Sahil Rajesh Dhayalkar

Sahil Rajesh Dhayalkar

We present a formal and constructive framework for simulating Alternating Finite Automata (AFAs) using Logic-Gated Time-Shared Feedforward Networks (LG-TS-FFNs). Unlike prior neural automata models limited to Nondeterministic Finite Automata (NFAs) and existential reachability, our architecture integrates learnable, state-dependent biases that function as differentiable logic gates, enabling the representation of both Existential \textsc{\textsc{OR}} and Universal \textsc{\textsc{AND}} aggregation within a shared-parameter linear recurrence. We prove that this architectural modification upgrades the network's computational class to be structurally isomorphic to AFAs, thereby inheriting their exponential succinctness: the network can represent regular languages requiring $2^n$ states in an NFA with only $n$ neurons. We rigorously establish that the forward pass of an LG-TS-FFN exactly simulates the reachability dynamics of an AFA, including instantaneous $\varepsilon$-closures. Furthermore, we demonstrate empirical learnability: a continuous relaxation of the logic gates allows the network to simultaneously recover the automaton's topology and logical semantics from binary labels via standard gradient descent. Extensive experiments confirm that our model achieves perfect recovery of ground-truth automata, bridging the gap between statistical learning and succinct, universal logical reasoning.

Sahil Rajesh Dhayalkar

We propose a combinatorial and graph-theoretic theory of dropout by modeling training as a random walk over a high-dimensional graph of binary subnetworks. Each node represents a masked version of the network, and dropout induces stochastic traversal across this space. We define a subnetwork contribution score that quantifies generalization and show that it varies smoothly over the graph. Using tools from spectral graph theory, PAC-Bayes analysis, and combinatorics, we prove that generalizing subnetworks form large, connected, low-resistance clusters, and that their number grows exponentially with network width. This reveals dropout as a mechanism for sampling from a robust, structured ensemble of well-generalizing subnetworks with built-in redundancy. Extensive experiments validate every theoretical claim across diverse architectures. Together, our results offer a unified foundation for understanding dropout and suggest new directions for mask-guided regularization and subnetwork optimization.

Sahil Rajesh Dhayalkar

We present a formal and constructive simulation framework for nondeterministic finite automata (NFAs) using time-shared, depth-unrolled feedforward networks (TS-FFNs), i.e., acyclic unrolled computations with shared parameters that are functionally equivalent to unrolled recurrent or state-space models. Unlike prior approaches that rely on explicit recurrent architectures or post hoc extraction methods, our formulation symbolically encodes automaton states as binary vectors, transitions as sparse matrix transformations, and nondeterministic branching-including $\varepsilon$-closures-as compositions of shared thresholded updates. We prove that every regular language can be recognized exactly by such a shared-parameter unrolled feedforward network, with parameter count independent of input length. Our construction yields a constructive equivalence between NFAs and neural networks and demonstrates \emph{empirical learnability}: these networks can be trained via gradient descent on supervised acceptance data to recover the target automaton behavior. This learnability, formalized in Proposition 5.1, is the crux of this work. Extensive experiments validate the theoretical results, achieving perfect or near-perfect agreement on acceptance, state propagation, and closure dynamics. This work clarifies the correspondence between automata theory and modern neural architectures, showing that unrolled feedforward networks can perform precise, interpretable, and trainable symbolic computation.

Sahil Rajesh Dhayalkar

Transformers have revolutionized various domains of artificial intelligence due to their unique ability to model long-range dependencies in data. However, they lack in nuanced, context-dependent modulation of features and information flow. This paper introduces two significant enhancements to the transformer architecture - the Evaluator Adjuster Unit (EAU) and Gated Residual Connections (GRC) - designed to address these limitations. The EAU dynamically modulates attention outputs based on the relevance of the input context, allowing for more adaptive response patterns. Concurrently, the GRC modifies the transformer's residual connections through a gating mechanism that selectively controls the information flow, thereby enhancing the network's ability to focus on contextually important features. We evaluate the performance of these enhancements across several benchmarks in natural language processing. Our results demonstrate improved adaptability and efficiency, suggesting that these modifications could set new standards for designing flexible and context-aware transformer models.

Mayur Sonawane, Sahil Rajesh Dhayalkar, Siddesh Waje et al.

Human Activity Recognition is a subject of great research today and has its applications in remote healthcare, activity tracking of the elderly or the disables, calories burnt tracking etc. In our project, we have created an Android application that recognizes the daily human activities and calculate the calories burnt in real time. We first captured labeled triaxial acceleration readings for different daily human activities from the smartphone's embedded accelerometer. These readings were preprocessed using a median filter. 42 features were extracted using various methods. We then tested various machine learning algorithms along with dimensionality reduction. Finally, in our Android application, we used the machine learning algorithm and a subset of features that provided maximum accuracy and minimum model building time. This is used for real-time activity recognition and calculation of calories burnt using a formula based on Metabolic Equivalent.

Sahil Rajesh Dhayalkar

We present a formal and constructive theory showing that probabilistic finite automata (PFAs) can be exactly simulated using symbolic feedforward neural networks. Our architecture represents state distributions as vectors and transitions as stochastic matrices, enabling probabilistic state propagation via matrix-vector products. This yields a parallel, interpretable, and differentiable simulation of PFA dynamics using soft updates-without recurrence. We formally characterize probabilistic subset construction, $\varepsilon$-closure, and exact simulation via layered symbolic computation, and prove equivalence between PFAs and specific classes of neural networks. We further show that these symbolic simulators are not only expressive but learnable: trained with standard gradient descent-based optimization on labeled sequence data, they recover the exact behavior of ground-truth PFAs. This learnability, formalized in Proposition 5.1, is the crux of this work. Our results unify probabilistic automata theory with neural architectures under a rigorous algebraic framework, bridging the gap between symbolic computation and deep learning.

Sahil Rajesh Dhayalkar

We extend the ReLU Transition Graph (RTG) framework into a comprehensive graph-theoretic model for understanding deep ReLU networks. In this model, each node represents a linear activation region, and edges connect regions that differ by a single ReLU activation flip, forming a discrete geometric structure over the network's functional behavior. We prove that RTGs at random initialization exhibit strong expansion, binomial degree distributions, and spectral properties that tightly govern generalization. These structural insights enable new bounds on capacity via region entropy and on generalization via spectral gap and edge-wise KL divergence. Empirically, we construct RTGs for small networks, measure their smoothness and connectivity properties, and validate theoretical predictions. Our results show that region entropy saturates under overparameterization, spectral gap correlates with generalization, and KL divergence across adjacent regions reflects functional smoothness. This work provides a unified framework for analyzing ReLU networks through the lens of discrete functional geometry, offering new tools to understand, diagnose, and improve generalization.