It is usual to represent the abstract structure of a document by a tree (derivation tree) and its model by an Extended Context Free grammars (ECFG)⁴. In an ECFG, the right member of each production is a regular expression as opposed to the sequence of terminal and non-terminal that constitute the right hand side of productions in classical context free grammar. More formally, an extended context free grammar is given by a finite set of syntactic categories, a finite set of production rules written as such that, and is a regular expression defined on.

The dependency graph of grammar is a graph whose set of node tags is included in and, for all rules in, there is an arrow from to, for all in a word belonging to the language denoted by and termed. An ECFG is said to be non recursive if and only if is acyclic, and recursive if not.

A document t conforms to a grammar and we write, if it is a derivation tree of this grammar: it’s the case if for any t node n labeled and with children nodes labeled respectively, , the word.

Let be an extended context free grammar, , , a view, t a derivation tree for (),the dependency graph of and a production rule.

is said to be finite type if and only if is non recursive. is said to be finite type with respect to if the restriction of dependency graph on symbols which belongs to is not recursive.

We note the t’s set nodes labels, and the t’s root node label.

The notation “” means that “p has the form”. We introduce function (resp.) which returns the symbol (resp. the symbols) in left hand side (resp. right hand side) of his argument p, a production rule. For example, if, (resp.). Also, means the substitution in the right hand side of p of all occurrences of each symbols by the corresponding. For example, with, ,.

is the language generated by grammar from symbol.

A grammatical structure is given as:

・ a set of non recursive grammatical symbols, and

・ a set of production rules. Each rule in has one of the two forms:

- (classical form of context free grammars rules),

- (i.e. is build up by a list of)

We recall that an equivalent ECFG can be evidently be derived from a grammatical structure.

Let be a grammatical structure, a view; let also be the complementary of in. The view projection on, termed is the grammatical structure where:

・ is obtained from by successive rewriting of symbols in in terms of those in, then, by substituting properly the result (of this rewriting) in the subset of rules having symbols in on the left hand side: or.

⁶For example a form of rule like can be obtained after successive rewriting of a rule; this is not an acceptable form of rule. So a new restructuring symbol is created and rule p is decompose in two new rules as follow and.

・ : syntactic categories of the projected grammar contains symbols of the view with enventually new symbols introduce for structuring purpose belonging to set. As the process of obtaining the production rules of the projected model proceed by successive rewriting of symbols which did not belong to the view, it can occur during the rewriting process of some symbols that, new symbols being added for format purpose (or decomposition) in order to bring some rules back to the form of the production rules adopted for the grammatical structures⁶ (cf. Section 3.1).

The algorithm for deriving and proceeds in two steps:

Step 1: consider the subset of’s rules which left hand side does not belongs to the view and transform

them by successive rewriting to rules like, an acceptable rule of grammatical structure, with, and containing only symbols. Hence set is given as:

Indeed, can be considered as production rules of a concrete context free grammar with as non terminal symbols and as terminal symbols; then.

From Equation (1), one easily deduces that is in fact the union of the rewriting of the productions of having a symbol belonging to in her left hand side. Thus, for every symbol belonging to, if we note the set obtained by rewriting rules of having as left hand side, we have with. Recall that, symbols in are considered as terminal symbols when rewriting.

Algorithm 1 describes the construction process of. Let’s emphasis that, for effective construction of, the different

Algorithm 1. Construction of.

sets should be built according to the topological sorting of the dependency graph: a symbol is evaluated after evaluation of symbols from which it depends.

⁷Note that.

Step 2: Consider the subset of’s rules, with view symbols in left hand side (⁷); for every rule in this set, replace all occurrences of elements in right hand side, by their right hand side counterpart in, this by all means; we finally obtain the set of production rules of the projected grammatical structure.

As for (Equation (1)), we deduce from Equation (2) that is the reunion of the sets obtained by rewriting the productions of having symbols belonging to in their left hand side, by using; that sets is denoted. Thus, with. The construction of is described in Algorithm 2 below.

Algorithm 3 purpose is to construct. It explicitly presents when restructuring symbols are created (line 5) and when they are explicitly used (line 5 and line 8) in generated productions rules.

Let be a grammatical structure, and a view; satisfies properties below:

Property 1: is a grammatical structure (stability property); this property is guaranteed by Algorithm 3.

Algorithm 2. Construction of.

Algorithm 3. Construction of.

Property 2: if then.

Property 3: if is a local update of a replica such that, then (consistency property).

We present below, the proof of the Property 2. The proof of Property 3 can be obtained from the proof of Theorem 3.3 given in [7] .

Proof. Let be such that; let’s show that.

In order to do this, if we consider an internal node n of labeled, with its k children, labeled; it suffices to show that the word belongs to the language denoted by the grammar, admitting the symbol axiom i.e..

Note that one can define a partition of so that, every tree (Figure 3(a)) can be uniquely partitioned into a finite set of maximal subtrees (Figure 3(b)) such as, for any subtree of the partition, either, and the labels of the successor nodes of the leaf nodes of in t if they exist do not belong to, or and the labels of the successor nodes of the leaf nodes of in t if they exist belong to. When, we say that is of type and when, we say that it is of type.

Considering the decomposition of t into subtrees of type and as described above (Figure 3), a node of t can be found either in a subtree of type or in a subtree of type. Moreover, by focusing on a node n of and his children, they can either: 1) all belong to the same subtree of type (Figure 4) or, 2) belong to different subtrees of type in t; in this case, n is a leaf in the subtree in which it appears, and the are labels of the root nodes (Figure 5) of other subtrees of type or, 3) n and some of his children are in the same subtree and the other are each one in their own subtree (Figure 6). Three case studies are therefore to be considered.

Case 1: belong to the same subtree such that. In this case, according to the construction algorithm of, and therefore to.

Case 2: Node n labeled is a leaf node of a subtree such that. Let be labels of m children of n in t. n has therefore been developed using a’s production rule of one of the two forms, with; or with. We develop below the second form, the treatment of the first being similar.

⁸Reminder: the non-terminals of are rewritten by the production rules of by considering the symbols of as terminals.

There is therefore m sub-terms of t, says, whose roots are respectively labeled by and such that. According to the

’s building process⁸, we can partition the word in m sub-words: with

As, according to the construction process of the productions rules of (modulo restructuring symbols).

Case 3: node n labeled is an internal node of a subtree such that and, there is some n’s children with labels not in. As previously, let’s termed labels of the children of n in t. n has therefore been developed using a production rule of the form in which at least one non-terminal belong to and at least one other belong to (Figure. 6). Let m be the number of non-terminals on the right-hand side of this production belonging to and named as. As before, there are m sub-terms of t, which we call, having respectively as their root labels nodes and such that. Similarly, according to the’s building process, we can partition the word in sub-words:

with and such that. As, , we have

and then, W

Some of the concepts and algorithms presented in the previous sections are illustrated in this section by considering a simplified case of cooperative writing of a small phone book.

Suppose that two employees of an organization want to cooperate in writing a phone book for their organization. One entry of the book is given by the name (Name), two first names (Fname1 and Fname2), the mails addresses (Emails) and phones numbers (Phones).

A corresponding grammatical structure describing this phone book is given in the Figure 7. Let us assume that there are two views: and for each of the respective employees. By applying the Algorithm 2, we have in Figure 8(a) (resp. Figure 8(b)), the grammatical structure (resp.) resulting from view (resp. view) projection on. Note that in, phone’ is a structuring symbol.