Humans are continually acquiring, representing, and reasoning with new facts about the world. To make sense of the vast quantity of information with which we are presented, we must compress, structure and generalize from what we experience. This allows us to quickly understand new concepts and make useful predictions about them. For example, we might represent our knowledge of animals in a taxonomic hierarchy like that shown in Figure 1. Using such a hierarchy coupled with an inheritance rule that specifies that the attributes of higher nodes are shared by lower ones, we can achieve exponential compression over a representation which just lists the facts. Even more exciting, it allows us to infer a whole range of new facts about an individual simply by observing where it fits in the hierarchy. For example, observing that a Harpy Eagle is a type of Eagle allows us to immediately deduce that a Harpy Eagle can fly and breathe.111There are some kinds of reasoning which are not easy to do with a taxonomy (for example, handling the exception that penguins are birds but don’t fly) but our proposal is not limited to taxonomic representations. But how can such representations be learned from raw observations? This has been a key problem in semantic knowledge acquisition going back to at least to the 1960’s in the work of collins69 collins69, with symbolic, Bayesian, and neural approaches proposed [Rogers and McClelland2004, Hinton1986, Yarden et al.2008]. In our view (and following katz2008 katz2008) there are three questions to be addressed in the development of a solution: (1) how can we induce logical rules from the observations? (2) how can we learn a small set of core facts (the taxonomy in the example) from which we can infer the observations (and more), and (3) how can this be done without explicit supervision on the structure of the rules?
In this paper we propose a model which can be used for both Inductive Logic Programming (ILP) and theory acquisition/compression. The network is neuro-symbolic in the sense that it represents predicates for both rules and facts using dense vectors which can be trained using gradient descent towards representations of known predicates (including a fixed set of anonymous invented predicates). The network implements forward chaining and soft unification.222Soft unification relaxes the requirement that two predicate symbols must be identical for the rule to be applicable instead favoring a measure of the degree of similarity.
For ILP problems, the network is given a set of “proto-rules” (rules with randomly initialized predicate parameters) and applies them using forward chaining to the background facts to produce consequent facts. After steps of forward chaining ( is a hyper-parameter) the consequent facts are compared to the labeled target facts and the rule predicate parameters are trained towards representations of the predicates which yield all the true target facts and none of the false ones. Representations can be learned either for the known predicates used in the facts or for auxiliary invented predicates.
In theory acquisition/compression, the network is given a set of fact observations and asked to learn a logical theory – a set of core facts and a set of rules – which together entail the observations. The ability to learn facts is an aspect that has not been emphasized in many ILP approaches but is present in the Bayesian literature. For example, when observing that salmon can swim, have fins and have gills, the model can learn the core fact that salmon are fish even though that is not deducible directly. By encouraging sparsity in the set of core learned facts with a penalty term in the loss, the model can be trained to try to minimize the size of the theory it learns.
The remainder of the paper is structured as follows. In the next section we provide background material and related work. Then we present our model and training procedure. In Section 4 we present experiments which investigate different capabilities of the model and demonstrate the efficacy of our approach on a variety of ILP rule induction and domain theory learning datasets. We conclude with a discussion of limitations and future directions.
Background and Related Work
There is a rich literature on neuro-symbolic induction to which our approach is related on two main lines: inductive logic programming (ILP) and semantic cognition. ILP systems try to learn a set of logical rules which can be used to deduce facts of interest while semantic cognition is broadly concerned with how human beings acquire, represent, and integrate knowledge.
Inductive Logic Programming
In ILP, the goal is to learn (induce) logical rules which can be chained to successfully answer queries about a target relation, given positive and negative examples of that relation and some background facts. Logical rules are of the form:
where is an atom called the head of the rule and are atoms which constitute the body. An atom is a positive or negative literal. A literal is a predicate applied to terms which may, in our case, be either variables or constants. For example is a (positive) atom whose predicate is and whose arguments (two in this case) are variables and . When the arguments of the atom are all constants (e.g. for constants and ) we call it a ground atom which we also refer to as a fact when it is given as true or its truth value is inferred from rules. Intuitively, the head of the rule is true if each of the in the body are. For example a rule might be:
Which is read: is the grandfather of if is the father of AND is the parent of . The head atom holds for any and as long there is some individual for which and are both true facts.
Given background facts such as and , logical rules can be chained together to prove a goal fact like . This is an example of a forward chaining deduction because it starts from a set of facts and unifies (matches) them with the body of a rule to derive the consequences. It is also possible to do backward chaining in which we start with a goal and work backwards by unifying it with the head of a rule, recursively trying to prove the body.
Symbolic ILP systems have a rich history dating back decades. A common approach to inducing rules is called learning from entailment [De Raedt1997], in which hypothesized rules are combined with background facts and trained to entail the positive and none of the negative concept examples. The FOIL algorithm [Quinlan and Cameron-Jones1995]
is an example of this approach. Our approach is also of this sort though we use neural networks to learn the rules. The classic ILP setting has been continually updated to handle richer knowledge. In[De Raedt and Kersting2008] for example they provide several formulations of probabilistic ILP. We do not consider probabilistic interpretations in our approach, though that is an interesting avenue for future research.
A related branch of research called Abductive Logic Programming attempts to learn consistent explanatory facts as well as rules [Kakas, Kowalski, and Toni1992]. For example, it might allow us to induce the fact that eagles are birds from the facts that eagles have wings and feathers together with the inheritance rule. Our approach for theory acquisition may be considered an example of this line of work.
Symbolic ILP systems do very well at generalizing from just a few examples. This is because they learn universal rules. They are, however, susceptible to noisy inputs and even a single bad fact can cause them to fail. On the other hand, neural systems generally are very robust to noisy input but are sample inefficient and prone to over-fitting on small amounts of data. Neuro-symbolic systems aim for the best of both worlds. They can be made robust to noisy inputs while still retaining some of the strong generalization properties typically associated with symbolic systems.
There is a long history of research in neural-symbolic systems from which we choose just a small set to present here. For a recent survey see [Besold et al.2017].
they introduce Logic Tensor Networks (LTN) and Real Logic whose semantics grounds the terms, atoms and clauses of the language as continuous functions. They demonstrate that the logic can be implemented using neural networks for the groundings of the symbols and apply it to solving a database completion problem. A follow-up paper applies LTN’s to semantic image segmentation[Donadello, Serafini, and D’Avila Garcez2017]. Neither work considers the problems of rule induction or theory acquisition.
A recent work [Manhaeve et al.2018]
starts from a probabilistic logic programming language (problog) and extends it to handle neural predicates which compute probabilities. Like Prolog and ProbLog, DeepProbLog is a backward chaining approach. It leverages the automatic differentiation system of ProbLog to incorporate neural predicates and trains with gradient descent. Like Logic Tensor Networks, ProbLog can train neural network implementations of relations. ProbLog does not do rule induction.
In Neural LP [Yang, Yang, and Cohen2017], the system can learn chaining-type rules. It uses a neural controller built on top of TensorLog [Cohen2016] and is trained to learn rules to compute a ranked list of entities which satisfy a partially specified query. It differs from our approach in several respects. It requires a partially specified query. It represents predicates as TensorLog operators (matrices) whereas we represent them as parameter embeddings which can be associated with constants and a valuation to represent an atom. And it is not obvious how it could be applied to learn fact representations.
[Sourek et al.2015] uses templates to create grounded networks that depend on the example. [Tran and D’Avila Garcez2016] studies the incorporation and extraction of knowledge into deep networks. [Kazemi and Poole2017] focuses on predicting the properties of objects.
Our approach is most directly related to two recent neural ILP approaches. In [Evans and Grefenstette2018] inference is done through the forward chained application of a set of logical rules. During learning, a set of all the possible candidate rules is generated according to a provided template. Parameters are weights associated with pairs of candidate rules. These weights are normalized to lie in and interpreted as probabilities associated to the rule pairs as possible definitions of the concept. When there are a large number of rules, this method may suffer scalability issues. In addition it requires a representation of the truth values of all possible facts and non-facts. By contrast, [Rocktäschel and Riedel2017] construct a function representing a backward-chained proof of the goal and require only a representation of the true facts. A more conceptual distinction arises in their parameterizations. In [Evans and Grefenstette2018] the parameters are weights on rule pairs. In [Rocktäschel and Riedel2017] they start with a set of parameterized rules which, as in our approach, acquire their meaning as the predicate embeddings of their head and body atoms are trained through unification with the predicates of the facts. It is not obvious how these approaches could be applied to theory acquisition requiring fact induction.
In our approach we follow [Rocktäschel and Riedel2017] in parameterizing with embeddings but use forward rather than backward chaining so that we don’t have to represent a proof tree explicitly. Unlike [Evans and Grefenstette2018], we don’t have to generate all the candidate rules. Instead, learning is at the level of individual atoms.
Semantic cognition concerns the acquisition and integration of knowledge. Previous work has modeled semantic cognition as a kind of logical dimensionality reduction [Yarden et al.2008, Ullman, Goodman, and Tenenbaum2012] which uses probabilistic generative models that can simultaneously learn logical rules and a set of core relations that form a theory underlying the observed data. Like ILP approaches, these models can make deductive inferences through the application of logical rules. But unlike traditional ILP algorithms, these Bayesian models are also able to induce facts.
This ability to apply both inductive and deductive reasoning at the level of both facts and rules provides humans with a rich space of techniques with which to tame the complexity of everyday experience. These approaches illuminated a promising direction but were severely challenged by scalability issues. With the success of neural techniques we believe it is useful to revisit these ideas.
As described above, our focus is on two tasks: rule induction in an ILP setting, and theory learning (learning both core facts and rules). In this section we describe a model which can be configured to perform either task and is trained using stochastic gradient descent. Figure2 illustrates the architecture common to both tasks. A rule is shown on the bottom left with a head predicate and two body predicates and . The model represents the rule as a triple . , , and are parameterized embeddings in corresponding to predicate symbols ; denote variables which are the subjects and objects that form the arguments of the atoms.
Facts are represented as quadruples , where is a parameter vector shared by all the facts associated with predicate ; is the subject constant; is the object constant; and is a valuation representing the model’s degree of belief in the truth of the atom . Constants are internally represented as integers and may be mapped to symbols for interpretation. The set of all current known facts (either initially given or inferred) is shown in the figure at the upper right. We maintain a set of all predicates and their associated embeddings which may include auxiliary predicates that can be used for predicate invention in the learning of rules and concepts. All predicates are randomly initialized.
To apply the rule to a pair of facts , where we first check for constant matching between the structure of the rules and the constants of the facts. This is just checking that the variable arguments of the rule body can be assigned to the corresponding constants in the facts. If they cannot, then then network construction stops. For example, as in the figure, the rule can be applied successfully to the pair of facts . However, if applied to , it would not match and would cause network construction to stop for this rule and fact pair. After a successful constant matching, a set of candidate output facts is generated, one for each predicate . The arguments of the fact are determined by the bindings of the rule body predicates to the input facts. The figure illustrates this with arrows flowing from the constants of the input facts to and forming a consequent fact at the top. The creation of separate facts for each predicate in is required because of our ignorance of the correct predicate for the head of the rule.
The valuation for each candidate output fact is determined through soft unification by (differentiably) combining the values of the input facts with measures of the degree of similarity between each input fact predicate and the corresponding rule body embedding, as well as with the similarity between the rule head embedding and the candidate fact predicate in . The values of the input facts are multiplied to implement a soft form of AND 333There are many other choices which are more theoretically well-grounded, such as the t-norm, but we found that simple multiplication works the best in practice for this application. Note that because we restrict ourselves to at most two atoms in the body of a rule, there is no issue with underflow here.. Specifically, we compute the value of an output fact resulting from the unification between a rule with head and body predicates , and facts and with values and corresponding predicates , , and as:
If the predicate and arguments of a consequent fact matches one in the set of known facts, its value is updated with the max between its previous and newly inferred values (implementing an OR); if it is not, the new fact is appended to the set of known facts. In this way the valuation is dynamically extended at each step of inference.
We have described how the network is constructed to perform one step of inference with a single rule. To construct the entire network, we start with an initial set of facts and perform this inference procedure for each rule and for each pair of facts. If a pair of facts fails to unify with the rule then that branch of the construction terminates. The number of steps of inference is a hyper-parameter as is the number of auxiliary predicates included in the predicate set .
To train the network we use a loss function that depends on the task (we describe the setup for each task below). The loss gradients are back-propagated to update the predicate embeddings for the rules and for the facts (the predicates of the facts can also be fixed, i.e as one-hot vectors). The orange arrows of Figure2 indicate the paths on which gradients flow (in the opposite direction). The rule and fact predicate embeddings are the parameters of the network, shown in green.
When a set of background facts is given, as in the case of the ILP tasks, we initialize the current valuation for the known facts to 1.0 and train the rules and predicates using a binary-cross-entropy loss to produce the correct values for the positive and negative target facts.
For theory learning, the aim is to learn a small theory which can recover the observations and generalize using the logical rules. Thus, we additionally learn a set of initial core facts that underlie the structure of the observations. In order to do that, unlike in the ILP setting, we additionally parameterize the valuations for all the facts and initialize them to 0.5 reflecting our initial ignorance of their truth. We train them towards values which allow the model to faithfully recover the observations. The model may produce additional facts not in the observations but the loss penalizes only the implied facts for whose predicates and arguments match those of an observed fact but for whose values differ. A regularization term controlled by penalizes the squared sum of the initial core valuations, encouraging compression. Using the notation for facts and to indicate that their predicates and arguments match exactly, and to denote the value of a fact , the loss can be written:
A wide range of previous work has focused on different aspects of logical induction. Here we test the capabilities of our algorithm in three different settings. First we test the capability of the algorithm to perform logical rule induction in the set of tasks covered by [Evans and Grefenstette2018]. Second, we test our algorithm in a bigger dataset and compare with [Rocktäschel and Riedel2017]. Finally, we test our model in the context of learning domain theories following [Yarden et al.2008, Ullman, Goodman, and Tenenbaum2012], the algorithm has to simultaneously learn the atoms of the logical rules, the representations of the facts of the predicates, and a set of core facts. To do that the algorithm learns to do deductive and inductive inferences of the facts.
Rule Learning ILP Tasks
We tested the model in the ILP problems from [Evans and Grefenstette2018]444We only skip the Husband and Uncle tasks which require the datasets from [Wang, Mazaitis, and Cohen2015].. The task of an ILP problem is to learn a target relation given a set of background knowledge facts and a set of positive and negative examples of a target relation.
As an example, consider the task of learning the predicate even(X). The background knowledge is defined using the zero(X) and succ(X,Y) predicates.
The target positive and negative predicate extensions are:
An example solution found by the algorithm is:
Where auxpred acquires the meaning of succ2 which is true of whenever .
. Since universal logical rules are perfectly generalizable, and to facilitate comparison, we use the same evaluation metric: the percentage of runs with different random weight initializations that successfully learn rules to solve the task with less thanmean squared error. To avoid local minima, we explored adding a decaying normal noise to the embeddings. This had a small positive effect in some of the tasks, reported results include the effect of the noise. Details of the problems are given in the Appendix.
|Adjacent to Red||2||No||50.5||100|
We see that our algorithm performs equally or better in most of the tasks. In contrast to theirs, search is not made at the level of rules but at the more compositional level of atoms. In fact, when the embeddings of the dictionary of predicates are fixed as one-hot vectors, our procedure is very similar to theirs: the parameters that form the predicates of the rules can be treated as weights that select the correct predicate. Thus, like in their model, training consists on selecting the weights that make the embeddings look like the right one-hot vectors and the procedure becomes a symbol search, except at the level of the atoms instead of the rules. The more general case where the embeddings are trained and dense, opens the interesting direction to be explored of studying the learned vector embedding semantic space, as has been done for standard NLP tasks [Mikolov et al.2013], which can potentially allow for similarity and analogical reasoning. In our table, we report the result with the trainable embeddings.
It is worth noting the failure of our model in the Graph Colouring and in the Two Children tasks. A quick exploration suggests that the global optima has a very sharp neighborhood while the local minima are attractors in most of the space. This is reminiscent of the Terpret problem [Selsam2018, Gaunt et al.2016] and the local minima can only be avoided when the random initialization is very close to the correct rules.
|Task||Model AUC-PR||Rule examples and confidences|
|S1||0.85 loc(X,Y) loc(X,Z), loc(Z,Y)|
|S2||0.57 loc(X,Y) neighbor(X,Z), loc(Z,Y)|
|neighbor(X,Z), loc(Z,W), loc(W,X)|
Notice that since our approach is differentiable, it is not prone to some of the problems of symbolic ILP, and like in DBLP:journals/corr/abs-1711-04574, it can handle ambiguity and noise.
We are not focused specifically on knowledge base completion but use the COUNTRIES dataset [Bouchard, Singh, and Trouillon2015] to evaluate the scalability of our algorithm, comparing to other neural logical approaches. The dataset contains 272 constants, 2 predicates and 1158 true facts and is designed to explicitly test the logical rule induction and reasoning capabilities of link prediction models.
We compare on the 3 tasks described in [Rocktäschel and Riedel2017], requiring reasoning steps of increasing length and difficulty (S1,S2,S3 in table 2). We report the Area Under the Precision-Recall-curve (AUC-PR) where results are comparable to the previous NTP approach. For completion, we also report from the same paper which uses an additional neural link network as an auxiliary loss. Like them, we also show some example rules and a confidence score by taking the minimum similarity between the atoms of the rule and their decoded predicates.
To perform the forward chaining during training, at each epoch we randomly sample from a section of the knowledge graph both the targets and a set of facts to form the background knowledge. Like the related work, our model can also suffer from scalability issues, as in forward chaining the size of the facts grows exponentially with the number of steps. Restricting the number of considered facts through sampling was sufficient for the task at consideration but this could show problems when scaling to much bigger datasets, we discuss some future directions in the conclusion.
|# Preds||# Const||# Facts||# Preds||# Const||# Facts|
|% Succ||% Acc||# Induced Facts||% Succ||% Acc||# Induced Facts|
We test the capability of our network to compress a set of observations in the form of a theory by learning a set of core facts in addition to the logical rules. We take the two domains considered by [Yarden et al.2008]: Taxonomy and Kinship.
A taxonomy is a set of observations structured into a tree where downstream nodes inherit the properties from the nodes above them in the tree. As shown in Figure 1, the tree constitutes a theory formed by two logical rules ; capturing inheritance and by a set of core facts represented with the edges. All observed facts can be recovered by iterative application of the rules (if salmon are fish, and fish have gills, then salmon have gills).
From a range of different possible taxonomies we report performance on the bigger original one from [Rogers and McClelland2004]. This data contains 145 facts composed of 4 predicates and 36 constants. The facts can be compressed into a tree structured theory as shown in the Appendix that contains only 40 core facts.
As shown in Table 3, the algorithm is able to learn the theory in 70% of the runs, achieving 99% accuracy and compressing close to the optimal level (average of 69 compared to the optimal of 40).
We also evaluated performance on the difficult kinship theory, which contains 10 constants and 6 observed predicates mother, father, daughter, wife, husband (see figure in the Appendix). In this case the compression of the theory consists of a set of 4 auxiliary core predicates with 28 facts. The algorithm has to learn the concepts female, male, spouse, child which acquire their meaning through their extensions and the 6 logical rules that generate the observations:
Table 3 shows the statistics for the observed and target compressed data. The algorithm’s performance is again quantified as the percentage of initializations where the rules are successfully learned, the accuracy of the recovered data and the number of learned core facts. The algorithm is able to perform this compression and learns a set of new predicates that conform the rules, recovering 96% of the data correctly (the algorithm sometimes deduces that some facts are true when they aren’t because it can induce incorrect core facts).
Conclusions and Future Work
We have presented a forward chaining inference network which is parameterized in the embeddings that form its rules and facts. Learning in this model means simultaneously learning the right sub-symbolic representations, and the right resulting symbolic conceptual relations implied through the logical rules; together constituting a Dual-Factor definition of the concepts [Carey2009].
We articulated a set of desiderata for models which learn logical theories from observations which include: compression, rule induction, and the ability to learn without direct supervision. We showed how our inference procedure satisfies these three conditions in two settings. Our model was able to learn a significant compression for the taxonomy and kinship datasets proposed by [Rogers and McClelland2004, Yarden et al.2008], learning interpretable representations not just for the parametrized rules but also for facts – a feature lacking in many traditional ILP solutions – using only the supplied observations as supervision. These are encouraging results for theory acquisition and point to the viability of this approach. As demonstrated on the ILP datasets from [Bouchard, Singh, and Trouillon2015, Cropper and Muggleton2016] and those from [Rocktäschel and Riedel2017], the method also provides an interesting alternative in the ILP setting.
As in previous work, our model is provided with rules that conform to templates, ideally this should not be necessary. The network needs to consider the set of all possible facts when doing core fact induction (which is not necessary for the rule induction problems), this is not be scalable in practice. Forward chaining grows exponentially in the number of facts considered at each step, this can also present a problem when scaling to bigger datasets. From a cognitive science perspective, the model is still more limited than its Bayesian symbolic counterparts, specifically, while those models provide graded measures of confidence in their inferences, our neural logical reasoner does not currently provide meaningful consistent estimates of uncertainty.
One straightforward attempt of learning the structural information of the rules provided by the templates (arity and variable order) would be to encode it by adding dimensions to the embeddings and have the algorithm interpret them by using independent unifications in the desired way. This would constitute a slightly more complicated learning task but would maintain the same structure and mechanism that could be trained through gradient descent. It would also be interesting to explore richer sampling procedures and the integration of forward with backward chaining, this could perhaps yield regimes more similar to those of humans and could help scale to larger datasets. We would also like to investigate ways of providing better estimates of uncertainty – from a full neural probabilistic formulation, to a heuristic metric based on the number of initializations and on the unification scores.
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Description of tasks and the proto-rule templates used during training. More details are in [Evans and Grefenstette2018]
In this task we aim to learn the predecessor relation, , from basic aritmetic facts . We use the following template:
In this task we aim to learn the predicate. Here the background knowledge is the same as in Predecessor (above). We must include an extra auxiliary predicate that learns to encode the relation . We use the following templates:
Described in Experiments.
Here we aim to learn the relation. Background knowledge is the same as in the tasks above. A possible solution would be:
We used these templates:
Following the same logic as in Fizz, we used the following templates:
Here we aim to learn , which is true if is an element of list . The background knowledge encodes values on a list by using two predicates: which is true if node is after list (lists are terminated with the null node 0); and which is true if the value of node is . One possible solution is:
We used these templates:
The relation is true if the length of list is . We represent lists in the same way as in the task. We required at least one extra intentional predicate . One possible solution would be:
We used these templates:
We aim to learn relation from family-related facts involving , and . We required at least one extra intentional predicate that learns the relation . One possible solution would be:
We used these templates:
The goal of this task is to infer the grandparent relation from observed mother-of and father-of facts. Our templates were:
is true if there is an undirected path between and . Background knowledge contains family related facts as in the tasks Son and Grandparent. We used these templates:
In this task we aim to learn the relation in challenging set up (incomplete background knowledge and irrelevant facts). For details please refer to [Evans and Grefenstette2018]. We used these template:
In this task the background knowledge is composed of several facts. The goal is to learn which is true if there is an edge between nodes and regardless of the direction. We used the templates:
Adjacent to Red
In this example we extend the background knowledge of the example above with color facts: , , as well as which is true if node is of colour . We included one auxiliary predicate that learns the relation . One possible solution would be:
We used these templates:
Here we aimed to learn the has-at-least-two-children(X) predicate, which is true if there are at least two facts of the form . The background knowledge includes and (not equals) relations. We included one auxiliary predicate . One possible solution would be:
We used these templates:
The task is to learn the adj-to-same(X,Y) which is true if nodes are of the same colour and there is an edge between them. The background knowledge is similar as in the task Adjacent to Red. We included an auxiliary predicate that should learn the relation same-colour(X,Y).One possible solution would be:
In this task we want to learn connected(X,Y) which is true if there is a sequence of edges connecting nodes and . We used the templates:
In this task the algorithm should learn the cocept of cyclicity. This is true of a node when there is a path departing from it and arriving to itself. The templates used were:
We mimicked some of the templates found in the appendix E of [Rocktäschel and Riedel2017]. In order to closely follow their approach, only for this task we enforced some predicates to be the same, so that #1 represents the same predicate across the rule. In detail for each task: