(*<*)
theory Paper
imports "../CpsG" "../ExtGG" "~~/src/HOL/Library/LaTeXsugar"
begin
(*
find_unused_assms CpsG
find_unused_assms ExtGG
find_unused_assms Moment
find_unused_assms Precedence_ord
find_unused_assms PrioG
find_unused_assms PrioGDef
*)
ML {*
open Printer;
show_question_marks_default := false;
*}
notation (latex output)
Cons ("_::_" [78,77] 73) and
vt ("valid'_state") and
runing ("running") and
birthtime ("last'_set") and
If ("(\<^raw:\textrm{>if\<^raw:}> (_)/ \<^raw:\textrm{>then\<^raw:}> (_)/ \<^raw:\textrm{>else\<^raw:}> (_))" 10) and
Prc ("'(_, _')") and
holding ("holds") and
waiting ("waits") and
Th ("T") and
Cs ("C") and
readys ("ready") and
depend ("RAG") and
preced ("prec") and
cpreced ("cprec") and
dependents ("dependants") and
cp ("cprec") and
holdents ("resources") and
original_priority ("priority") and
DUMMY ("\<^raw:\mbox{$\_\!\_$}>")
(*abbreviation
"detached s th \<equiv> cntP s th = cntV s th"
*)
(*>*)
section {* Introduction *}
text {*
Many real-time systems need to support threads involving priorities and
locking of resources. Locking of resources ensures mutual exclusion
when accessing shared data or devices that cannot be
preempted. Priorities allow scheduling of threads that need to
finish their work within deadlines. Unfortunately, both features
can interact in subtle ways leading to a problem, called
\emph{Priority Inversion}. Suppose three threads having priorities
$H$(igh), $M$(edium) and $L$(ow). We would expect that the thread
$H$ blocks any other thread with lower priority and the thread itself cannot
be blocked indefinitely by threads with lower priority. Alas, in a naive
implementation of resource locking and priorities this property can
be violated. For this let $L$ be in the
possession of a lock for a resource that $H$ also needs. $H$ must
therefore wait for $L$ to exit the critical section and release this
lock. The problem is that $L$ might in turn be blocked by any
thread with priority $M$, and so $H$ sits there potentially waiting
indefinitely. Since $H$ is blocked by threads with lower
priorities, the problem is called Priority Inversion. It was first
described in \cite{Lampson80} in the context of the
Mesa programming language designed for concurrent programming.
If the problem of Priority Inversion is ignored, real-time systems
can become unpredictable and resulting bugs can be hard to diagnose.
The classic example where this happened is the software that
controlled the Mars Pathfinder mission in 1997 \cite{Reeves98}.
Once the spacecraft landed, the software shut down at irregular
intervals leading to loss of project time as normal operation of the
craft could only resume the next day (the mission and data already
collected were fortunately not lost, because of a clever system
design). The reason for the shutdowns was that the scheduling
software fell victim to Priority Inversion: a low priority thread
locking a resource prevented a high priority thread from running in
time, leading to a system reset. Once the problem was found, it was
rectified by enabling the \emph{Priority Inheritance Protocol} (PIP)
\cite{Sha90}\footnote{Sha et al.~call it the \emph{Basic Priority
Inheritance Protocol} \cite{Sha90} and others sometimes also call it
\emph{Priority Boosting} or \emph{Priority Donation}.} in the scheduling software.
The idea behind PIP is to let the thread $L$ temporarily inherit
the high priority from $H$ until $L$ leaves the critical section
unlocking the resource. This solves the problem of $H$ having to
wait indefinitely, because $L$ cannot be blocked by threads having
priority $M$. While a few other solutions exist for the Priority
Inversion problem, PIP is one that is widely deployed and
implemented. This includes VxWorks (a proprietary real-time OS used
in the Mars Pathfinder mission, in Boeing's 787 Dreamliner, Honda's
ASIMO robot, etc.), but also the POSIX 1003.1c Standard realised for
example in libraries for FreeBSD, Solaris and Linux.
One advantage of PIP is that increasing the priority of a thread
can be dynamically calculated by the scheduler. This is in contrast
to, for example, \emph{Priority Ceiling} \cite{Sha90}, another
solution to the Priority Inversion problem, which requires static
analysis of the program in order to prevent Priority
Inversion. However, there has also been strong criticism against
PIP. For instance, PIP cannot prevent deadlocks when lock
dependencies are circular, and also blocking times can be
substantial (more than just the duration of a critical section).
Though, most criticism against PIP centres around unreliable
implementations and PIP being too complicated and too inefficient.
For example, Yodaiken writes in \cite{Yodaiken02}:
\begin{quote}
\it{}``Priority inheritance is neither efficient nor reliable. Implementations
are either incomplete (and unreliable) or surprisingly complex and intrusive.''
\end{quote}
\noindent
He suggests avoiding PIP altogether by not allowing critical
sections to be preempted. Unfortunately, this solution does not
help in real-time systems with hard deadlines for high-priority
threads.
In our opinion, there is clearly a need for investigating correct
algorithms for PIP. A few specifications for PIP exist (in English)
and also a few high-level descriptions of implementations (e.g.~in
the textbook \cite[Section 5.6.5]{Vahalia96}), but they help little
with actual implementations. That this is a problem in practice is
proved by an email by Baker, who wrote on 13 July 2009 on the Linux
Kernel mailing list:
\begin{quote}
\it{}``I observed in the kernel code (to my disgust), the Linux PIP
implementation is a nightmare: extremely heavy weight, involving
maintenance of a full wait-for graph, and requiring updates for a
range of events, including priority changes and interruptions of
wait operations.''
\end{quote}
\noindent
The criticism by Yodaiken, Baker and others suggests another look
at PIP from a more abstract level (but still concrete enough
to inform an implementation), and makes PIP a good candidate for a
formal verification. An additional reason is that the original
presentation of PIP~\cite{Sha90}, despite being informally
``proved'' correct, is actually \emph{flawed}.
Yodaiken \cite{Yodaiken02} points to a subtlety that had been
overlooked in the informal proof by Sha et al. They specify in
\cite{Sha90} that after the thread (whose priority has been raised)
completes its critical section and releases the lock, it ``returns
to its original priority level.'' This leads them to believe that an
implementation of PIP is ``rather straightforward''~\cite{Sha90}.
Unfortunately, as Yodaiken points out, this behaviour is too
simplistic. Consider the case where the low priority thread $L$
locks \emph{two} resources, and two high-priority threads $H$ and
$H'$ each wait for one of them. If $L$ releases one resource
so that $H$, say, can proceed, then we still have Priority Inversion
with $H'$ (which waits for the other resource). The correct
behaviour for $L$ is to switch to the highest remaining priority of
the threads that it blocks. The advantage of formalising the
correctness of a high-level specification of PIP in a theorem prover
is that such issues clearly show up and cannot be overlooked as in
informal reasoning (since we have to analyse all possible behaviours
of threads, i.e.~\emph{traces}, that could possibly happen).\medskip
\noindent
{\bf Contributions:} There have been earlier formal investigations
into PIP \cite{Faria08,Jahier09,Wellings07}, but they employ model
checking techniques. This paper presents a formalised and
mechanically checked proof for the correctness of PIP (to our
knowledge the first one). In contrast to model checking, our
formalisation provides insight into why PIP is correct and allows us
to prove stronger properties that, as we will show, can help with an
efficient implementation of PIP in the educational PINTOS operating
system \cite{PINTOS}. For example, we found by ``playing'' with the
formalisation that the choice of the next thread to take over a lock
when a resource is released is irrelevant for PIP being correct---a
fact that has not been mentioned in the literature and not been used
in the reference implementation of PIP in PINTOS. This is important
for an efficient implementation of PIP, because we can give the lock
to the thread with the highest priority so that it terminates more
quickly.
*}
section {* Formal Model of the Priority Inheritance Protocol *}
text {*
The Priority Inheritance Protocol, short PIP, is a scheduling
algorithm for a single-processor system.\footnote{We shall come back
later to the case of PIP on multi-processor systems.}
Following good experience in earlier work \cite{Wang09},
our model of PIP is based on Paulson's inductive approach to protocol
verification \cite{Paulson98}. In this approach a \emph{state} of a system is
given by a list of events that happened so far (with new events prepended to the list).
\emph{Events} of PIP fall
into five categories defined as the datatype:
\begin{isabelle}\ \ \ \ \ %%%
\mbox{\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{7mm}}l}
\isacommand{datatype} event
& @{text "="} & @{term "Create thread priority"}\\
& @{text "|"} & @{term "Exit thread"} \\
& @{text "|"} & @{term "Set thread priority"} & {\rm reset of the priority for} @{text thread}\\
& @{text "|"} & @{term "P thread cs"} & {\rm request of resource} @{text "cs"} {\rm by} @{text "thread"}\\
& @{text "|"} & @{term "V thread cs"} & {\rm release of resource} @{text "cs"} {\rm by} @{text "thread"}
\end{tabular}}
\end{isabelle}
\noindent
whereby threads, priorities and (critical) resources are represented
as natural numbers. The event @{term Set} models the situation that
a thread obtains a new priority given by the programmer or
user (for example via the {\tt nice} utility under UNIX). As in Paulson's work, we
need to define functions that allow us to make some observations
about states. One, called @{term threads}, calculates the set of
``live'' threads that we have seen so far:
\begin{isabelle}\ \ \ \ \ %%%
\mbox{\begin{tabular}{lcl}
@{thm (lhs) threads.simps(1)} & @{text "\<equiv>"} &
@{thm (rhs) threads.simps(1)}\\
@{thm (lhs) threads.simps(2)[where thread="th"]} & @{text "\<equiv>"} &
@{thm (rhs) threads.simps(2)[where thread="th"]}\\
@{thm (lhs) threads.simps(3)[where thread="th"]} & @{text "\<equiv>"} &
@{thm (rhs) threads.simps(3)[where thread="th"]}\\
@{term "threads (DUMMY#s)"} & @{text "\<equiv>"} & @{term "threads s"}\\
\end{tabular}}
\end{isabelle}
\noindent
In this definition @{term "DUMMY # DUMMY"} stands for list-cons.
Another function calculates the priority for a thread @{text "th"}, which is
defined as
\begin{isabelle}\ \ \ \ \ %%%
\mbox{\begin{tabular}{lcl}
@{thm (lhs) original_priority.simps(1)[where thread="th"]} & @{text "\<equiv>"} &
@{thm (rhs) original_priority.simps(1)[where thread="th"]}\\
@{thm (lhs) original_priority.simps(2)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
@{thm (rhs) original_priority.simps(2)[where thread="th" and thread'="th'"]}\\
@{thm (lhs) original_priority.simps(3)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
@{thm (rhs) original_priority.simps(3)[where thread="th" and thread'="th'"]}\\
@{term "original_priority th (DUMMY#s)"} & @{text "\<equiv>"} & @{term "original_priority th s"}\\
\end{tabular}}
\end{isabelle}
\noindent
In this definition we set @{text 0} as the default priority for
threads that have not (yet) been created. The last function we need
calculates the ``time'', or index, at which time a process had its
priority last set.
\begin{isabelle}\ \ \ \ \ %%%
\mbox{\begin{tabular}{lcl}
@{thm (lhs) birthtime.simps(1)[where thread="th"]} & @{text "\<equiv>"} &
@{thm (rhs) birthtime.simps(1)[where thread="th"]}\\
@{thm (lhs) birthtime.simps(2)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
@{thm (rhs) birthtime.simps(2)[where thread="th" and thread'="th'"]}\\
@{thm (lhs) birthtime.simps(3)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
@{thm (rhs) birthtime.simps(3)[where thread="th" and thread'="th'"]}\\
@{term "birthtime th (DUMMY#s)"} & @{text "\<equiv>"} & @{term "birthtime th s"}\\
\end{tabular}}
\end{isabelle}
\noindent
In this definition @{term "length s"} stands for the length of the list
of events @{text s}. Again the default value in this function is @{text 0}
for threads that have not been created yet. A \emph{precedence} of a thread @{text th} in a
state @{text s} is the pair of natural numbers defined as
\begin{isabelle}\ \ \ \ \ %%%
@{thm preced_def[where thread="th"]}
\end{isabelle}
\noindent
The point of precedences is to schedule threads not according to priorities (because what should
we do in case two threads have the same priority), but according to precedences.
Precedences allow us to always discriminate between two threads with equal priority by
taking into account the time when the priority was last set. We order precedences so
that threads with the same priority get a higher precedence if their priority has been
set earlier, since for such threads it is more urgent to finish their work. In an implementation
this choice would translate to a quite natural FIFO-scheduling of processes with
the same priority.
Next, we introduce the concept of \emph{waiting queues}. They are
lists of threads associated with every resource. The first thread in
this list (i.e.~the head, or short @{term hd}) is chosen to be the one
that is in possession of the
``lock'' of the corresponding resource. We model waiting queues as
functions, below abbreviated as @{text wq}. They take a resource as
argument and return a list of threads. This allows us to define
when a thread \emph{holds}, respectively \emph{waits} for, a
resource @{text cs} given a waiting queue function @{text wq}.
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm cs_holding_def[where thread="th"]}\\
@{thm cs_waiting_def[where thread="th"]}
\end{tabular}
\end{isabelle}
\noindent
In this definition we assume @{text "set"} converts a list into a set.
At the beginning, that is in the state where no thread is created yet,
the waiting queue function will be the function that returns the
empty list for every resource.
\begin{isabelle}\ \ \ \ \ %%%
@{abbrev all_unlocked}\hfill\numbered{allunlocked}
\end{isabelle}
\noindent
Using @{term "holding"} and @{term waiting}, we can introduce \emph{Resource Allocation Graphs}
(RAG), which represent the dependencies between threads and resources.
We represent RAGs as relations using pairs of the form
\begin{isabelle}\ \ \ \ \ %%%
@{term "(Th th, Cs cs)"} \hspace{5mm}{\rm and}\hspace{5mm}
@{term "(Cs cs, Th th)"}
\end{isabelle}
\noindent
where the first stands for a \emph{waiting edge} and the second for a
\emph{holding edge} (@{term Cs} and @{term Th} are constructors of a
datatype for vertices). Given a waiting queue function, a RAG is defined
as the union of the sets of waiting and holding edges, namely
\begin{isabelle}\ \ \ \ \ %%%
@{thm cs_depend_def}
\end{isabelle}
\noindent
Given four threads and three resources, an instance of a RAG can be pictured
as follows:
\begin{center}
\newcommand{\fnt}{\fontsize{7}{8}\selectfont}
\begin{tikzpicture}[scale=1]
%%\draw[step=2mm] (-3,2) grid (1,-1);
\node (A) at (0,0) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>0"}};
\node (B) at (2,0) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>1"}};
\node (C) at (4,0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>1"}};
\node (D) at (4,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>2"}};
\node (E) at (6,-0.7) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>2"}};
\node (E1) at (6, 0.2) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>3"}};
\node (F) at (8,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>3"}};
\draw [<-,line width=0.6mm] (A) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (B);
\draw [->,line width=0.6mm] (C) to node [pos=0.4,sloped,above=-0.5mm] {\fnt{}waiting} (B);
\draw [->,line width=0.6mm] (D) to node [pos=0.4,sloped,below=-0.5mm] {\fnt{}waiting} (B);
\draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,below=-0.5mm] {\fnt{}holding} (E);
\draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (E1);
\draw [->,line width=0.6mm] (F) to node [pos=0.45,sloped,below=-0.5mm] {\fnt{}waiting} (E);
\end{tikzpicture}
\end{center}
\noindent
The use of relations for representing RAGs allows us to conveniently define
the notion of the \emph{dependants} of a thread using the transitive closure
operation for relations. This gives
\begin{isabelle}\ \ \ \ \ %%%
@{thm cs_dependents_def}
\end{isabelle}
\noindent
This definition needs to account for all threads that wait for a thread to
release a resource. This means we need to include threads that transitively
wait for a resource being released (in the picture above this means the dependants
of @{text "th\<^isub>0"} are @{text "th\<^isub>1"} and @{text "th\<^isub>2"}, which wait for resource @{text "cs\<^isub>1"},
but also @{text "th\<^isub>3"},
which cannot make any progress unless @{text "th\<^isub>2"} makes progress, which
in turn needs to wait for @{text "th\<^isub>0"} to finish). If there is a circle of dependencies
in a RAG, then clearly
we have a deadlock. Therefore when a thread requests a resource,
we must ensure that the resulting RAG is not circular. In practice, the
programmer has to ensure this.
Next we introduce the notion of the \emph{current precedence} of a thread @{text th} in a
state @{text s}. It is defined as
\begin{isabelle}\ \ \ \ \ %%%
@{thm cpreced_def2}\hfill\numbered{cpreced}
\end{isabelle}
\noindent
where the dependants of @{text th} are given by the waiting queue function.
While the precedence @{term prec} of a thread is determined statically
(for example when the thread is
created), the point of the current precedence is to let the scheduler increase this
precedence, if needed according to PIP. Therefore the current precedence of @{text th} is
given as the maximum of the precedence @{text th} has in state @{text s} \emph{and} all
threads that are dependants of @{text th}. Since the notion @{term "dependants"} is
defined as the transitive closure of all dependent threads, we deal correctly with the
problem in the informal algorithm by Sha et al.~\cite{Sha90} where a priority of a thread is
lowered prematurely.
The next function, called @{term schs}, defines the behaviour of the scheduler. It will be defined
by recursion on the state (a list of events); this function returns a \emph{schedule state}, which
we represent as a record consisting of two
functions:
\begin{isabelle}\ \ \ \ \ %%%
@{text "\<lparr>wq_fun, cprec_fun\<rparr>"}
\end{isabelle}
\noindent
The first function is a waiting queue function (that is, it takes a
resource @{text "cs"} and returns the corresponding list of threads
that lock, respectively wait for, it); the second is a function that
takes a thread and returns its current precedence (see
the definition in \eqref{cpreced}). We assume the usual getter and setter methods for
such records.
In the initial state, the scheduler starts with all resources unlocked (the corresponding
function is defined in \eqref{allunlocked}) and the
current precedence of every thread is initialised with @{term "Prc 0 0"}; that means
\mbox{@{abbrev initial_cprec}}. Therefore
we have for the initial shedule state
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm (lhs) schs.simps(1)} @{text "\<equiv>"}\\
\hspace{5mm}@{term "(|wq_fun = all_unlocked, cprec_fun = (\<lambda>_::thread. Prc 0 0)|)"}
\end{tabular}
\end{isabelle}
\noindent
The cases for @{term Create}, @{term Exit} and @{term Set} are also straightforward:
we calculate the waiting queue function of the (previous) state @{text s};
this waiting queue function @{text wq} is unchanged in the next schedule state---because
none of these events lock or release any resource;
for calculating the next @{term "cprec_fun"}, we use @{text wq} and
@{term cpreced}. This gives the following three clauses for @{term schs}:
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm (lhs) schs.simps(2)} @{text "\<equiv>"}\\
\hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
\hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Create th prio # s)|)"}\smallskip\\
@{thm (lhs) schs.simps(3)} @{text "\<equiv>"}\\
\hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
\hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Exit th # s)|)"}\smallskip\\
@{thm (lhs) schs.simps(4)} @{text "\<equiv>"}\\
\hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
\hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Set th prio # s)|)"}
\end{tabular}
\end{isabelle}
\noindent
More interesting are the cases where a resource, say @{text cs}, is locked or released. In these cases
we need to calculate a new waiting queue function. For the event @{term "P th cs"}, we have to update
the function so that the new thread list for @{text cs} is the old thread list plus the thread @{text th}
appended to the end of that list (remember the head of this list is assigned to be in the possession of this
resource). This gives the clause
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm (lhs) schs.simps(5)} @{text "\<equiv>"}\\
\hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
\hspace{5mm}@{text "let"} @{text "new_wq = wq(cs := (wq cs @ [th]))"} @{text "in"}\\
\hspace{8mm}@{term "(|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|)"}
\end{tabular}
\end{isabelle}
\noindent
The clause for event @{term "V th cs"} is similar, except that we need to update the waiting queue function
so that the thread that possessed the lock is deleted from the corresponding thread list. For this
list transformation, we use
the auxiliary function @{term release}. A simple version of @{term release} would
just delete this thread and return the remaining threads, namely
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}lcl}
@{term "release []"} & @{text "\<equiv>"} & @{term "[]"}\\
@{term "release (DUMMY # qs)"} & @{text "\<equiv>"} & @{term "qs"}\\
\end{tabular}
\end{isabelle}
\noindent
In practice, however, often the thread with the highest precedence in the list will get the
lock next. We have implemented this choice, but later found out that the choice
of which thread is chosen next is actually irrelevant for the correctness of PIP.
Therefore we prove the stronger result where @{term release} is defined as
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}lcl}
@{term "release []"} & @{text "\<equiv>"} & @{term "[]"}\\
@{term "release (DUMMY # qs)"} & @{text "\<equiv>"} & @{term "SOME qs'. distinct qs' \<and> set qs' = set qs"}\\
\end{tabular}
\end{isabelle}
\noindent
where @{text "SOME"} stands for Hilbert's epsilon and implements an arbitrary
choice for the next waiting list. It just has to be a list of distinctive threads and
contain the same elements as @{text "qs"}. This gives for @{term V} the clause:
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm (lhs) schs.simps(6)} @{text "\<equiv>"}\\
\hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
\hspace{5mm}@{text "let"} @{text "new_wq = release (wq cs)"} @{text "in"}\\
\hspace{8mm}@{term "(|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|)"}
\end{tabular}
\end{isabelle}
Having the scheduler function @{term schs} at our disposal, we can ``lift'', or
overload, the notions
@{term waiting}, @{term holding}, @{term depend} and @{term cp} to operate on states only.
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}rcl}
@{thm (lhs) s_holding_abv} & @{text "\<equiv>"} & @{thm (rhs) s_holding_abv}\\
@{thm (lhs) s_waiting_abv} & @{text "\<equiv>"} & @{thm (rhs) s_waiting_abv}\\
@{thm (lhs) s_depend_abv} & @{text "\<equiv>"} & @{thm (rhs) s_depend_abv}\\
@{thm (lhs) cp_def} & @{text "\<equiv>"} & @{thm (rhs) cp_def}
\end{tabular}
\end{isabelle}
\noindent
With these abbreviations in place we can introduce
the notion of a thread being @{term ready} in a state (i.e.~threads
that do not wait for any resource) and the running thread.
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm readys_def}\\
@{thm runing_def}\\
\end{tabular}
\end{isabelle}
\noindent
In the second definition @{term "DUMMY ` DUMMY"} stands for the image of a set under a function.
Note that in the initial state, that is where the list of events is empty, the set
@{term threads} is empty and therefore there is neither a thread ready nor running.
If there is one or more threads ready, then there can only be \emph{one} thread
running, namely the one whose current precedence is equal to the maximum of all ready
threads. We use sets to capture both possibilities.
We can now also conveniently define the set of resources that are locked by a thread in a
given state
\begin{isabelle}\ \ \ \ \ %%%
@{thm holdents_def}
\end{isabelle}
\noindent
and also when a thread is detached
\begin{isabelle}\ \ \ \ \ %%%
@{thm detached_def}
\end{isabelle}
\noindent
which means that @{text th} neither holds nor waits for a resource in @{text s}.
Finally we can define what a \emph{valid state} is in our model of PIP. For
example we cannot expect to be able to exit a thread, if it was not
created yet.
These validity constraints on states are characterised by the
inductive predicate @{term "step"} and @{term vt}. We first give five inference rules
for @{term step} relating a state and an event that can happen next.
\begin{center}
\begin{tabular}{c}
@{thm[mode=Rule] thread_create[where thread=th]}\hspace{1cm}
@{thm[mode=Rule] thread_exit[where thread=th]}
\end{tabular}
\end{center}
\noindent
The first rule states that a thread can only be created, if it is not alive yet.
Similarly, the second rule states that a thread can only be terminated if it was
running and does not lock any resources anymore (this simplifies slightly our model;
in practice we would expect the operating system releases all locks held by a
thread that is about to exit). The event @{text Set} can happen
if the corresponding thread is running.
\begin{center}
@{thm[mode=Rule] thread_set[where thread=th]}
\end{center}
\noindent
If a thread wants to lock a resource, then the thread needs to be
running and also we have to make sure that the resource lock does
not lead to a cycle in the RAG. In practice, ensuring the latter
is the responsibility of the programmer. In our formal
model we brush aside these problematic cases in order to be able to make
some meaningful statements about PIP.\footnote{This situation is
similar to the infamous \emph{occurs check} in Prolog: In order to say
anything meaningful about unification, one needs to perform an occurs
check. But in practice the occurs check is omitted and the
responsibility for avoiding problems rests with the programmer.}
\begin{center}
@{thm[mode=Rule] thread_P[where thread=th]}
\end{center}
\noindent
Similarly, if a thread wants to release a lock on a resource, then
it must be running and in the possession of that lock. This is
formally given by the last inference rule of @{term step}.
\begin{center}
@{thm[mode=Rule] thread_V[where thread=th]}
\end{center}
\noindent
A valid state of PIP can then be conveniently be defined as follows:
\begin{center}
\begin{tabular}{c}
@{thm[mode=Axiom] vt_nil}\hspace{1cm}
@{thm[mode=Rule] vt_cons}
\end{tabular}
\end{center}
\noindent
This completes our formal model of PIP. In the next section we present
properties that show our model of PIP is correct.
*}
section {* The Correctness Proof *}
(*<*)
context extend_highest_gen
begin
(*>*)
text {*
Sha et al.~state their first correctness criterion for PIP in terms
of the number of low-priority threads \cite[Theorem 3]{Sha90}: if
there are @{text n} low-priority threads, then a blocked job with
high priority can only be blocked a maximum of @{text n} times.
Their second correctness criterion is given
in terms of the number of critical resources \cite[Theorem 6]{Sha90}: if there are
@{text m} critical resources, then a blocked job with high priority
can only be blocked a maximum of @{text m} times. Both results on their own, strictly speaking, do
\emph{not} prevent indefinite, or unbounded, Priority Inversion,
because if a low-priority thread does not give up its critical
resource (the one the high-priority thread is waiting for), then the
high-priority thread can never run. The argument of Sha et al.~is
that \emph{if} threads release locked resources in a finite amount
of time, then indefinite Priority Inversion cannot occur---the high-priority
thread is guaranteed to run eventually. The assumption is that
programmers must ensure that threads are programmed in this way. However, even
taking this assumption into account, the correctness properties of
Sha et al.~are
\emph{not} true for their version of PIP---despite being ``proved''. As Yodaiken
\cite{Yodaiken02} pointed out: If a low-priority thread possesses
locks to two resources for which two high-priority threads are
waiting for, then lowering the priority prematurely after giving up
only one lock, can cause indefinite Priority Inversion for one of the
high-priority threads, invalidating their two bounds.
Even when fixed, their proof idea does not seem to go through for
us, because of the way we have set up our formal model of PIP. One
reason is that we allow critical sections, which start with a @{text P}-event
and finish with a corresponding @{text V}-event, to arbitrarily overlap
(something Sha et al.~explicitly exclude). Therefore we have
designed a different correctness criterion for PIP. The idea behind
our criterion is as follows: for all states @{text s}, we know the
corresponding thread @{text th} with the highest precedence; we show
that in every future state (denoted by @{text "s' @ s"}) in which
@{text th} is still alive, either @{text th} is running or it is
blocked by a thread that was alive in the state @{text s} and was waiting
for or in the possession of a lock in @{text s}. Since in @{text s}, as in
every state, the set of alive threads is finite, @{text th} can only
be blocked a finite number of times. This is independent of how many
threads of lower priority are created in @{text "s'"}. We will actually prove a
stronger statement where we also provide the current precedence of
the blocking thread. However, this correctness criterion hinges upon
a number of assumptions about the states @{text s} and @{text "s' @
s"}, the thread @{text th} and the events happening in @{text
s'}. We list them next:
\begin{quote}
{\bf Assumptions on the states {\boldmath@{text s}} and
{\boldmath@{text "s' @ s"}:}} We need to require that @{text "s"} and
@{text "s' @ s"} are valid states:
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{l}
@{term "vt s"}\\
@{term "vt (s' @ s)"}
\end{tabular}
\end{isabelle}
\end{quote}
\begin{quote}
{\bf Assumptions on the thread {\boldmath@{text "th"}:}}
The thread @{text th} must be alive in @{text s} and
has the highest precedence of all alive threads in @{text s}. Furthermore the
priority of @{text th} is @{text prio} (we need this in the next assumptions).
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{l}
@{term "th \<in> threads s"}\\
@{term "prec th s = Max (cprec s ` threads s)"}\\
@{term "prec th s = (prio, DUMMY)"}
\end{tabular}
\end{isabelle}
\end{quote}
\begin{quote}
{\bf Assumptions on the events in {\boldmath@{text "s'"}:}} We want to prove that @{text th} cannot
be blocked indefinitely. Of course this can happen if threads with higher priority
than @{text th} are continuously created in @{text s'}. Therefore we have to assume that
events in @{text s'} can only create (respectively set) threads with equal or lower
priority than @{text prio} of @{text th}. We also need to assume that the
priority of @{text "th"} does not get reset and also that @{text th} does
not get ``exited'' in @{text "s'"}. This can be ensured by assuming the following three implications.
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{l}
{If}~~@{text "Create th' prio' \<in> set s'"}~~{then}~~@{text "prio' \<le> prio"}\\
{If}~~@{text "Set th' prio' \<in> set s'"}~~{then}~~@{text "th' \<noteq> th"}~~{and}~~@{text "prio' \<le> prio"}\\
{If}~~@{text "Exit th' \<in> set s'"}~~{then}~~@{text "th' \<noteq> th"}\\
\end{tabular}
\end{isabelle}
\end{quote}
\noindent
The locale mechanism of Isabelle helps us to manage conveniently such assumptions~\cite{Haftmann08}.
Under these assumptions we shall prove the following correctness property:
\begin{theorem}\label{mainthm}
Given the assumptions about states @{text "s"} and @{text "s' @ s"},
the thread @{text th} and the events in @{text "s'"},
if @{term "th' \<in> running (s' @ s)"} and @{text "th' \<noteq> th"} then
@{text "th' \<in> threads s"}, @{text "\<not> detached s th'"} and
@{term "cp (s' @ s) th' = prec th s"}.
\end{theorem}
\noindent
This theorem ensures that the thread @{text th}, which has the
highest precedence in the state @{text s}, can only be blocked in
the state @{text "s' @ s"} by a thread @{text th'} that already
existed in @{text s} and requested or had a lock on at least
one resource---that means the thread was not \emph{detached} in @{text s}.
As we shall see shortly, that means there are only finitely
many threads that can block @{text th} in this way and then they
need to run with the same current precedence as @{text th}.
Like in the argument by Sha et al.~our
finite bound does not guarantee absence of indefinite Priority
Inversion. For this we further have to assume that every thread
gives up its resources after a finite amount of time. We found that
this assumption is awkward to formalise in our model. Therefore we
leave it out and let the programmer assume the responsibility to
program threads in such a benign manner (in addition to causing no
circularity in the @{text RAG}). In this detail, we do not
make any progress in comparison with the work by Sha et al.
However, we are able to combine their two separate bounds into a
single theorem improving their bound.
In what follows we will describe properties of PIP that allow us to prove
Theorem~\ref{mainthm} and, when instructive, briefly describe our argument.
It is relatively easy to see that
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{text "running s \<subseteq> ready s \<subseteq> threads s"}\\
@{thm[mode=IfThen] finite_threads}
\end{tabular}
\end{isabelle}
\noindent
The second property is by induction of @{term vt}. The next three
properties are
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm[mode=IfThen] waiting_unique[of _ _ "cs\<^isub>1" "cs\<^isub>2"]}\\
@{thm[mode=IfThen] held_unique[of _ "th\<^isub>1" _ "th\<^isub>2"]}\\
@{thm[mode=IfThen] runing_unique[of _ "th\<^isub>1" "th\<^isub>2"]}
\end{tabular}
\end{isabelle}
\noindent
The first property states that every waiting thread can only wait for a single
resource (because it gets suspended after requesting that resource); the second
that every resource can only be held by a single thread;
the third property establishes that in every given valid state, there is
at most one running thread. We can also show the following properties
about the @{term RAG} in @{text "s"}.
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{text If}~@{thm (prem 1) acyclic_depend}~@{text "then"}:\\
\hspace{5mm}@{thm (concl) acyclic_depend},
@{thm (concl) finite_depend} and
@{thm (concl) wf_dep_converse},\\
\hspace{5mm}@{text "if"}~@{thm (prem 2) dm_depend_threads}~@{text "then"}~@{thm (concl) dm_depend_threads}
and\\
\hspace{5mm}@{text "if"}~@{thm (prem 2) range_in}~@{text "then"}~@{thm (concl) range_in}.
\end{tabular}
\end{isabelle}
\noindent
The acyclicity property follows from how we restricted the events in
@{text step}; similarly the finiteness and well-foundedness property.
The last two properties establish that every thread in a @{text "RAG"}
(either holding or waiting for a resource) is a live thread.
The key lemma in our proof of Theorem~\ref{mainthm} is as follows:
\begin{lemma}\label{mainlem}
Given the assumptions about states @{text "s"} and @{text "s' @ s"},
the thread @{text th} and the events in @{text "s'"},
if @{term "th' \<in> threads (s' @ s)"}, @{text "th' \<noteq> th"} and @{text "detached (s' @ s) th'"}\\
then @{text "th' \<notin> running (s' @ s)"}.
\end{lemma}
\noindent
The point of this lemma is that a thread different from @{text th} (which has the highest
precedence in @{text s}) and not holding any resource, cannot be running
in the state @{text "s' @ s"}.
\begin{proof}
Since thread @{text "th'"} does not hold any resource, no thread can depend on it.
Therefore its current precedence @{term "cp (s' @ s) th'"} equals its own precedence
@{term "prec th' (s' @ s)"}. Since @{text "th"} has the highest precedence in the
state @{text "(s' @ s)"} and precedences are distinct among threads, we have
@{term "prec th' (s' @s ) < prec th (s' @ s)"}. From this
we have @{term "cp (s' @ s) th' < prec th (s' @ s)"}.
Since @{text "prec th (s' @ s)"} is already the highest
@{term "cp (s' @ s) th"} can not be higher than this and can not be lower either (by
definition of @{term "cp"}). Consequently, we have @{term "prec th (s' @ s) = cp (s' @ s) th"}.
Finally we have @{term "cp (s' @ s) th' < cp (s' @ s) th"}.
By defintion of @{text "running"}, @{text "th'"} can not be running in state
@{text "s' @ s"}, as we had to show.\qed
\end{proof}
\noindent
Since @{text "th'"} is not able to run in state @{text "s' @ s"}, it is not able to
issue a @{text "P"} or @{text "V"} event. Therefore if @{text "s' @ s"} is extended
one step further, @{text "th'"} still cannot hold any resource. The situation will
not change in further extensions as long as @{text "th"} holds the highest precedence.
From this lemma we can deduce Theorem~\ref{mainthm}: that @{text th} can only be
blocked by a thread @{text th'} that
held some resource in state @{text s} (that is not @{text "detached"}). And furthermore
that the current precedence of @{text th'} in state @{text "(s' @ s)"} must be equal to the
precedence of @{text th} in @{text "s"}.
We show this theorem by induction on @{text "s'"} using Lemma~\ref{mainlem}.
This theorem gives a stricter bound on the threads that can block @{text th} than the
one obtained by Sha et al.~\cite{Sha90}:
only threads that were alive in state @{text s} and moreover held a resource.
This means our bound is in terms of both---alive threads in state @{text s}
and number of critical resources. Finally, the theorem establishes that the blocking threads have the
current precedence raised to the precedence of @{text th}.
We can furthermore prove that under our assumptions no deadlock exists in the state @{text "s' @ s"}
by showing that @{text "running (s' @ s)"} is not empty.
\begin{lemma}
Given the assumptions about states @{text "s"} and @{text "s' @ s"},
the thread @{text th} and the events in @{text "s'"},
@{term "running (s' @ s) \<noteq> {}"}.
\end{lemma}
\begin{proof}
If @{text th} is blocked, then by following its dependants graph, we can always
reach a ready thread @{text th'}, and that thread must have inherited the
precedence of @{text th}.\qed
\end{proof}
%The following lemmas show how every node in RAG can be chased to ready threads:
%\begin{enumerate}
%\item Every node in RAG can be chased to a ready thread (@{text "chain_building"}):
% @ {thm [display] chain_building[rule_format]}
%\item The ready thread chased to is unique (@{text "dchain_unique"}):
% @ {thm [display] dchain_unique[of _ _ "th\<^isub>1" "th\<^isub>2"]}
%\end{enumerate}
%Some deeper results about the system:
%\begin{enumerate}
%\item The maximum of @{term "cp"} and @{term "preced"} are equal (@{text "max_cp_eq"}):
%@ {thm [display] max_cp_eq}
%\item There must be one ready thread having the max @{term "cp"}-value
%(@{text "max_cp_readys_threads"}):
%@ {thm [display] max_cp_readys_threads}
%\end{enumerate}
%The relationship between the count of @{text "P"} and @{text "V"} and the number of
%critical resources held by a thread is given as follows:
%\begin{enumerate}
%\item The @{term "V"}-operation decreases the number of critical resources
% one thread holds (@{text "cntCS_v_dec"})
% @ {thm [display] cntCS_v_dec}
%\item The number of @{text "V"} never exceeds the number of @{text "P"}
% (@ {text "cnp_cnv_cncs"}):
% @ {thm [display] cnp_cnv_cncs}
%\item The number of @{text "V"} equals the number of @{text "P"} when
% the relevant thread is not living:
% (@{text "cnp_cnv_eq"}):
% @ {thm [display] cnp_cnv_eq}
%\item When a thread is not living, it does not hold any critical resource
% (@{text "not_thread_holdents"}):
% @ {thm [display] not_thread_holdents}
%\item When the number of @{text "P"} equals the number of @{text "V"}, the relevant
% thread does not hold any critical resource, therefore no thread can depend on it
% (@{text "count_eq_dependents"}):
% @ {thm [display] count_eq_dependents}
%\end{enumerate}
%The reason that only threads which already held some resoures
%can be runing and block @{text "th"} is that if , otherwise, one thread
%does not hold any resource, it may never have its prioirty raised
%and will not get a chance to run. This fact is supported by
%lemma @{text "moment_blocked"}:
%@ {thm [display] moment_blocked}
%When instantiating @{text "i"} to @{text "0"}, the lemma means threads which did not hold any
%resource in state @{text "s"} will not have a change to run latter. Rephrased, it means
%any thread which is running after @{text "th"} became the highest must have already held
%some resource at state @{text "s"}.
%When instantiating @{text "i"} to a number larger than @{text "0"}, the lemma means
%if a thread releases all its resources at some moment in @{text "t"}, after that,
%it may never get a change to run. If every thread releases its resource in finite duration,
%then after a while, only thread @{text "th"} is left running. This shows how indefinite
%priority inversion can be avoided.
%All these assumptions are put into a predicate @{term "extend_highest_gen"}.
%It can be proved that @{term "extend_highest_gen"} holds
%for any moment @{text "i"} in it @{term "t"} (@{text "red_moment"}):
%@ {thm [display] red_moment}
%From this, an induction principle can be derived for @{text "t"}, so that
%properties already derived for @{term "t"} can be applied to any prefix
%of @{text "t"} in the proof of new properties
%about @{term "t"} (@{text "ind"}):
%\begin{center}
%@ {thm[display] ind}
%\end{center}
%The following properties can be proved about @{term "th"} in @{term "t"}:
%\begin{enumerate}
%\item In @{term "t"}, thread @{term "th"} is kept live and its
% precedence is preserved as well
% (@{text "th_kept"}):
% @ {thm [display] th_kept}
%\item In @{term "t"}, thread @{term "th"}'s precedence is always the maximum among
% all living threads
% (@{text "max_preced"}):
% @ {thm [display] max_preced}
%\item In @{term "t"}, thread @{term "th"}'s current precedence is always the maximum precedence
% among all living threads
% (@{text "th_cp_max_preced"}):
% @ {thm [display] th_cp_max_preced}
%\item In @{term "t"}, thread @{term "th"}'s current precedence is always the maximum current
% precedence among all living threads
% (@{text "th_cp_max"}):
% @ {thm [display] th_cp_max}
%\item In @{term "t"}, thread @{term "th"}'s current precedence equals its precedence at moment
% @{term "s"}
% (@{text "th_cp_preced"}):
% @ {thm [display] th_cp_preced}
%\end{enumerate}
%The main theorem of this part is to characterizing the running thread during @{term "t"}
%(@{text "runing_inversion_2"}):
%@ {thm [display] runing_inversion_2}
%According to this, if a thread is running, it is either @{term "th"} or was
%already live and held some resource
%at moment @{text "s"} (expressed by: @{text "cntV s th' < cntP s th'"}).
%Since there are only finite many threads live and holding some resource at any moment,
%if every such thread can release all its resources in finite duration, then after finite
%duration, none of them may block @{term "th"} anymore. So, no priority inversion may happen
%then.
*}
(*<*)
end
(*>*)
section {* Properties for an Implementation\label{implement} *}
text {*
While our formalised proof gives us confidence about the correctness of our model of PIP,
we found that the formalisation can even help us with efficiently implementing it.
For example Baker complained that calculating the current precedence
in PIP is quite ``heavy weight'' in Linux (see the Introduction).
In our model of PIP the current precedence of a thread in a state @{text s}
depends on all its dependants---a ``global'' transitive notion,
which is indeed heavy weight (see Def.~shown in \eqref{cpreced}).
We can however improve upon this. For this let us define the notion
of @{term children} of a thread @{text th} in a state @{text s} as
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm children_def2}
\end{tabular}
\end{isabelle}
\noindent
where a child is a thread that is only one ``hop'' away from the thread
@{text th} in the @{term RAG} (and waiting for @{text th} to release
a resource). We can prove the following lemma.
\begin{lemma}\label{childrenlem}
@{text "If"} @{thm (prem 1) cp_rec} @{text "then"}
\begin{center}
@{thm (concl) cp_rec}.
\end{center}
\end{lemma}
\noindent
That means the current precedence of a thread @{text th} can be
computed locally by considering only the children of @{text th}. In
effect, it only needs to be recomputed for @{text th} when one of
its children changes its current precedence. Once the current
precedence is computed in this more efficient manner, the selection
of the thread with highest precedence from a set of ready threads is
a standard scheduling operation implemented in most operating
systems.
Of course the main work for implementing PIP involves the
scheduler and coding how it should react to events. Below we
outline how our formalisation guides this implementation for each
kind of event.\smallskip
*}
(*<*)
context step_create_cps
begin
(*>*)
text {*
\noindent
\colorbox{mygrey}{@{term "Create th prio"}:} We assume that the current state @{text s'} and
the next state @{term "s \<equiv> Create th prio#s'"} are both valid (meaning the event
is allowed to occur). In this situation we can show that
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm eq_dep},\\
@{thm eq_cp_th}, and\\
@{thm[mode=IfThen] eq_cp}
\end{tabular}
\end{isabelle}
\noindent
This means in an implementation we do not have recalculate the @{text RAG} and also none of the
current precedences of the other threads. The current precedence of the created
thread @{text th} is just its precedence, namely the pair @{term "(prio, length (s::event list))"}.
\smallskip
*}
(*<*)
end
context step_exit_cps
begin
(*>*)
text {*
\noindent
\colorbox{mygrey}{@{term "Exit th"}:} We again assume that the current state @{text s'} and
the next state @{term "s \<equiv> Exit th#s'"} are both valid. We can show that
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm eq_dep}, and\\
@{thm[mode=IfThen] eq_cp}
\end{tabular}
\end{isabelle}
\noindent
This means again we do not have to recalculate the @{text RAG} and
also not the current precedences for the other threads. Since @{term th} is not
alive anymore in state @{term "s"}, there is no need to calculate its
current precedence.
\smallskip
*}
(*<*)
end
context step_set_cps
begin
(*>*)
text {*
\noindent
\colorbox{mygrey}{@{term "Set th prio"}:} We assume that @{text s'} and
@{term "s \<equiv> Set th prio#s'"} are both valid. We can show that
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm[mode=IfThen] eq_dep}, and\\
@{thm[mode=IfThen] eq_cp_pre}
\end{tabular}
\end{isabelle}
\noindent
The first property is again telling us we do not need to change the @{text RAG}.
The second shows that the @{term cp}-values of all threads other than @{text th}
are unchanged. The reason is that @{text th} is running; therefore it is not in
the @{term dependants} relation of any other thread. This in turn means that the
change of its priority cannot affect other threads.
%The second
%however states that only threads that are \emph{not} dependants of @{text th} have their
%current precedence unchanged. For the others we have to recalculate the current
%precedence. To do this we can start from @{term "th"}
%and follow the @{term "depend"}-edges to recompute using Lemma~\ref{childrenlem}
%the @{term "cp"} of every
%thread encountered on the way. Since the @{term "depend"}
%is assumed to be loop free, this procedure will always stop. The following two lemmas show, however,
%that this procedure can actually stop often earlier without having to consider all
%dependants.
%
%\begin{isabelle}\ \ \ \ \ %%%
%\begin{tabular}{@ {}l}
%@{thm[mode=IfThen] eq_up_self}\\
%@{text "If"} @{thm (prem 1) eq_up}, @{thm (prem 2) eq_up} and @{thm (prem 3) eq_up}\\
%@{text "then"} @{thm (concl) eq_up}.
%\end{tabular}
%\end{isabelle}
%
%\noindent
%The first lemma states that if the current precedence of @{text th} is unchanged,
%then the procedure can stop immediately (all dependent threads have their @{term cp}-value unchanged).
%The second states that if an intermediate @{term cp}-value does not change, then
%the procedure can also stop, because none of its dependent threads will
%have their current precedence changed.
\smallskip
*}
(*<*)
end
context step_v_cps_nt
begin
(*>*)
text {*
\noindent
\colorbox{mygrey}{@{term "V th cs"}:} We assume that @{text s'} and
@{term "s \<equiv> V th cs#s'"} are both valid. We have to consider two
subcases: one where there is a thread to ``take over'' the released
resource @{text cs}, and one where there is not. Let us consider them
in turn. Suppose in state @{text s}, the thread @{text th'} takes over
resource @{text cs} from thread @{text th}. We can prove
\begin{isabelle}\ \ \ \ \ %%%
@{thm depend_s}
\end{isabelle}
\noindent
which shows how the @{text RAG} needs to be changed. The next lemma suggests
how the current precedences need to be recalculated. For threads that are
not @{text "th"} and @{text "th'"} nothing needs to be changed, since we
can show
\begin{isabelle}\ \ \ \ \ %%%
@{thm[mode=IfThen] cp_kept}
\end{isabelle}
\noindent
For @{text th} and @{text th'} we need to use Lemma~\ref{childrenlem} to
recalculate their current precedence since their children have changed. *}(*<*)end context step_v_cps_nnt begin (*>*)text {*
\noindent
In the other case where there is no thread that takes over @{text cs}, we can show how
to recalculate the @{text RAG} and also show that no current precedence needs
to be recalculated.
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm depend_s}\\
@{thm eq_cp}
\end{tabular}
\end{isabelle}
*}
(*<*)
end
context step_P_cps_e
begin
(*>*)
text {*
\noindent
\colorbox{mygrey}{@{term "P th cs"}:} We assume that @{text s'} and
@{term "s \<equiv> P th cs#s'"} are both valid. We again have to analyse two subcases, namely
the one where @{text cs} is not locked, and one where it is. We treat the former case
first by showing that
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm depend_s}\\
@{thm eq_cp}
\end{tabular}
\end{isabelle}
\noindent
This means we need to add a holding edge to the @{text RAG} and no
current precedence needs to be recalculated.*}(*<*)end context step_P_cps_ne begin(*>*) text {*
\noindent
In the second case we know that resource @{text cs} is locked. We can show that
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm depend_s}\\
@{thm[mode=IfThen] eq_cp}
\end{tabular}
\end{isabelle}
\noindent
That means we have to add a waiting edge to the @{text RAG}. Furthermore
the current precedence for all threads that are not dependants of @{text th}
are unchanged. For the others we need to follow the edges
in the @{text RAG} and recompute the @{term "cp"}. However, like in the
case of @{text Set}, this operation can stop often earlier, namely when intermediate
values do not change.
*}
(*<*)
end
(*>*)
text {*
\noindent
As can be seen, a pleasing byproduct of our formalisation is that the properties in
this section closely inform an implementation of PIP, namely whether the
@{text RAG} needs to be reconfigured or current precedences need to
be recalculated for an event. This information is provided by the lemmas we proved.
We confirmed that our observations translate into practice by implementing
our version of PIP on top of PINTOS, a small operating system written in C and used for teaching at
Stanford University \cite{PINTOS}. To implement PIP, we only need to modify the kernel
functions corresponding to the events in our formal model. The events translate to the following
function interface in PINTOS:
\begin{center}
\begin{tabular}{|l@ {\hspace{2mm}}|l@ {\hspace{2mm}}|}
\hline
{\bf Event} & {\bf PINTOS function} \\
\hline
@{text Create} & @{text "thread_create"}\\
@{text Exit} & @{text "thread_exit"}\\
@{text Set} & @{text "thread_set_priority"}\\
@{text P} & @{text "lock_acquire"}\\
@{text V} & @{text "lock_release"}\\
\hline
\end{tabular}
\end{center}
\noindent
Our assumption that every event is an atomic operation is ensured by the architecture of
PINTOS. The case where an unlocked resource is given next to the waiting thread with the
highest precedence is realised in our implementation by priority queues. We implemented
them as \emph{Braun trees} \cite{Paulson96}, which provide efficient @{text "O(log n)"}-operations
for accessing and restructuring. Apart from having to implement complex datastructures in C
using pointers, our experience with the implementation has been very positive: our specification
and formalisation of PIP translates smoothly to an efficent implementation.
*}
section {* Conclusion *}
text {*
The Priority Inheritance Protocol (PIP) is a classic textbook
algorithm used in many real-time operating systems in order to avoid the problem of
Priority Inversion. Although classic and widely used, PIP does have
its faults: for example it does not prevent deadlocks in cases where threads
have circular lock dependencies.
We had two goals in mind with our formalisation of PIP: One is to
make the notions in the correctness proof by Sha et al.~\cite{Sha90}
precise so that they can be processed by a theorem prover. The reason is
that a mechanically checked proof avoids the flaws that crept into their
informal reasoning. We achieved this goal: The correctness of PIP now
only hinges on the assumptions behind our formal model. The reasoning, which is
sometimes quite intricate and tedious, has been checked by Isabelle/HOL.
We can also confirm that Paulson's
inductive method for protocol verification~\cite{Paulson98} is quite
suitable for our formal model and proof. The traditional application
area of this method is security protocols.
The second goal of our formalisation is to provide a specification for actually
implementing PIP. Textbooks, for example \cite[Section 5.6.5]{Vahalia96},
explain how to use various implementations of PIP and abstractly
discuss their properties, but surprisingly lack most details important for a
programmer who wants to implement PIP (similarly Sha et al.~\cite{Sha90}).
That this is an issue in practice is illustrated by the
email from Baker we cited in the Introduction. We achieved also this
goal: The formalisation gives the first author enough data to enable
his undergraduate students to implement PIP (as part of their OS course)
on top of PINTOS, a simple instructional operating system for the x86
architecture \cite{PINTOS}. A byproduct of our formalisation effort is that nearly all
design choices for the PIP scheduler are backed up with a proved
lemma. We were also able to establish the property that the choice of
the next thread which takes over a lock is irrelevant for the correctness
of PIP.
PIP is a scheduling algorithm for single-processor systems. We are
now living in a multi-processor world. So the question naturally
arises whether PIP has any relevance in such a world beyond
teaching. Priority Inversion certainly occurs also in
multi-processor systems. However, the answer is that
there is very little work about PIP and multi-processors in the literature.
We are not aware of any proofs in this area, not even informal ones. There
is an implementation of PIP on multi-processors in Linux as part of the ``real-time'' effort,
with an informal description given in \cite{LINUX}.
We estimate that the formal verification of this algorithm, involving more
fine-grained events, is a magnitude harder than the one we presented here, but
still within reach of current theorem proving technology. We leave this for future
work.
The most closely related work to ours is the formal verification in
PVS of the Priority Ceiling Protocol done by Dutertre
\cite{dutertre99b}---another solution to the Priority Inversion
problem, which however needs static analysis of programs in order to
avoid it. There have been earlier formal investigations
into PIP \cite{Faria08,Jahier09,Wellings07}, but they employ model
checking techniques. In this way they are limited to validating
one particular implementation. In contrast, our paper is a good
witness for one of the major reasons to be interested in machine checked
reasoning: gaining deeper understanding of the subject matter.
Our formalisation
consists of around 210 lemmas and overall 6950 lines of readable Isabelle/Isar
code with a few apply-scripts interspersed. The formal model of PIP
is 385 lines long; the formal correctness proof 3800 lines. Some auxiliary
definitions and proofs span over 770 lines of code. The properties relevant
for an implementation require 2000 lines. The code of our formalisation
can be downloaded from
\url{http://www.inf.kcl.ac.uk/staff/urbanc/pip.html}.\medskip
\noindent
{\bf Acknowledgements:}
We are grateful for the comments we received from anonymous
referees.
\bibliographystyle{plain}
\bibliography{root}
*}
(*<*)
end
(*>*)