imported>Nicolas |
imported>Billaude |
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| − | Currently the operators and basic predicates of the Event-B mathematical language supported by Rodin are fixed.
| + | = Objective = |
| − | We propose to extend Rodin to define basic predicates, new operators or new algebraic types.
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| | | | |
| − | == Requirements ==
| + | This page describes the design of the reasoner MembershipGoal and its associated tactic MembershipGoalTac.<br> |
| | + | This reasoner discharges sequent whose goal denotes a membership which can be inferred with hypotheses. Here an basic example of what it discharges :<br> |
| | + | <math>H,\quad x\in S,\quad S\subset T,\quad T\subseteq U \quad\vdash x\in U</math> |
| | | | |
| − | === User Requirements === | + | = Analysis = |
| − | * Binary operators (prefix form, infix form or suffix form).
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| − | * Operators on boolean expressions.
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| − | * Unary operators, such as absolute values.
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| − | : Note that the pipe, which is already used for set comprehension, cannot be used to enter absolute values (''in fact, as in the new design the pipe used in set comprehension is only a syntaxic sugar, the same symbol may be used for absolute value. To be confirmed with the prototyped parser. It may however be not allowed in a first time until backtracking is implemented, due to use of lookahead making assumptions on how the pipe symbol is used. -Mathieu ).
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| − | * Basic predicates (e.g., the symmetry of relations <math>sym(R) \defi R=R^{-1}</math>).
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| − | : Having a way to enter such predicates may be considered as syntactic sugar, because it is already possible to use sets (e.g., <math>R \in sym</math>, where <math>sym \defi \{R \mid R=R ^{-1}\}</math>) or functions (e.g., <math>sym(R) = \True</math>, where <math>sym \defi (\lambda R \qdot R \in A \rel B \mid \bool(R = R^{-1}))</math>).
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| − | * Quantified expressions (e.g., <math>\sum x \qdot P \mid y</math>, <math>\prod x \qdot P \mid y</math>, <math>~\min(S)</math>, <math>~\max(S)</math>).
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| − | * Types.
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| − | ** Enumerated types.
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| − | ** Scalar types.
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| | | | |
| − | === User Input ===
| + | Such sequent are proved by PP and ML. But, these provers have both drawbacks : |
| − | The end-user shall provide the following information:
| + | *All the visible are added as needed hypotheses, which is most of the time not the case. |
| − | * keyboard input | + | *They take quite consequent time to prove it (even with the basic example given here above, the difference in time execution is noticeable). |
| − | * Lexicon and Syntax. <br/>More precisely, it includes the symbols, the form (prefix, infix, postfix), the grammar, associativity (left-associative or right associative), commutativity, priority, the mode (flattened or not), ... | + | *If there are too many hypotheses, or if the expression of the <math>x</math> is too complicated, they may not prove it. |
| − | * Pretty-print. <br/>Alternatively, the rendering may be determined from the notation parameters passed to the parser. | + | 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> |
| − | * Typing rules.
| + | <p> |
| − | * Well-definedness.
| + | 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. |
| | | | |
| − | === Development Requirements === | + | = Tactic = |
| − | * Scalability.
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| | | | |
| − | == Towards a generic AST == | + | 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><goal>\quad:\quad expresion\quad\in\quad<set></math> |
| | + | *<math><member>\quad:\quad expresion\quad\mid\quad FA</math> |
| | + | *<math><set>\quad:\quad expresion\otimes expresion\quad\mid\quad expresion\parallel expresion\quad\mid\quad <union>\quad\left[\cup <union>\right]</math> |
| | + | *<math><union>\quad:\quad<cprod>\quad\left[\cprod<cprod>\right]</math> |
| | + | *<math><cprod>\quad:\quad\left(<set>\right)\quad\mid\quad expresion</math> |
| | + | For example those examples matches with the definition here above : |
| | + | *<math>f(x)\in g\otimes h</math> |
| | + | *<math>x\in A\cprod\left(B\cup C\right)</math> |
| | + | Every other goals denoting a membership should be re- |
| | + | == Hypotheses == |
| | + | The hard part is to find the hypotheses leading to discharge the sequent. To do so, the tactic recovers three 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\}\subseteq\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> |
| | + | #the one from which we can infer inclusion : |
| | + | #*<math>f\in A\;op\;B \quad\vdash\quad f\subseteq A\cprod B\quad\bigl(</math> where <math>op</math> is one of :<math>\quad\rel, \trel, \srel, \strel, \pfun, \tfun, \pinj, \tinj, \psur, \tsur, \tbij\bigr)</math> |
| | + | #*<math>f\subseteq A\cprod B \quad\vdash\quad f\otimes f\subseteq A\cprod\left(B\cprod B\right)</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 <i>just</i> 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, <i>positions</i> 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> |
| | + | #*<math>x\in f,\quad f\otimes g\subseteq A\cprod\left(B\cprod C\right)\quad\vdash\quad x\in A\cprod B</math> |
| | + | #*<math>x\in f,\quad f\parallel g\subseteq \left(A\cprod C\right)\cprod\left(B\cprod D\right)\quad\vdash\quad x\in A\cprod B</math> |
| | + | #Those rewrites can be useful when the Russian dolls system does not succeed. |
| | + | #*<math>dom(f)\cup dom(g) = dom(f\ovl g)</math> |
| | + | #*<math>ran(f)\cup ran(g)\subseteq ran(f\ovl g)</math> |
| | + | #*<math>dom(A\domres f) = dom(f)\cap A</math> |
| | + | #*<math>ran(f\ranres A) = ran(f)\cap A</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B\quad\vdash\quad f\subseteq A\cprod B</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B\quad\vdash\quad f^{-1}\subseteq B\cprod A</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B\quad\vdash\quad f[A]\subseteq B</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B\quad\vdash\quad f\domres prj1 = prj1\ranres f\subseteq A\cprod B\cprod A</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B\quad\vdash\quad f\domres prj2 = prj2\ranres f\subseteq A\cprod B\cprod B</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B\quad\vdash\quad f\domres id = id\ranres f\subseteq (A\cprod B)\cprod(A\cprod B)</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B,\quad g\in B\;op_2\;C\;\lor\; g\subseteq B\cprod C \quad\vdash\quad f;g = g\circ f \subseteq A\cprod C</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B,\quad g\in A\;op_2\;C\;\lor\; g\subseteq A\cprod C \quad\vdash\quad f\otimes g \subseteq A\cprod (B\cprod C)</math> |
| | + | #*<math>f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B,\quad g\in C\;op_2\;D\;\lor\; g\subseteq C\cprod D \quad\vdash\quad f\parallel g \subseteq (A\cprod C)\cprod (B\cprod D)</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> |
| | | | |
| − | The following AST parts are to become generic, or at least parameterised:
| + | By using these inclusion and rewrites, it tries to find a path among the recovered hypotheses. Every hypotheses denoting an inclusion should be used only once. Every hypotheses from which new hypothesis can be inferred could be used as many times as wanted. 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. |
| − | * [[Constrained_Dynamic_Lexer | Lexer]]
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| − | * [[Constrained Dynamic Parser | Parser]]
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| − | * Nodes ( Formula class hierarchy ): parameters needed for:
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| − | ** Type Solve (type rule needed to synthesize the type)
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| − | ** Type Check (type rule needed to verify constraints on children types)
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| − | ** WD (WD predicate)
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| − | ** PrettyPrint (tag image + notation (prefix, infix, postfix) + needs parentheses (deduced from compatibility graph))
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| − | ** Visit Formula (getting children + visitor callback mechanism)
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| − | ** Rewrite Formula (associative formulæ have a specific flattening treatment)
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| − | * Types (Type class hierarchy): parameters needed for:
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| − | ** Building the type expression (type rule needed)
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| − | ** PrettyPrint (set operator image)
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| − | ** getting Base / Source / Target type (type rule needed)
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| − | * Verification of preconditions (see for example <tt>AssociativeExpression.checkPreconditions</tt>)
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| | | | |
| − | === Vocabulary === | + | = Reasoner = |
| | | | |
| − | An '''extension''' is to be understood as a single additional operator definition.
| + | This part describes how the reasoner works. |
| | | | |
| − | === Tags === | + | == Goal == |
| | + | The goal must match the same kind of goal as describes in the tactic part. |
| | | | |
| − | Every extension is associated with an integer tag, just like existing operators. Thus, questions arise about how to allocate new tags and how to deal with existing tags.<br />
| + | == Input == |
| − | The solution proposed here consists in keeping existing tags 'as is'. They are already defined and hard coded in the <tt>Formula</tt> class, so this choice is made with backward compatibility in mind.
| + | Because we do not want to do the job of re-writing, every added hypotheses used to discharge the sequent should be contained in the input (the reasoner must extend HypothesesReasoner) as well as hypotheses used to perform the re-writing and hypotheses used to discharge the sequent. Thus we get those obligations : |
| | + | *Every added hypotheses must be re-writable with hypotheses contained in the sequent and in the input. |
| | + | *Every added hypotheses should be used to discharge the sequent. |
| | + | *Hypotheses used to perform re-writing could be also used to discharge a sequent and may be used several times to prove that added hypotheses are re-writable. |
| | + | *Hypotheses used to discharge the sequent should be used only once in the path (no loop allowed). |
| | + | *It checks that with all these hypotheses can actually discharge the sequent |
| | | | |
| − | Now concerning extension tags, we will first introduce a few hypotheses:
| + | If all those tests pass, then the sequent is discharged. The added hypotheses are set as AddedHyps, and all the other hypotheses of the input are set as NeededHps. |
| − | * Tags_Hyp1: tags are never persisted across sessions on a given platform
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| − | * Tags_Hyp2: tags are never referenced for sharing purposes across various platforms
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| − | In other words, cross-platform/session formula references are always made through their ''String'' representation. These assumptions, which were already made and verified for Rodin before extensions, lead us to restrict further considerations to the scope of a single session on a single platform.
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| − | | |
| − | The following definitions hold at a given instant <math>t</math> and for the whole platform.<br />
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| − | Let <math>\mathit{EXTENSION}_t</math> be the set of extensions supported by the platform at instant <math>t</math>;<br /> let <math>\mathit{tag}_t</math> denote the affectation of tags to an extension at instant <math>t</math> (<math>\mathit{tag}_t \in \mathit{EXTENSION}_t \pfun \intg</math>);<br /> let <math>\mathit{COMMON}</math> be the set of existing tags defined by the <tt>Formula</tt> class (<math>\mathit{COMMON} \sub \intg</math>).<br /> The following requirements emerge:
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| − | * Tags_Req1: <math>\forall t \qdot \mathit{tag}_t \in \mathit{EXTENSION}_t \tinj \intg</math>
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| − | * Tags_Req2: <math>\forall e, t_1,t_2 \qdot \mathit{tag}_{t_1}(e)=\mathit{tag}_{t_2}(e)</math> where <math>t_1, t_2</math> are two instants during a given session
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| − | * Tags_Req3: <math>\forall t \qdot \ran(\mathit{tag}_t) \cap \mathit{COMMON} = \empty</math>
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| − | | |
| − | The above-mentioned scope-restricting hypothesis can be reformulated into: <math>\mathit{tag}</math> needs not be stable across sessions nor across platforms.
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| − | | |
| − | === Formula Factory ===
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| − | | |
| − | The requirements about tags give rise to a need for centralising the <math>\mathit{tag}</math> relation in order to enforce tag uniqueness.
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| − | The Formula Factory appears to be a convenient and logical candidate for playing this role. Each time an extension is used to make a formula, the factory is called and it can check whether its associated tag exists, create it if needed, then return the new extended formula while maintaining tag consistency.
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| − | | |
| − | The factory can also provide API for requests about tags and extensions: getting the tag from an extension and conversely.
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| − | | |
| − | We also need additional methods to create extended formulæ. A first problem to address is: which type should these methods return ?
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| − | We could define as many extended types as the common AST API does, namely <tt>ExtendedUnaryPredicate</tt>, <tt>ExtendedAssociativeExpression</tt>, and so on, but this would lead to a large number of new types to deal with (in visitors, filters, …), together with a constraint about which types extensions would be forced to fit into. It is thus preferable to have as few extended types as possible, but with as much parameterisation as can be. Considering that the two basic types <tt>Expression</tt> and <tt>Predicate</tt> have to be extensible, we come to add two extended types <tt>ExtendedExpression</tt> and <tt>ExtendedPredicate</tt>.
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| − | | |
| − | ExtendedExpression makeExtendedExpression( ? )
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| − | ExtendedPredicate makeExtendedPredicate( ? )
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| − | | |
| − | Second problem to address: which arguments should these methods take ?
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| − | Other factory methods take the tag, a collection of children where applicable, and a source location. In order to discuss what can be passed as argument to make extended formulæ, we have to recall that the <tt>make…</tt> factory methods have various kinds of clients, namely:
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| − | * parser
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| − | * POG
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| − | * provers
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| − | (other factory clients use the parsing or identifier utility methods: SC modules, indexers, …)
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| − | | |
| − | Thus, the arguments should be convenient for clients, depending on which information they have at hand.
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| − | The source location does not obviously seem to require any adaptation and can be taken as argument the same way. Concerning the tag, it depends on whether clients have a tag or an extension at hand. Both are intended to be easily retrieved from the factory. As a preliminary choice, we can go for the tag and adjust this decision when we know more about client convenience. | |
| − | | |
| − | As for children, the problem is more about their types. We want to be able to handle as many children as needed, and of all possible types. Looking to existing formula children configurations, we can find:
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| − | * expressions with predicate children: <math>\mathit{bool}(P)</math>
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| − | * expressions with expression children: <math>E_1 + E_2</math>
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| − | * predicates with predicate children: <math>P_1 \limp P_2</math>
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| − | * predicates with expression children: <math>\mathit{partition}(S, E_1, E_2)</math>
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| − | * mixed operators: <math>\{x \qdot P(x) \mid E(x)\}</math>, but it is worth noting that the possibility of introducing bound variables in extended formulæ is not established yet.
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| − | Thus, for the sake of generality, children of both types should be supported for both extended predicates and extended expressions.
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| − | | |
| − | ExtendedExpression makeExtendedExpression(int tag, Expression[] expressions, Predicate[] predicates, SourceLocation location)
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| − | ExtendedPredicate makeExtendedPredicate(int tag, Expression[] expressions, Predicate[] predicates, SourceLocation location)
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| − | | |
| − | == Defining Extensions ==
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| − | | |
| − | An extension is meant to contain every information and behaviour required by:
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| − | * Keyboard
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| − | * (Extensible Fonts ?)
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| − | * Lexer
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| − | * Parser
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| − | * AST
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| − | * Static Checker
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| − | * Proof Obligation Generator
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| − | * Provers
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| − | | |
| − | === Keyboard requirements ===
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| − | | |
| − | '''Kbd_req1''': an extension must provide an association combo/translation for every uncommon symbol involved in the notation.
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| − | | |
| − | === Lexer requirements ===
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| − | | |
| − | '''Lex_req1''': an extension must provide an association lexeme/token for every uncommon symbol involved in the notation.
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| − | | |
| − | === Parser requirements ===
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| − | | |
| − | According to the [[Constrained_Dynamic_Parser| Parser]] page, the following informations are required by the parser in order to be able to parse a formula containing a given extension.
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| − | | |
| − | * symbol compatibility
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| − | * group compatibility
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| − | * symbol precedence
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| − | * group precedence
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| − | * notation:
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| − | ** prefix, infix, …
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| − | ** configuration (sub-parsers, symbols and their relative position, child types)
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| − | | |
| − | === AST requirements ===
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| − | | |
| − | ==== Extended Formulæ ====
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| − | | |
| − | [[Image:AST_Formulae.png]]
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| − | | |
| − | ==== Extension API ====
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| − | | |
| − | Extensions are required to implement the following APIs in order to be supported in the AST. It is the expression of the missing information for a <tt>ExtendedExpression</tt> (resp. <tt>ExtendedPredicate</tt>) to behave as a <tt>Expression</tt> (resp. <tt>Predicate</tt>). It can also be viewed as the parametrization of the AST.
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| − | | |
| − | [[Image:AST_Extensions.png]]
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| − | | |
| − | ==== Mediator API ====
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| − | | |
| − | The mediators provided as arguments to these APIs have the following specification:
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| − | | |
| − | [[Image:AST_Mediators.png]]
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| − | | |
| − | ==== Example: Binary PLUS ====
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| − | | |
| − | public class BinaryPlus implements IExpressionExtension {
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| − |
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| − | public Type getType(ITypeMediator mediator, ExtendedExpression expression) {
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| − | final Type resultType = mediator.getFormulaFactory().makeIntegerType();
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| − | for (Expression child : expression.getChildExpressions()) {
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| − | final Type childType = child.getType();
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| − | if (!childType.equals(resultType)) {
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| − | return null;
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| − | }
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| − | }
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| − | return resultType;
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| − | }
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| − |
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| − | public Type typeCheck(ITypeCheckMediator mediator,
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| − | ExtendedExpression expression) {
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| − | final Type resultType = mediator.getFormulaFactory().makeIntegerType();
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| − | for (Expression child : expression.getChildExpressions()) {
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| − | mediator.sameType(child.getType(), resultType);
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| − | }
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| − | return resultType;
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| − | }
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| − |
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| − | public void checkPreconditions(Expression[] expressions,
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| − | Predicate[] predicates) {
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| − | Assert.isTrue(expressions.length >= 2);
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| − | Assert.isTrue(predicates.length == 0);
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| − | }
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| − |
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| − | public String getSyntaxSymbol() {
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| − | return "+";
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| − | }
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| − |
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| − | public Predicate getWDPredicate(IWDMediator mediator,
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| − | IExtendedFormula formula) {
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| − | return mediator.makeChildWDConjunction(formula);
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| − | }
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| − |
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| − | public void toString(IToStringMediator mediator, IExtendedFormula formula) {
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| − | final Expression[] childExpressions = formula.getChildExpressions();
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| − | mediator.append(childExpressions[0], false);
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| − | mediator.append(getSyntaxSymbol());
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| − | mediator.append(childExpressions[1], true);
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| − | }
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| − |
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| − | public boolean isFlattenable() {
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| − | return false;
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| − | }
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| − |
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| − | }
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| − | | |
| − | === Static Checker requirements ===
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| − | | |
| − | The static checker is involved in:
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| − | * static checking an extension definition (pre deployment)
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| − | : {{TODO}}
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| − | * static checking a model that uses extensions (post deployment);
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| − | : to that purpose, (deployed) extensions need to be placed in the build dependency graph before models that use them
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| − | : it must check that referenced (deployed) extensions are present in the project
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| − | : it needs to get the appropriate FormulaFactory for parsing (with referenced extensions enabled)
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| − | : it must be aware of extensions being added/deleted/replaced to reprocess dependent models
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| − | | |
| − | === Proof Obligation Generator requirements ===
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| − | | |
| − | Likewise, the POG may run on
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| − | * an extension definition (pre deployment)
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| − | : {{TODO}}
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| − | * a model that uses extensions (post deployment)
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| − | : nothing to do for WD POs (relies on the fact that extension WD is transmitted to the AST)
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| − | | |
| − | === Prover requirements ===
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| − | | |
| − | Theorems about extensions must be converted to additional inference rules.
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| − | | |
| − | {{TODO}}
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| − | | |
| − | === Extension compatibility issues ===
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| − | {{TODO}}
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| − | | |
| − | == User Input Summarization ==
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| − | | |
| − | Identified required data entail the following user input:
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| − | | |
| − | * '''id''':
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| − | :a String id
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| − | | |
| − | * '''type''':
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| − | :either EXPRESSION or PREDICATE
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| − | | |
| − | * '''keyboard''':
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| − | :a list of associations 'symbol <-> ASCII shortcut' for uncommon symbols
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| − | | |
| − | * '''notation''':
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| − | :a choice between
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| − | :* fixed-size:
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| − | ::* prefix: <math>\operatorname{op}(a,b,c)</math>
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| − | ::* infix: <math>a ~ \lozenge ~ b ~ \lozenge ~ c</math>
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| − | :: the number of children (3 here) is given
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| − | :: for each child, the type (''Expression'' or ''Predicate'') is given
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| − | :: only one operator symbol is provided
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| − | :or
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| − | :*variable-size notation:
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| − | ::* prefix: <math>\operatorname{op}(x_1,\ldots,x_n)</math>
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| − | ::* infix: <math>x_1 ~ \lozenge ~ \ldots ~ \lozenge ~ x_n</math>
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| − | :: the minimum number of children is given (for infix operators, the minimum must be 2 or more)
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| − | :: all children bear the same type, one of ''Expression'' or ''Predicate''
| |
| − | :: only one operator symbol is provided
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| − | | |
| − | * '''parsing''':
| |
| − | :compatibility:
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| − | :* a compatibility group id (an existing one or a new one)
| |
| − | :* a list of compatibility group ids compatible with this group
| |
| − | :* a list of other extension ids in this compatibility group compatible with this extension
| |
| − | | |
| − | :precedence:
| |
| − | :* a precedence group id (an existing one or a new one)
| |
| − | :* a list of precedence group ids with higher priority than this group
| |
| − | :* a list of precedence group ids with lower priority than this group
| |
| − | :* a list of other extension ids in this precedence group with higher priority than this extension
| |
| − | :* a list of other extension ids in this precedence group with lower priority than this extension
| |
| − | | |
| − | :As these informations have a global scope, we may think about more convenient ways to input/review them, avoiding a manual scanning of all individual extensions.
| |
| − | | |
| − | * '''type rule''':
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| − | :* a type predicate
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| − | :* (for EXPRESSION) a type expression
| |
| − |
| |
| − | * '''well-definedness''':
| |
| − | :a choice between:
| |
| − | :* the pre-defined WD condition: 'children WD conjunction'
| |
| − | :or
| |
| − | :* a custom well-definedness predicate
| |
| − | | |
| − | == Impact on other tools ==
| |
| − | | |
| − | Impacted plug-ins (use a factory to build formulæ):
| |
| − | * <tt>org.eventb.core</tt>
| |
| − | : In particular, the static checker and proof obligation generator are impacted.
| |
| − | * <tt>org.eventb.core.seqprover</tt>
| |
| − | * <tt>org.eventb.pp</tt>
| |
| − | * <tt>org.eventb.pptrans</tt>
| |
| − | * <tt>org.eventb.ui</tt>
| |
| − | | |
| − | == Identified Problems ==
| |
| − | The static checker shall enforce verifications to detect the following situations:
| |
| − | * Two mathematical extensions are not compatible (the extensions define symbols with the same name but with a different semantics).
| |
| − | * A mathematical extension is added to a model and there is a conflict between a symbol and an identifier.
| |
| − | * An identifier which conflicts with a symbol of a visible mathematical extension is added to a model.
| |
| − | | |
| − | Beyond that, the following situations are problematic:
| |
| − | * A formula has been written with a given parser configuration and is read with another parser configuration.
| |
| − | : As a consequence, it appears as necessary to remember the parser configuration.
| |
| − | : The static checker will then have a way to invalid the sources of conflicts (e.g., priority of operators, etc).
| |
| − | :: ''The static checker will then have a way to invalid the formula upon detecting a conflict (name clash, associativity change, semantic change...) [[User:Mathieu|mathieu]]
| |
| − | | |
| − | | |
| − | * A proof may free a quantified expression which is in conflict with a mathematical extension.
| |
| − | : SOLUTION #1: Renaming the conflicting identifiers in proofs?
| |
| − | | |
| − | * A theorem library changes: all proofs made using this library are potentially invalidated
| |
| − | : it is just as when a reasoner version changes, proof status should be updated to show broken proofs
| |
| − | | |
| − | == Open Questions ==
| |
| − | | |
| − | === New types ===
| |
| − | Which option should we prefer for new types?
| |
| − | * OPTION #1: Transparent mode.
| |
| − | :In transparent mode, it is always referred to the base type. As a consequence, the type conversion is implicitly supported (weak typing).
| |
| − | :For example, it is possible to define the <tt>DISTANCE</tt> and <tt>SPEED</tt> types, which are both derived from the <math>\intg</math> base type, and to multiply a value of the former type with a value of the latter type.
| |
| − | | |
| − | * OPTION #2: Opaque mode.
| |
| − | :In opaque mode, it is never referred to the base type. As a consequence, values of one type cannot be converted to another type (strong typing).
| |
| − | :Thus, the above multiplication is not allowed.
| |
| − | :This approach has at least two advantages:
| |
| − | :* Stronger type checking.
| |
| − | :* Better prover performances.
| |
| − | :It also has some disadvantages:
| |
| − | :* need of ''extractors'' to convert back to base types.
| |
| − | :* need of extra circuitry to allow things like <math>x:=d*2</math> where <math>x, d</math> are of type <tt>DISTANCE</tt>
| |
| − | | |
| − | * OPTION #3: Mixed mode.
| |
| − | :In mixed mode, the transparent mode is applied to scalar types and the opaque mode is applied to other types.
| |
| − | | |
| − | === Scope of the mathematical extensions ===
| |
| − | * OPTION #1: Project scope.
| |
| − | :The mathematical extensions are implicitly visible to all components of the project that has imported them.
| |
| − | * OPTION #2: Component scope.
| |
| − | :The mathematical extensions are only visible to the components that have explicitly imported them. However, note that this visibility is propagated through the hierarchy of contexts and machines (<tt>EXTENDS</tt>, <tt>SEES</tt> and <tt>REFINES</tt> clauses).
| |
| − | :An issue has been identified. Suppose that <tt>ext1</tt> extension is visible to component <tt>C1</tt> and <tt>ext2</tt> is visible to component <tt>C2</tt>, and there is no compatibility issue between <tt>ext1</tt> and <tt>ext2</tt>. It is not excluded that an identifier declared in <tt>C1</tt> conflict with a symbol in <tt>ext2</tt>. As a consequence, a global verification is required when adding a new mathematical extension.
| |
| − | | |
| − | == Bibliography ==
| |
| − | * J.R. Abrial, M.Butler, M.Schmalz, S.Hallerstede, L.Voisin, [http://deploy-eprints.ecs.soton.ac.uk/80 ''Proposals for Mathematical Extensions for Event-B''], 2009.
| |
| − | :This proposal consists in considering three kinds of extension:
| |
| − | # Extensions of set-theoretic expressions or predicates: example extensions of this kind consist in adding the transitive closure of relations or various ordered relations.
| |
| − | # Extensions of the library of theorems for predicates and operators.
| |
| − | # Extensions of the Set Theory itself through the definition of algebraic types such as lists or ordered trees using new set constructors.
| |
| | | | |
| | [[Category:Design proposal]] | | [[Category:Design proposal]] |
is too complicated, they may not prove it.
![<set>\quad:\quad expresion\otimes expresion\quad\mid\quad expresion\parallel expresion\quad\mid\quad <union>\quad\left[\cup <union>\right]](/images/math/5/2/7/52716b854ca9d6957fb9a1aba0c2fbed.png)
![<union>\quad:\quad<cprod>\quad\left[\cprod<cprod>\right]](/images/math/4/8/8/488b043c8999faf9503ba1f02d3a6444.png)
(this is the start point):
where 
is one of :
always represent an expression (may be a domain, a range, etc.).
.
.
where 
and 
are ones of :
![f\in A\;op_1\;B\;\lor\; f\subseteq A\cprod B\quad\vdash\quad f[A]\subseteq B](/images/math/d/0/f/d0f87e70ea31bc0378c96cb36a9b3453.png)
