Sequential digital circuits, characterized by feedback loops and state-dependent behavior, pose unique challenges for web-based simulation systems. This paper presents novel mathematical models and algorithms for real-time simulation of sequential circuits using reactive programming paradigms. We extend the DAG-based circuit representation from combinational logic to handle cyclic dependencies through clock-driven evaluation and state machine modeling. The proposed approach successfully simulates flip-flops, latches, registers, and counters with sub-millisecond response times in browser environments. We introduce a formal framework for feedback loop detection, cycle breaking through temporal separation, and state consistency guarantees. Performance analysis demonstrates that the clock-synchronized evaluation model achieves 99.7% timing accuracy compared to hardware simulation while maintaining real-time interactivity. Case studies include SR latch, D flip-flop, and 4-bit counter implementations with detailed analysis of propagation behavior and metastability handling. This work extends the virtualization framework presented in our previous research to encompass stateful digital systems, laying groundwork for complex finite state machine simulation relevant to robotic control systems.

Sequential Logic, State Machines, Feedback Loops, Clock Synchronization, Web-Based Simulation, Reactive Programming, Temporal Logic.

Ergashev, A. “Web-Based Virtualization Framework for Digital Logic Systems.” TUIT, 2024.

Mano, M. M., & Ciletti, M. D. “Digital Design: With an Introduction to the Verilog HDL.” Pearson, 5th edition, 2012.

Wakerly, J. F. “Digital Design: Principles and Practices.” Prentice Hall, 4th edition, 2005.

IEEE Standard 1364-2005. “IEEE Standard for Verilog Hardware Description Language.” 2005.

Tarjan, R. “Depth-first search and linear graph algorithms.” SIAM Journal on Computing, vol. 1, no. 2, pp. 146-160, 1972.

Harel, D. “Statecharts: A visual formalism for complex systems.” Science of Computer Programming, vol. 8, no. 3, pp. 231-274, 1987.

Vue.js Team. “Vue 3 Reactivity System.” 2024. https://vuejs.org/guide/extras/reactivity-in-depth.html

Falstad, P. “Circuit Simulator: Sequential Circuits.” 2024. https://www.falstad.com/circuit/

Burch, C. “Logisim: A graphical system for logic circuit design and simulation.” Journal of Educational Resources in Computing, vol. 2, no. 1, 2002.

Lee, E. A., & Sangiovanni-Vincentelli, A. “A framework for comparing models of computation.” IEEE Transactions on CAD, vol. 17, no. 12, pp. 1217-1229, 1998.

Marino, M. D. “Principles of Metastability.” Proceedings of IEEE Custom Integrated Circuits Conference, 1981.

Dally, W. J., & Poulton, J. W. “Digital Systems Engineering.” Cambridge University Press, 1998.

Pnueli, A. “The temporal logic of programs.” Proceedings of 18th Annual Symposium on Foundations of Computer Science, pp. 46-57, 1977.

Clarke, E. M., Grumberg, O., & Peled, D. A. “Model Checking.” MIT Press, 1999.

ROS.org. “Robot Operating System.” 2024. https://www.ros.org/