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Title: Logic and the 2-Simplicial Transformer

Paper content:
We introduce the 2-simplicial Transformer, an extension of the Transformer which includes a form of higher-dimensional attention generalising the dot-product attention, and uses this attention to update entity representations with tensor products of value vectors.
We show that this architecture is a useful inductive bias for logical reasoning in the context of deep reinforcement learning.
Deep learning contains many differentiable algorithms for computing with learned representations.
These representations form vector spaces, sometimes equipped with additional structure.
A recent example is the Transformer (Vaswani et al., 2017) in which there is a vector space V of value vectors and an inner product space H of query and key vectors.
This structure supports a kind of messagepassing, where a value vector v j ∈ V derived from entity j is propagated to update an entity i with weight q i · k j , where q i ∈ H is a query vector derived from entity i, k j ∈ H is a key vector derived from entity j, and the inner product on H is written as a dot product.
The Transformer therefore represents a relational inductive bias, where a relation from entity j to entity i is perceived to the extent that q i · k j is large and positive.
However, the real world has structure beyond entities and their direct relationships: for example, the three blocks in Figure 1 are arranged in such a way that if either of the supporting blocks is removed, the top block will fall.
This is a simple 3-way relationship between entities i, j, k that is complex to represent as a system of 2-way relationships.
It is natural to make the hypothesis that such higher-order relationships are essential to extracting the full predictive power of data, across many domains.
In accordance with this hypothesis, we introduce a generalisation of the Transformer architecture, the 2-simplicial Transformer, which incorporates both 2-and 3-way interactions.
Mathematically, the key observation is that higher-order interactions between entities can be understood using algebras.
This is nothing but Boole's insight (Boole, 1847) which set in motion the development of modern logic.
In our situation, an appropriate algebra is the Clifford algebra Cl(H) of the space H of queries and keys, which contains that space H ⊆ Cl(H) and in which queries and keys can be multiplied.
To represent a 3-way interaction we map each entity i to a triple (p i , l k ) using a natural continuous function η : Cl(H) −→ R associated to the Z-grading of Cl(H).
This scalar measures how strongly the network perceives a 3-way interaction involving i, j, k.
In summary, the 2-simplicial Transformer learns how to represent entities in its environment as vectors v ∈ V , and how to transform those entities to queries and (pairs of) keys in H, so that the signals provided by the scalars q i · k j and η(p i l 1 j l 2 k ) are informative about higher-order structure in the environment.
As a toy example of higher-order structure, we consider the reinforcement learning problem in a variant of the BoxWorld environment from (Zambaldi et al., 2019) .
The original BoxWorld is played on a rectangular grid populated by keys and locked boxes of varying colours, with the goal being to open the box containing the "Gem".
In our variant of the BoxWorld environment, bridge BoxWorld, the agent must use two keys simultaneously to obtain the Gem; this structure in the environment creates many 3-way relationships between entities, including for example the relationship between the locked boxes j, k providing the two keys and the Gem entity i.
This structure in the environment is fundamentally logical in nature, and encodes a particular kind of conjunction; see Appendix I.
The architecture of our deep reinforcement learning agent largely follows (Zambaldi et al., 2019) and the details are given in Section 4.
The key difference between our simplicial agent and the relational agent of (Zambaldi et al., 2019) is that in place of a standard Transformer block we use a 2-simplicial Transformer block.
Our experiments show that the simplicial agent confers an advantage over the relational agent as an inductive bias in our reasoning task.
Motivation from neuroscience for a simplicial inductive bias for abstract reasoning is contained in Appendix J.
Our use of tensor products of value vectors is inspired by the semantics of linear logic in vector spaces (Girard, 1987; Mellis, 2009; Clift & Murfet, 2017; Wallbridge, 2018) in which an algorithm with multiple inputs computes on the tensor product of those inputs, but this is an old idea in natural language processing, used in models including the second-order RNN (Giles et al., 1989; Pollack, 1991; Goudreau et al., 1994; Giles et al., 1991) , multiplicative RNN (Sutskever et al., 2011; Irsoy & Cardie, 2015) , Neural Tensor Network (Socher et al., 2013 ) and the factored 3-way Restricted Boltzmann Machine (Ranzato et al., 2010) , see Appendix A. Tensors have been used to model predicates in a number of neural network architectures aimed at logical reasoning (Serafini & Garcez, 2016; Dong et al., 2019) .
The main novelty in our model lies in the introduction of the 2-simplicial attention, which allows these ideas to be incorporated into the Transformer architecture.
On general grounds one might expect that in the limit of infinite experience, any reinforcement learning agent with a sufficiently deep neural network will be able to solve any environment, in-cluding those like bridge BoxWorld that involve higher-order relations between entities.
In practice, however, we do not care about the infinite computation limit.
In the regime of bounded computation it is reasonable to introduce biases towards learning representations of structures that are found in a wide range of environments that we consider important.
We argue that higher-order relations between entities are an important example of such structures, and that the 2-simplicial Transformer is a natural inductive bias for 3-way interactions between entities.
We have given preliminary evidence for the utility of this bias by showing that in the bridge BoxWorld environment the simplicial agent has better performance than a purely relational agent, and that this performance involves in a meaningful way the prediction of 3-way interactions (or 2-simplices).
We believe that simplicial Transformers may be useful for any problem in which higher-order relations between entities are important.
The long history of interactions between logic and algebra is a natural source of inspiration for the design of inductive biases in deep learning.
In this paper we have exhibited one example: Boole's idea, that relationships between entities can be modeled by multiplication in an algebra, may be realised in the context of deep learning as an augmentation to the Transformer architecture using Clifford algebras of spaces of representations.

assistant: text: We introduce the 2-simplicial Transformer and show that this architecture is a useful inductive bias for logical reasoning in the context of deep reinforcement learning.