Towards a Mathematical Theory of Abstraction

by   Beren Millidge, et al.

While the utility of well-chosen abstractions for understanding and predicting the behaviour of complex systems is well appreciated, precisely what an abstraction is has so far has largely eluded mathematical formalization. In this paper, we aim to set out a mathematical theory of abstraction. We provide a precise characterisation of what an abstraction is and, perhaps more importantly, suggest how abstractions can be learnt directly from data both for static datasets and for dynamical systems. We define an abstraction to be a small set of `summaries' of a system which can be used to answer a set of queries about the system or its behaviour. The difference between the ground truth behaviour of the system on the queries and the behaviour of the system predicted only by the abstraction provides a measure of the `leakiness' of the abstraction which can be used as a loss function to directly learn abstractions from data. Our approach can be considered a generalization of classical statistics where we are not interested in reconstructing `the data' in full, but are instead only concerned with answering a set of arbitrary queries about the data. While highly theoretical, our results have deep implications for statistical inference and machine learning and could be used to develop explicit methods for learning precise kinds of abstractions directly from data.


page 1

page 2

page 3

page 4


Data-driven Abstractions for Verification of Deterministic Systems

A common technique to verify complex logic specifications for dynamical ...

Guidelines For Pursuing and Revealing Data Abstractions

Many data abstraction types, such as networks or set relationships, rema...

Towards Computing an Optimal Abstraction for Structural Causal Models

Working with causal models at different levels of abstraction is an impo...

Likelihood Computations Using Value Abstractions

In this paper, we use evidence-specific value abstraction for speeding B...

LEMMA: Bootstrapping High-Level Mathematical Reasoning with Learned Symbolic Abstractions

Humans tame the complexity of mathematical reasoning by developing hiera...

Symbolic Abstractions From Data: A PAC Learning Approach

Symbolic control techniques aim to satisfy complex logic specifications....

Towards Global Neural Network Abstractions with Locally-Exact Reconstruction

Neural networks are a powerful class of non-linear functions. However, t...

Please sign up or login with your details

Forgot password? Click here to reset