Line numbers added.
(*<*)
theory Paper
imports "../CpsG" "../ExtGG" "~~/src/HOL/Library/LaTeXsugar"
begin
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{$\_\!\_$}>")
(*>*)
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 itself cannot
be blocked by any thread with lower priority. Alas, in a naive
implementation of resource looking and priorities this property can
be violated. Even worse, $H$ can be delayed indefinitely by
threads with lower priorities. For this let $L$ be in the
possession of a lock for a resource that also $H$ 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 of 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}.} 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 to avoid 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 practise is
proved by an email from 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 to us to look
again at PIP from a more abstract level (but still concrete enough
to inform an implementation), and makes PIP an ideal candidate for a
formal verification. One reason, of course, 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 revert 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; the earlier informal proof by Sha et
al.~\cite{Sha90} is flawed). 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 inform an
efficient implementation. 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. Something
which has not been mentioned in the relevant literature.
*}
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.} Our model of
PIP is based on Paulson's inductive approach to protocol
verification \cite{Paulson98}, where the \emph{state} of a system is
given by a list of events that happened so far. \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 three 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 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.
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 by the programmer
(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
\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 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 we can introduce
the notion of threads being @{term readys} 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 this 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}
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 does not yet exists.
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 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 ommited 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
print_locale extend_highest_gen
thm extend_highest_gen_def
thm extend_highest_gen_axioms_def
thm highest_gen_def
(*>*)
text {*
Sha et al.~\cite[Theorem 6]{Sha90} state their correctness criterion for PIP in terms
of the number of critical resources: if there are @{text m} critical
resources, then a blocked job can only be blocked @{text m} times---that is
a bounded number of times.
For their version of PIP, this property is \emph{not} true (as pointed out by
Yodaiken \cite{Yodaiken02}) as a high-priority thread can be
blocked an unbounded number of times by creating medium-priority
threads that block a thread, which in turn locks a critical resource and has
too low priority to make progress. In the way we have set up our formal model of PIP,
their proof idea, even when fixed, does not seem to go through.
The idea behind our correctness criterion of PIP 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}. 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. We will actually prove a stricter bound below. 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 @{text s} and @{text "s' @ s"}:} In order to make
any meaningful statement, we need to require that @{text "s"} and
@{text "s' @ s"} are valid states, namely
\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 @{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 @{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 continously 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
Under these assumptions we will 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"}.
\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}. As we shall see shortly,
that means by only finitely many threads. Consequently, indefinite wait of
@{text th}---which would be Priority Inversion---cannot occur.
In what follows we will describe properties of PIP that allow us to prove
Theorem~\ref{mainthm}. It is relatively easily 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
where 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 one states that every waiting thread can only wait for a single
resource (because it gets suspended after requesting that resource and having
to wait for it); 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 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}\\
\hspace{5mm}@{text "if"}~@{thm (prem 2) range_in}~@{text "then"}~@{thm (concl) range_in}
\end{tabular}
\end{isabelle}
TODO
\noindent
The following lemmas show how RAG is changed with the execution of events:
\begin{enumerate}
\item Execution of @{term "Set"} does not change RAG (@{text "depend_set_unchanged"}):
@{thm[display] depend_set_unchanged}
\item Execution of @{term "Create"} does not change RAG (@{text "depend_create_unchanged"}):
@{thm[display] depend_create_unchanged}
\item Execution of @{term "Exit"} does not change RAG (@{text "depend_exit_unchanged"}):
@{thm[display] depend_exit_unchanged}
\item Execution of @{term "P"} (@{text "step_depend_p"}):
@{thm[display] step_depend_p}
\item Execution of @{term "V"} (@{text "step_depend_v"}):
@{thm[display] step_depend_v}
\end{enumerate}
*}
text {* \noindent
These properties are used to derive the following important results about RAG:
\begin{enumerate}
\item RAG is loop free (@{text "acyclic_depend"}):
@{thm [display] acyclic_depend}
\item RAGs are finite (@{text "finite_depend"}):
@{thm [display] finite_depend}
\item Reverse paths in RAG are well founded (@{text "wf_dep_converse"}):
@{thm [display] wf_dep_converse}
\item The dependence relation represented by RAG has a tree structure (@{text "unique_depend"}):
@{thm [display] unique_depend[of _ _ "n\<^isub>1" "n\<^isub>2"]}
\item All threads in RAG are living threads
(@{text "dm_depend_threads"} and @{text "range_in"}):
@{thm [display] dm_depend_threads range_in}
\end{enumerate}
*}
text {* \noindent
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}
*}
text {* \noindent
Properties about @{term "next_th"}:
\begin{enumerate}
\item The thread taking over is different from the thread which is releasing
(@{text "next_th_neq"}):
@{thm [display] next_th_neq}
\item The thread taking over is unique
(@{text "next_th_unique"}):
@{thm [display] next_th_unique[of _ _ _ "th\<^isub>1" "th\<^isub>2"]}
\end{enumerate}
*}
text {* \noindent
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}
*}
text {* \noindent
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}
*}
(*<*)
end
(*>*)
subsection {* Proof idea *}
(*<*)
context extend_highest_gen
begin
print_locale extend_highest_gen
thm extend_highest_gen_def
thm extend_highest_gen_axioms_def
thm highest_gen_def
(*>*)
text {*
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.
So, the key of the proof is to establish the correctness of @{text "moment_blocked"}.
We are going to show how this lemma is proved. At the heart of this proof, is
lemma @{text "pv_blocked"}:
@{thm [display] pv_blocked}
This lemma says: for any @{text "s"}-extension {text "t"}, if thread @{text "th'"}
does not hold any resource, it can not be running at @{text "t@s"}.
\noindent Proof:
\begin{enumerate}
\item Since thread @{text "th'"} does not hold any resource, no thread may depend on it,
so its current precedence @{text "cp (t@s) th'"} equals to its own precedence
@{text "preced th' (t@s)"}. \label{arg_1}
\item Since @{text "th"} has the highest precedence in the system and
precedences are distinct among threads, we have
@{text "preced th' (t@s) < preced th (t@s)"}. From this and item \ref{arg_1},
we have @{text "cp (t@s) th' < preced th (t@s)"}.
\item Since @{text "preced th (t@s)"} is already the highest in the system,
@{text "cp (t@s) th"} can not be higher than this and can not be lower neither (by
the definition of @{text "cp"}), we have @{text "preced th (t@s) = cp (t@s) th"}.
\item Finally we have @{text "cp (t@s) th' < cp (t@s) th"}.
\item By defintion of @{text "running"}, @{text "th'"} can not be runing at
@{text "t@s"}.
\end{enumerate}
Since @{text "th'"} is not able to run at state @{text "t@s"}, it is not able to
make either {text "P"} or @{text "V"} action, so if @{text "t@s"} is extended
one step further, @{text "th'"} still does not hold any resource.
The situation will not unchanged in further extensions as long as
@{text "th"} holds the highest precedence. Since this @{text "t"} is arbitarily chosen
except being constrained by predicate @{text "extend_highest_gen"} and
this predicate has the property that if it holds for @{text "t"}, it also holds
for any moment @{text "i"} inside @{text "t"}, as shown by lemma @{text "red_moment"}:
@{thm [display] "extend_highest_gen.red_moment"}
so @{text "pv_blocked"} can be applied to any @{text "moment i t"}.
From this, lemma @{text "moment_blocked"} follows.
*}
(*<*)
end
(*>*)
section {* Properties for an Implementation\label{implement} *}
text {*
While a formal correctness proof for our model of PIP is certainly
attractive (especially in light of the flawed proof by Sha et
al.~\cite{Sha90}), we found that the formalisation can even help us
with efficiently implementing PIP.
For example Baker complained that calculating the current precedence
in PIP is quite ``heavy weight'' in Linux (see our Introduction).
In our model of PIP the current precedence of a thread in a state s
depends on all its dependants---a ``global'' transitive notion,
which is indeed heavy weight (see Def.~shown in \eqref{cpreced}).
We can however prove how to improve upon this. For this let us
define the notion of @{term children} of a thread as
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm children_def2}
\end{tabular}
\end{isabelle}
\noindent
where a child is a thread that is one ``hop'' away in the @{term RAG} from the
tread @{text th} (and waiting for @{text th} to release a resource). We can prove that
\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 change their 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 implementation work for PIP involves the scheduler
and coding how it should react to the events, for example which
datastructures need to be modified (mainly @{text RAG} and @{text cprec}).
Below we outline how our formalisation guides this implementation for each
event.\smallskip
*}
(*<*)
context step_create_cps
begin
(*>*)
text {*
\noindent
@{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}\\
@{thm[mode=IfThen] eq_cp}
\end{tabular}
\end{isabelle}
\noindent
This means 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 is just its precedence, that is the pair @{term "(prio, length (s::event list))"}.
\smallskip
*}
(*<*)
end
context step_exit_cps
begin
(*>*)
text {*
\noindent
@{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}\\
@{thm[mode=IfThen] eq_cp}
\end{tabular}
\end{isabelle}
\noindent
This means also we do not have to recalculate the @{text RAG} and
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
@{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}\\
@{thm[mode=IfThen] eq_cp}
\end{tabular}
\end{isabelle}
\noindent
The first is again telling us we do not need to change the @{text RAG}. The second
however states that only threads that are \emph{not} dependent on @{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"}-chains to recompute the @{term "cp"} of every
thread encountered on the way using Lemma~\ref{childrenlem}. Since the @{term "depend"}
is loop free, this procedure always stop. The the following two lemmas show this
procedure can actually stop often earlier.
\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 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
@{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 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 show
\begin{isabelle}\ \ \ \ \ %%%
@{thm depend_s}
\end{isabelle}
\noindent
which shows how the @{text RAG} needs to be changed. This also 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 prcedence 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, except for @{text th} (like in the case above).
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm depend_s}\\
@{thm eq_cp}
\end{tabular}
\end{isabelle}
*}
(*<*)
end
context step_P_cps_e
begin
(*>*)
text {*
\noindent
@{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 locked, and where it is not. We treat the second case
first by showing that
\begin{isabelle}\ \ \ \ \ %%%
\begin{tabular}{@ {}l}
@{thm depend_s}\\
@{thm eq_cp}
\end{tabular}
\end{isabelle}
\noindent
This means we do not need to add a holding edge to the @{text RAG} and no
current precedence must be recalculated (including that for @{text th}).*}(*<*)end context step_P_cps_ne begin(*>*) text {*
\noindent
In the second case we know that resouce @{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 dependent on @{text th}
are unchanged. For the others we need to follow the @{term "depend"}-chains
in the @{text RAG} and recompute the @{term "cp"}. However, like in the
@{text Set}-event, this operation can stop often earlier, namely when intermediate
values do not change.
*}
(*<*)
end
(*>*)
text {*
\noindent
TO DO a few sentences summarising what has been achieved.
*}
section {* Conclusion *}
text {*
The Priority Inheritance Protocol (PIP) is a classic textbook
algorithm used in 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 beyond any
reasonable doubt by Isabelle/HOL. We can also confirm that Paulson's
inductive method to protocol verification~\cite{Paulson98} is quite
suitable for our formal model and proof. The traditional application
area of this method is security protocols. The only other
application of Paulson's method we know of outside this area is
\cite{Wang09}.
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 for a
programmer who wants to implement PIP. 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 small operating system for teaching
purposes. 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. Earlier model checking approaches which verified implementations
of PIP \cite{Faria08,Jahier09,Wellings07} cannot
provide this kind of ``deep understanding'' about the principles behind
PIP and its correctness.
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 surprising answer, according to
\cite{Steinberg10}, is that except for one unsatisfactory proposal
nobody has a good idea for how PIP should be modified to work
correctly on multi-processor systems. The difficulties become clear
when considering that locking and releasing a resource always
requires a some small amount of time. If processes work independently, then a
low priority process can ``steal'' in this unguarded moment a lock for a
resource that was supposed allow a high-priority process to run next. Thus
the problem of Priority Inversion is not really prevented. It seems
difficult to design a PIP-algorithm with a meaningful correctness
property on independent multi-processor systems. We can imagine PIP
to be of use in a situation where
processes are not independent, but coordinated via a master
process that distributes work over some slave processes. However a
formal investigation of this is beyond the scope of this paper.
We are not aware of any proofs in this area, not even informal ones.
Our formalisation consists of 6894 of readable Isabelle/Isar code, with some
apply-scripts interspersed. The formal model is 385 lines long; the
formal correctness proof 3777 lines. The properties relevant for an
implementation are 1964 lines long; Auxlliary definitions and notions are
768 lines.
*}
section {* Key properties \label{extension} *}
(*<*)
context extend_highest_gen
begin
(*>*)
text {*
The essential of {\em Priority Inheritance} is to avoid indefinite priority inversion. For this
purpose, we need to investigate what happens after one thread takes the highest precedence.
A locale is used to describe such a situation, which assumes:
\begin{enumerate}
\item @{term "s"} is a valid state (@{text "vt_s"}):
@{thm vt_s}.
\item @{term "th"} is a living thread in @{term "s"} (@{text "threads_s"}):
@{thm threads_s}.
\item @{term "th"} has the highest precedence in @{term "s"} (@{text "highest"}):
@{thm highest}.
\item The precedence of @{term "th"} is @{term "Prc prio tm"} (@{text "preced_th"}):
@{thm preced_th}.
\end{enumerate}
*}
text {* \noindent
Under these assumptions, some basic priority can be derived for @{term "th"}:
\begin{enumerate}
\item The current precedence of @{term "th"} equals its own precedence (@{text "eq_cp_s_th"}):
@{thm [display] eq_cp_s_th}
\item The current precedence of @{term "th"} is the highest precedence in
the system (@{text "highest_cp_preced"}):
@{thm [display] highest_cp_preced}
\item The precedence of @{term "th"} is the highest precedence
in the system (@{text "highest_preced_thread"}):
@{thm [display] highest_preced_thread}
\item The current precedence of @{term "th"} is the highest current precedence
in the system (@{text "highest'"}):
@{thm [display] highest'}
\end{enumerate}
*}
text {* \noindent
To analysis what happens after state @{term "s"} a sub-locale is defined, which
assumes:
\begin{enumerate}
\item @{term "t"} is a valid extension of @{term "s"} (@{text "vt_t"}): @{thm vt_t}.
\item Any thread created in @{term "t"} has priority no higher than @{term "prio"}, therefore
its precedence can not be higher than @{term "th"}, therefore
@{term "th"} remain to be the one with the highest precedence
(@{text "create_low"}):
@{thm [display] create_low}
\item Any adjustment of priority in
@{term "t"} does not happen to @{term "th"} and
the priority set is no higher than @{term "prio"}, therefore
@{term "th"} remain to be the one with the highest precedence (@{text "set_diff_low"}):
@{thm [display] set_diff_low}
\item Since we are investigating what happens to @{term "th"}, it is assumed
@{term "th"} does not exit during @{term "t"} (@{text "exit_diff"}):
@{thm [display] exit_diff}
\end{enumerate}
*}
text {* \noindent
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}
*}
text {* \noindent
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 {* Related works \label{related} *}
text {*
\begin{enumerate}
\item {\em Integrating Priority Inheritance Algorithms in the Real-Time Specification for Java}
\cite{Wellings07} models and verifies the combination of Priority Inheritance (PI) and
Priority Ceiling Emulation (PCE) protocols in the setting of Java virtual machine
using extended Timed Automata(TA) formalism of the UPPAAL tool. Although a detailed
formal model of combined PI and PCE is given, the number of properties is quite
small and the focus is put on the harmonious working of PI and PCE. Most key features of PI
(as well as PCE) are not shown. Because of the limitation of the model checking technique
used there, properties are shown only for a small number of scenarios. Therefore,
the verification does not show the correctness of the formal model itself in a
convincing way.
\item {\em Formal Development of Solutions for Real-Time Operating Systems with TLA+/TLC}
\cite{Faria08}. A formal model of PI is given in TLA+. Only 3 properties are shown
for PI using model checking. The limitation of model checking is intrinsic to the work.
\item {\em Synchronous modeling and validation of priority inheritance schedulers}
\cite{Jahier09}. Gives a formal model
of PI and PCE in AADL (Architecture Analysis \& Design Language) and checked
several properties using model checking. The number of properties shown there is
less than here and the scale is also limited by the model checking technique.
\item {\em The Priority Ceiling Protocol: Formalization and Analysis Using PVS}
\cite{dutertre99b}. Formalized another protocol for Priority Inversion in the
interactive theorem proving system PVS.
\end{enumerate}
There are several works on inversion avoidance:
\begin{enumerate}
\item {\em Solving the group priority inversion problem in a timed asynchronous system}
\cite{Wang:2002:SGP}. The notion of Group Priority Inversion is introduced. The main
strategy is still inversion avoidance. The method is by reordering requests
in the setting of Client-Server.
\item {\em A Formalization of Priority Inversion} \cite{journals/rts/BabaogluMS93}.
Formalized the notion of Priority
Inversion and proposes methods to avoid it.
\end{enumerate}
{\em Examples of inaccurate specification of the protocol ???}.
*}
(*<*)
end
(*>*)