| { | |
| "paper_id": "1991", | |
| "header": { | |
| "generated_with": "S2ORC 1.0.0", | |
| "date_generated": "2023-01-19T07:34:36.756131Z" | |
| }, | |
| "title": "PARSING 2-D LANGUAGES WITH POSITIONAL GRAMMARS", | |
| "authors": [ | |
| { | |
| "first": "Gennaro", | |
| "middle": [], | |
| "last": "Costagliola", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "University of Pittsburgh Pittsburgh", | |
| "location": { | |
| "postCode": "15260", | |
| "region": "PA" | |
| } | |
| }, | |
| "email": "" | |
| }, | |
| { | |
| "first": "Shi-Kuo", | |
| "middle": [], | |
| "last": "Chang", | |
| "suffix": "", | |
| "affiliation": { | |
| "laboratory": "", | |
| "institution": "University of Pittsburgh Pittsburgh", | |
| "location": { | |
| "postCode": "15260", | |
| "region": "PA" | |
| } | |
| }, | |
| "email": "" | |
| } | |
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| "year": "", | |
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| "abstract": "In this paper we will present a way 10 parse two-dimensional languages using LR parsing tables. To do this we describe two-dimensional (positional) grammars as a generalization of the context-free string grammars. The main idea behind this is to allow a traditional LR parser to choose the next symbol to parse from a two-dimensional space. Cases of ambiguity are analyzed and some ways to avoid them are presented. Finally, we consrruct a parser fo r the two-dimensional arithmetic expression language and implement it by using the tool Yacc.", | |
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| "text": "In this paper we will present a way 10 parse two-dimensional languages using LR parsing tables. To do this we describe two-dimensional (positional) grammars as a generalization of the context-free string grammars. The main idea behind this is to allow a traditional LR parser to choose the next symbol to parse from a two-dimensional space. Cases of ambiguity are analyzed and some ways to avoid them are presented. Finally, we consrruct a parser fo r the two-dimensional arithmetic expression language and implement it by using the tool Yacc.", | |
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| "text": "In this paper, we present an extension of a context-free grammar by explicitly describing the positional relations between the elements (terminals and non-terminals) in the right hand-side of each production rule of the grammar. As these relations can be very general, the resulting grammar can be seen as a generalization of Tomita's 2-D Chomsky Normal Form grammar where only horizontal and vertical relations are allowed.", | |
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| "section": "One of the latest approaches in parsing 2-D languages has been presented by Tomita in [37), where he introduces a 2-D Chomsky Normal Form grammar and constructs extensions to the two-dimensional case of Earley\u2022 s and LR parsing algo rithms.", | |
| "sec_num": null | |
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| "text": "The resulting parser for such a positional grammar is con structed by simply adding a column to the LR parsing table. This column contains the position of the next symbol to be shifted, for each state. Unlikely from the 2-D LR parsing algo rithms given in [37] , our parser slightly modifies the original LR parsing algorithm, so that the tool Y ace can be easily used to construct a two-dimensional parser for a positional gram", | |
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| "section": "One of the latest approaches in parsing 2-D languages has been presented by Tomita in [37), where he introduces a 2-D Chomsky Normal Form grammar and constructs extensions to the two-dimensional case of Earley\u2022 s and LR parsing algo rithms.", | |
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| "text": "Furthermore, we analyze cases of ambiguity, give some ways to avoid them and then present a general methodology to parse two-dimensional patterns applying it to the case of the two-dimensional arithmetic expressions.", | |
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| "section": "mar.", | |
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| "text": "tion. Each of them is based on the particular data structure used for representing the pictures: a string,-an array, a tree, a graph, and a plex.", | |
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| "section": "Many other approaches have been proposed till now in high dimensional syntactic pattern representation and recogni-", | |
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| "text": "One of the first approaches is given by a traditional string grammar in which more general relations (HOR, VER, ABOVE, LEFT, etc.), other than concatenation, are allowed among primitives in the pattern [2, 8, 16] . Shaw, by attaching a \"head\" and a \"tail\" to each primitive, has used four binary operators for defining binary concatenation relations between primitives. A context-free string grammar is used to generate the resulting Picture Description Language (POL) [16, 31] .", | |
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| "text": "Another interesting approach using a string grammar, has been given in [5] where each primitive has associated spatial attributes.", | |
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| "text": "A simple two-dimensional generalization of string gram mars is to extend grammars for one-dimensional strings to two-dimensional arrays [23, 28, 35, 38] . The primitives arc the array elements and the relation between primitives is the two dimensional concatenation.", | |
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| "text": "Pf alz and Rosenberg have extended the concept of string grammar to grammars for labeled graphs called webs [16, 17, 26, 27, 29] . These grammars were originally suggested as a syntactical formalism for data structure useful in image analysis. An application of graph languages for describing scenes is of frequent occurrence in the literature dealing with image processing, whereas the use of graph grammars for pat tern recognition is rare (for this purpose tree grammars are applied inste\ufffd [3, 17, 18, 22, 30, 32] ). Difficulties concerning building a syntax analyzcr for graph grammars are causes of \ufffdis situation. Recently, however, parsing methods _ for a par ocular kind of graph grammar have been proposed, and an efficient parsing, close to the parsing efficiency of tree languages, has been obtained [15, 21, 33) .", | |
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| "text": "Based on an idea in the work of Narasimhan [24] , Feder [14 ) has formalized a \"plex\" grammar which generates languages with terminals having an arbitrary number of attach ing points in order to connect to other primitives or sub patterns. The primitives of the plex grammar are called N Attaching Point Entities (NAPEs). Plex structures defined by a plex grammar may be viewed as a hypergraph, with each NAPE corresponding to a hypcredge. Therefore this kind of plex grammar is a more general model than that of graph gram-mar. Until recently, however, very little was known about the parsing method for plex grammars. Recently, a parsing method has been developed (25] to achieve more efficient parsing of plex grammars, by adapting Earley parsing algorithm, (13] .", | |
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| "text": "The paper is organized as follows. In Section 2 the posi tional grammar is defined, and some examples are given. In Section 3 the extension of the LR parser, named positional LR (pLR) parser, is prescnte.d along with a description of the pLR parsing tables and of the parsing algorithm. In Section 4 con siderations of ambiguity are given along with the construction of a pLR parser for the arithmetic expression grammar. In Sec tion 5 we present the general methodology for parsing 2-D languages generate.d by a positional grammar. The conclusions are in Section 6. The following definitions are understood to be with respect to a particular positional grammar G.", | |
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| "section": "Many other approaches have been proposed till now in high dimensional syntactic pattern representation and recogni-", | |
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| "text": "We write TT => l: if there exist \ufffd. r, A, 11 such that TT = r M, A \u2794 11 is a production and l: = r11\ufffd.", | |
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| "section": "POSITIONAL GRAMMARS", | |
| "sec_num": null | |
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| "text": "We write n =>* l: (l: is derived from TT) if there exist strings flo ,", | |
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| "section": "POSITIONAL GRAMMARS", | |
| "sec_num": null | |
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| "text": "Il 1 \u2022 \u2022 \u2022 n ,,. (m 2: 0) such that n =no=> n 1 => \u2022 .", | |
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| "section": "POSITIONAL GRAMMARS", | |
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| "text": ". => n ,,. = 1: The sequence flo . ... , n ,,. is called a derivation of l: from n. A positional sentential fo rm is a string n such that S =>* n. where pos is the spatial position of the token w[i ], and a stan ing index that points to the first token to parse. The association between a position and a token allows us to reach a token in w each time its spatial position has been given and viceversa. The input tape is, then, no longer required to be accessed sequentially but rather, according to the positional require ments given by the parser.", | |
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| "section": "POSITIONAL GRAMMARS", | |
| "sec_num": null | |
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| "text": "In this context, the definition of the sequential end-of string marker must be extented. In fact, the end-of-string marker hides an operational aspect: when parsed, it signals that no symbols to parse are left. While in a sequential scanning nothing must be done other than recognizing the '$' character, in a non-sequential scanning such operational aspect must be made explicit Before returning an end-of-input symbol, the scanner has to check whether all the symbols have been parsed.", | |
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| "section": "POSITIONAL GRAMMARS", | |
| "sec_num": null | |
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| "text": "In where o is the distance between each couple of dots. in \u2022 the column \" po sition\" can be considered as pointers to the code implementing the operators. As the construction of the \" po sition\" column does not affect the other entries of the original LR parsing table, we can use the traditional three techniques (with some variations) for having Simple pLR, canonical pLR and LookAhead pLR parsers.", | |
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| "section": "POSITIONAL GRAMMARS", | |
| "sec_num": null | |
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| "text": "A pLR(0) item of a positional grammar PG is a produc tion of PG with a dot at some po sition of the right side. A dot. however\ufffd can never be between a po sitional operator identifier and either a terminal or a non terminal, in this order. Thus, a production A \u2794 SP X REL 1 Y REL 2 Z yields the four items:", | |
| "cite_spans": [], | |
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| "section": "3.2) For the arithmetic expression grammar the operators", | |
| "sec_num": null | |
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| { | |
| "text": "A \u2794 .SP X REL 1 YREL 2 Z A \u2794 SP X .REL 1 Y REL 2 Z A \u2794 SP X REL 1 Y .REL 2 Z A \u2794 _ SP X REL . 1 YREL 2 Z.", | |
| "cite_spans": [], | |
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| "section": "3.2) For the arithmetic expression grammar the operators", | |
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| "text": "Intuitively, an item indicates how much of a production we have seen at a given point in the parsing process. For example, the first item above indicates that we hope next to see a pattern derivable from XYZ starting from position SP. The second item indicates that we have just seen on the input a pat tern derivable from X and that we hope next to .see a pattei:n derivable from YZ starting from the po sition specified by the operator associated to REL 1 .", | |
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| "section": "3.2) For the arithmetic expression grammar the operators", | |
| "sec_num": null | |
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| { | |
| "text": "If PG is a grammar with starting symbol S, then PG', the aug mented positional grammar for PG, is PG with a new starting symbol S' and production S' := SP S. array of tokens w, a starting index in w, and a list Q of couples (pos , i ) ; the specification of a set of positional operators, and the pLR parsing table with functions \"action\", \"goto\" and \"position\" for a positional grammar PG.", | |
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| { | |
| "start": 158, | |
| "end": 236, | |
| "text": "array of tokens w, a starting index in w, and a list Q of couples (pos , i )", | |
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| "section": "3.2) For the arithmetic expression grammar the operators", | |
| "sec_num": null | |
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| { | |
| "text": "Each time the pLr parser reaches a state in the recognition of the pattern, .the next symbol to parse is determined by using the positional operator associated to that state. As in LR pars ing, a same symbol cannot be considered more than once.", | |
| "cite_spans": [], | |
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| "section": "3.2) For the arithmetic expression grammar the operators", | |
| "sec_num": null | |
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| { | |
| "text": "Details on the Positional LR parsing algorithm can be found in (9, 10). Figure 3. 3 and applying the pLR parsing algo rithm, it can be verified that the following picture", | |
| "cite_spans": [], | |
| "ref_spans": [ | |
| { | |
| "start": 72, | |
| "end": 81, | |
| "text": "Figure 3.", | |
| "ref_id": null | |
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| "section": "3.2) For the arithmetic expression grammar the operators", | |
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| { | |
| "text": "=> \u25a1\u25a1 \u25a1\u25a1 is accepted. In particular, note that the parser drives the scanning of the input such that the first block is visited by columns, and the second block by rows, according to the productions of the grammar. All the other ways of scanning this input are not taken into consideration.", | |
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| "section": "\u25a1\u25a1 no", | |
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| "text": "In Section 2 we gave a two-dimensional version of the grammar given in [1] for arithmetic expressions. We will show now that this grammar is not pLR( 1) from the fact that it has conflicts regar\ufffding the position of the next symbol. Let us con sider the following pictorial form:", | |
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| "section": "AMBIGUITY CONSIDERATIONS In this Section we will show that conflicts in positions can lead to conflicts in the \"action\" part of the parsing table even if it has no multiple entries.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "T ' d -+, id", | |
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| "section": "AMBIGUITY CONSIDERATIONS In this Section we will show that conflicts in positions can lead to conflicts in the \"action\" part of the parsing table even if it has no multiple entries.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "assuming that T has already been reduced.", | |
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| "section": "AMBIGUITY CONSIDERATIONS In this Section we will show that conflicts in positions can lead to conflicts in the \"action\" part of the parsing table even if it has no multiple entries.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "After reducing T, the parser has to decide whether to choose 'hbar ' in vertical reading, or '+' in horizontal reading. Both the alternatives are valid: if 'hbar ' is chosen, then the parser has to shift, otherwise it has to reduce. One possibility for avoiding this conflict is to assign priority to each positional operator. In this example we could decide that the vertical reading has always higher priority than the horizontal one. This would respect the priority between 'hbar ' and '+' implicitly given in the grammar. But, if this other example is considered", | |
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| "section": "AMBIGUITY CONSIDERATIONS In this Section we will show that conflicts in positions can lead to conflicts in the \"action\" part of the parsing table even if it has no multiple entries.", | |
| "sec_num": null | |
| }, | |
| { | |
| "text": "the priority resolution will fail. In fact, in this case, after read ing T, we want to move horizontally because of the parenthesis, and not vertically. Another possibility for avoiding this conflict is to give a \"smart\" representation of the two-dimensional pattern deriving it from techniques of image analysis like dominancy (4, 12]. Last but not least, we can construct an equivalent pLR( 1) grammar as it is normally done for solving conflicts in LR parsers. Following these ideas, the pLR( 1) grammar for the arithmetic expressions has been constructed: The new grammar, then, has a particular section dedicated to the generation of the numerator of any division.", | |
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| "section": "...... ( T _+ _ id __ ) + id id", | |
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| "text": "During the recognition, this allows us to decide whether the expression to be parsed is the numerator of a division or not. In particular, the new terminals i and l mark the beginning and the end of any complex numerator, respectively, and the terminal kl. is the simple numerator.", | |
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| "text": "EQUATION", | |
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| "text": ".. IV. Construct the parser. Point I requires a general data structure to represent the original symbolic picture input. This can be a matrix of sym bols, or an iconic index, i. e., an analogous linear representa tion based on the projections of the symbols: the 2-D string as defined in [6] , or, for high dimensional symbolic patterns, the Gen_string, [ 11] . As the whole parsing model presented is extensible to the n-D case ( n >2) just considering positional relations and operators for the n-dimensional space, \ufffde will make use of the Gen_string iconic index. The characteristics of it and the algorithms to derive it from a high dimensional pattern are given in [ 11] . In the proposed implementation, each element of the Gen_string is a-token. A lexical analyzer to construct such a Gen_string can be obtained by using the same actions described above, but allowing the elements of the gen eral data structure (another Gen_string) to be elementary items or pixels.", | |
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| "start": 288, | |
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| "text": "[ 11]", | |
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| { | |
| "text": "Point II requires the construction of the pLR linear posi tional grammar along with the positional operators.", | |
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| { | |
| "text": "Point III requires routines for the conversion of the gen eral data structure into an array of tokens w, a starting index in w, SP, and an association list Q of positions and tokens. In par ticular the list Q must be implemented such that the positional operators can be executed efficiently.", | |
| "cite_spans": [], | |
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| { | |
| "text": "Finally, Point IV requires the construction of the parser. As a result of Theorem 7.1 in[9], this can be done by translat ing the positional LR grammar into an LR grammar with actions and then by using the tool Yacc, [20] .", | |
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| { | |
| "start": 217, | |
| "end": 221, | |
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| "sec_num": null | |
| }, | |
| { | |
| "text": "As an example of the construction given in that Theorem, let us consider the the positional LR grammar for the arith metic expressions. The resulting LR context free grammar with actions is: ", | |
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| "section": "16", | |
| "sec_num": null | |
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| { | |
| "text": "In this paper we constructed a parser for \u2022 a subclass of symbolic\u2022 pictorial languages. We showed that this class con tains the context-free string languages and th\ufffdt a complex language like the two-dimensional arithmetic ex p ression language can be parsed by the proposed model. -W \ufffd \ufffd showed that this class has a real nice property: the posStbility to be parsed in a very simple way by using an existing tool. ([7, 12, 19, 34] ).", | |
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| { | |
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| "eq_spans": [], | |
| "section": "CONCLUSIONS", | |
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| "text": "In the future we intend to extend the subclass of pictorial languages parseable by constructing more powerful parsers. A first approach regards the extension of the concept of symbol to an N-Attaching Point Entity as defined in [14] . \u2022 A second approach regards instead the possibility to have more than one positional relation between two symbols. In this way a symbol can be connected to non-adjacent symbols, too.", | |
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| "FIGREF0": { | |
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| "text": "The parser we are .going to present recognizes pictorial languages generate.d by positional grammars. Definition 2.1 A context-free positional grammar PG can be represente.d by a six-tuple (N, T, S, P, POS, PE) where: . N is a finite non-empty set of non-terminal symbols, T is a finite non-empty set of terminal symbols, NnT=0, S e N is the staning symbol, P is a finite set of productions POS is a finite set of positional relation identifiers POS n (N u T) = 0, PE is an evaluation rule Each production in P has the following form: m \ufffd 1 where A e N, each a i is in Nu T and each REl j is in POS. I Each positional relation REl i gives infonnation about the relative position of a i + 1 with respect to a; . In the following, the words \"positional grammar\" will always refer to a context free positional grammar. While in a string grammar the only possible positional relation_ is the string concatenation, in a positional grammar other positional relations can be define.d and then use.d for describing high dimensional languages. When parsing, this positional information will be useful for letting the scanner know where the next symbol to parse is. Some simple examples of positional relations on a Carte sian plane: String concatenation or adjacent horizontal concatenation AHOR = ( (p 1 , pi) : p 1 and p 2 are pictures horizontally concatenate.d with alignment of their centroids } Adjacent vertical concatenation A VER = ( (p 1 , pi) : p 1 and p 2 are pictures vertically con-catenate.d with alignment of their centroids } Upper horizontal concatenation UHOR = { (p 1 , pi) : p 1 and p 2 are pictures horizontally con catenate.d with alignment of the centroid of p 1 and the up-most element of p 2 } Horizontal concatenation HOR = { (p 1 , p 2 ) : p 1 and p 2 are pictures and location(p 1 ) = (x , y ) and location' (pi) = (x' , y' ) and the position (x' , y' ) is fe asible and x' > x } Vertical concatenation VER = { (p 1 , p 2 ) : p 1 and p 2 are pictures and location(p 1 ) = (x , y ) and location' (p 2 ) = (x' , y' ) and the position (x' , y' ) is fe asible and y' < y and x' s x } where a picture is a spatial arrangement of one or more . symbols, location(p) is a function returning the position of a symbol of the picture p and a feasible location is a location that has not been made unfeasible by another symbol or by the side effect of an evaluation rule, as it will be seen in the following. An evaluation rule PE is a function whose input is a string P 1 R EL 1P2 RE L 2 \u2022 \u2022 \u2022 RE L ,,. -1 P m m \ufffd p 2. ... , P m are dispose.d in the space such that <Pi , p \u2022 i + l ) e REl i The evaluation of the positional relations is meant to be sequential from left to right. As side effects can be generated for any evaluation, an evaluation rule is simple if no side effects are involved. I A possible side effect of the evaluation of a relation is co make certain positions in the space unfeasible. As the evalua tion is sequential, each evaluation inherites the side effects generate.d by the previous evaluations. Some examples of applications of the simple evaluation rule", | |
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| "FIGREF1": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "A positional sentence is a positional sentential fonn with only terminal symbols. A pictorial f onn is the evaluation of a positional sentential fonn. A picture is a pictorial fonn with only terminal symbols. The pictorial language defined by a positional grammar L(G) is the set of its pictures. Some examples of positional grammars: The following grammar generates the strings of the fonn a\u2022\u2022\u2022 ab \u2022 \u2022 \u2022 b with equal number of a's and b's. The positional operator . is defined as above. A posi tional sentence of this grammar is: a . a . a . b . b . b and the corresponding picture is: aaabbb. This example shows that every context-free string language can be represented by a positional grammar. The following grammar generates an upper-right corner with variable length of the edges. N = { Comer, IIl.,ine, VLine } T = (dot} S =Comer POS= (UHOR , AHOR ,AVER } PE is the simple evaluation rule p = { Comer := fil.,ine UHOR VLine IIl.,ine := fil.,ine AHOR dot I dot VLine := VLine A VER dot I dot } where UHOR , AHOR and AVER are defined as above. A positional sentence of this grammar is: dot AHOR dot AHOR dot AHOR dot UHOR dot AVER dot AVER dot AVER dot Replacing dot with the character '. ', the corresponding picture is: The following grammar generates two-dimensional arithmetic expressions using the binary operations addi tion and division: 237 N = {E, T, F} S =E T = { +, hbar , (, ), id} POS= (HOR , VER } PE is the evaluation rule defined below p = { E := E HOR + HOR TI T T := T VER hbar VER F I F F := \u2022 c HOR E HOR ) I id } The evaluation rule is so defined (see Figure 2. 1): PE(p 1 HOR p i.): The evaluation of HOR will give coordinates (x, y) to location(p 1 ) and (x', y') to location(p 2 ) such that (p 1 , p 2 ) e HOR . Moreover it will make unfeasible each position belonging to any of the following sets: { (x, y 1 ): y S y 1 Sm} {(x 1 , yi) : x < x 1 < x' and OS y 2 Sm} {(x', y3): y' Sy 3 Sm} where m \ufffd1 is an upper bound on the y-coordinate in the two-dimensional space. PE(p 1 VER p 2 ): The evaluation of VER will give coordinates (x, y) to location(p 1 ) and (x', y') to location(p 2 ) such that (p 1 , p 2 ) e VER . Moreover it will make unfeasible each position belonging to any of the following sets: {(x 1 ,y):0Sx 1 Sx} { (x 2 , y 1 ) : 0 S x 2 S x and y' < y 1 < y} {(x 3 , y') : 0 S x 3 S x' } {p 1 HOR P 2l and {p 1 VER P2l A positional sentence of this grammar is: id HOR + HOR ( HOR id HOR + HOR id HOR ) VER hbar VER id HOR + HOR id Replacing hbar with an horizontal bar, according to the definitions of HOR , VER and PE, there are many possible pic tures corresponding to the evaluation of this positional sen tence, but all of them can be mapped into the following one: . d (id + id ) . d l +...;.... -\ufffd+l id that is still a picture of this language. POSITIONAL LR PARSERS Positional LR parsers (pLR parsers) are nothing else but a generalization of the .LR parsers. The model of a pLR parser is given byThe model of a pLR Parser The input Ouput The input to a pLR parser is a spatial arrangement of tokens, or, in other words, a symbolic picture where each symbol is a token. Such an input is represented by an array w (the input tape) where eachtoken is stored, a list Q of couples (pos , i)", | |
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| "type_str": "figure", | |
| "num": null, | |
| "text": "a pLR parser, the end-of-input marking is implemented by storing the symbol '$' in location O of the input tape, and defining the end-of-iripur operator ANY as a function whose return value is O if all the symbols in the input tape have been parsed and 'error' otherwise. The positional operators For each positional relation we define a positional operator with the same name. Such an operator is a function that takes in input the index in the tape of the last token parsed, calcu lates a new position and then returns the index of the next token to parse, by consulting the list Q. Definition 3.1 Given a positional grammar PG= (N, T, S, P, POS, PE) and a relation REL e POS, then for all a,\ufffd e \u2022 Nu T such that \"a REL \ufffd\" occurs on the \u2022right hand-side of a production rule in PG, the corresponding positional operator REL is defined as follows: REL(i) = j iff i is the index in w of 'a\u2022, the last token parsed to reduce a, and j is the index in w of 'b', the first token to parse to reduce \ufffd-I Examples: In the grammar of Example 2.2, the corres po nding operators for POS can be defined as follows: UHOR(i) = AHOR(i) = j iff location(w[i]) = (x, y) and location(wUD = (x+o, y). A VER(i) = j iff location(w[i]) = (x, y) and location(w[j]) = ( x, y-0).", | |
| "uris": null | |
| }, | |
| "FIGREF3": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "HOR and VER can be defined as follow: HOR(i) =j iff location(wU]) is the highest spatial posi tion in the first non-empty column on the right of location(w[i]). VER(i) = j iff location(wUD is the spatial po sition on the left of location(w[i]) such that it is the leftmost position in the first non empty row below location(w[i]). The Positional LR Parsing Table Besides the \"action\" and \"goto\" \u2022 columns of . an LR parsing table, . the pLR parsing table contains an additional column called \" po sition\". The positional operators SP, ANY and the names of the positional operators are the elements of this new column. SP returns the staning index given in input with the picture and ANY is the operator defined above. All the names", | |
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| }, | |
| "FIGREF4": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "Let us consider the following positional grammar generat ing an horizontal concatenation of a block of squares, an arrow and another block of squares , (1) S := BI HOR '=> HOR B2 (2) BI := CHOR -C (3) C :=. square VER square (4) B2 := R VER R (5) R := square HOR square Here the definition of PE is as in Example 2.3. The canonical collection of sets of pLR(0) items for this grammar follows next, along with the position values. The goto function for this set of items is shown as the transition diagram of a determinis tic finite automaton in Figure 3.2 and the . resulting Positional LR parsing table is given in Figure 3.3. I O : S' := .SP S position[0] = { SP} S := .Bl HOR => HOR B2 BI := .C HOR C C := .square VER square / 1 : S' := SP S. /2 : S :=BI .HOR => HOR B2 / 3 : B 1 := C .HOR C C := .square VER square I 4 : C := square . VER square position[ l] = {ANY} position[2] = (HOR} position[3] = {HOR} position[4] = (VER} I 5 : S := B 1 HOR => .HOR B2 B2 := .R VER R \ufffd := .square HOR square I 6 : B 1 := C HOR C . 1 7 : C := square VER square . / s : S := B 1 HOR => HOR B2 . B2 := R . VER R R := .square HOR square / 1o : R := square _HOR square / 11 : R := square HOR square . f 1 2 : B2 := R VER R . position[5] = (HOR } position[6] = (HOR} position[?] = {HOR} position[8] = {ANY} position[9] = {VER } po sition[lO] = (HOR } po sition[ 11] = (VER.ANY } position[12] = {ANY } Note that in the construction of each closure, the posi tional operators HOR and VER are ignored by the dot. This information is instead caught by the po sition array. A Simple pLR parsing table Details on the algorithm for the construction of a Positional LR parsing table can be found in (9, 10]. The Positional LR Parsing Algorithm The pLR algorithm is a simple extension of Algorithm 4.7 in [ 1 ]; the only differences regard the form of the input and the setting of the pointer to the next symbol.The input is now given by a picture p represented by an", | |
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| }, | |
| "FIGREF5": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "Figure 3.4 shows the parsing action, goto and position of a canonical pLR parsing table for the following linear positional grammar for the vertical concatenation of two strings both of the type \"c \u2022 \u2022 \u2022 cd\". rule is simple when applyed to AHOR and defined as in Example 2.3 when applied to VER . Using the parsing table in Figure 3.4 and applying the pLR parsing algorithm, it can be verified that the Given the grammar in Example 0 using the parsing table in", | |
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| }, | |
| "FIGREF6": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "(0) E' := SP E (l) E := E HOR +HOR T F' := {HOR E HOR l (10) F':= kl.", | |
| "uris": null | |
| }, | |
| "FIGREF7": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "1 shows the resulting pLR( 1) parsing table for this grammar. Note that the terminals id , (, and ) have been duplicated as well as the non-terminals T and F. Moreover, rules (3), (4), (5) and (6) have been duplicated in rules (7), (8), (9) and (10).", | |
| "uris": null | |
| }, | |
| "FIGREF8": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "a general data structure to represent the two dimensional symbolic pictures. II. Define the positional relations and operators meant to relate objects in the patterns, and construct the pLR posi tional grammar, if possible, to describe the language. III. Convert the general data structure into the input to the parser as defined in Section 3.", | |
| "uris": null | |
| }, | |
| "FIGREF9": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "F':= kJ. (VER()} An implementation by Yacc for this grammar, using rhe Gen_string representation, has been developed at the Depart ment of Computer Science of the University of Pittsburgh.The implementation consists of the following:The function get _gs( ): the Gen_string representing a two-dimensional arithmetic expression is stored in a global data structure \"gs\". The Gen_string can be taken from a data base or derived from the original pattern.The function gs _ir( ): the Gen_string is converted into an internal representation (data structure \"spg\", and others).The functions read_hor() and read_ver(): the spatial operators HOR and VER are implemented, respectively. The yacc specifications for the grammar. the functions read_hor() and read_ ver() are insened in the rules as actions. Both of them update a global variable \"current\" used by the function yylex() to select the next token to be par\ufffded.In the following, the results of the execu\u2022 tion of such specifications are given. Note that the array \"spg\" represents the set of tokens occurring in the expression . while the values of \"cumnt\" give the order in which the tokens are parsed.For each token spg[i], the (x, y)coordinates are also given (the list Q). In this implementation x represents the column index in left-right progression, and y the row index in top-down pro gres", | |
| "uris": null | |
| }, | |
| "FIGREF10": { | |
| "type_str": "figure", | |
| "num": null, | |
| "text": "miversal parsers like Earl\ufffdy\u2022s ([13]) and Tomita's ([36]) algo rithms by applying the same technique used for extending the LR parser. Moreover we are considering applications of the model proposed to graphics and to \u2022 visual languages", | |
| "uris": null | |
| } | |
| } | |
| } | |
| } |