Difference between pages "Current Developments" and "Membership in Goal"

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{{TOCright}}
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= Objective =
This page sum up the known developments that are being done around or for the [[Rodin Platform]]. ''Please contributes informations about your own development to keep the community informed''
 
  
== Deploy Tasks ==
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This page describes the design of the reasoner MembershipGoal and its associated tactic MembershipGoalTac.<br>
The following tasks were planned at some stage of the [[Deploy]] project.
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This reasoner discharges sequent whose goal denotes a membership which can be inferred from hypotheses. Here an basic example of what it discharges :<br>
=== Core Platform ===
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<math>H,\quad x\in S,\quad S\subset T,\quad T\subseteq U \quad\vdash x\in U</math>
==== New Mathematical Language ====
 
==== Rodin Index Manager ====
 
[[Systerel]] is in charge of this task.
 
{{details|Rodin Index Design|Rodin index design}}
 
  
The purpose of the Rodin index manager is to store in a uniform way the entities that are declared in the database together with their occurrences. This central repository of declarations and occurrences will allow for fast implementations of various refactoring mechanisms (such as renaming) and support for searching models or browsing them.
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= Analysis =
  
==== Text Editor ====
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Such sequent are proved by PP and ML. But, these provers have both drawbacks :
[[Düsseldorf]] has a prototype text-based editor for Event-B (courtesy of [[User:Fabian|Fabian]]). As of end of sempteber 2008, it still needs more work to fully integrate into Rodin.
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*All the visible are added as needed hypotheses, which is most of the time not the case.
 +
*They take quite consequent time to prove it (even with the basic example given here above, the difference in time execution is noticeable).
 +
*If there are too many hypotheses, or if the expression of the <math>x</math> is too complicated, they may not prove it.
 +
This is particularly true when in the list of inclusion expressions of each side of the relation are not equal. For example : <math>H,\quad a\in S,\quad S\subset T_1\cap T_2,\quad T_1\cup T_3\subseteq  U\quad\vdash a\in U</math>
 +
<p>
 +
Such a reasoner contributes to prove more Proof Obligations automatically, faster and with fewer needed hypotheses which makes proof rule more legible and proof replay less sensitive to modifications.
  
[[Newcastle]] has another text editor based on EMF.  Among other things, it defines an EMF model of Event-B machines and contexts. At some point, the editor code is to be split into two plugins - an EMF adapater to rodin and the editor itself. Source code is currently available from [http://iliasov.org/editor-source.zip].
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= Design Decision =
  
=== Plug-ins ===
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== Tactic ==
==== Requirement Management Plug-in ====
 
[[User:Jastram|Michael]] at [[Düsseldorf]] is in charge of the [[:Category:Requirement Plugin|Requirements Management Plug-in]].
 
  
{{See also|ReqsManagement|Requirements Tutorial|l1=Requirements Management Plug-in}}
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This part explains how the tactic (MembershipGoalTac) associated to the reasoner MembershipGoal is working.
 +
=== Goal ===
 +
The tactic (as the reasoner) should works only on goals such as :
 +
*<math>\cdots~\in~\cdots</math>
 +
For examples :
 +
*<math>f(x)\in g\otimes h</math>
 +
*<math>x\in A\cprod\left(B\cup C\right)</math>
 +
*<math>x\mapsto y\in A\cprod B</math>
 +
In the latter case, the reasoner won't try to prove that ''x'' belongs to ''A'' and ''y'' belongs to ''B'', but that the mapplet belong to the Cartesian product.
 +
=== Hypotheses ===
 +
Now we have to find hypotheses leading to discharge the sequent. To do so, the tactic recovers two kinds of hypotheses :
 +
#the ones related to the left member of the goal <math>\left( x\in S\right)</math> (this is the start point):
 +
#*<math>x\in \cdots</math>
 +
#*<math>\cdots\mapsto x\mapsto\cdots\in\cdots</math>
 +
#*<math>\left\{\cdots, x,\cdots\right\}=\cdots</math>
 +
#*<math>\left\{\cdots, \cdots\mapsto x\mapsto\cdots,\cdots\right\}=\cdots</math>
 +
#the ones denoting inclusion (all but the ones matching the description of the first point) :
 +
#*<math>\cdots\subset\cdots</math>
 +
#*<math>\cdots\subseteq\cdots</math>
 +
#*<math>\cdots=\cdots</math>
 +
Then, it will search a link between those hypotheses so that the sequent can be discharged.
 +
=== Find a path ===
 +
Now that we recovered all the hypotheses that could be useful for the reasoner, it remains to find a path among the hypotheses leading to discharge the sequent. Depending on the relations on each side of the inclusion, we will act differently. <math>f</math> always represent an expression (may be a domain, a range, etc.).
 +
#The following sequent is provable because <math>f\subseteq \varphi (f)</math>.
 +
#*<math>x\in f,\quad \varphi (f)\subseteq g\quad\vdash\quad x\in g</math>
 +
#*<math>\varphi (f) = f\quad\mid\quad f\cup h \quad\mid\quad h\cup f \quad\mid\quad h\ovl f</math>
 +
#The following sequent is provable because <math>\psi (f)\subseteq f</math>.
 +
#*<math>x\in \psi (f),\quad f\subseteq g\quad\vdash\quad x\in g</math>
 +
#*<math>\psi (f) = f\quad\mid\quad f\cap h \quad\mid\quad h\cap f \quad\mid\quad f\setminus h \quad\mid\quad f\ransub A \quad\mid\quad f\ranres A \quad\mid\quad A\domsub f \quad\mid\quad A\domres f</math>
 +
#We can generalized the first two points. This is the Russian dolls system. We can easily prove a sequent with multiple inclusions by going from hypothesis to hypothesis.
 +
#*<math>x\in \psi (f),\quad \varphi (f)\subseteq g\quad\vdash\quad x\in g</math>
 +
#For some relations, [[#positions|positions]] are needed to be known to continue to find hypotheses, but it is not always necessary.
 +
#*<math>x\mapsto y\in f,\quad f\subseteq A\cprod B\quad\vdash\quad x\in A</math>
 +
#*<math>x\in dom(f),\quad f\subseteq A\cprod B\quad\vdash\quad x\in A</math>
 +
#*<math>x\in ran(f),\quad f\subseteq A\cprod B\quad\vdash\quad x\in B</math>
  
This plug-in allows:
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By using these inclusion and rewrites, it tries to find a path among the recovered hypotheses. Every one of those should only be used once, avoiding possible infinite loop <math>\left(A\subseteq B,\quad B\subseteq A\right)</math>. Among all paths that lead to discharge the sequent, the tactic give the first it finds. Moreover, so that the reasoner does not do the same work as the tactic of writing new hypothesis, it gives all needed hypotheses and added hypotheses in the input.
* Requirements to be edited in a set of documents (independently from Rodin)
 
* Requirements to be viewed within Rodin
 
* Individual Requirements to be linked to individual Event-B-Entities
 
* A basic completion test to be performed
 
  
==== UML-B Plug-in ====
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== Reasoner ==
[[Southampton]] is in charge of [[UML-B]] plug-in.
 
  
* Support for synchronisation of transitions from different statemachines. This feature will allow two or more transitions in different statemachines to contribute to a single event. This feature is needed because a single event can alter several variables (in this case statemachines) simultaneously.
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The way the reasoner must work is still in discussion.
  
*Allow user to allocate the name of the 'implicit contextual instance' used in a class. Events and Transitions owned by a class are implicitly acting upon an instance of the class which has formerly been denoted by the reserved word 'self'. This modification allows the modeller to override 'self' (which is now the default name) with any other identifier. This feature is needed to avoid name clashes when synchronising transitions into a single event. It also allows events to be moved between different classes (or outside of all classes) during refinement without creating name clashes.
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= Implementation =
  
* Better support for state machine refinement in UML-B. This revision to UML-B allows a statemachine to be recognised as a refinement of another one and to be treated in an appropriate way during translation to Event-B. The states and transitions of a refined statemachine can be elaborated by adding more detailed hierarchical statemachines.
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This section explain how the tactic has bee implemented.
  
==== ProB Plug-in ====
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=== Positions ===
[[Düsseldorf]] is in charge of [[ProB]].
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As it was said, we may sometimes need the position. It is represented by an array of integer. Here are the possible values the array contains (atomic positions) :
<!-- {{details|ProB current developments|ProB current developments}} -->
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* '''''kdom''''' : it corresponds to the domain.
 +
**<math>\left[A\cprod B\right]_{pos\;=\;kdom} = A</math>
 +
**<math>\left[x\mapsto y\right]_{pos\;=\;kdom} = x</math>
 +
**<math>\left[g\right]_{pos\;=\;kdom} = dom(g)</math>
 +
* '''''kran''''' : it corresponds to the domain.
 +
**<math>\left[A\cprod B\right]_{pos\;=\;kran} = B</math>
 +
**<math>\left[x\mapsto y\right]_{pos\;=\;kran} = y</math>
 +
**<math>\left[g\right]_{pos\;=\;kran} = ran(g)</math>
 +
* '''''leftDProd''''' : it corresponds to the left member of a direct product.
 +
**<math>\left[f\otimes g\right]_{pos\;=\;leftDProd} = f</math>
 +
**<math>\left[A\cprod \left(B\cprod C\right)\right]_{pos\;=\;leftDProd} = A\cprod B</math>
 +
* '''''rightDProd''''' : it corresponds to the right member of a direct product
 +
**<math>\left[f\otimes g\right]_{pos\;=\;rightDProd} = g</math>
 +
**<math>\left[A\cprod\left(B\cprod C\right)\right]_{pos\;=\;rightDProd} = A\cprod C</math>
 +
* '''''leftPProd''''' : it corresponds to the left member of a parallel product
 +
**<math>\left[f\parallel g\right]_{pos\;=\;leftPProd} = f</math>
 +
**<math>\left[\left(A\cprod B\right)\cprod\left(C\cprod D\right)\right]_{pos\;=\;leftPProd} = A\cprod C</math>
 +
* '''''rightPProd''''' : it corresponds to the right member of a parallel product
 +
**<math>\left[f\parallel g\right]_{pos\;=\;rightPProd} = g</math>
 +
**<math>\left[\left(A\cprod B\right)\cprod\left(C\cprod D\right)\right]_{pos\;=\;rightPProd} = B\cprod D</math>
 +
* '''''converse''''' : it corresponds to the child of an inverse
 +
**<math>\left[f^{-1}\right]_{pos\;=\;converse}=f</math>
 +
**<math>\left[A\cprod B\right]_{pos\;=\;converse} = B\cprod A</math>
 +
For example, the following expressions at the given positions are equivalent.
 +
:<math>\left[f\otimes\left(\left(A\cprod dom(g)\right)\parallel g\right)\right]_{pos\;=\;\left[rightDProd,\;leftPProd,\;kran\right]} = \left[\left(A\cprod dom(g)\right)\parallel g\right]_{pos\;=\;\left[leftPProd,\;kran\right]} = \left[A\cprod dom(g)\right]_{pos\;=\;\left[kran\right]} = \left[dom(g)\right]_{pos\;=\;\left[~\right]} = \left[g\right]_{pos\;=\;\left[kdom\right]}</math>
  
===== Work already performed =====
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Some combinations of atomic positions are equivalent. In order to simplify comparison between positions, they are normalized :
 +
*<math>ran(f^{-1}) = dom(f)\quad\limp\quad \left[f\right]_{pos \;=\; \left[converse,~kran\right]} = \left[f\right]_{pos\;=\;\left[kdom\right]}</math>
 +
*<math>dom(f^{-1}) = ran(f)\quad\limp\quad \left[f\right]_{pos \;=\; \left[converse,~kdom\right]} = \left[f\right]_{pos\;=\;\left[kran\right]}</math>
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*<math>\left(f^{-1}\right)^{-1} = f\quad\limp\quad \left[f\right]_{pos \;=\; \left[converse,~converse\right]} = \left[f\right]_{pos\;=\;\left[~\right]}</math>
  
We have now ported ProB to work directly on the Rodin AST. Animation is working and the user can now set a limited number of preferences.
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=== Goal ===
The model checking feature is now also accessible.
 
It is also possible to create CSP and classical B specification files. These files can be edited with BE4 and animated/model checked with ProB.
 
On the classical B side we have moved to a new, more robust parser (which is now capable of parsing some of the more complicated AtelierB
 
specifications from Siemens).
 
  
On the developer side, we have moved to a continuous integration infrastructure using CruiseControl. Rodin is also building from CVS in that infrastructure.
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As explained in the design decision part, goal is checked. If it matches the description here above <math>\left(x\in S\right)</math> then ''x'' is stored in an attribute. Moreover, from the set ''S'', we compute every pair ''expression'' & ''position'' equivalent to it. For example, from the set <math>dom(ran(ran(g)))</math>, the map will be computed :
 +
*<math>dom(ran(ran(g)))\;\mapsto\;[\;]</math>
 +
*<math>ran(ran(g))\;\mapsto\;[0]</math>
 +
*<math>ran(g)\;\mapsto\;[1,~0]</math>
 +
*<math>g\;\mapsto\;[1,~1,~0]</math>
 +
Only ''range'', ''domain'' and ''converse'' can be taken into account to get all the possibles goals.
  
===== Ongoing and future developments=====
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A pair (expression ; position) is said equal to the goal if and only if there exists a pair equivalent to the goal (GoalExp ; GoalPos) and a pair equivalent to the given pair (Exp ; Pos) such as ''Pos = GoalPos'' and ''Exp'' is contained in ''GoalExp''.
  
We are currently developing a new, better user interface.
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=== Hypotheses ===
We also plan to support multi-level animation with checking of the gluing invariant.
 
  
We have prototypes for several extensions working, but they need to be fully tested and integrated into the plugin:
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As explained in the design decision part, there are two kinds of hypotheses which are recovered. But when hypotheses related to the left member of the goal <math>\left(x\in S\right)</math> are stored, the position of ''x'' is also record. Then, if there is an hypothesis such as <math>\left\{\cdots\;,\;y\mapsto x\mapsto z\;,\;m\mapsto x\;,\;\cdots\right\} = A</math>, then this hypothesis is mapped to the positions <math>\left\{\left[0,~1\right],~\left[1\right]\right\}</math>.
* an inspector that allows the user to inspect complex predicates (such as invariants or guards) as well as expressions in a tree-like manner
 
* a graphical animator based on SWT that allows the user to design his/her own animations easily within the tool
 
* a 2D viewer to inspect the state space of the specification
 
  
==== B2Latex Plug-in ====
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=== Find a path ===
[[Southampton]] is in charge of [[B2Latex]].
 
  
Kriangsak Damchoom will update the plug-in to add [[Event Extension|extensions of events]].
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Let's considered the sequent with the following goal : <math>x\in V</math>.
 +
We start with the hypotheses which have a connection with the goal's member. Such a hypothesis gives two informations : the position <math>pos</math> and the set <math>S</math> as explained in [[#Hypotheses|hypotheses]]. Then, for each equivalent pair to these one <math>\left(S', pos'\right)</math>, we compute set containing <math>S'</math> ([[#Design Decision#Tactic#Find a path| Find a path 2.]]). For every new pair, we test if it is contained in the goal.
  
==== Parallel Composition Plug-in ====
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To be continued.
[[Southampton]] is in charge of the [[Parallel Composition using Event-B]] .
 
  
The intention of the plug-in is to allow the parallel composition of events using Event-B syntax. The composition uses a value-passing style (shared event composition), where parameters can be shared/merged.
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= Untreated cases =
  
This plug-in allows:
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Some cases are not treated. Further enhancement may be provided for some.
* Selection of machines that will be part of the composition (''Includes'' Section)
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*<math>x\in A\cup B,\quad A\cup B\cup C\subseteq D\quad\vdash\quad x\in D</math>
* Possible selection of an abstract machine (''Refines'' Section)
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*<math>x\in f,\quad f\in A\;op\;B\quad\vdash\quad x\in A\cprod B</math>
* Possible inclusion of invariants that relate the included machines (''Invariant'' Section and use of the monotonicity )
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*<math>f\ovl \left\{x\mapsto y\right\}\subseteq A\cprod B\quad\vdash\quad x\in A</math>
* Invariants of included machines are conjoined.
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*<math>x\in f\otimes g,\quad f\subseteq A\cprod B,\quad g\subseteq C\cprod D\quad\vdash\quad x\in (A\cprod C)\cprod(B\cprod D)</math>
* Selection of events that will be merged. The event(s) must come from different machines. At the moment, events with parameters with same name are merged. If it is a refinement composition, it is possible to choose the abstract event that is being refined.
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*<math>x\in f\otimes g,\quad f\subseteq h\quad\vdash\quad x\in h\otimes g</math>
* Initialisation event is the parallel composition of all the included machines' initialisations.
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*<math>x\in \left\{a,~b,~c\right\},\quad\left\{a,~b,~c,~d,~e,~f\right\}\subseteq D\quad\vdash\quad x\in D</math>
* For a composed event, the guards are conjoined and the all the actions are composed in parallel.
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*<math>x\in A\cprod B,\quad A\subseteq C\quad\vdash\quad x\in C\cprod B</math>
 +
*<math>x\in dom(f)\cap A\quad\vdash\quad x\in dom(A\domres f)</math>
 +
*<math>x\in ran(f)\cap A\quad\vdash\quad x\in ran(f\ranres A)</math>
 +
*<math>x\in \quad\vdash\quad</math>
 +
*<math>\quad\vdash\quad</math>
 +
*<math>\quad\vdash\quad</math>
 +
*<math>\quad\vdash\quad</math>
 +
*:<math>\bigl(</math> where <math>op_1</math> and <math>op_2</math> are ones of :<math>\quad\rel, \trel, \srel, \strel, \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij\bigr)</math>
  
Currently, after the conclusion of the composition machine, a new machine can be generated, resulting from the properties defined on the composition file. This allows proofs to be generated as well as a visualisation of the composition machine file. In the future, the intention is to make the validation directly on the composition machine file directly where proofs would be generated ( and discharged) - the new machine generation would be optional. An event-b model for the validation/generation of proofs in currently being developed. Another functionality which should be quite useful for the composition (but not restricted to that) is '''renaming''':
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[[Category:Design proposal]]
 
 
* while composing, two machines may have variables with the same name for instance (which is not allowed for this type of composition). In order to solve this problem, one would have to rename one of the variables in order to avoid the clash, which would mean change the original machine. A possible solution for that would be to rename the variable but just on composition machine file, keeping the original machine intact. A renaming framework designed and developed by Stefan Hallerstede and Sonja Holl exists currently although still on a testing phase. The framework was developed to be used in a general fashion (not constrained to event-b syntax). The idea is to extend the development of this framework and apply to Event-B syntax (current development).
 
 
 
There is a prototype for the composition plug-in available, but only works for Rodin 0.8.2. A release for the Rodin 0.9 is concluded and will be available from the Rodin Main Update Site soon, under the update 'Shared Event Composition'.
 
 
 
==== Refactoring Framework Plug-in ====
 
[[Southampton]] is in charge of the [[Refactoring Framework]].
 
 
 
The intention of the plug-in is to allow the renaming/refactoring of elements on a file (and possible related files). Although created to be used in a general way, the idea is to embed this framework on the Rodin platform, using Event-B syntax. This plug-in was initially designed and developed by Stefan Hallerstede and Sonja Holl.
 
 
 
This plug-in allows:
 
* Defining extensions that can be used to select related files.
 
* Defining extensions that can be used to rename elements based on the type of file.
 
* Renaming of elements on a file and possible occurrences on related files.
 
* Generating of a report of possible problems (clashes) that can occur while renaming.
 
 
 
== Exploratory Tasks ==
 
=== One Single View ===
 
[[Maria]] is in charge of this exploratory work during is internship.
 
{{details|Single View Design|Single View Design}}
 
The goal of this project is to present everything in a single view in Rodin. So the user won't have to switch perspectives.
 
 
 
 
 
 
 
== Others ==
 
 
 
=== AnimB ===
 
[[Christophe]] devotes some of its spare time for this plug-in.
 
{{details|AnimB Current Developments|AnimB Current Developments}}
 
The current developments around the [[AnimB]] plug-in encompass the following topics:
 
;Live animation update
 
:where the modification of the animated event-B model is instantaneously taken into account by the animator, without the need to restart the animation.
 
;Collecting history
 
:The history of the animation will be collected.
 
 
 
=== Team-Based Development ===
 
 
 
; Usage Scenarios
 
: In order to understand the problem properly, [http://www.stups.uni-duesseldorf.de/ Düsseldorf] created a number of usage [[Scenarios for Team-based Development]].
 
: A page as also been opened for [[Scenarios for Merging Proofs|merging proofs scenarios]].
 
[[Category:Work in progress]]
 
 
 
=== B2C ===
 
This plug-in translates Event-B models to C source code, which may then be compiled using external C development tools. [[Steve]] wrote B2C with the specific purpose of translating the  [http://dx.doi.org/10.1007/978-3-540-87603-8_21 MIDAS] model, an Event-B implementation of a Virtual Machine instruction set.
 
 
 
B2C supports a sub-set of Event-B that can be easily translated to C form. The user provides a final refinement step that does nothing except restate the model in this translatable form: symbolic constants must be replaced by their literal values, range membership guards are replaced by greater-than and less-than guards, and actions are restated not to use global statments on their left-sides (this because the variable may have been modified by an earlier action, and may not be valid). The manipulations are done within EventB where they can be checked by the Proof Obligation system, and B2C made as simple as possible to maximise reliability. This re-write process is currently a manual step, but could in principle be done by another plug-in
 
 
 
B2C source code is not currently available for download: contact [[Steve]] directly if it is required.
 

Revision as of 13:28, 9 August 2011

Objective

This page describes the design of the reasoner MembershipGoal and its associated tactic MembershipGoalTac.
This reasoner discharges sequent whose goal denotes a membership which can be inferred from hypotheses. Here an basic example of what it discharges :
H,\quad x\in S,\quad S\subset T,\quad T\subseteq U \quad\vdash x\in U

Analysis

Such sequent are proved by PP and ML. But, these provers have both drawbacks :

  • All the visible are added as needed hypotheses, which is most of the time not the case.
  • They take quite consequent time to prove it (even with the basic example given here above, the difference in time execution is noticeable).
  • If there are too many hypotheses, or if the expression of the x is too complicated, they may not prove it.

This is particularly true when in the list of inclusion expressions of each side of the relation are not equal. For example : H,\quad a\in S,\quad S\subset T_1\cap T_2,\quad T_1\cup T_3\subseteq  U\quad\vdash a\in U

Such a reasoner contributes to prove more Proof Obligations automatically, faster and with fewer needed hypotheses which makes proof rule more legible and proof replay less sensitive to modifications.

Design Decision

Tactic

This part explains how the tactic (MembershipGoalTac) associated to the reasoner MembershipGoal is working.

Goal

The tactic (as the reasoner) should works only on goals such as :

  • \cdots~\in~\cdots

For examples :

  • f(x)\in g\otimes h
  • x\in A\cprod\left(B\cup C\right)
  • x\mapsto y\in A\cprod B

In the latter case, the reasoner won't try to prove that x belongs to A and y belongs to B, but that the mapplet belong to the Cartesian product.

Hypotheses

Now we have to find hypotheses leading to discharge the sequent. To do so, the tactic recovers two kinds of hypotheses :

  1. the ones related to the left member of the goal \left( x\in S\right) (this is the start point):
    • x\in \cdots
    • \cdots\mapsto x\mapsto\cdots\in\cdots
    • \left\{\cdots, x,\cdots\right\}=\cdots
    • \left\{\cdots, \cdots\mapsto x\mapsto\cdots,\cdots\right\}=\cdots
  2. the ones denoting inclusion (all but the ones matching the description of the first point) :
    • \cdots\subset\cdots
    • \cdots\subseteq\cdots
    • \cdots=\cdots

Then, it will search a link between those hypotheses so that the sequent can be discharged.

Find a path

Now that we recovered all the hypotheses that could be useful for the reasoner, it remains to find a path among the hypotheses leading to discharge the sequent. Depending on the relations on each side of the inclusion, we will act differently. f always represent an expression (may be a domain, a range, etc.).

  1. The following sequent is provable because f\subseteq \varphi (f).
    • x\in f,\quad \varphi (f)\subseteq g\quad\vdash\quad x\in g
    • \varphi (f) = f\quad\mid\quad f\cup h \quad\mid\quad h\cup f \quad\mid\quad h\ovl f
  2. The following sequent is provable because \psi (f)\subseteq f.
    • x\in \psi (f),\quad f\subseteq g\quad\vdash\quad x\in g
    • \psi (f) = f\quad\mid\quad f\cap h \quad\mid\quad h\cap f \quad\mid\quad f\setminus h \quad\mid\quad f\ransub A \quad\mid\quad f\ranres A \quad\mid\quad A\domsub f \quad\mid\quad A\domres f
  3. We can generalized the first two points. This is the Russian dolls system. We can easily prove a sequent with multiple inclusions by going from hypothesis to hypothesis.
    • x\in \psi (f),\quad \varphi (f)\subseteq g\quad\vdash\quad x\in g
  4. For some relations, positions are needed to be known to continue to find hypotheses, but it is not always necessary.
    • x\mapsto y\in f,\quad f\subseteq A\cprod B\quad\vdash\quad x\in A
    • x\in dom(f),\quad f\subseteq A\cprod B\quad\vdash\quad x\in A
    • x\in ran(f),\quad f\subseteq A\cprod B\quad\vdash\quad x\in B

By using these inclusion and rewrites, it tries to find a path among the recovered hypotheses. Every one of those should only be used once, avoiding possible infinite loop \left(A\subseteq B,\quad B\subseteq A\right). Among all paths that lead to discharge the sequent, the tactic give the first it finds. Moreover, so that the reasoner does not do the same work as the tactic of writing new hypothesis, it gives all needed hypotheses and added hypotheses in the input.

Reasoner

The way the reasoner must work is still in discussion.

Implementation

This section explain how the tactic has bee implemented.

Positions

As it was said, we may sometimes need the position. It is represented by an array of integer. Here are the possible values the array contains (atomic positions) :

  • kdom : it corresponds to the domain.
    • \left[A\cprod B\right]_{pos\;=\;kdom} = A
    • \left[x\mapsto y\right]_{pos\;=\;kdom} = x
    • \left[g\right]_{pos\;=\;kdom} = dom(g)
  • kran : it corresponds to the domain.
    • \left[A\cprod B\right]_{pos\;=\;kran} = B
    • \left[x\mapsto y\right]_{pos\;=\;kran} = y
    • \left[g\right]_{pos\;=\;kran} = ran(g)
  • leftDProd : it corresponds to the left member of a direct product.
    • \left[f\otimes g\right]_{pos\;=\;leftDProd} = f
    • \left[A\cprod \left(B\cprod C\right)\right]_{pos\;=\;leftDProd} = A\cprod B
  • rightDProd : it corresponds to the right member of a direct product
    • \left[f\otimes g\right]_{pos\;=\;rightDProd} = g
    • \left[A\cprod\left(B\cprod C\right)\right]_{pos\;=\;rightDProd} = A\cprod C
  • leftPProd : it corresponds to the left member of a parallel product
    • \left[f\parallel g\right]_{pos\;=\;leftPProd} = f
    • \left[\left(A\cprod B\right)\cprod\left(C\cprod D\right)\right]_{pos\;=\;leftPProd} = A\cprod C
  • rightPProd : it corresponds to the right member of a parallel product
    • \left[f\parallel g\right]_{pos\;=\;rightPProd} = g
    • \left[\left(A\cprod B\right)\cprod\left(C\cprod D\right)\right]_{pos\;=\;rightPProd} = B\cprod D
  • converse : it corresponds to the child of an inverse
    • \left[f^{-1}\right]_{pos\;=\;converse}=f
    • \left[A\cprod B\right]_{pos\;=\;converse} = B\cprod A

For example, the following expressions at the given positions are equivalent.

\left[f\otimes\left(\left(A\cprod dom(g)\right)\parallel g\right)\right]_{pos\;=\;\left[rightDProd,\;leftPProd,\;kran\right]} = \left[\left(A\cprod dom(g)\right)\parallel g\right]_{pos\;=\;\left[leftPProd,\;kran\right]} = \left[A\cprod dom(g)\right]_{pos\;=\;\left[kran\right]} = \left[dom(g)\right]_{pos\;=\;\left[~\right]} = \left[g\right]_{pos\;=\;\left[kdom\right]}

Some combinations of atomic positions are equivalent. In order to simplify comparison between positions, they are normalized :

  • ran(f^{-1}) = dom(f)\quad\limp\quad \left[f\right]_{pos \;=\; \left[converse,~kran\right]} = \left[f\right]_{pos\;=\;\left[kdom\right]}
  • dom(f^{-1}) = ran(f)\quad\limp\quad \left[f\right]_{pos \;=\; \left[converse,~kdom\right]} = \left[f\right]_{pos\;=\;\left[kran\right]}
  • \left(f^{-1}\right)^{-1} = f\quad\limp\quad \left[f\right]_{pos \;=\; \left[converse,~converse\right]} = \left[f\right]_{pos\;=\;\left[~\right]}

Goal

As explained in the design decision part, goal is checked. If it matches the description here above \left(x\in S\right) then x is stored in an attribute. Moreover, from the set S, we compute every pair expression & position equivalent to it. For example, from the set dom(ran(ran(g))), the map will be computed :

  • dom(ran(ran(g)))\;\mapsto\;[\;]
  • ran(ran(g))\;\mapsto\;[0]
  • ran(g)\;\mapsto\;[1,~0]
  • g\;\mapsto\;[1,~1,~0]

Only range, domain and converse can be taken into account to get all the possibles goals.

A pair (expression ; position) is said equal to the goal if and only if there exists a pair equivalent to the goal (GoalExp ; GoalPos) and a pair equivalent to the given pair (Exp ; Pos) such as Pos = GoalPos and Exp is contained in GoalExp.

Hypotheses

As explained in the design decision part, there are two kinds of hypotheses which are recovered. But when hypotheses related to the left member of the goal \left(x\in S\right) are stored, the position of x is also record. Then, if there is an hypothesis such as \left\{\cdots\;,\;y\mapsto x\mapsto z\;,\;m\mapsto x\;,\;\cdots\right\} = A, then this hypothesis is mapped to the positions \left\{\left[0,~1\right],~\left[1\right]\right\}.

Find a path

Let's considered the sequent with the following goal : x\in V. We start with the hypotheses which have a connection with the goal's member. Such a hypothesis gives two informations : the position pos and the set S as explained in hypotheses. Then, for each equivalent pair to these one \left(S', pos'\right), we compute set containing S' ( Find a path 2.). For every new pair, we test if it is contained in the goal.

To be continued.

Untreated cases

Some cases are not treated. Further enhancement may be provided for some.

  • x\in A\cup B,\quad A\cup B\cup C\subseteq D\quad\vdash\quad x\in D
  • x\in f,\quad f\in A\;op\;B\quad\vdash\quad x\in A\cprod B
  • f\ovl \left\{x\mapsto y\right\}\subseteq A\cprod B\quad\vdash\quad x\in A
  • x\in f\otimes g,\quad f\subseteq A\cprod B,\quad g\subseteq C\cprod D\quad\vdash\quad x\in (A\cprod C)\cprod(B\cprod D)
  • x\in f\otimes g,\quad f\subseteq h\quad\vdash\quad x\in h\otimes g
  • x\in \left\{a,~b,~c\right\},\quad\left\{a,~b,~c,~d,~e,~f\right\}\subseteq D\quad\vdash\quad x\in D
  • x\in A\cprod B,\quad A\subseteq C\quad\vdash\quad x\in C\cprod B
  • x\in dom(f)\cap A\quad\vdash\quad x\in dom(A\domres f)
  • x\in ran(f)\cap A\quad\vdash\quad x\in ran(f\ranres A)
  • x\in \quad\vdash\quad
  • \quad\vdash\quad
  • \quad\vdash\quad
  • \quad\vdash\quad
    \bigl( where op_1 and op_2 are ones of :\quad\rel, \trel, \srel, \strel, \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij\bigr)