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\title{Core SC DOM\\\medskip \Large 
  A Formal Model of the Document Object Model for Safe Components}%
\author{%
  \href{https://www.brucker.ch/}{Achim~D.~Brucker}\footnotemark[1]
  \and
  \href{https://www.michael-herzberg.de/}{Michael Herzberg}\footnotemark[2]
}  

\publishers{
  \footnotemark[1]~Department of Computer Science, University of Exeter, Exeter, UK\texorpdfstring{\\}{, }
  \texttt{a.brucker@exeter.ac.uk}\\[2em]
  %
  \footnotemark[2]~  Department of Computer Science, The University of Sheffield, Sheffield, UK\texorpdfstring{\\}{, }
   \texttt{msherzberg1@sheffield.ac.uk}
}
\begin{document}
  \maketitle
  \begin{abstract}
    \begin{quote}
      In this AFP entry, we formalize the core of the \emph{Safely
        Composable Document Object Model} (SC DOM). The SC DOM improve
      the standard DOM by strengthening the tree boundaries set by
      shadow roots: in the SC DOM, the shadow root is a sub-class of
      the document class (instead of a base class). 

      This modifications also results in changes to some API methods
      (e.g., getOwnerDocument) to return the nearest shadow root
      rather than the document root. As a result, many API methods
      that, when called on a node inside a shadow tree, would
      previously ``break out'' and return or modify nodes that are
      possibly outside the shadow tree, now stay within its
      boundaries. This change in behavior makes programs that operate
      on shadow trees more predictable for the developer and allows
      them to make more assumptions about other code accessing the
      DOM. 

      \bigskip
    \noindent{\textbf{Keywords:}} 
      Document Object Model, DOM, SC DOM, Safely Composable DOM, Formal Semantics, Isabelle/HOL      
    \end{quote}
  \end{abstract}


\tableofcontents
\cleardoublepage

\chapter{Introduction}
In a world in which more and more applications are offered as services
on the internet, web browsers start to take on a similarly central
role in our daily IT infrastructure as operating systems. Thus, web
browsers should be developed as rigidly and formally as operating
systems. While formal methods are a well-established technique in the
development of operating systems (see,
\eg,~\citet{klein:operating:2009} for an overview of formal
verification of operating systems), there are few proposals for
improving the development of web browsers using formal
approaches~\cite{gardner.ea:dom:2008,raad.ea:dom:2016,jang.ea:establishing:2012,bohannon.ea:featherweight:2010}.

As a first step towards a verified client-side web application stack,
we model and formally verify the Document Object Model (DOM) in
Isabelle/HOL\@. The DOM~\cite{whatwg:dom:2017,w3c:dom:2015} is
\emph{the} central data structure of all modern web browsers.  At its
core, the Document Object Model (DOM), defines a tree-like data
structure for representing documents in general and HTML documents in
particular. Thus, the correctness of a DOM implementation is crucial
for ensuring that a web browser displays web pages correctly.
Moreover, the DOM is the core data structure underlying client-side
JavaScript programs, \ie, client-side JavaScript programs are mostly
programs that read, write, and update the DOM.

In more detail, we formalize the core core of the \emph{Safely
  Composable Document Object Model} (SC DOM) a shallow
embedding~\cite{joyce.ea:higher:1994} in Isabelle/HOL\@. Our
formalization is based on a typed data model for the \emph{node-tree},
\ie, a data structure for representing XML-like documents in a tree
structure. Furthermore, we formalize a typed heap for storing
(partial) node-trees together with the necessary consistency
constraints. Finally, we formalize the operations (as described in the
DOM standard~\cite{whatwg:dom:2017}) on this heap that allow
manipulating node-trees.

Our machine-checked formalization of the DOM node
tree~\cite{whatwg:dom:2017} has the following desirable properties:
\begin{itemize}
\item It provides a \emph{consistency guarantee.} Since all
  definitions in our formal semantics are conservative and all rules
  are derived, the logical consistency of the DOM node-tree is reduced
  to the consistency of HOL.
\item It serves as a \emph{technical basis for a proof system.}  Based
  on the derived rules and specific setup of proof tactics over
  node-trees, our formalization provides a generic proof environment
  for the verification of programs manipulating node-trees.
\item It is \emph{executable}, which allows to validate its compliance
  to the standard by evaluating the compliance test suite on the
  formal model and
\item It is \emph{extensible} in the sense
  of~\cite{brucker.ea:extensible:2008-b,brucker:interactive:2007},
  \ie, properties proven over the core DOM do not need to be re-proven
  for object-oriented extensions such as the HTML document model.
\end{itemize}

The rest of this document is automatically generated from the
formalization in Isabelle/HOL, i.e., all content is checked by
Isabelle.\footnote{For a brief overview of the work, we refer the
  reader to~\cite{brucker.ea:core-dom:2018,herzberg:web-components:2020}.} 
The structure follows the theory dependencies (see \autoref{fig:session-graph}): we start
with introducing the technical preliminaries of our formalization
(\autoref{cha:perliminaries}).  Next, we introduce the concepts of
pointers (\autoref{cha:pointers}) and classes (\autoref{cha:classes}),
i.e., the core object-oriented datatypes of the DOM. On top of this
data model, we define the functional behavior of the DOM classes,
i.e., their methods (\autoref{cha:monads}). In \autoref{cha:dom}, we
introduce the formalization of the functionality of the core DOM,
i.e., the \emph{main entry point for users} that want to use this AFP
entry. Finally, we formalize the relevant compliance test cases in
\autoref{cha:tests}.

\paragraph{Important Note:} This document describes the formalization
of the \emph{Safely Composable Document Object Model} (SC DOM), which
deviated in one important aspect from the official DOM standard: in
the SC DOM, the shadow root is a sub-class of the document class
(instead of a base class). This modification results in a stronger
notion of web components that provide improved safety properties for
the composition of web components. While the SC DOM still passes the
compliance test suite as provided by the authors of the DOM standard,
its data model is different. We refer readers interested in a
formalisation of the standard compliant DOM to the AFP entry
``Core\_DOM''~\cite{brucker.ea:afp-core-dom:2018}.


\begin{figure}
  \centering
  \includegraphics[width=.8\textwidth]{session_graph}
  \caption{The Dependency Graph of the Isabelle Theories.\label{fig:session-graph}}
\end{figure}

\clearpage

\chapter{Preliminaries}
\label{cha:perliminaries}
In this chapter, we introduce the technical preliminaries of our
formalization of the core DOM, namely a mechanism for hiding type
variables and the heap error monad.
\input{Hiding_Type_Variables}
\input{Heap_Error_Monad}

\chapter{References and Pointers}
\label{cha:pointers}
In this chapter, we introduce a generic type for object-oriented
references and typed pointers for each class type defined in the DOM
standard. 
\input{Ref}
\input{ObjectPointer}
\input{NodePointer}
\input{ElementPointer} brucker.ea:afp-core-dom:2018
\input{CharacterDataPointer}
\input{DocumentPointer}
\input{ShadowRootPointer}

\chapter{Classes}
\label{cha:classes}
In this chapter, we introduce the classes of our DOM model. 
The definition of the class types follows closely the one of the
pointer types.  Instead of datatypes, we use records for our classes.
a generic type for object-oriented references and typed pointers for
each class type defined in the DOM standard.
\input{BaseClass}
\input{ObjectClass}
\input{NodeClass}
\input{ElementClass}
\input{CharacterDataClass}
\input{DocumentClass}

\chapter{Monadic Object Constructors and Accessors}
\label{cha:monads}
In this chapter, we introduce the moandic method definitions for the
classes of our DOM formalization. Again the overall structure follows
the same structure as for the class types and the pointer types.
\input{BaseMonad}
\input{ObjectMonad}
\input{NodeMonad}
\input{ElementMonad}
\input{CharacterDataMonad}
\input{DocumentMonad}

\chapter{The Core SC DOM}
\label{cha:dom}
In this chapter, we introduce the formalization of the core DOM, i.e.,
the most important algorithms for querying or modifying the DOM, as
defined in the standard. For more details, we refer the reader to
\cite{brucker.ea:core-dom:2018}.
\input{Core_DOM_Basic_Datatypes}
\input{Core_DOM_Functions}
\input{Core_DOM_Heap_WF}
\input{Core_DOM}

\chapter{Test Suite}
\label{cha:tests}
In this chapter, we present the formalized compliance test cases for
the core DOM. As our formalization is executable, we can
(symbolically) execute the test cases on top of our model. Executing
these test cases successfully shows that our model is compliant to the
official DOM standard. As future work, we plan to generate test cases
from our formal model (e.g.,
using~\cite{brucker.ea:interactive:2005,brucker.ea:theorem-prover:2012})
to improve the quality of the official compliance test suite. For more
details on the relation of test and proof in the context of web
standards, we refer the reader to
\cite{brucker.ea:standard-compliance-testing:2018}.
\input{Core_DOM_BaseTest} \input{Document_adoptNode}
\input{Document_getElementById} \input{Node_insertBefore}
\input{Node_removeChild} \input{Core_DOM_Tests}

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