A semiotic perspective on polysemy

This paper extends Semiotic-Conceptual Analysis (SCA) to provide a means for comparing and evaluating semiotic relations (i.e. sets of signs) with respect to their ability of and efficiency of expressing conceptual structures. One contributing factor for efficiency is polysemy which refers to reusing representamens of signs in different contexts. Two different types of polysemy are identified: ambiguous polysemy which encodes a view from part signs to compound signs and simultaneous polysemy for analysing how compound signs simultaneously denote more than one meaning. Two detailed examples are included.


Introduction
Semiotic-Conceptual Analysis (SCA) is a mathematical formalisation of basic semiotic notions. SCA defines a sign as a triple of three components which are called representamen, denotation and interpretation (Priss [6]). A set of signs using shared sets of representamens, denotations and interpretations is called a semiotic relation. The definition of a sign was inspired by the semiotics of the American philosopher Charles S. Peirce but SCA does not intend to describe an exact formalisation of his ideas. Furthermore SCA reduces each notion to an abstract mathematical role. SCA can be applied to linguistic data in which case representamens might correspond to words (or lexemes), denotations to word meanings and interpretations to a mapping from words to their meanings. But SCA can also be applied to non-linguistic data, for example graphical representations, in which case "representamen", "denotation" and so on are structural labels. The only condition for a semiotic relation is that interpretations are partial functions (as explained below) that map representamens onto denotations. These notions are thus abstractions of their usual philosophical or linguistic meanings. In the same manner, the notion of polysemy presented in this paper focuses on structural properties and is an abstraction of its linguistic meaning. More traditional linguistic applications of SCA have been discussed elsewhere (Priss [7]) and are not the main topic of this paper.
Being a partial function for an interpretation means that it does not need to be applicable to all representamens but if it is applicable to a representamen then it must yield a unique denotation. A semiotic relation usually contains not just one but a set of interpretations. Two different interpretations applied to the same representamen can yield different denotations. But each pair of an interpretation and a representamen uniquely identifies a denotation. Altogether this implies that many kinds of triadic relations might be considered semiotic relations. Therefore, in order for SCA to be "semiotic", a further condition can be added stating that the data should be collected in some kind of communication context. But such a condition cannot be mathematically formalised. SCA is meant to be used as a data analysis method. Users can decide which sets to use as representamens and denotations and how to model interpretations. The manner of defining a sign in SCA is similar to how Formal Concept Analysis (FCA) as described by Ganter & Wille [3] defines concepts and conceptual hierarchies. These notions are more abstract and maybe shallower than their philosophical or linguistic counterparts, but they result in mathematical precision.
SCA is intended to provide a means for evaluating and comparing semiotic relations. Sets of representamens must be sufficiently small so that sign users can memorise them (which might depend on the users' age: natural languages learnt by children have larger sets of representamens than traffic signs learnt by adults). A set of n representamens together with m interpretations can be mapped onto or express a maximum of m×n denotations. If simple representamens are combined to form compound representamens according to some syntax or grammar, then even more denotations can be expressed. For example, if compounds of exactly one noun and verb are formed using k nouns, j verbs and m interpretations, then a maximum of m×k ×j denotations can be expressed. Such numbers can be calculated using basic combinatorics. Formal languages and information theory address similar questions, but are more aimed at calculating the expressiveness of types of languages whereas SCA concentrates on calculating values for actual semiotic data. Furthermore SCA starts with a distinction between representamens and interpretations.
The focus of this paper is on reusing representamens with different interpretations which is called polysemy. Two different kinds of polysemy are identified. The notion of ambiguous polysemy is coined for reusing representamens or their parts in different contexts whereas the notion of simultaneous polysemy refers to compound signs that simultaneously have different meanings within a single context similar to the proverbial "picture being worth a 1000 words". Ambiguous polysemy considers how parts can be combined to form different compounds whereas simultaneous polysemy describes what partial meanings can be observed from a compound sign.
Pictures and diagrams are examples of compound signs which often engage simultaneous polysemy. In the field of diagrams research the notion of observational advantage has been studied (e.g. Stapleton et al. [9]), for example with respect to how in mathematics, a single diagram sometimes displays a whole proof. It is not easy to define the exact mechanisms of such proofs but Barwise & Etchemendy [1] present a systematic approach for some types of visual proofs. The focus of SCA is not on proof theory but on the relationships between representamens and denotations which are usually modelled as concepts belonging to a conceptual structure, for example in the sense of FCA. SCA then facilitates to investigate what conceptual structure belongs to a semiotic relation, how signs are mapped into a conceptual structure and how much of the conceptual structure is covered by a semiotic relation.
Measures of polysemy provide additional information for determining the efficiency of a representation of concepts with respect to a semiotic relation.
Some further research that is relevant for SCA should be mentioned: Barwise & Seligman's [2] information flow theory and Goguen's [4] algebraic semiotics have somewhat related goals to SCA. But information flow and algebraic semiotics do not start with a distinction between representamens and interpretations as discussed in this paper. Moody [5] proposes a "scientific basis for constructing visual notations". He provides a detailed list of characteristics of visualisations using perceptual, cognitive and structural features. Some of his notions can be mapped onto SCA, but at least in this paper, SCA is mainly applied to quantitative features whereas many of Moody's features are of a qualitative nature. A more detailed comparison of Moody's and similar work and SCA will be left for a separate paper.
The next section summarises basic SCA notions. Section 3 defines compound signs and some measures for evaluating semiotic relations. Section 4 discusses observational advantages. Section 5 provides an example. While Section 2 repeats notions from Priss [7], most of the material of Sections 3-5 is new and has not been previously published.

Basic notions
All relevant SCA definitions are repeated in this paper. Further details, motivating examples and explanations can be found in Priss ([6]). As mentioned in the introduction, SCA can be applied to a variety of linguistic or non-linguistic data. The following examples of four signs are used in this section: s 1 = (i 1 , "pickup", "small van") s 2 = (i 2 , "PICKUP","a casual encounter with a stranger") s 3 = (i 3 , "pickup", "an act of collecting goods in a small van") s 4 = (i 3 , "truck", "heavy road vehicle") The sign s 1 consists of the string "pickup" as representamen, the string "small van" as denotation and an interpretation i 1 with i 1 ("pickup")= "small van" corresponding to some context and sign user. The sign s 3 has the same representamen as s 1 but a different interpretation and denotation. Data collection usually involves some abstraction which requires equivalence relations on the data. For example "pickup" and "PICKUP" might be considered equivalent and not be further distinguished. The interpretations of s 1 , s 2 and s 3 have to be different because they have equivalent representamens but different denotations. But s 4 can have the same interpretation (in this case i 3 ) as one of the other signs. In general, anything that is stated about a sign and its components requires some judgement and modelling. Therefore from a philosophical viewpoint, signs start an infinite chain of interpretation where interpretations themselves can be further interpreted and so on. From an SCA viewpoint, such an infinite chain has been stopped at the point at which the data is collected and modelled. SCA focuses on analysing provided data and not on how the data has been collected. Thus a semiotic relation involves a structure (S, I, R, D, ≈ I , ≈ R , ∼ D ). Additionally, a non-mathematical condition is assumed stating that the data for SCA has been collected in some kind of communication context. For each interpretation some further parameters about context, time, place, sign user and so on might be available. In the examples above, i k might correspond to sign user u k at time t k in location l k . If i 1 and i 2 refer to the same time and location, then s 1 and s 2 might indicate a miscommunication where a sign user u 1 has a totally different understanding of "pickup" from sign user u 2 . For denotations an equivalence relation or even equality could be defined but quite often a weaker tolerance relation is sufficient. A tolerance relation expresses similarity (in this case synonymy) because it is reflexive and symmetric, but need not be transitive. For example "small van" might be similar to "an act of collecting goods in a small van" which might be similar to "a casual encounter with a stranger" but the last one is unlikely to be similar to "small van". The originally linguistic definitions have been abstracted in SCA as follows: Thus continuing with the example, s 1 and s 4 might be synonyms. The signs s 1 and s 3 might be polysemous, but s 1 and s 2 are most likely homographs instead of polysemous. It should be noted that contrary to standard linguistic terminology, in SCA it is not required for synonyms to have different representamens. Thus a sign is both polysemous and synonymous to itself and, in SCA, polysemy is a special kind of synonymy.
The question remains as to how one decides whether "small van" ∼ D "heavy road vehicle" and so on. In the SCA, the denotations are not usually strings as in the example above, but concepts that are modelled according to some kind of formalisation (formal ontology, FCA or other logical conceptual formalism). SCA does not prescribe which formalisation should be used but only defines conceptual classes as a means for describing some aspects that a modelling of conceptual structures must provide. Figure 1 presents an overview of how the sign components are mapped into conceptual classes. Representamens are mapped onto formal objects of a conceptual class. If human sign users are involved, it is not possible to obtain exact information about the denotations because they only exist in the minds of the sign users. It is therefore preferable not to map from denotations to concepts but the other way around in order to approximate the hypothetical denotations.  Sign components and how they can be investigated A conceptual class is a formalisation of conceptual knowledge containing extensional information (some elements, sets, relations, functions, ...) and intensional information (some logical statements, axioms, implications, clauses, algorithms, ...). Because a purpose of SCA is to analyse data, all sets are assumed to be finite. Only some aspects of a conceptual class are required: Because n-ary relations and functions are also sets and operations can be modelled as n-ary functions, formal objects can be of a variety of complex types. An example of a conceptual class is a formal context in the sense of FCA where attributes are considered as predicates. Other examples of conceptual classes are using description logic for the statements in A and set-theoretical models for O or a relational database as a conceptual class where each O i is a table and A might be empty.

Compound signs
Ideally, sign users should be able to produce a maximal number of signs with a minimal amount of information that they need to memorise about a semiotic relation. This section discusses the use of compound signs as a means for reducing the cognitive load of a semiotic relation. Measures similar to the ones provided in this section most likely already exist in other fields. But, as far as we know, they have not yet been discussed in a similar setting of triadic signs.

Definition 4
Let L(S) denote a measure for the cognitive load involved in using the signs of a semiotic relation S, such as memorising, parsing, selecting, understanding and so on.
There are many possibilities for specifying L(S). Because R and I together determine denotations, they need to be memorised. In analogy to (but not identical to) Peirce's triad of symbol, index and icon, in SCA interpretations can be divided into arbitrary 1 , algorithmic and observational interpretations. Observations use perceptual features of representamens. For example, the meaning of onomatopoeic words (such as "cuckoo") can sometimes be directly observed without having to learn the word. Arbitrary interpretations need to be fully memorised whereas for algorithmic interpretations only some rules need to be memorised from which other interpretations can be derived. For example, if "a" is interpreted as 1 and "b" as 2, then it is possible to guess what "c" might be interpreted as. The distinctions between arbitrary, algorithmic and observational are not mathematically precise and depend on the socio-cultural background of sign users. Even for arbitrary interpretations contextual clues are relevant, for example in order to express metaphors. Thus the actual cognitive load can differ for different types of interpretations.
The complexity of parsing and of applying interpretations is also relevant. From a perceptual viewpoint, it may be impossible to parse representamens that are too small or too quietly uttered. In this paper a constant c(S) is assumed to summarise perceptual and other qualitative aspects of a semiotic relation. If the set of representamens is large, then c(S) will be insignificant compared to |R|. Further details of c(S) are left for future research. The importance of Theorem 1 is that the cognitive load is calculated using addition whereas the number of possible signs is calculated using multiplication which tends to be multitudes larger than addition at least for large numbers. Proof: For a maximal number of possible signs |S| = |R| × |I | the term |I | + |R| is supposed to be minimal. After substituting |I | by |S|/|R| the minimum of |I | + |R| = |S|/|R|+|R| with respect to |R| can be calculated as a point where its derivative is zero. The derivative is 1 − |S|/|R| 2 which leads to 1 = |S|/|R| 2 ⇐⇒ |R| 2 = |S| ⇐⇒ |R| = √ |S|. Thus |I | = |R| = √ |S|.
A single representamen being interpreted onto different denotations is a case of polysemy as defined above. Theorem 2 indicates that the cognitive load would be minimal if the polysemy of signs were to correspond to the square root of the number of signs of a semiotic relation. The English language has 200.000 -500.000 entries 2 or word-sense pairs. The most polysemous word "run" has about 650 different senses 3 which is approximately √ |S|. But most natural language words only have a few different senses. According to WordNet 4 the average polysemy of words is only between 1 and 3 senses. Polysemy of natural language words also depends on how it is counted. A word such as "today" would be infinitely polysemous if each day was a different denotation. Considering denotations as concepts lowers the count because then the denotation of "today" is not an actual day but a concept for "current day" which has actual days in its extension.
Two interpretations can be considered equivalent (≈ I ) if they encode the same usage context, for example, the same time, place, sign user and physical sign context. Again deciding which interpretations are equivalent is a modelling decision made by a user of SCA with respect to an application.
This section develops a measure based on ambiguous polysemy. Simultaneous polysemy is discussed in the next section. Both types of polysemy often involve compound signs: parts are ambiguously polysemous with respect to their use in compounds, for example a letter used in different words or a word in different sentences. The different meanings of a compound sign, however, can be explained as simultaneous polysemy.

Definition 6
For a set R of representamens with a partial order (R, ≤), a representamen r is called a compound representamen if ∃r p ∈ R with r p < r. The set of minimal elements of (R, ≤) is denoted by R π (set of part representamens) and R ω := R \ R π . For each r ∈ R ω , a set φ(r) is defined with (r p < r ⇐⇒ ∃n ∈ N, i ∈ I π : (n, i, r p ) ∈ φ(r)) where n is a unique index number within φ(r) and I π is a set of interpretations. The set R ω is constructable if each r ∈ R ω is uniquely identified by its set φ(r).
It follows that R = R π ∪ R ω and R π ∩ R ω = {}. In Def. 6 the interpretations indicate roles that the parts have in the compound, for example, parts of speech of a sentence. The set φ(r) represents a result of parsing r into a list of parts with their interpretations. Each pair (i, r p ) has a distinct denotation but can occur more than once for a compound. Therefore, the purpose of the index numbers is to distinguish duplicate pairs. Otherwise index numbers are not intended to provide denotational information. Table 1 shows some examples: 48 as a decimal number or Roman numeral and 9:16 as a time on an analogue clock where the minute hand is meant to have 60 different interpretations and the hour hand 12 interpretations.
For a constructable R ω , the compounds can be parsed into their parts and the parts then be put together again to form the compound. Graphical representamens such as paintings may not be constructable because it may be difficult to define how the parts constitute the compound. In the remainder of this paper it is assumed that R ω is constructable. It is also assumed that I π has been designed by a user of SCA to contain as few interpretations as needed.
Definition 7 For a semiotic relation S with a constructable R ω , the following measures are defined: • μ φ (r) := |{i | ∃r p , n : (n, i, r p ) ∈ φ(r)}| The measure μ φ counts the number of different interpretations per compound. In natural languages μ φ (r) might correspond to counting the different parts of speech that a piece  of text contains. If R = R π then μ φ (S) = 0 and I π := {} can be assumed. The measure rpt (r p ) (for "repetition") detects if a part r p is used more than once with the same interpretation in a compound. For example, I is used 3 times with the interpretation + in the Roman numeral for 48. Presumably reusing a part many times with the same interpretation can render a compound difficult to perceptually parse. Therefore ideally rpt (r p ) = 0 and rpt (S) = 0. The next definition extends the partial order amongst representamens to a partial order amongst signs.  = (i, r, d) is called a compound sign if φ(r) = {} and ∀(n p , i p , r p ) ∈ φ(r) : a) s p := (i p , r p , i p (r p )) ∈ S; b) ∃i 1 (i 1 , r, i p (r p )) ∈ S; c) i ≈ I i p and d ∼ D i p (r p ). In that case the notation is s p < S s and, as usual, s p ≤ S s :⇐⇒ s p < S s or s p = s.
The conditions mean that a) each part representamen yields a sign itself; b) the denotations of part signs can also be extracted from the compound sign; and c) compound and part signs are simultaneously polysemous. It follows that a compound sign has at least as many interpretations as it has parts. For example a compound sign for the Roman numeral in Table  1 is (i, XLVIII, 48). It has a part (-,X,-10) and the -10 can also be read from the compound sign: (i 1 , XLVIII, -10). The denotational similarity required for simultaneous polysemy can be quite weak. For Roman numerals all numbers might be considered similar.

Theorem 3 (S, ≤ S ) is a partial order if ∼ D is an equivalence relation.
Proof: (S, ≤ S ) has to be shown to be reflexive, antisymmetric and transitive. For s 1 = (i 1 , r 1 , d 1 ) and s 2 = (i 2 , r 2 , d 2 ): Reflexive because s 1 = s 2 ⇒ s 1 ≤ S s 2 . Antisymmetric: if s 1 ≤ S s 2 , then either s 1 < S s 2 ⇒ r 1 < r 2 ⇒ r 2 < r 2 or s 1 = s 2 . Transitive because conditions a) and b) follow from the transitivity of (R, ≤) and c) is fulfilled if ∼ D is an equivalence relation.

Theorem 4 For a semiotic relation S with a constructable R ω , an upper threshold for the cognitive load is L(S) ≤ |R π | + |I π | + |I | + c(S)
Proof: The threshold holds because according to Theorem 1: L(S) ≤ |R| + |I | + c(S) and every representamen in R ω can be constructed from representamens in R π and interpretations in I π .
For decimal numbers |I π | = μ φ (S) according to Table 1 because the largest number uses every interpretation. For analogue clocks |I π | = 12 + 60 but μ φ (S) = 2. Analogue clocks involve algorithmic interpretations. It is not necessary to memorise 60 interpretations for the minute hand, but only the principle of how to read one of these interpretation. Thus μ φ (S) may be more relevant for the cognitive load than |I π |. On the other hand, a high rpt (S)value is not an indication of ambiguous polysemy but of repetition. The following measure intends to improve the threshold of Theorem 4 by an estimation which is demonstrated in the examples below. A small |R π | with a large |I π | but small μ φ (S) results in a particularly low value of the measure which is desirable.
The example of the first rows of Table 1 is discussed further in Table 2 for the integers from 1 to 100. S 1 is a semiotic relation for decimal numbers (1, 2, ..., 100). S 2 uses tally marks: (I, II, III, IIII, Z Z IIII, ...). S 3 utilises Roman numerals (I, II, III, IV, V, ..., C). The semiotic relations S 1 -S 3 contain compound signs; the semiotic relations S 4 and S 5 do not. In S 4 all numbers are considered non-compound representamens which all share a single interpretation. S 5 contains 10 representamens with 10 interpretations. Table 2 calculates μ ap for all five semiotic relations. In S 1 , the digit 1 can occur in 3 positions which are counted as 3 interpretations (or log(|S|) + 1 positions/interpretations). For S 2 , the tally mark I can occur at most 4 times. The mark Z Z IIII can occur up to |S|/5 = 20 times. Each mark has only 1 interpretation, but rpt (I) = 3 and rpt (Z Z IIII) = 19. Tally marks can become difficult to visually parse as indicated by the high rpt (S) value. The next row of the table shows Roman numerals. The numerals I and X have 2 interpretations (subtractively before or additively after) which can be applied up to 3 times. Table 2 shows that for numbers below 100, Roman numerals have the best measure of ambiguous polysemy. For larger numbers, decimal numbers outperform Roman numerals even if one were to invent more representamens because one would need 2log(|S|) + 1 representamens.

Observational advantages
The polysemy measure μ ap (S) is just one type of measure for semiotic relations. While μ ap (S) focuses on a view from parts to compounds, the opposite view is to consider how much can be represented with a single compound sign. In both views it is an advantage for a semiotic relation to express many denotations with few representamens. In this section, the number of denotations for a fixed set of signs is counted. As mentioned before, observational interpretations need not be learned but are derived from contextual clues and prior knowledge. The term "observation" in SCA was chosen in analogy to Stapleton et al.'s, ( [9]) "observation" as a generalisation of "free rides" in the context of diagrams. While parts of a compound sign are ambiguously polysemous with respect to the different manners in which they can be employed, a compound sign is simultaneously polysemous with respect to the information that can be observed from it.
First, equivalence classes of signs are defined. Denotational similarity and thus synonymy and polysemy are only tolerance relations, not equivalence relations. But it is always possible to generate an equivalence relation from a tolerance relation by assigning all elements that can be reached from an element (through any chain of pairs of the tolerance relation) to the same equivalence class. For synonymy this might not be useful because in the worst case, all signs of a semiotic relation might belong to a single equivalence class.
But for polysemy, the equivalence classes are much smaller because they are constrained by the fact that their representamens are equivalent. Theorem 5 For each s ∈ S, the sets S p (s) and S sp (s) are equivalence classes on S of equivalence relations ≈ S p and ≈ S sp . The equivalence relation ≈ S sp is a finer grained partition of ≈ S p which is finer grained than ≈ R . If homographs do not exist in S, then each equivalence class of ≈ S p corresponds to an equivalence class of ≈ R .

Definition 10
Proof: ∼ D is not an equivalence relation, but the construction of S p (s) generates the transitive closure of ∼ D with respect to a selected s and the transitive closure of a tolerance relation is an equivalence relation. S sp is an intersection of equivalence relations, S p (s) and ≈ I , and thus an equivalence relation.
Next, for each class of equivalent signs, the set of denotations belonging to that class is determined. For two semiotic relations S 1 and S 2 whose interpretations all belong to the same equivalence class, S 1 has a higher observational efficiency over S 2 if D sp (S 1 ) = D sp (S 2 ) and |S sp (S 1 )| < |S sp (S 2 )|. A semiotic relation S has maximal observational advantage over a conceptual class if D sp (S) contains the set of true statements of the intensional description of the conceptual class.

Definition 11
A non-mathematical condition for Def. 12 is that only signs with observational interpretations should be considered. The formulas of Def. 12 are still reasonable for other types of interpretations, but the word observational might then be misleading. Single words or mathematical formulas tend to not produce any observations at all (other than perceiving the sign itself). In exceptional cases, they produce a small number of observations. A formula  Observability also depends on what conceptual class corresponds to a semiotic relation and how well the semiotic relation covers the content of a conceptual class. Table 3 shows three different conceptual classes for the semiotic relations of Table 2. They all contain numbers and an iterator function which retrieves the next number for any number (by adding 1) as formal objects. The usual operations (addition, subtraction), comparison operations (<, =) and properties of integers can be assumed for these conceptual classes. The only difference is a constructor method: in Int 10 numbers are represented as base-ten digits and in Int 5 numbers are represented as a sum of a number ≤ 4 plus the remainder as a multiple of 5. Each conceptual class is assumed to provide functions to convert numbers between its specific representation and a plain integer representation. The first column of Table 4 shows that each of the conceptual classes belongs to one of the semiotic relations S 1 -S 4 .
With respect to observability, someone who has never seen tally marks, may be able to deduce their denotations directly from the representamens. This is also the case for the first three Roman numerals. For tally marks the functions for progressing from n to n + 1, addition, subtraction and size comparisons are also easy to observe. The construction mechanisms of decimal numbers and tally marks correspond to their representamen structures. With some background knowledge, it is possible to observe whether a decimal number is divisible by 2, 5 or 10 based on the last digit. For all four semiotic relations one can observe whether two numbers (of the same semiotic relation) are equal or not. Table 4 shows that tally marks support more observations than signs from the other semiotic relations. But larger numbers expressed by tally marks are difficult to perceptually distinguish (resulting in a high value for μ ap (S)). The results from Tables 2-4 taken together, each semiotic relation has some advantages and disadvantages. Further investigations can be conducted with respect to the complexity of functions for operations and properties. For example, decimal numbers are better suited for multiplication than the other semiotic relations.  shows the time as a string of hours, a colon and minutes. A unary clock (C5) prints one character "x" per minute on a computer terminal. We are also considering analogue clocks (C3 and C4) which were not part of Goguen's original analysis. Goguen discusses the differences between the clocks mainly by constructing morphisms which corresponds to an analysis of conceptual classes in SCA. We would argue that other semiotic aspects are missing from Goguen's modelling. Table 5 shows three different conceptual classes that are relevant for this example. How to chose conceptual classes is a matter of judgement in SCA. Whether or not these classes are appropriately described is a question that is external to SCA. SCA focuses on how well a semiotic relation fits to a given conceptual class. The conceptual class T1 only counts minutes, T2 and T3 also have concepts for hours and maybe for parts of hours. T2 repeats after 12 hours, T3 after 24 hours. Table 6 calculates the measure of ambiguous polysemy for the five different types of clocks. C5 is clearly poorly designed compared to the other 4 and not commonly used as a clock at all. C1, C2 and C4 are all similar but C3 is best. Table 7 shows that the clocks C2, C3 and C4 outperform the other clocks with respect to how many concepts can be observed from them. The only advantage of C1 and C5 is that they allow an easy comparison of length of time (<) because it is not necessary to convert between different units. For C5 it is possible to observe whether one clock shows a larger amount of time, but it is  difficult to visually determine the exact amount of time. Whether two clocks show exactly the same time (with respect to the timeframe that can be expressed in each conceptual class), is observable for all types of clocks. For digital clocks, concepts of quarter/half/three quarter hour can only be calculated and not observed. But a progress of 10 minutes or 1 hour is easily observable. In total, C3 may be more advantageous than C2 because both clocks share a conceptual class but C3 uses fewer part representamens. But C3 requires users to memorise where the hour digits exist on an analogue clock whereas they can be read from C2. It is probably not surprising that clocks such as C1 and C5 do not actually exist for use by humans. C1 is used internally by computers.

Conclusion
SCA as described in this paper attempts to provide quantitative measures for comparing representations of signs and qualitative means for comparing efficiency of semiotic relations with respect to underlying conceptual classes. The main application domain for SCA are graphical representations because they contain more observational interpretations than textual representations. Graphical mathematical representations are particularly suitable because their concepts are defined with mathematical precision. It is intended to apply the results from this paper to a further development of methods for analysing and structuring teaching materials in mathematics education as suggested by Priss [8]. Other applications, for example, in the domain of usability and user experience would also be possible applications of SCA in the future.
Funding Open Access funding enabled and organized by Projekt DEAL. Partial financial support was received from the German Bundesministerium für Bildung und Forschung under grant number: 01PL16066H.

Conflicts of interest/Competing interests
The author has no conflicts of interest to declare that are relevant to the content of this article.
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