Logic Tensor Networks (2012.13635v4)

Abstract: Artificial Intelligence agents are required to learn from their surroundings and to reason about the knowledge that has been learned in order to make decisions. While state-of-the-art learning from data typically uses sub-symbolic distributed representations, reasoning is normally useful at a higher level of abstraction with the use of a first-order logic language for knowledge representation. As a result, attempts at combining symbolic AI and neural computation into neural-symbolic systems have been on the increase. In this paper, we present Logic Tensor Networks (LTN), a neurosymbolic formalism and computational model that supports learning and reasoning through the introduction of a many-valued, end-to-end differentiable first-order logic called Real Logic as a representation language for deep learning. We show that LTN provides a uniform language for the specification and the computation of several AI tasks such as data clustering, multi-label classification, relational learning, query answering, semi-supervised learning, regression and embedding learning. We implement and illustrate each of the above tasks with a number of simple explanatory examples using TensorFlow 2. Keywords: Neurosymbolic AI, Deep Learning and Reasoning, Many-valued Logic.

• The paper introduces Logic Tensor Networks (LTN) as a neurosymbolic AI framework that integrates fuzzy logic semantics with neural network processing.
• The paper presents Real Logic, a differentiable logic language that combines first-order fuzzy logic with tensor operations for effective learning and reasoning.
• The paper demonstrates LTN's practical applications in classification, semi-supervised learning, and clustering, achieving improved accuracy and logical satisfaction.

Logic Tensor Networks

Introduction

"Logic Tensor Networks" (LTN) proposes an innovative approach to neurosymbolic AI, integrating logic reasoning with neural network processing. The paper introduces Real Logic, a differentiable logic language combining first-order fuzzy logic semantics with neural computational graphs. The framework supports a wide range of AI tasks, demonstrating efficient querying, learning, and reasoning with symbolic knowledge and raw data. Implementing LTN with TensorFlow 2, the paper showcases several examples of practical applications, suggesting the versatility and capability of LTN in various AI settings.

Real Logic

Real Logic forms the backbone of LTN, combining elements of first-order logic with differentiable operations suited for neural processing. Essential components include:

• Syntax: Objects, functions, predicates, and variables form a symbolic language grounded in real values via tensors.
• Semantics: In Real Logic, symbols are interpreted as tensor operations, fostering a seamless connection between symbolic representation and neural network implementation.
• Connectives and Quantifiers: Fuzzy logic operators define conjunctions, disjunctions, implications, and negations, providing flexibility in logical operations.
• Extensions: Features like guarded and diagonal quantification enable efficient handling of complex logical relationships.

  Figure 1: Illustration of an aggregation operation implementing quantification $(\forall y \exists x)$ over variables $x$ and $y$. Assumes differing domains for $x$ and $y$, resulting in a number in [0,1].

Learning and Reasoning

LTN extends the concept of logic-based reasoning by enabling learning and inference on logical knowledge through differentiable mathematical operations. Key features include:

• Learning: Optimization processes derive parameter values that maximize the satisfaction of formulated logic theory, promoting embedding learning, generative modeling, and classification tasks.
• Querying: Truth, value, and generalization queries enable assessment and validation of logical expressions and relationships within the AI model.
• Reasoning: LTN incorporates logical consequence definitions adapted for Real Logic, offering techniques for evaluating grounded theories and exploring potential counterexamples to logical statements.

Practical Applications

The paper illustrates LTN's capacities through varied AI scenarios:

• Classification Tasks: Binary and multi-class classification showcases LTN's ability to utilize logical constraints effectively, demonstrating superior satisfaction and accuracy metrics compared to baseline neural networks.
• Semi-Supervised Pattern Recognition: Highlighted by MNIST digit addition tasks, LTN leverages logic knowledge to enhance learning performance with minimal labeled data.
• Regression and Clustering: Unsupervised learning tasks illustrate how logical constraints guide the clustering process, achieving notable satisfaction levels despite limited knowledge.
• Embedding Learning: LTN's approach to learning embeddings, exemplified by smokers-friends-cancer models, underscores its capabilities in handling incomplete and inconsistent knowledge.

  Figure 2: Binary Classification task performance displays average accuracy (left) and satisfiability (right), indicating rapid improvement post few epochs.

Conclusion

The paper demonstrates that Logic Tensor Networks offer a robust framework for integrating logic reasoning with neural network models, achieving efficient learning and inference across diverse AI tasks. The introduction of Real Logic provides a flexible and expressive language for defining rich knowledge, bridging the gap between symbolic reasoning and deep learning capabilities. Future work could expand on continual learning frameworks, higher-order logic integration, and broader applications requiring sophisticated reasoning mechanisms.

As a robust tool for neurosymbolic AI, LTN leverages computational graphs and fuzzy logic semantics, ensuring that logical reasoning is conducted efficiently within practical AI models. Its versatility and depth position LTN as a promising direction for future advancements in AI research and applications.