Graph Parsing as Graph Transformation

Abstract

Hyperedge replacement (HR) allows to define context-free graph languages, but parsing is NP-hard in the general case. Predictive top-down (PTD) is an efficient, backtrack-free parsing algorithm for subclasses of HR and contextual HR grammars, which has been described and implemented in earlier work, based on a representation of graphs and grammar productions as strings. In this paper, we define PTD parsers for HR grammars by graph transformation rules and prove that they are correct.

1 Introduction

Hyperedge replacement (HR, [8]) is one of the best-studied mechanisms for generating graphs. Being context-free, HR grammars inherit most of the favorable structural and computational properties of context-free string grammars. Unfortunately, simplicity of parsing is not one of these, as there are NP-complete HR languages [1, 14]. Hence, efficient parsing can only be done for suitable subclasses. The authors have devised predictive top-down (PTD, [4]) and predictive shift-reduce (PSR, [6]) parsing for subclasses of HR grammars and, in fact, for subclasses of contextual HR grammars (CHR grammars, [2, 3]), which are a modest extension of HR grammars that allows to overcome some of the structural limitations of HR languages.

Although the concepts and implementation of PTD parsers have been described at depth in [4], their correctness has not yet been formally established. We show in this paper how PTD parsing can be defined by graph transformation rules and use this in order to prove the correctness of PTD parsers. Our experience with the correctness proof for PSR parsing in [6] seems to indicate that a graph- and rule-based definition of parsers can make this task easier.

Related work on using graph transformation for defining parsers has dealt with LR string grammars [11] and two-level string grammars [12]. For a broader discussion of related work on parsing algorithms for graph grammars in general we refer to [6, Sect. 10.1].

The paper is structured as follows. After recalling graph transformation concepts (Sect. 2) and HR grammars (Sect. 3), we introduce threaded HR grammars (Sect. 4), which impose a total order on the edges of their derived graphs, which in turn induces a dependency relation on their nodes. In Sect. 5, we define a general top-down parser for HR grammars that respects edge order and node dependencies, and prove it correct. Since this parser is nondeterministic and hence inefficient, we introduce properties that make the parser predictive, and backtrack-free (Sect. 6) and show that this yields correct parsers that terminate for grammars without left recursion.¹ We conclude the paper by indicating some future work (Sect. 7).

2 Preliminaries

In this paper, $\mathbb {N}$ denotes the set of non-negative integers and [n] denotes $\{1,\dots ,n\}$ for all $n\in \mathbb {N}$. $A^*$ denotes the set of all finite sequences over a set A; the empty sequence is denoted by $\varepsilon$, and the length of a sequence $\alpha$ by $|\alpha |$. As usual, and denote the transitive and the transitive reflexive closure of a binary relation . For a function , its extension to sequences is defined by $f^*(a_1 \cdots a_n) = f(a_1) \cdots f(a_n)$, for all $n\in \mathbb {N}$ and $a_1,\dots ,a_n \in A$. The composition of functions and is denoted as $g\circ f$ and defined by $(g \circ f) (x) = g(f(x))$ for $x \in A$. The restriction of f to some subset $X \subseteq A$ is denoted as $f|_X$.

Definition 1

(Hypergraph). An alphabet $\varSigma$ is a finite set of symbols that comes with an arity function . A hypergraph (over $\varSigma$) is a tuple , where and are finite sets of nodes and hyperedges, respectively, the function attaches hyperedges to sequences of nodes, and the function labels hyperedges so that for every , i.e., the number of attached nodes of hyperedges is dictated by the arity of their labels.

$\mathcal {G}_\varSigma$ denotes the class of hypergraphs over $\varSigma$; denotes the empty hypergraph, with empty sets of nodes and hyperedges. A set of hyperedges induces the subgraph consisting of these hyperedges and their attached nodes.

For brevity, we omit the prefix “hyper” in the sequel. Instead of “ or ”, we often write “$x \in G$”. We often refer to the functions of a graph G by $att _G$ and $lab _G$. An edge carrying a label in an alphabet $\varSigma$ is also called a $\varSigma$-edge. And a node is called isolated if no edge is attached to it.

Definition 2

(Graph Morphism). Given graphs G and H, a graph morphism (morphism, for short) is a pair of functions and that preserve attachments and labels, i.e., and . The morphism is injective or surjective if both and are, and a subgraph inclusion of G in H if $m (x) = x$ for every $x \in G$; then we write $G \subseteq H$. If m is surjective and injective, it is called an isomorphism, and G and H are called isomorphic, written as .

For transforming graphs, we use the classical approach of [7], with injective matching and non-injective rules [9], but without rules that delete nodes.

Definition 3

(Rule). A graph transformation rule $r=(P, R, r^\circ )$ consists of a pattern graph P, a replacement graph R, and a mapping .² We briefly call r a rule and denote it as . An injective morphism into a graph G is a match of r, and r transforms G at m to a graph H as follows:

• Remove all edges m(e), , from G to obtain a graph K.

• Construct H from the disjoint union of K and R by identifying m(x) with $r^\circ (x)$ for every .

Then we write , but may omit m if it is irrelevant, and write if $\mathcal {R}$ is a set of rules such that for some $r \in \mathcal {R}$.

Sometimes it is necessary to restrict the application of a rule by requiring the existence or non-existence of certain graphs in the context of its match. Our definition of application conditions is based on [10].

Definition 4

(Conditional Rule). For a graph P, the set of conditions over P is defined inductively as follows: (i) a subgraph relation $P \subseteq C$ defines a basic condition $\exists C$ over P. (ii) if $c,c'$ are conditions over P, then $\lnot c$, $(c \wedge c')$, and $(c \vee c')$ are conditions over P.³

An injective morphism satisfies a condition c, written $m \vDash c$, if

• $c = \exists C$ and there is an injective morphism so that $m'|_P = m$;

• $c = \lnot c'$ and ;

• $c = (c' \wedge c'')$ and both $m\vDash c'$ and $m\vDash c''$;

• $c = (c' \vee c'')$ and $m\vDash c'$ or $m\vDash c''$.

A conditional rule $r'=(r,c)$ consists of a rule and a condition c over P, and is denoted as . We let if $m \vDash c$ and . Note that each rule without a condition can also be seen as a conditional rule . If ${\mathcal {C}}$ is a finite set of conditional rules, denotes the conditional transformation relation using these rules.

Examples of graphs and rules, with and without conditions, will be shown below.

3 Hyperedge Replacement Graph Grammars

We recall graph grammars based on hyperedge replacement [8].⁴

Definition 5

(Hyperedge Replacement Grammar). Consider a finite alphabet $\varSigma$ and a subset $\mathcal {N}\subseteq \varSigma$ of nonterminals. Edges with labels in $\mathcal {N}$ are accordingly nonterminal edges; those with labels in $\varSigma \setminus \mathcal {N}$ are terminal edges.

A rule is a hyperedge replacement production (production, for short) over $\varSigma$ if the pattern P consists of a single edge e and its attached nodes, where $lab _P(e) \in \mathcal {N}$, and the mapping is injective.

A hyperedge-replacement grammar (HR grammar) consists of $\varSigma$ and $\mathcal {N}\subseteq \varSigma$ as above, a finite set $\mathcal {P}$ of productions over $\varSigma$, and a start graph $Z \in \mathcal {G}_\varSigma$.

The language generated by $\varGamma$ is given by .

Example 1

(HR Grammars for Trees). As a running example for the constructions in this paper, we use the productions in Fig. 1. They derive n-ary trees like the one in Fig. 2, if the pattern of production is the start graph. We draw nodes as circles, and nonterminal edges as boxes that contain their labels. Edges are connected to their attached nodes by lines, called tentacles. Tentacles are ordered counter-clockwise around the edge, starting in the north.

Fig. 1.

HR productions for trees

Fig. 2.

A tree

For the purpose of this paper, we restrict ourselves to this simple example because illustrations would otherwise become too complex. Further examples of well-known HR languages for which PTD parsers can be built include string graph languages such as palindromes, non-context-free ones like $a^nb^nc^n$, arithmetic expressions, and Nassi-Shneiderman diagrams.

In our running example, edges of shape with designate root nodes, whereas edges of shape with connect parent nodes to their children.

In productions (and later in other rules), nodes of the pattern P have the same identifier ascribed in P as their images in R under $p^\circ$, like x in our example. In the following, the letters , and under the arrows in Fig. 1 are used as identifiers that refer to the corresponding production.

Assumption 1

Throughout the remainder of this paper, we consider only HR grammars that satisfy the following conditions:

• 1. Z consists of a single edge e of arity 0.

• 2. ${\mathcal {L}}(\varGamma )$ does not contain graphs with isolated nodes.

These assumptions imply no loss of generality: a new initial nonterminal with a single start production according to Assumption 1 can be added easily. A grammar that violates Assumption 1 and produces isolated nodes can be transformed easily into an equivalent grammar that attaches virtual unary edges to those nodes.

4 Threaded HR Grammars

We now prepare HR grammars for parsing. The edges in graphs, productions and derivations will be ordered linearly with the idea that the parser is instructed to process the symbols of a grammar in this order when it attempts to construct a derivation for a given input graph. The edge order induces a dependency relation between nodes of a graph as follows: for an edge, an attached node is “known” if it is also attached to some preceeding edge, which will be processed earlier by the parser; it is “unknown” otherwise. This defines what we call the profile of an edge: a node is classified as incoming if it is known, and as outgoing otherwise.

Technically, edge order and profiles are represented by extending the structure and labels of a graph: Every edge is equipped with two additional tentacles by which edges are connected to a thread, and the label $\ell$ of an edge is equipped with a profile $\nu \subseteq \mathbb {N}$ indicating the positions of its incoming nodes. Unary hyperedges labeled with a fresh symbol distinguish thread nodes from kernel nodes of a graph.

Definition 6

(Threaded Graph). The profiled alphabet of an alphabet $\varSigma$ is with and . The profile of an edge labelled by is $\nu$.

Let . A node is called a thread node if a -edge is attached to it and a kernel node otherwise. and denote the sets of all kernel nodes and thread nodes of G, respectively. An edge is a profiled edge if . The set of all profiled edges of G is denoted by . The profile $\nu$ divides the set of attached kernel nodes of e into sets $\textit{in}_G(e) = \{ v_i \mid i \in \nu \}$ and $\textit{out}_G(e) = \{ v_i \mid i\in [ arity ( lab _G(e))] \setminus \nu \}$ of incoming and outgoing nodes, respectively.

A graph is threaded if the following hold:

• 1. Each node of G has at most one attached -edge.

• 2. For every with $lab _G(e) = \ell ^\nu$ and $att _G(e) = v_1 \dots v_k v_{k+1} v_{k+2}$, the nodes $v_1,\dots ,v_k$ are kernel nodes of G and $v_{k+1},v_{k+2}$ are thread nodes of G. (Hence, $\textit{in}_G(e)$ and $\textit{out}_G(e)$ partition the kernel nodes of e into incoming and outgoing nodes.)

• 3. The profiled edges and thread nodes of G can be ordered as and so that, for $i\in [n]$,

  • (a) $att _G(e_i)$ ends in $v_{i-1} v_i$ and

  • (b) no edge $e_j$ with $j\in [i-1]$ is attached to any node in $\textit{out}_G(e_i)$.

We call $v_0$ the first and $v_n$ the last thread node of G, and define furthermore .

The kernel graph of G is the graph obtained by removing the profiles of edge labels, the -edges, the thread nodes and their attached tentacles. denotes the set of threaded graphs over ; denotes the empty threaded graph that consists of a single thread node with its attached -edge.

Remark 1

It is important to note that the profiles of the (profiled) edges of a threaded graph G are uniquely determined by $\textit{in}(G)$ and the structure of G. To see this, let , threaded in this order. For every , let

$$ first (v)=\left\{ \begin{array}{ll} 0 &{}\text {if } v\in \textit{in}(G)\\ i &{}\text {if } v\notin \textit{in}(G) \text { and } i=\min \{j\in [n]\mid att _G(e_j) \text { contains } v\}. \end{array}\right. $$

Then $v\in \textit{in}_G(e_i)$ if $v\in att _G(e_i)$ and $first (v)<i$.

Let the concatenation $H = G \circ G'$ of two threaded graphs G and $G'$ with be the threaded graph H that is constructed from the union of G and $G'$ by identifying the last thread node of G with the first thread node of $G'$ (and removing one of their attached -edges). Note that kernel nodes of G may also occur in $G'$.

Definition 7

(Threaded Production and HR grammar). A rule is a threaded production if P and R are threaded and the following conditions are satisfied:

• 1. the rule , where is the restriction of $p^\circ$ to , is a production, called kernel production of p,

• 2. $p^\circ$ maps the first and last thread nodes of P onto the first and last thread nodes of R, respectively, and

• 3. $p^\circ (\textit{in}(P))=\textit{in}(R)$.

An application of a threaded production p to a threaded graph G is called leftmost, written , if it replaces the first nonterminal on the thread of G.

A HR grammar over a profiled alphabet is threaded if all its productions are threaded.

As in the case of context-free string grammars, the context-freeness of hyperedge replacement implies that derivations can be restricted to leftmost ones:

Fact 1

For every threaded HR grammar and every , there is a leftmost derivation , i.e., a derivation in which all applications of productions are leftmost.

This fact will be important, as top-down parsers for HR grammars attempt to construct leftmost derivations of a graph.

It follows from Remark 1 and condition 3 of Definition 7 that the profiles of edges in the replacement graph of a threaded production are uniquely determined by the profile of the pattern. Hence, given a HR grammar and an order on for each of its productions , a unique threaded version of $\varGamma$ is obtained as follows:

• 1. The threaded start graph of is given by (recall that $arity (Z)=0$).

• 2. Every production of $\varGamma$ is turned into all threaded productions where , , and the edges of are threaded according to the chosen order on (which defines the profiles of edges in uniquely).

While the procedure above creates an exponential number of profiles and thus productions, in most cases many of them will be useless. A more efficient way of constructing is thus to choose the threading order and then construct the useful threaded productions inductively. The procedure would initially construct the threaded start production (in which $\textit{in}(P)=\varnothing$) and then, as long as a replacement graph of one of the constructed productions contains a hitherto unseen profiled nonterminal, continue by constructing the threaded productions for this nonterminal. This leads to the following definition:

Definition 8

(Threaded Version of a HR Grammar). Let be a HR grammar. A threaded version of $\varGamma$ is a threaded grammar , such that

• 1. and ,

• 2. all threaded productions with the same kernel production order the edges of R identically, and

• 3. is reduced, i.e., every production can participate in the generation of a graph in : there is a derivation such that .

Fig. 3.

Threaded tree productions

Example 2

(Threaded Tree Grammar). We consider a threaded version of the tree grammar, given by the threaded productions in Fig. 3. In examples such as this one we draw thread nodes in gray and omit the attached -edges, and we write profiles as ascending sequences of numbers rather than as sets. The profiles of profiled terminal edges are inscribed into the label symbols, i.e., for and for Moreover, we distinguish threaded productions with the same kernel productions by the profile of the (unique edge in the) pattern in the production name. The profiled symbols , , , , and do not appear as they occur only in useless productions.

It is worthwhile to note that production merges thread nodes t and n, which we indicate in the drawing by annotating the corresponding node in the replacement graph with “$t{=}n$”.

We arrange thread nodes from left to right and draw thread tentacles in gray so that the kernel graph can be better identified. To make it easier to distinguish incoming from outgoing attached nodes, we draw the former to the left of an edge and the latter to the right of it.

In production , left-recursion was avoided by choosing the terminal edge to be the first one on the thread. Figure 4 shows a threaded derivation of the tree in Fig. 2, which is leftmost.

Fig. 4.

A leftmost threaded derivation of the tree in Fig. 2

Threaded productions derive threaded graphs to threaded graphs.

Fact 2

If and G is a threaded graph, H is a threaded graph as well, and $\textit{in}(H)=\textit{in}(G)$.

Threaded derivations and unthreaded ones correspond to each other.

Lemma 1

Let be a HR grammar, a threaded version of $\varGamma$, and G a threaded graph such that . Then it holds for all graphs $G'$ that if and only if there is a threaded graph H with and .

Thus the threaded and unthreaded version of a HR grammar generate the same language of kernel graphs.

Theorem 1

If is a HR grammar and is a threaded version of $\varGamma$, then .

Proof

Easy induction on the length of derivations, using Lemma 1.   $\square$

5 General Top-Down Parsing for HR Grammars

We define top-down parsers for HR grammars as stack automata, which perform transitions of configurations that represent the input graph and a stack. Configurations are graphs, and transitions are described by graph transformation rules. This definition is more precise than the original definition of PTD parsing in [4], but avoids the technical complications occuring in the precise definition of PSR parsing for HR grammars [6], where graphs are represented textually as sequences of literals, and transitions are defined by the transformation of literal sequences, involving substitution and renaming operations on node identifiers. The use of graph transformation and graph morphisms avoids the explicit handling of these technical issues.

A configuration consists of a threaded graph as in Definition 6, which represents its stack and its read input, edges without profile that induce its unread input, and further edges that serve as flags, distinguishing different types of nodes.

Definition 9

(Configuration). Given a HR grammar and its profiled alphabet , let , $\otimes$, and be fresh symbols of arity 1. A graph G without isolated nodes is a configuration (of $\varGamma$) if the following hold:

• The subgraph induced by its -edges is a threaded graph.

• Exactly one thread node h of is attached to a -edge, representing the top of the stack.

• Every kernel node of every profiled edge between the start node of the thread and h is attached to a -edge, marking it as read.

• Every node of every $\varSigma$-edge that is not attached to a profiled edge at the same time is attached to a $\otimes$-edge, marking it as unread.

• No node is attached to several edges with labels in .

We let , the read input, denote the subgraph of induced by the profiled edges between the first thread node and h (including the -edges attached to those nodes). The (threaded) subgraph of induced by the profiled edges between h and the last node of the thread (again including the -edges attached to those nodes) represents the stack , and the subgraph induced by the $\varSigma$-edges represents the unread input. The union of and the kernel of is the input represented by G, denoted by .

A configuration G is

• initial if and , and

• accepting if and .

Definition 10

(Top-Down Parser). Let $\varGamma$ be a HR grammar and $\mathcal {R}$ a set of conditional rules. A derivation is a parse if G is an initial configuration. A parse is successful if H is an accepting configuration. A configuration G is promising (with respect to $\mathcal {R}$) if there is an accepting configuration H so that . $\mathcal {R}$ is a top-down parser for $\varGamma$ if, for each initial configuration G, if and only if G is promising. $\mathcal {R}$ terminates if there is no infinite parse.

Consider in the following a threaded version of a HR grammar . We define two types of general top-down parsing rules, called match and expand rules.

Definition 11

(Match and Expand Rules). For every terminal symbol , the match rule is given as follows:

• The pattern P is a configuration where

  • ,

  • consists of one a-edge with $a\in \varSigma \setminus \mathcal {N}$ and (where $arity (a)=k$), with a -edge attached to every with $i \in \nu$ and a $\otimes$-edge attached to every with $i \not \in \nu$, and

  • consists of one $a^\nu$-edge e with $att _P(e)=v_1 \dots v_k v_{k+1}v_{k+2}$ such that if $i \in \nu$. If $i\notin \nu$, then $v_i$ is not attached to .

• The replacement R is a configuration where

  • , with a -edge attached to every $v_i$, for $i\in [k]$,

  • ,

  • .

• The mapping $t^\circ _{a^\nu }$ identifies node $v_i$ with if and only if $i \notin \nu$.

For each of the threaded productions in , the expand rule is given as follows:

• ,

• ,

• and ,

• the mapping $t^\circ _p$ is the same as in p;

We let $\mathcal {R}^\mathrm {M}_{\tilde{\varGamma }}$ denote the set of all match rules for terminal symbols, and $\mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$ the set of all expand rules for productions of ${\tilde{\varGamma }}$. In the following, we will show that $\mathcal {R}_{\tilde{\varGamma }}= \mathcal {R}^\mathrm {M}_{\tilde{\varGamma }}\cup \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$ is in fact a top-down parser for $\varGamma$, hence we call a general top-down parser of ${\tilde{\varGamma }}$ (for $\varGamma$).

Example 3

(General Top-Down Parser for Trees). The expand rules of the general top-down parser for trees in Fig. 5 differ from the threaded productions only in the -edge marking the top of the stack. (We draw - and -edges around the nodes to which they are attached, so that they look like distinguished kinds of nodes. Nodes with an attached $\otimes$-edge are drawn as $\otimes$, omitting the attached edge in the drawing.) The match rules for the two edge patterns needed are shown in Fig. 6.

Fig. 5.

Expand rules of the general top-down parser for trees

Fig. 6.

Two match rules of the general top-down parser for trees

Fig. 7.

A top-down parse of the tree in Fig. 2

Figure 7 shows snapshots of a successful parse of the tree in Fig. 2 with these rules, where five configurations are omitted for brevity. The parse constructs the leftmost derivation in Fig. 4.

Note that match rules do not change the thread, but just “move” the matched terminal from the unread to the read subgraph of the configuration. In contrast, expand rules do not modify the unread or read subgraphs of the configuration, but just replace the first nonterminal on the thread by the replacement graph of a threaded production for this nonterminal. We can summarize these observations in the following fact:

Fact 3

For a parse (where ), the following hold:

• 1. ;

• 2. if $r = t_{a^\nu }$ is a match for some $a \in \varSigma \setminus \mathcal {N}$, then ;

• 3. if $r = t_p$ for some threaded production , then .

Thus constitutes a top-down parser: there is a successful parse if and only if its input graph is in the language of the grammar.

Theorem 2

For every HR grammar $\varGamma$ and each threaded version ${\tilde{\varGamma }}$ of $\varGamma$, $\mathcal {R}_{\tilde{\varGamma }}$ is a top-down parser for $\varGamma$.

Proof Sketch

Let be a successful parse. and hold by Fact 3; is the kernel of because H is accepting, and hence by Lemma 1.

In order to show the opposite direction, let us consider any configuration G with terminal read input and $H'$ a terminal threaded graph with kernel . It is easy to prove, by induction on the length of the derivation, that implies where H is an accepting configuration obtained from by adding a -edge to the last thread node and -edges to all kernel nodes, i.e., G is promising. Now let G be an initial configuration with . By Lemma 1, there is a threaded graph $H'$ with kernel and . Hence, G must be promising.    $\square$

If is not left-recursive, the general top-down parser terminates. Here, we say that is left-recursive if there is a threaded graph G consisting of a single nonterminal edge labeled A (for some nonterminal A) and there is a derivation for some graph H such that the first profiled edge of H is also labeled with A.

Theorem 3

(Termination). Let ${\tilde{\varGamma }}$ be a threaded version of a HR grammar. The general top-down parser $\mathcal {R}_{\tilde{\varGamma }}$ terminates unless ${\tilde{\varGamma }}$ is left-recursive.

Proof

Assume that there is an infinite parse with $t_i \in \mathcal {R}_{\tilde{\varGamma }}$ for $i \in \mathbb {N}$. Since is finite and each match operation “removes” an unread edge, there must be a $k \in \mathbb {N}$ such that $t_i$ is an expand rule for all $i > k$. As their number is finite, there must be numbers i and j, $k< i < j$, such that and start with edges labeled with the same nonterminal. By Fact 3, , which proves that ${\tilde{\varGamma }}$ is left-recursive.    $\square$

Inconveniently, the steps of the general top-down parser are nondeterministic:

• 1. The expansion of a nonterminal $A^\nu$ may choose any of its productions.

• 2. The match of an edge $a^\nu$ may choose any unread edge fitting the profile $\nu$.

We consider a parse as a blind alley if the configuration H is not accepting, but does not allow further steps (using $\mathcal {R}_{\tilde{\varGamma }}$). This is the case if

• starts with an edge $a^\nu$, but $t_{a^\nu }$ does not apply (edge mismatch), or

• but (input too big).

Due to nondeterminism, a successful parse may nevertheless exist in such a situation. Exploring the entire search space of parses to determine whether a successful one exists is very inefficient.

6 Predictive Top-Down Parsing for HR Grammars

The aim of predictive top-down parsing for threaded HR grammars is to avoid backtracking, the major source of inefficiency of a straightforward implementation of the general top-down parser. So we have to cope with the nondeterminism identified in the previous section. In every configuration of a parse, it must efficiently be possible to predict which choices of moves are wrong in the sense that they lead into a blind alley, whereas other moves could still lead to a successful parse if there is any. However, this is most likely not achievable for every threaded HR grammar ${\tilde{\varGamma }}$ because Theorem 2 in combination with the known NP-completeness of some HR languages would otherwise imply that P=NP. For such a grammar, certain configurations will allow more than one expansion, and it may be the case that any of them is promising, or just some of them (or none).

Thus backtrack-free parsing only seems to be possible for HR grammars that make correct moves of their top-down parsers predictable.

Let us first define predictive expand rules that will prevent a parser from running into blind alleys by additionally checking so-called lookahead conditions. Henceforth, given a rule and a condition c over P, we denote the conditional rule by r[c].

Definition 12

(Predictive expand rules). Let $\varGamma$ be a HR grammar, ${\tilde{\varGamma }}$ a threaded version of $\varGamma$, and $\mathcal {R}_{\tilde{\varGamma }}= \mathcal {R}^\mathrm {M}_{\tilde{\varGamma }}\cup \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$ its general top-down parser. For an expand rule , a condition c over P is a lookahead condition for $t_{p^\nu }$ if the following holds:

For every derivation where G is an initial configuration and H is promising,⁵ if $m\vDash c$ then $H'$ is promising.

A set $\mathcal {R}= \{t_{p^\nu }[c_{p^\nu }] \mid t_{p^\nu } \in \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}\}$ of conditional rules is a set of predictive expand rules for ${\tilde{\varGamma }}$ if $c_{p^\nu }$ is a lookahead condition for every .

In the following, we briefly describe a simple way to check whether a set of predictive expand rules can be obtained from $\mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$. For this purpose, let G be any initial configuration and any expand rule so that where $H'$ is promising, i.e., there is an accepting configuration F such that

(1)

(2)

Consider case (1) first. There is an isomorphism because K is obtained from H by expand rules only. Let e be the edge of that is read by the match operation and E the subgraph of K induced by e. Clearly, m(P) as well as $iso (E)$ are both subgraphs of H. Now select a graph C and an injective morphism $m'$ so that $P \subseteq C$, $m = m'|_P$, and $m'(C) = m(P) \cup iso (E)$. By definition, $m \vDash \exists C$. In case (2), is empty and $m \vDash \exists P$.

We can make use of this as follows. For an expand rule $t_{p^\nu }$, performing the above analysis for all derivations of types (1) and (2) yields only finitely many distinct graphs C (up to isomorphism). These graphs $C_1,\dots ,C_n$ can be computed by procedures similar to the construction of FIRST and FOLLOW sets for LL(k) parsing [15, Sect. 5.5]. Defining $\hat{c}_{p^\nu } = \exists C_1 \vee \exists C_2 \vee \cdots \vee \exists C_n$ we thus obtain for all promising graphs $H,H'$ that implies $m \vDash \hat{c}_{p^\nu }$. Thus, by contraposition, if H is promising and but , then $H'$ cannot be promising.

Note, however, that $m \vDash \hat{c}_{p^\nu }$ does not necessarily imply that $H'$ is promising if and H is promising. Therefore, $\hat{c}_{p^\nu }$ can in general not directly serve as a lookahead condition. To solve this problem, we define a relation $\sqsubset$ on expand rules. For this purpose, let us consider two different expand rules $t_{p_a^\nu }, t_{p_b^\nu } \in \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$ with isomorphic left-hand sides. Without loss of generality, we assume that the left-hand sides are identical. We define $t_{p_a^\nu } \sqsubset t_{p_b^\nu }$ if there is an initial configuration G and a derivation where $H'$ is promising and $m \vDash \hat{c}_{p_b^\nu }$. In fact, relation $\sqsubset$ can be defined while conditions $\hat{c}_{p_i^\nu }$ are constructed.⁶

Note that $\sqsubset$ is in general not an ordering and that it may even contain cycles $t_{p_a^\nu } \sqsubset t_{p_b^\nu } \sqsubset \cdots \sqsubset t_{p_a^\nu }$. But if there are no such cycles, one can create (by topological sorting) a linear ordering $\prec$ on all expand rules with isomorphic left-hand sides (where we again assume that they have in fact identical left-hand sides) so that $t_{p_a^\nu } \sqsubset t_{p_b^\nu }$ always implies $t_{p_a^\nu } \prec t_{p_b^\nu }$. We then define, for each expand rule $t_{p^\nu }$, the condition $c_{p^\nu } \equiv \hat{c}_{p^\nu } \wedge \lnot c_1 \wedge \lnot c_2 \wedge \cdots \wedge \lnot c_n$ where $\{c_1, c_2, \ldots c_n\} = \{\hat{c}_{\bar{p}^\nu } \mid t_{\bar{p}^\nu } \prec t_{p^\nu }\}$. The following lemma states that these conditions can serve as lookahead conditions for predictive expand rules:

Lemma 2

Let $\varGamma$ be a HR grammar, ${\tilde{\varGamma }}$ a threaded version of $\varGamma$, and $\mathcal {R}_{\tilde{\varGamma }}= \mathcal {R}^\mathrm {M}_{\tilde{\varGamma }}\cup \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$ its general top-down parser. If $\sqsubset$ is acyclic and the condition $c_{p^\nu }$ is defined as above for each expand rule $t_{p^\nu } \in \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$, then $\{t_{p^\nu }[c_{p^\nu }] \mid t_{p^\nu } \in \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}\}$ is a set of predictive expand rules for ${\tilde{\varGamma }}$.

Proof

Consider any derivation where G is an initial configuration, and H is promising. Then there is an expand rule $t_{p^\nu }$ so that and K is promising. By construction, $m \vDash \hat{c}_{p^\nu }$. If there were a smaller expand rule $t_{\bar{p}^\nu } \prec t_{p^\nu }$ with $m \vDash \hat{c}_{\bar{p}^\nu }$, then this would imply $t_{p^\nu } \sqsubset t_{\bar{p}^\nu }$ because K is promising, and therefore, $t_{p^\nu } \prec t_{\bar{p}^\nu }$, contradicting the linearity of $\prec$. Therefore, $m \vDash \lnot \hat{c}_{\bar{p}^\nu }$ for $t_{\bar{p}^\nu } \prec t_{p^\nu }$ and $m \vDash \hat{c}_{p^\nu }$, i.e., $t_{p^\nu }$ is the only expand rule that satisfies its lookahead condition for H, i.e., $m \vDash c_{p^\nu }$.    $\square$

The proof shows that these lookahead conditions always select a unique expand rule. Clearly, this cannot succeed for situations where expand rules can turn a promising configuration into two or more promising successor configurations.

However, the existence of a set of predictive expand rules is not sufficient for obtaining a predictive top-down parser. The threaded HR grammar must satisfy the following property as well:

Definition 13

(Free edge choice property). Let $\varGamma$ be a HR grammar, ${\tilde{\varGamma }}$ a threaded version of $\varGamma$, and $\mathcal {R}_{\tilde{\varGamma }}= \mathcal {R}^\mathrm {M}_{\tilde{\varGamma }}\cup \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$ its general top-down parser. ${\tilde{\varGamma }}$ is said to possess the free edge choice property if, for every derivation where G is an initial configuration and H is promising, $H'$ is promising as well.

Theorem 4

Let $\varGamma$ be a HR grammar, ${\tilde{\varGamma }}$ a threaded version of $\varGamma$ without left-recursion, and $\mathcal {R}_{\tilde{\varGamma }}= \mathcal {R}^\mathrm {M}_{\tilde{\varGamma }}\cup \mathcal {R}^\mathrm {E}_{\tilde{\varGamma }}$ its general top-down parser. $\mathcal {R}^\mathrm {ptd} = \mathcal {R}^\mathrm {M}_{\tilde{\varGamma }}\cup \mathcal {R}$ is a terminating top-down parser for $\varGamma$ that cannot run into blind alleys if $\mathcal {R}$ is a set of predictive expand rules for ${\tilde{\varGamma }}$ and ${\tilde{\varGamma }}$ has the free edge choice property.

Proof

Let $\varGamma$, ${\tilde{\varGamma }}$, and $\mathcal {R}^\mathrm {ptd}$ be as in the theorem. Moreover, let ${\tilde{\varGamma }}$ satisfy the free edge choice property, and let $\mathcal {R}$ be a set of predictive expand rules for ${\tilde{\varGamma }}$. Each derivation where G and H are initial and accepting configurations, resp., is also a successful parse in $\mathcal {R}_{\tilde{\varGamma }}$, i.e., by Theorem 2.

Now let G be any initial configuration with , i.e., G is promising. Any infinite derivation would also be an infinite parse , contradicting Theorem 3.

Finally assume that $\mathcal {R}^\mathrm {ptd}$ runs into a blind alley starting at G, i.e., there is a derivation , H is not accepting, and there is no configuration $H'$ so that . By the free edge choice property and $\mathcal {R}$ being a set of predictive expand rules, H must be promising, i.e., there is a configuration $H''$ so that or . In either case, there is a configuration $H'$ so that , contradicting the assumption.    $\square$

This theorem justifies to call a threaded HR grammar ${\tilde{\varGamma }}$ predictively top-down parsable (PTD for short) if ${\tilde{\varGamma }}$ satisfies the free edge choice property and there is a set of predictive expand rules for ${\tilde{\varGamma }}$.

Example 4

(A Predictive Top-Down Tree Parser). The threaded tree grammar in Example 2 is PTD. To see this, let us construct lookahead conditions for expand rule and as described above.

Inspection of expand rule shows that choosing this rule cannot produce a promising configuration if the unread part of the input does not contain a -edge starting at node x. The existence of this edge is hence requested by the graph condition , defined by the supergraph of the pattern of (see Fig. 8). No such edge can be requested for expand rule ; each match of satisfies since is just the pattern of . Condition is in particular satisfied if choosing produces a promising configuration, and therefore . By Lemma 2, we can choose lookahead conditions and .

Fig. 8.

Graphs defining and for expand rule and , resp.

Fig. 9.

Predictive expand operations of the tree parser

Figure 9 shows the resulting predictive expand rules for the nonterminal T of the tree parser. For brevity, lookahead conditions show only those subgraphs that must or must not exist in order to apply or . The match rules and the expand rule for the start production remain the same as in Example 3. Moreover, it is easy to see that match rule produces a promising configuration for each of its matches, i.e., the threaded tree grammar has the free edge choice property. With these modified expand rules, the predictive parser can select the same parse as in Fig. 7. As mentioned earlier, other well-known examples that allow for predictive parsing include palindromes, $a^nb^nc^n$, arithmetic expressions, and Nassi-Shneiderman diagrams.

7 Conclusions

In this paper, we have defined PTD parsers for HR grammars by graph transformation rules, and shown their correctness. The definition is consistent with the implementation of PTD parsers in the graph parser distiller grappa⁷ described in [4], but some features are still missing: First, productions that merge nodes of the left-hand side have been omitted. Such productions may occur when a HR grammar is “left-factorized” in order to allow for predictive expansion. (This corresponds to left-factorization of CF string grammars for LL-parsing.) Second, PTD parsing for contextual HR grammars [2, 3] has not been considered. Finally, a more sophisticated way of calculating lookahead conditions, by approximating Parikh images, has been ignored.

So our next step will be to extend our definitions and proofs to cover these concepts as well. Our ultimate goal ist to use this definition to relate the power of PTD parsing to that of PSR parsing, probably by using a definition of PSR parsing that is based on graph transformation as well.