 |
Description  |
|
|
FIELD OF THE INVENTION
The present invention relates to a method and apparatus for determining whether a labelled, ordered parse tree can be extended to a valid tree and, also, to a method and apparatus for determining whether or not a labelled, ordered parse tree can
still be extended to a valid tree after at least one editing step.
BACKGROUND OF THE INVENTION
Documents can be described by internal computer structures in accordance with varying computer languages. For example, a document represented by one computer may use the Word Perfect language while a document represented by a different computer
may use the Microsoft Word language. In order to make document processing more efficient, it is often desirable to enable interchange between documents represented using different languages. Thus, there should be consistency between interrelated
documents.
In order to provide such consistency enabling interchange of documents, the Standard Generalized Markup Language (SGML) for example, was developed as an external representation making document interchange possible.
There are many editing applications wherein objects to be edited have hierarchical structures. Thus, any editing step must conform to a formal specification of possible content. The editing of documents described in SGML document type
definitions is a specific instance of structured document editing.
Structured document editing has been deemed by many users as an excessively restrictive way of constructing structured objects (e.g., programs or documents) because of the strong ordering constraints (i.e., top-bottom, and sometimes left-right,
as well) that grammar-based editors typically impose. Thus, a more relaxed version of syntax-directed editing is desirable. Often the class of hierarchical structures of interest can be made to correspond to incomplete parse trees of a particular
context-free grammar (set of rules), i.e., any tree remaining after having deleted zero or more proper subtrees of a parse tree of the same grammar.
For example, given the following grammar, it would be desirable to easily and efficiently determine whether a user could freely build a hierarchical structure;
Doc.fwdarw.(Front, Body, Back)
Body.fwdarw.(Intro, Mn. Sections.sup.+, App..sup.+)
A parse tree corresponding to the above grammar is illustrated in FIG. 1.
An SGML document class is defined by a document type definition or DTD (context-free grammar). In accordance with SGML, a document is classified as weakly valid if it can be made valid solely by the insertion of additional structural mark-up in
appropriate places. For example, if "Front" was missing from the parse tree of FIG. 1 as shown in FIG. 2, the document would be classified as weakly valid since the simple insertion of "Front" would validate the parse tree. Similarly, if "Body" was
missing from the parse tree of FIG. 1 as in FIG. 3, the document would also be classified as weakly valid since the insertion of begin and end tags around the [Intro, Sections.sup.+, Appendix] would result in a valid parse tree. A document is valid if
and only if the structure is a member of language defined by the DTD.
Thus, if, for example, a user wanted to add a node "x" beneath node "Body", it would be highly desirable to enable one to easily check for validity of this action, thus determining whether the action is legalized if some action can be taken. It
is further desirable to perform such a check in an arbitrary, convenient order to enable the user to freely work on document segments.
SUMMARY OF THE INVENTION
The method and apparatus of the present invention determines whether or not an SGML document is valid after at least one editing step. In addition, if the resulting document is weakly valid but not valid, then a possible completion of the
document is indicated that will make the document valid. In accordance with the present invention, the document is checked to see whether components are thus far legal according to structure compatibility. If any component is determined to not be
legal, the method and parsing/editing apparatus of the present invention look at the point of failure and determine what could be done to render the document legal.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be described in detail with reference to the following figures, wherein:
FIG. 1 is an exemplary parse tree corresponding to given grammar;
FIG. 2 illustrates an example of a horizontally gapped parse tree;
FIG. 3 illustrates an example of a vertically gapped parse tree;
FIG. 4 is a block diagram of an editing apparatus in accordance with the present invention;
FIG. 5 is an exemplary parse tree corresponding to an example grammar;
FIG. 6 is an exemplary horizontally gapped parse tree corresponding to an example grammar;
FIG. 7 is an exemplary vertically gapped parse tree corresponding to an example grammar;
FIG. 8 is a reachability graph with strongly connected components and ranks indicated for an example grammar; and
FIGS. 9-13 are exemplary finite state machines useful for weakly validating a parse tree for an example grammar.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENT
The use of document rules to form a document can be illustrated in the form of a parse tree, the parse tree having nodes labelled with symbols of the document rules (grammar productions). When the document is edited, the location of arcs and
nodes of the original parse tree change. The method and apparatus of the present invention enable rapid validation of the newly created parse tree as described herein below.
For a context-free grammar G=<V, .SIGMA., S, P>, where V is the set of symbols (both terminal and non-terminal) of the grammar (nodes), .SIGMA. is the set of terminal symbols of the grammar, S is the start symbol of the grammar and P is
the set of productions of the grammar with general regular expressions permitted for right-hand-sides of productions, T is assumed to be a parse tree of grammar G. A further assumption is made that the only productions of G having terminal symbols are of
the form A.fwdarw.a, wherein a is a terminal symbol.
For example, let the grammar be <V, .SIGMA., S, P> with:
v={A,B,C,D,E,F,G,H,P,Q,R,S,T,U,V,W,X,Y,Z,p,q,r,s,t,u,v,w,x,y,z}
.SIGMA.={p,q,r,s,t,u,v,w,x,y,z}
S=H
P=
{
H.fwdarw.A B C
A.fwdarw.(D A E A F) .vertline. P
B.fwdarw.(X G Y) .vertline. Q
C.fwdarw.Z
D.fwdarw.W
E.fwdarw.V
F.fwdarw.U
G.fwdarw.(T B S) .vertline. R
P.fwdarw.p
Q.fwdarw.q
R.fwdarw.r
S.fwdarw.s
T.fwdarw.t
U.fwdarw.u
V.fwdarw.v
W.fwdarw.w
X.fwdarw.x
Y.fwdarw.y
Z.fwdarw.z
}
A typical parse tree T for this grammar is in FIG. 5.
Upon editing, the parse tree T corresponding to grammar G becomes parse tree T', an ordered, labelled parse tree, having nodes labelled with elements of V, the root node of parse tree T' being labelled with S. An edited structured document can
have a corresponding parse tree which is "gapped". A gapped parse tree can be in the form of a horizontally gapped tree or a vertically gapped tree. FIG. 6 illustrates an example of a horizontal gapped tree, the gap being created with the deletion of
the entire subtree rooted in the topmost "A".
FIG. 7 illustrates a vertically gapped tree, the vertical gap being created with the deletion of the topmost "A" and putting its children in its place.
The ordered, labelled tree T' is a gapped tree of G if and only if there is a one-to-one mapping of the subtrees of T' into subtrees of a T such that any subtree of T' is mapped to a subtree of T having the same label on its root, and the mapping
preserves the horizontal and vertical ordering of nodes as the nodes appeared in tree T' and T is a valid parse tree of G.
A piecewise-structured editor for grammar G enables the transformation of one gapped tree of G into another such tree. While a gapped tree may not be a complete parse tree of G, the gapped tree can be made a complete parse tree purely by the
insertion of new nodes and arcs in the tree. Hence, various "gaps" in T' can be "filled" to yield a valid parse tree T. Thus, the editing is piecewise, as the user is permitted to create any gapped tree T' via insertions/deletions of arcs and nodes. In
order to make a piecewise-structured editing plausible, it must be efficiently determinable that a parse tree T' is a gapped tree of G. The method and apparatus of the present invention make this determination.
For example, given the example grammar, the complete parse tree illustrated in FIG. 5 could result.
The following definitions are provided as background of the present invention.
Definition 1
A gap grammar is a context-free grammar G=<V, .SIGMA., S, P> such that for every A contained in V-.SIGMA. (nonterminal symbols), A.fwdarw. empty is contained in P, where empty (.epsilon.) is the empty string. This captures the idea of a
horizontally-gapped tree.
Thus, for the preceeding example grammar, the following productions must be added to make it a gap grammar:
H.fwdarw..epsilon.
A.fwdarw..epsilon.
B.fwdarw..epsilon.
C.fwdarw..epsilon.
D.fwdarw..epsilon.
E.fwdarw..epsilon.
F.fwdarw..epsilon.
G.fwdarw..epsilon.
P.fwdarw..epsilon.
Q.fwdarw..epsilon.
R.fwdarw..epsilon.
S.fwdarw..epsilon.
T.fwdarw..epsilon.
U.fwdarw..epsilon.
V.fwdarw..epsilon.
W.fwdarw..epsilon.
X.fwdarw..epsilon.
Y.fwdarw..epsilon.
Z.fwdarw..epsilon.
Definition 2
axb.fwdarw.adb if there is a production x.fwdarw.y in G having d as a possible string in the language defined by the regular expression y. a.fwdarw.* b if either a=b or there is a c such that a.fwdarw.c and c.fwdarw.* b.
Definition 3
The reachability graph of the context-free grammar G=<V, .SIGMA., S, P> is a directed graph whose set of nodes is V and whose arcs are the set of all ordered pairs <x, y> such that x,y are in V and x.fwdarw.ayb is in P. For symbols
x,y in V, y is said to be reachable from x if there is a directed path from x to y of one or more arcs. This definition defines whether one node (e.g., y) is reachable from another node (e.g., x).
The reachability graph for the example grammar is shown in FIG. 8.
TABLE OF REACHABILITY FOR EXAMPLE GRAMMAR:
H can reach A B C D E F G P p Q q R r S s T t U u V v W w X x Y y Z z
A can reach A D E F P p U u V v W w
B can reach B G Q q R r S s T t X x Y y
C can reach Z z
D can reach W w
E can reach V v
F can reach U u
G can reach B G Q q R r S s T t X x Y y
P can reach p
Q can reach q
R can reach r
S can reach s
T can reach t
U can reach u
V can reach v
W can reach w
X can reach x
Y can reach y
Z can reach z
Definition 4
A strongly connected component of the reachability graph of G=<V, .SIGMA., S, P> is a set of nodes in the reachability graph of G with the property that all nodes in the set are reachable from all other nodes of the set, and no node of the
set is reachable from any node not in the set that is reachable from some node in the set.
Consider the partition of the reachability graph of G=<V, .SIGMA., S, P> into strongly connected components as illustrated in FIG. 8, for the example grammar. In FIG. 8, a set of regions are illustrated wherein a node located in one region
goes to a node in another region when the two nodes are connected by an arrow.
A strongly connected component having no outgoing (labelled) arcs has rank 0. A strongly connected component U having outgoing arcs has a rank one greater than that of the maximum of the ranks of the strongly connected components to which U is
connected by outgoing arcs (from U). The rank of x in V is the rank of the strongly connected component in which x is found. FIG. 8 also labels the ranks of the strongly connected components.
Definition 5
A symbol x in V of a context-free grammar G=<V, .SIGMA., S, P> is recursive if and only if it is reachable from itself. A recursive symbol S is multiply-recursive if there exists a production in P from a symbol R in the strongly connected
component containing S that can produce more than one instance of any symbol in the strongly connected component containing S. A recursive symbol that is not multiply-recursive is singly-recursive.
In the example grammar, A is multiply recursive and B and G are singly-recursive. All other symbols are not recursive.
Definition 6
Let x be a symbol of the context-free grammar G=<V, .SIGMA., S, P>. We associate with every symbol x a regular expression called gap(x). We inductively define the regular expression gap(x) as follows:
1. If x in .SIGMA. is a terminal symbol (of rank 0), gap(x)=(x).
2. If x in V-.SIGMA. is a non-recursive symbol of rank 0, gap(x)=(.epsilon..vertline.x).
3. If x in V is a non-recursive symbol of nonzero rank, gap(x)=(.epsilon..vertline.x.vertline.R1.vertline.R2 . . . ) where R1, R2 . . . are the right-hand-sides of productions for x with gap(w) substituted for each symbol w in the production.
4. If x in V is a multiply-recursive symbol, gap(x)=(x.vertline.S1.vertline.S2 . . . )* where S1, S2 . . . is the complete set of symbols reachable from x.
5. Let x in V be a singly-recursive symbol of G. Construct the sets of regular expressions L, M, and R as follows:
Define SPLIT(exp) as:
If SPLIT(exp) has already been evaluated for the same exp then return.
Otherwise, if exp does not contain any symbol in the same strongly connected component as x then add exp with gap(w) substituted for each symbol w in exp to the set M;
Otherwise, if exp is a symbol, then it must be in the same strongly connected component as x so do nothing.
Otherwise, if the top-level operator in exp is * (or STAR), then there is an error as x could not possibly be singly-recursive.
Otherwise, if the top-level operator in exp is .vertline. (or OR), then call SPLIT(alt) where alt is each of the alternatives of .vertline. (or OR) in turn.
Otherwise, if the top-level operator in exp is CONCATENATE (juxtaposition), then there must exactly one term of the CONCATENATE that contains a symbol that is in the strongly connected component containing x. Add all of the symbols in terms to
the left of this term to the set L; Add all of the symbols in terms to the right of this term to the set R; Call SPLIT(term) with the term containing a symbol that is in the strongly connected component containing x.
Otherwise, if the top level operator is + (or PLUS), then there is an error as x could not possibly be singly-recursive.
Otherwise, if the top level operator is ? (or OPTIONAL), rewrite the expression using OR and EMPTY (e.g., x? rewrites to (.epsilon..vertline.x) ). Then call SPLIT on the resulting expression.
End of definition for SPLIT.
Initialize L, M, and R to be empty sets;
Call SPLIT(RHS) for every production for x in G where RHS is the right hand side of the production;
Then left(x)=(L1 .vertline. L2 .vertline. . . . ) where L1, L2 . . . is the set of all symbols appearing in the set L or reachable from a symbol in the set L; middle(x)=(.epsilon..vertline.S1 .vertline.S2 . . . .vertline. M1 .vertline. M2
.vertline. . . . ) where S1, S2 . . . are the symbols in the strongly connected component containing x, M1, M2 is the set of regular expressions in the set M; right(x)=(R1 .vertline. R2 .vertline. . . . ) where R1, R2 . . . is the set of all symbols
appearing in the set R or reachable from a symbol in the set R. Then gap(x)=(left(x))* middle(x) (right(x))*.
For the example grammar, SPLIT(B) will construct the following sets:
L={T,X}
M={gap(Q),gap(R)}
R={S,Y}and
left(B)=(T.vertline.t.vertline.X.vertline.x)
middle(B)=(.epsilon..vertline.B.vertline.G.vertline.gap(Q).vertline.gap(R)) =(.epsilon..vertline.B.vertline.G.vertline.Q.vertline.q.vertline.R.vertline .r)
right(B)=(S.vertline.s.vertline.Y.vertline.y)
The value of gap(x) for all of the nonterminals in the example grammar are:
gap(H)=(.epsilon..vertline.H.vertline.gap(A)gap(B)gap(C))
gap(A)=(A.vertline.D.vertline.E.vertline.F.vertline.P.vertline.U.vertline.V .vertline.W.vertline.p.vertline.u.vertline.v.vertline.w)*
gap(B)=((T.vertline.t.vertline.X.vertline.x)*(.epsilon..vertline.B.vertline .G?51 gap(Q)gap(R))(S.vertline.s.vertline.Y.vertline.y)*)
gap(C)=(.epsilon..vertline.C.vertline.gap(Z))
gap(D)=(.epsilon..vertline.D.vertline.gap(W))
gap(E)=(.epsilon..vertline.E.vertline.gap(V))
gap(F)=(.epsilon..vertline.F.vertline.gap(U))
gap(G)=((T.vertline.t.vertline.X.vertline.x)*(.epsilon..vertline.B.vertline .G.vertline.gap(Q).vertline.gap(R))(S.vertline.s.vertline.Y.vertline.y)*)
gap(P)=(.epsilon..vertline.P.vertline.gap(p))
gap(Q)=(.epsilon..vertline.Q.vertline.gap(q))
gap(R)=(.epsilon..vertline.R.vertline.gap(r))
gap(S)=(.epsilon..vertline.S.vertline.gap(s))
gap(T)=(.epsilon..vertline.T.vertline.gap(t))
gap(U)=(.epsilon..vertline.U.vertline.gap(u))
gap(V)=(.epsilon..vertline.V.vertline.gap(v))
gap(W)=(.epsilon..vertline.W.vertline.gap(w))
gap(X)=(.epsilon..vertline.X.vertline.gap(x))
gap(Y)=(.epsilon..vertline.Y.vertline.gap(y))
gap(Z)=(.epsilon..vertline.Z.vertline.gap(z))
The value of gap(x) expanded and simplified:
gap(H)=(.epsilon..vertline.H.vertline.(A.vertline.D.vertline.E.vertline.F.v ertline.P.vertline.U.vertline.V.vertline.W.vertline.p.vertline.u.vertline.v .vertline.w)*(T.vertline.t.vertline.X.vertline.x.vertline.)*(.epsilon..vert
line.B.vertline.G.vertline.Q.vertline.q.vertline.R.vertline.r)(S.vertline.s .vertline.Y.vertline.y)*(.epsilon..vertline.C.vertline.Z.vertline.z))
gap(A)=(A.vertline.D.vertline.E.vertline.F.vertline.P.vertline.U.vertline.V .vertline.W.vertline.p.vertline.u.vertline.v.vertline.w)*
gap(B)=(T.vertline.t.vertline.X.vertline.x)*(.epsilon..vertline.B.vertline. G.vertline.Q.vertline.q.vertline.R.vertline.r)(S.vertline.s.vertline.Y.vert line.y)*
gap(C)=.epsilon..vertline.C.vertline.Z.vertline.z
gap(D)=.epsilon..vertline.D.vertline.W.vertline.w
gap(E)=.epsilon..vertline.E.vertline.V.vertline.v
gap(F)=.epsilon..vertline.F.vertline.U.vertline.u
gap(G)=(T.vertline.t.vertline.X.vertline.x)*(.epsilon..vertline.B.vertline. G.vertline.Q.vertline.q.vertline.R.vertline.r)(S.vertline.s.vertline.Y.vert line.y)*
gap(P)=.epsilon..vertline.P.vertline.p
gap(Q)=.epsilon..vertline.Q.vertline.q
gap(R)=.epsilon..vertline.R.vertline.r
gap(S)=.epsilon..vertline.S.vertline.s
gap(T)=.epsilon..vertline.T.vertline.t
gap(U)=.epsilon..vertline.U.vertline.u
gap(V)=.epsilon..vertline.V.vertline.v
gap(W)=.epsilon..vertline.W.vertline.w
gap(X)=.epsilon..vertline.X.vertline.x
gap(Y)=.epsilon..vertline.Y.vertline.y
gap(Z)=.epsilon..vertline.Z.vertline.z
As is well-known, a finite-state machine can be constructed for any regular expression that efficiently checks whether or not a given string is a member of the language defined by the regular expression [see paper by R. McNaughton and H. Yamada,
"Regular Expressions and State Graphs for Automata" in Sequential Machines: Selected Papers, Addison Wesley, 1964]. An exemplary finite state machine for recognizing weakly valid children of node H is in FIG. 9, A in FIG. 10, B in FIG. 11, C in FIG. 12,
and S in FIG. 13. In these diagrams, accept states are shown as double circles while fail states are shown as single circles. Theorem 1
Let G=<V, .SIGMA., S, P> be a gap grammar, A in V-.SIGMA. and a in V*. A.fwdarw.* a if and only if a is in gap(A).
In accordance with the present invention, the editor enables determination of whether tree T' is valid.
In a first embodiment, the validity of the entire tree is checked by the editor in accordance with the following:
Let R be a regular expression. It is well-known that the strings of a language defined by a regular expression can be recognized in deterministic time of the order of the string length. Thus, given a regular expression R, a procedure
recognize(R) is compiled which is parameterized by strings (ordered lists of symbols in V) S that return true if and only if S is in R where recognize(R) runs in deterministic time of order length(S). By virtue of the previous theorem, the deterministic
procedure below returns true if and only if T' is a gapped tree of G. Moreover, it runs in time linear in the size of the tree (gapped or not).
For each node labelled A in T' with immediate descendant nodes labelled B1, . . . ,Bk do
If recognize (R.sub.1 .vertline.R.sub.2 . . . ) (B.sub.1, . . . , B.sub.k), where R.sub.1, R.sub.2 . . . are the right-hand-sides of productions for A with gap(w) substituted for each symbol w in the right-hand-sides for A, then true else
return(false)
Return(true)
A may have no immediate descendants, e.g., B1, . . . ,Bk=.epsilon..
The finite state machines constructed by methods taught by the present invention determine whether or not the sub-structures of a document fed them as input can comprise an instance of a parse tree of a gap grammar. The names of the children of
each and every nonterminal node (in order, as a string) must be accepted by the finite state machine named by the parent node for the tree to be an instance of a parse tree of a gap grammar. If this condition holds, then the parse tree is that of a
weakly valid document, otherwise, it is the parse tree of neither a valid nor weakly valid document.
In accordance with another embodiment of the invention, the editor needs only check certain subtrees of tree T' created by the insertion or deletion of nodes and arcs in tree T. In this embodiment, for an inserted node, only the parent of a newly
inserted node with its new children and the newly inserted node and its children are checked. For a deleted node, only the parent of a newly deleted node with its new children are checked. This alternative procedure is performed in accordance with the
following:
parent(node) takes a node as a parameter and returns its parent node in the tree T';
childnames(node) takes a node as a parameter and returns the ordered list of child labels below the node in the tree T';
fsm(node) takes a node as a parameter and returns the finite-state machine (regular expression) corresponding to the regular expression (R.sub.1 .vertline.R.sub.2 . . . ) where R.sub.1, R.sub.2 . . . are the right-hand-sides of productions for
the label of that node with gap(w) substituted for each symbol w in the right-hand-sides;
For each new node n in T' do
If not(recognize(fsm(parent(n)))(childnames(parent(n)))) then return false
If not(recognize(fsm(n))(childnames(n))) then return false
For each parent node n (that still exists) of a deleted node in T' do
If not(recognize(fsm(n))(childnames(n))) then return false Return true.
The latter procedure runs in time proportional to the sum of the number of nodes below the old parents of inserted nodes and the number of nodes below the parents of deleted nodes.
For example, to check if the nodes below the node "H" of the tree in FIG. 7 are allowed in the parse tree of a weakly valid document, the labels of the children "DAEAFBC" are formed into a string and attempted to be recognized by the finite state
machine labelled by H in FIG. 9. Starting in state 1 the machine will make the following transitions for the given string:
state 1 scans D resulting in
state 2 scans A resulting in
state 2 scans E resulting in
state 2 scans A resulting in
state 2 scans F resulting in
state 2 scans B resulting in
state 4 scans C resulting in state 5.
Since state 5 is marked as an accept state (double circle) the sequence of children is possible in a parse tree of a weakly valid document.
Accordingly, the editor/parser of the present invention enables efficient determination of whether a given labelled, ordered tree can be extended to be a parse tree of given context-free grammar. The editor/parser can be a programmed computer
which determines whether a given parse tree can be extended to a valid parse tree. The determination can be made either in time linear to the number of nodes in the newly created tree or can be modified to make the determination in time linear to the
size of only the changed portion of the parse tree.
While this invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. For example, while the method and
apparatus are described in conjunction with the SGML grammar-based specification, the present invention could be applied to any document representation described by a formal context-free grammar-based specification, such as open document architecture
(ODA). Accordingly, the preferred embodiments of the invention as set forth herein are intended to be illustrative, not limiting. Various changes may be made without departing from the spirit and scope of the invention as defined in the following
claims.
* * * * *
|
|
 |
|
 |
Description |
|
