--- a/prio/Paper/Paper.thy Mon Dec 03 08:16:58 2012 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1345 +0,0 @@
-(*<*)
-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