\section{State of the Art and Limits of Current Practice}

The principal aim of this research is to create autonomous reactor control
systems that are tractably safe. But, to understand what exactly is being
automated, it is important to understand how nuclear reactors are operated
today. First, the reactor operator themselves is discussed. Then, operating
procedures that we aim to leverage later are examined. Next, limitations of
human-based operation are investigated, while finally we discuss current formal
methods based approaches to building reactor control systems.

\subsection{Current Reactor Procedures and Operation}

Current generation nuclear power plants employ 3,600+ active NRC-licensed
reactor operators in the United States. These operators are divided into Reactor
Operators (ROs) who manipulate reactor controls and Senior Reactor Operators
(SROs) who direct plant operations and serve as shift
supervisors~\cite{10CFR55}. Staffing typically requires 2+ ROs with at least one
SRO for current generation units. To become a reactor operator, an individual
might spend up to six years to pass required training~\cite{princeton}.

The role of human operators is paradoxically both critical and
problematic. Operators hold legal authority under 10 CFR Part 55 to make
critical decisions including departing from normal regulations during
emergencies. The Three Mile Island (TMI) accident demonstrated how
``combination of personnel error, design deficiencies, and component
failures'' led to partial meltdown when operators ``misread confusing
and contradictory readings and shut off the emergency water
system''~\cite{Kemeny1979}. The President's Commission on TMI identified
a fundamental ambiguity: placing ``responsibility and accountability for
safe power plant operations...on the licensee in all circumstances''
without formal verification that operators can fulfill this
responsibility under all conditions~\cite{Kemeny1979}. This tension
between operational flexibility and safety assurance remains unresolved
in current practice.

Nuclear plant procedures exist in a hierarchy: normal operating procedures for
routine operations, abnormal operating procedures for off-normal conditions,
Emergency Operating Procedures (EOPs) for design-basis accidents, Severe
Accident Management Guidelines (SAMGs) for beyond-design-basis events, and
Extensive Damage Mitigation Guidelines (EDMGs) for catastrophic damage
scenarios. These procedures must comply with 10 CFR 50.34(b)(6)(ii) and are
developed using guidance from NUREG-0899~\cite{NUREG-0899}, but their
development process relies fundamentally on expert judgment and simulator
validation rather than formal verification. Procedures undergo technical
evaluation, simulator validation testing, and biennial review as part of
operator requalification under 10 CFR 55.59~\cite{10CFR55}. Despite these
rigorous development processes, procedures fundamentally lack formal
verification of key safety properties. There is no mathematical proof that
procedures cover all possible plant states, that required actions can be
completed within available timeframes under all scenarios, or that transitions
between procedure sets maintain safety invariants. 

\textbf{LIMITATION:} \textit{Procedures lack formal verification of correctness
and completeness.} Current procedure development relies on expert judgment and
simulator validation. No mathematical proof exists that procedures cover all
possible plant states, that required actions can be completed within available
timeframes, or that transitions between procedure sets maintain safety
invariants. Paper-based procedures cannot ensure correct application, and even
computer-based procedure systems lack the formal guarantees that automated
reasoning could provide.

Nuclear plants operate with multiple control modes: automatic control where the
reactor control system maintains target parameters through continuous rod
adjustment, manual control where operators directly manipulate control rods, and
various intermediate modes. In typical pressurized water reactor operation, the
reactor control system automatically maintains a floating average temperature,
compensating for changes in power demand with reactivity feedback loops alone.
Safety systems instead operate with implemented automation. Reactor
Protection Systems trip automatically on safety signals with millisecond
response times, and engineered safety features actuate automatically on accident
signals without operator action required.

The current division between automated and human-controlled functions
reveals the fundamental challenge of hybrid control. Highly
automated systems handle reactor protection like automatic trips on safety
parameters, emergency core cooling actuation, containment isolation,
and basic process control. Human operators, however, retain control of
strategic decision-making such as power level changes, startup/shutdown
sequences, mode transitions, and procedure implementation. %%%NEED MORE

\textbf{LIMITATION:} \textit{Current practice treats continuous plant
dynamics and discrete control logic separately.} No application of
hybrid control theory exists that could provide mathematical guarantees
across mode transitions, verify timing properties formally, or optimize
the automation-human interaction trade-off with provable safety bounds.

\subsection{Human Factors in Nuclear Accidents}
The persistent role of human error in nuclear safety incidents, despite
decades of improvements in training and procedures, provides perhaps the
most compelling motivation for formal automated control with
mathematical safety guarantees.

Multiple independent analyses converge on a striking statistic: \textbf{70--80\%
of all nuclear power plant events are attributed to human error} versus
approximately 20\% to equipment failures~\cite{DOE-HDBK-1028-2009,WNA2020}. More
significantly, the International Atomic Energy Agency concluded that ``human
error was the root cause of all severe accidents at nuclear power plants''---a
categorical statement spanning Three Mile Island, Chernobyl, and Fukushima
Daiichi~\cite{IAEA-severe-accidents}. A detailed analysis of 190 events at
Chinese nuclear power plants from 2007--2020~\cite{Wang2025} found that 53\% of
events involved active errors while 92\% were associated with latent errors
(organizational and systemic weaknesses that create conditions for failure). The
persistence of this 70--80\% human error contribution despite four decades of
continuous improvements in operator training, control room design, procedures,
and human factors engineering. This suggests fundamental cognitive limitations
rather than remediable deficiencies.

The Three Mile Island Unit 2 accident on March 28, 1979 remains the definitive
case study in human factors failures in nuclear operations. The accident began
at 4:00 AM with a routine feedwater pump trip, escalating when a
pressure-operated relief valve (PORV) stuck open---draining reactor
coolant---but control room instrumentation showed only whether the valve had
been commanded to close, not whether it actually closed. When Emergency Core
Cooling System pumps automatically activated as designed, operators made the
fateful decision to shut them down based on their incorrect assessment of plant
conditions. The result was a massive loss of coolant accident and the core
quickly began to overheat. During the emergency, operators faced more than 100
simultaneous alarms, overwhelming their cognitive capacity~\cite{Kemeny1979}.
The core suffered partial meltdown with \textbf{44\% of the fuel melting} before
the situation was stabilized.

Quantitative risk analysis revealed the magnitude of failure in existing
safety assessment methods: the actual core damage probability was
approximately 5\% per year while Probabilistic Risk Assessment
had predicted 0.01\% per year---a \textbf{500-fold underestimation}.
This dramatic failure demonstrated that human reliability could not be
adequately assessed through expert judgment and historical data alone.
%%%SOURCE??? Human Reliability Analysis (HRA) methods developed over four decades
quantify human error probabilities and performance shaping factors. The
SPAR-H method represents current best practice,
providing nominal Human Error Probabilities (HEPs) of \textbf{0.01 (1\%)
for diagnosis tasks} and \textbf{0.001 (0.1\%) for action tasks} under
optimal conditions~\cite{NUREG-CR-6883}.

However, these nominal error rates degrade dramatically under realistic
accident conditions: inadequate available time increases HEP by
\textbf{10-fold}, extreme stress by \textbf{5-fold}, high complexity by
\textbf{5-fold}, missing procedures by \textbf{50-fold}, and poor
ergonomics by \textbf{50-fold}. Under combined adverse conditions
typical of severe accidents, human error probabilities can approach
\textbf{0.1 to 1.0 (10\% to 100\%)}---essentially guaranteed failure for
complex diagnosis tasks~\cite{NUREG-2114}.

Rasmussen's influential 1983 taxonomy~\cite{Rasmussen1983} divides human errors
into skill-based (highly practiced responses, HEP $10^{-3}$ to $10^{-4}$),
rule-based (following procedures, HEP $10^{-2}$ to $10^{-1}$), and
knowledge-based (novel problem solving, HEP $10^{-1}$ to 1). Severe accidents
inherently require knowledge-based responses where human reliability is lowest.
Miller's classic 1956 finding~\cite{Miller1956} that working memory capacity is
limited to 7$\pm$2 chunks explains why Three Mile Island's 100+
%WHAT IS A CHUNK?
simultaneous alarms exceeded operators' processing capacity.

\textbf{LIMITATION:} \textit{Human factors impose fundamental reliability
limits that cannot be overcome through training alone.} Response time
limitations constrain human effectiveness---reactor protection systems
must respond in milliseconds, 100--1000 times faster than human
operators. Cognitive biases systematically distort judgment:
confirmation bias, overconfidence, and anchoring bias are inherent
features of human cognition, not individual failings~\cite{Reason1990}.
The persistent 70--80\% human error contribution despite four decades of
improvements demonstrates that these limitations are fundamental
rather than remediable.

\subsection{HARDENS and Formal Methods}

The High Assurance Rigorous Digital Engineering for Nuclear Safety
(HARDENS) project, completed by Galois, Inc. for the U.S. Nuclear
Regulatory Commission in 2022, represents the most advanced application
of formal methods to nuclear reactor control systems to
date---and simultaneously reveals the critical gaps that remain.

\subsubsection{Rigorous Digital Engineering Demonstrated Feasibility}

HARDENS aimed to address the nuclear industry's fundamental dilemma:
existing U.S. nuclear control rooms rely on analog technologies from the
1950s--60s, making construction costs exceed \$500 million and timelines
stretch to decades. The NRC contracted Galois to demonstrate that
Model-Based Systems Engineering and formal methods could design, verify,
and implement a complex protection system meeting regulatory criteria at
a fraction of typical cost.

The project delivered far beyond its scope, creating what Galois
describes as ``the world's most advanced, high-assurance protection
system demonstrator.'' Completed in \textbf{nine months at a tiny
fraction of typical control system costs}~\cite{Kiniry2022}, the project
produced a complete Reactor Trip System (RTS) implementation with full
traceability from NRC Request for Proposals and IEEE standards through
formal architecture specifications to formally verified binaries and
hardware running on FPGA demonstrator boards.

Principal Investigator Joseph Kiniry led the team in applying Galois's
Rigorous Digital Engineering methodology combining model-based
engineering, digital twins with measurable fidelity, and applied formal
methods. The approach integrates multiple abstraction levels---from
semi-formal natural language requirements through formal specifications
to verified implementations---all maintained as integrated artifacts
rather than separate documentation prone to divergence.

\subsubsection{Comprehensive Formal Methods Toolkit Provided Verification}

HARDENS employed an impressive array of formal methods tools and
techniques across the verification hierarchy. High-level specifications
used Lando, SysMLv2, and FRET (NASA JPL's Formal Requirements
Elicitation Tool) to capture stakeholder requirements, domain
engineering, certification requirements, and safety requirements.
Requirements were formally analyzed for \textbf{consistency,
completeness, and realizability} using SAT and SMT solvers---verification
that current procedure development methods lack.

Executable formal models employed Cryptol to create an executable
behavioral model of the entire RTS including all subsystems, components,
and formal digital twin models of sensors, actuators, and compute
infrastructure. Automatic code synthesis generated formally verifiable C
implementations and System Verilog hardware implementations directly
from Cryptol models---eliminating the traditional gap between
specification and implementation where errors commonly arise.

Formal verification tools included SAW (Software Analysis Workbench) for
proving equivalence between models and implementations, Frama-C for C
code verification, and Yosys for hardware verification. HARDENS verified
both automatically synthesized and hand-written implementations against
their models and against each other, providing redundant assurance
paths.

This multi-layered verification approach represents a quantum leap
beyond current nuclear I\&C verification practices, which rely primarily
on testing and simulation. HARDENS demonstrated that \textbf{complete
formal verification from requirements to implementation is technically
feasible} for safety-critical nuclear control systems.

\subsubsection{Critical Limitation: Discrete Control Logic Only}

Despite its impressive accomplishments, HARDENS has a fundamental
limitation directly relevant to hybrid control synthesis: \textbf{the
project addressed only discrete digital control logic without modeling
or verifying continuous reactor dynamics}. The Reactor Trip System
specification and formal verification covered discrete state transitions
(trip/no-trip decisions), digital sensor input processing through
discrete logic, and discrete actuation outputs (reactor trip commands).
The system correctly implements the digital control logic for reactor
protection with mathematical guarantees.

However, the project did not address continuous dynamics of nuclear
reactor physics including neutron kinetics, thermal-hydraulics, xenon
oscillations, fuel temperature feedback, coolant flow dynamics, and heat
transfer---all governed by continuous differential equations. Real
reactor safety depends on the interaction between continuous processes
(temperature, pressure, neutron flux evolving according to differential
equations) and discrete control decisions (trip/no-trip, valve
open/close, pump on/off). HARDENS verified the discrete controller in
isolation but not the closed-loop hybrid system behavior.

\textbf{LIMITATION:} \textit{HARDENS addressed discrete control logic
without continuous dynamics or hybrid system verification.} Hybrid
automata, differential dynamic logic, or similar hybrid systems
formalisms would be required to specify and verify properties like ``the
controller maintains core temperature below safety limits under all
possible disturbances''---a property that inherently spans continuous and
discrete dynamics. Verifying discrete control logic alone provides no
guarantee that the closed-loop system exhibits desired continuous
behavior such as stability, convergence to setpoints, or maintained
safety margins.

\subsubsection{Experimental Validation Gap Limits Technology Readiness}

The second critical limitation is \textbf{absence of experimental
validation} in actual nuclear facilities or realistic operational
environments. HARDENS produced a demonstrator system at Technology
Readiness Level 3--4 (analytical proof of concept with laboratory
breadboard validation) rather than a deployment-ready system validated
through extended operational testing. The NRC Final Report explicitly
notes~\cite{Kiniry2022}: ``All material is considered in development and
not a finalized product'' and ``The demonstration of its technical
soundness was to be at a level consistent with satisfaction of the
current regulatory criteria, although with no explicit demonstration of
how regulatory requirements are met.''

The project did not include deployment in actual nuclear facilities,
testing with real reactor systems under operational conditions,
side-by-side validation with operational analog RTS systems, systematic
failure mode testing (radiation effects, electromagnetic interference,
temperature extremes), actual NRC licensing review, or human factors
validation with licensed nuclear operators in realistic control room
scenarios.

\textbf{LIMITATION:} \textit{HARDENS achieved TRL 3--4 without experimental
validation.} While formal verification provides mathematical correctness
guarantees for the implemented discrete logic, the gap between formal
verification and actual system deployment involves myriad practical
considerations: integration with legacy systems, long-term reliability
under harsh environments, human-system interaction in realistic
operational contexts, and regulatory acceptance of formal methods as
primary assurance evidence.

\subsection{Research Imperative: Formal Hybrid Control Synthesis}

Three converging lines of evidence establish an urgent research
imperative for formal hybrid control synthesis applied to nuclear
reactor systems.

\textbf{Current reactor control practices} reveal fundamental gaps in
verification. Procedures lack mathematical proofs of completeness or
timing adequacy. Mode transitions preserve safety properties only
informally. Operator decision-making relies on training rather than
verified algorithms. The divide between continuous plant dynamics and
discrete control logic has never been bridged with formal methods.
Despite extensive regulatory frameworks developed over six decades,
\textbf{no mathematical guarantees exist} that current control approaches
maintain safety under all possible scenarios.

\textbf{Human factors in nuclear accidents} demonstrate that human error
contributes to 70--80\% of nuclear incidents despite four decades of
systematic improvements. The IAEA's categorical statement that ``human
error was the root cause of all severe accidents'' reveals fundamental
cognitive limitations: working memory capacity of 7$\pm$2 chunks,
response times of seconds to minutes versus milliseconds required,
cognitive biases immune to training, stress-induced performance
degradation. Human Reliability Analysis methods document error
probabilities of 0.001--0.01 under optimal conditions degrading to
0.1--1.0 under realistic accident conditions. These limitations
\textbf{cannot be overcome through human factors improvements alone}.

\textbf{The HARDENS project} proved that formal verification is
technically feasible and economically viable for nuclear control
systems, achieving complete verification from requirements to
implementation in nine months at a fraction of typical costs. However,
HARDENS addressed only discrete control logic without considering
continuous reactor dynamics or hybrid system verification, and the
demonstrator achieved only TRL 3--4 without experimental validation in
realistic nuclear environments. These limitations directly define the
research frontier: \textbf{formal synthesis of hybrid controllers that
provide mathematical safety guarantees across both continuous plant
dynamics and discrete control logic}.

The research opportunity is clear. Nuclear reactors are quintessential
hybrid cyber-physical systems where continuous neutron kinetics,
thermal-hydraulics, and heat transfer interact with discrete control
mode decisions, trip logic, and valve states. Current practice treats
these domains separately---reactor physics analyzed with simulation,
control logic verified through testing, human operators expected to
integrate everything through procedures. \textbf{Hybrid control
synthesis offers the possibility of unified formal treatment} where
controllers are automatically generated from high-level safety
specifications with mathematical proofs that guarantee safe operation
across all modes, all plant states, and all credible disturbances.

Recent advances in hybrid systems theory---including reachability
analysis, barrier certificates, counterexample-guided inductive
synthesis, and satisfiability modulo theories for hybrid systems---provide
the theoretical foundation. Computational advances enable verification of
systems with continuous state spaces that were intractable a decade ago.
The confluence of mature formal methods, powerful verification tools
demonstrated by HARDENS, urgent safety imperatives documented by
persistent human error statistics, and fundamental gaps in current
hybrid dynamics treatment creates a compelling and timely research
opportunity.
