Difference between pages "Event Model Decomposition" and "The Use of Theories in Code Generation"

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= Defining Translations Using The Theory Plug-in =
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The theory plug-in is used to add mathematical extensions to Rodin. The theories are created, and deployed, and can then be used in any models in the workspace. When dealing with implementation level models, such as in Tasking Event-B, we need to consider how to translate newly added types and operators into code. We have augmented the theory interface with a Translation Rules section. This enables a user to define translation rules that map Event-B formulas to code.
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== Translation Rules==
 +
<div id="fig:Translation Rules">
 +
<br/>
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[[Image:TheoryCGRules.png|center||caption text]]
 +
<center>'''Figure 1''': Translation Rules</center>
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<br/>
 +
</div>
  
== Introduction ==
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Figure 1 shows the interface, and some translations rules of the mapping to Ada.
One of the most important feature of the Event-B approach is the ability to introduce new events during refinement steps, but a consequence is an increasing complexity of the refinement process when having to deal with many events and many state variables.
 
  
The main idea of the decomposition is to cut a model <math>A</math> into sub-models <math>A_1, ..., A_n</math>, which can be refined separately and more confortably than the whole.  
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The theory is given a name, and may import some other theories. Type parameters can be added, and we use them here to type the meta-variables. The meta-variable ''a'' is restricted to be an integer type, but meta-variable ''c'' can be any type. Meta-variables are used in the translator rules for pattern matching.
  
The constraint that shall be satisfied by the decomposition is that these refined models might - the recomposition will never be performed in practice - be recomposed into a whole model <math>C</math> in a way that guarantees that <math>C</math> refines <math>A</math>. An event-based decomposition of a model is detailed in [[#ancre_1|Event Model Decomposition]]: the events of a model are partitioned to form the events of the sub-models. In parallel, the variables on which these events act are distributed among the sub-models.  
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Translator rules are templates, which are used in pattern matching. Event-B formulas are defined on the left hand side of the rule, and the code to be output (as text) appears on the right hand side of the matching rule. During translation an abstract syntax tree (AST) representation of the formula is used. The theory plug-in attempts to match the formulas in the rules with each syntactic element of the AST. As it does so it builds the textual output as a string, until the whole AST has been successfully matched. When a complete tree is matched, the target code is returned. If the AST is not matched, a warning is issued, and a string representation of the original formula is returned.
  
The purpose is here to precisely describe what is required at the Rodin platform level to integrate this event model decomposition, and to explain why. The details of how it could be implemented are out of scope.
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== Type Rules for Code Generation ==
  
Other [[#ancre_2|decomposition structures for Event-B]] are not considered here.
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The type rules section, shown in Figure 1, is where the relationship is defined, between Event-B types and the type system of the implementation.
  
== Terminology ==
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= Adding New (implementation-level) Types =
 +
When we are working at abstraction levels close to the implementation level, we may make an implementation decision which requires the introduction of a new type to the development. We give an example of our approach, where we add a new array type, shown in Figure 2, and then define its translation to code.
  
* Event model decomposition: The decomposition of a model, as defined in [[Machines_and_Contexts_(Modelling_Language)|the modelling language]], in sub-models.
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== An Array Type Definition ==
A model can contain contexts, machines, or both. The notion of model decomposition covers on the one hand the machine decomposition, and on the other hand the context decomposition, both being interdependent.
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<div id="fig:Extension with an Array Type">
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<br/>
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[[Image:ArrayDef.png|center||caption text]]
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<center>'''Figure 2''': Array Definition</center>
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<br/>
 +
</div>
  
[[Image:Models.png]]
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The array operator notation is defined in the expression array(s: P(T)); and the semantics is defined in the direct definition. arrayN constrains the arrays to be of fixed length. Array lookup, update, and constructor operators are subsequently defined. In the next step we need to define any translations required to implement the array in code.
* ''Shared'' variable: A variable of a given machine which is accessed by events distributed in distinct sub-machines (by opposition to ''private'' variable).  
 
* ''Private'' variable: A variable of a given machine which is only accessed by events of the same sub-machine (by opposition to ''shared'' variable).
 
* ''External'' event: An event of a sub-machine which is built from an event of the non-decomposed machine, and which simulates the way the ''shared'' variables (between this sub-machine and another sub-machine) are handled in the non-decomposed machine (by opposition to ''internal'' event).  
 
* ''Internal'' event: An event copied from the non-decomposed machine to a sub-machine, according to the end-user specified distribution (by opposition to ''external'' event).
 
  
Note that a variable is said to be ''accessed'' when it is read or written. More precisely, such an access may be performed by a predicate (invariant, guard, witness) or in an assignment (action).
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== Translation Rules ==
  
== Architecture ==
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<div id="Translation Rules for the Array Type">
<font color="red">TODO</font>
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<br/>
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[[Image:ArrayTrans.png|center||caption text]]
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<center>'''Figure 3''': Translation Rules for the Array Type</center>
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<br/>
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</div>
  
== Low-level Specification ==
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Figure 3 shows the Ada translation; beginning with the meta-variable definitions that are used for pattern matching in the translation rules. Each of the operators; ''newArray'', and ''update'', and an expression using the ''lookup'' operator, are mapped to their implementations on the right hand side of the rule. The ''Type Rules'' section describes the implementation's description of the ''arrayN'' type.
=== Decomposition of a machine in sub-machines ===
 
<font color="red">What about the hierarchy of machines? Is it necessary to flatten this hierarchy?</font>
 
 
 
==== About the variables ====
 
Some variables are needed by several sub-machines of the decomposition. As a consequence, these variables shall be replicated in the sub-machines. Beyond that, since it is not possible to ensure that such a variable will be refined in the same way in each sub-machine, they shall be given a special status (''shared'' variable), with the limitation that they cannot be refined.
 
 
 
We will specify in this section how to introduce the notion of ''shared'' variable in the Rodin platform, and how to check the associated rules.
 
 
 
===== Defining a variable as shared =====
 
The following DTD excerpt describes the structure of a variable in the Rodin database:
 
 
 
<pre>
 
<!ENTITY % NameAttDecl "name CDATA #REQUIRED">
 
<!ENTITY % CommentAttDecl "org.eventb.core.comment CDATA #IMPLIED">
 
<!ENTITY % IdentAttDecl "org.eventb.core.identifier CDATA #REQUIRED">
 
 
 
<!ELEMENT %variable; EMPTY>
 
<!ATTLIST %variable;
 
  %NameAttDecl;
 
  %CommentAttDecl;
 
  %IdentAttDecl;
 
  >
 
</pre>
 
 
 
A first possibility to tag a variable as ''shared'' would be to add a <tt>shared</tt> specific attribute, which would be set to <tt>true</tt> if and only if the variable is ''shared'':
 
 
 
<pre>
 
<ENTITY % shared "org.eventb.core.shared CDATA #REQUIRED">
 
 
 
<!ELEMENT %variable; EMPTY>
 
<!ATTLIST %variable;
 
  ...
 
  %shared; (false|true) #REQUIRED
 
>
 
</pre>
 
 
 
Another possibility would be to define a more generic attribute, which could take different values, according to the nature of the variable:
 
 
 
<pre>
 
<ENTITY % nature "org.eventb.core.nature CDATA #REQUIRED">
 
 
 
<!ATTLIST %variable;
 
  ...
 
  %nature; (0|1) #REQUIRED
 
>
 
 
 
<!-- The nature attribute encodes the kind of variable:
 
      0: private variable
 
      1: shared variable
 
-->
 
</pre>
 
 
 
The second option, which has the main advantage to be more evolutive, is retained here.
 
 
 
===== Ensuring that a shared variable is not data-refined =====
 
A shared variable shall always be present in the state space of any refinement of the component.
 
The verification shall be added to those already performed by the static checker.
 
 
 
===== <font id="var_partition">Partitionning the variables in the sub-components of the decomposition</font> =====
 
It is assumed in this section that the events have been previously partitionned among sub-machines.
 
 
 
The following cases have to be taken into consideration when dealing with the variable distribution:
 
* If <math>v</math> is a variable that is only accessed by events of a given sub-machine <math>m</math>, then <math>v</math> is a ''private'' variable of <math>m</math>, and it shall be deplaced to <math>m</math>.
 
* If <math>v</math> is a variable that is accessed by events of distinct sub-machines <math>m_1, ..., m_n</math>, then <math>v</math> is a ''shared'' variable, and it shall be duplicated in all sub-machines.
 
 
 
If all the variables are ''shared'' at the conclusion of the partition, the end user shall be notified (it certainly means that the decomposition was not judicious).
 
 
 
==== About the events ====
 
It shall be possible to simulate the way the ''shared'' variables are handled in the non-decomposed machine. This is precisely the purpose of the so-called ''external'' events.
 
 
 
We will examine in this section how to define such events in the Rodin platform, how to construct them, and how to enforce the rules that apply (in particular, these events cannot be refined).
 
 
 
===== Identifying an event as external =====
 
An attribute is already defined, which is introduced below, to precise the nature of an event. A first solution would be to add another masked value (''eg.'' 4) to encode the ''external'' status.
 
 
 
<pre>
 
<!ENTITY % convergence "org.eventb.core.convergence">
 
 
 
<!ATTLIST %event;
 
  ...
 
  %convergence; (0|1|2) #REQUIRED
 
  ...
 
  >
 
  <!--
 
    The convergence attribute specifies which PO should be generated
 
    for the combination event / variant:
 
      0: ordinary event, no PO
 
      1: convergent event, PO to show event decreases variant
 
      2: anticipated event, PO to show event doesn't increase variant
 
  -->
 
</pre>
 
 
 
Another solution would be to add a distinct <tt>external</tt> attribute, which would be set to <tt>true</tt> if and only if the event is ''external'':
 
 
 
<pre>
 
<ENTITY % external "org.eventb.core.external CDATA #REQUIRED">
 
 
 
<!ATTLIST %event;
 
  ...
 
  %external; (false|true) #REQUIRED
 
>
 
</pre>
 
 
 
This solution is preferred because the notion of ''external'' event is totally orthogonal to the notion of convergence.
 
 
 
===== <font id="build_external">Constructing an external event</font> =====
 
An ''external'' event shall be built for each event of the non-decomposed machine modifying ''shared'' variables.
 
 
 
As explained in [[Actions_(Modelling_Language)|the modelling language]], a non-deterministic action expressed as a variable identifier, followed by <math>\bcmin</math>, followed by a set expression, can always be translated to a non-deterministic action made of a list of distinct variable identifiers, followed by <math>\bcmsuch</math>, followed by a before-after predicate.
 
 
 
In the same manner, a deterministic action can always be translated to a non-deterministic action, as shown in the following example. Let's consider a machine with variables <math>x</math>, and <math>y</math>. Here is an action:
 
<center>
 
{{SimpleHeader}}
 
|<math>x \bcmeq y</math> is the same as <math>x \bcmsuch\ x' = y</math>
 
|}
 
</center>
 
 
 
As a consequence, we will first focus on the construction of an external event from an event of a sub-machine <math>m</math> whose action relies on <math>\bcmsuch</math>, and then derivate the construction from an event of <math>m</math> whose action relies on <math>\bcmeq</math> or <math>\bcmin</math>.
 
<br>
 
<math>e</math> is an event of <math>m</math>, <math>v_i</math> are ''private'' variables of <math>m</math>, <math>s_i</math> are ''shared'' variables between <math>m</math> and the destination sub-machine (''i.e.'' the sub-machine where the ''external'' event will be dispatched), <math>P,~P_i,~Q_i</math> are before-after predicates of <math>m</math>, and <math>G</math> is a predicate of <math>m</math>.
 
 
 
'''Generic construction for <math>\bcmsuch</math>'''
 
 
 
  e
 
  '''WHERE''' <math>G(v_1,...,v_n,s_1,...,s_m)</math>     
 
  '''THEN'''                 
 
    <math>v_1,...,v_n,s_1,...,s_m \bcmsuch P(v1,...,v_n,s_1,...,s_m, v_1',...,v_n',s_1',...,s_m')</math>
 
 
 
The first step of the construction is to replace the ''private'' variables by parameters. Note that this step is purely fictive, because assigning an event parameter is not allowed!
 
 
 
  e
 
  '''ANY''' <math>x_1,...,x_n</math>
 
  '''WHERE''' <math>G(x_1,...,x_n,s_1,...,s_m)</math>     
 
  '''THEN'''                 
 
    <math>v_1,...,v_n,s_1,...,s_m \bcmsuch P(x1,...,x_n,s_1,...,s_m, x_1',...,x_n',s_1',...,s_m')</math>
 
 
 
The second step consists of adding guards to define the types of the parameters, if necessary. More precisely, a theorem shall be added for each parameter for which typing is required. The .bcm file associated to the non-decomposed machine shall be parsed in order to retrieve the typing information.
 
<font color="red">TODO: Some precisions and examples should be added here</font>.
 
 
 
The third and last step of the construction is to introduce an existantial quantifier to resolve the invalid assigment. <math>external\_e</math> is the newly built external event.
 
 
 
  external_e
 
  '''ANY''' <math>x_1,...,x_n</math>
 
  '''WHERE''' <math>G(x_1,...,x_n,s_1,...,s_m)</math>     
 
  '''THEN'''                 
 
    <math>s_1,...,s_m \bcmsuch \exists y_1,...y_n.P(x1,...,x_n,s_1,...,s_m, y_1,...,y_n,s_1',...,s_m')</math>
 
 
 
'''Derived construction for <math>\bcmsuch</math>'''
 
 
 
  e
 
  '''WHERE''' <math>G(v_1,...,v_n,s_1,...,s_m)</math>     
 
  '''THEN'''                 
 
    <math>v_1 \bcmsuch P_1(v_1,...,v_n,s_1,...,s_m,v_1')</math>
 
    ...
 
    <math>v_n \bcmsuch P_n(v_1,...,v_n,s_1,...,s_m,v_n')</math>
 
    <math>s_1 \bcmsuch P_{n+1}(v_1,...,v_n,s_1,...,s_m,s_1')</math>
 
    ...
 
    <math>s_m \bcmsuch P_{n+m}(v_1,...,v_n,s_1,...,s_m,s_m')</math>
 
 
 
It may alternatively be represented with a single predicate, which is an intersection of the P_1,...,P_{n+m} predicates:
 
 
 
  e
 
  '''WHERE''' <math>G(v_1,...,v_n,s_1,...,s_m)</math>     
 
  '''THEN'''                 
 
    <math>v_1,...,v_n,s_1,...,s_m \bcmsuch P_1(v1,...,v_n,s_1,...,s_m,v_1') \land ... \land P_n(v_1,...,v_n,s_1,...,s_m,v_n') \land P_{n+1}(v_1,...,v_n,s_1,...,s_m,s_1') \land ... \land P_{n+m}(v_1,...,v_n,s_1,...,s_m,s_m') </math>
 
 
 
It is then possible to build the external event, by following the same steps of construction than for the generic case:
 
 
 
  external_e
 
  '''ANY''' <math>x_1,...,x_n</math>
 
  '''WHERE''' <math>G(x_1,...,x_n,s_1,...,s_m)</math>     
 
  '''THEN'''                 
 
    <math>s_1,...,s_m \bcmsuch \exists y_1,...y_n.(P_1(x1,...,x_n,s_1,...,s_m,y_1) \land ... \land P_n(x_1,...,x_n,s_1,...,s_m,y_n) \land P_{n+1}(x_1,...,x_n,s_1,...,s_m,s_1') \land ... \land P_{n+m}(x_1,...,x_n,s_1,...s_m,s_m'))</math>
 
 
 
Or, after applying some simplification rules:
 
 
 
external_e
 
  '''ANY''' <math>x_1,...,x_n</math>
 
  '''WHERE''' <math>G(x_1,...,x_n,s_1,...,s_m)</math>     
 
  '''THEN'''                 
 
    <math>s_1 \bcmsuch \exists y_1.(P_1(x1,...,x_n,s_1,...,s_m,y_1)) \land ... \land \exists y_n.(P_n(x_1,...,x_n,s_1,...,s_m,y_n)) \land P_{n+1}(x_1,...,x_n,s_1,...,s_m,s_1')</math>
 
    ...
 
    <math>s_m \bcmsuch \exists y_1.(P_1(x1,...,x_n,s_1,...,s_m,y_1)) \land ... \land \exists y_n.(P_n(x_1,...,x_n,s_1,...,s_m,y_n)) \land P_{n+m}(x_1,...,x_n,s_1,...s_m,s_m')</math>
 
 
 
'''Construction for <math>\bcmeq</math>'''<br>
 
Additional simplification rules apply:
 
If <math>P_i(x_1,...,x_n,s_1,...,s_m,y_i)</math> is equal to  <math>y_i = Q_i(x_1,...,x_n,s_1,...,s_m)</math>, then <math>\exists y_i.P_i(x1,...,x_n,s_1,...,s_m,y_i)</math> is always true and shall be deleted.
 
 
 
If <math>P_i(x_1,...,x_n,s_1,...,s_m,s_i')</math> is equal to <math>s_i' = Q_i(x_1,...,x_n,s_1,...,s_m)</math>, and if there is no private variable (''i.e.'' there is no existantial quantifier on the right-hand side of the assigments), then <math>s_i \bcmsuch P_{n+i}(x_1,...,x_n,s_1,...s_m,s_i')</math> shall be rewritten as <math>s_i \bcmeq Q_i(x_1,...,x_n,s_1,...,s_m)</math>.
 
<br>
 
Proof obligations generated for deterministic actions are indeed more suitable than those generated for non-deterministic actions.
 
 
 
'''Construction for <math>\bcmin</math>'''<br>
 
Additional simplication rules apply:
 
If <math>P_i(x_1,...,x_n,s_1,...,s_m,s_i')</math> is equal to <math>s_i' \in Q_i(x_1,...,x_n,s_1,...,s_m)</math>, and if there is no private variable (''i.e.'' there is no existantial quantifier on the right-hand side of the assigments), then <math>s_i \bcmsuch P_{n+i}(x_1,...,x_n,s_1,...s_m,s_i')</math> shall be rewritten as <math>s_i \bcmin Q_i(x_1,...,x_n,s_1,...,s_m)</math>.
 
<br>
 
For a given set <math>S</math>, proving that <math>\exists x.x \in S</math> (proof obligation generated from <math>x \bcmsuch x' \in S</math>) is indeed not as "simple" as proving that <math>S \neq \emptyset</math> (proof obligation generated from <math>x \bcmin S</math>).
 
 
 
===== Partitionning the events in the sub-components of the decomposition =====
 
Let's first assume that the variables have been previously partitionned among sub-machines. The case where <math>e</math> is an event that accesses a ''private'' variable <math>v1</math> associated to a sub-machine <math>m1</math> and a ''private'' variable <math>v2</math> associated to a sub-machine <math>m2</math> cannot be successfully handled. As a consequence, the following sequence shall be followed:
 
* The events shall be first partitionned, as specified by the end user.
 
* The Rodin platform shall then be able to distribute the variables, according to the event partition (see [[#var_partition|Variable partition]]).
 
* The Rodin platform shall be able to distribute the invariants, according to the variable partition (see [[#inv_partition|Invariant partition]]).
 
* If <math>e</math> is an event that modifies a ''shared'' variable <math>s</math> (''i.e.'' <math>s</math> is listed among the free identifiers on the left-hand side of an assignment), then an ''external'' event that modifies <math>s</math> shall be built from <math>e</math> in each sub-machine where <math>s</math> is accessed.
 
 
 
''N.B.'' : Note that the construction of an ''external'' event depends on the ''source'' sub-machine (''i.e.'' the sub-machine containing the ''internal'' event <math>e</math> from which the ''external'' event is to be built) and on the ''destination'' sub-machine (''i.e.'' the sub-machine where the ''external'' event is to be built).
 
<br>
 
Building an ''external'' event from a given event <math>e</math> modifying a ''shared'' variable <math>s</math> and duplicating it in each sub-machine where <math>s</math> is accessed does not indeed entirely fit the requirements, as illustrated below: the sub-machine <math>M3</math> does not know the ''shared'' variable <math>s2</math> and the sub-machine <math>M1</math> does not know the ''shared'' variable <math>s4</math>.
 
 
 
<center>
 
[[Image:External_events.png]]
 
</center>
 
 
 
===== <font id="convergence">Propagating the convergence status</font> =====
 
A sub-machine can be seen as a new abstract machine. As a consequence, the convergence status of a given event shall be propagated in the sub-machines as described below:
 
* An event tagged as ordinary in the non-decomposed machine shall remain ordinary in the sub-machine.
 
* An event tagged as convergent in the non-decomposed machine shall become ordinary in the sub-machine.  
 
* An event tagged as anticipated in the non-decomposed machine shall remain anticipated in the sub-machine.
 
* An ''external'' event shall always be declared as ordinary.
 
 
 
See [[Events_(Modelling_Language)|the modelling language]] for precisions on the convergence status.
 
 
 
===== Propagating the inheritance status =====
 
An event (''external'' or not) of a sub-machine shall always be declared as non-extended.
 
 
 
===== Ensuring that an external event cannot be refined =====
 
The verification shall be performed by the static checker.
 
 
 
===== Decomposing the initialization event =====
 
An initialization event shall be built in each sub-machine from the initialization event of the non-decomposed machine, and according the distribution of the variables among these sub-machines. The construction is detailed below. <math>initialization</math> is the initial event and <math>e</math> the built event, <math>x_i</math> are variables (''private'' or ''shared'') of the sub-machine containing <math>e</math>, <math>y_i</math> are variables of other sub-machines, P is a before-after predicate and G is a predicate.
 
 
 
  initialization
 
  '''WHERE''' <math>G(x_1,...,x_n,y_1,...,y_m)</math>     
 
  '''THEN'''                 
 
    <math>x_1,...,x_n,y_1,...,y_m \bcmsuch P(x1,...,x_n,y_1,...,y_m,x_1',...,x_n',y_1',...,y_m')</math>
 
 
 
Only the variables of the considered sub-machine shall appear in the built initialization event; other variables shall become bound:
 
 
 
  e
 
  '''WHERE''' <math>G(x_1,...,x_n)</math>     
 
  '''THEN'''                 
 
    <math>x_1,...,x_n \bcmsuch \exists z_1,...,z_m.P(x1,...,x_n,y_1,...,y_m,x_1',...,x_n',y_1,...,y_m')</math>
 
 
 
The derivated cases and simplification rules introduced during [[#build_external|the construction of the external events]] apply here as well.
 
 
 
==== <font id="inv_partition">About the invariants</font> ====
 
We will see in this section how to distribute the invariants among the sub-machines, once the variables have been partitionned.
 
 
 
* Case 1: If <math>i</math> is an invariant only involving ''private'' variables, then it shall be copied in the sub-machines containing these variables.
 
* Case 2: If <math>i</math> is an invariant only involving ''shared'' variables, then it shall be copied in the sub-machines containing these variables.
 
* Case 3: If <math>i</math> is an invariant involving ''private'' variables and ''shared'' variables, then it shall not be copied.
 
 
 
==== About the variants ====
 
As mentionned [[#convergence|before]], there is no convergent event in sub-machines. As a consequence, there is no need to take the variants into consideration when performing the decomposition.
 
 
 
=== Decomposition of a context in sub-contexts ===
 
The purpose of this paragraph is to specify how to decompose the contexts, according to the decomposition of the machines, and to establish how to link the sub-contexts to the sub-machines.
 
 
 
The hierarchy of contexts shall be first accumulated in a single context. More precisely, a new context shall be built (virtually or not), which contains all the carrier sets, constants and axioms of the hierarchy. This context is assumed to be the non-decomposed context from which the sub-contexts shall be built.
 
<br>Note that it may be necessary to rename some axioms when flattening the hierarchy.
 
 
 
<center>
 
[[Image:Flattening_contexts.png]]
 
</center>
 
 
 
==== About the carrier sets ====
 
 
 
==== About the constants ====
 
<font color="red">See decomposition of the invariants.</font>
 
 
 
==== About the axioms ====
 
 
 
==== Linking the sub-contexts to the sub-machines ====
 
 
 
== High-level Specification ==
 
 
 
* Configuring the decomposition
 
** Identifying the machine to be decomposed
 
** Identifying the sub-machines to be created
 
** Specifying how to perform the decomposition (event partition)
 
** Asking for simplifications
 
** Asking for invariants and theorems that can be "forgotten" during the decomposition (because they are not required anylonger by user)
 
* Performing the decomposition
 
** Generating the sub-machines
 
** Linking the sub-machines to the initial machine
 
** Generating the sub-contexts
 
** Linking the sub-contexts to the sub-machines
 
* Generating the useful proof obligations
 
* Not "propagating" useless proof obligations
 
* Checking the decomposition
 
 
 
== Mathematical Approach ==
 
The purpose of this section is to mathematically formalize the Event-B decomposition previously specified, and by the way to remove the possible remaining ambiguity.
 
 
 
Let's define <math>\mathit{MACHINE}</math> as the set of all machine handles, <math>\mathit{EVENT}</math> the set of all events, and <math>\mathit{VAR}</math> the set of all variables.
 
 
 
* The distribution of the events of the non-decomposed machine among the different sub-machines (according to the end-user configuration) can be represented as with a partial function:
 
<math>\mathit{partition}\in\mathit{EVENT}\pfun\mathit{MACHINE}</math>
 
<br>For a given sub-machine <math>m</math>, <math>partition^{-1}[\{m\}]</math> is then the set of ''internal'' events of <math>m</math>.
 
* The access of a variable by a given event (according to the static-checker) can be expressed as:
 
<math>\mathit{access}\in\mathit{EVENT}\rel\mathit{VAR}</math>
 
<br>For a given sub-machine <math>m</math>, <math>(access;partition^{-1})[\{m\}]</math> is then the set of variables accessed by the events contained in <math>m</math>.
 
* The association of a variable with the events modifying this variable (according to the static-checker) can be specified as:
 
<math>\mathit{modify}\in\mathit{VAR}\rel\mathit{EVENT}</math>
 
<br>For a given sub-machine <math>m</math> and a variable <math>v \in (access;partition^{-1})[\{m\}]</math>, <math>modify[\{v\}]</math> is then the set of the events modifying <math>v</math>.
 
* The construction of the ''external'' events for a sub-machine can be represented with a relation:
 
<math>\mathit{extern}\in\mathit{MACHINE}\rel\mathit{EVENT}</math>
 
<br>It is computed as follows: <math>extern = (modify;access;partition^{-1}) \setminus partition^{-1}</math>
 
<br>Thus, the ''external'' events of a given sub-machine <math>m</math> are events modifying the variables accessed by the ''internal'' events of <math>m</math>, but they are not ''internal'' events of <math>m</math>.
 
 
 
=== Example ===
 
The following example is taken from the [[#ancre_1|Event Model Decomposition]].
 
 
 
<center>
 
[[Image:Decomposition.png]]
 
</center>
 
 
 
A non-decomposed machine has been decomposed in two sub-machines <math>M1</math> and <math>M2</math>, as illustrated by the figure.
 
<br>According to the terminology, <math>in\_a</math> and <math>a\_2\_b</math> are ''internal'' events of <math>M1</math>, and <math>b\_2\_c</math> and <math>out\_c</math> are ''internal'' events of <math>M2</math>. Concerning the variables, <math>a</math> and <math>m</math> are ''private'' variables of <math>M1</math>, <math>c</math> and <math>p</math> are ''private'' variables of <math>M2</math>, and <math>b</math>, <math>r</math> and <math>s</math> are ''shared'' variables.
 
<br>The variables accessed by the ''internal'' events of <math>M1</math> are <math>a</math>, <math>m</math>, <math>b</math>, <math>r</math> and <math>s</math>. The events modifying these variables are <math>in\_a</math>, <math>a\_2\_b</math>, which both are ''internal'' events of <math>M1</math>, and <math>b\_2\_c</math>, which is an ''internal'' event of <math>M2</math>. Thus, according to the definition given before, <math>b\_2\_c</math> is an ''external'' event for <math>M1</math>. In the same manner, <math>a\_2\_b</math> is an ''external'' event for <math>M2</math>.
 
 
 
''N.B.'': Note that the expression "is an ''external'' event for" is an extrapolation, and shall be literally interpreted as "should lead to the construction of an ''external'' event in".
 
 
 
== Bibliography ==
 
 
 
* J.R. Abrial, Mathematical Models for Refinement and Decomposition, in ''The Event-B Book'', to be published in 2009 ([http://www.event-b.org/abook.html lien externe]).
 
* J.R. Abrial, <font id="ancre_1">''Event Model Decomposition''</font>, Version 1.3, April 2009.
 
* M. Butler, <font id="ancre_2">Decomposition Structures for Event-B</font>, in ''Integrated Formal Methods iFM2009'', Springer, LNCS 5423, 2009 ([http://eprints.ecs.soton.ac.uk/16965/1/butler.pdf lien externe]).
 
 
 
[[Category:Design]]
 
[[Category:Work in progress]]
 

Revision as of 15:51, 15 May 2012

Defining Translations Using The Theory Plug-in

The theory plug-in is used to add mathematical extensions to Rodin. The theories are created, and deployed, and can then be used in any models in the workspace. When dealing with implementation level models, such as in Tasking Event-B, we need to consider how to translate newly added types and operators into code. We have augmented the theory interface with a Translation Rules section. This enables a user to define translation rules that map Event-B formulas to code.

Translation Rules


caption text
Figure 1: Translation Rules


Figure 1 shows the interface, and some translations rules of the mapping to Ada.

The theory is given a name, and may import some other theories. Type parameters can be added, and we use them here to type the meta-variables. The meta-variable a is restricted to be an integer type, but meta-variable c can be any type. Meta-variables are used in the translator rules for pattern matching.

Translator rules are templates, which are used in pattern matching. Event-B formulas are defined on the left hand side of the rule, and the code to be output (as text) appears on the right hand side of the matching rule. During translation an abstract syntax tree (AST) representation of the formula is used. The theory plug-in attempts to match the formulas in the rules with each syntactic element of the AST. As it does so it builds the textual output as a string, until the whole AST has been successfully matched. When a complete tree is matched, the target code is returned. If the AST is not matched, a warning is issued, and a string representation of the original formula is returned.

Type Rules for Code Generation

The type rules section, shown in Figure 1, is where the relationship is defined, between Event-B types and the type system of the implementation.

Adding New (implementation-level) Types

When we are working at abstraction levels close to the implementation level, we may make an implementation decision which requires the introduction of a new type to the development. We give an example of our approach, where we add a new array type, shown in Figure 2, and then define its translation to code.

An Array Type Definition


caption text
Figure 2: Array Definition


The array operator notation is defined in the expression array(s: P(T)); and the semantics is defined in the direct definition. arrayN constrains the arrays to be of fixed length. Array lookup, update, and constructor operators are subsequently defined. In the next step we need to define any translations required to implement the array in code.

Translation Rules


caption text
Figure 3: Translation Rules for the Array Type


Figure 3 shows the Ada translation; beginning with the meta-variable definitions that are used for pattern matching in the translation rules. Each of the operators; newArray, and update, and an expression using the lookup operator, are mapped to their implementations on the right hand side of the rule. The Type Rules section describes the implementation's description of the arrayN type.