be the set of labels and T = {T1 , . . . , Tn } be the set of types. Thus, we can define the organization structure A with domain dom(A) = {li : 1 ≤ i ≤ k}, subsets T1 , . . . , Tn of L and the relation E. In what follows, we write “α” = β to mean that the symbol β is a formal repre- sentation of the expression α. Besides, Xβ is the interpretation of β in the structure A, where X is a set over the domain of A. The elements of the domain dom(A) of a structure A are indicated by bars above letters. Example 2.1. Let S = C ∪ P ∪ R be the signature such that C = {i, f, o, r, s, v}, P = {E, A} and R = {L}, 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) = {ī, f¯, ō, r̄, s̄, v̄}; 2. iA = ī, f A = f¯, oA = ō, rA = r̄, sA = s̄, v A = v̄; 3. E A = {ī, f¯} and AA = {ō, r̄, s̄, v̄}; 4. LA = {(ī, ō), (ō, r̄), (r̄, s̄), (r̄, v̄), (s̄, f¯), (v̄, f¯)}. The organizational structure AT defined above represents the bussiness process in Figure 1. Figure 2. Extended bussiness process 3. Structural information We turn now to the fundamental notion associated to knowledge, namely, information. The concept of information is polysemantic [11]. In this work we think of information in semantic terms. Since we are going to define a notion of information about organiza- tional 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 ω into AT is an organizational structure ATi such that ATi is an structure over S 0 = S ∪ {ω} with the following properties: 1. ATi (τ ) = A(τ ) for all τ 6= ω such that τ ∈ S; 2. If ω is a constant in S, then dom(ATi ) = dom(A) and ATi (ω) 6= A(ω), but if ω is a constant not in S, then dom(ATi ) = dom(A) ∪ {a} and ATi (ω) = a; 3. If ω is a property symbol, then dom(ATi ) = dom(A) ∪ {a} and ATi (ω) = A(ω) ∪ {a}; 4. If ω is a relation symbol, then dom(ATi ) = dom(A) ∪ {a1 , a2 } and ATi (ω) = A(ω) ∪ {(a1 , a2 )}. Example 3.1. Consider the organizational structure AT over S from example 2.1. Define the signature S 0 = C 0 ∪ P 0 ∪ R0 equals to S except by the fact that C 0 = C ∪ {t}, where “transfer goods” = t. Thus, the organizational structure ATi defined below is an insertion of t into AT : 1. dom(Ai ) = dom(A) ∪ {t̄}; 2. tA = t̄ and τ Ai = τ A for τ ∈ C; 3. E Ai = E A and AAi = AA ∪ {t̄}; 4. LAi = LA − {(ō, r̄)} ∪ {(ō, t̄), (t̄, r̄)}. The organizational structure ATi represents the business process in Figure 2. Definition 3.2. Let A be an S-structure. An element a ∈ dom(A) is called free for the symbol ω ∈ S if there is no constant τ ∈ S with A(τ ) = A(ω) neither a property symbol α such that a ∈ A(α) and a = A(ω) nor a relation symbol β such that (a1 , a2 ) ∈ A(β) and ai = A(ω) for i ∈ {1, 2}. If δ is a relation symbol, we write A(δ)i to denote element ai of (a1 , a2 ) ∈ A(δ). Definition 3.3. Let AT be an organization structure over S. An extraction of the symbol ω from AT is a database ATe such that ATe is an structure over S 0 = S − {ω} with the following properties: 1. ATe (τ ) = A(τ ) for all τ 6= ω such that τ ∈ S 0 ; Figure 3. Contracted bussiness process 2. If ω is a constant not in S, then dom(ATe ) = dom(A), but if ω is a constant in S, dom(ATe ) = dom(A) − {A(ω)} in the case of A(ω) being free for ω, otherwise, dom(ATe ) = dom(A); 3. If ω is a property symbol not in S, then dom(ATe ) = dom(A), but if ω is a property symbol in S, then dom(ATe ) = dom(A) − {A(ω)}, where A(ω) is an element free for ω, and ATe (ω) = A(ω) − {A(ω)}; 4. If ω is a relational symbol not in S, then dom(ATe ) = dom(A), but if ω is a relational symbol in S, then dom(ATe ) = dom(A)−{A(ω)1 , A(ω)2 }, where A(ω)i is an element free for ω, and ATe (ω) = A(ω) − {(A(ω)1 , A(ω)2 )}. Example 3.2. Consider the organizational structure AT over S from example 2.1. Define the signature S 00 = C 00 ∪ P 00 ∪ R00 equals to S except by the fact that C 0 = C − {r}. Thus, the organizational structure ATe defined below is an extraction of r from AT : 1. dom(Ae ) = dom(A) − {r̄}; 2. τ Ae = τ A for τ ∈ C 00 ; 3. E Ae = E A and AAe = AA − {r̄}; 4. LAe = LA − {(ō, r̄)} ∪ {(ō, s̄), (ō, v̄)}. The organizational structure AeT represents the business process in Figure 3. Strictly speaking, the organizational structure ATe in example 3.2 is not an extrac- e e tion from AT . For example, LA = LA − {(ō, r̄)} ∪ {(ō, s̄), (ō, v̄)}, which means that LA was made of insertions in AT as well. Since we are interested here in practical applica- tions, 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 se- quence U A = (ATj : 0 ≤ j ≤ n) such that AT0 = AT and each ATj+1 is an insertion into or an extraction from ATj . An update U A = (ATj : 0 ≤ j ≤ n) is satisfactory for a formula φ if, and only if, either An (φ) = 1 or An (φ) = 0. In the case of a satisfactory update U A for φ, we write U A (φ) = 1 to denote that An (φ) = 1 and U A (φ) = 0 to designate that An (φ) = 0. A recipient over in organizational structure AT for a formula φ is a non-empty collection of updates U of AT satisfactory for φ. Example 3.3. Given the organizational structures AT , ATi and ATe from the previous examples. The sequences (AT , ATi ) and (AT , ATe ) are updates of AT that generates, re- spectively, the business processes in Figures 2 and 3. Definition 3.5. Given a recipient U over a fixed organizational structure AT , the (struc- tural) information of a sentence φ is the set IU (φ) = {U A ∈ U : U A (φ) = 1}. Besides that, for a finite set of sentences Γ = {φ0 , φ1 , . . . , φn }, the (structural) information of Γ is the set n [ IU (Γ) = IU (φi ). i=0 Example 3.4. Consider the recipient U = {(AT , ATi ), (AT , AT2 )}. In this case, we have the following: 1. IU (Lrs ∨ Lrv) = {(AT , ATi )}; 2. IU (Lio) = U. 4. Organizational knowledge Since we have a precise definition of information about organizational structures, we can now define mathematically what is organizational knowledge. The intuition behind our formal definition is that knowledge is information plus something else [12]. To be specific, we defined organizational knowledge as justified relevant information about or- ganizational structures. Definition 4.1. Given an organizational structure AT over S = C ∪ P ∪ R such that C = {c1 , . . . , ck }, P = {P1 , . . . , Pm }, and R = {R1 , . . . , Rn }, the organizational graph associated to AT is the multi-graph GA = (V, {El }l