🤖 AI Summary
This work proposes a training-free method for constructing neural networks that precisely simulate arbitrary Turing machines. Given a formal description of a Turing machine, the system automatically compiles it into a corresponding neural network whose forward pass exactly mirrors a single step of the machine’s execution. Grounded in first principles, the approach employs ReLU networks to implement Boolean logic and adders, leverages Cantor set encoding together with hard attention mechanisms to manage tape read–write operations, and demonstrates Turing completeness within both Transformer architectures—incorporating self-attention and cross-attention—and recurrent neural network frameworks. This study establishes a reproducible and verifiable theoretical and practical foundation for integrating neural and symbolic computation.
📝 Abstract
We present a PyTorch package that compiles neural networks and their weights from Turing machine descriptions, producing models that exactly simulate the specified machine without any training. Given a transition function and a set of terminal states, the package constructs a model whose forward pass corresponds to one step of the Turing machine. Two architectures are implemented, each realizing a different theoretical result: (1) a transformer with self-attention, cross-attention, and feedforward layers based on Wei, Chen, and Ma (2021), and (2) a recurrent network based on Siegelmann and Sontag (1995) that encodes the stack in a Cantor set. We develop the constructions from first principles, showing how ReLU networks implement Boolean circuits (AND, OR, NOT, XOR gates and their composition into DNF formulas and binary adders) and how hard attention implements positional lookup on the tape. The package serves as a concrete, runnable reference for the symbolic-neural bridge, and as a foundation for future work on the stability of constructed solutions under gradient-based optimization. Code is available at https://github.com/jonrbates/turing.