\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. \oldt{To understand what is being automated,
we must first understand how nuclear reactors are operated today. This section
examines reactor operators and the operating procedures we aim to leverage,
then investigates limitations of human-based operation, and concludes with
current formal methods approaches to reactor control systems.}
\newt{Understanding what is being automated requires understanding how
nuclear reactors are operated today. This section examines reactor operating
procedures, investigates limitations of human-based operation, and reviews
current formal methods approaches to reactor control
systems.}

\subsection{Current Reactor Procedures and Operation}

\oldt{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.} \newt{Nuclear plant operations
are governed by a hierarchy of written procedures, ranging from normal
operating procedures for routine operations to Emergency Operating
Procedures (EOPs) for design-basis accidents and Severe Accident Management
Guidelines (SAMGs) for beyond-design-basis events. These procedures exist
because reactor operation requires deterministic responses to a wide range
of plant conditions, from routine power changes to catastrophic
failures.}These procedures must comply with 10
CFR 50.34(b)(6)(ii) and are developed using guidance from
NUREG-0899~\cite{NUREG-0899, 10CFR50.34}, but their development 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{10CFR55.59}. Despite this rigor, procedures fundamentally lack
\oldt{formal verification of key safety properties.} \newt{exhaustive
verification of key safety properties.}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.

\textbf{LIMITATION:} \textit{Procedures lack exhaustive verification of
correctness and completeness.} \oldt{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.} \newt{Paper-based
procedures cannot ensure correct application, and even computer-based
procedure systems lack the 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
reactivity adjustment; manual control, where operators directly manipulate
the reactor; and various intermediate modes. In typical pressurized water
reactor operation, the reactor control system automatically maintains a
floating average temperature and compensates for power demand changes through
reactivity feedback loops alone. Safety systems, by contrast, 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.

\oldt{The division between automated and human-controlled functions reveals
the fundamental challenge of hybrid control. Highly automated systems handle
reactor protection---automatic trips on safety parameters, emergency core
cooling actuation, containment isolation, and basic process
control~\cite{WRPS.Description, gentillon_westinghouse_1999}. Human
operators, however, retain control of strategic decision-making: power level
changes, startup/shutdown sequences, mode transitions, and procedure
implementation.} \newt{This division between automated and human-controlled
functions is itself the hybrid control problem. Automated systems handle
reactor protection: trips on safety parameters, emergency core cooling
actuation, containment isolation, and basic process
control~\cite{WRPS.Description, gentillon_westinghouse_1999}. Human
operators retain control of everything else: power level changes,
startup/shutdown sequences, mode transitions, and procedure
implementation.}

\subsection{Human Factors in Nuclear Accidents}

\oldt{Current-generation nuclear power plants employ over 3,600 active
NRC-licensed reactor operators in the United
States~\cite{operator_statistics}. These operators divide 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 at least two ROs and
one SRO for current-generation units~\cite{10CFR50.54}. Becoming a reactor
operator requires several years of training.} \newt{Nuclear plant staffing
requires at least two Reactor Operators (ROs) and one Senior Reactor
Operator (SRO) per unit~\cite{10CFR50.54}, with operators requiring several
years of training and NRC licensing under 10 CFR Part
55~\cite{10CFR55}.}

The persistent role of human error in nuclear safety incidents---despite
decades of improvements in training and procedures---provides the most
compelling motivation for formal automated control with mathematical safety
guarantees.Operators hold
legal authority under 10 CFR Part 55 to make critical
decisions~\cite{10CFR55},including departing from normal
regulations during emergencies. \oldt{The Three Mile Island (TMI) accident}
\newt{This authority itself introduces risk. The Three Mile Island (TMI)
accident}demonstrated how a
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 for safe power plant operations on the licensee without
formal verification that operators can fulfill this responsibility does not
guarantee safety. This tension between operational flexibility and safety
assurance remains unresolved: the person responsible for reactor safety is
often the root cause of failures.

Multiple independent analyses converge on a striking statistic: 70--80\% of
nuclear power plant events are attributed to human error, versus
approximately 20\% to equipment failures~\cite{WNA2020}. More significantly,
the root cause of all severe accidents at nuclear power plants---Three Mile
Island, Chernobyl, and Fukushima Daiichi---has been identified as poor
safety management and safety culture: primarily human
factors~\cite{hogberg_root_2013}. A detailed analysis of 190 events at
Chinese nuclear power plants from
2007--2020~\cite{zhang_analysis_2025} found that 53\% of events involved
active errors, while 92\% were associated with latent
errors---organizational and systemic weaknesses that create conditions for
failure.

\textbf{LIMITATION:} \textit{Human factors impose fundamental reliability
limits that cannot be overcome through training alone.} The persistent human
error contribution despite four decades of improvements demonstrates that
these limitations are fundamental rather than a remediable part of
human-driven control.

\subsection{Formal Methods}
\subsubsection{HARDENS}

The High Assurance Rigorous Digital Engineering for Nuclear Safety (HARDENS)
project represents the most advanced application of formal methods to
nuclear reactor control systems to date~\cite{Kiniry2024}.

HARDENS aimed to address a fundamental dilemma: existing U.S. nuclear
control rooms rely on analog technologies from the
1950s--60s.This technology is obsolete compared to
modern control systems and incurs significant risk and cost. The NRC
contracted Galois, a formal methods firm, 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 a Reactor Trip System (RTS)
implementation with full traceability from \oldt{NRC Request for Proposals
and IEEE standards through formal architecture specifications to verified
software.} \newt{regulatory requirements through formal architecture
specifications to verified software.}

\oldt{HARDENS employed formal methods tools and techniques across the
verification hierarchy.} \newt{HARDENS employed formal methods tools at
every level of system development, from high-level requirements through
executable models to generated code.}High-level specifications used
Lando, SysMLv2, and FRET (NASA Formal Requirements Elicitation Tool) to
capture stakeholder requirements, domain engineering, certification
requirements, and safety requirements. Requirements were analyzed for
consistency, completeness, and realizability using SAT and SMT solvers.
Executable formal models used Cryptol to create a behavioral model of the
entire RTS, including all subsystems, components, and limited digital twin
models of sensors, actuators, and compute infrastructure. Automatic code
synthesis generated verifiable C implementations and SystemVerilog hardware
implementations directly from Cryptol models---eliminating the traditional
gap between specification and implementation where errors commonly
arise.

\oldt{Despite its accomplishments, HARDENS has a fundamental limitation
directly relevant to hybrid control synthesis: the project addressed only
discrete digital control logic without modeling or verifying continuous
reactor dynamics. The Reactor Trip System specification and verification
covered discrete state transitions (trip/no-trip decisions), digital sensor
input processing through discrete logic, and discrete actuation outputs
(reactor trip commands). The project did not address continuous dynamics of
nuclear reactor physics. Real reactor safety depends on the interaction
between continuous processes---temperature, pressure, neutron flux---evolving
in response to discrete control decisions. HARDENS verified the discrete
controller in isolation but not the closed-loop hybrid system behavior.}
\newt{Despite these accomplishments, HARDENS addressed only discrete digital
control logic. The Reactor Trip System verification covered discrete state
transitions, digital sensor processing, and discrete actuation outputs. It
did not model or verify continuous reactor dynamics. Real reactor safety
depends on the interaction between continuous processes---temperature,
pressure, neutron flux---evolving in response to discrete control decisions.
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.} 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.

HARDENS produced a demonstrator system at Technology Readiness Level 2--3
(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{Kiniry2024} that all
material is considered in development, not a finalized product, and that
``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, NRC licensing review, or human
factors validation with licensed operators in realistic control room
scenarios.

\textbf{LIMITATION:} \textit{HARDENS achieved TRL 2--3 without experimental
validation.} \oldt{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.} \newt{The gap between formal verification and actual system
deployment remains wide: integration with legacy systems, long-term
reliability under harsh environments, human-system interaction, and
regulatory acceptance of formal methods as primary assurance
evidence.}

\subsubsection{Sequent Calculus and Differential Dynamic Logic}
\oldt{There has been additional work to do verification of hybrid systems by
extending the temporal logics directly.} \newt{A separate line of work
extends temporal logics to verify hybrid systems
directly.}
The result has been the field of differential dynamic logic (dL). dL
introduces two additional operators into temporal logic: the box operator and
the diamond operator. The box operator \([\alpha]\phi\) states that for some
region \(\phi\), the hybrid system \(\alpha\) always remains within that
region. In this way, it is a safety \oldt{ivariant} \newt{invariant} being
enforced for the system.The second operator, the diamond operator
\(<\alpha>\phi\) says that for the region \(\phi\), there is at least one
trajectory of \(\alpha\) that enters that region. This is a declaration of a
liveness property.

%source: https://symbolaris.com/logic/dL.html

While dL allows for the specification of these liveness and safety
properties, actually proving them for a given hybrid system is quite
difficult. Automated proof assistants such as KeYmaera X exist to help
develop proofs of systems using dL, but so far have been insufficient for
reasonably complex hybrid systems. The main issue behind creating system
proofs using dL is state space explosion and non-terminating solutions.
%Source: that one satellite tracking paper that has the problem with the
%gyroscopes overloding and needing to dump speed all the time
Approaches have been made to alleviate these issues for nuclear power
contexts using contract and decomposition based methods, \oldt{but are far
from a complete methodology to design systems with.} \newt{but do not yet
constitute a complete design methodology.}
%source: Manyu's thesis.
Instead, these approaches have been used on systems that have been designed a
priori, and require expert knowledge to create the system proofs.

%Maybe a limitation here? Something about dL doesn't scale or help in design
%very much, so the limitation is that logic based hybrid system approaches
%have not been used in the DESIGN of autonomous controllers, only in the
%analysis of systems that already exist.

\textbf{LIMITATION:} \textit{Differential dynamic logic has been used for
post-hoc analysis of existing systems, not for the constructive design of
autonomous controllers.} Current dL-based approaches verify systems that
have already been designed, requiring expert knowledge to construct proofs.
They have not been applied to the synthesis of new controllers from
operational requirements.
