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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}