Quantifying The Limits of AI Reasoning: Systematic Neural Network Representations of Algorithms
A main open question in contemporary AI research is quantifying the forms of reasoning neural networks can perform when perfectly trained. This paper answers this by interpreting reasoning tasks as circuit emulation, where the gates define the type of reasoning; e.g. Boolean gates for predicate logic, tropical circuits for dynamic programming, arithmetic and analytic gates for symbolic mathematical representation, and hybrids thereof for deeper reasoning; e.g. higher-order logic. We present a systematic meta-algorithm that converts essentially any circuit into a feedforward neural network (NN) with ReLU activations by iteratively replacing each gate with a canonical ReLU MLP emulator. We show that, on any digital computer, our construction emulates the circuit exactly--no approximation, no rounding, modular overflow included--demonstrating that no reasoning task lies beyond the reach of neural networks. The number of neurons in the resulting network (parametric complexity) scales with the circuit's complexity, and the network's computational graph (structure) mirrors that of the emulated circuit. This formalizes the folklore that NNs networks trade algorithmic run-time (circuit runtime) for space complexity (number of neurons). We derive a range of applications of our main result, from emulating shortest-path algorithms on graphs with cubic--size NNs, to simulating stopped Turing machines with roughly quadratically--large NNs, and even the emulation of randomized Boolean circuits. Lastly, we demonstrate that our result is strictly more powerful than a classical universal approximation theorem: any universal function approximator can be encoded as a circuit and directly emulated by a NN.