FueL: Representing function structure and function dependencies with a UML profile for function modeling

4.1.1. Function

The notion of function has a rich literature across many scientific and engineering fields, yet there is no consensus on its single reading. Surveys (including Burek, 2006; Erden et al., 2008; Perlman, 2004; Wouters, 2005) witness the many definitions of function proposed in engineering, philosophy and ontology research. A popular family of accounts of function is formulated in terms of the notion of role. For example, Kitamura et al. define a function as a role played by a behavior and specified in some context (Kitamura and Mizoguchi, 2004; Kitamura et al., 2006). Similarly, Lind represents a function as a role that an item plays in the achievement of the goals of the system(s) of which the item is a part. Chandrasekaran and Josephson (1998; 2000) define the object O to have a role F in the world W if and only if (iff) O causes F to be satisfied in W. If F is intended or desired by an agent A, who is a designer or a user of O, then O is said to have or perform the function F in W for A.

In the context of biological systems functions are also defined in terms of roles. For example, Karp (2000) indicates that a biological function of a molecule can be described in terms of the role that a molecule plays in a biological process. He recognizes four roles that can be identified for biological functions, namely substrate, catalyst, modulator and cofactor (Karp, 2000, esp. p. 275f. and p. 284).

Following the above intuitions, the notion of function in the presented profile is constructed based upon the notion of role. We follow the very general understanding of a role presented in (Loebe, 2007), where a role individual is defined as a relational entity that links any entity, called the role player (or filler), with some context, in which the entity plays that role. Hence, one can say that a role is that which is played by some entity in a given context, formally captured as ∀ x . Role I ( x ) ↔ ∃ y z ( plays II ( y , x ) ∧ has - context II ( x , z ) ) .

We take the notion of context as a primitive here and, following Loebe (2007), we understand it as “a more comprehensive whole in which a role may be interpreted as a part”. In that sense, the has - context II relation7

At the categorial level we adopt the following two axioms stated by Loebe (2007) as weak restrictions on role categories in connection with role playing and having contexts, respectively. ∀ x y . Role C ( x ) ∧ ¬ plays CC ( y , x ) → ¬ ∃ u v ( is - instance - of IC ( u , x ) ∧ is - instance - of IC ( v , y ) ∧ plays II ( v , u ) ) . ∀ x y . Role C ( x ) ∧ has - context CC ( x , y ) → ∀ z ( is - instance - of IC ( z , y ) → ∃ u ( is - instance - of IC ( u , x ) ∧ has - context II ( u , z ) ) ) .

The former formula expresses the constraint that if y’s instances cannot play roles of type x (the meaning of ¬ plays CC ( y , x ) ), then indeed no instance of y plays any individual role instantiating x. The latter formula states that if a role type x is linked to a type of contexts y at the categorial level, then each context instance must have an individual role of that type x.

Based on the observations from the literature above and the discussed role understanding, we are almost prepared to introduce the notion of function. A missing component, which is elucidated in detail in Section 4.1.2, is the notion of goal achievement. It draws its inspiration directly from Lind’s (1994) representation of function as a role that an item plays in achieving some goal. To provide a rough, first intuition, at this stage one may think of a goal achievement as a process or a situation (categorially) that is associated with a goal, which is to be brought about or maintained.

Before we provide a specification for goal achievement, let us illustrate this characterization with the example of the molecular function GO:0015144: carbohydrate transmembrane transporter activity, taken from the Gene Ontology. This function represents the catalysis of the transfer of carbohydrate from one side of the membrane (under consideration, which is implicitly assumed) to the other. In terms of the above function specification the function GO:0015144 is understood as a role (category) of the kinds “doer” or “executor” in the teleological process of transferring carbohydrate through a membrane. Notably, roles of these kinds are predestined candidates for being functions, cf. Sections 4.1.6 and 4.2, whereas not every role within a goal achievement may be seen as a function. This explains why the above characterization of function cannot be formulated as an equivalence. Moreover, functions are specified as role categories and as such they can have instances. In our example, any such instance is a catalyst role individual, played by, e.g., a protein, in the context of an individual carbohydrate transmembrane transfer process. Figure 5 presents the FueL model of GO:0015144. The constructs utilized in the figure are expounded in the remainder of the section.

4.1.2. Goal achievement

The notion of goal achievement is intended to grasp the teleological character of a function – its orientation at a goal. This aspect of functions is stressed in many approaches to function representation. For instance, Sasajima et al. (1995) define a function as a teleological interpretation of B1 behavior, where B1 behavior is considered as a change of an attribute value of an operand from that at the input port of a device to that at the output port of the device. Chandrasekaran and Josephson (1998) identify the function of a device with the effect “that the object under discussion has on its environment”. In Functional Representation (Chandrasekaran, 1994) and in the Causal Functional Representational Language (CFRL) (Iwasaki et al., 1995) a goal, understood as the state in which function realization results, is an explicit part of function specification. Last but not least, Gero (1990) and colleagues (Qian and Gero, 1996) consider a function as the result of a behavior, as its effect or product.

Those intuitions lead us to the notion of goal achievement, which is meant to capture teleological, i.e., goal-oriented, transitions from one situation to another. In order to capture this notion more precisely, we provide characterizations of a notion of goal, in order to account for the teleological aspect, as well as of transition. Both are introduced as starting points for the purposes of this paper. Especially the case of goal certainly constitutes only a first and rough approximation of an actual ontological analysis of concepts labeled with the term ‘goal’.

This notion of goal is based on utility as a primitive, denoted by Util C . The latter can be expected to be related to some further context, however, for the purposes of FueL models we assume that such context applies to the overall model and therefore leave it implicit. Apart from that, it is interesting to note that a goal is first established on the categorial level, before an individual situation can be identified as a goal or, put differently, as a realization of a goal (in the former categorial sense). Further discussion on goals follows in Section 4.1.4 below, including an anchoring in the function modeling literature.

A situation is a state of affairs that is composed of facts and can be comprehended as a whole. We interpret situations very broadly here – they can vary in their complexity depending on the number of objects that they contain, as well as on the features and interrelations of those objects. In our running example we consider ‘carbohydrate being located on the one side of the membrane’ as a situation.8

The second notion required for capturing goal achievement is the idea of transitions. At the level of individuals we capture this aspect of the transition from one situation (or state) to another by means of the primitive ternary relation is - transition - from - to III ( x , y , z ) that applies if y and z are situations and x is an entity that mediates between or transitions from y to z.

A special class of transitions are causal transitions, where much background intuition on the notion of causality rests on the work by Michalek (2009), which is compatible with GFO. We consider two kinds of such transitions. A causal transition x from y to z is either a causally mediated process9

Although we utilize the ontological understanding of causality and situation aligned with GFO, notably FueL as such is agnostic about the ontological interpretation of those notions and can also be employed if different theories of situations and causality are utilized.

With goals and transitions available we are prepared for the formal introduction of goal achievements.

Overall, a goal achievement category accounts for a teleological specification of transitioning to what is intended to be achieved, hence, to what is expected. It describes the context in which an item has (and possibly realizes) a function. Typically, GAs are realized by processes, in which case a GA category can be understood as a process category. In the running example, the GA is a teleological process category describing the process of carbohydrate transfer from one side of the membrane to the other. Accordingly, the process exhibits the causal transition from the situation of carbohydrate being on one side of the membrane to the situation in which carbohydrate is on the other side of the membrane. However, as we will demonstrate in Section 4.3, GA is a broader concept, the instances of which cannot be reduced to temporally extended processes only.

4.1.3. Input–output perspective

GAs support the modeling of two aspects of functions: the input–output perspective and the participant perspective. Defining a function in terms of input–output pairs has a long tradition in modeling technical artifacts. For instance, Rodenacker (1971) defines a function as a relationship between input and output of energy, material and information. Borgo et al. (2011) indicate that an overall artifact function is often given in terms of input and output flows and Goel et al. (2009) characterize a function as a schema that specifies its preconditions and its expected postconditions. Similarly, Wirth and O’Rorke (1993) define a function, among others, by the input flux and the output flux. For example, the function of a hot–cold water facet can be understood as follows: the input fluxes are hot and cold water, while the mixture of both is the output flux. In the area of software engineering and systems modeling, especially in the framework of structured methods, a function is considered as a transformation of an input object into an output object (Yourdon, 1993).

It is worth mentioning that input–output pairs are not only used for modeling functions, but likewise for modeling processes. For example, the Business Process Modeling Notation (OMG, 2011c), a widely used business process modeling formalism, enables the modeling of inputs and outputs of a process with data input and data output constructs. The Eriksson–Penker UML extension for business process modeling (Eriksson and Penker, 2000), in turn, models a process with input and output objects. In this context, the introduced GA concept can be considered as a general teleological component which is common for both processes and functions.

In the UML profile presented herein the input–output perspective of goal achievements is modeled with its two attributes: input and output. Since this aspect of GAs originates from their understanding as transitions, these attributes are defined more generically for transitions as follows, whereby they apply directly to GAs, as well.

For example, the goal achievement carbohydrate transmembrane transport has an associated input-category the instances of which are situations of carbohydrate being on the one side of the membrane, and an output-category of situations of carbohydrate being on the other side of the membrane. This specifies that every instance of carbohydrate transmembrane transport exhibits a transition from an instance of the input-category to an instance of the output-category, i.e. from an individual situation of carbohydrate being located on one side of the membrane to another individual situation of carbohydrate being located on the other side of the membrane.

We note that inputs and outputs of GAs are in general not assumed to be maximal, but they can be partial situations, in the sense that there may be even more comprehensive situations that a GA transitions from or to and that embed the “smaller” situations, possibly identified as input or output. This appears useful for modeling, e.g. in order to focus on relevant aspects only. As a consequence, a GA may be linked to multiple inputs and/or outputs. Regarding outputs, this further allows for distinguishing some of them as intended, namely goals. Yet also unwanted effects play a role in modeling functions. Next we turn to both of these elements of function representation in more detail.

4.1.4. Goals and side effects

The notion of output in the context of function analysis is intimately tied to the notion of goal. Indeed, the term ‘goal’ occurs commonly in function modeling frameworks, however, its meaning is diverse. In Functional Representation (Chandrasekaran, 1994), for example, the goal of a function is considered to be a causal process of function realization, whereas McDowell et al. (1996) understand a goal as a goal state, which is the result of a realization. In FueL both of these intuitions behind ‘goal’ are reflected. Goals comprehended as realizing processes are represented by the notion of goal achievement. Goals understood as resulting states are based on the notion of output. But before strengthening the link between goals and outputs, we turn to the observation of side effects.

According to Britton et al. (2000), one of the limitations of (pure) input–output approaches is their focus on useful outcomes only, thereby ignoring negative side effects. A side effect is basically an output, as well, in the sense that it indicates a class of situations that result from realizing goal achievements. The remaining question is why some resulting outputs are considered to be goals, whereas others are not. In some functional modeling frameworks outputs or goals are identified with desired states of reality (e.g. Chandrasekaran and Josephson, 1998). For instance, in the context of the goal achievement to generate circular motion, realized by an internal combustion engine, overheating could be considered as a side effect since it is not what is intended. Side effects can be illustrated even more clearly when it comes to drug design. For instance, antibiotics have the goal to fight some kind(s) of bacteria that cause some disease. However, antibiotics administered per oral have a side effect of killing bacteria in the intestine. Typically, as in the above examples, the desires or intentions of a designer are taken into account when delimiting goals from side effects. However, also the intentions of users (as in Mizoguchi et al., 2012), or other parties, such as stakeholders or researchers investigating an object (cf. Burek and Herre, 2013), can be taken into account.

To some extent, interpreting outputs as goals or side effects based on a designer’s desire is problematic, especially for natural systems. However, the establishment of a goal, in our understanding, can be driven not only by a desire, but likewise by the belief in the utility of a given output or in the role that an output plays in the explanation of a given phenomenon. For example, the state in which blood is being pumped is useful for the human organism (in order to stay alive; as an autopoietic system) and as such it is acknowledged as a goal by some agent, like a physician or biologist.

A particular type of a goal’s utility is its epistemic usefulness. A state of the world may be crucial for some theory acknowledged by an agent as it may provide an explanation of some phenomenon and, therefore, it may be recognized as the goal of some function. For instance, the fact that hemoglobin transports oxygen from the lungs to other parts of the body may be considered as an answer to the question “Why does blood contain hemoglobin?”, relevant in the context of some biological theory. This kind of explanation, called functional, is common in biology.

In the light of those considerations, we distinguish between two types of outputs in FueL: goals and side effects. Formally, this distinction in FueL traces back to the primitive predicate Util C , as already employed in the characterization of goal (with predicates Goal C and Goal I ) prior to GA above, with the reading that x is considered to be of utility. On this basis we define goals and side effects of goal achievements.

It should be mentioned that the usefulness of an output does not only depend on the context, but likewise on the agent who recognizes that utility. The very same output may be considered useful by one person but may lack utility for another. However, it is worth noting that the rationale behind the establishment of the Util C predicate is hardly relevant from the modeling point of view, at least if a shared position regarding utility can be assumed within one and the same model. We assume the latter as the common case and, therefore, we view further analysis of ‘utility’ to be out of scope of the current paper.12

Function specification can transcend the specification of mere input–output pairs. In particular, a function can be characterized by constraints on the method of function realization. For instance, for Chandrasekaran (1994) one component of function specification is a causal process, which is a particular way of function realization. In the ontology of Kitamura et al. (2002), the way-of-function-achievement represents the abstraction of key information about how to implement a function, such as physical laws and intended phenomena. A mode of realization, often called the implementation, is a common component of modeling formalisms also present in UML. For instance, as discussed in Section 3, UML use cases are specified by behaviors implementing them.

Following these views we think that often it is insufficient to describe a function only in terms of its input and output. For example, the molecular functions GO:0015399: primary active transmembrane transporter activity and GO:0015291: secondary active transmembrane transporter activity share the same input: solute is on one side of the membrane and the same output: solute is on the other side of the membrane. Thus, in terms of the pure input–output view, one could say that both functions are the same because they are based on goal achievements with equal inputs and equal outputs. In that sense their realizations do the same, namely, they transport a solute from the one side of the membrane to the other. However, in fact, those functions differ significantly. What distinguishes them is not their goal, but instead the way in which it is to be achieved. The first function (GO:0015399) reaches its goal by means of some primary energy source such as a chemical, electrical or solar source, whereas the latter (GO:0015291) utilizes a uniporter, symporter or antiporter protein. Consequently, both functions share the same goal and therefore provide the same answer to the question of what they achieve, however they provide different answers to the question of how they do so.

In order to enable a clear separation of the goal from the way of realization we introduce the mode of realization as an additional and optional component of GAs.

For the function GO:0015399: primary active transmembrane transporter activity the mode could be specified as follows: the transport is realized by means of some primary energy source such as a chemical, electrical or solar source. The mode for the function GO:0015291: secondary active transmembrane transporter activity would be: the transport is realized by means of a uniporter, symporter or antiporter protein.

The mode is a constraint on the function realization which does not affect the input or the output. For instance, if one adds to the function of transmembrane transport the constraint that the transport should be realized by a uniporter protein, then the input and the output remain unchanged. However, the function will change and not every transportation process can realize the new one but only those processes driven by a uniporter protein.

We provide only a very weak formal specification of mode above. It is difficult to determine an appropriate kind of categories that could be used instead of Category C itself and that excludes categories that affect or interfere with the structural aspects of goal achievements, as required in the previous paragraph. In the literature, a mode is often presented as a specification of a process, e.g. in Functional Representation (Chandrasekaran, 1994). However, as the above examples demonstrate, it can constrain the function realization not by depicting the whole process of realization but only some of its aspects as, for instance, the entity which drives the realization. Altogether, it remains future work to further characterize a kind of categories by means of which the notion of mode of realization can be grasped more precisely and more specifically formally.

4.1.6. Participants perspective

A function together with its GA cannot only be depicted in teleological terms. Another perspective on function representation does not focus on the goal of the function, but on the entities involved in the realization of the function. In device modeling a common component of function specification is the device realizing the function (Chandrasekaran, 1994), also called a function carrier (Wirth and O’Rorke, 1993). Moreover, in structured methods of software engineering the common part of function specification is not only the entity realizing the function, i.e. the system or its component, but also the objects on which the function operates.

In FueL, the entities occurring in a goal achievement13

The latter relationship yields the most commonly used participant type in the Molecular Functions Ontology (MFO), namely operand. An operand is affected by the GA it participates in, for example, resulting in changes of its shape or state. In FueL the notion is defined as follows, employing operates - on II as a primitive relation.

The notion of operand in FueL is inspired by the concept of ‘objective entity’ in (Shimomura et al., 1998; Takeda et al., 1996). An objective entity is an entity that a function “occurs on or to”. Typically, in a noun–verb phrase depicting a function it is represented by a noun, e.g. carbohydrate in the phrase to transport carbohydrate. In that sense, the objective entity and the operand correspond to the thematic roles of theme and patient, as introduced in the linguistic analysis of action sentences (cf. Harley, 2010).

We have constructed the notion of operand based on the primitive predicate operates - on II . This leaves the question open of what it means that a function operates on an entity. For example, Keuneke distinguishes ToMake, ToMaintain, ToPrevent and ToControl as ways in which a function operates on entities (Keuneke, 1989). Another useful categorization of operations may be found in the categorization of basic operations of persistent storage (Martin, 1983) into create, read, update, delete, known as CRUD. In accordance with this one may say that a function operates on an entity if it either creates, reads, updates/modifies or deletes/destroys the entity. In general, however, we doubt that there exists an ultimate, domain-independent classification of operations. Instead, different classifications of operations could be useful, depending on the modeling context or domain.

Besides ‘operand’, two further types of participants in goal achievements are declared above for FueL. Next we focus on the notion of doer, which basically captures those participants of a function that are responsible for or cause GA execution. executes II is the corresponding primitive predicate.

Although, as discussed at the beginning of the present section, a doer is a common component of function modeling frameworks in mechanical engineering, in MFO it is not as typical to specify a goal achievement with a doer as with an operand. For example, in the carbohydrate transmembrane transport process discussed so far no doer is indicated. Typically, the doer is a part of the GA in cases where the mode of realization is provided. For instance, the functions GO:0015293: symporter activity and GO:0015292: uniporter activity both specify a mode of realization and each indicates its doer, namely symporter protein and uniporter protein, respectively.

In addition to doers, entities can contribute to goal achievements in other ways. In FueL we call such entities contributors, which are defined as follows.

Besides the general characterization, we identify two more specific patterns of contribution. Firstly, there are instruments, which contribute to the goal achievements in which they are used. Thereby it is assumed that instruments are used by a doer of the GA under consideration.

A doer can use an instrument to achieve a goal. In this sense resources such as tools are contributors. For instance, a hammer is a contributor for the GA of hammering nails.

Beside instruments, contributors can be understood in terms of sub-components. For example, in (Mizoguchi et al., 2012) the behavior of sub-components participating in the realization of a function of a certain system s is seen to be contributing to the function of s. In FueL, a corresponding conception of contribution can be defined by utilizing the sequence-part relationship seq-part. This relation is introduced and discussed only in Section 4.4.5 below, yet we specify a characterization of this kind of contribution here: Such a contributor to the GA of a function x refers to a doer of the GA of another function, which is a subfunction of x in virtue of being a sequence-part of x. The following axiom provides a weak reflection of these connections. ∀ uvxyz . Function C ( u ) ∧ Function C ( v ) ∧ has - goal - achievement CC ( u , x ) ∧ has - goal - achievement CC ( v , y ) ∧ seq - part CC ( u , v ) ∧ has - doer CC ( x , z ) → has - contrib CC ( y , z ) .

4.4.1. Function decomposition

Function decomposition is a popular technique used for the modeling of complex systems, which comprise high numbers of functions. In software and systems engineering it is present not only in structured methods, where functions serve as the main elements of abstraction, but also in other paradigms, e.g. in the Object Process Methodology (OPM) (Dori, 2002) or in Axiomatic Design (Suh, 2001). In the area of information and enterprise architecture, modeling function decomposition plays a significant role in TOGAF (The Open Group, 2011), for example. In mechanical engineering taxonomic links between functions are employed in numerous approaches, cf., among others, links of function-is-a and method-of-function-achievement (Kitamura et al., 2002), functional decomposition (Pahl et al., 2007), degree of complexity (Hubka and Eder, 1988) or the part-whole relation in Multilevel Flow Modeling (Lind, 1994; 2011).

OPEN IN VIEWER

Fig. 7.

FueL stereotypes enabling the modeling of interdependencies between functions, and of function decomposition in particular.

Figure 7 shows the FueL constructs designed for handling interdependencies between functions, and function decomposition in particular. These constructs are the result of an earlier ontological analysis (Burek et al., 2009), which can be summarized by the statement that behind functional decomposition actually several distinct relations are hidden implicitly. We illustrate this claim with an example of the Business Goal Decomposition Pattern (Eriksson and Penker, 2000).17

The goal for FueL is to support, within a single framework, the modeling of the rich family of function decomposition types, which is currently scattered across numerous approaches. Hence we introduce the relation of function decomposition as a common umbrella relation for various ways of decomposing functions. This does not only enable the modeling of function decomposition by means of a single relation if there is no need for more precise models (as in the example reported above). Moreover, it allows for the refinement of links already introduced in a model in later steps.

As an umbrella relation, function decomposition is a relation between two functions, denoted by fu - decompose CC . It immediately generalizes the three relations of function subsumption, sequence part and operand part, which are introduced in the subsequent sections and which yields the following definition. Figure 7 illustrates the FueL stereotypes designed for modeling these types of function decomposition and more specific ones.

4.4.2. Function subsumption

We introduce the notion of function subsumption based on a generic understanding of the is-a relation, which we denote by is - a CC ( x , y ) . The is-a relation is the main taxonomic link, often considered as the backbone of an ontology. It is common across formalisms for conceptual modeling and for ontology representation. For example, in UML there is the generalization relation between classifiers, whereas in OWL the RDFS property rdfs:subClassOf is the basic taxonomic constructor that can be applied to OWL classes.

The semantics of the is-a relation is, however, not clear-cut.18

Taking a subsumption relation as a primitive, using the terms ‘generic is-a relation’ or ‘generic subsumption’, we specify function subsumption in the following way.

In FueL the notion of function subsumption is founded on the subsumption of goal achievements. Accordingly, GAs can be utilized in order to detect patterns of function subsumption. Much of Section 4.1 above elaborates a structural analysis of GAs, including in particular their operands as well as modes of realization. On this basis the next two subsections introduce three specializations of function subsumption, namely operand specialization, mode addition and mode specialization.

4.4.3. Operand specialization

Functions are often specified with the help of operands occurring in their realizations. Therefore it is very common to construct subsumption hierarchies of functions in accordance with the taxonomic structure among their operands. This leads us to the relation of operand specialization, corresponding to the first pattern for function subsumption.

The pattern of operand specialization is applied frequently in the Molecular Functions Ontology of Gene Ontology. For instance, consider the functions GO:0015075: ion transmembrane transporter activity and GO:0008324 cation transmembrane transporter activity linked by the GO is_a relation. The is-a relation between those two functions is justified by the same relation applying to their operands, where cation is a subtype of ion. Analogously, the function GO:0051119: sugar transmembrane transporter activity is a subfunction of GO:0015144: carbohydrate transmembrane transporter activity because sugar is subsumed by carbohydrate.

4.4.4. Mode addition and specialization

Another pattern of function subsumption is not based on the operand, but on the mode of realization. This pattern can be found, e.g., in work of Bracewell and Wallace (2001), where the link of function-means-context provides for function decomposition driven by means of function realization. As an example of the pattern let us consider two functions, GO:0022857: transmembrane transporter activity and GO:0022804: active transmembrane transporter activity. Both functions have the same type, i.e. doer, they share the same operand, namely substance, and the same input–output pair, respectively – operand is on the one side of the membrane and operand is on the other side of the membrane. Regarding these aspects both functions are equal. However, they differ since the former does not define any particular mode of realization, whereas the latter has the following mode defined: the transporter binding the solute undergoes a series of conformational changes. Additionally, it is clear that each instance of GO:0022804 is an instance of GO:0022857. Therefore, one can say that GO:0022804 specializes GO:0022857 by the addition of a mode.

Furthermore, modes can serve as a principle of decomposition also in cases where a parent function has already a mode assigned. Consider, for instance, the function GO:0022804: active transmembrane transporter activity having the mode: transporter binds the solute and undergoes a series of conformational changes and the function GO:0015291: secondary active transmembrane transporter activity with the mode: transporter binds the solute and undergoes a series of conformational changes by uniport, symport or antiport. Clearly uniport, symport and antiport are the particular modes of active transmembrane transport. Thus it seems intuitive to say that GO:0015291 is a specialization of GO:0022804. We define that pattern of function decomposition as follows:

Since the mode of realization could be a complex structure, e.g. a complex process, it is not trivial to determine if a mode is a subclass of another mode. In this connection the notion of participant may be helpful, since it is often the case that the specialization of a mode of a GA is related to the specialization of its participants. That is the case in the decomposition of GO:0015291, which is based on the specialization of transport protein into uniport, symport and antiport protein.

4.4.5. Function decomposition and part-of relations

There are two contexts in which the interrelation between the notions of function, function decomposition and part-of relation are discussed in literature. One focuses on the interpretation of the part-of relation between objects or, more precisely, of doers and contributors, in terms of their functions. The second concerns representing the interconnection of functions in terms of the part-of relation.

The former approach is applied in several frameworks, e.g. in Axiomatic Design, Object Process Methodology or OntoUML. The general idea is that an entity can be decomposed into its parts based on the functions of those parts and their contribution to the overall function. In Axiomatic Design that pattern is present in zigzagging, which is a basic routine of decomposition. In zigzagging each design parameter is decomposed into its sub-parameters by dissecting the corresponding function requirement. Analogously, in OPM objects modeled in function hierarchies are decomposed into their parts by the analysis of their functions and their dependencies. The pattern of a function-oriented part-of relationship is further formalized by Vieu and Aurnague (2005) and is adopted in OntoUML. Vieu and Aurnague (2005) introduce functional parthood together with distinct flavors as a relation between a whole and its part, where there is some type of functional dependency between them. For direct functional parthood, for instance, for x is to be a part of y to function as an X, then y must function as a Y.

In the present paper we are not primarily interested in function-driven part-of relations, but instead we focus on the second interplay mentioned above, namely between the notion of functional dependency and part-of relationship. This addresses the question to what extent functional dependency, as a relation between functions, can be expressed in terms of the part-of relation. It is worth mentioning that some authors, for instance Vermaas (2010), question the adequacy of interpreting function decomposition purely in terms of a part-of relationship between functions. On the other hand, numerous authors interpret function decomposition in that way. For example, Kitamura et al. (2004) introduce the notion of decomposition of functions by means of the part-of relation: “the part-of relation between functions represents how a function is achieved by finer grained functions” (Kitamura et al., 2004, p. 116). Analogously, regarding the function hierarchy introduced in (Dori, 2002), lower level functions answer the question of how the superordinate function is achieved.

It can be observed that those how-questions can be already tackled by the mode-related subsumption introduced in the previous section. For instance, the function uniport transmembrane transport reveals how the function transmembrane transport is achieved. Nevertheless, the way of realization cannot only be specified by means of subsumption. Let us illustrate that with the function F of a purchase system: to purchase an item. The input of F is an item is available for sale, and the output of F is a customer possesses an item. This function may be realized by the sequence of the following functions: search item, view item, choose item, buy item, pack item, post item and deliver item. None of the functions of this sequence is subsumed by the root function, e.g., because their inputs and outputs do not match. However, it seems intuitive to say that the whole sequence provides an answer to the question of how the root function F is achieved. Therefore one may say that each of the above functions is a part of F, in the sense that together they explain how F is achieved. That sort of relation between a function and the elements of the sequence realizing the function we call sequence-part.

In order to introduce sequence-part formally, we utilize three additional notions: that of a function sequence, denoted by FuSeq C ( x ) , in which each function supports its successor (captured by the binary relation fu - support CC ( x , y ) ), and a relation to link a function to a sequence that can realize it, denoted by has - realizing - sequence CC ( x , y ) . An extensive axiomatic characterization of sequences is beyond the scope of this paper, thus we make only two very basic conditions on function sequences explicit: they have a first and a last element (using primitive relations has - first CC and has - last CC ) and they are interlinked by fu - support CC . The primitive relation append - into CC ( x , y , z ) denotes the fact that a sequence x with sequence y appended to it yields a new sequence z. The has - member CC ( x , y ) relation is defined for auxiliary purposes. ∀ x . FuSeq C ( x ) → ∃ y z ( has - first CC ( x , y ) ∧ has - last CC ( x , z ) ) . ∀ x . FuSeq C ( x ) → ∀ uvyz ( Function C ( u ) ∧ Function C ( v ) ∧ FuSeq C ( y ) ∧ FuSeq C ( z ) ∧ append - into CC ( y , z , x ) ∧ has - last CC ( y , u ) ∧ has - first CC ( z , v ) → fu - support CC ( u , v ) ) . ∀ x y . has - member CC ( x , y ) ↔ FuSeq C ( x ) ∧ Function C ( y ) ∧ ( has - first CC ( x , y ) ∨ ∃ u v ( FuSeq C ( u ) ∧ FuSeq C ( v ) ∧ append - into CC ( u , v , x ) ∧ has - first CC ( v , y ) ) ) .

The notion of function support rests on the relationship of the inputs and outputs of two functions.

For example, one can say that the function F 1 : pack item supports function F 2 : post item, since the output of F 1 : item is packed is a part of the input of F 2 .

Returning to function sequences, those can realize functions, which is formalized by means of has - realizing - sequence CC , satisfying the condition that all inputs and outputs of the realized function must be covered by inputs and outputs of elements of the realizing sequence. ∀ x y . has - realizing - sequence CC ( x , y ) → Function C ( x ) ∧ FuSeq C ( y ) ∧ ∀ z ( has - goal - achievement CC ( x , z ) → ( ∀ u ( has - input CC ( z , u ) → ∃ v w ( has - member CC ( y , v ) ∧ ( has - goal - achievement CC ( v , w ) ∧ has - input CC ( w , u ) ) ) ∧ ( ∀ u ( has - output CC ( z , u ) → ∃ v w ( has - member CC ( y , v ) ∧ ( has - goal - achievement CC ( v , w ) ∧ has - output CC ( w , u ) ) ) ) ) .

On this overall basis the relationship of sequence part can be characterized by a necessary condition.

The specification of function decomposition based on function sequences realizing the goal of the decomposed function is clearly present in the literature. For instance, in the work of Stone and Wood (2000) function decomposition is on chains of subfunctions that operate on the flow of the super function. Vermaas and Garbacz (2009) interpret such chains in terms of an ordering relation between functions, such that a functional ordering φ → ψ exists between two functions if the functional output of φ is functional input to ψ. The FueL support relation presented above grasps intuitions similar to those of functional ordering.

This section on function decomposition concludes with yet another dimension that can be utilized for dissecting functions, namely by means of the part-of relation applied to operands. This is especially useful in cases where an operand is a complex entity or where it is a collection of entities that are difficult to be operated on at once. Consider, for instance, the transportation of big building constructions like bridge elements or wind turbines. Usually these objects are transported in parts, thus the root function transport wind turbine is split into a series of functions of transposing parts of the turbine. This type of function decomposition is called operand decomposition-based decomposition. It constitutes the final building block of FueL as presented herein, before the next section turns to exemplary applications of the set of notions introduced in Section 4.

Specification (Operand decomposition-based decomposition).

For a function x that is a component of a function y by operand decomposition, the goal achievement of x has an operand part that is a part of an operand regarding y. ∀ x y . fu - operand - part CC ( x , y ) → Function C ( x ) ∧ Function C ( y ) ∧ ∃ rsuv ( has - goal - achievement CC ( x , u ) ∧ has - goal - achievement CC ( y , v ) ∧ has - operand CC ( u , r ) ∧ has - operand CC ( v , s ) ∧ is - part - of CC ( r , s ) ) .