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% Reviewer 1.
\section*{Comments by Reviewer \#1}
\begin{itemize}
\item {\it 1.) I don't see the point of working with a more general
library of possibly infinite graphs when it is clear that no
(necessarily finite) evolution of the state could ever generate an
infinite graph.}
It seemed the most convenient approach for us and we added the following
comment on page 8 about justifying our choice. There was already a
comment about having finiteness as a property, rather than a built-in
assumption.
\begin{quote}
It seems for our purposes the most convenient representation of
graphs are binary relations given by sets of pairs. The pairs
stand for the edges in graphs. This relation-based representation
has the advantage that the notions \textit{waiting} and
\textit{holding} are already defined in terms of relations
amongst threads and resources. Also, we can easily re-use the
standard notions of transitive closure operations $\_*$ and $\_+$,
as well as relation composition for our graphs.
\end{quote}
\item {\it 2.) Later, the release function is defined using Hilbert
choice (Isabelle/HOL's \texttt{SOME} function) as a method for emulating
non-determinism. This is horrid. The highest level generation of
traces handles non-determinism beautifully; using choice at a
lower level lets you keep things functional, but hides the fact
that you want to model non-determinism at the lower level as
well. I agree that the final theorem does imply that the way in
which the new waiting list is chosen is irrelevant, but this
implication is only apparent through a close reading of that
definition. Better I think to lift the non-determinism and make it
more apparent at the top level.}
\begin{quote}
We generally agree with the reviewer about the use of
\texttt{SOME}, but in this instance we cannot see a way to make
this nondeterminism explicit without complicating the
toplevel rules about \textit{PIP} on page 12. By using
\texttt{SOME}, we can leave implicit the order of the waiting
queue returned by release (which corresponds to the original
system call from Sha et al). If we represent the non-determinism
explicitly, we would need to add another argument to $V$
specifying which thread is the next one that obtains the lock and
add another premise to the \textit{PIP} rule for ensuring that
this thread is member of the waiting queue.
\end{quote}
\item {\it 3.) In \S 4, there is an assumption made about the number of
threads allowed to be created. Given the form of theorem 2, this
seems to me to be unnecessary. It's certainly extremely weird and
ugly because it says that there are at most BC many thread
creation events in all possible future traces (of which there are
of course infinitely many). \ldots}
\begin{quote}
We think this mis-reads our theorem: Although the constrains we
put on thread creation prohibit the creation of higher priority
threads, the creation of lower priority threads may still consume
CPU time and prevent \textit{th} from accomplishing its task. That
is the purpose we put in the \textit{BC} constraint to limit the
overall number of thread creations. We slightly augmented the
sentence on page 15 by:
\begin{quote}
Otherwise our PIP scheduler could be ``swamped'' with
\textit{Create}-requests of lower priority threads.
\end{quote}
\end{quote}
\item {\it Why are variables corresponding to resources given the name
\textit{cs}? This is confusing. \textit{rsc} or \textit{r} would be
better.}
\begin{quote}
\textit{cs} stands for ``critical section'', which is the original name
used by Sha et al to represent ``critical resources''.
\end{quote}
\item All other comments of the reviewer have been implemented.
\end{itemize}
%%%%
% Reviever 2.
\section*{Comments by Reviewer \#2}
\begin{itemize}
\item {\it Well-founded comment:} We updated the paragraph where
\textit{acyclic} and \textit{well-founded} are used for the first
time.
\begin{quote}
Note that forests can have trees with infinite depth and containing
nodes with infinitely many children. A \emph{finite forest} is a
forest whose underlying relation is well-founded and every node
has finitely many children (is only finitely branching).
We also added a footnote that we are using the standard definition of
well-foundedness from Isabelle.
\end{quote}
\item {\it Comment about graph-library:} This point was also raised by
Reviewer~\#1 and we gave a better justification on page 8 (see
answer to first point of Reviewer~\#1).
\item {\it 20.) In the case of "Set th prio:", it is not declared what
the cprec (e::s) th should be (presumably it is something like
Max((prio,|s|),prec th s) or possibly just prec th (e::s) given
the assumptions on Set events listed before).}
\begin{quote}
We note the concern of the reviewer about the effect of the
Set-operation on the \textit{cprec} value of a thread. According to
equation (6) on page 11, the \textit{cprec} of a thread is
determined by the precedence values in its subtree, while the Set
operation only changes the precedence of the `Set' thread. If the
reviewer thinks we should add this explanation again, then we are
happy to do so.
\end{quote}
\end{itemize}
\section*{Comments by Reviewer \#3}
\begin{itemize}
\item {\it The verification is at the model level, instead of code level:...}
\begin{quote}
The verification is indeed on the level of the algorithm. A
verification of the C-code is *well* beyond the scope of the
paper. For example, it would require a formalised semantics for C
(as for example given in the seL4-project). This and interfacing
with it would be a tremendous amount of work and we are not sure
whether their results would actually sufficient for the code we
wrote.
In light of this, we have made it clearer in the abstract and in a
footnote on the first page that our C-code is unverified. Additionally
we added the following sentence in Section 5 before we describe the
C-code:
\begin{quote}
While there is no formal connection between our formalisation
and the C-code shown below, the results of the formalisation
clearly shine through in the design of the code.
\end{quote}
\end{quote}
\item{\it The model cannot express the execution of instructions. As a
result, it is difficult to express the case that the critical
region does infinite loops without generating events for system
calls.}
\begin{quote}
Yes, we discuss this limitation in the first paragraph of section
4 and already wrote that in this aspect we do not
improve the limitations of the original paper by Sha et al.
\end{quote}
\item{\it The authors should have provided URL of their mechanized
proofs for the review process.}
\begin{quote}
As is custom, we have given a link to the sources.
It is mentioned in the last
sentence of the conclusion.
The code of our formalisation can be downloaded from the Mercurial
repository at \url{http://talisker.inf.kcl.ac.uk/cgi-bin/repos.cgi/pip}.
\end{quote}
\item{\it It is more like engineering work. It's unclear what general
principles or theories are proposed.}
\begin{quote}
The general point we are making is that a `proof-by-hand' is
generally worthless in this area for ensuring the correctness of
an algorithm. We underline this point by listing the references
[13, 14, 15, 20, 24, 25], which all repeat the error from the
original paper.
More specifically we extend the claims of Sha at al and give
a machine-checked formalisation of our claims (the first such
formalisation for PIP). We also wrote about our experiences:
\begin{quote}
Following good experience in earlier work [27], our model of PIP
is based on Paulson’s inductive approach for protocol
verification [18].
\end{quote}
\end{quote}
\item{\it 1) There are some places that the authors follow the Isabelle
syntax, which require more explanation. For instance, in the
definitions of "threads" and "priority" in page 5, I had trouble
understanding the underscore until I realized that the 4 cases are
pattern matching and they have orders.}
\begin{quote}
We have added the sentence after the definition of \textit{threads}:
\begin{quote}
We use \_ to match any pattern, like in functional programming.
\end{quote}
\end{quote}
\item{\it 2) The last assumption on page 13: the assumption seems very
strong. If you prove Theorem 1 inductively on all s (which means
you have to consider s of length 1), then following the assumption
you essentially require that the highest priority task is the
first task created.}
\begin{quote}
We feel like this mis-reads what we are proving and how we
organised the statements. The thread \textit{th} is the thread
with the highest priority in the state \textit{s}. It can be in
\textit{s} at any `place'---it does not need to be the first
one. What we prove is what happens to \textit{th} in the state
\textit{$s' @ s$} which happens after \textit{s}. We only require
that threads created after \textit{s} need to have a lower
priority.
\end{quote}
\item{\it 3) 2nd line of the proofs of Lemma 1: th should be th' ?}
\begin{quote}
Yes, we corrected this error.
\end{quote}
\item{\it 4) Is your state (event traces) finite or infinite?}
\begin{quote}
Yes, they are always finite, but can be arbitrary long.
\end{quote}
\item{\it 5) Assumption at the bottom of page 15: I don't understand
why this assumption is necessary. First, is es a finite or
infinite trace? If es is finite, of course there's limited number
of Create-requests. Even if it can be infinite, I don't see how
the PIP scheduler could be "swamped" with create requests. You
already assumed that there are no threads of precedence higher
than th created in es. If all the newly created threads have lower
precedence, they have no chance to run anyway (because of the low
precedence). Then why should we care?}
\begin{quote}
Yes, \textit{es} and \textit{$es @ s$} are finite traces (finite
list of events), but they can be arbitrarily long.
This means that knowing that \text{es} is finite, does *not* bound
the number of create events.
We also understand that newly created threads have no chance to
run (because of the lower priority), but the process of creating
any processes consumes according to our model some time and
therefore needs to be bounded. A Create event is not assumed to be
`instantaneously'.
\end{quote}
\item {\it 6) The next assumption in page 16 does not look right to me
either. What if there are no actions of th' in es at all, which
may be caused by infinite loop inside the critical region of th'
(but the loop does not generate any events)? In this case, the
assumption is still satisfied because the length is 0, which is
less than BND(th').}
\begin{quote}
We made this point clearer (it was also requested by another reviewer).
We discuss this limitation in more depth in the first paragraph of section
4 and already wrote that in this aspect we do not
improve the limitations of the original paper by Sha et al.
\end{quote}
\item All other comments have been implemented.
\end{document}