[Paper Review] Derivatives of Turing machines in Linear Logic

James Clift, Daniel Murfet|arXiv (Cornell University)|May 30, 2018

Computability, Logic, AI Algorithms38 references3 citations

TL;DR

This paper computes the Ehrhard-Regnier derivatives of Turing machine encodings in linear logic using Sweedler semantics, showing that these derivatives represent rates of change in naive probability—defined as Bayesian degrees of belief under conditional independence assumptions. The key contribution is a formal link between differential linear logic and gradient-based learning, demonstrating that derivatives of Turing machines compute sensitivity to input uncertainty.

ABSTRACT

We calculate denotations under the Sweedler semantics of the Ehrhard-Regnier derivatives of various encodings of Turing machines into linear logic. We show that these derivatives calculate the rate of change of probabilities naturally arising in the Sweedler semantics of linear logic proofs. The resulting theory is applied to the problem of synthesising Turing machines by gradient descent.

Motivation & Objective

• To compute the Ehrhard-Regnier derivatives of Turing machine encodings in linear logic using Sweedler semantics.
• To investigate whether these derivatives have computational meaning, particularly in relation to uncertainty propagation.
• To establish a formal connection between differential linear logic and gradient descent in machine learning.
• To show that derivatives of proofs compute rates of change of naive probability under conditional independence assumptions.
• To demonstrate that the same uncertainty propagation arises from different encodings (e.g., boolstep vs. relstep), despite structural differences.

Proposed method

• Uses Sweedler semantics to interpret linear logic proofs in vector spaces via cofree coalgebras.
• Applies the Ehrhard-Regnier derivative calculus to plain proofs, particularly those encoding Turing machines.
• Defines 'naive probability' as the denotation of a proof applied to a group-like element representing a probability distribution.
• Models uncertainty propagation using distributions over tapes and states, with denotations mapping to linear maps on distribution spaces.
• Constructs patch and fill functions to model tape movement and region-specific propagation in derivative computations.
• Proves commutativity of diagrams involving derivative maps and projections, showing consistency across encodings (e.g., boolstep and relstep).

Experimental results

Research questions

• RQ1 What do the derivatives of algorithms compute in the context of linear logic and differential semantics?
• RQ2 How can the derivative of a Turing machine encoding be interpreted as a rate of change in a probabilistic setting?
• RQ3 Do structurally different encodings of the same computational behavior (e.g., boolstep vs. relstep) yield equivalent uncertainty propagation?
• RQ4 Can the derivative of a proof in linear logic be used to implement gradient descent in algorithm synthesis?
• RQ5 What is the role of conditional independence assumptions in defining naive probability in this framework?

Key findings

• The derivative of a proof in linear logic computes the rate of change of naive probability under the Sweedler semantics.
• Naive probability is defined as the Bayesian degree of belief in an output given input uncertainty and conditional independence assumptions.
• The derivative of a Turing machine encoding computes sensitivity to input distribution changes, analogous to gradients in backpropagation.
• The same uncertainty propagation is achieved by different encodings (e.g., boolstep and relstep), as shown by commutative diagrams in the derivative framework.
• The derivative of the step function in the boolstep encoding matches the derivative of the relstep encoding under appropriate dimensional constraints.
• The theory supports gradient descent synthesis of Turing machines by interpreting derivatives as directional updates to minimize error in probabilistic outputs.