Logic of Differentiable Logics: Towards a Uniform Semantics of DL
This work addresses a foundational problem for researchers in AI and machine learning by providing a unified framework to compare and analyze differentiable logics, though it is incremental as it builds on existing DLs rather than introducing a new paradigm.
The paper tackles the lack of a systematic framework for comparing differentiable logics (DLs) used to train neural networks with logical specifications by proposing a meta-language called LDL, which generalizes syntax to first-order logic and provides a uniform semantics for defining loss functions, enabling theoretical analysis and empirical study in neural network verification.
Differentiable logics (DL) have recently been proposed as a method of training neural networks to satisfy logical specifications. A DL consists of a syntax in which specifications are stated and an interpretation function that translates expressions in the syntax into loss functions. These loss functions can then be used during training with standard gradient descent algorithms. The variety of existing DLs and the differing levels of formality with which they are treated makes a systematic comparative study of their properties and implementations difficult. This paper remedies this problem by suggesting a meta-language for defining DLs that we call the Logic of Differentiable Logics, or LDL. Syntactically, it generalises the syntax of existing DLs to FOL, and for the first time introduces the formalism for reasoning about vectors and learners. Semantically, it introduces a general interpretation function that can be instantiated to define loss functions arising from different existing DLs. We use LDL to establish several theoretical properties of existing DLs, and to conduct their empirical study in neural network verification.