(*<*)+ −
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, inform an+ −
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+ −
+ −
\begin{isabelle}\ \ \ \ \ %%%+ −
@{thm cs_depend_def}+ −
\end{isabelle}+ −
+ −
\noindent+ −
Given three threads and three resources, an instance of a RAG is 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. This is defined as+ −
+ −
\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"}, 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>1"} 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+ −
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 scheduler increase this+ −
priority, 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 + −
processes 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 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); @{term "schs"} 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 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 (see \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+ −
+ −
\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 resources; + −
for calculating the next @{term "cprec_fun"}, we use @{text wq} and the function + −
@{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 when 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 old thread list plus the thread @{text th} + −
appended to the end of that list (remember the head of this list is seen to be in the possession of this+ −
resource).+ −
+ −
\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 rest, 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+ −
@{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 "f ` S"} stands for the set @{text S} under the image of the + −
function @{text f}. + −
Note that in the initial case, that is where the list of events is empty, the set + −
@{term threads} is empty and therefore there is no thread ready nor a 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 the set-comprehension to capture both possibilities.+ −
We can now also define the set of resources that are locked by a thread in any+ −
given state.+ −
+ −
\begin{isabelle}\ \ \ \ \ %%%+ −
@{thm holdents_def}+ −
\end{isabelle}+ −
+ −
\noindent+ −
These resources are given by the holding edges in the RAG.+ −
+ −
Finally we can define what a \emph{valid state} is in our PIP. For+ −
example we cannot expect to be able to exit a thread, if it was not+ −
created yet. These validity constraints are characterised by the+ −
inductive predicate @{term "step"}. We give five inference rules+ −
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 (to simplify ). 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+ −
of course the responsibility of the programmer. Here in our formal+ −
model we just exclude such problematic cases in order 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, we need to perform an occurs+ −
check; but in practice the occurs check is ommited and the+ −
responsibility to avoid problems rests with the programmer.}+ −
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 two inference rules of @{term step}.+ −
+ −
\begin{center}+ −
\begin{tabular}{c}+ −
@{thm[mode=Rule] thread_P[where thread=th]}\medskip\\+ −
@{thm[mode=Rule] thread_V[where thread=th]}\\+ −
\end{tabular}+ −
\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 version of PIP is correct.+ −
*}+ −
+ −
section {* 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 {* + −
Main lemma+ −
+ −
\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}.+ −
+ −
\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}+ −
+ −
\begin{lemma}+ −
@{thm[mode=IfThen] moment_blocked}+ −
\end{lemma}+ −
+ −
\begin{theorem}+ −
@{thm[mode=IfThen] runing_inversion_2}+ −
\end{theorem}+ −
+ −
\begin{theorem}+ −
@{thm[mode=IfThen] runing_inversion_3}+ −
\end{theorem}+ −
+ −
+ −
+ −
TO DO + −
+ −
*}+ −
+ −
(*<*)+ −
end+ −
(*>*)+ −
+ −
section {* Properties for an Implementation *}+ −
+ −
text {* TO DO *}+ −
+ −
section {* Conclusion *}+ −
+ −
text {* + −
The Priority Inheritance Protocol is a classic textbook algorithm+ −
used in real-time systems in order to avoid the problem of Priority+ −
Inversion.+ −
+ −
A clear and simple understanding of the problem at hand is both a+ −
prerequisite and a byproduct of such an effort, because everything+ −
has finally be reduced to the very first principle to be checked+ −
mechanically.+ −
+ −
Our formalisation and the one presented+ −
in \cite{Wang09} are the only ones that employ Paulson's method for+ −
verifying protocols which are \emph{not} security related. + −
+ −
TO DO + −
+ −
no clue about multi-processor case according to \cite{Steinberg10} + −
+ −
*}+ −
+ −
text {*+ −
\bigskip+ −
The priority inversion phenomenon was first published in+ −
\cite{Lampson80}. The two protocols widely used to eliminate+ −
priority inversion, namely PI (Priority Inheritance) and PCE+ −
(Priority Ceiling Emulation), were proposed in \cite{Sha90}. PCE is+ −
less convenient to use because it requires static analysis of+ −
programs. Therefore, PI is more commonly used in+ −
practice\cite{locke-july02}. However, as pointed out in the+ −
literature, the analysis of priority inheritance protocol is quite+ −
subtle\cite{yodaiken-july02}. A formal analysis will certainly be+ −
helpful for us to understand and correctly implement PI. All+ −
existing formal analysis of PI+ −
\cite{Jahier09,Wellings07,Faria08} are based on the+ −
model checking technology. Because of the state explosion problem,+ −
model check is much like an exhaustive testing of finite models with+ −
limited size. The results obtained can not be safely generalized to+ −
models with arbitrarily large size. Worse still, since model+ −
checking is fully automatic, it give little insight on why the+ −
formal model is correct. It is therefore definitely desirable to+ −
analyze PI using theorem proving, which gives more general results+ −
as well as deeper insight. And this is the purpose of this paper+ −
which gives a formal analysis of PI in the interactive theorem+ −
prover Isabelle using Higher Order Logic (HOL). The formalization+ −
focuses on on two issues:+ −
+ −
\begin{enumerate}+ −
\item The correctness of the protocol model itself. A series of desirable properties is + −
derived until we are fully convinced that the formal model of PI does + −
eliminate priority inversion. And a better understanding of PI is so obtained + −
in due course. For example, we find through formalization that the choice of + −
next thread to take hold when a + −
resource is released is irrelevant for the very basic property of PI to hold. + −
A point never mentioned in literature. + −
\item The correctness of the implementation. A series of properties is derived the meaning + −
of which can be used as guidelines on how PI can be implemented efficiently and correctly. + −
\end{enumerate} + −
+ −
The rest of the paper is organized as follows: Section \ref{overview} gives an overview + −
of PI. Section \ref{model} introduces the formal model of PI. Section \ref{general} + −
discusses a series of basic properties of PI. Section \ref{extension} shows formally + −
how priority inversion is controlled by PI. Section \ref{implement} gives properties + −
which can be used for guidelines of implementation. Section \ref{related} discusses + −
related works. Section \ref{conclusion} concludes the whole paper.+ −
+ −
The basic priority inheritance protocol has two problems:+ −
+ −
It does not prevent a deadlock from happening in a program with circular lock dependencies.+ −
+ −
A chain of blocking may be formed; blocking duration can be substantial, though bounded.+ −
+ −
+ −
Contributions+ −
+ −
Despite the wide use of Priority Inheritance Protocol in real time operating+ −
system, it's correctness has never been formally proved and mechanically checked. + −
All existing verification are based on model checking technology. Full automatic+ −
verification gives little help to understand why the protocol is correct. + −
And results such obtained only apply to models of limited size. + −
This paper presents a formal verification based on theorem proving. + −
Machine checked formal proof does help to get deeper understanding. We found + −
the fact which is not mentioned in the literature, that the choice of next + −
thread to take over when an critical resource is release does not affect the correctness+ −
of the protocol. The paper also shows how formal proof can help to construct + −
correct and efficient implementation.\bigskip + −
+ −
*}+ −
+ −
section {* An overview of priority inversion and priority inheritance \label{overview} *}+ −
+ −
text {*+ −
+ −
Priority inversion refers to the phenomenon when a thread with high priority is blocked + −
by a thread with low priority. Priority happens when the high priority thread requests + −
for some critical resource already taken by the low priority thread. Since the high + −
priority thread has to wait for the low priority thread to complete, it is said to be + −
blocked by the low priority thread. Priority inversion might prevent high priority + −
thread from fulfill its task in time if the duration of priority inversion is indefinite + −
and unpredictable. Indefinite priority inversion happens when indefinite number + −
of threads with medium priorities is activated during the period when the high + −
priority thread is blocked by the low priority thread. Although these medium + −
priority threads can not preempt the high priority thread directly, they are able + −
to preempt the low priority threads and cause it to stay in critical section for + −
an indefinite long duration. In this way, the high priority thread may be blocked indefinitely. + −
+ −
Priority inheritance is one protocol proposed to avoid indefinite priority inversion. + −
The basic idea is to let the high priority thread donate its priority to the low priority + −
thread holding the critical resource, so that it will not be preempted by medium priority + −
threads. The thread with highest priority will not be blocked unless it is requesting + −
some critical resource already taken by other threads. Viewed from a different angle, + −
any thread which is able to block the highest priority threads must already hold some + −
critical resource. Further more, it must have hold some critical resource at the + −
moment the highest priority is created, otherwise, it may never get change to run and + −
get hold. Since the number of such resource holding lower priority threads is finite, + −
if every one of them finishes with its own critical section in a definite duration, + −
the duration the highest priority thread is blocked is definite as well. The key to + −
guarantee lower priority threads to finish in definite is to donate them the highest + −
priority. In such cases, the lower priority threads is said to have inherited the + −
highest priority. And this explains the name of the protocol: + −
{\em Priority Inheritance} and how Priority Inheritance prevents indefinite delay.+ −
+ −
The objectives of this paper are:+ −
\begin{enumerate}+ −
\item Build the above mentioned idea into formal model and prove a series of properties + −
until we are convinced that the formal model does fulfill the original idea. + −
\item Show how formally derived properties can be used as guidelines for correct + −
and efficient implementation.+ −
\end{enumerate}+ −
The proof is totally formal in the sense that every detail is reduced to the + −
very first principles of Higher Order Logic. The nature of interactive theorem + −
proving is for the human user to persuade computer program to accept its arguments. + −
A clear and simple understanding of the problem at hand is both a prerequisite and a + −
byproduct of such an effort, because everything has finally be reduced to the very + −
first principle to be checked mechanically. The former intuitive explanation of + −
Priority Inheritance is just such a byproduct. + −
*}+ −
+ −
section {* Formal model of Priority Inheritance \label{model} *}+ −
text {*+ −
\input{../../generated/PrioGDef}+ −
*}+ −
+ −
section {* General properties of Priority Inheritance \label{general} *}+ −
+ −
text {*+ −
The following are several very basic prioprites:+ −
\begin{enumerate}+ −
\item All runing threads must be ready (@{text "runing_ready"}):+ −
@{thm[display] "runing_ready"} + −
\item All ready threads must be living (@{text "readys_threads"}):+ −
@{thm[display] "readys_threads"} + −
\item There are finite many living threads at any moment (@{text "finite_threads"}):+ −
@{thm[display] "finite_threads"} + −
\item Every waiting queue does not contain duplcated elements (@{text "wq_distinct"}): + −
@{thm[display] "wq_distinct"} + −
\item All threads in waiting queues are living threads (@{text "wq_threads"}): + −
@{thm[display] "wq_threads"} + −
\item The event which can get a thread into waiting queue must be @{term "P"}-events+ −
(@{text "block_pre"}): + −
@{thm[display] "block_pre"} + −
\item A thread may never wait for two different critical resources+ −
(@{text "waiting_unique"}): + −
@{thm[display] waiting_unique[of _ _ "cs\<^isub>1" "cs\<^isub>2"]}+ −
\item Every resource can only be held by one thread+ −
(@{text "held_unique"}): + −
@{thm[display] held_unique[of _ "th\<^isub>1" _ "th\<^isub>2"]}+ −
\item Every living thread has an unique precedence+ −
(@{text "preced_unique"}): + −
@{thm[display] preced_unique[of "th\<^isub>1" _ "th\<^isub>2"]}+ −
\end{enumerate}+ −
*}+ −
+ −
text {* \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 There can only be one running thread (@{text "runing_unique"}):+ −
@{thm [display] runing_unique[of _ "th\<^isub>1" "th\<^isub>2"]}+ −
\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}+ −
*}+ −
+ −
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 {* Properties to guide implementation \label{implement} *}+ −
+ −
text {*+ −
The properties (especially @{text "runing_inversion_2"}) convinced us that the model defined + −
in Section \ref{model} does prevent indefinite priority inversion and therefore fulfills + −
the fundamental requirement of Priority Inheritance protocol. Another purpose of this paper + −
is to show how this model can be used to guide a concrete implementation. As discussed in+ −
Section 5.6.5 of \cite{Vahalia96}, the implementation of Priority Inheritance in Solaris + −
uses sophisticated linking data structure. Except discussing two scenarios to show how+ −
the data structure should be manipulated, a lot of details of the implementation are missing. + −
In \cite{Faria08,Jahier09,Wellings07} the protocol is described formally + −
using different notations, but little information is given on how this protocol can be + −
implemented efficiently, especially there is no information on how these data structure + −
should be manipulated. + −
+ −
Because the scheduling of threads is based on current precedence, + −
the central issue in implementation of Priority Inheritance is how to compute the precedence+ −
correctly and efficiently. As long as the precedence is correct, it is very easy to + −
modify the scheduling algorithm to select the correct thread to execute. + −
+ −
First, it can be proved that the computation of current precedence @{term "cp"} of a threads+ −
only involves its children (@{text "cp_rec"}):+ −
@{thm [display] cp_rec} + −
where @{term "children s th"} represents the set of children of @{term "th"} in the current+ −
RAG: + −
\[+ −
@{thm (lhs) children_def} @{text "\<equiv>"} @{thm (rhs) children_def}+ −
\]+ −
where the definition of @{term "child"} is: + −
\[ @{thm (lhs) child_def} @{text "\<equiv>"} @{thm (rhs) child_def}+ −
\]+ −
+ −
The aim of this section is to fill the missing details of how current precedence should+ −
be changed with the happening of events, with each event type treated by one subsection,+ −
where the computation of @{term "cp"} uses lemma @{text "cp_rec"}.+ −
*}+ −
+ −
subsection {* Event @{text "Set th prio"} *}+ −
+ −
(*<*)+ −
context step_set_cps+ −
begin+ −
(*>*)+ −
+ −
text {*+ −
The context under which event @{text "Set th prio"} happens is formalized as follows:+ −
\begin{enumerate}+ −
\item The formation of @{term "s"} (@{text "s_def"}): @{thm s_def}.+ −
\item State @{term "s"} is a valid state (@{text "vt_s"}): @{thm vt_s}. This implies + −
event @{text "Set th prio"} is eligible to happen under state @{term "s'"} and+ −
state @{term "s'"} is a valid state.+ −
\end{enumerate}+ −
*}+ −
+ −
text {* \noindent+ −
Under such a context, we investigated how the current precedence @{term "cp"} of + −
threads change from state @{term "s'"} to @{term "s"} and obtained the following+ −
conclusions:+ −
\begin{enumerate}+ −
%% \item The RAG does not change (@{text "eq_dep"}): @{thm "eq_dep"}.+ −
\item All threads with no dependence relation with thread @{term "th"} have their+ −
@{term "cp"}-value unchanged (@{text "eq_cp"}):+ −
@{thm [display] eq_cp}+ −
This lemma implies the @{term "cp"}-value of @{term "th"}+ −
and those threads which have a dependence relation with @{term "th"} might need+ −
to be recomputed. The way to do this is to start from @{term "th"} + −
and follow the @{term "depend"}-chain to recompute the @{term "cp"}-value of every + −
encountered thread using lemma @{text "cp_rec"}. + −
Since the @{term "depend"}-relation is loop free, this procedure + −
can always stop. The the following lemma shows this procedure actually could stop earlier.+ −
\item The following two lemma shows, if a thread the re-computation of which+ −
gives an unchanged @{term "cp"}-value, the procedure described above can stop. + −
\begin{enumerate}+ −
\item Lemma @{text "eq_up_self"} shows if the re-computation of+ −
@{term "th"}'s @{term "cp"} gives the same result, the procedure can stop:+ −
@{thm [display] eq_up_self}+ −
\item Lemma @{text "eq_up"}) shows if the re-computation at intermediate threads+ −
gives unchanged result, the procedure can stop:+ −
@{thm [display] eq_up}+ −
\end{enumerate}+ −
\end{enumerate}+ −
*}+ −
+ −
(*<*)+ −
end+ −
(*>*)+ −
+ −
subsection {* Event @{text "V th cs"} *}+ −
+ −
(*<*)+ −
context step_v_cps_nt+ −
begin+ −
(*>*)+ −
+ −
text {*+ −
The context under which event @{text "V th cs"} happens is formalized as follows:+ −
\begin{enumerate}+ −
\item The formation of @{term "s"} (@{text "s_def"}): @{thm s_def}.+ −
\item State @{term "s"} is a valid state (@{text "vt_s"}): @{thm vt_s}. This implies + −
event @{text "V th cs"} is eligible to happen under state @{term "s'"} and+ −
state @{term "s'"} is a valid state.+ −
\end{enumerate}+ −
*}+ −
+ −
text {* \noindent+ −
Under such a context, we investigated how the current precedence @{term "cp"} of + −
threads change from state @{term "s'"} to @{term "s"}. + −
+ −
+ −
Two subcases are considerted, + −
where the first is that there exits @{term "th'"} + −
such that + −
@{thm [display] nt} + −
holds, which means there exists a thread @{term "th'"} to take over+ −
the resource release by thread @{term "th"}. + −
In this sub-case, the following results are obtained:+ −
\begin{enumerate}+ −
\item The change of RAG is given by lemma @{text "depend_s"}: + −
@{thm [display] "depend_s"}+ −
which shows two edges are removed while one is added. These changes imply how+ −
the current precedences should be re-computed.+ −
\item First all threads different from @{term "th"} and @{term "th'"} have their+ −
@{term "cp"}-value kept, therefore do not need a re-computation+ −
(@{text "cp_kept"}): @{thm [display] cp_kept}+ −
This lemma also implies, only the @{term "cp"}-values of @{term "th"} and @{term "th'"}+ −
need to be recomputed.+ −
\end{enumerate}+ −
*}+ −
+ −
(*<*)+ −
end+ −
+ −
context step_v_cps_nnt+ −
begin+ −
(*>*)+ −
+ −
text {*+ −
The other sub-case is when for all @{text "th'"}+ −
@{thm [display] nnt}+ −
holds, no such thread exists. The following results can be obtained for this + −
sub-case:+ −
\begin{enumerate}+ −
\item The change of RAG is given by lemma @{text "depend_s"}:+ −
@{thm [display] depend_s}+ −
which means only one edge is removed.+ −
\item In this case, no re-computation is needed (@{text "eq_cp"}):+ −
@{thm [display] eq_cp}+ −
\end{enumerate}+ −
*}+ −
+ −
(*<*)+ −
end+ −
(*>*)+ −
+ −
+ −
subsection {* Event @{text "P th cs"} *}+ −
+ −
(*<*)+ −
context step_P_cps_e+ −
begin+ −
(*>*)+ −
+ −
text {*+ −
The context under which event @{text "P th cs"} happens is formalized as follows:+ −
\begin{enumerate}+ −
\item The formation of @{term "s"} (@{text "s_def"}): @{thm s_def}.+ −
\item State @{term "s"} is a valid state (@{text "vt_s"}): @{thm vt_s}. This implies + −
event @{text "P th cs"} is eligible to happen under state @{term "s'"} and+ −
state @{term "s'"} is a valid state.+ −
\end{enumerate}+ −
+ −
This case is further divided into two sub-cases. The first is when @{thm ee} holds.+ −
The following results can be obtained:+ −
\begin{enumerate}+ −
\item One edge is added to the RAG (@{text "depend_s"}):+ −
@{thm [display] depend_s}+ −
\item No re-computation is needed (@{text "eq_cp"}):+ −
@{thm [display] eq_cp}+ −
\end{enumerate}+ −
*}+ −
+ −
(*<*)+ −
end+ −
+ −
context step_P_cps_ne+ −
begin+ −
(*>*)+ −
+ −
text {*+ −
The second is when @{thm ne} holds.+ −
The following results can be obtained:+ −
\begin{enumerate}+ −
\item One edge is added to the RAG (@{text "depend_s"}):+ −
@{thm [display] depend_s}+ −
\item Threads with no dependence relation with @{term "th"} do not need a re-computation+ −
of their @{term "cp"}-values (@{text "eq_cp"}):+ −
@{thm [display] eq_cp}+ −
This lemma implies all threads with a dependence relation with @{term "th"} may need + −
re-computation.+ −
\item Similar to the case of @{term "Set"}, the computation procedure could stop earlier+ −
(@{text "eq_up"}):+ −
@{thm [display] eq_up}+ −
\end{enumerate}+ −
+ −
*}+ −
+ −
(*<*)+ −
end+ −
(*>*)+ −
+ −
subsection {* Event @{text "Create th prio"} *}+ −
+ −
(*<*)+ −
context step_create_cps+ −
begin+ −
(*>*)+ −
+ −
text {*+ −
The context under which event @{text "Create th prio"} happens is formalized as follows:+ −
\begin{enumerate}+ −
\item The formation of @{term "s"} (@{text "s_def"}): @{thm s_def}.+ −
\item State @{term "s"} is a valid state (@{text "vt_s"}): @{thm vt_s}. This implies + −
event @{text "Create th prio"} is eligible to happen under state @{term "s'"} and+ −
state @{term "s'"} is a valid state.+ −
\end{enumerate}+ −
The following results can be obtained under this context:+ −
\begin{enumerate}+ −
\item The RAG does not change (@{text "eq_dep"}):+ −
@{thm [display] eq_dep}+ −
\item All threads other than @{term "th"} do not need re-computation (@{text "eq_cp"}):+ −
@{thm [display] eq_cp}+ −
\item The @{term "cp"}-value of @{term "th"} equals its precedence + −
(@{text "eq_cp_th"}):+ −
@{thm [display] eq_cp_th}+ −
\end{enumerate}+ −
+ −
*}+ −
+ −
+ −
(*<*)+ −
end+ −
(*>*)+ −
+ −
subsection {* Event @{text "Exit th"} *}+ −
+ −
(*<*)+ −
context step_exit_cps+ −
begin+ −
(*>*)+ −
+ −
text {*+ −
The context under which event @{text "Exit th"} happens is formalized as follows:+ −
\begin{enumerate}+ −
\item The formation of @{term "s"} (@{text "s_def"}): @{thm s_def}.+ −
\item State @{term "s"} is a valid state (@{text "vt_s"}): @{thm vt_s}. This implies + −
event @{text "Exit th"} is eligible to happen under state @{term "s'"} and+ −
state @{term "s'"} is a valid state.+ −
\end{enumerate}+ −
The following results can be obtained under this context:+ −
\begin{enumerate}+ −
\item The RAG does not change (@{text "eq_dep"}):+ −
@{thm [display] eq_dep}+ −
\item All threads other than @{term "th"} do not need re-computation (@{text "eq_cp"}):+ −
@{thm [display] eq_cp}+ −
\end{enumerate}+ −
Since @{term th} does not live in state @{term "s"}, there is no need to compute + −
its @{term cp}-value.+ −
*}+ −
+ −
(*<*)+ −
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 ???}.+ −
+ −
*}+ −
+ −
section {* Conclusions \label{conclusion} *}+ −
+ −
text {*+ −
The work in this paper only deals with single CPU configurations. The+ −
"one CPU" assumption is essential for our formalisation, because the+ −
main lemma fails in multi-CPU configuration. The lemma says that any+ −
runing thead must be the one with the highest prioirty or already held+ −
some resource when the highest priority thread was initiated. When+ −
there are multiple CPUs, it may well be the case that a threads did+ −
not hold any resource when the highest priority thread was initiated,+ −
but that thread still runs after that moment on a separate CPU. In+ −
this way, the main lemma does not hold anymore.+ −
+ −
+ −
There are some works deals with priority inversion in multi-CPU+ −
configurations[???], but none of them have given a formal correctness+ −
proof. The extension of our formal proof to deal with multi-CPU+ −
configurations is not obvious. One possibility, as suggested in paper+ −
[???], is change our formal model (the defiintion of "schs") to give+ −
the released resource to the thread with the highest prioirty. In this+ −
way, indefinite prioirty inversion can be avoided, but for a quite+ −
different reason from the one formalized in this paper (because the+ −
"mail lemma" will be different). This means a formal correctness proof+ −
for milt-CPU configuration would be quite different from the one given+ −
in this paper. The solution of prioirty inversion problem in mult-CPU+ −
configurations is a different problem which needs different solutions+ −
which is outside the scope of this paper.+ −
+ −
*}+ −
+ −
(*<*)+ −
end+ −
(*>*)+ −