--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/prio/PrioGDef.thy Tue Jan 24 00:20:09 2012 +0000
@@ -0,0 +1,401 @@
+(*<*)
+theory PrioGDef
+imports Precedence_ord Moment
+begin
+(*>*)
+
+text {*
+ In this section, the formal model of Priority Inheritance is presented.
+ The model is based on Paulson's inductive protocol verification method, where
+ the state of the system is modelled as a list of events happened so far with the latest
+ event put at the head.
+
+ 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"}:
+*}
+
+type_synonym thread = nat -- {* Type for thread identifiers. *}
+type_synonym priority = nat -- {* Type for priorities. *}
+type_synonym cs = nat -- {* Type for critical sections (or critical resources). *}
+
+text {*
+ \noindent
+ Every event in the system corresponds to a system call, the formats of which are
+ defined as follows:
+ *}
+
+datatype event =
+ Create thread priority | -- {* Thread @{text "thread"} is created with priority @{text "priority"}. *}
+ Exit thread | -- {* Thread @{text "thread"} finishing its execution. *}
+ P thread cs | -- {* Thread @{text "thread"} requesting critical resource @{text "cs"}. *}
+ V thread cs | -- {* Thread @{text "thread"} releasing critical resource @{text "cs"}. *}
+ Set thread priority -- {* Thread @{text "thread"} resets its priority to @{text "priority"}. *}
+
+text {*
+\noindent
+ Resource Allocation Graph (RAG for short) is used extensively in our formal analysis.
+ The following type @{text "node"} is used to represent nodes in RAG.
+ *}
+datatype node =
+ Th "thread" | -- {* Node for thread. *}
+ Cs "cs" -- {* Node for critical resource. *}
+
+text {*
+ In Paulson's inductive method, the states of system are represented as lists of events,
+ which is defined by the following type @{text "state"}:
+ *}
+type_synonym state = "event list"
+
+text {*
+ \noindent
+ The following function
+ @{text "threads"} is used to calculate the set of live threads (@{text "threads s"})
+ in state @{text "s"}.
+ *}
+fun threads :: "state \<Rightarrow> thread set"
+ where
+ -- {* At the start of the system, the set of threads is empty: *}
+ "threads [] = {}" |
+ -- {* New thread is added to the @{text "threads"}: *}
+ "threads (Create thread prio#s) = {thread} \<union> threads s" |
+ -- {* Finished thread is removed: *}
+ "threads (Exit thread # s) = (threads s) - {thread}" |
+ -- {* Other kind of events does not affect the value of @{text "threads"}: *}
+ "threads (e#s) = threads s"
+text {* \noindent
+ Functions such as @{text "threads"}, which extract information out of system states, are called
+ {\em observing functions}. A series of observing functions will be defined in the sequel in order to
+ model the protocol.
+ Observing function @{text "original_priority"} calculates
+ the {\em original priority} of thread @{text "th"} in state @{text "s"}, expressed as
+ : @{text "original_priority th s" }. The {\em original priority} is the priority
+ assigned to a thread when it is created or when it is reset by system call
+ @{text "Set thread priority"}.
+*}
+
+fun original_priority :: "thread \<Rightarrow> state \<Rightarrow> priority"
+ where
+ -- {* @{text "0"} is assigned to threads which have never been created: *}
+ "original_priority thread [] = 0" |
+ "original_priority thread (Create thread' prio#s) =
+ (if thread' = thread then prio else original_priority thread s)" |
+ "original_priority thread (Set thread' prio#s) =
+ (if thread' = thread then prio else original_priority thread s)" |
+ "original_priority thread (e#s) = original_priority thread s"
+
+text {*
+ \noindent
+ In the following,
+ @{text "birthtime th s"} is the time when thread @{text "th"} is created,
+ observed from state @{text "s"}.
+ The time in the system is measured by the number of events happened so far since the very beginning.
+*}
+fun birthtime :: "thread \<Rightarrow> state \<Rightarrow> nat"
+ where
+ "birthtime thread [] = 0" |
+ "birthtime thread ((Create thread' prio)#s) =
+ (if (thread = thread') then length s else birthtime thread s)" |
+ "birthtime thread ((Set thread' prio)#s) =
+ (if (thread = thread') then length s else birthtime thread s)" |
+ "birthtime thread (e#s) = birthtime thread s"
+
+text {*
+ \noindent
+ The {\em precedence} is a notion derived from {\em priority}, where the {\em precedence} of
+ a thread is the combination of its {\em original priority} and {\em birth time}. The intention is
+ to discriminate threads with the same priority by giving threads whose priority
+ is assigned earlier higher precedences, becasue such threads are more urgent to finish.
+ This explains the following definition:
+ *}
+definition preced :: "thread \<Rightarrow> state \<Rightarrow> precedence"
+ where "preced thread s = Prc (original_priority thread s) (birthtime thread s)"
+
+
+text {*
+ \noindent
+ A number of important notions are defined here:
+ *}
+
+consts
+ holding :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ waiting :: "'b \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> bool"
+ depend :: "'b \<Rightarrow> (node \<times> node) set"
+ dependents :: "'b \<Rightarrow> thread \<Rightarrow> thread set"
+
+text {*
+ \noindent
+ In the definition of the following several functions, it is supposed that
+ the waiting queue of every critical resource is given by a waiting queue
+ function @{text "wq"}, which servers as arguments of these functions.
+ *}
+defs (overloaded)
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ We define that the thread which is at the head of waiting queue of resource @{text "cs"}
+ is holding the resource. This definition is slightly different from tradition where
+ all threads in the waiting queue are considered as waiting for the resource.
+ This notion is reflected in the definition of @{text "holding wq th cs"} as follows:
+ \end{minipage}
+ *}
+ cs_holding_def:
+ "holding wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread = hd (wq cs))"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ In accordance with the definition of @{text "holding wq th cs"},
+ a thread @{text "th"} is considered waiting for @{text "cs"} if
+ it is in the {\em waiting queue} of critical resource @{text "cs"}, but not at the head.
+ This is reflected in the definition of @{text "waiting wq th cs"} as follows:
+ \end{minipage}
+ *}
+ cs_waiting_def:
+ "waiting wq thread cs \<equiv> (thread \<in> set (wq cs) \<and> thread \<noteq> hd (wq cs))"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ @{text "depend wq"} represents the Resource Allocation Graph of the system under the waiting
+ queue function @{text "wq"}.
+ \end{minipage}
+ *}
+ cs_depend_def:
+ "depend (wq::cs \<Rightarrow> thread list) \<equiv>
+ {(Th t, Cs c) | t c. waiting wq t c} \<union> {(Cs c, Th t) | c t. holding wq t c}"
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ The following @{text "dependents wq th"} represents the set of threads which are depending on
+ thread @{text "th"} in Resource Allocation Graph @{text "depend wq"}:
+ \end{minipage}
+ *}
+ cs_dependents_def:
+ "dependents (wq::cs \<Rightarrow> thread list) th \<equiv> {th' . (Th th', Th th) \<in> (depend wq)^+}"
+
+text {*
+ The data structure used by the operating system for scheduling is referred to as
+ {\em schedule state}. It is represented as a record consisting of
+ a function assigning waiting queue to resources and a function assigning precedence to
+ threads:
+ *}
+record schedule_state =
+ waiting_queue :: "cs \<Rightarrow> thread list" -- {* The function assigning waiting queue. *}
+ cur_preced :: "thread \<Rightarrow> precedence" -- {* The function assigning precedence. *}
+
+text {* \noindent
+ The following
+ @{text "cpreced s th"} gives the {\em current precedence} of thread @{text "th"} under
+ state @{text "s"}. The definition of @{text "cpreced"} reflects the basic idea of
+ Priority Inheritance that the {\em current precedence} of a thread is the precedence
+ inherited from the maximum of all its dependents, i.e. the threads which are waiting
+ directly or indirectly waiting for some resources from it. If no such thread exits,
+ @{text "th"}'s {\em current precedence} equals its original precedence, i.e.
+ @{text "preced th s"}.
+ *}
+definition cpreced :: "state \<Rightarrow> (cs \<Rightarrow> thread list) \<Rightarrow> thread \<Rightarrow> precedence"
+ where "cpreced s wq = (\<lambda> th. Max ((\<lambda> th. preced th s) ` ({th} \<union> dependents wq th)))"
+
+text {* \noindent
+ The following function @{text "schs"} is used to calculate the schedule state @{text "schs s"}.
+ It is the key function to model Priority Inheritance:
+ *}
+fun schs :: "state \<Rightarrow> schedule_state"
+ where "schs [] = \<lparr>waiting_queue = \<lambda> cs. [], cur_preced = cpreced [] (\<lambda> cs. [])\<rparr>" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ \begin{enumerate}
+ \item @{text "ps"} is the schedule state of last moment.
+ \item @{text "pwq"} is the waiting queue function of last moment.
+ \item @{text "pcp"} is the precedence function of last moment.
+ \item @{text "nwq"} is the new waiting queue function. It is calculated using a @{text "case"} statement:
+ \begin{enumerate}
+ \item If the happening event is @{text "P thread cs"}, @{text "thread"} is added to
+ the end of @{text "cs"}'s waiting queue.
+ \item If the happening event is @{text "V thread cs"} and @{text "s"} is a legal state,
+ @{text "th'"} must equal to @{text "thread"},
+ because @{text "thread"} is the one currently holding @{text "cs"}.
+ The case @{text "[] \<Longrightarrow> []"} may never be executed in a legal state.
+ the @{text "(SOME q. distinct q \<and> set q = set qs)"} is used to choose arbitrarily one
+ thread in waiting to take over the released resource @{text "cs"}. In our representation,
+ this amounts to rearrange elements in waiting queue, so that one of them is put at the head.
+ \item For other happening event, the schedule state just does not change.
+ \end{enumerate}
+ \item @{text "ncp"} is new precedence function, it is calculated from the newly updated waiting queue
+ function. The dependency of precedence function on waiting queue function is the reason to
+ put them in the same record so that they can evolve together.
+ \end{enumerate}
+ \end{minipage}
+ *}
+ "schs (e#s) = (let ps = schs s in
+ let pwq = waiting_queue ps in
+ let pcp = cur_preced ps in
+ let nwq = case e of
+ P thread cs \<Rightarrow> pwq(cs:=(pwq cs @ [thread])) |
+ V thread cs \<Rightarrow> let nq = case (pwq cs) of
+ [] \<Rightarrow> [] |
+ (th'#qs) \<Rightarrow> (SOME q. distinct q \<and> set q = set qs)
+ in pwq(cs:=nq) |
+ _ \<Rightarrow> pwq
+ in let ncp = cpreced (e#s) nwq in
+ \<lparr>waiting_queue = nwq, cur_preced = ncp\<rparr>
+ )"
+
+text {*
+ \noindent
+ The following @{text "wq"} is a shorthand for @{text "waiting_queue"}.
+ *}
+definition wq :: "state \<Rightarrow> cs \<Rightarrow> thread list"
+ where "wq s = waiting_queue (schs s)"
+
+text {* \noindent
+ The following @{text "cp"} is a shorthand for @{text "cur_preced"}.
+ *}
+definition cp :: "state \<Rightarrow> thread \<Rightarrow> precedence"
+ where "cp s = cur_preced (schs s)"
+
+text {* \noindent
+ Functions @{text "holding"}, @{text "waiting"}, @{text "depend"} and
+ @{text "dependents"} still have the
+ same meaning, but redefined so that they no longer depend on the
+ fictitious {\em waiting queue function}
+ @{text "wq"}, but on system state @{text "s"}.
+ *}
+defs (overloaded)
+ s_holding_def:
+ "holding (s::state) thread cs \<equiv> (thread \<in> set (wq s cs) \<and> thread = hd (wq s cs))"
+ s_waiting_def:
+ "waiting (s::state) thread cs \<equiv> (thread \<in> set (wq s cs) \<and> thread \<noteq> hd (wq s cs))"
+ s_depend_def:
+ "depend (s::state) \<equiv>
+ {(Th t, Cs c) | t c. waiting (wq s) t c} \<union> {(Cs c, Th t) | c t. holding (wq s) t c}"
+ s_dependents_def:
+ "dependents (s::state) th \<equiv> {th' . (Th th', Th th) \<in> (depend (wq s))^+}"
+
+text {*
+ The following function @{text "readys"} calculates the set of ready threads. A thread is {\em ready}
+ for running if it is a live thread and it is not waiting for any critical resource.
+ *}
+definition readys :: "state \<Rightarrow> thread set"
+ where "readys s = {thread . thread \<in> threads s \<and> (\<forall> cs. \<not> waiting s thread cs)}"
+
+text {* \noindent
+ The following function @{text "runing"} calculates the set of running thread, which is the ready
+ thread with the highest precedence.
+ *}
+definition runing :: "state \<Rightarrow> thread set"
+ where "runing s = {th . th \<in> readys s \<and> cp s th = Max ((cp s) ` (readys s))}"
+
+text {* \noindent
+ The following function @{text "holdents s th"} returns the set of resources held by thread
+ @{text "th"} in state @{text "s"}.
+ *}
+definition holdents :: "state \<Rightarrow> thread \<Rightarrow> cs set"
+ where "holdents s th = {cs . (Cs cs, Th th) \<in> depend s}"
+
+text {* \noindent
+ @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in
+ state @{text "s"}:
+ *}
+definition cntCS :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntCS s th = card (holdents s th)"
+
+text {* \noindent
+ The fact that event @{text "e"} is eligible to happen next in state @{text "s"}
+ is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as
+ follows:
+ *}
+inductive step :: "state \<Rightarrow> event \<Rightarrow> bool"
+ where
+ -- {*
+ A thread can be created if it is not a live thread:
+ *}
+ thread_create: "\<lbrakk>thread \<notin> threads s\<rbrakk> \<Longrightarrow> step s (Create thread prio)" |
+ -- {*
+ A thread can exit if it no longer hold any resource:
+ *}
+ thread_exit: "\<lbrakk>thread \<in> runing s; holdents s thread = {}\<rbrakk> \<Longrightarrow> step s (Exit thread)" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ A thread can request for an critical resource @{text "cs"}, if it is running and
+ the request does not form a loop in the current RAG. The latter condition
+ is set up to avoid deadlock. The condition also reflects our assumption all threads are
+ carefully programmed so that deadlock can not happen:
+ \end{minipage}
+ *}
+ thread_P: "\<lbrakk>thread \<in> runing s; (Cs cs, Th thread) \<notin> (depend s)^+\<rbrakk> \<Longrightarrow>
+ step s (P thread cs)" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ A thread can release a critical resource @{text "cs"}
+ if it is running and holding that resource:
+ \end{minipage}
+ *}
+ thread_V: "\<lbrakk>thread \<in> runing s; holding s thread cs\<rbrakk> \<Longrightarrow> step s (V thread cs)" |
+ -- {*
+ A thread can adjust its own priority as long as it is current running:
+ *}
+ thread_set: "\<lbrakk>thread \<in> runing s\<rbrakk> \<Longrightarrow> step s (Set thread prio)"
+
+text {* \noindent
+ With predicate @{text "step"}, the fact that @{text "s"} is a legal state in
+ Priority Inheritance protocol can be expressed as: @{text "vt step s"}, where
+ the predicate @{text "vt"} can be defined as the following:
+ *}
+inductive vt :: "(state \<Rightarrow> event \<Rightarrow> bool) \<Rightarrow> state \<Rightarrow> bool"
+ for cs -- {* @{text "cs"} is an argument representing any step predicate. *}
+ where
+ -- {* Empty list @{text "[]"} is a legal state in any protocol:*}
+ vt_nil[intro]: "vt cs []" |
+ -- {*
+ \begin{minipage}{0.9\textwidth}
+ If @{text "s"} a legal state, and event @{text "e"} is eligible to happen
+ in state @{text "s"}, then @{text "e#s"} is a legal state as well:
+ \end{minipage}
+ *}
+ vt_cons[intro]: "\<lbrakk>vt cs s; cs s e\<rbrakk> \<Longrightarrow> vt cs (e#s)"
+
+text {* \noindent
+ It is easy to see that the definition of @{text "vt"} is generic. It can be applied to
+ any step predicate to get the set of legal states.
+ *}
+
+text {* \noindent
+ The following two functions @{text "the_cs"} and @{text "the_th"} are used to extract
+ critical resource and thread respectively out of RAG nodes.
+ *}
+fun the_cs :: "node \<Rightarrow> cs"
+ where "the_cs (Cs cs) = cs"
+
+fun the_th :: "node \<Rightarrow> thread"
+ where "the_th (Th th) = th"
+
+text {* \noindent
+ The following predicate @{text "next_th"} describe the next thread to
+ take over when a critical resource is released. In @{text "next_th s th cs t"},
+ @{text "th"} is the thread to release, @{text "t"} is the one to take over.
+ *}
+definition next_th:: "state \<Rightarrow> thread \<Rightarrow> cs \<Rightarrow> thread \<Rightarrow> bool"
+ where "next_th s th cs t = (\<exists> rest. wq s cs = th#rest \<and> rest \<noteq> [] \<and>
+ t = hd (SOME q. distinct q \<and> set q = set rest))"
+
+text {* \noindent
+ The function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"}
+ in list @{text "l"}:
+ *}
+definition count :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> nat"
+ where "count Q l = length (filter Q l)"
+
+text {* \noindent
+ The following @{text "cntP s"} returns the number of operation @{text "P"} happened
+ before reaching state @{text "s"}.
+ *}
+definition cntP :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntP s th = count (\<lambda> e. \<exists> cs. e = P th cs) s"
+
+text {* \noindent
+ The following @{text "cntV s"} returns the number of operation @{text "V"} happened
+ before reaching state @{text "s"}.
+ *}
+definition cntV :: "state \<Rightarrow> thread \<Rightarrow> nat"
+ where "cntV s th = count (\<lambda> e. \<exists> cs. e = V th cs) s"
+(*<*)
+end
+(*>*)
+