--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Paper/Paper.thy Thu Dec 06 15:12:49 2012 +0000
@@ -0,0 +1,1345 @@
+(*<*)
+theory Paper
+imports "../CpsG" "../ExtGG" "~~/src/HOL/Library/LaTeXsugar"
+begin
+
+(*
+find_unused_assms CpsG
+find_unused_assms ExtGG
+find_unused_assms Moment
+find_unused_assms Precedence_ord
+find_unused_assms PrioG
+find_unused_assms PrioGDef
+*)
+
+ML {*
+ open Printer;
+ show_question_marks_default := false;
+ *}
+
+notation (latex output)
+ Cons ("_::_" [78,77] 73) and
+ vt ("valid'_state") and
+ runing ("running") and
+ birthtime ("last'_set") and
+ If ("(\<^raw:\textrm{>if\<^raw:}> (_)/ \<^raw:\textrm{>then\<^raw:}> (_)/ \<^raw:\textrm{>else\<^raw:}> (_))" 10) and
+ Prc ("'(_, _')") and
+ holding ("holds") and
+ waiting ("waits") and
+ Th ("T") and
+ Cs ("C") and
+ readys ("ready") and
+ depend ("RAG") and
+ preced ("prec") and
+ cpreced ("cprec") and
+ dependents ("dependants") and
+ cp ("cprec") and
+ holdents ("resources") and
+ original_priority ("priority") and
+ DUMMY ("\<^raw:\mbox{$\_\!\_$}>")
+
+(*abbreviation
+ "detached s th \<equiv> cntP s th = cntV s th"
+*)
+(*>*)
+
+section {* Introduction *}
+
+text {*
+ Many real-time systems need to support threads involving priorities and
+ locking of resources. Locking of resources ensures mutual exclusion
+ when accessing shared data or devices that cannot be
+ preempted. Priorities allow scheduling of threads that need to
+ finish their work within deadlines. Unfortunately, both features
+ can interact in subtle ways leading to a problem, called
+ \emph{Priority Inversion}. Suppose three threads having priorities
+ $H$(igh), $M$(edium) and $L$(ow). We would expect that the thread
+ $H$ blocks any other thread with lower priority and the thread itself cannot
+ be blocked indefinitely by threads with lower priority. Alas, in a naive
+ implementation of resource locking and priorities this property can
+ be violated. For this let $L$ be in the
+ possession of a lock for a resource that $H$ also needs. $H$ must
+ therefore wait for $L$ to exit the critical section and release this
+ lock. The problem is that $L$ might in turn be blocked by any
+ thread with priority $M$, and so $H$ sits there potentially waiting
+ indefinitely. Since $H$ is blocked by threads with lower
+ priorities, the problem is called Priority Inversion. It was first
+ described in \cite{Lampson80} in the context of the
+ Mesa programming language designed for concurrent programming.
+
+ If the problem of Priority Inversion is ignored, real-time systems
+ can become unpredictable and resulting bugs can be hard to diagnose.
+ The classic example where this happened is the software that
+ controlled the Mars Pathfinder mission in 1997 \cite{Reeves98}.
+ Once the spacecraft landed, the software shut down at irregular
+ intervals leading to loss of project time as normal operation of the
+ craft could only resume the next day (the mission and data already
+ collected were fortunately not lost, because of a clever system
+ design). The reason for the shutdowns was that the scheduling
+ software fell victim to Priority Inversion: a low priority thread
+ locking a resource prevented a high priority thread from running in
+ time, leading to a system reset. Once the problem was found, it was
+ rectified by enabling the \emph{Priority Inheritance Protocol} (PIP)
+ \cite{Sha90}\footnote{Sha et al.~call it the \emph{Basic Priority
+ Inheritance Protocol} \cite{Sha90} and others sometimes also call it
+ \emph{Priority Boosting} or \emph{Priority Donation}.} in the scheduling software.
+
+ The idea behind PIP is to let the thread $L$ temporarily inherit
+ the high priority from $H$ until $L$ leaves the critical section
+ unlocking the resource. This solves the problem of $H$ having to
+ wait indefinitely, because $L$ cannot be blocked by threads having
+ priority $M$. While a few other solutions exist for the Priority
+ Inversion problem, PIP is one that is widely deployed and
+ implemented. This includes VxWorks (a proprietary real-time OS used
+ in the Mars Pathfinder mission, in Boeing's 787 Dreamliner, Honda's
+ ASIMO robot, etc.), but also the POSIX 1003.1c Standard realised for
+ example in libraries for FreeBSD, Solaris and Linux.
+
+ One advantage of PIP is that increasing the priority of a thread
+ can be dynamically calculated by the scheduler. This is in contrast
+ to, for example, \emph{Priority Ceiling} \cite{Sha90}, another
+ solution to the Priority Inversion problem, which requires static
+ analysis of the program in order to prevent Priority
+ Inversion. However, there has also been strong criticism against
+ PIP. For instance, PIP cannot prevent deadlocks when lock
+ dependencies are circular, and also blocking times can be
+ substantial (more than just the duration of a critical section).
+ Though, most criticism against PIP centres around unreliable
+ implementations and PIP being too complicated and too inefficient.
+ For example, Yodaiken writes in \cite{Yodaiken02}:
+
+ \begin{quote}
+ \it{}``Priority inheritance is neither efficient nor reliable. Implementations
+ are either incomplete (and unreliable) or surprisingly complex and intrusive.''
+ \end{quote}
+
+ \noindent
+ He suggests avoiding PIP altogether by designing the system so that no
+ priority inversion may happen in the first place. However, such ideal designs may
+ not always be achievable in practice.
+
+ In our opinion, there is clearly a need for investigating correct
+ algorithms for PIP. A few specifications for PIP exist (in English)
+ and also a few high-level descriptions of implementations (e.g.~in
+ the textbook \cite[Section 5.6.5]{Vahalia96}), but they help little
+ with actual implementations. That this is a problem in practice is
+ proved by an email by Baker, who wrote on 13 July 2009 on the Linux
+ Kernel mailing list:
+
+ \begin{quote}
+ \it{}``I observed in the kernel code (to my disgust), the Linux PIP
+ implementation is a nightmare: extremely heavy weight, involving
+ maintenance of a full wait-for graph, and requiring updates for a
+ range of events, including priority changes and interruptions of
+ wait operations.''
+ \end{quote}
+
+ \noindent
+ The criticism by Yodaiken, Baker and others suggests another look
+ at PIP from a more abstract level (but still concrete enough
+ to inform an implementation), and makes PIP a good candidate for a
+ formal verification. An additional reason is that the original
+ presentation of PIP~\cite{Sha90}, despite being informally
+ ``proved'' correct, is actually \emph{flawed}.
+
+ Yodaiken \cite{Yodaiken02} points to a subtlety that had been
+ overlooked in the informal proof by Sha et al. They specify in
+ \cite{Sha90} that after the thread (whose priority has been raised)
+ completes its critical section and releases the lock, it ``returns
+ to its original priority level.'' This leads them to believe that an
+ implementation of PIP is ``rather straightforward''~\cite{Sha90}.
+ Unfortunately, as Yodaiken points out, this behaviour is too
+ simplistic. Consider the case where the low priority thread $L$
+ locks \emph{two} resources, and two high-priority threads $H$ and
+ $H'$ each wait for one of them. If $L$ releases one resource
+ so that $H$, say, can proceed, then we still have Priority Inversion
+ with $H'$ (which waits for the other resource). The correct
+ behaviour for $L$ is to switch to the highest remaining priority of
+ the threads that it blocks. The advantage of formalising the
+ correctness of a high-level specification of PIP in a theorem prover
+ is that such issues clearly show up and cannot be overlooked as in
+ informal reasoning (since we have to analyse all possible behaviours
+ of threads, i.e.~\emph{traces}, that could possibly happen).\medskip
+
+ \noindent
+ {\bf Contributions:} There have been earlier formal investigations
+ into PIP \cite{Faria08,Jahier09,Wellings07}, but they employ model
+ checking techniques. This paper presents a formalised and
+ mechanically checked proof for the correctness of PIP (to our
+ knowledge the first one). In contrast to model checking, our
+ formalisation provides insight into why PIP is correct and allows us
+ to prove stronger properties that, as we will show, can help with an
+ efficient implementation of PIP in the educational PINTOS operating
+ system \cite{PINTOS}. For example, we found by ``playing'' with the
+ formalisation that the choice of the next thread to take over a lock
+ when a resource is released is irrelevant for PIP being correct---a
+ fact that has not been mentioned in the literature and not been used
+ in the reference implementation of PIP in PINTOS. This fact, however, is important
+ for an efficient implementation of PIP, because we can give the lock
+ to the thread with the highest priority so that it terminates more
+ quickly.
+*}
+
+section {* Formal Model of the Priority Inheritance Protocol *}
+
+text {*
+ The Priority Inheritance Protocol, short PIP, is a scheduling
+ algorithm for a single-processor system.\footnote{We shall come back
+ later to the case of PIP on multi-processor systems.}
+ Following good experience in earlier work \cite{Wang09},
+ our model of PIP is based on Paulson's inductive approach to protocol
+ verification \cite{Paulson98}. In this approach a \emph{state} of a system is
+ given by a list of events that happened so far (with new events prepended to the list).
+ \emph{Events} of PIP fall
+ into five categories defined as the datatype:
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \mbox{\begin{tabular}{r@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{7mm}}l}
+ \isacommand{datatype} event
+ & @{text "="} & @{term "Create thread priority"}\\
+ & @{text "|"} & @{term "Exit thread"} \\
+ & @{text "|"} & @{term "Set thread priority"} & {\rm reset of the priority for} @{text thread}\\
+ & @{text "|"} & @{term "P thread cs"} & {\rm request of resource} @{text "cs"} {\rm by} @{text "thread"}\\
+ & @{text "|"} & @{term "V thread cs"} & {\rm release of resource} @{text "cs"} {\rm by} @{text "thread"}
+ \end{tabular}}
+ \end{isabelle}
+
+ \noindent
+ whereby threads, priorities and (critical) resources are represented
+ as natural numbers. The event @{term Set} models the situation that
+ a thread obtains a new priority given by the programmer or
+ user (for example via the {\tt nice} utility under UNIX). As in Paulson's work, we
+ need to define functions that allow us to make some observations
+ about states. One, called @{term threads}, calculates the set of
+ ``live'' threads that we have seen so far:
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \mbox{\begin{tabular}{lcl}
+ @{thm (lhs) threads.simps(1)} & @{text "\<equiv>"} &
+ @{thm (rhs) threads.simps(1)}\\
+ @{thm (lhs) threads.simps(2)[where thread="th"]} & @{text "\<equiv>"} &
+ @{thm (rhs) threads.simps(2)[where thread="th"]}\\
+ @{thm (lhs) threads.simps(3)[where thread="th"]} & @{text "\<equiv>"} &
+ @{thm (rhs) threads.simps(3)[where thread="th"]}\\
+ @{term "threads (DUMMY#s)"} & @{text "\<equiv>"} & @{term "threads s"}\\
+ \end{tabular}}
+ \end{isabelle}
+
+ \noindent
+ In this definition @{term "DUMMY # DUMMY"} stands for list-cons.
+ Another function calculates the priority for a thread @{text "th"}, which is
+ defined as
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \mbox{\begin{tabular}{lcl}
+ @{thm (lhs) original_priority.simps(1)[where thread="th"]} & @{text "\<equiv>"} &
+ @{thm (rhs) original_priority.simps(1)[where thread="th"]}\\
+ @{thm (lhs) original_priority.simps(2)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
+ @{thm (rhs) original_priority.simps(2)[where thread="th" and thread'="th'"]}\\
+ @{thm (lhs) original_priority.simps(3)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
+ @{thm (rhs) original_priority.simps(3)[where thread="th" and thread'="th'"]}\\
+ @{term "original_priority th (DUMMY#s)"} & @{text "\<equiv>"} & @{term "original_priority th s"}\\
+ \end{tabular}}
+ \end{isabelle}
+
+ \noindent
+ In this definition we set @{text 0} as the default priority for
+ threads that have not (yet) been created. The last function we need
+ calculates the ``time'', or index, at which time a process had its
+ priority last set.
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \mbox{\begin{tabular}{lcl}
+ @{thm (lhs) birthtime.simps(1)[where thread="th"]} & @{text "\<equiv>"} &
+ @{thm (rhs) birthtime.simps(1)[where thread="th"]}\\
+ @{thm (lhs) birthtime.simps(2)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
+ @{thm (rhs) birthtime.simps(2)[where thread="th" and thread'="th'"]}\\
+ @{thm (lhs) birthtime.simps(3)[where thread="th" and thread'="th'"]} & @{text "\<equiv>"} &
+ @{thm (rhs) birthtime.simps(3)[where thread="th" and thread'="th'"]}\\
+ @{term "birthtime th (DUMMY#s)"} & @{text "\<equiv>"} & @{term "birthtime th s"}\\
+ \end{tabular}}
+ \end{isabelle}
+
+ \noindent
+ In this definition @{term "length s"} stands for the length of the list
+ of events @{text s}. Again the default value in this function is @{text 0}
+ for threads that have not been created yet. A \emph{precedence} of a thread @{text th} in a
+ state @{text s} is the pair of natural numbers defined as
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{thm preced_def[where thread="th"]}
+ \end{isabelle}
+
+ \noindent
+ The point of precedences is to schedule threads not according to priorities (because what should
+ we do in case two threads have the same priority), but according to precedences.
+ Precedences allow us to always discriminate between two threads with equal priority by
+ taking into account the time when the priority was last set. We order precedences so
+ that threads with the same priority get a higher precedence if their priority has been
+ set earlier, since for such threads it is more urgent to finish their work. In an implementation
+ this choice would translate to a quite natural FIFO-scheduling of processes with
+ the same priority.
+
+ Next, we introduce the concept of \emph{waiting queues}. They are
+ lists of threads associated with every resource. The first thread in
+ this list (i.e.~the head, or short @{term hd}) is chosen to be the one
+ that is in possession of the
+ ``lock'' of the corresponding resource. We model waiting queues as
+ functions, below abbreviated as @{text wq}. They take a resource as
+ argument and return a list of threads. This allows us to define
+ when a thread \emph{holds}, respectively \emph{waits} for, a
+ resource @{text cs} given a waiting queue function @{text wq}.
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm cs_holding_def[where thread="th"]}\\
+ @{thm cs_waiting_def[where thread="th"]}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ In this definition we assume @{text "set"} converts a list into a set.
+ At the beginning, that is in the state where no thread is created yet,
+ the waiting queue function will be the function that returns the
+ empty list for every resource.
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{abbrev all_unlocked}\hfill\numbered{allunlocked}
+ \end{isabelle}
+
+ \noindent
+ Using @{term "holding"} and @{term waiting}, we can introduce \emph{Resource Allocation Graphs}
+ (RAG), which represent the dependencies between threads and resources.
+ We represent RAGs as relations using pairs of the form
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{term "(Th th, Cs cs)"} \hspace{5mm}{\rm and}\hspace{5mm}
+ @{term "(Cs cs, Th th)"}
+ \end{isabelle}
+
+ \noindent
+ where the first stands for a \emph{waiting edge} and the second for a
+ \emph{holding edge} (@{term Cs} and @{term Th} are constructors of a
+ datatype for vertices). Given a waiting queue function, a RAG is defined
+ as the union of the sets of waiting and holding edges, namely
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{thm cs_depend_def}
+ \end{isabelle}
+
+ \noindent
+ Given four threads and three resources, an instance of a RAG can be pictured
+ as follows:
+
+ \begin{center}
+ \newcommand{\fnt}{\fontsize{7}{8}\selectfont}
+ \begin{tikzpicture}[scale=1]
+ %%\draw[step=2mm] (-3,2) grid (1,-1);
+
+ \node (A) at (0,0) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>0"}};
+ \node (B) at (2,0) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>1"}};
+ \node (C) at (4,0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>1"}};
+ \node (D) at (4,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>2"}};
+ \node (E) at (6,-0.7) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>2"}};
+ \node (E1) at (6, 0.2) [draw, circle, very thick, inner sep=0.4mm] {@{text "cs\<^isub>3"}};
+ \node (F) at (8,-0.7) [draw, rounded corners=1mm, rectangle, very thick] {@{text "th\<^isub>3"}};
+
+ \draw [<-,line width=0.6mm] (A) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (B);
+ \draw [->,line width=0.6mm] (C) to node [pos=0.4,sloped,above=-0.5mm] {\fnt{}waiting} (B);
+ \draw [->,line width=0.6mm] (D) to node [pos=0.4,sloped,below=-0.5mm] {\fnt{}waiting} (B);
+ \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,below=-0.5mm] {\fnt{}holding} (E);
+ \draw [<-,line width=0.6mm] (D) to node [pos=0.54,sloped,above=-0.5mm] {\fnt{}holding} (E1);
+ \draw [->,line width=0.6mm] (F) to node [pos=0.45,sloped,below=-0.5mm] {\fnt{}waiting} (E);
+ \end{tikzpicture}
+ \end{center}
+
+ \noindent
+ The use of relations for representing RAGs allows us to conveniently define
+ the notion of the \emph{dependants} of a thread using the transitive closure
+ operation for relations. This gives
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{thm cs_dependents_def}
+ \end{isabelle}
+
+ \noindent
+ This definition needs to account for all threads that wait for a thread to
+ release a resource. This means we need to include threads that transitively
+ wait for a resource being released (in the picture above this means the dependants
+ of @{text "th\<^isub>0"} are @{text "th\<^isub>1"} and @{text "th\<^isub>2"}, which wait for resource @{text "cs\<^isub>1"},
+ but also @{text "th\<^isub>3"},
+ which cannot make any progress unless @{text "th\<^isub>2"} makes progress, which
+ in turn needs to wait for @{text "th\<^isub>0"} to finish). If there is a circle of dependencies
+ in a RAG, then clearly
+ we have a deadlock. Therefore when a thread requests a resource,
+ we must ensure that the resulting RAG is not circular. In practice, the
+ programmer has to ensure this.
+
+
+ Next we introduce the notion of the \emph{current precedence} of a thread @{text th} in a
+ state @{text s}. It is defined as
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{thm cpreced_def2}\hfill\numbered{cpreced}
+ \end{isabelle}
+
+ \noindent
+ where the dependants of @{text th} are given by the waiting queue function.
+ While the precedence @{term prec} of a thread is determined statically
+ (for example when the thread is
+ created), the point of the current precedence is to let the scheduler increase this
+ precedence, if needed according to PIP. Therefore the current precedence of @{text th} is
+ given as the maximum of the precedence @{text th} has in state @{text s} \emph{and} all
+ threads that are dependants of @{text th}. Since the notion @{term "dependants"} is
+ defined as the transitive closure of all dependent threads, we deal correctly with the
+ problem in the informal algorithm by Sha et al.~\cite{Sha90} where a priority of a thread is
+ lowered prematurely.
+
+ The next function, called @{term schs}, defines the behaviour of the scheduler. It will be defined
+ by recursion on the state (a list of events); this function returns a \emph{schedule state}, which
+ we represent as a record consisting of two
+ functions:
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{text "\<lparr>wq_fun, cprec_fun\<rparr>"}
+ \end{isabelle}
+
+ \noindent
+ The first function is a waiting queue function (that is, it takes a
+ resource @{text "cs"} and returns the corresponding list of threads
+ that lock, respectively wait for, it); the second is a function that
+ takes a thread and returns its current precedence (see
+ the definition in \eqref{cpreced}). We assume the usual getter and setter methods for
+ such records.
+
+ In the initial state, the scheduler starts with all resources unlocked (the corresponding
+ function is defined in \eqref{allunlocked}) and the
+ current precedence of every thread is initialised with @{term "Prc 0 0"}; that means
+ \mbox{@{abbrev initial_cprec}}. Therefore
+ we have for the initial shedule state
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm (lhs) schs.simps(1)} @{text "\<equiv>"}\\
+ \hspace{5mm}@{term "(|wq_fun = all_unlocked, cprec_fun = (\<lambda>_::thread. Prc 0 0)|)"}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ The cases for @{term Create}, @{term Exit} and @{term Set} are also straightforward:
+ we calculate the waiting queue function of the (previous) state @{text s};
+ this waiting queue function @{text wq} is unchanged in the next schedule state---because
+ none of these events lock or release any resource;
+ for calculating the next @{term "cprec_fun"}, we use @{text wq} and
+ @{term cpreced}. This gives the following three clauses for @{term schs}:
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm (lhs) schs.simps(2)} @{text "\<equiv>"}\\
+ \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
+ \hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Create th prio # s)|)"}\smallskip\\
+ @{thm (lhs) schs.simps(3)} @{text "\<equiv>"}\\
+ \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
+ \hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Exit th # s)|)"}\smallskip\\
+ @{thm (lhs) schs.simps(4)} @{text "\<equiv>"}\\
+ \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
+ \hspace{8mm}@{term "(|wq_fun = wq\<iota>, cprec_fun = cpreced wq\<iota> (Set th prio # s)|)"}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ More interesting are the cases where a resource, say @{text cs}, is locked or released. In these cases
+ we need to calculate a new waiting queue function. For the event @{term "P th cs"}, we have to update
+ the function so that the new thread list for @{text cs} is the old thread list plus the thread @{text th}
+ appended to the end of that list (remember the head of this list is assigned to be in the possession of this
+ resource). This gives the clause
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm (lhs) schs.simps(5)} @{text "\<equiv>"}\\
+ \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
+ \hspace{5mm}@{text "let"} @{text "new_wq = wq(cs := (wq cs @ [th]))"} @{text "in"}\\
+ \hspace{8mm}@{term "(|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|)"}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ The clause for event @{term "V th cs"} is similar, except that we need to update the waiting queue function
+ so that the thread that possessed the lock is deleted from the corresponding thread list. For this
+ list transformation, we use
+ the auxiliary function @{term release}. A simple version of @{term release} would
+ just delete this thread and return the remaining threads, namely
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}lcl}
+ @{term "release []"} & @{text "\<equiv>"} & @{term "[]"}\\
+ @{term "release (DUMMY # qs)"} & @{text "\<equiv>"} & @{term "qs"}\\
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ In practice, however, often the thread with the highest precedence in the list will get the
+ lock next. We have implemented this choice, but later found out that the choice
+ of which thread is chosen next is actually irrelevant for the correctness of PIP.
+ Therefore we prove the stronger result where @{term release} is defined as
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}lcl}
+ @{term "release []"} & @{text "\<equiv>"} & @{term "[]"}\\
+ @{term "release (DUMMY # qs)"} & @{text "\<equiv>"} & @{term "SOME qs'. distinct qs' \<and> set qs' = set qs"}\\
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ where @{text "SOME"} stands for Hilbert's epsilon and implements an arbitrary
+ choice for the next waiting list. It just has to be a list of distinctive threads and
+ contain the same elements as @{text "qs"}. This gives for @{term V} the clause:
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm (lhs) schs.simps(6)} @{text "\<equiv>"}\\
+ \hspace{5mm}@{text "let"} @{text "wq = wq_fun (schs s)"} @{text "in"}\\
+ \hspace{5mm}@{text "let"} @{text "new_wq = release (wq cs)"} @{text "in"}\\
+ \hspace{8mm}@{term "(|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|)"}
+ \end{tabular}
+ \end{isabelle}
+
+ Having the scheduler function @{term schs} at our disposal, we can ``lift'', or
+ overload, the notions
+ @{term waiting}, @{term holding}, @{term depend} and @{term cp} to operate on states only.
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}rcl}
+ @{thm (lhs) s_holding_abv} & @{text "\<equiv>"} & @{thm (rhs) s_holding_abv}\\
+ @{thm (lhs) s_waiting_abv} & @{text "\<equiv>"} & @{thm (rhs) s_waiting_abv}\\
+ @{thm (lhs) s_depend_abv} & @{text "\<equiv>"} & @{thm (rhs) s_depend_abv}\\
+ @{thm (lhs) cp_def} & @{text "\<equiv>"} & @{thm (rhs) cp_def}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ With these abbreviations in place we can introduce
+ the notion of a thread being @{term ready} in a state (i.e.~threads
+ that do not wait for any resource) and the running thread.
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm readys_def}\\
+ @{thm runing_def}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ In the second definition @{term "DUMMY ` DUMMY"} stands for the image of a set under a function.
+ Note that in the initial state, that is where the list of events is empty, the set
+ @{term threads} is empty and therefore there is neither a thread ready nor running.
+ If there is one or more threads ready, then there can only be \emph{one} thread
+ running, namely the one whose current precedence is equal to the maximum of all ready
+ threads. We use sets to capture both possibilities.
+ We can now also conveniently define the set of resources that are locked by a thread in a
+ given state and also when a thread is detached that state (meaning the thread neither
+ holds nor waits for a resource):
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm holdents_def}\\
+ @{thm detached_def}
+ \end{tabular}
+ \end{isabelle}
+
+ %\noindent
+ %The second definition states that @{text th} in @{text s}.
+
+ Finally we can define what a \emph{valid state} is in our model of PIP. For
+ example we cannot expect to be able to exit a thread, if it was not
+ created yet.
+ These validity constraints on states are characterised by the
+ inductive predicate @{term "step"} and @{term vt}. We first give five inference rules
+ for @{term step} relating a state and an event that can happen next.
+
+ \begin{center}
+ \begin{tabular}{c}
+ @{thm[mode=Rule] thread_create[where thread=th]}\hspace{1cm}
+ @{thm[mode=Rule] thread_exit[where thread=th]}
+ \end{tabular}
+ \end{center}
+
+ \noindent
+ The first rule states that a thread can only be created, if it is not alive yet.
+ Similarly, the second rule states that a thread can only be terminated if it was
+ running and does not lock any resources anymore (this simplifies slightly our model;
+ in practice we would expect the operating system releases all locks held by a
+ thread that is about to exit). The event @{text Set} can happen
+ if the corresponding thread is running.
+
+ \begin{center}
+ @{thm[mode=Rule] thread_set[where thread=th]}
+ \end{center}
+
+ \noindent
+ If a thread wants to lock a resource, then the thread needs to be
+ running and also we have to make sure that the resource lock does
+ not lead to a cycle in the RAG. In practice, ensuring the latter
+ is the responsibility of the programmer. In our formal
+ model we brush aside these problematic cases in order to be able to make
+ some meaningful statements about PIP.\footnote{This situation is
+ similar to the infamous \emph{occurs check} in Prolog: In order to say
+ anything meaningful about unification, one needs to perform an occurs
+ check. But in practice the occurs check is omitted and the
+ responsibility for avoiding problems rests with the programmer.}
+
+
+ \begin{center}
+ @{thm[mode=Rule] thread_P[where thread=th]}
+ \end{center}
+
+ \noindent
+ Similarly, if a thread wants to release a lock on a resource, then
+ it must be running and in the possession of that lock. This is
+ formally given by the last inference rule of @{term step}.
+
+ \begin{center}
+ @{thm[mode=Rule] thread_V[where thread=th]}
+ \end{center}
+
+ \noindent
+ A valid state of PIP can then be conveniently be defined as follows:
+
+ \begin{center}
+ \begin{tabular}{c}
+ @{thm[mode=Axiom] vt_nil}\hspace{1cm}
+ @{thm[mode=Rule] vt_cons}
+ \end{tabular}
+ \end{center}
+
+ \noindent
+ This completes our formal model of PIP. In the next section we present
+ properties that show our model of PIP is correct.
+*}
+
+section {* The Correctness Proof *}
+
+(*<*)
+context extend_highest_gen
+begin
+(*>*)
+text {*
+ Sha et al.~state their first correctness criterion for PIP in terms
+ of the number of low-priority threads \cite[Theorem 3]{Sha90}: if
+ there are @{text n} low-priority threads, then a blocked job with
+ high priority can only be blocked a maximum of @{text n} times.
+ Their second correctness criterion is given
+ in terms of the number of critical resources \cite[Theorem 6]{Sha90}: if there are
+ @{text m} critical resources, then a blocked job with high priority
+ can only be blocked a maximum of @{text m} times. Both results on their own, strictly speaking, do
+ \emph{not} prevent indefinite, or unbounded, Priority Inversion,
+ because if a low-priority thread does not give up its critical
+ resource (the one the high-priority thread is waiting for), then the
+ high-priority thread can never run. The argument of Sha et al.~is
+ that \emph{if} threads release locked resources in a finite amount
+ of time, then indefinite Priority Inversion cannot occur---the high-priority
+ thread is guaranteed to run eventually. The assumption is that
+ programmers must ensure that threads are programmed in this way. However, even
+ taking this assumption into account, the correctness properties of
+ Sha et al.~are
+ \emph{not} true for their version of PIP---despite being ``proved''. As Yodaiken
+ \cite{Yodaiken02} pointed out: If a low-priority thread possesses
+ locks to two resources for which two high-priority threads are
+ waiting for, then lowering the priority prematurely after giving up
+ only one lock, can cause indefinite Priority Inversion for one of the
+ high-priority threads, invalidating their two bounds.
+
+ Even when fixed, their proof idea does not seem to go through for
+ us, because of the way we have set up our formal model of PIP. One
+ reason is that we allow critical sections, which start with a @{text P}-event
+ and finish with a corresponding @{text V}-event, to arbitrarily overlap
+ (something Sha et al.~explicitly exclude). Therefore we have
+ designed a different correctness criterion for PIP. The idea behind
+ our criterion is as follows: for all states @{text s}, we know the
+ corresponding thread @{text th} with the highest precedence; we show
+ that in every future state (denoted by @{text "s' @ s"}) in which
+ @{text th} is still alive, either @{text th} is running or it is
+ blocked by a thread that was alive in the state @{text s} and was waiting
+ for or in the possession of a lock in @{text s}. Since in @{text s}, as in
+ every state, the set of alive threads is finite, @{text th} can only
+ be blocked a finite number of times. This is independent of how many
+ threads of lower priority are created in @{text "s'"}. We will actually prove a
+ stronger statement where we also provide the current precedence of
+ the blocking thread. However, this correctness criterion hinges upon
+ a number of assumptions about the states @{text s} and @{text "s' @
+ s"}, the thread @{text th} and the events happening in @{text
+ s'}. We list them next:
+
+ \begin{quote}
+ {\bf Assumptions on the states {\boldmath@{text s}} and
+ {\boldmath@{text "s' @ s"}:}} We need to require that @{text "s"} and
+ @{text "s' @ s"} are valid states:
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{l}
+ @{term "vt s"}, @{term "vt (s' @ s)"}
+ \end{tabular}
+ \end{isabelle}
+ \end{quote}
+
+ \begin{quote}
+ {\bf Assumptions on the thread {\boldmath@{text "th"}:}}
+ The thread @{text th} must be alive in @{text s} and
+ has the highest precedence of all alive threads in @{text s}. Furthermore the
+ priority of @{text th} is @{text prio} (we need this in the next assumptions).
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{l}
+ @{term "th \<in> threads s"}\\
+ @{term "prec th s = Max (cprec s ` threads s)"}\\
+ @{term "prec th s = (prio, DUMMY)"}
+ \end{tabular}
+ \end{isabelle}
+ \end{quote}
+
+ \begin{quote}
+ {\bf Assumptions on the events in {\boldmath@{text "s'"}:}} We want to prove that @{text th} cannot
+ be blocked indefinitely. Of course this can happen if threads with higher priority
+ than @{text th} are continuously created in @{text s'}. Therefore we have to assume that
+ events in @{text s'} can only create (respectively set) threads with equal or lower
+ priority than @{text prio} of @{text th}. We also need to assume that the
+ priority of @{text "th"} does not get reset and also that @{text th} does
+ not get ``exited'' in @{text "s'"}. This can be ensured by assuming the following three implications.
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{l}
+ {If}~~@{text "Create th' prio' \<in> set s'"}~~{then}~~@{text "prio' \<le> prio"}\\
+ {If}~~@{text "Set th' prio' \<in> set s'"}~~{then}~~@{text "th' \<noteq> th"}~~{and}~~@{text "prio' \<le> prio"}\\
+ {If}~~@{text "Exit th' \<in> set s'"}~~{then}~~@{text "th' \<noteq> th"}\\
+ \end{tabular}
+ \end{isabelle}
+ \end{quote}
+
+ \noindent
+ The locale mechanism of Isabelle helps us to manage conveniently such assumptions~\cite{Haftmann08}.
+ Under these assumptions we shall prove the following correctness property:
+
+ \begin{theorem}\label{mainthm}
+ Given the assumptions about states @{text "s"} and @{text "s' @ s"},
+ the thread @{text th} and the events in @{text "s'"},
+ if @{term "th' \<in> running (s' @ s)"} and @{text "th' \<noteq> th"} then
+ @{text "th' \<in> threads s"}, @{text "\<not> detached s th'"} and
+ @{term "cp (s' @ s) th' = prec th s"}.
+ \end{theorem}
+
+ \noindent
+ This theorem ensures that the thread @{text th}, which has the
+ highest precedence in the state @{text s}, can only be blocked in
+ the state @{text "s' @ s"} by a thread @{text th'} that already
+ existed in @{text s} and requested or had a lock on at least
+ one resource---that means the thread was not \emph{detached} in @{text s}.
+ As we shall see shortly, that means there are only finitely
+ many threads that can block @{text th} in this way and then they
+ need to run with the same current precedence as @{text th}.
+
+ Like in the argument by Sha et al.~our
+ finite bound does not guarantee absence of indefinite Priority
+ Inversion. For this we further have to assume that every thread
+ gives up its resources after a finite amount of time. We found that
+ this assumption is awkward to formalise in our model. Therefore we
+ leave it out and let the programmer assume the responsibility to
+ program threads in such a benign manner (in addition to causing no
+ circularity in the @{text RAG}). In this detail, we do not
+ make any progress in comparison with the work by Sha et al.
+ However, we are able to combine their two separate bounds into a
+ single theorem improving their bound.
+
+ In what follows we will describe properties of PIP that allow us to prove
+ Theorem~\ref{mainthm} and, when instructive, briefly describe our argument.
+ It is relatively easy to see that
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{text "running s \<subseteq> ready s \<subseteq> threads s"}\\
+ @{thm[mode=IfThen] finite_threads}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ The second property is by induction of @{term vt}. The next three
+ properties are
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm[mode=IfThen] waiting_unique[of _ _ "cs\<^isub>1" "cs\<^isub>2"]}\\
+ @{thm[mode=IfThen] held_unique[of _ "th\<^isub>1" _ "th\<^isub>2"]}\\
+ @{thm[mode=IfThen] runing_unique[of _ "th\<^isub>1" "th\<^isub>2"]}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ The first property states that every waiting thread can only wait for a single
+ resource (because it gets suspended after requesting that resource); the second
+ that every resource can only be held by a single thread;
+ the third property establishes that in every given valid state, there is
+ at most one running thread. We can also show the following properties
+ about the @{term RAG} in @{text "s"}.
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{text If}~@{thm (prem 1) acyclic_depend}~@{text "then"}:\\
+ \hspace{5mm}@{thm (concl) acyclic_depend},
+ @{thm (concl) finite_depend} and
+ @{thm (concl) wf_dep_converse},\\
+ \hspace{5mm}@{text "if"}~@{thm (prem 2) dm_depend_threads}~@{text "then"}~@{thm (concl) dm_depend_threads}
+ and\\
+ \hspace{5mm}@{text "if"}~@{thm (prem 2) range_in}~@{text "then"}~@{thm (concl) range_in}.
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ The acyclicity property follows from how we restricted the events in
+ @{text step}; similarly the finiteness and well-foundedness property.
+ The last two properties establish that every thread in a @{text "RAG"}
+ (either holding or waiting for a resource) is a live thread.
+
+ The key lemma in our proof of Theorem~\ref{mainthm} is as follows:
+
+ \begin{lemma}\label{mainlem}
+ Given the assumptions about states @{text "s"} and @{text "s' @ s"},
+ the thread @{text th} and the events in @{text "s'"},
+ if @{term "th' \<in> threads (s' @ s)"}, @{text "th' \<noteq> th"} and @{text "detached (s' @ s) th'"}\\
+ then @{text "th' \<notin> running (s' @ s)"}.
+ \end{lemma}
+
+ \noindent
+ The point of this lemma is that a thread different from @{text th} (which has the highest
+ precedence in @{text s}) and not holding any resource, cannot be running
+ in the state @{text "s' @ s"}.
+
+ \begin{proof}
+ Since thread @{text "th'"} does not hold any resource, no thread can depend on it.
+ Therefore its current precedence @{term "cp (s' @ s) th'"} equals its own precedence
+ @{term "prec th' (s' @ s)"}. Since @{text "th"} has the highest precedence in the
+ state @{text "(s' @ s)"} and precedences are distinct among threads, we have
+ @{term "prec th' (s' @s ) < prec th (s' @ s)"}. From this
+ we have @{term "cp (s' @ s) th' < prec th (s' @ s)"}.
+ Since @{text "prec th (s' @ s)"} is already the highest
+ @{term "cp (s' @ s) th"} can not be higher than this and can not be lower either (by
+ definition of @{term "cp"}). Consequently, we have @{term "prec th (s' @ s) = cp (s' @ s) th"}.
+ Finally we have @{term "cp (s' @ s) th' < cp (s' @ s) th"}.
+ By defintion of @{text "running"}, @{text "th'"} can not be running in state
+ @{text "s' @ s"}, as we had to show.\qed
+ \end{proof}
+
+ \noindent
+ Since @{text "th'"} is not able to run in state @{text "s' @ s"}, it is not able to
+ issue a @{text "P"} or @{text "V"} event. Therefore if @{text "s' @ s"} is extended
+ one step further, @{text "th'"} still cannot hold any resource. The situation will
+ not change in further extensions as long as @{text "th"} holds the highest precedence.
+
+ From this lemma we can deduce Theorem~\ref{mainthm}: that @{text th} can only be
+ blocked by a thread @{text th'} that
+ held some resource in state @{text s} (that is not @{text "detached"}). And furthermore
+ that the current precedence of @{text th'} in state @{text "(s' @ s)"} must be equal to the
+ precedence of @{text th} in @{text "s"}.
+ We show this theorem by induction on @{text "s'"} using Lemma~\ref{mainlem}.
+ This theorem gives a stricter bound on the threads that can block @{text th} than the
+ one obtained by Sha et al.~\cite{Sha90}:
+ only threads that were alive in state @{text s} and moreover held a resource.
+ This means our bound is in terms of both---alive threads in state @{text s}
+ and number of critical resources. Finally, the theorem establishes that the blocking threads have the
+ current precedence raised to the precedence of @{text th}.
+
+ We can furthermore prove that under our assumptions no deadlock exists in the state @{text "s' @ s"}
+ by showing that @{text "running (s' @ s)"} is not empty.
+
+ \begin{lemma}
+ Given the assumptions about states @{text "s"} and @{text "s' @ s"},
+ the thread @{text th} and the events in @{text "s'"},
+ @{term "running (s' @ s) \<noteq> {}"}.
+ \end{lemma}
+
+ \begin{proof}
+ If @{text th} is blocked, then by following its dependants graph, we can always
+ reach a ready thread @{text th'}, and that thread must have inherited the
+ precedence of @{text th}.\qed
+ \end{proof}
+
+
+ %The following lemmas show how every node in RAG can be chased to ready threads:
+ %\begin{enumerate}
+ %\item Every node in RAG can be chased to a ready thread (@{text "chain_building"}):
+ % @ {thm [display] chain_building[rule_format]}
+ %\item The ready thread chased to is unique (@{text "dchain_unique"}):
+ % @ {thm [display] dchain_unique[of _ _ "th\<^isub>1" "th\<^isub>2"]}
+ %\end{enumerate}
+
+ %Some deeper results about the system:
+ %\begin{enumerate}
+ %\item The maximum of @{term "cp"} and @{term "preced"} are equal (@{text "max_cp_eq"}):
+ %@ {thm [display] max_cp_eq}
+ %\item There must be one ready thread having the max @{term "cp"}-value
+ %(@{text "max_cp_readys_threads"}):
+ %@ {thm [display] max_cp_readys_threads}
+ %\end{enumerate}
+
+ %The relationship between the count of @{text "P"} and @{text "V"} and the number of
+ %critical resources held by a thread is given as follows:
+ %\begin{enumerate}
+ %\item The @{term "V"}-operation decreases the number of critical resources
+ % one thread holds (@{text "cntCS_v_dec"})
+ % @ {thm [display] cntCS_v_dec}
+ %\item The number of @{text "V"} never exceeds the number of @{text "P"}
+ % (@ {text "cnp_cnv_cncs"}):
+ % @ {thm [display] cnp_cnv_cncs}
+ %\item The number of @{text "V"} equals the number of @{text "P"} when
+ % the relevant thread is not living:
+ % (@{text "cnp_cnv_eq"}):
+ % @ {thm [display] cnp_cnv_eq}
+ %\item When a thread is not living, it does not hold any critical resource
+ % (@{text "not_thread_holdents"}):
+ % @ {thm [display] not_thread_holdents}
+ %\item When the number of @{text "P"} equals the number of @{text "V"}, the relevant
+ % thread does not hold any critical resource, therefore no thread can depend on it
+ % (@{text "count_eq_dependents"}):
+ % @ {thm [display] count_eq_dependents}
+ %\end{enumerate}
+
+ %The reason that only threads which already held some resoures
+ %can be runing and block @{text "th"} is that if , otherwise, one thread
+ %does not hold any resource, it may never have its prioirty raised
+ %and will not get a chance to run. This fact is supported by
+ %lemma @{text "moment_blocked"}:
+ %@ {thm [display] moment_blocked}
+ %When instantiating @{text "i"} to @{text "0"}, the lemma means threads which did not hold any
+ %resource in state @{text "s"} will not have a change to run latter. Rephrased, it means
+ %any thread which is running after @{text "th"} became the highest must have already held
+ %some resource at state @{text "s"}.
+
+
+ %When instantiating @{text "i"} to a number larger than @{text "0"}, the lemma means
+ %if a thread releases all its resources at some moment in @{text "t"}, after that,
+ %it may never get a change to run. If every thread releases its resource in finite duration,
+ %then after a while, only thread @{text "th"} is left running. This shows how indefinite
+ %priority inversion can be avoided.
+
+ %All these assumptions are put into a predicate @{term "extend_highest_gen"}.
+ %It can be proved that @{term "extend_highest_gen"} holds
+ %for any moment @{text "i"} in it @{term "t"} (@{text "red_moment"}):
+ %@ {thm [display] red_moment}
+
+ %From this, an induction principle can be derived for @{text "t"}, so that
+ %properties already derived for @{term "t"} can be applied to any prefix
+ %of @{text "t"} in the proof of new properties
+ %about @{term "t"} (@{text "ind"}):
+ %\begin{center}
+ %@ {thm[display] ind}
+ %\end{center}
+
+ %The following properties can be proved about @{term "th"} in @{term "t"}:
+ %\begin{enumerate}
+ %\item In @{term "t"}, thread @{term "th"} is kept live and its
+ % precedence is preserved as well
+ % (@{text "th_kept"}):
+ % @ {thm [display] th_kept}
+ %\item In @{term "t"}, thread @{term "th"}'s precedence is always the maximum among
+ % all living threads
+ % (@{text "max_preced"}):
+ % @ {thm [display] max_preced}
+ %\item In @{term "t"}, thread @{term "th"}'s current precedence is always the maximum precedence
+ % among all living threads
+ % (@{text "th_cp_max_preced"}):
+ % @ {thm [display] th_cp_max_preced}
+ %\item In @{term "t"}, thread @{term "th"}'s current precedence is always the maximum current
+ % precedence among all living threads
+ % (@{text "th_cp_max"}):
+ % @ {thm [display] th_cp_max}
+ %\item In @{term "t"}, thread @{term "th"}'s current precedence equals its precedence at moment
+ % @{term "s"}
+ % (@{text "th_cp_preced"}):
+ % @ {thm [display] th_cp_preced}
+ %\end{enumerate}
+
+ %The main theorem of this part is to characterizing the running thread during @{term "t"}
+ %(@{text "runing_inversion_2"}):
+ %@ {thm [display] runing_inversion_2}
+ %According to this, if a thread is running, it is either @{term "th"} or was
+ %already live and held some resource
+ %at moment @{text "s"} (expressed by: @{text "cntV s th' < cntP s th'"}).
+
+ %Since there are only finite many threads live and holding some resource at any moment,
+ %if every such thread can release all its resources in finite duration, then after finite
+ %duration, none of them may block @{term "th"} anymore. So, no priority inversion may happen
+ %then.
+ *}
+(*<*)
+end
+(*>*)
+
+section {* Properties for an Implementation\label{implement} *}
+
+text {*
+ While our formalised proof gives us confidence about the correctness of our model of PIP,
+ we found that the formalisation can even help us with efficiently implementing it.
+
+ For example Baker complained that calculating the current precedence
+ in PIP is quite ``heavy weight'' in Linux (see the Introduction).
+ In our model of PIP the current precedence of a thread in a state @{text s}
+ depends on all its dependants---a ``global'' transitive notion,
+ which is indeed heavy weight (see Def.~shown in \eqref{cpreced}).
+ We can however improve upon this. For this let us define the notion
+ of @{term children} of a thread @{text th} in a state @{text s} as
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm children_def2}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ where a child is a thread that is only one ``hop'' away from the thread
+ @{text th} in the @{term RAG} (and waiting for @{text th} to release
+ a resource). We can prove the following lemma.
+
+ \begin{lemma}\label{childrenlem}
+ @{text "If"} @{thm (prem 1) cp_rec} @{text "then"}
+ \begin{center}
+ @{thm (concl) cp_rec}.
+ \end{center}
+ \end{lemma}
+
+ \noindent
+ That means the current precedence of a thread @{text th} can be
+ computed locally by considering only the children of @{text th}. In
+ effect, it only needs to be recomputed for @{text th} when one of
+ its children changes its current precedence. Once the current
+ precedence is computed in this more efficient manner, the selection
+ of the thread with highest precedence from a set of ready threads is
+ a standard scheduling operation implemented in most operating
+ systems.
+
+ Of course the main work for implementing PIP involves the
+ scheduler and coding how it should react to events. Below we
+ outline how our formalisation guides this implementation for each
+ kind of events.\smallskip
+*}
+
+(*<*)
+context step_create_cps
+begin
+(*>*)
+text {*
+ \noindent
+ \colorbox{mygrey}{@{term "Create th prio"}:} We assume that the current state @{text s'} and
+ the next state @{term "s \<equiv> Create th prio#s'"} are both valid (meaning the event
+ is allowed to occur). In this situation we can show that
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm eq_dep},\\
+ @{thm eq_cp_th}, and\\
+ @{thm[mode=IfThen] eq_cp}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ This means in an implementation we do not have recalculate the @{text RAG} and also none of the
+ current precedences of the other threads. The current precedence of the created
+ thread @{text th} is just its precedence, namely the pair @{term "(prio, length (s::event list))"}.
+ \smallskip
+ *}
+(*<*)
+end
+context step_exit_cps
+begin
+(*>*)
+text {*
+ \noindent
+ \colorbox{mygrey}{@{term "Exit th"}:} We again assume that the current state @{text s'} and
+ the next state @{term "s \<equiv> Exit th#s'"} are both valid. We can show that
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm eq_dep}, and\\
+ @{thm[mode=IfThen] eq_cp}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ This means again we do not have to recalculate the @{text RAG} and
+ also not the current precedences for the other threads. Since @{term th} is not
+ alive anymore in state @{term "s"}, there is no need to calculate its
+ current precedence.
+ \smallskip
+*}
+(*<*)
+end
+context step_set_cps
+begin
+(*>*)
+text {*
+ \noindent
+ \colorbox{mygrey}{@{term "Set th prio"}:} We assume that @{text s'} and
+ @{term "s \<equiv> Set th prio#s'"} are both valid. We can show that
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm[mode=IfThen] eq_dep}, and\\
+ @{thm[mode=IfThen] eq_cp_pre}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ The first property is again telling us we do not need to change the @{text RAG}.
+ The second shows that the @{term cp}-values of all threads other than @{text th}
+ are unchanged. The reason is that @{text th} is running; therefore it is not in
+ the @{term dependants} relation of any other thread. This in turn means that the
+ change of its priority cannot affect other threads.
+
+ %The second
+ %however states that only threads that are \emph{not} dependants of @{text th} have their
+ %current precedence unchanged. For the others we have to recalculate the current
+ %precedence. To do this we can start from @{term "th"}
+ %and follow the @{term "depend"}-edges to recompute using Lemma~\ref{childrenlem}
+ %the @{term "cp"} of every
+ %thread encountered on the way. Since the @{term "depend"}
+ %is assumed to be loop free, this procedure will always stop. The following two lemmas show, however,
+ %that this procedure can actually stop often earlier without having to consider all
+ %dependants.
+ %
+ %\begin{isabelle}\ \ \ \ \ %%%
+ %\begin{tabular}{@ {}l}
+ %@{thm[mode=IfThen] eq_up_self}\\
+ %@{text "If"} @{thm (prem 1) eq_up}, @{thm (prem 2) eq_up} and @{thm (prem 3) eq_up}\\
+ %@{text "then"} @{thm (concl) eq_up}.
+ %\end{tabular}
+ %\end{isabelle}
+ %
+ %\noindent
+ %The first lemma states that if the current precedence of @{text th} is unchanged,
+ %then the procedure can stop immediately (all dependent threads have their @{term cp}-value unchanged).
+ %The second states that if an intermediate @{term cp}-value does not change, then
+ %the procedure can also stop, because none of its dependent threads will
+ %have their current precedence changed.
+ \smallskip
+ *}
+(*<*)
+end
+context step_v_cps_nt
+begin
+(*>*)
+text {*
+ \noindent
+ \colorbox{mygrey}{@{term "V th cs"}:} We assume that @{text s'} and
+ @{term "s \<equiv> V th cs#s'"} are both valid. We have to consider two
+ subcases: one where there is a thread to ``take over'' the released
+ resource @{text cs}, and one where there is not. Let us consider them
+ in turn. Suppose in state @{text s}, the thread @{text th'} takes over
+ resource @{text cs} from thread @{text th}. We can prove
+
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{thm depend_s}
+ \end{isabelle}
+
+ \noindent
+ which shows how the @{text RAG} needs to be changed. The next lemma suggests
+ how the current precedences need to be recalculated. For threads that are
+ not @{text "th"} and @{text "th'"} nothing needs to be changed, since we
+ can show
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ @{thm[mode=IfThen] cp_kept}
+ \end{isabelle}
+
+ \noindent
+ For @{text th} and @{text th'} we need to use Lemma~\ref{childrenlem} to
+ recalculate their current precedence since their children have changed. *}(*<*)end context step_v_cps_nnt begin (*>*)text {*
+ \noindent
+ In the other case where there is no thread that takes over @{text cs}, we can show how
+ to recalculate the @{text RAG} and also show that no current precedence needs
+ to be recalculated.
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm depend_s}\\
+ @{thm eq_cp}
+ \end{tabular}
+ \end{isabelle}
+ *}
+(*<*)
+end
+context step_P_cps_e
+begin
+(*>*)
+text {*
+ \noindent
+ \colorbox{mygrey}{@{term "P th cs"}:} We assume that @{text s'} and
+ @{term "s \<equiv> P th cs#s'"} are both valid. We again have to analyse two subcases, namely
+ the one where @{text cs} is not locked, and one where it is. We treat the former case
+ first by showing that
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm depend_s}\\
+ @{thm eq_cp}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ This means we need to add a holding edge to the @{text RAG} and no
+ current precedence needs to be recalculated.*}(*<*)end context step_P_cps_ne begin(*>*) text {*
+ \noindent
+ In the second case we know that resource @{text cs} is locked. We can show that
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ @{thm depend_s}\\
+ @{thm[mode=IfThen] eq_cp}
+ \end{tabular}
+ \end{isabelle}
+
+ \noindent
+ That means we have to add a waiting edge to the @{text RAG}. Furthermore
+ the current precedence for all threads that are not dependants of @{text th}
+ are unchanged. For the others we need to follow the edges
+ in the @{text RAG} and recompute the @{term "cp"}. To do this we can start from @{term "th"}
+ and follow the @{term "depend"}-edges to recompute using Lemma~\ref{childrenlem}
+ the @{term "cp"} of every
+ thread encountered on the way. Since the @{term "depend"}
+ is loop free, this procedure will always stop. The following lemma shows, however,
+ that this procedure can actually stop often earlier without having to consider all
+ dependants.
+
+ \begin{isabelle}\ \ \ \ \ %%%
+ \begin{tabular}{@ {}l}
+ %%@ {t hm[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
+ This lemma 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.
+ *}
+(*<*)
+end
+(*>*)
+text {*
+ \noindent
+ As can be seen, a pleasing byproduct of our formalisation is that the properties in
+ this section closely inform an implementation of PIP, namely whether the
+ @{text RAG} needs to be reconfigured or current precedences need to
+ be recalculated for an event. This information is provided by the lemmas we proved.
+ We confirmed that our observations translate into practice by implementing
+ our version of PIP on top of PINTOS, a small operating system written in C and used for teaching at
+ Stanford University \cite{PINTOS}. To implement PIP, we only need to modify the kernel
+ functions corresponding to the events in our formal model. The events translate to the following
+ function interface in PINTOS:
+
+ \begin{center}
+ \begin{tabular}{|l@ {\hspace{2mm}}|l@ {\hspace{2mm}}|}
+ \hline
+ {\bf Event} & {\bf PINTOS function} \\
+ \hline
+ @{text Create} & @{text "thread_create"}\\
+ @{text Exit} & @{text "thread_exit"}\\
+ @{text Set} & @{text "thread_set_priority"}\\
+ @{text P} & @{text "lock_acquire"}\\
+ @{text V} & @{text "lock_release"}\\
+ \hline
+ \end{tabular}
+ \end{center}
+
+ \noindent
+ Our implicit assumption that every event is an atomic operation is ensured by the architecture of
+ PINTOS. The case where an unlocked resource is given next to the waiting thread with the
+ highest precedence is realised in our implementation by priority queues. We implemented
+ them as \emph{Braun trees} \cite{Paulson96}, which provide efficient @{text "O(log n)"}-operations
+ for accessing and updating. Apart from having to implement relatively complex data\-structures in C
+ using pointers, our experience with the implementation has been very positive: our specification
+ and formalisation of PIP translates smoothly to an efficent implementation in PINTOS.
+*}
+
+section {* Conclusion *}
+
+text {*
+ The Priority Inheritance Protocol (PIP) is a classic textbook
+ algorithm used in many real-time operating systems in order to avoid the problem of
+ Priority Inversion. Although classic and widely used, PIP does have
+ its faults: for example it does not prevent deadlocks in cases where threads
+ have circular lock dependencies.
+
+ We had two goals in mind with our formalisation of PIP: One is to
+ make the notions in the correctness proof by Sha et al.~\cite{Sha90}
+ precise so that they can be processed by a theorem prover. The reason is
+ that a mechanically checked proof avoids the flaws that crept into their
+ informal reasoning. We achieved this goal: The correctness of PIP now
+ only hinges on the assumptions behind our formal model. The reasoning, which is
+ sometimes quite intricate and tedious, has been checked by Isabelle/HOL.
+ We can also confirm that Paulson's
+ inductive method for protocol verification~\cite{Paulson98} is quite
+ suitable for our formal model and proof. The traditional application
+ area of this method is security protocols.
+
+ The second goal of our formalisation is to provide a specification for actually
+ implementing PIP. Textbooks, for example \cite[Section 5.6.5]{Vahalia96},
+ explain how to use various implementations of PIP and abstractly
+ discuss their properties, but surprisingly lack most details important for a
+ programmer who wants to implement PIP (similarly Sha et al.~\cite{Sha90}).
+ That this is an issue in practice is illustrated by the
+ email from Baker we cited in the Introduction. We achieved also this
+ goal: The formalisation allowed us to efficently implement our version
+ of PIP on top of PINTOS \cite{PINTOS}, a simple instructional operating system for the x86
+ architecture. It also gives the first author enough data to enable
+ his undergraduate students to implement PIP (as part of their OS course).
+ A byproduct of our formalisation effort is that nearly all
+ design choices for the PIP scheduler are backed up with a proved
+ lemma. We were also able to establish the property that the choice of
+ the next thread which takes over a lock is irrelevant for the correctness
+ of PIP.
+
+ PIP is a scheduling algorithm for single-processor systems. We are
+ now living in a multi-processor world. Priority Inversion certainly
+ occurs also there. However, there is very little ``foundational''
+ work about PIP-algorithms on multi-processor systems. We are not
+ aware of any correctness proofs, not even informal ones. There is an
+ implementation of a PIP-algorithm for multi-processors as part of the
+ ``real-time'' effort in Linux, including an informal description of the implemented scheduling
+ algorithm given in \cite{LINUX}. We estimate that the formal
+ verification of this algorithm, involving more fine-grained events,
+ is a magnitude harder than the one we presented here, but still
+ within reach of current theorem proving technology. We leave this
+ for future work.
+
+ The most closely related work to ours is the formal verification in
+ PVS of the Priority Ceiling Protocol done by Dutertre
+ \cite{dutertre99b}---another solution to the Priority Inversion
+ problem, which however needs static analysis of programs in order to
+ avoid it. There have been earlier formal investigations
+ into PIP \cite{Faria08,Jahier09,Wellings07}, but they employ model
+ checking techniques. The results obtained by them apply,
+ however, only to systems with a fixed size, such as a fixed number of
+ events and threads. In contrast, our result applies to systems of arbitrary
+ size. Moreover, our result is a good
+ witness for one of the major reasons to be interested in machine checked
+ reasoning: gaining deeper understanding of the subject matter.
+
+ Our formalisation
+ consists of around 210 lemmas and overall 6950 lines of readable Isabelle/Isar
+ code with a few apply-scripts interspersed. The formal model of PIP
+ is 385 lines long; the formal correctness proof 3800 lines. Some auxiliary
+ definitions and proofs span over 770 lines of code. The properties relevant
+ for an implementation require 2000 lines.
+ %The code of our formalisation
+ %can be downloaded from
+ %\url{http://www.inf.kcl.ac.uk/staff/urbanc/pip.html}.\medskip
+
+ \noindent
+ {\bf Acknowledgements:}
+ We are grateful for the comments we received from anonymous
+ referees.
+
+ \bibliographystyle{plain}
+ \bibliography{root}
+*}
+
+
+(*<*)
+end
+(*>*)
\ No newline at end of file