(*<*)

theory Paper

imports CpsG ExtGG LaTeXsugar

begin

ML {*

  *}

(*>*)

section {* Introduction *}

text {*

  Many realtime systems need to support processes with priorities and

  Priority inversion referrers to the phenomena where tasks with higher 

  priority are blocked by ones with lower priority. If priority inversion 

  is not controlled, there will be no guarantee the urgent tasks will be 

  processed in time. As reported in \cite{Reeves-Glenn-1998}, 

  priority inversion used to cause software system resets and data lose in 

  JPL's Mars pathfinder project. Therefore, the avoiding, detecting and controlling 

  of priority inversion is a key issue to attain predictability in priority 

  based real-time systems. 

  The priority inversion phenomenon was first published in \cite{Lampson:Redell:cacm:1980}. 

  The two protocols widely used to eliminate priority inversion, namely 

  PI (Priority Inheritance) and PCE (Priority Ceiling Emulation), were proposed 

  in \cite{journals/tc/ShaRL90}. PCE is less convenient to use because it requires 

  static analysis of programs. Therefore, PI is more commonly used in 

  practice\cite{locke-july02}. However, as pointed out in the literature, 

  the analysis of priority inheritance protocol is quite subtle\cite{yodaiken-july02}. 

  A formal analysis will certainly be helpful for us to understand and correctly 

  implement PI. All existing formal analysis of PI

  \cite{conf/fase/JahierHR09,WellingsBSB07,Faria08} are based on the model checking 

  technology. Because of the state explosion problem, model check 

  is much like an exhaustive testing of finite models with limited size. 

  The results obtained can not be safely generalized to models with arbitrarily 

  large size. Worse still, since model checking is fully automatic, it give little 

  insight on why the formal model is correct. It is therefore 

  definitely desirable to analyze PI using theorem proving, which gives 

  more general results as well as deeper insight. And this is the purpose 

  of this paper which gives a formal analysis of PI in the interactive 

  theorem prover Isabelle using Higher Order Logic (HOL). The formalization 

  focuses on on two issues:

  \begin{enumerate}

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

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

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

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

    next thread to take hold when a 

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

    A point never mentioned in literature. 

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

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

  \end{enumerate} 

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

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

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

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

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

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

  Contributions

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

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

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

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

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

  This paper presents a formal verification based on theorem proving. 

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

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

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

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

  correct and efficient implementation.\bigskip 

*}

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

text {*

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

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

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

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

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

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

  and unpredictable. Indefinite priority inversion happens when indefinite number 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  The objectives of this paper are:

  \begin{enumerate}

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

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

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

    and efficient implementation.

  \end{enumerate}

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

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

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

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

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

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

  Priority Inheritance is just such a byproduct. 

  *}

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

text {*

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

*}

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

text {*

  The following are several very basic prioprites:

  \begin{enumerate}

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

          @{thm[display] "runing_ready"}  

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

          @{thm[display] "readys_threads"} 

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

          @{thm[display] "finite_threads"} 

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

          @{thm[display] "wq_distinct"} 

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

          @{thm[display] "wq_threads"} 

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

         (@{text "block_pre"}): 

          @{thm[display] "block_pre"}   

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

         (@{text "waiting_unique"}): 

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

  \item Every resource can only be held by one thread

         (@{text "held_unique"}): 

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

  \item Every living thread has an unique precedence

         (@{text "preced_unique"}): 

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

  \end{enumerate}

*}

text {* \noindent

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

  \begin{enumerate}

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

    @{thm[display] depend_set_unchanged}

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

    @{thm[display] depend_create_unchanged}

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

    @{thm[display] depend_exit_unchanged}

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

    @{thm[display] step_depend_p}

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

    @{thm[display] step_depend_v}

  \end{enumerate}

  *}

text {* \noindent

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

  \begin{enumerate}

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

  @{thm [display] acyclic_depend}

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

  @{thm [display] finite_depend}

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

  @{thm [display] wf_dep_converse}

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

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

  \item All threads in RAG are living threads 

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

    @{thm [display] dm_depend_threads range_in}

  \end{enumerate}

  *}

text {* \noindent

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

  \begin{enumerate}

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

    @{thm [display] chain_building[rule_format]}

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

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

  \end{enumerate}

  *}

text {* \noindent

  Properties about @{term "next_th"}:

  \begin{enumerate}

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

  (@{text "next_th_neq"}):

  @{thm [display] next_th_neq}

  \item The thread taking over is unique

  (@{text "next_th_unique"}):

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

  \end{enumerate}

  *}

text {* \noindent

  Some deeper results about the system:

  \begin{enumerate}

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

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

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

  @{thm [display] max_cp_eq}

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

  (@{text "max_cp_readys_threads"}):

  @{thm [display] max_cp_readys_threads}

  \end{enumerate}

  *}

text {* \noindent

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

  critical resources held by a thread is given as follows:

  \begin{enumerate}

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

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

     @{thm [display]  cntCS_v_dec}

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

    (@{text "cnp_cnv_cncs"}):

    @{thm [display]  cnp_cnv_cncs}

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

    the relevant thread is not living:

    (@{text "cnp_cnv_eq"}):

    @{thm [display]  cnp_cnv_eq}

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

    (@{text "not_thread_holdents"}):

    @{thm [display] not_thread_holdents}

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

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

    (@{text "count_eq_dependents"}):

    @{thm [display] count_eq_dependents}

  \end{enumerate}

  *}

section {* Key properties \label{extension} *}

(*<*)

context extend_highest_gen

begin

(*>*)

text {*

  The essential of {\em Priority Inheritance} is to avoid indefinite priority inversion. For this 

  purpose, we need to investigate what happens after one thread takes the highest precedence. 

  A locale is used to describe such a situation, which assumes:

  \begin{enumerate}

  \item @{term "s"} is a valid state (@{text "vt_s"}):

    @{thm  vt_s}.

  \item @{term "th"} is a living thread in @{term "s"} (@{text "threads_s"}):

    @{thm threads_s}.

  \item @{term "th"} has the highest precedence in @{term "s"} (@{text "highest"}):

    @{thm highest}.

  \item The precedence of @{term "th"} is @{term "Prc prio tm"} (@{text "preced_th"}):

    @{thm preced_th}.

  \end{enumerate}

  *}

text {* \noindent

  Under these assumptions, some basic priority can be derived for @{term "th"}:

  \begin{enumerate}

  \item The current precedence of @{term "th"} equals its own precedence (@{text "eq_cp_s_th"}):

    @{thm [display] eq_cp_s_th}

  \item The current precedence of @{term "th"} is the highest precedence in 

    the system (@{text "highest_cp_preced"}):

    @{thm [display] highest_cp_preced}

  \item The precedence of @{term "th"} is the highest precedence 

    in the system (@{text "highest_preced_thread"}):

    @{thm [display] highest_preced_thread}

  \item The current precedence of @{term "th"} is the highest current precedence 

    in the system (@{text "highest'"}):

    @{thm [display] highest'}

  \end{enumerate}

  *}

text {* \noindent

  To analysis what happens after state @{term "s"} a sub-locale is defined, which 

  assumes:

  \begin{enumerate}

  \item @{term "t"} is a valid extension of @{term "s"} (@{text "vt_t"}): @{thm vt_t}.

  \item Any thread created in @{term "t"} has priority no higher than @{term "prio"}, therefore

    its precedence can not be higher than @{term "th"},  therefore

    @{term "th"} remain to be the one with the highest precedence

    (@{text "create_low"}):

    @{thm [display] create_low}

  \item Any adjustment of priority in 

    @{term "t"} does not happen to @{term "th"} and 

    the priority set is no higher than @{term "prio"}, therefore

    @{term "th"} remain to be the one with the highest precedence (@{text "set_diff_low"}):

    @{thm [display] set_diff_low}

  \item Since we are investigating what happens to @{term "th"}, it is assumed 

    @{term "th"} does not exit during @{term "t"} (@{text "exit_diff"}):

    @{thm [display] exit_diff}

  \end{enumerate}

*}

text {* \noindent

  All these assumptions are put into a predicate @{term "extend_highest_gen"}. 

  It can be proved that @{term "extend_highest_gen"} holds 

  for any moment @{text "i"} in it @{term "t"} (@{text "red_moment"}):

  @{thm [display] red_moment}

  From this, an induction principle can be derived for @{text "t"}, so that 

  properties already derived for @{term "t"} can be applied to any prefix 

  of @{text "t"} in the proof of new properties 

  about @{term "t"} (@{text "ind"}):

  \begin{center}

  @{thm[display] ind}

  \end{center}

  The following properties can be proved about @{term "th"} in @{term "t"}:

  \begin{enumerate}

  \item In @{term "t"}, thread @{term "th"} is kept live and its 

    precedence is preserved as well

    (@{text "th_kept"}): 

    @{thm [display] th_kept}

  \item In @{term "t"}, thread @{term "th"}'s precedence is always the maximum among 

    all living threads

    (@{text "max_preced"}): 

    @{thm [display] max_preced}

  \item In @{term "t"}, thread @{term "th"}'s current precedence is always the maximum precedence

    among all living threads

    (@{text "th_cp_max_preced"}): 

    @{thm [display] th_cp_max_preced}

  \item In @{term "t"}, thread @{term "th"}'s current precedence is always the maximum current 

    precedence among all living threads

    (@{text "th_cp_max"}): 

    @{thm [display] th_cp_max}

  \item In @{term "t"}, thread @{term "th"}'s current precedence equals its precedence at moment 

    @{term "s"}

    (@{text "th_cp_preced"}): 

    @{thm [display] th_cp_preced}

  \end{enumerate}

  *}

text {* \noindent

  The main theorem of this part is to characterizing the running thread during @{term "t"} 

  (@{text "runing_inversion_2"}):

  @{thm [display] runing_inversion_2}

  According to this, if a thread is running, it is either @{term "th"} or was

  already live and held some resource 

  at moment @{text "s"} (expressed by: @{text "cntV s th' < cntP s th'"}).

  Since there are only finite many threads live and holding some resource at any moment,

  if every such thread can release all its resources in finite duration, then after finite

  duration, none of them may block @{term "th"} anymore. So, no priority inversion may happen

  then.

  *}

(*<*)

end

(*>*)

section {* Properties to guide implementation \label{implement} *}

text {*

  The properties (especially @{text "runing_inversion_2"}) convinced us that the model defined 

  in Section \ref{model} does prevent indefinite priority inversion and therefore fulfills 

  the fundamental requirement of Priority Inheritance protocol. Another purpose of this paper 

  is to show how this model can be used to guide a concrete implementation. As discussed in

  Section 5.6.5 of \cite{Vahalia:1996:UI}, the implementation of Priority Inheritance in Solaris 

  uses sophisticated linking data structure. Except discussing two scenarios to show how

  the data structure should be manipulated, a lot of details of the implementation are missing. 

  In \cite{Faria08,conf/fase/JahierHR09,WellingsBSB07} the protocol is described formally 

  using different notations, but little information is given on how this protocol can be 

  implemented efficiently, especially there is no information on how these data structure 

  should be manipulated. 

  Because the scheduling of threads is based on current precedence, 

  the central issue in implementation of Priority Inheritance is how to compute the precedence

  correctly and efficiently. As long as the precedence is correct, it is very easy to 

  modify the scheduling algorithm to select the correct thread to execute. 

  First, it can be proved that the computation of current precedence @{term "cp"} of a threads

  only involves its children (@{text "cp_rec"}):

  @{thm [display] cp_rec} 

  where @{term "children s th"} represents the set of children of @{term "th"} in the current

  RAG: 

  \[

  @{thm (lhs) children_def} @{text "\<equiv>"} @{thm (rhs) children_def}

  \]

  where the definition of @{term "child"} is: 

  \[ @{thm (lhs) child_def} @{text "\<equiv>"}  @{thm (rhs) child_def}

  \]

  The aim of this section is to fill the missing details of how current precedence should

  be changed with the happening of events, with each event type treated by one subsection,

  where the computation of @{term "cp"} uses lemma @{text "cp_rec"}.

  *}

subsection {* Event @{text "Set th prio"} *}

(*<*)

context step_set_cps

begin

(*>*)

text {*

  The context under which event @{text "Set th prio"} happens is formalized as follows:

  \begin{enumerate}

    \item The formation of @{term "s"} (@{text "s_def"}): @{thm s_def}.

    \item State @{term "s"} is a valid state (@{text "vt_s"}): @{thm vt_s}. This implies 

      event @{text "Set th prio"} is eligible to happen under state @{term "s'"} and

      state @{term "s'"} is a valid state.

  \end{enumerate}

  *}

text {* \noindent

  Under such a context, we investigated how the current precedence @{term "cp"} of 

  threads change from state @{term "s'"} to @{term "s"} and obtained the following

  conclusions:

  \begin{enumerate}

  %% \item The RAG does not change (@{text "eq_dep"}): @{thm "eq_dep"}.

  \item All threads with no dependence relation with thread @{term "th"} have their

    @{term "cp"}-value unchanged (@{text "eq_cp"}):

    @{thm [display] eq_cp}

    This lemma implies the @{term "cp"}-value of @{term "th"}

    and those threads which have a dependence relation with @{term "th"} might need

    to be recomputed. The way to do this is to start from @{term "th"} 

    and follow the @{term "depend"}-chain to recompute the @{term "cp"}-value of every 

    encountered thread using lemma @{text "cp_rec"}. 

    Since the @{term "depend"}-relation is loop free, this procedure 

    can always stop. The the following lemma shows this procedure actually could stop earlier.

  \item The following two lemma shows, if a thread the re-computation of which

    gives an unchanged @{term "cp"}-value, the procedure described above can stop. 

    \begin{enumerate}

      \item Lemma @{text "eq_up_self"} shows if the re-computation of

        @{term "th"}'s @{term "cp"} gives the same result, the procedure can stop:

        @{thm [display] eq_up_self}

      \item Lemma @{text "eq_up"}) shows if the re-computation at intermediate threads

        gives unchanged result, the procedure can stop:

        @{thm [display] eq_up}

  \end{enumerate}

  \end{enumerate}

  *}

(*<*)

end

(*>*)