(*<*)

theory Paper

imports CpsG ExtGG (* "~~/src/HOL/Library/LaTeXsugar" *) LaTeXsugar

begin

ML {*

  open Printer;

  show_question_marks_default := false;

  *}

(*>*)

section {* Introduction *}

text {*

  locking of resources. Locking of resources ensures mutual exclusion

  when accessing shared data or devices that cannot be

  finish their work within deadlines.  Unfortunately, both features

  can interact in subtle ways leading to a problem, called

  implementation of resource looking and priorities this property can

  be violated. Even worse, $H$ can be delayed indefinitely by

  possession of a lock for a resource that also $H$ needs. $H$ must

  therefore wait for $L$ to exit the critical section and release this

  lock. The problem is that $L$ might in turn be blocked by any

  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

  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

  \begin{quote}

  \it{}``Priority inheritance is neither efficient nor reliable. Implementations

  are either incomplete (and unreliable) or surprisingly complex and intrusive.''

  \end{quote}

  \noindent

  He suggests to avoid PIP altogether by not allowing critical

  sections to be preempted. While this might have been a reasonable

  solution in 2002, in our modern multiprocessor world, this seems out

  of date. The reason is that this precludes other high-priority 

  resource.

  However, there is clearly a need for investigating correct

  algorithms for PIP. A few specifications for PIP exist (in English)

  and also a few high-level descriptions of implementations (e.g.~in

  the textbook \cite[Section 5.6.5]{Vahalia96}), but they help little

  with actual implementations. That this is a problem in practise is

  proved by an email from Baker, who wrote on 13 July 2009 on the Linux

  Kernel mailing list:

  \begin{quote}

  \it{}``I observed in the kernel code (to my disgust), the Linux PIP

  implementation is a nightmare: extremely heavy weight, involving

  maintenance of a full wait-for graph, and requiring updates for a

  range of events, including priority changes and interruptions of

  wait operations.''

  \end{quote}

  \noindent

  The criticism by Yodaiken, Baker and others suggests to us to look

  again at PIP from a more abstract level (but still concrete enough

  to inform an implementation) and makes PIP an ideal candidate for a

  formal verification. One reason, of course, is that the original

  ``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

  completes its critical section and releases the lock, it ``returns

  to its original priority level.'' This leads them to believe that an

  $H'$ each wait for one of them.  If $L$ then releases one resource

  so that $H$, say, can proceed, then we still have Priority Inversion

  with $H'$ (which waits for the other resource). The correct

  behaviour for $L$ is to revert to the highest remaining priority of

  There have been earlier formal investigations into PIP, but ...\cite{Faria08}

*}

section {* Formal Model of the Priority Inheritance Protocol *}

text {*

  \begin{isabelle}\ \ \ \ \ %%%

  \end{isabelle}

  \noindent

  To define events, the identifiers of {\em threads},

  {\em priority} and {\em critical resources } (abbreviated as @{text "cs"}) 

  need to be represented. All three are represetned using standard 

  Isabelle/HOL type @{typ "nat"}:

*}

text {*

  \bigskip

  focuses on on two issues:

  \begin{enumerate}

  \item The correctness of the protocol model itself. A series of desirable properties is 

    derived until we are fully convinced that the formal model of PI does 

    eliminate priority inversion. And a better understanding of PI is so obtained 

    in due course. For example, we find through formalization that the choice of 

    next thread to take hold when a 

    resource is released is irrelevant for the very basic property of PI to hold. 

    A point never mentioned in literature. 

  \item The correctness of the implementation. A series of properties is derived the meaning 

    of which can be used as guidelines on how PI can be implemented efficiently and correctly. 

  \end{enumerate} 

  The rest of the paper is organized as follows: Section \ref{overview} gives an overview 

  of PI. Section \ref{model} introduces the formal model of PI. Section \ref{general} 

  discusses a series of basic properties of PI. Section \ref{extension} shows formally 

  how priority inversion is controlled by PI. Section \ref{implement} gives properties 

  which can be used for guidelines of implementation. Section \ref{related} discusses 

  related works. Section \ref{conclusion} concludes the whole paper.

  The basic priority inheritance protocol has two problems:

  It does not prevent a deadlock from happening in a program with circular lock dependencies.

  A chain of blocking may be formed; blocking duration can be substantial, though bounded.

  Contributions

  Despite the wide use of Priority Inheritance Protocol in real time operating

  system, it's correctness has never been formally proved and mechanically checked. 

  All existing verification are based on model checking technology. Full automatic

  verification gives little help to understand why the protocol is correct. 

  And results such obtained only apply to models of limited size. 

  This paper presents a formal verification based on theorem proving. 

  Machine checked formal proof does help to get deeper understanding. We found 

  the fact which is not mentioned in the literature, that the choice of next 

  thread to take over when an critical resource is release does not affect the correctness

  of the protocol. The paper also shows how formal proof can help to construct 

  correct and efficient implementation.\bigskip 

*}

section {* An overview of priority inversion and priority inheritance \label{overview} *}

text {*

  Priority inversion refers to the phenomenon when a thread with high priority is blocked 

  by a thread with low priority. Priority happens when the high priority thread requests 

  for some critical resource already taken by the low priority thread. Since the high 

  priority thread has to wait for the low priority thread to complete, it is said to be 

  blocked by the low priority thread. Priority inversion might prevent high priority 

  thread from fulfill its task in time if the duration of priority inversion is indefinite 

  and unpredictable. Indefinite priority inversion happens when indefinite number 

  of threads with medium priorities is activated during the period when the high 

  priority thread is blocked by the low priority thread. Although these medium 

  priority threads can not preempt the high priority thread directly, they are able 

  to preempt the low priority threads and cause it to stay in critical section for 

  an indefinite long duration. In this way, the high priority thread may be blocked indefinitely. 

  Priority inheritance is one protocol proposed to avoid indefinite priority inversion. 

  The basic idea is to let the high priority thread donate its priority to the low priority 

  thread holding the critical resource, so that it will not be preempted by medium priority 

  threads. The thread with highest priority will not be blocked unless it is requesting 

  some critical resource already taken by other threads. Viewed from a different angle, 

  any thread which is able to block the highest priority threads must already hold some 

  critical resource. Further more, it must have hold some critical resource at the 

  moment the highest priority is created, otherwise, it may never get change to run and 

  get hold. Since the number of such resource holding lower priority threads is finite, 

  if every one of them finishes with its own critical section in a definite duration, 

  the duration the highest priority thread is blocked is definite as well. The key to 

  guarantee lower priority threads to finish in definite is to donate them the highest 

  priority. In such cases, the lower priority threads is said to have inherited the 

  highest priority. And this explains the name of the protocol: 

  {\em Priority Inheritance} and how Priority Inheritance prevents indefinite delay.

  The objectives of this paper are:

  \begin{enumerate}

  \item Build the above mentioned idea into formal model and prove a series of properties 

    until we are convinced that the formal model does fulfill the original idea. 

  \item Show how formally derived properties can be used as guidelines for correct 

    and efficient implementation.

  \end{enumerate}

  The proof is totally formal in the sense that every detail is reduced to the 

  very first principles of Higher Order Logic. The nature of interactive theorem 

  proving is for the human user to persuade computer program to accept its arguments. 

  A clear and simple understanding of the problem at hand is both a prerequisite and a 

  byproduct of such an effort, because everything has finally be reduced to the very 

  first principle to be checked mechanically. The former intuitive explanation of 

  Priority Inheritance is just such a byproduct. 

  *}

section {* Formal model of Priority Inheritance \label{model} *}

text {*

  \input{../../generated/PrioGDef}

*}

section {* General properties of Priority Inheritance \label{general} *}

text {*

  The following are several very basic prioprites:

  \begin{enumerate}

  \item All runing threads must be ready (@{text "runing_ready"}):

          @{thm[display] "runing_ready"}  

  \item All ready threads must be living (@{text "readys_threads"}):

          @{thm[display] "readys_threads"} 

  \item There are finite many living threads at any moment (@{text "finite_threads"}):

          @{thm[display] "finite_threads"} 

  \item Every waiting queue does not contain duplcated elements (@{text "wq_distinct"}): 

          @{thm[display] "wq_distinct"} 

  \item All threads in waiting queues are living threads (@{text "wq_threads"}): 

          @{thm[display] "wq_threads"} 

  \item The event which can get a thread into waiting queue must be @{term "P"}-events

         (@{text "block_pre"}): 

          @{thm[display] "block_pre"}   

  \item A thread may never wait for two different critical resources

         (@{text "waiting_unique"}): 

          @{thm[display] waiting_unique[of _ _ "cs\<^isub>1" "cs\<^isub>2"]}

  \item Every resource can only be held by one thread

         (@{text "held_unique"}): 

          @{thm[display] held_unique[of _ "th\<^isub>1" _ "th\<^isub>2"]}

  \item Every living thread has an unique precedence

         (@{text "preced_unique"}): 

          @{thm[display] preced_unique[of "th\<^isub>1" _ "th\<^isub>2"]}

  \end{enumerate}

*}

text {* \noindent

  The following lemmas show how RAG is changed with the execution of events:

  \begin{enumerate}

  \item Execution of @{term "Set"} does not change RAG (@{text "depend_set_unchanged"}):

    @{thm[display] depend_set_unchanged}

  \item Execution of @{term "Create"} does not change RAG (@{text "depend_create_unchanged"}):

    @{thm[display] depend_create_unchanged}

  \item Execution of @{term "Exit"} does not change RAG (@{text "depend_exit_unchanged"}):

    @{thm[display] depend_exit_unchanged}

  \item Execution of @{term "P"} (@{text "step_depend_p"}):

    @{thm[display] step_depend_p}

  \item Execution of @{term "V"} (@{text "step_depend_v"}):

    @{thm[display] step_depend_v}

  \end{enumerate}

  *}

text {* \noindent

  These properties are used to derive the following important results about RAG:

  \begin{enumerate}

  \item RAG is loop free (@{text "acyclic_depend"}):

  @{thm [display] acyclic_depend}

  \item RAGs are finite (@{text "finite_depend"}):

  @{thm [display] finite_depend}

  \item Reverse paths in RAG are well founded (@{text "wf_dep_converse"}):

  @{thm [display] wf_dep_converse}

  \item The dependence relation represented by RAG has a tree structure (@{text "unique_depend"}):

  @{thm [display] unique_depend[of _ _ "n\<^isub>1" "n\<^isub>2"]}

  \item All threads in RAG are living threads 

    (@{text "dm_depend_threads"} and @{text "range_in"}):

    @{thm [display] dm_depend_threads range_in}

  \end{enumerate}

  *}

text {* \noindent

  The following lemmas show how every node in RAG can be chased to ready threads:

  \begin{enumerate}

  \item Every node in RAG can be chased to a ready thread (@{text "chain_building"}):

    @{thm [display] chain_building[rule_format]}

  \item The ready thread chased to is unique (@{text "dchain_unique"}):

    @{thm [display] dchain_unique[of _ _ "th\<^isub>1" "th\<^isub>2"]}

  \end{enumerate}

  *}

text {* \noindent

  Properties about @{term "next_th"}:

  \begin{enumerate}

  \item The thread taking over is different from the thread which is releasing

  (@{text "next_th_neq"}):

  @{thm [display] next_th_neq}

  \item The thread taking over is unique

  (@{text "next_th_unique"}):

  @{thm [display] next_th_unique[of _ _ _ "th\<^isub>1" "th\<^isub>2"]}  

  \end{enumerate}

  *}

text {* \noindent

  Some deeper results about the system:

  \begin{enumerate}

  \item There can only be one running thread (@{text "runing_unique"}):

  @{thm [display] runing_unique[of _ "th\<^isub>1" "th\<^isub>2"]}

  \item The maximum of @{term "cp"} and @{term "preced"} are equal (@{text "max_cp_eq"}):

  @{thm [display] max_cp_eq}

  \item There must be one ready thread having the max @{term "cp"}-value 

  (@{text "max_cp_readys_threads"}):

  @{thm [display] max_cp_readys_threads}

  \end{enumerate}

  *}

text {* \noindent

  The relationship between the count of @{text "P"} and @{text "V"} and the number of 

  critical resources held by a thread is given as follows:

  \begin{enumerate}

  \item The @{term "V"}-operation decreases the number of critical resources 

    one thread holds (@{text "cntCS_v_dec"})

     @{thm [display]  cntCS_v_dec}

  \item The number of @{text "V"} never exceeds the number of @{text "P"} 

    (@{text "cnp_cnv_cncs"}):

    @{thm [display]  cnp_cnv_cncs}

  \item The number of @{text "V"} equals the number of @{text "P"} when 

    the relevant thread is not living:

    (@{text "cnp_cnv_eq"}):

    @{thm [display]  cnp_cnv_eq}

  \item When a thread is not living, it does not hold any critical resource 

    (@{text "not_thread_holdents"}):

    @{thm [display] not_thread_holdents}

  \item When the number of @{text "P"} equals the number of @{text "V"}, the relevant 

    thread does not hold any critical resource, therefore no thread can depend on it