A Semantic Framework for Neuro-Symbolic Computing

TL;DR

A formal definition of semantic encoding is provided, specifying the components and conditions under which a knowledge-base can be encoded correctly by a neural network, and it is shown that many neuro-symbolic approaches are accounted for by this definition.

Abstract

The field of neuro-symbolic AI aims to benefit from the combination of neural networks and symbolic systems. A cornerstone of the field is the translation or encoding of symbolic knowledge into neural networks. Although many neuro-symbolic methods and approaches have been proposed, and with a large increase in recent years, no common definition of encoding exists that can enable a precise, theoretical comparison of neuro-symbolic methods. This paper addresses this problem by introducing a semantic framework for neuro-symbolic AI. We start by providing a formal definition of semantic encoding, specifying the components and conditions under which a knowledge-base can be encoded correctly by a neural network. We then show that many neuro-symbolic approaches are accounted for by this definition. We provide a number of examples and correspondence proofs applying the proposed framework to the neural encoding of various forms of knowledge representation. Many, at first sight disparate, neuro-symbolic methods, are shown to fall within the proposed formalization. This is expected to provide guidance to future neuro-symbolic encodings by placing them in the broader context of semantic encodings of entire families of existing neuro-symbolic systems. The paper hopes to help initiate a discussion around the provision of a theory for neuro-symbolic AI and a semantics for deep learning.

Figures (10)

• Figure 1: A simple feed-forward neural network encoding a knowledge-base containing rules C if A, written $C\leftarrow A$, C if B, written $C\leftarrow B$, and fact A, written $A\leftarrow$. The parameters of the network (weights and biases) are shown next to the arrows in the diagram. With bias $1$, neuron A will always produce output $1$ (we say that A is activated in this case) for any input in $\{0,1\}$, given a step function as activation function. With bias $-1$, neuron B will output $0$ for every input. Activating either A or B will always activate neuron h, since the weight ($1$) from either A or B to h is equal to (or larger than) the negative of the bias of h. Finally, activating h also activates C, for the same reason as above.
• Figure 2: A simple recurrent neural network with $3$ neurons
• Figure 3: A block diagram of a semantic encoding. The stable states of a neural network are mapped to sets of interpretations which are aggregated into a single set of interpretations, $\mathcal{M}_N$. If these interpretations are models of a knowledge-base then the neural network is said to be a neural model of the knowledge-base. If these interpretations represent all models of the knowledge base (or a sufficient number of them to determine the logical entailment relation) then the neural network is said to be a semantic encoding.
• Figure 4: A recurrent neural network that semantically encodes the propositional knowledge-base $\{(A \wedge B) \vee (\neg A \wedge \neg B)\}.$
• Figure 5: A neural network that semantically encodes the first-order knowledge-base $L=\{\forall x.(R_1(x)\leftrightarrow R_2(x)), R_1(a),R_1(b), R_1(c),\neg R_1(d)\}$ under $I_{DAT}$ and $Agg=\cap$.

Theorems & Definitions (53)

• Example 2.1
• Definition 2.1
• Example 2.2
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Example 2.3
• Definition 3.1