Since the shift in philosophy of language from logical positivism to behaviourism and pragmatism, it is widely accepted that most natural language terms do not have precise universally agreed definitions that fix their meanings. Even when conversation participants share the same vocabulary and agree on taxonomic relationships (such as subsumption and mutual exclusivity, which might be encoded in an ontology), they may differ greatly in the specific semantics they give to the terms in a particular situation. Moreover, except for certain technical terms, individuals do not hold permanent and precise interpretations of the meaning of terms themselves [1].
Humans generally cope with these imprecisions of language by using context and other pragmatic information to narrow the semantic variability of the terms. We say, they 'take a standpoint' on the semantics of the terms. If conflicts occur during the conversation, participants may adapt dynamically their standpoints in order to maintain the cooperation principles (Grice [2]) and achieve successful information exchange. In settings where precision is necessary, such as in scientific or policy making domains, an explicit concept negotiation may be needed if the participants expect or detect conflicting standpoints.
To support human-machine interactions, ontologies are aimed at providing a common vocabulary in which shared knowledge can be represented [3]. However, in most ontology-driven systems, concepts have rigid and static semantics that do not take account of vagueness or context dependence, and in this respect fail to reproduce the conditions of human conversation. Meanwhile, the extensive and increasing amount of data accessible through the internet encourages research on the remaining core challenges facing ontology construction, among them semantic heterogeneity.
In this paper we take the example of negotiating the interpretation of the term 'forest', and propose a framework based on supervaluation semantics to allow for semantic variability within the ontology. We first introduce what we understand as vagueness and how it affects geographical objects in general, followed by the specific case of forest. We then summarise different approaches to vagueness and how they relate to our research. We propose a framework based on standpoint semantics and its tools for supporting share of information, concept negotiation and querying and reasoning within an agent's standpoint.
Vagueness is pervasive in language [4] and arises whenever a concept or linguistic expression admits of borderline cases of application [5]. We adopt here the distinction proposed in [6] between sorites vagueness and conceptual vagueness. While Sorites vagueness occurs when the applicability of a predicate depends on specific measurable parameters but their thresholds are undetermined, conceptual vagueness arises when there is a lack of clarity on which attributes or conditions are essential to fix the meaning of a given term, so that it is controversial how it should be defined [7,6]. This is different from 'simple ambiguity', where a term has more than one distinct meaning.
Moreover, the geographic domain is particularly characterised by objects lacking bona fide boundaries and by the inherent vagueness in the definition of geographic descriptors, frames of reference and context [8,9]. When thinking about geographic space, people typically employ several different concepts, and change between them frequently depending on the scale and the perceptual and geometrical properties of the space [10] in addition to the contextual and pragmatic information.
While the value of ontologies is now well established [11], their support of vagueness and semantic heterogeneity1 remains challenging. One of the main advantages of ontologies is that they improve the interoperability acting to enforce a consensus view reached by a community regarding a certain domain [7]. This is done by formalising the semantics of the terminology of the domain of discourse in some logic formalism. Typically this process involves the cooperation of domain experts which results in a unified decision on the formalization of the semantics of the terminology. However, as a result of the semantic heterogeneity and vagueness of the concepts to define, strong semantic commitments favour specific interpretations of language and involve a loss of generality, thus restricting the opportunities for interoperability. On the other hand, approaches with shallower semantics rely fundamentally on taxonomic relationships, such as subsumption and mutual exclusivity, thus leaving uncertainty on the specific semantics of instances of these terms and potentially compromising the sound reuse of information.
In this research we examine the term 'forest', for which a broad range of definitions (more than 600 were reported in [12]) have been specified for different purposes. Beyond the spatio-temporal context dependence and fuzziness of their boundaries, many factors contribute to the semantic variability of 'forest'. They arise from: fundamental differences between the land cover / land use2 perspectives, specific uses of the term in different disciplines (ecology, forestry, environmental science, ...) and for different management objectives [14], pragmatic differences between conceptualizations created for science and policy [15], different aspects involved in classifying land as opposed to individuating and demarcating geographic objects [7], and the modelling from endurant or perdurant perspectives on fundamental ontology [16,14].
This scenario poses challenges both for the acquisition of global forest knowledge, particularly its extent and spatial distribution [17,18,19], and for the sound reuse of information and knowledge extraction from the many resources that are increasingly publicly available [20,21]. Moreover, different pieces of research point at cases of crossdisciplinary information reuse among semantically non-interoperable datasets in science, resulting in misleading results [15,14]. At the same time, the option of a centralised and standard definition of 'forest' (the definition of the FAO3 plays that role de facto) only supports the discourse of certain institutions and is reported to be unsuitable to capture the needs of different contexts, disciplines and agents [15,14]. In these cases, the lack of recognition for alternate definitions may distort the understanding of certain scenarios such as in the case of contested spaces [16].
Despite the call for addressing the variability of interpretations and definitions of 'forest' in the literature [19,14,15,20], most public ontologies containing the concept 'forest' either avoid any semantic commitments beyond those embedded in the taxonomies or reduce them to the FAO definition. In the best scenarios, the forest concept has a fuzzy boundary thus not committing to that fixed by the FAO. In this paper we propose a framework that explicitly recognises a variety of acceptable definitions of 'forest' and we outline how such a system can support different interactions in a context of semantic heterogeneity. Specifically, we support agents to discover the semantic heterogeneity of a concept in the ontology, to analyse it and to take their standpoint. Subsequently, they can reason and query the ontology according to their standpoint.
Our intent is to provide a novel way of tackling the prior issues by approaching them from the perspective of vagueness in ontology, more specifically conceptual vagueness. In this section, we briefly review some approaches to vagueness and concept creation and negotiation, and how they relate to our research on forest definitions.
The most broadly used logic-based techniques to model vagueness in information systems are multivalued logics [22], in particular fuzzy logic [23]. Fuzzy logic works by assigning degrees of truth to statements rather than making truth valuation a binary choice. As a result, this approach provides a reasonably intuitive model for sorites vagueness [7], assigning a gradually increasing value to borderline cases as they transition from less to more likely. However, fuzzy sets do not fully characterise the different precise overlapping meanings that a term can adopt, which can be sharp but diverse, and fails to incorporate penumbral connections [24] among them. For these reasons, we consider that Fuzzy logic and similar formalisms are not suitable for the problem under consideration.
Conceptual spaces (Gardenfors,[25]), have received great attention in the last years. Although they are not a formal theory of vagueness, they are relevant for this research as they deal with concept formation and representation, expressed within a geometrical space. A concept, say vanilla flavour, is then defined as some region of a four dimensional space of taste, where the dimensions are salt, sour, sweet and bitter. This approach shows interesting features for concept comparison through geometrical operations within a specific n-dimensional space, conceptual adaptation to context (location) in GIS [26] and an account of prototypicality and borderline cases [27]. However, it remains unclear how to tackle the issue of conceptual vagueness, as every concept is defined within a fixed set of quality dimensions; choosing the relevant ones and thus describing the underlying conceptual space for complex and ambiguous categories is not trivial [25].
Semantic heterogeneity and concept negotiation in information systems have been mainly investigated as a separate phenomenon from that of concept creation and vagueness. Instead, they are approached as a phenomenon that arises in the context of the need for interoperability of two systems, ontologies or agents, and the necessary conciliations for their successful interoperation. In the area of ontology matching, a wide literature review is provided in [28]. It is our intention to provide a complementary approach to the existing work on the topic, based on the explicit support for semantic variation within an ontology. Thus, the framework aims to support the semantic negotiation of the meaning of its concepts, by providing agents with expressive power to represent their interpretation of the terms of the ontology both when instantiating its concepts and when querying and reasoning within it.
Standpoint Semantics is our theoretical framework for representing, interpreting and reasoning about information expressed using vague terminology. This theory is an elaboration of the Supervaluation approach, in which the semantics of a vague language is modelled by a set of precise interpretations called precisifications, where each precisification corresponds to a precise and coherent interpretation of all vocabulary of the language.
Consider a formal language based on classical first order logic, with a vocabulary V = N ∪ C ∪ R, where N is a set of name symbols (these may be used as constants or as variables when used with a quantifier), C a set of unary concept terms and R a set of binary relation symbols. The set L of formulae of the language is the smallest set that contains Ca and Rab for every C ∈ C every R ∈ R and every a, b ∈ N , and, for every ϕ and ψ in L , formulae ¬ϕ, ϕ ∧ ψ and ϕ ∨ ψ are also in L . 4We define an interpretation structure I = P, D, V , δ , where:
• D is a set of individuals (the domain of discourse),
• δ is a denotation function, which, for each precisification and each vocabulary symbol in V , gives the semantic value of that symbol. It is a compound of the following sub-functions: δ N : (P × N ) → D maps name symbols to elements of the domain, δ C :
maps concept terms to subsets of the domain, δ R : (P × R) → 2 D×D maps relation symbols to sets of ordered pairs of elements of the domain.
The only difference from the usual classical logic semantics is that the denotation of each symbol is relative to a precisification p ∈ P, which determines a precise interpretation of that predicate. We write p I ϕ to mean that the formula ϕ is true for precisification p, in interpretation structure I . This determines the truth of propositions as follows:
that is identical to I , except that the value of δ (p, x) may be any element of D. 5Notice that this semantics not only accounts for variability in the meaning of vocabulary terms but also allows one to enforce dependencies between the terms occurring in a formula. For example, precisifications might vary in the set of people considered tall. If precisification p 1 sets a greater height threshold than p 2 for applicability of the vague property Tall, we might have δ (p 1 , Tall) = {sally, tom}, δ (p 2 , Tall) = {sally, tom, uli}. Hence, we would have p 1 I Tall(uli) and p 2 I ¬Tall(uli). But no precisification can make the formula Tall(uli) ∧ ¬Tall(uli) true. Moreover, we can also enforce constraints between different terms in a formula. For instance, we might require that the denotations of Tall and Short are disjoint for all precisifications. Thus, no formula of the form Tall(x) ∧ ¬Short(x) would be true in any precisification.
The supervaluation semantics allows us to to augment the logical language with operators that are interpreted relative to the set of precisifications. In particular, we can specify a semantics for Uϕ, meaning that ϕ is unequivocally true, and Sϕ, meaning that ϕ is in some sense true:
• p I Uϕ iff q I ϕ for every precisification q ∈ P, • p I Sϕ iff q I ϕ for some precisification q ∈ P.
Standpoint semantics adds detail to the basic supervaluation approach in several ways. The semantic choices that determine each particular precisification are modelled explicitly. These consist of choices of (a) threshold values that determine the applicability of 'sorites vague' predicates such as 'tall' and 'bald'; and (b) choices of definitions that resolve conceptual ambiguities, such as which kinds of vegetation species can be considered as constituting a forest.
Standpoint semantics explicitly models the variability of vague terminology in relation to precise observable measurements and properties. Such measurements could be heights, weights, distances and the properties might relate to physical composition, topological relationships. This is the kind of information one might store in a database (or compute directly from the information in a database). Of course, in practice such information might be inaccurate for various reasons such as limitations of measuring equipment or errors in data capture. But these issues are not due to vagueness of terminology, so in formulating the standpoint theory we need not worry about the correctness of the measurement data: we take it as we find it.
We now specify a classical first order language L o for describing entities in terms of objective measurements, properties and relations. Let its vocabulary be the set V o = N ∪D ∪F ∪G ∪C ∪R, where N is a set of names, D is a set of numerical expressions (e.g. standard decimals), F is a set of unary function symbols, G is a set of binary function symbols, C is a set of (precise) unary concepts and R is a set of (precise) binary relations. So in a forestry related domain a vocabulary might be something like: {tree1, ..., boris, ...}, {height, radius}, {distance}, {Oak, Beech, ...}, {Owns, ...} .
Here we have identifiers for individual trees, names of people, unary functions giving height and (canopy) radius measurements of the trees, a binary function giving the distance between any two trees, predicates specifying the species of trees and an owenership relation between trees and people.
In addition to the predications, Ca and Rab, of L , the vocabulary of The truth conditions of the language L o relative to interpretation I o are as follows:
The interpretation of formulae formed by the Boolean connectives and quantifiers will be exactly the same as for standard first order logic, so we do not specify it here.
We now specify a language L v in which we can specify possible precise interpretations of vague predicates. In adition to V o , we also have the vocabulary
where C v is a set of vague unary predicates, R a set of vague binary relations T v a set of threshold variables. The specification is technically complex, although it has a relatively simple informal explanation. Each precisification determines a mapping of vague conceptual terms and relations to possible definitions and also determiens a valuation threshold variables that fix the limits of applicability of vague graded predicates. Thus each precisification provides a translation from formulae containing vague terms of V v into formulae containing only the precise objective vocabulary of V o .
Let L d be the set of possible defining formulae. These are any well formed formulae over the vocabulary V o ∪ T v , where as well as measurement functions and decimals, the terms that can occur in atomic formulae of L d formed with the = and ≤ relations also include threshold variables. So for example, we could have height(x) ≤ thresh1. We write L n d to refer to the subset of L n d containing those formulae that have exactly n distinct free variables -i.e. formulae in which exactly n symbols in N occur outside the scope of any quantifier.
We now give an explicit specification of the particular interpretation associated with a precisification. Let each precisification p be associated with a tuple p T , p C , p R , where:
d maps each vague unary predicate to a precise unary definition, • p R : R v → L 2 d maps each vague binary predicate to a precise binary definition. So for example, for precisification p we could have: p C (Tree) = Plant(x) ∧ Woody(x) ∧ min tree height thresh ≤ height(x) and p T (min tree height) = 10, so precisification p defines Tree to be a woody plant of height greater than or equal to 10 (metres).
We can now give an interpretation function for the full language L v in which formulae containing vague terms are interpreted relative to a precisification, which maps them into precise formulae of L o . The interpretation is specified by
where the interpretation of symbols in V o is given as for the structure V o given above, and the interpretation of formulae in L v is given by:
, where δ (p, τ) = δ o (τ) for all terms in the precise language L o (for whose interpretation p is not relevant) and δ (p, τ) = p T (p, τ), where τ is a threshold variable (i.e. τ ∈ T v ),
, where δ (τ) is defined the same way as in the previous clause,
, where p C (C)(a/x) is the result of substituting a for the free variable x occurring in the precise definition of C determined by the function p C of precisification p,
x, b/y), where p R (R)(a/x, b/y) is the result of substituting a and b for the free variables x and y, in order of their occurrence in the definition of R determined by the function p R of precisification p,
In this section we explore the potential of the proposed system to enable communication between agents holding diverse standpoints, by providing a shared ontology that supports vague semantics. Our framework enables agents to both link data to the ontology according to their precise interpretation and also to query and reason within the ontology according to a specific standpoint, which can be modified during the interaction. In a context of sharing information, the proposed framework offers tools that enable both the analysis of the relations that hold between precisifications and the execution of modifying operations such as calculating the intersection or union of a pair of precisifications. This, together with information on the instances satisfying these precisifications, serves as a support to the agent for the specification of sound standpoints that guarantee integrity and enable interoperation with the ontology.
Assuming that a concept negotiation is necessary, we support the analysis of the relations that hold between precisifications. We identify five main relations, analogous to RCC5, to be inferred by the system. These are:
1. Equivalence: p 1 ↔ p 2 . Anything that is a 'forest' in p 1 must be a 'forest' in p 2 and vice versa. 2. Subsumption: p 2 → p 1 . Anything that is a 'forest' in p 2 must be a 'forest' in p 1 . 3. Inverse subsumption: p 1 → p 2 . Anything that is a 'forest' in p 1 must also be in p 2 . 4. Disjunction: p 1 → ¬p 2 . No entity can be a 'forest' in both p 1 and p 2 5. Overlap: None of the previous (NP) relations hold between p 1 and p 2 . Thus there may be entities that are classified as 'forest' in both p 1 and p 2 , although neither precisification has a stronger definition that the other.
Relations between precisifications show, to some degree, the connections among them. While overlap is the most common scenario, other relations such as subsumption and disjunction provide valuable information to the agent, to the point of interoperation becoming trivial (such as when information is linked to a definition that is subsumed by the agent's definition) or not feasible because of explicitly conflicting commitments (in the case of disjoint definitions).
It must be noted that the relation between precisifications is purely logical, i.e. is a relationship that holds between the formal representation of two interpretations of the semantics of the ontology, p1 and p2, which is not dependent on the real world objects that may instantiate them. Consequently, the relation between the logical relations of the precisifications and that of the sets of the real instances (in this world) may not match up. However, the former do constrain the possibilities of the latter. This is shown in Table 2 and discussed in more detail in Section 5.4.
Table 1. The operations (columns) than can be performed between p1 and p2 and the result with respect to the relations holding between them
Logic definitions
In order to support both the analysis of the differences and variation among the existing instantiations of the concepts of the ontology and also to facilitate the specification of standpoints that comprise the appropriate subset of precisifications, we define four basic operations between precisifications.
1. Union: p1 ∪ p2. All the instances that satisfy either p1 or p2 (or both). E.g. Both p1 and p2 are considered admissible precisifications for a specific use of the ontology by a certain agent. 2. Intersection: p1 ∩ p2 . All the instances that satisfy both p1 and p2. E.g. An agent is interested on the intersection between their own interpretation p1 and a precisification in the ontology p2. 3. Complement: p2 \ p1. All instances that satisfy p2 but not p1. E.g. An agent finds p1 conflicting with its interpretation and wants to prevent instances of p1 from his query on p2. 4. Symmetric difference: p1 ∆ p2. All instances that satisfy either p1 or p2 but not both. E.g. An agent is interested in exploring the borderline scenarios in which p1 and p2 disagree.
Table 1 shows the analysis of these operations with respect to the relations holding between p1 and p2, which allows for the simplification of the result in some scenarios and shows the operation to be trivial or the empty set (ø) in others. In the table, two cases are marked with a star (*), the union of two disjoint precisifications and the complement of a precisification p1 subsumed in p2. Albeit legal, both operations are potentially problematic and the agent should be aware of the explicit inconsistencies between p1 and p2 in the former and the subsumption of p1 in p2 in the latter.
In the context of knowledge sharing, operations between precisifications are expected to be used for two main purposes, namely concept negotiation and specification of the standpoints. In practice, concept negotiation not only involves the analysis of the objective meaning and ontological commitments formalised on the different precisifications, but also of the real world implications of such commitments. This analysis is possible by querying the ontology with the previous operations, as the agent can gain insight on which instances fall within the borderline scenarios, which instances comply with all the desired precisifications and so on. In the case of GIS, this is enhanced because a spa-Table 2. Relations between the logical relations among two definitions, p1 and p2 (columns), and the RCC5 spatial relations between the projections of the instances satisfying those definitions, s1 and s2 (rows).
Logic definitions
The weakness of p1 with respect to p2 does not manifest in any instance.
The weakness of p2 with respect to p1 does not manifest in any instance.
The only instances lay in the intersection of p1 and p2.
The only instances of p2 lay in the intersection of p2 and p1.
The only instances of p1 lay in the intersection of p2 and p1.
Despite the definitions overlapping, there are no known instances on the intersection between them.
Impossible Impossible Impossible Impossible Expected tial projection is available, thus providing powerful means for understanding the way definitions perform in specific areas of interest. Moreover, once the process of concept negotiation is complete for a particular use of the ontology, operations can be used to support the formalization of the agent's standpoint s1. During further interactions of the agent with the ontology, queries will be associated with this standpoint. s1 acts by restricting the admissible precisifications of the ontology to those which are consistent with s1 and enabling reasoning and information retrieval within the subset.
In the case of geographical information, the spatial projection of the instances can be particularly relevant to complement the analysis of the relations that hold between precisifications. One might expect that the possible relations holding between precisifications (equivalence, subsumption, inverse subsumption, overlap and disjunction) would map to the analogous RCC5 relations (equal, proper part, inverse proper part, partial overlap and discrete) between the total spatial area covered by sets of instances satisfying each precisification. However, that does not necessarily need to be the case. Instead, relations between precisifications only restrict the space of possibilities but leave some variation open. In Table 2 we show the possible relations holding in the different scenarios. While the analogous to RCC5 is the expected relation to hold, other possibilities are allowed. These may be symptomatic of different circumstances. Take, for example, the case where the spatial projection of p1 is equal to that of p2. It may be the case that p1 and p2 are equivalent. p1 could also be weaker than p2 by allowing some more scenarios, but they never manifest in the available data about the state of the world. Even, it could be that p1 and p2 describe forest in a logically different way (e.g. one uses tree proximity and the other canopy cover) that is highly correlated, and therefore both map to the same objects. In other cases it may be due to mere exemplar clustering 6 . Finally, even when the expected spatial relation holds between the projections, it is expected to be useful for the agent to know how populated are the intersections or borderline scenarios of the definitions under consideration.
The diagrams in Figure 1 give some output examples from our prototype standpoint semantics based software, which enables visualisation of forest data in accordance with different standpoints. In Figure 1(a) we see forest extensions according to three different precisifications in different shades of green. The darker green corresponds to a stricter definition, which requires trees to be closer together to count as constituents of a forest. None of these precisifications consider the shrubs (small brown discs) to be forest constituents. In Figure 1(b) another precisification (demarcated by a black dashed line) is overlayed over the map. According to this precisification the shrubs are counted as forest constituents, so the forest takes on a very different shape, overlapping the extensions associated with the other precisifications.
It is widely agreed that, while domain specific ontologies can be used within a community of users for achieving a consensus about conceptualizing, structuring and sharing domain knowledge [29], it is unrealistic to expect that a fixed interpretation could satisfy the needs of different communities in real-world applications. In that scenario, semantic heterogeneity challenges the successful interoperation of systems and agents within decentralized environments. In this paper we have analysed the needs of the multidisciplinary forest community. The semantics of the key concepts of their ontology need to be negotiated depending on the context of use, and systems should be semantically rich to guarantee the sound reuse of information between different domains.
We present a novel approach, consisting on a framework based on 'Standpoint Semantics', that makes explicit the semantic variability of the terms of an ontology. We suggest that our framework can support agents with different interpretations and standpoints on their interoperation with a common ontology. Our aim is to provide a description of such a theory and to outline the particular features that can support the conceptual negotiation between agents.
Further work remains to be done to provide a fully operable standpoint semantics ontology, so, in a sense, this paper is only a preliminary step. However, we expect to open the possibility for a complementary approach to the research on semantic alignment in distributed systems. We consider that, in the current context, it is key to support ontologies that can both bring together the views of interdisciplinary domains and that are expressive enough to prevent reasoning and inference from what can be inconsistent interpretations. Moreover, we believe that the explicit acknowledgement of the semantic variability of the terms of the ontology is key for that purpose.