Motivation
Knowledge about the domain is essential to performing any task. Representations of this knowledge have ranged over a broad spectrum in terms of the features and tradeoffs. Recently, with the increased interest in deep neural networks, work has focussed on developing knowledge representations based on the kind of structures used in such networks. In this paper, we discuss some of the representational expressiveness tradeoffs that are made, often implicitly. In particular we focus on the loss of the ability to encode partial knowledge and explore two different paths to regain this ability.
Logic Based Representations
Beginning with McCarthy’s Advice Taker [1], logic has been the formal foundation for a wide range of knowledge representations. These have ranged across the spectrum of formal models to the more experimental languages created as part of working systems. The more formal work started with McCarthy and Hayes’s Situation Calculus [2], the various flavors of nonmonotonic logics [3], [4] and included proposed axiomatizations such as those suggested in the ‘Naive physics manifesto’ [5] and Allen’s temporal representation [6]. There were a number of less formal approaches, that also had their roots in logic. Notable amongst these include Minsky’s frames [7], EMycin [8], KRL [9] and others. The representation language used by Cyc, CycL [10] is a hybrid of these approaches.
Most of these representation systems can be formalized as variants of first order logic. Inference is done via some form of theorem proving. One of the main design issues in these systems is the tradeoff between expressiveness and inferential complexity.
More recently, systems such as Google’s ‘Knowledge Graph’ have started finding use, though their extremely limited expressiveness makes them much more like simple databases than knowledge bases.
Though a survey of the wide range of KR systems that have been built is beyond the scope of this paper, we note that there are some very basic abilities all of these systems have. In particular,

They are all relational, i.e., are built around the concept of entities that have relations between them

They can be updated incrementally

The language for representing ’ground facts’ is the same as the language for representing generalizations. There is no clear line demarcating the two.

They can represent partial knowledge. It is this feature we focus on in this paper.
Feature Vectors
One of the shortcomings of the logic based representations was that they largely assumed that everything the system knew was manually given to the system, i.e., learning from ground data was not a central part of the design. The complexity of the representational formalisms and wide range of possible functions or expressions have made machine learning rather difficult in traditional knowledge representation systems.
The rise of machine learning as the primary mechanism for creating models of the domain have lead to the use of much simpler representations. In particular, we notice,

The language for representing ground facts is distinct from the language for representing generalizations or models. The ground facts about the domain, i.e., the training data, is usually represented as a set of feature vectors.

The language for representing generalizations or models is also usually highly restricted. Each family of models (e.g., linear regression, logistic regression, support vector machines, has a function template with a number of parameters that the learning algorithm computes. Recent work on neural networks attempts to capture the generality of Turing machines with deep networks, but the structure of the function learnt by these systems is still uniform.

The language for representing ground facts is propositional, i.e., doesn’t have the concept of entities or relations. This constraint makes it very difficult to use these systems for modeling situations that are relational. Many problems, especially those that involve reasoning about people, places, events, etc. need the ability to represent these entities and the relations between them.

Most of these systems allow for partial knowledge in their representation of ground facts. i.e., some of the features in the training data may be missing for some of the instances of the training data.
The inability of feature vectors to represent entities and relations between them has lead to work in embeddings, which try to represent entities and relations in a language that is more friendly to learning systems. However, as we note below, these embedding based representations leave out an important feature of classical logic based representations — a feature we argue is very important.
We first review embedding based representations, show how they are incapable of capturing partial information
Embeddings
Recent work on distributed representations [
[11], [12], [13], [14], [15]] has explored the use of embeddings as a representation tool. These approaches typically ’learn an embedding’, which maps terms and statements in a knowledge base (such as Freebase [16]) to points in an Ndimensional vector space. Vectors between points can then be interpreted as relations between the terms. A very attractive property of these distributed representations is the fact that they are learnt from a set of examples.Weston, Bordes, et. al. [13] proposed a simple model (TransE) wherein each entity is mapped to a point in the Ndimensional space and each relation is mapped to a vector. So, given the triple , we have the algebraic constraint , where is an error term. Given a set of ground facts, TransE picks coordinates for each entity and vectors for each relation so as to minimize the cumulative error. This simple formulation has some problems (e.g., it cannot represent many to many relations), which has been fixed by subsequent work ([17]). However, the core representational deficiency of TransE has been retained by these subsequent systems.
The goal of systems such as TransE is to learn an embedding that can predict new ground facts from old ones. They do this by dimensionality reduction, i.e., by using a low number of dimensions into which the the ground facts are embedded. Each triple is mapped into an algebraic constraint of the form and an optimization algorithm is used determine the vectors for the objects and relations such that the er ror is minimized. If the number of dimensions is sufficiently large, the s are all zero and no generalizations are made. As the number of dimensions is reduced, the objects and relation vectors get values that minimize the εs, in effect learning generalizations. Some of the generalizations learnt may be wrong, which contribute to the nonzero s. As the number of dimensions is reduced further, the number of wrong generalizations increases. This is often referred to as ‘KB completion’.
We believe that the value of such embeddings goes beyond learning simple generalizations. These embeddings are a representation of the domain and should be usable by an agent to encode its knowledge of the domain. Further, any learning task that takes descriptions of situations that are best represented using a graph now has a uniform representation in terms of this embedding. When it is used for this purpose, it is very important that the embedding accurately capture what is in the training data (i.e., the input graph). In such a case, we are willing to forgo learning in order to minimize the overall error and can pick the smallest number of dimensions that accomplishes this. In the rest of this paper, we will focus on this case.
Ignorance or Partial Knowledge
In a logic based system that is capable of representing some proposition (relational or propositional), it is trivial for the system to not know whether is true or not. I.e., its knowledge about the world is partial with respect to .
However, when a set of triples is converted to an embedding, this ability is lost. Consider a KB with the entities , , , , . It has a single relation . The KB specifies that and are friends and that and are friends and that and are friends. It does not say anything about whether and are friends or not friends. This KB can be said to have partial knowledge about the relation between and . When this KB is converted into an embedding, to represent the agent’s knowledge about the domain, it is important that this aspect of the KB be preserved.
Unfortunately, in an embedding, and are assigned particular coordinates. Either is equal to or it is not. If it is, then, according to the embedding, they are friends and if it is not, then, according to the embedding, they are not friends. The embedding is a complete world, i.e., every proposition is either true or false. There is no way of encoding ’unknown’ in an embedding.
If the task is knowledge base completion, especially of an almost complete knowledge base, then it may be argued that this deficiency is excusable. However, if the KB is very incomplete (as most real world KBs are) and if such as KB is being used as input training data, or as the basis for an agents engagement with the world, this forced completion could be problematic.
We now explore two alternatives for encoding partial knowledge in embeddings.
Encoding Partial Knowledge
Logic based formalisms distinguish between a knowledge base (a set of statements) and what it might denote. The object of the denotation is some kind of structure (set of entities and ntuples in the case of first order logic or truth assignments in the case of propositional logic). The KB corresponds not to a single denotation, but set of denotations. Something is true in the KB if it holds in every possible denotation and false if it does not hold in any of the possible denotations. If it holds in some denotations and does not hold in some, then it is neither true nor false in the KB.
Ensemble of Embeddings
In other words, the key in logic based systems to partial knowledge is the distinction between the KB and its denotation and the use of a set of possible denotations of a KB.
Note that in logic based KR systems, these possible denotations (or possible worlds) are almost always in the metatheory of the system. They are rarely actually instantiated. We could also imagine a KR system which does instantiate an ensemble (presumbaly representative) of possible denotations and determine if a proposition is true, false or unknown based on whether it holds in all, none of some of these models. We follow this approach with embeddings to encode partial knowledge. Instead of a single embedding, we can use an ensemble of embeddings to capture a KB. If we use a sufficient number of dimensions, we should be able to create embeddings that have zero cumulative error. Further, different initial conditions for the network will give us different embeddings. These different embeddings correspond to the different possible denotations. A ensemble of such embeddings can be used to capture partial knowledge, to the extent desired.
While this approach is technically correct in some sense, it also defeats the purpose. The reason for creating the embeddings was in part to create an encoding that could be used as input to a learning system. When we go from a single embedding to an ensemble of embeddings, the learning algorithm gets substantially complicated.
One approach to solving this problem is to develop a more compact encoding for an ensemble of embeddings. Remember that we don’t need to capture every single possible embedding corresponding to the given KB. All we need is sample that is enough to capture the essential aspects of the partiality of the KB.
Aggregate Models
Consider the set of points across different models corresponding to a particular term. Consider a cluster of these points (from an ensemble of embeddings) which are sufficiently close to each other. This cluster or cloud of points (each of which is in a different embeddingl), corresponds to
aggregate of possible interpretations of the term. We can extend this approach for all the terms in the language. We pick a subset of models where every term forms such a cluster. The set of clusters and the vectors between them gives us the aggregate model. Note that in vectors corresponding to relations will also allow amount of variation. If a model satisfies the KB, any linear transform of the model will also satisfy the KB. In order to keep these transforms from taking over, no two models that form an aggregate should be linear transforms of each other.
In both aggregate models, each object corresponds to a cloud in the Ndimensional space and the relation between objects is captured by their approximate relative positions. The size of the cloud corresponds to the vagueness/approximateness (i.e., range of possible meanings) of the concept.
Partial knowledge is captured by the fact that while some of the points in the clouds corresponding to a pair of terms may have coordinates that maps to a given relation, other points in the clouds might not. In the earlier example, we now have a set of points corresponding to and . Some of these are such that , while others are such that . Thus, these aggregates can encode partial knowledge.
Conclusions
The ability to encode partial knowledge is a very important aspect of knowledge representation systems. While recent advances to KR using embeddings offer many attractions, the current approaches are lacking in this important aspect. We argue that an agent should be aware of what it doesn’t know and should use representations that are capable of capturing this. We described one possible approach to extending embeddings to capture partial knowledge. While there is much work to be done before embeddings can be used as practical knowledge representation systems, we believe that with additions like the one described here, embeddings may turn out to be another useful addition to the knowledge representation tool chest.
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