The symbol grounding problem refers to the necessity of grounding abstract symbolic knowledge (such as encoded in formal ontologies) in the real world through perception and action. The cognitive framework of conceptual spaces provides a potential way for solving the symbol grounding problem by proposing an intermediate representational layer: Concepts are represented by regions in low-dimensional similarity spaces, which are in turn grounded in subsymbolic processing. Logic tensor networks provide a general mechanism for learning membership functions in the presence of both bottom-up information (i.e., training examples) and top-down constraints in the form of logical rules. In this paper, we propose to combine logic tensor networks with conceptual spaces in order to ground predicates from the symbolic layer in conceptual regions while taking into account logical constraints from abstract background knowledge. We discuss several potential membership functions for concepts and argue that this approach can be used to provide a cognitive grounding for formal ontologies.
Formal ontologies provide us with ways of encoding knowledge in a logical format. Large-scale
applications such as Google’s knowledge graph1 illustrate the usefulness of such a structured
representation for encoding entities, classes, and their respective relations. However, ontologies
as used in current technical systems sufer from the symbol grounding problem [
Deep neural networks have become the predominant approach in many machine learning
tasks, including the areas of computer vision [
Neural-symbolic integration [
The cognitive framework of conceptual spaces [
Logic tensor networks [
In this paper, we propose to use logic tensor networks to ground ontologies in conceptual
spaces. Since conceptual spaces are based on meaningful dimensions, this aids the
interpretability of the resulting embedding. Conceptual spaces can be grounded in psychological dissimilarity
ratings [
The remainder of this paper is structured as follows: In Section 2, we describe both conceptual spaces and logic tensor networks in more detail. In Section 3, we then argue why a combination of these two frameworks seems promising and discuss possible implementations of diferent membership functions. Section 4 then concludes this paper.
In the following, we will first give a general overview of the conceptual spaces framework (Section 2.1), before introducing logic tensor networks (Section 2.2).
A conceptual space as proposed by Gärdenfors [
The overall conceptual space can be structured into so-called domains, which represent, for
example, diferent perceptual modalities such as color, shape, taste, and sound. The color domain,
for instance, can be represented by the three dimensions hue, saturation, and brightness, while
the sound domain is spanned by the dimensions pitch and loudness. Based on psychological
evidence [
Gärdenfors defines properties like red, round, and sweet as convex regions within a single domain (namely, color, shape, and taste, respectively). A property thus corresponds to a set of observations from a single perceptual modality. Concept hierarchies are an emergent property of this spatial representation: If the sky blue region is a subset of the blue region, this implicitly encodes that sky blue is a special shade of blue. Based on properties, Gärdenfors now defines full-fleshed concepts like apple or dog by using one convex region per domain, a set of salience weights (which represent the relevance of the given domain to the given concept), and information about cross-domain correlations. The apple concept may thus be represented by the regions red, sweet, and round in the domains of color, taste, and shape, respectively.
There are in principle three ways of obtaining the dimensions of a conceptual space [9, Sections 1.7, 1.9, and 6.5]: Firstly, if the domain of interest is well understood, one can manually define the dimensions of the similarity space.
A second approach is based on machine learning algorithms for dimensionality reduction. For instance, unsupervised artificial neural networks (ANNs) such as autoencoders or selforganizing maps can be used to find a compressed representation for a given set of input stimuli. This task is however solved by minimizing a mathematical error function which seems to be not satisfactory from a psychological point of view.
A third popular way of obtaining a conceptual similarity space is based on psychological
dissimilarity ratings. These dissimilarity ratings are collected for a fixed set of stimuli in a
psychological experiment. They are then converted into an -dimensional geometric
representation of the stimulus set by using a technique called “multidimensional scaling” (MDS),
which ensures that geometric distances between pairs of stimuli reflect their psychological
dissimilarity [
Recently, a hybrid approach has been proposed [
Gärdenfors argues that the convexity requirement relates conceptual spaces to the prototype
theory of concepts [
Since conceptual spaces can be interpreted as an intermediate layer of representation between the traditional symbolic and subsymbolic layers, they can also help to solve the symbol grounding problem: Individual observations, which correspond to high-dimensional activation vectors in the subsymbolic layer, are represented by points in the lower-dimensional conceptual space and can be mapped onto constants and variables from the symbolic layer. Predicates from the symbolic layer (such as apple and red) can be mapped onto concepts and properties in the conceptual layer. The symbols from the symbolic layer can therefore be indirectly grounded in subsymbolic perception through the conceptual layer.
Conceptual spaces in their original formulation focus mostly on concepts that can be defined based on perceptual properties. Relations between objects can be represented using product spaces, for instance by defining longerThan as a convex region in R+ × R+ (where each dimension represents the length of one individual object) [9, Section 3.10.1]. Relational concepts like robber or seat can be represented based on their respective roles in events like robbing and sitting (which involve agent, patient, theme, action, and result) [24, Sections 6.7 and 8.4].
Logic Tensor Networks (LTNs) [
Logic Tensor Networks integrate knowledge representation, learning, and reasoning using
a diferentiable fuzzy first-order logic language called “Real Logic”. Real Logic is a first-order
language containing constant symbols (representing individual observations such as Bob or
Paris), function symbols (representing mappings between observations, e.g., homeTownOf ),
predicate symbols (representing concepts and relations such as lawyer and livesIn), and variable
symbols (representing lists of observations) [
In order to relate the semantics of the logical language to actual data points, Real Logic
makes use of a so-called grounding function , which maps terms (i.e., constants, variables,
and results of function applications) onto points in a feature space, and both functions and
predicates onto neural networks. The networks implementing predicates are required to return
a value from the interval [
This is done as follows (see Figure 1): First, all terms (such as Bob) are grounded into vectors,
and all function and predicate symbols (such as homeTownOf an livesIn) are grounded into
their respective neural networks. The structure of the symbolic formula then determines which
neural network is applied to which feature vector. If predicates such as married(, ) are applied
to variables such as = (Bob, John) and = (Alice, Mary, Susan), the resulting grounding is a
matrix containing the degree of truth for each possible combination of observations [
In order to ground logical connectives such as ∧, ∨, ¬, and →, the corresponding operators
from fuzzy logic are used. Badreddine et al. [
Since Real Logic is a first-order language, it also needs to provide a grounding for the universal
and the existential quantifier. Mapping ∀ to the minimum over all entries of the variable may
be a straightforward choice, but does not tolerate exceptions and may thus not be suitable for
real-life applications, if one assumes a certain amount of noise both in the background
knowledge and in the empirical data (e.g., incorrect labels) [
3Note that 1 corresponds to the standard arithmetic mean and − 1 to the harmonic mean. If = (1, . . . , ) is a diference vector, then () is equivalent to a Minkowski metric.
Satisfiability specifies the degree to which a grounding satisfies a given set of formulas
by simply aggregating the truth values of all formulas ∈ [
The knowledge represented in logic tensor networks consists of both the formulas in the
logical language (corresponding to symbolic top-down information) and the grounding
obtained from observations (corresponding to subsymbolic bottom-up information) [
Learning in logic tensor networks takes place through gradient descent on the parameter
values of the grounding in order to maximize the satisfiability of the overall set of formulas
[
Since LTNs unify subsymbolic machine learning aspects with symbolic logical constraints,
they can be applied to a variety of problems. Badreddine et al. [
Logic tensor networks ofer the possibility to combine bottom-up information in the form of training examples with top-down information in the form of general rules. They thus make an ideal candidate for closing the gap between the conceptual and the symbolic layer.
In Section 3.1, we show how logic tensor networks can reflect the general properties of the conceptual spaces framework. Afterwards, we investigate diferent membership functions, using a distinction into partitional (Section 3.2) and nonpartitional (Section 3.3) approaches.
The knowledge approach to concepts from psychology [34, Chapter 6] emphasizes the crucial
role of world knowledge in the learning and application of concepts and can thus be linked
to both formal ontologies and the influence of logical rules on the learning process in LTNs.
Moreover, LTNs take into account information about points in a feature space, and the grounding
of predicates usually gives rise to a membership function with one or more receptive fields.
LTNs can therefore also be related to the prototype [
These observations and the combination of bottom-up and top-down information make LTNs quite interesting from the perspective of conceptual spaces: If we use a conceptual space as a feature space, then the LTN can implement a two-way interaction between the conceptual and the symbolic layer. Just as with the conceptual spaces framework, observations can be represented by points and concepts can be represented as regions in the feature space. Moreover, LTNs are able to encode the domain structure of a conceptual space and they use a similar way of encoding simple relations as regions in a product space. Finally, the usage of fuzzy sets and fuzzy logic allows us to represent vague conceptual boundaries.
How exactly can we apply LTNs to conceptual spaces? Both properties and concepts can be represented by predicates with a convex membership function. While properties refer only to a single domain, concepts are defined on a concatenation of domains. We can require that the predicate red() is defined on the three-dimensional color domain, while the predicate apple( = (, , )) involves the domains of color, taste, and shape. Individual observations can then be represented by one point per domain. Function symbols could potentially be used to represent actions and changes: Applying a function symbol like lift could for instance translate into a simple vector addition in the location domain that increases the altitude of the given object. Finally, basic relations are defined by considering regions in product spaces of multiple domains. In order to represent more complex relational knowledge, one could try to use the event structure proposed by Gärdenfors [24, Chapter 9].
An advantage of using logic tensor networks for grounding ontologies in conceptual spaces is their large variety of inference and learning methods. In addition to learning conceptual regions from observations, they can for instance also generate an embedding of an unobserved object based on a symbolic description. For example, “object is a red apple” can be translated into a point in the conceptual space by maximizing the satisfiability of apple() ∧ red(). This spatial representation of object can then in turn be used to make further inferences, for instance about the taste domain (e.g., by evaluating sweet()). Thus, the geometric embedding can give rise to common-sense inferences not easily realizable within the symbolic layer.
Moreover, LTNs do not only provide an embedding of entities and classes, but they are also able to enforce the validity of general rules, which may reduce the required number of training examples. This makes them especially attractive for bridging the conceptual and the symbolic layer, since they can harness the whole expressivity of formal ontologies in order to guide the machine learning process. This also related to embodied and enactivist approaches to cognition [36], which assume that top-down information strongly influences bottom-up perception and conceptualization of the environment. One may furthermore speculate that the enforcement of general logical rules can help to prevent catastrophic interference, where continued learning causes a neural network to forget previously learned knowledge [37].
While LTNs have already been used in the context of ontologies [
Finally, we would like to mention the recent work by Singh et al. [38], who combine the computational power of deep ANNs with a psychological model of categorization. Their endto-end model learns both a similarity space and a prototype-based categorization model at once. One could envision a similar application of LTNs: The input domain contains raw images, which are then mapped by a function symbol (implemented as deep neural network) into a low-dimensional conceptual space. In this conceptual space, one can then define membership functions for the diferent concepts under consideration. The whole system could then be trained based on labeled examples, but also using additional background knowledge based on human similarity ratings and general rules from the symbolic layer. In the terms of cognitive psychology, this would result in a combination of prototype theory (represented by convex membership functions) with the knowledge view on concepts (represented by the presence of constraints from background knowledge), spanning all three layers of representation.
When viewed from the perspective of cognitive science, logic tensor networks can of course not be labeled as a cognitively plausible learning mechanism: They rely on batch-processing large amounts of (typically labeled) data with gradient descent. One can of course argue that LTNs are not used to model the human concept acquisition process itself, but rather to take a shortcut to the resulting concept inventory. However, it would certainly also be interesting to extend LTNs such that they can work in an incremental way. A potential example application in this context are language games [39], where a population of agents needs to negotiate a common conceptualization of the world, receiving only indirect feedback through the success or failure of their interactions.
In the following, we will take a look at diferent membership functions from the conceptual spaces literature and discuss their applicability in logic tensor networks. For illustration purposes, we will consider a one-dimensional conceptual space with two concepts 1 and 2 as well as three data points 1, 2, 3, which are supposed to belong to 1, but not to 2. Since the parameters of the membership functions are optimized through gradient descent, we will especially focus on their derivatives.
Let us first consider membership functions which partition the underlying conceptual space, i.e., which aim to assign each point to exactly one concept. We start with Gärdenfors’ approach of identifying concepts with a prototypical point and creating a Voronoi tessellation of the space [9, Section 3.9]: One starts from a set of prototypical points 1, . . . , for the concepts under consideration. Each point in the conceptual space is then assigned to its closest prototype based on the distances (, ). As Gärdenfors [9, Section 3.9] argues, a Voronoi tessellation based on the Euclidean metric partitions the overall space into convex regions.
Since the Voronoi tessellation gives us a partitioning of the overall space, the membership function of each concept is constant almost everywhere and undefined on the border line to a neighboring conceptual region (see Figure 2a). Therefore, the derivative of this membership function with respect to any variable is either zero or undefined, which is highly problematic for gradient descent. For example, consider the point 3, which is currently misclassified as belonging to 2 instead of 1. In gradient-based optimization, the prototypes 1 and 2 are updated by computing the derivative (3) and then slightly increasing or decreasing the value of , depending on the sign of the derivative and whether we want to increase or decrease (3). In the case of Figure 2a, we however note that both derivatives are zero – small changes to 1 and 2 do not result in any changes to (3). Thus, gradient descent is incapable of making any update to the prototypes.
In order to make gradient-based learning possible, we need a soft version of the Voronoi approach. We can express the classification decision of the Voronoi tessellation as follows (where > 0 is a sensitivity parameter):
= (, ) = (− · (, )) Instead of the crisp function (which results in flat membership values), we can now apply the so called softmax function, which is commonly used in neural networks to provide an output probability distribution over a set of mutually exclusive classes: () = ∑︀ Here, () gives the probability for class , given a vector of raw confidence values . If we combine this with the Voronoi tessellation approach, we obtain a soft Voronoi tessellation with the following membership function: − · (,) () = (− · (, )) = ∑︀ − · (,) As we can see, the numerator reflects the semantic similarity of and , while the denominator is the sum over all similarities to all prototypes. The resulting membership function can thus also be interpreted as a normalized version of semantic similarity. Figure 2b illustrates this membership function: We now have a continuous transition from high membership values to low membership values. Moreover, the derivative of this membership function is defined on the whole conceptual space and nonzero in all cases - even points such as 1 have a very small, but nonzero derivative.
We should highlight at this point that we assume that the same sensitivity parameter is used for all concepts. If we allow diferent values 1 ̸= 2, we can control the size of the respective conceptual regions (smaller values of leading to larger regions). However, these diferent values may cause some unintended efects. For instance, Figure 3a illustrates the case where 1 ≫ 2, which causes the membership function 2() to be no longer convex.
Generalized Voronoi tessellations [9, Section 4.9] allow to encode diferently sized conceptual regions by considering prototypical regions instead of prototypical points . These prototypical regions are usually represented as disks with a central point and a radius . Based on these prototypical regions, one can now generate a generalized Voronoi tessellation by assigning each point in the conceptual space to the concept whose prototypical region is closest. In the case of disks, this corresponds to finding the concept for which (, ) = max(0, (, ) − ) is smallest. Concepts with larger prototypical regions (as reflected through a larger value of ) thus result in larger conceptual regions in the generalized Voronoi tessellation (see Figure 2c). Again, by using the instead of the function, this can be generalized to a soft notion of concept membership (see Figure 2d).
Also Douven et al. [40] consider prototypical regions instead of prototypical points. However, they create all possible Voronoi diagrams by picking a single point ∈ for all prototypical regions . These individual Voronoi tessellations are then aggregated into a so called “collated Voronoi diagram”: A point is assigned to concept if and only if it has been assigned to in all individual Voronoi diagrams. Douven et al. identify borderline cases as points that belong to diferent conceptual regions for diferent Voronoi diagrams. These borderline points are not assigned to any concept and represent vagueness in concept boundaries. In Figure 2e, we again note that the derivative is zero within the conceptual regions and undefined in the border area.
Decock and Douven [41] extend the work of Douven et al. [40] by providing a degree of 1 ≫ 2. (b) Workaround for zero gradient inside prototypical regions. membership for borderline cases. They define the membership of a point to a concept as the fraction of individual Voronoi diagrams for which belongs to the conceptual region of . Decock and Douven note that if the prototypical regions have an infinite number of points, then the membership function is s-shaped (cf. Figure 2f). However, we can observe that the membership function is flat for large parts of the conceptual space, namely, for all non-borderline points. This is again highly problematic for gradient descent.
The usage of Voronoi tessellations for conceptual spaces has not been without challenge in the literature. For instance, Lewis and Lawry [42] argue that partitioning the conceptual space may be adequate for individual domains such as color, but that it is not suitable for a combination of multiple domains. It seems implausible that every single point in a high-dimensional space has to be assigned to exactly one category: On the one hand, some regions in the overall conceptual space may not be covered by any existing concept. Points in such regions should be recognized as outliers or members of a novel, previously unknown category. On the other hand, conceptual regions may also overlap, for instance in order to represent conceptual hierarchies.
Lewis and Lawry [42] have also made a general proposal for nonpartitional membership functions: A point in the conceptual space is said to belong to concept if its distance to the prototypical region is not greater than a threshold distance . Lewis and Lawry assume that the threshold is not known, but that a probability distribution over its possible values is available. The degree of membership of a point to a concept is then given by the probability of (, ) being smaller than : () = P ((, ) ≤ ) =
( ) ∫︁ ∞ (,) Lewis and Lawry are in general open to diferent forms for the probability distribution . If we use ( ) = · − · , then concept membership reflects similarity to the prototypical region: () = ∫︁ ∞ (,) · − · = [︀ − − · ]︀ →∞ =(,) = 0 − ︁( − − · (,)︁) = − · (,) The shape of the resulting similarity function for = {} (i.e., prototypical points) is illustrated in Figure 4a. As we can see, all points in the similarity space receive a non-zero membership value. Moreover, the derivative of the membership function is defined for all points except for the prototypes 1 and 2. In practical applications, this theoretical shortcoming can be overcome by defining the derivative in this point to equal zero. Furthermore, we are able to control the size of the conceptual regions by choosing diferent sensitivity parameters 1 ̸= 2.
However, we can also note that the derivative of the membership function is proportional to the membership value itself: The largest derivatives are observed for the points with the highest membership in the concept. Since gradient descent algorithms typically take into account not only the direction, but also the magnitude of the gradient, this can lead to undesired efects. Consider for instance the point 2 in Figure 4a, which has a fairly high membership to 1. The derivative 1(2) is quite large and will thus cause the gradient descent algorithm to increase 1 1 considerably. In the resulting configuration, 1(2) may however be smaller than before, since 1 may have moved considerably past 2. This seems to be a major shortcoming of this similarity-based approach to concept membership.
The examples by Lewis and Lawry [42] often make use of uniform distributions ( ) = (0, ). As we can see in Figure 4b, the membership curve has in this case a triangular shape and its derivative is therefore constant for all points with a partial membership. However, both concept membership and its derivative are zero for most parts of the similarity space.
These considerations can of course also be generalized to prototypical regions , which subsumes our own formalization of the conceptual spaces framework [43, 44, 45]: There exists a well-defined region with full membership, which in our case is based on the union of axis-aligned cuboids. Membership is then defined as similarity to this prototypical region.
In Figure 4c, we can see two problems with this approach: On the one hand, we again have the problem of large derivatives for large membership values as already discussed for Figure 4a. On the other hand, the membership function is constant for all points in the prototypical region, hence, the derivative is zero. If an observation such as 3 is confidently misclassified as belonging to 2, then gradient descent is not able to move 2 away from 3.
The problem of a zero derivative could be circumvented as follows: We define a new membership function ′() := (1 − ) · () for some small > 0. Furthermore, we identify the central point ∈ . The membership value for ∈ is then increased based on its distance to , such that ′() = 1 and that ′() = 1 − for points on the border of . This provides a small slope for the membership function inside the prototypical region and thus a nonzero derivative (cf. Figure 3b). However, it remains to be seen whether such a workaround is useful in practice.
Despite these shortcomings, there are however reasonably strong arguments for using a membership function like the one proposed in our formalization: Firstly, by using a union of axis-aligned cuboids, our formalization is able to represent correlations between domains. This is an important aspect of human conceptualization [46, 47] which is not captured by any of the aforementioned approaches. Secondly, one can apply a variety of operations defined in the context of our formalization in order to reason on the learned concepts. For instance, relations such as conceptual similarity and conceptual betweenness are not defined in LTNs, but they become immediately available with the use of our proposed formalization of concepts. Thirdly, logical formulas in LTNs always have to be evaluated on a set of data points which requires that one keeps all examples in memory. Our formalization on the other hand provides closed formulas for computing the validity of such logical formulas – the original data points are not needed any more and the computation can potentially be faster. However, the operations defined in our formalization are based on the minimum norm, while LTNs are commonly used with the product norm. Therefore, the numeric results of the computations might difer.
Motivated by the problem of large gradients for large membership values, we also consider multivariate Gaussian functions, whose membership value can be defined as follows with a symmetric, positive semi-definite matrix Σ :
() = − 21 (− ) Σ− 1(− )
Figure 4d illustrates the usage of such Gaussian functions in our one-dimensional similarity space.4 As one can see, this type of membership function does not sufer from the gradient size problem as identified in Figure 4a: The derivative is small both for points with a very low and for points with a very high membership. It is largest for points with an intermediate level of membership, i.e., points that are currently treated as borderline members. Another advantage of multivariate Gaussian functions is that they are able to encode correlations between dimensions as well as diferent distribution widths through their covariance matrix Σ .
2 4We can model this with ( ) = 2 · − 2 2 in the one-dimensional case using the approach by Lewis and Lawry [42]
However, the usage of Gaussians in the context of conceptual spaces is somewhat unsatisfactory from a theoretical standpoint. The notion of similarity is not based on the Euclidean distance (, ) = √︀∑︀( − )2, but on the squared Mahalanobis distance (, ) = √︀( − ) Σ − 1( − ). Applying the Mahalanobis distance corresponds to transforming the similarity space with the covariance matrix, and then computing the Euclidean metric in the transformed space. This implicit transformation of the similarity space would in our opinion cause a major modification of the original framework. Nevertheless, the simplicity and computational attractiveness of multivariate Gaussians make them an interesting candidate for experimental investigations, such that one should not hastily dismiss them.
In this paper, we have introduced both conceptual spaces and logic tensor networks. We have argued that a combination of the two frameworks is a promising direction of research: Conceptual spaces can provide a grounding for the feature spaces considered in logic tensor networks and allow us to use relatively simple membership functions for representing predicates. Logic tensor networks on the other hand can help us to bridge the gap between the conceptual and the symbolic layer by learning concepts not only based on labeled examples, but also based on general logical top-down constraints. Moreover, the resulting system can provide a cognitive grounding for formal ontologies: Individual concepts from the ontology can be grounded in regions of a conceptual space, whose dimensions are grounded in psychological data and/or perceptual sensor information. Moreover, this grounding can take into account the most important part of ontologies, namely the relations between concepts. Since the envisioned system unifies both bottom-up and top-down processes, the information from the conceptual layer can furthermore give rise to additional rules for the symbolic ontology.
Moreover, we have also discussed several possible membership functions for concepts in conceptual spaces and their applicability to gradient-based optimization methods. If we consider partitional approaches, a soft version of generalized Voronoi tessellations seems to be most promising: It is capable of representing conceptual regions of diferent sizes and comes with a derivative that is guaranteed to be non-zero everywhere. If we are however interested in nonpartitional membership functions, multivariate Gaussians seem to be preferable from a computational point of view: They are able to explicitly encode correlations between dimensions, they can take into account concepts of varying size, and they provide a meaningful non-zero gradient everywhere. Nevertheless, also the membership function of our own formalization of the conceptual spaces framework should be explored, since it provides us with a large number of operations for downstream reasoning processes.
Our proposal has so far been only a theoretical one. In order to evaluate its actual merit, practical experiments need to be conducted. Ideally, these experiments should consider all membership functions discussed in this paper in order to confirm or refute our theoretical analyses. There are several data sets that can serve as test beds for a first study, including the conceptual spaces extracted by Banaee et al. [48] and Derrac and Schockaert [49], as well as the robotics data set by Spranger et al. [50]. Since the strength of LTNs stems from their ability to incorporate top-down rules to compensate for scarce training data, especially the movie spaces from Derrac and Schockaert [49] are relevant: Each movie is annotated with its genres, a set of plot keywords, at its age restriction. Using techniques such as the apriori algorithm [51], one can extract rules from the co-occurrence statistics of the labels and then simulate few-shot learning [52] by showing only a small part of the available examples, but providing the general rules as additional constraints to the system. After such initial experiments, studies with actual ontologies are needed in order to ensure that all relevant pieces of ontological information (especially relations of varying complexity) can be adequately encoded by the proposed approach. Rules and Reasoning, Springer International Publishing, Cham, 2019, pp. 161–170. [30] S. Auer, C. Bizer, G. Kobilarov, J. Lehmann, R. Cyganiak, Z. Ives, DBpedia: A Nucleus for a Web of Open Data, in: K. Aberer, K.-S. Choi, N. Noy, D. Allemang, K.-I. Lee, L. Nixon, J. Golbeck, P. Mika, D. Maynard, R. Mizoguchi, G. Schreiber, P. Cudré-Mauroux (Eds.), The Semantic Web, Springer Berlin Heidelberg, Berlin, Heidelberg, 2007, pp. 722–735. [31] I. Donadello, L. Serafini, A. d’Avila Garcez, Logic Tensor Networks for Semantic Image Interpretation, in: Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17), 2017. [32] B. Wagner, A. d’Avila Garcez, Neural-Symbolic Integration for Fairness in AI, in: A. Martin, K. Hinkelmann, H.-G. Fill, A. Gerber, D. Lenat, R. Stolle, F. van Harmelen (Eds.), Proceedings of the AAAI 2021 Spring Symposium on Combining Machine Learning and Knowledge Engineering (AAAI-MAKE 2021), 2021. [33] S. Badreddine, M. Spranger, Injecting Prior Knowledge for Transfer Learning into Reinforcement Learning Algorithms using Logic Tensor Networks (2019). arXiv:1906.06576. [34] G. Murphy, The Big Book of Concepts, MIT Press, 2002. [35] D. L. Medin, M. M. Schafer, Context Theory of Classification Learning, Psychological
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