diff -r b4bcd1edbb6d -r 633b1fc8631b PIPDefs.thy~ --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/PIPDefs.thy~ Thu Jan 07 08:33:13 2016 +0800 @@ -0,0 +1,614 @@ +chapter {* Definitions *} +(*<*) +theory PIPDefs +imports Precedence_ord Moment +begin +(*>*) + +text {* + In this section, the formal model of Priority Inheritance Protocol (PIP) 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. +*} + +text {* + 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 + The abstraction of Priority Inheritance Protocol (PIP) is set at the system call level. + Every system call is represented as an event. The format of events is defined + 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 {* + As mentioned earlier, 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 + 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 {* + \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 \ 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} \ 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 + The function @{text "threads"} defined above is one of + the so called {\em observation function}s which forms + the very basis of Paulson's inductive protocol verification method. + Each observation function {\em observes} one particular aspect (or attribute) + of the system. For example, the attribute observed by @{text "threads s"} + is the set of threads living in state @{text "s"}. + The protocol being modelled + The decision made the protocol being modelled is based on the {\em observation}s + returned by {\em observation function}s. Since {\observation function}s forms + the very basis on which Paulson's inductive method is based, there will be + a lot of such observation functions introduced in the following. In fact, any function + which takes event list as argument is a {\em observation function}. + *} + +text {* \noindent + Observation @{text "priority th s"} is + the {\em original priority} of thread @{text "th"} in state @{text "s"}. + The {\em original priority} is the priority + assigned to a thread when it is created or when it is reset by system call + (represented by event @{text "Set thread priority"}). +*} + +fun priority :: "thread \ state \ priority" + where + -- {* @{text "0"} is assigned to threads which have never been created: *} + "priority thread [] = 0" | + "priority thread (Create thread' prio#s) = + (if thread' = thread then prio else priority thread s)" | + "priority thread (Set thread' prio#s) = + (if thread' = thread then prio else priority thread s)" | + "priority thread (e#s) = priority thread s" + +text {* + \noindent + Observation @{text "last_set th s"} is the last time when the priority of thread @{text "th"} is set, + 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 last_set :: "thread \ state \ nat" + where + "last_set thread [] = 0" | + "last_set thread ((Create thread' prio)#s) = + (if (thread = thread') then length s else last_set thread s)" | + "last_set thread ((Set thread' prio)#s) = + (if (thread = thread') then length s else last_set thread s)" | + "last_set thread (_#s) = last_set 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 time} the priority is set. + 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 \ state \ precedence" + where "preced thread s \ Prc (priority thread s) (last_set thread s)" + + +text {* + \noindent + A number of important notions in PIP are represented as the following functions, + defined in terms of the waiting queues of the system, where the waiting queues + , as a whole, is represented by the @{text "wq"} argument of every notion function. + The @{text "wq"} argument is itself a functions which maps every critical resource + @{text "cs"} to the list of threads which are holding or waiting for it. + The thread at the head of this list is designated as the thread which is current + holding the resrouce, which is slightly different from tradition where + all threads in the waiting queue are considered as waiting for the resource. + *} + +consts + holding :: "'b \ thread \ cs \ bool" + waiting :: "'b \ thread \ cs \ bool" + RAG :: "'b \ (node \ node) set" + dependants :: "'b \ thread \ thread set" + +defs (overloaded) + -- {* + \begin{minipage}{0.9\textwidth} + This meaning of @{text "wq"} is reflected in the following definition of @{text "holding wq th cs"}, + where @{text "holding wq th cs"} means thread @{text "th"} is holding the critical + resource @{text "cs"}. This decision is based on @{text "wq"}. + \end{minipage} + *} + + cs_holding_def: + "holding wq thread cs \ (thread \ set (wq cs) \ 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 \ (thread \ set (wq cs) \ thread \ hd (wq cs))" + -- {* + \begin{minipage}{0.9\textwidth} + @{text "RAG wq"} generates RAG (a binary relations on @{text "node"}) + out of waiting queues of the system (represented by the @{text "wq"} argument): + \end{minipage} + *} + cs_RAG_def: + "RAG (wq::cs \ thread list) \ + {(Th th, Cs cs) | th cs. waiting wq th cs} \ {(Cs cs, Th th) | cs th. holding wq th cs}" + -- {* + \begin{minipage}{0.9\textwidth} + The following @{text "dependants wq th"} represents the set of threads which are RAGing on + thread @{text "th"} in Resource Allocation Graph @{text "RAG wq"}. + Here, "RAGing" means waiting directly or indirectly on the critical resource. + \end{minipage} + *} + cs_dependants_def: + "dependants (wq::cs \ thread list) th \ {th' . (Th th', Th th) \ (RAG wq)^+}" + + +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 dependants, 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 :: "(cs \ thread list) \ state \ thread \ precedence" + where "cpreced wq s = (\th. Max ((\th'. preced th' s) ` ({th} \ dependants wq th)))" + +text {* + Notice that the current precedence (@{text "cpreced"}) of one thread @{text "th"} can be boosted + (becoming larger than its own precedence) by those threads in + the @{text "dependants wq th"}-set. If one thread get boosted, we say + it inherits the priority (or, more precisely, the precedence) of + its dependants. This is how the word "Inheritance" in + Priority Inheritance Protocol comes. +*} + +(*<*) +lemma + cpreced_def2: + "cpreced wq s th \ Max ({preced th s} \ {preced th' s | th'. th' \ dependants wq th})" + unfolding cpreced_def image_def + apply(rule eq_reflection) + apply(rule_tac f="Max" in arg_cong) + by (auto) +(*>*) + + +text {* \noindent + Assuming @{text "qs"} be the waiting queue of a critical resource, + the following abbreviation "release qs" is the waiting queue after the thread + holding the resource (which is thread at the head of @{text "qs"}) released + the resource: +*} +abbreviation + "release qs \ case qs of + [] => [] + | (_#qs') => (SOME q. distinct q \ set q = set qs')" +text {* \noindent + It can be seen from the definition that the thread at the head of @{text "qs"} is removed + from the return value, and the value @{term "q"} is an reordering of @{text "qs'"}, the + tail of @{text "qs"}. Through this reordering, one of the waiting threads (those in @{text "qs'"} } + is chosen nondeterministically to be the head of the new queue @{text "q"}. + Therefore, this thread is the one who takes over the resource. This is a little better different + from common sense that the thread who comes the earliest should take over. + The intention of this definition is to show that the choice of which thread to take over the + release resource does not affect the correctness of the PIP protocol. +*} + +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 + (to be used as the @{text "wq"} argument in @{text "holding"}, @{text "waiting"} + and @{text "RAG"}, etc) and a function assigning precedence to threads: + *} + +record schedule_state = + wq_fun :: "cs \ thread list" -- {* The function assigning waiting queue. *} + cprec_fun :: "thread \ precedence" -- {* The function assigning precedence. *} + +text {* \noindent + The following two abbreviations (@{text "all_unlocked"} and @{text "initial_cprec"}) + are used to set the initial values of the @{text "wq_fun"} @{text "cprec_fun"} fields + respectively of the @{text "schedule_state"} record by the following function @{text "sch"}, + which is used to calculate the system's {\em schedule state}. + + Since there is no thread at the very beginning to make request, all critical resources + are free (or unlocked). This status is represented by the abbreviation + @{text "all_unlocked"}. + *} +abbreviation + "all_unlocked \ \_::cs. ([]::thread list)" + + +text {* \noindent + The initial current precedence for a thread can be anything, because there is no thread then. + We simply assume every thread has precedence @{text "Prc 0 0"}. + *} + +abbreviation + "initial_cprec \ \_::thread. Prc 0 0" + + +text {* \noindent + The following function @{text "schs"} is used to calculate the system's schedule state @{text "schs s"} + out of the current system state @{text "s"}. It is the central function to model Priority Inheritance: + *} +fun schs :: "state \ schedule_state" + where + -- {* + \begin{minipage}{0.9\textwidth} + Setting the initial value of the @{text "schedule_state"} record (see the explanations above). + \end{minipage} + *} + "schs [] = (| wq_fun = all_unlocked, cprec_fun = initial_cprec |)" | + + -- {* + \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 (NOT USED). + \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 "[] \ []"} may never be executed in a legal state. + the @{text "(SOME q. distinct q \ 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 RAGency 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} + + + The calculation of @{text "cprec_fun"} depends on the value of @{text "wq_fun"}. + Therefore, in the following cases, @{text "wq_fun"} is always calculated first, in + the name of @{text "wq"} (if @{text "wq_fun"} is not changed + by the happening event) or @{text "new_wq"} (if the value of @{text "wq_fun"} is changed). + \end{minipage} + *} + "schs (Create th prio # s) = + (let wq = wq_fun (schs s) in + (|wq_fun = wq, cprec_fun = cpreced wq (Create th prio # s)|))" +| "schs (Exit th # s) = + (let wq = wq_fun (schs s) in + (|wq_fun = wq, cprec_fun = cpreced wq (Exit th # s)|))" +| "schs (Set th prio # s) = + (let wq = wq_fun (schs s) in + (|wq_fun = wq, cprec_fun = cpreced wq (Set th prio # s)|))" + -- {* + \begin{minipage}{0.9\textwidth} + Different from the forth coming cases, the @{text "wq_fun"} field of the schedule state + is changed. So, the new value is calculated first, in the name of @{text "new_wq"}. + \end{minipage} + *} +| "schs (P th cs # s) = + (let wq = wq_fun (schs s) in + let new_wq = wq(cs := (wq cs @ [th])) in + (|wq_fun = new_wq, cprec_fun = cpreced new_wq (P th cs # s)|))" +| "schs (V th cs # s) = + (let wq = wq_fun (schs s) in + let new_wq = wq(cs := release (wq cs)) in + (|wq_fun = new_wq, cprec_fun = cpreced new_wq (V th cs # s)|))" + +lemma cpreced_initial: + "cpreced (\ cs. []) [] = (\_. (Prc 0 0))" +apply(simp add: cpreced_def) +apply(simp add: cs_dependants_def cs_RAG_def cs_waiting_def cs_holding_def) +apply(simp add: preced_def) +done + +lemma sch_old_def: + "schs (e#s) = (let ps = schs s in + let pwq = wq_fun ps in + let nwq = case e of + P th cs \ pwq(cs:=(pwq cs @ [th])) | + V th cs \ let nq = case (pwq cs) of + [] \ [] | + (_#qs) \ (SOME q. distinct q \ set q = set qs) + in pwq(cs:=nq) | + _ \ pwq + in let ncp = cpreced nwq (e#s) in + \wq_fun = nwq, cprec_fun = ncp\ + )" +apply(cases e) +apply(simp_all) +done + + +text {* + \noindent + The following @{text "wq"} is a shorthand for @{text "wq_fun"}. + *} +definition wq :: "state \ cs \ thread list" + where "wq s = wq_fun (schs s)" + +text {* \noindent + The following @{text "cp"} is a shorthand for @{text "cprec_fun"}. + *} +definition cp :: "state \ thread \ precedence" + where "cp s \ cprec_fun (schs s)" + +text {* \noindent + Functions @{text "holding"}, @{text "waiting"}, @{text "RAG"} and + @{text "dependants"} still have the + same meaning, but redefined so that they no longer RAG on the + fictitious {\em waiting queue function} + @{text "wq"}, but on system state @{text "s"}. + *} +defs (overloaded) + s_holding_abv: + "holding (s::state) \ holding (wq_fun (schs s))" + s_waiting_abv: + "waiting (s::state) \ waiting (wq_fun (schs s))" + s_RAG_abv: + "RAG (s::state) \ RAG (wq_fun (schs s))" + s_dependants_abv: + "dependants (s::state) \ dependants (wq_fun (schs s))" + + +text {* + The following lemma can be proved easily, and the meaning is obvious. + *} +lemma + s_holding_def: + "holding (s::state) th cs \ (th \ set (wq_fun (schs s) cs) \ th = hd (wq_fun (schs s) cs))" + by (auto simp:s_holding_abv wq_def cs_holding_def) + +lemma s_waiting_def: + "waiting (s::state) th cs \ (th \ set (wq_fun (schs s) cs) \ th \ hd (wq_fun (schs s) cs))" + by (auto simp:s_waiting_abv wq_def cs_waiting_def) + +lemma s_RAG_def: + "RAG (s::state) = + {(Th th, Cs cs) | th cs. waiting (wq s) th cs} \ {(Cs cs, Th th) | cs th. holding (wq s) th cs}" + by (auto simp:s_RAG_abv wq_def cs_RAG_def) + +lemma + s_dependants_def: + "dependants (s::state) th \ {th' . (Th th', Th th) \ (RAG (wq s))^+}" + by (auto simp:s_dependants_abv wq_def cs_dependants_def) + +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 \ thread set" + where "readys s \ {th . th \ threads s \ (\ cs. \ waiting s th 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 \ thread set" + where "runing s \ {th . th \ readys s \ cp s th = Max ((cp s) ` (readys s))}" + +text {* \noindent + Notice that the definition of @{text "running"} reflects the preemptive scheduling strategy, + because, if the @{text "running"}-thread (the one in @{text "runing"} set) + lowered its precedence by resetting its own priority to a lower + one, it will lose its status of being the max in @{text "ready"}-set and be superseded. +*} + +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 \ thread \ cs set" + where "holdents s th \ {cs . holding s th cs}" + +lemma holdents_test: + "holdents s th = {cs . (Cs cs, Th th) \ RAG s}" +unfolding holdents_def +unfolding s_RAG_def +unfolding s_holding_abv +unfolding wq_def +by (simp) + +text {* \noindent + Observation @{text "cntCS s th"} returns the number of resources held by thread @{text "th"} in + state @{text "s"}: + *} +definition cntCS :: "state \ thread \ nat" + where "cntCS s th = card (holdents s th)" + +text {* \noindent + According to the convention of Paulson's inductive method, + the decision made by a protocol that event @{text "e"} is eligible to happen next under state @{text "s"} + is expressed as @{text "step s e"}. The predicate @{text "step"} is inductively defined as + follows (notice how the decision is based on the {\em observation function}s + defined above, and also notice how a complicated protocol is modeled by a few simple + observations, and how such a kind of simplicity gives rise to improved trust on + faithfulness): + *} +inductive step :: "state \ event \ bool" + where + -- {* + A thread can be created if it is not a live thread: + *} + thread_create: "\thread \ threads s\ \ step s (Create thread prio)" | + -- {* + A thread can exit if it no longer hold any resource: + *} + thread_exit: "\thread \ runing s; holdents s thread = {}\ \ 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: "\thread \ runing s; (Cs cs, Th thread) \ (RAG s)^+\ \ + 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: "\thread \ runing s; holding s thread cs\ \ step s (V thread cs)" | + -- {* + \begin{minipage}{0.9\textwidth} + A thread can adjust its own priority as long as it is current running. + With the resetting of one thread's priority, its precedence may change. + If this change lowered the precedence, according to the definition of @{text "running"} + function, + \end{minipage} + *} + thread_set: "\thread \ runing s\ \ step s (Set thread prio)" + +text {* + In Paulson's inductive method, every protocol is defined by such a @{text "step"} + predicate. For instance, the predicate @{text "step"} given above + defines the PIP protocol. So, it can also be called "PIP". +*} + +abbreviation + "PIP \ step" + + +text {* \noindent + For any protocol defined by a @{text "step"} predicate, + the fact that @{text "s"} is a legal state in + the protocol is expressed as: @{text "vt step s"}, where + the predicate @{text "vt"} can be defined as the following: + *} +inductive vt :: "state \ bool" + where + -- {* Empty list @{text "[]"} is a legal state in any protocol:*} + vt_nil[intro]: "vt []" | + -- {* + \begin{minipage}{0.9\textwidth} + If @{text "s"} a legal state of the protocol defined by predicate @{text "step"}, + and event @{text "e"} is allowed to happen under state @{text "s"} by the protocol + predicate @{text "step"}, then @{text "e#s"} is a new legal state rendered by the + happening of @{text "e"}: + \end{minipage} + *} + vt_cons[intro]: "\vt s; step s e\ \ vt (e#s)" + +text {* \noindent + It is easy to see that the definition of @{text "vt"} is generic. It can be applied to + any specific protocol specified by a @{text "step"}-predicate to get the set of + legal states of that particular protocol. + *} + +text {* + The following are two very basic properties of @{text "vt"}. +*} + +lemma step_back_vt: "vt (e#s) \ vt s" + by(ind_cases "vt (e#s)", simp) + +lemma step_back_step: "vt (e#s) \ step s e" + by(ind_cases "vt (e#s)", simp) + +text {* \noindent + The following two auxiliary 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 \ cs" + where "the_cs (Cs cs) = cs" + +fun the_th :: "node \ 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. + Notice how this definition is backed up by the @{text "release"} function and its use + in the @{text "V"}-branch of @{text "schs"} function. This @{text "next_th"} function + is not needed for the execution of PIP. It is introduced as an auxiliary function + to state lemmas. The correctness of this definition will be confirmed by + lemmas @{text "step_v_hold_inv"}, @{text " step_v_wait_inv"}, + @{text "step_v_get_hold"} and @{text "step_v_not_wait"}. + *} +definition next_th:: "state \ thread \ cs \ thread \ bool" + where "next_th s th cs t = (\ rest. wq s cs = th#rest \ rest \ [] \ + t = hd (SOME q. distinct q \ set q = set rest))" + +text {* \noindent + The aux function @{text "count Q l"} is used to count the occurrence of situation @{text "Q"} + in list @{text "l"}: + *} +definition count :: "('a \ bool) \ 'a list \ nat" + where "count Q l = length (filter Q l)" + +text {* \noindent + The following observation @{text "cntP s"} returns the number of operation @{text "P"} happened + before reaching state @{text "s"}. + *} +definition cntP :: "state \ thread \ nat" + where "cntP s th = count (\ e. \ cs. e = P th cs) s" + +text {* \noindent + The following observation @{text "cntV s"} returns the number of operation @{text "V"} happened + before reaching state @{text "s"}. + *} +definition cntV :: "state \ thread \ nat" + where "cntV s th = count (\ e. \ cs. e = V th cs) s" +(*<*) + +end +(*>*) +