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<Paper uid="C88-2150">
  <Title>Generation as Structure Driven Derivation</Title>
  <Section position="3" start_page="0" end_page="733" type="metho">
    <SectionTitle>
2 Generation from functional structures
</SectionTitle>
    <Paragraph position="0"> In this section we describe an algorithm which constructs a generator for an arbitrary given LFG 3 which generates terminal strings from functional structures. 4 Such an algorithm has to define for every LFG a relation FC/(~,s) (s is generable from (I)) between directed acyclic graphs (DAGs) and terminal strings.</Paragraph>
    <Paragraph position="1"> VCeDAG VseV,~ (r,/,(##, s) /fie(... { ..... )) Depending on what the adequacy condition C(... 4~, s...) for this relation is, one will get different adequacy criteria for possible explications of what 'generation' can mean within the LFG-framework. We started from the - perhaps too idealized - condition that is normally used for the relation between strings and f-structures At, specified by an adequate parsing algorithm for LFG. s V~,eDAG VseV~,(AC/(s, ~) iffs is derivable from the start symbol S and s has f-structure (I)) If we use this condition and insert it into the schema C(... q~, s...), (*) V(~eDAG VseV~(FC/(C/, s) iffs is derivable from the start symbol S and s has f-structure ~) then the relation ~X~ is simply defined inversely by an adequate generation algorithm (F~ 1 = At and also A~ 1 = FC/). This means that a generator for an LFG G must accept an input structure (I) by building up a string s, iff s is derivable with f-structure (P in G. Thus, the generator for an LFG constructed bythc algorithm which satisfies this condition is simply a parser or transducer for the set of well-formed f-structures, which constructs for an input structure the set of all sentences which have this structure as their f-structure. This implies that the generator produces no output, if the input structure is not a well-formed f-structure.</Paragraph>
    <Section position="1" start_page="0" end_page="732" type="sub_section">
      <SectionTitle>
2.1 Derivational background
</SectionTitle>
      <Paragraph position="0"> The algorithm can he based on derivation concepts for LFGs which can be strengthened in such a way that the derivation can be driven by an input structure (structure driven derivation), tIowever, the derivability conditions formulated in \[Kaplan/Bresnan 82\] cannot he used directly. According to \[Kaplan/Bresnan 82\] a terminal string s is regarded as well-formed iff it satisfies the following conditions:  (WFF) 1. There is a c-structure c for s that can be derived by the context-free base of the grammar.</Paragraph>
      <Paragraph position="1"> 2. There is an f-structure ~ and a mapping C/ from the c-structure nodes to the nodes of C/ such that q' is the unique minimal f-structure that satisfies the annotations associated with the e-structure nodes. (The f-description solution algorithm (fds-algorithm) constructs both C/ and ~.)~ 3. All constraints in the f-description are satisfied by C/.</Paragraph>
      <Paragraph position="2"> 4. (~ is complete and coherent.</Paragraph>
      <Paragraph position="3">  These conditions are tested in the following order: 1. -&lt; 2. -&lt; 3.,4.. Thus, if the f-structure is built'up after the derivation of the e-structure, ~ it is impossible to use the functional information contained in an input structure for the control of the derivation of the c-structure. A decidable generation procedure presupposes the possibility of comparing the input structure with the partial f-structure of a derived partial e-structure. Thus, in order to drive the c-structure derivation by a given input structure it is necessary to derive the partial f-structure in parallel to a partial c-structure. This means that one can use only those derivation concepts which make f-description solutions for partial c-structures available in each step of the derivation. The concept on which the following algorithm is based is described in more detail in \[Wedekind 86\]. According to this concept, a derivation is a sequence of quadruples (e, (I), C/, C% Each quadruple consists of c a partial c-structure, q~ a partial f-structure, C/ a mapping from the c-structure nodes into the set of nodes of (~, and C C/ a constraint set.</Paragraph>
      <Paragraph position="4"> (C/ and C/ would be the result of the fds-algorithm if it were applied to the corresponding annotated c-structure. C ~ corresponds to. tile set of instantiated constraining equations contained in the f-description of c. See the example in fig. 1.) I follow the usual convention of identifying the c-structure nodes with sequences of integers. The linear order of the edges of a tree is normally encoded by numbering the arcs, and every node is identified with the sequence of integers numbering the arcs along the path from the root to that node.</Paragraph>
      <Paragraph position="6"> Fig. 1 The corresponding annotated c-structure is derivable by the following rules:</Paragraph>
      <Paragraph position="8"> The initial quadruple consists of an S-labeled root node cs, an f-structure root node 4)s, to which the S node is mapped by ~s and an empty constraint set C~.</Paragraph>
      <Paragraph position="9"> (es, 4)s,C/~,c::~) cs Cs 4)s c.$ #o ---*.</Paragraph>
      <Paragraph position="10"> A grammar rule also introduces a quadruple.</Paragraph>
      <Paragraph position="11"> v -- (~,, 4),.,,V, c,.*) The time(tonal part (4)r,qF) is obtained by applying the fds-algorithm to the annotated local tree represented by the rule, and by instantiating the metavariables in the constraining equations with the node indices of the local tree introduced by the rule. (The constraint set of a rule contains the constraints of the f-description of the local tree.) Fig. 2 gives an example.</Paragraph>
      <Paragraph position="13"> This quadruple corresponds to the rule:</Paragraph>
      <Paragraph position="15"> Suppose that we have derived from the initial quadruple the quadruple given in fig. 1; then we can apply the V-rule, since the leaf 2.1 is labeled with V. The derived quadruple (d, 4)', ~b ~, C ~) consists of a e-structure which is the result of expanding 2.1 by or. The partial f-structure 4)~ is the minimal extension of 4) which results from 4) by unifying the DAG (I) r introduced by the rule with the substructure rooted by qt2A. Since the new DAG 4)J is a homomorphie extension of 4), the values of ~b * for the old nodes (of c) are given by 5 and the homomorphism. The values of ~J for the new nodes result from ~b ~ and from the value of ~b for 2.1 (of. the definition below)* C C/~ contains besides C ~ (which is empty) the constraint (2.1 NUM)=~(2.1 SUBJ NUM) which is con.structed from (O NUM) =c (0 SUBJ NUM) by attaching 2.1 ;Is a prefix to each node index within that constraint. The result is the constraint set of the f-description of the derived tree. Fig. 3 illustrates the result of the rule application.</Paragraph>
      <Paragraph position="17"> If we reconstruct directed labeled connected rooted aeyelic graphs (DAGs) as transition graphs of connected rooted acyelic finite state automata, whose leaves (states without leaving transitions) are labeled by a partial function rrt with atoms of a set A (4) = (Q,L,6, qo,A,m)), s then we can state the definition of the derivability relation A~ as follows.</Paragraph>
      <Paragraph position="18">  DEFINITION 2.1 A terminal string s is derivable with f-structure 4)1 (AC/(s, 4)I)) i~there is a sequence wo... wn such that  - coo = (cs, 4)s, ~s, C~) and - for all wi = (c, 4), ~b, C*), wi+l = (c', 4)', ~b', C ~') (0 &lt; i &lt; n) there is a rule V -~ {e~, 4). ~, Ct) and - Visalabelofaleafpofc - d is the result of expanding # in c by er - 4)~ is the minimal extension of 4) which results from 4) by unifying 4)r with the substructure rooted by C/~ - ifv is anode oft, whose c-value is ~*(qo,P) in 4), then its C/Cvalne in ~' is ~*(q'o,P) if/~.J is a node of c', not contained in c, the value of q~ is 6~* (q~, q) and the value of q~ for p is 6*(qo,P), then the value of qV for ,~.j is 6'*(qto,p.q) in 4)' - C C/' contains besides C C/ the constaints (#.j p) =c (#.i q) etc) for all (j p) =~ (i q) etc. in C~ - s is the terminal string of the c-structure ofw,~ - 4)1 is equal to the DAG ofw, (q~,) - 4),~ satisfies all constraints in C,~ - 4), is complete and coherent, tdeg</Paragraph>
    </Section>
    <Section position="2" start_page="732" end_page="733" type="sub_section">
      <SectionTitle>
2.2 Generation as f-structure driven derivation
</SectionTitle>
      <Paragraph position="0"> Ill principle we could use this derivation concept for generation if we substituted the DAG in the initial quadruple by an input structure and mapped the S node to the root of the input structure. IIowever, such a concept of generation would not satisfy the adequacy condition mentioned above.</Paragraph>
      <Paragraph position="1"> The derivation would not be adequately controlled by tbe input structure because it is not guaranteed that i) the information contained in the input structure is completely derived and it) no additional information is introduced during the derivation.</Paragraph>
      <Paragraph position="2"> It is possible, for example, to derive additional adjuncts or not to derive all adjuncts represented in the input structure. Due to tbe unification part of the derivation process, it is only guaranteed that the f-structure of the generated sentence is compatible with the input structure. The requirements i) and it), which will be referred to as completeness and coherence, n show that the input structure is in fact a complex constraint with a positive and a negative part. The positive part (compleleness), which requires that the input structure (4)in) is subsumed by the derived structure (4),) (4)i,, E 4),), can be made explicit by two kinds of constraints: existential constraints, which demand that COMPa: all paths of the input structure are derived, and reentrancy constraints, whicb demand that COMPb: all reentraueies of the input strneture are derived) 2 Tim negative part (coherence) which demands that COH: the derived structure is subsumed by the input structure (4),, L (bl,) ensures that the f-structure of the generated string is the unique minimal structure that satisfies the completeness constraints expressed by the input structure.</Paragraph>
      <Paragraph position="3"> * The central problem of generation designed o.s structure driven deriwv tion is the control of the fidfillment of these conditions. Since this problem also occurs within other formalisms which build up DAG-strnctures during the derivation process, the solutions proposed here for LFG can also be applied in more or less the same way within the other formalisms.</Paragraph>
      <Paragraph position="4"> i.) COMPb. This condition is controllable if the input structure (I)i~ is unfolded. The functional structure of the initial tuple is then an unordered tree 4)t. Since the input structure is a (homomorphic) extension of the unordered tree (4)t ~ chin) and both structures have the same path set, the relating homomorphism is an 'onto'-mapping and therefore called epimorphism. Part B of fig. 4 gives an example. 13 Now, since coherence has to be ensured during generation, the derived structure will never become an extension of the input structure and each generation step induces a new epimorphism frmn the derived structure to the input structure. The coherence condition guarantees that the epimorphisms induced in the generation steps always approximate an isomorphism. When the derived structure and the input structure are isomorphic, all rcentrancies are derived.</Paragraph>
      <Paragraph position="5"> it.) COMPs. The fnlfillment of this condition can be controlled, if, apart from the root, all nodes of the DAG introduced by a rule are labeled by a '+'-marker. This additional labeling distinguishes the generator rules from the grammar rules. Fig. 5 shows the generator rule corresponding to the grammar rule of fig. 2. If the root of the unordered tree 4)~ is also T-labeled and all nodes of the strncture that is derived from 4)( are +-labeled, then all paths of the input structure are derived) 4 The condition that all leaves of a well-formed f-structure are labeled by atomic values ensures that all atomic values of the input structure are derived.</Paragraph>
      <Paragraph position="7"> Fig. 5 ' iii.) COIt It is possible to check this condition in each step of the derivation since the input structure is accessible by the epimorphism induced in each particular step. It is guaranteed that no additional information is introduced by the rule application, if the substructure to which the expanded c-structure node is mapped by C/ and the epimorphism h ~ is (asid e from the +-labels) an extension of the DAG introduced by the rule. 1~ If, for example, the constellation shown in fig. 4 (A and B) is generated and the rule in fig. 6 were to be applied to node 2.2, condition COtI will be violated, since the substructure rooted by h~(~b~.2) is not an extension of the (unlabeled) structure introduced by that rule. On the other hand, the substructure rooted by h C/ (C/2.~) is an extension of the (unlabeled) structure introduced by the V-rule of fig. 5 and fulfills COH with respect to node 2.1. vm~~o Fig. 6 i Since the (functional) structure which is to be derived from &amp;~ is equal (or isomorphic) to the input structure itself, it is possible to check - the f-completeness and f-coherence of the input structure before the generation starts, and - the constraints expressed by the rules simultaneously during the generation. null Although the V-rule would satisfy the coherence condition with respect to node 2.1, its application is ruled out, since the substructure rooted by hC/(C/2.1) does not satisfy the constraint expressed by the rule. is Thus, the sequence of tuples which constitute the generation of a terminal string need not contain a constraint set.</Paragraph>
      <Paragraph position="8"> The start entity of a generation is then a quintuple /c~, ~,, C/~, ~,., h~).</Paragraph>
      <Paragraph position="9"> ~in is an f-complete and f-coherent input structure, C/~ is the unfolded input structure and h~ is the relating epimorphism. The fleaerabillty relation F~  is then defined as follows./~ DEFINITION 2.2 A terminal string s is gencn~ble from an input structure ~n (rC/(~b~n, s)) iffthere is a sequence We...wn such that o Wo = (cs,~,C/S,C/~,h~s) and o for allw~ = (c,~,~,~,h~), w~+~ = {d,~b',~',~n,h ~') (0 &lt;_ i &lt; n) there is a generator rule V --* (cr, ~r, ~r, C~) and - V is the label of a leaf/~ of c o the substructure rooted by h#(~b~)is (aside from the +-labels) an extension of ~ o for all (j p) =~ (i q) etc. in C~, 6~,, (h4'(~bp.j), p) = $~(h~(C/,.i), q) etc.</Paragraph>
      <Paragraph position="10"> ,734 - c p is the result of expanding p in e by cr - ~' is the minimal extension of @ which results from ~ by unifying * r with the substructure rooted by. ~bp - if ~ is a node of c, whose e-value is ,5*(qo,p) in C/, then its ~.bt-value  in q,' is ~'*(qto,p ).</Paragraph>
      <Paragraph position="11"> if p.j is a node of c I, not contained in c, the value of C/~ is ~* (q~, q) and the value ofC/ for p is 6*(qo,P), then the value of ~t tbr #.j</Paragraph>
      <Paragraph position="13"> o C/. is isomorphic to ~{n (h~ is an isomorphism) - s is the terminal string of the c-structure ofwn.</Paragraph>
      <Paragraph position="14"> Lemma 1 follows from the above.</Paragraph>
      <Paragraph position="15"> LEMMA 1 V~'eDAG Vsev~(r~(~, s) ~ A~(s, ~)) This lemma can easily be proved, since in each step of the generation of a sentence the applied rule can be applied exactly in the same way in the corresponding derivation step of a derivation of that sentence (and vice versa). So the substructure which includes all +-labeled nodes of a generated functional structure corresponds exactly to that partial f-structure which is derived up to that step (and vice versa). Thus, the derived c-structure is identical to the generated c-stmcture, the derived f-structure is equal to the generated f-structure and thus identical (isomorphic) to the input structure. Since the constraints in the constraint set of a derivation must in fact be the (instantiated) constraints of all rules applied during the generation, the input structure satisfies all constraints iff the derived structure does.</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="733" end_page="736" type="metho">
    <SectionTitle>
3 Generation from semantic structures
</SectionTitle>
    <Paragraph position="0"> In this section we use the ideas described in section 2 to develop an algorithm that constructs generators which generate terminal strings from semantic structures. These ideas are applicable if we can ensure, that a) the semantic structures are representable as DAGe, b) the only operation which is used to construct the semantic structures is the unification operation, and c) the semantic structure of a sentence can be built up simultaneously with the derivation of the c- and f-structure of that sentence.</Paragraph>
    <Paragraph position="1"> That a) and b) can be ensured for most of the current semantic theories, like Montague Semantics ~MS), Discourse Representation Theory (D1Tr) and Situation Semantics (as), is illustrated, for example, in \[Reyle 88\] (MS, Dffr), \[Halvorsen 87\] (SS) and in works concerning eategorial grammars which are augmented by a unification component. Is C0ndltion c) is satisfiable if we follow a proposal by Halvorsen who describes a possible extension of LFGs such that the semantic representation of a sentence can he &amp;quot;simultaneously described (co-described) with the functional structure&amp;quot; (\[Halvorsen 87\], p.9). Halvorsen extends the formalism to include a new type of equation, which is used to build up a semantic representation and establishes an additional (partial) mapping from the f-structure nodes into the node set of the semantic structure.</Paragraph>
    <Paragraph position="2"> Since the semantic structures are represented as DAGs, we can use for the generation from semantic structures, a condition llke (*) as an adequacy condition, which refers'to a semantic (a-) structure instead of an f-structure. (**) VI?,eDAG VseV~ (r~ (~, s) iff s is derivable from the start symbol S and s has a-structure ~) Since this condition implicitly assmnes that F~ -1 is determined by an adequate parsing algorithm, this extension of the formalism is neutral with respect to the problem of the creativity 19 of the extension. This neutrality is desirable, since the algorithm should be definable independent of a specific semantic theory. The question whether a semantic component is (or should be) a creative or a conservative extension oftbe syntactic theory (LFG), on the other hand, depends crucially on the specific semantic theory and on the format of the rules which prescribe how the compositionality principles of the chosen theory are to he translated into this new .type of equation.</Paragraph>
    <Paragraph position="3"> According to condition (**), the generator constructed by the algorithm for an (extended) LFG is a parser or transducer for well-formed semantic structures which, in principle, constructs for an input structure all sentences which have this structure as its semantic structure, analogously to section 2. A semantic structure alone, however, is usually regarded as too poor with respect to the syntactic information relevant for 'adequate generation results' within a natural language system. We therefore integrate the additional possibility of driving the derivation by syntactic (functional) information. This is possible because the f-structure of a sentence is built up in parallel to the a-structure driven derivation of that sentence. If we assume that sentences % and f-structures @i are related to an input structure according to the following schema:</Paragraph>
    <Paragraph position="5"> v~e can enm~re by additional constraints on the f-structures that only those sentences are generable whose f-structures satisfy these constraints. Thus, we can drive the derivation in a way which is more sensitive to syntactic information. We can ensure, for example, that only a pusive realization or a realization with a specific topic/focus structure is generable. Since not every input structure has a surface realization with an f-structure that fulfills these additional constraints, these constraints can be highly creative from a formal point of view. Therefore, a generator which drives the derivation by a semantic structure with additional constraints and also satisfies condition (**), must be capable to loosen the f-structure constraints if no output sentence can be derived wh~e f-structure satisfies the constraints, i.e., to drop constraints successively until an output sentence can be derived. The control of such a dropping procedure is again dependent on the application domain us and can be determined only by empirical investigations.</Paragraph>
    <Paragraph position="6"> 3oi The derivation of semantic structures According to I\[alvorsen's proposal, the semantic structures are consh'uctable by means of additional equations which are formulated with a projector a which &amp;quot;can be prefixed to any expression denoting a functional structure&amp;quot; (\[Halvorsen 87\], p.8). Fig. 7 A gives a slightly simplified example of a lexical entry from \[Halvorsen 87\], written as the expansion of a lexical category symbol.</Paragraph>
    <Paragraph position="7"> v ~ kicks ~~tr ~-~ (1 PRED) = kick' Vo</Paragraph>
    <Paragraph position="9"> A ~a Tit. 7 In Part A tim two equations at the hottom are called inter-modular equations.</Paragraph>
    <Paragraph position="10"> We will first give the syntax of this new type of equation. Let us assume a distinction between functional attributes (L~) and values (A~) and semantic attributes (L~) and values (Ao) (with all four sets pairwise disjoint). Then the set of semantic (a-) designators contains the simple a-designators Aa and the complex a-designators. A complex a-designator is a term a(t# q) consisting of a complex f-designator (a metavariable (\]&amp;quot; or l) followed by a possibly empty sequence of functional attributes), and a possibly empty ~equence q of semantic attributes. The a-equations, then, are those equations which have a complex ~r-designator in the first argument position and complex or simple ~-designator in the second.</Paragraph>
    <Paragraph position="11"> ff we assume that i. the fds-algotithm is used to solve the projector equations and to construet th(' semantic structure, and it. const, rain~s are expressable, s! then we can add the following two conditions to the WFF-condition:  (W~'F~) (WFF) ~uld 5. There is a ~-structure i\] and a partial mapping a from the f-structure nodes to the nodes of E, such that ~, is the only minimal a-structure that satisfies the projector equations a~ociated with the c-structure nodes. (The fds-algorithm constructs also a and ~.) 6. All w conl~traints are satisfied by ~, a and ~.=~  These addition~d couditions must be tested in the order 3.4 5. -~ 6.. Since the description-solution algorithm is also used for solving the projector equations, it is easy to simulate the projector-mechanism by using an additional singular ~ttribute. The solution algorithm enforces and preserves the fanc.ti0n proper~y (uuiqu~ness) of the projector, i.e., if (q~ p) and (~, p') a~'e terms designating some nodes in the f-structure which are \[napped by to some nodes in the a-structure (a(~b~ p), a(~bv f)), then the identity of (~b~ p) and (~ f), which might be established by some other equations, will also enforce the identity of (u(~b~ p) and a(~bv f)). Thus, we can simulate the projector by using a singular attribute a (not contained in LC/, At, La, and A~) that is inserted between the f-designator and the semantic attributes of a semantic designator according to the following rewrite schemata.</Paragraph>
    <Paragraph position="13"> The identity of (~b~ p) and (q~ p') will then also enforce the identity of the values of (~ p o') and (~v ff o') .93 If we simulate the projector mechanism in this way, the derivation of a tr-strueture in parallel to a c- and f-structure closely resembles the derivation process described in section 2.1. A rule introduces a quadruple, where ~r is replaced by the DAG ~r which is obtained by solving the functional and a-annotations of the local tree introduced by the rule. The constraint set C~r contains the instantiated f- and a-constraints expressed by that rule.</Paragraph>
    <Paragraph position="14"> Fig. 7 shows in part B the solution of the equation system for the local tree introduced by the rule of part A. The initial quadruple consists of a DAG ~s which has one a transition.</Paragraph>
    <Paragraph position="16"> The only additional condition that has to be satisfied by the derived DAG-structures results from the fact that tbe semantic substructures of a derived DAG are not necessarily connected, i.e. that not every substructure which is a value of a a-attribute must necessarily be a substructure of the topmost a-attribute of 9. On the one hand, the syntax for the a-equations permits us to formulate rules which introduce unconnected semantic structures (fig. 8.1 gives an example). On the other hand, rules can introduce grammatical functions without a a-attribute (eL fig. 8.2 and 8.3). This can lead to unconnected semantic structures, if the expansion of a c-structure node which is associated with such a grammatical function introduces a semantic structure. The motivation for not excluding these two sources for the unconneetedness is that the semantic function of a constituent or grammatieal function can be uncertain within the local context given by a rule and has to be determined by another constituent not introduced in that rule. ~4 In traditional LFG it is usually assumed that the semantic function of subeategorized constituents such as SUBJ and OBJ is determined by the governing verb (PRED) and that the assignment of grammatical functions to semantic functions may be aflhcted by lexical rules. These assignments can be established by inter-modular equations in the lexical entry for the verb, as illustrated in fig. 7, while at the same time leaving open the assignments of semantic functions to the NPs in the rules of fig. 8.2 and 8.3 by not annotating them with a-equations.</Paragraph>
    <Paragraph position="18"> Thus, unconnected semantic structures can become connected through inter-modular equations which take into account the semantic structure of those constituents whose semantic function is not determined by the context represented by the local tree of a rule.</Paragraph>
    <Paragraph position="19"> Since the DAG of a derived sentence has to contain a connected semantic structure, we have to add to the well-formeduess conditions of that DAG, that the value of each or-transition must be accessible from the e-value of the  root of that DAG. The derived semantic structure is the substructure of the derived DAG which is rooted by the value of the uppermost a-transition. This leads immediately to the definition of the derivability relation.  DEFINITION 3.1 A terminal string s is derivable with a-structure ~C (A~(s, ~.)) i~there is a sequence To...wn such that  o Wo = (cs,~s,C/S,Ces) and o for all C/0~ = (c,~,C/,c+), ~+~ = (e',~',C/',c&amp;quot;') (0 &lt; i &lt; ~) there i~ arule V --+ (c,,%,C/LCg) and - V is a label of a leaf it of c - c' is the result of expanding it in e by cr - ~/~ is the minimal extension of * which results from * by unifying @,. with the substructure rooted by Ct, - if v is a node of c, whose C-value is 5&amp;quot; (qo, P) in q', then its C/'-value in ~' is 6~*(q~,p) if #.j is a node of d, not contained in e, the value of C/~ is ~f~ (qD, q) and the value of C/ for it is 8*(qo,p), then the value of C/~ for p.j is ~'*(q~o,P.q) in ~' - C '/'' contains besides C C/ the constraints (it.j p) =~ (it.i q) etc. for all (j p) =~ (i q) etc. in C~ - ~,~ satisfies all constraints in C~ - ~,, is f-complete and f-coherent o VpeL*c((q~,p.a)eDom(g*) -4 ~qeL*(g*(q~,a.q) = g*n(q~,p.a))) ( conncctedness) o E is the substructure of ~n rooted by ~n(q~', a) - s is the terminal string of the c-structure ofwn.</Paragraph>
    <Section position="1" start_page="735" end_page="736" type="sub_section">
      <SectionTitle>
3.2 Generation as semantic structure driven and con-
</SectionTitle>
      <Paragraph position="0"> strained derivation The mechanisms for controlling completeness and coherence, which have been developed in section 2, together with the defining conditions of the different types of derivation concepts can now be used to design an adequate algorithm for the generation from semantic structures, which in addition can be directe d by functional constraints.</Paragraph>
      <Paragraph position="1"> The process of generating from a semantic input structure ~in has to start with a complex entity (e~, ,~,, C/~, s,,,, h~, c$) where ~t is an unordered tree consisting of only one a-transition, whose value is +-labeled and dominates the unfolded input structure ~'in. Fig. 9 shows the start entity for the generation from a simplified semantic input structure, h~ is the induced epimorphism from the substructure rooted by the a-value of k~t onto I;~,. The start entity contains an empty set of constraints, since not all constraints expressed by the applied rules can be checked immediately (e.g. pure functional constraints) and have to be added to this set until they can be checked.</Paragraph>
      <Paragraph position="2"> So Fig. 9 The derivation can be constrained by a set C ~deg~ of pure functional and/or inter-modular constraints of the form (g p) =, (0 f) et .... d (~ a q) =, (0 p a) which enforce the derivation of those sentences, whose f- and a-structures fulfill these constraints. (These constraints must eventually be retracted if no sentence is generable.) E.g. the set C:e. { (,SUBJa)=~(OaARG2) } -- (0 BYOBJ a) ---C/ (~ a ARGI) will enforce a passive realization of the input structure given in fig. 9 ('Peter is kicked by John'), while the set  enforces a passive realization with the BYOBJ in the TOPIC-position ('By John, Peter is kicked').</Paragraph>
      <Paragraph position="3"> The generator rules are constructed analogously to section 2 by an additional +-labeling of all nodes which are values of semantic attributes (not of the a-attribute !) within the DAG-structure introduced by the rule. Fig. 10 gives an example.</Paragraph>
      <Paragraph position="4"> C/ tr Fig. 10 Again a crucial restriction follows from the fact that these rules permit the derivation of structures with an unconnected semantic substructure, which becomes connected in later steps of the derivation by means of inter-modular equations. If a constellation is derived by means of these rules, as it is given schematically in fig. 11, Fig. 11 the generation by expansion of V proceeds without control by the input structure as long as the semantic structure introduced by V remains unaccessible. The coherence-condition is undermined and sentences can be generated whose semantic structure is not subsumed by the input structure. null The coherence-condition can be checked, however, if only those rules are applied which preserve the connectedness of the semantic structure derived so far. This restriction on the order-freeness of the rule application does not affect the adequacy condition (**), because in each derivation step of a sentence with a connected semantic structure there must he at least one node of the partial c-structure which is expandable in such a way that the expansion preserves the connectedness of the partial semantic structure. If the expansion of every terminal node violated this connectedness condition, the semantic structure of that sentence could not be connected. Hence, at least one rule must be applicable in such a way.</Paragraph>
      <Paragraph position="5"> Since it is possible that pure semantic constraints of the rule's constraint set refer to semantic substructures whose semantic function is uncertain up to that generation step, not all pure semantic constraints expressed by the rules can be tested immediately. This applies to pure semantic constraints which are constructable from inter-modular constraints by conversion. These constraints contain terms of the form (i a p) (which correspond to (~ a p) annotations) and they are not testable immediately if i is the value for (~ q) (q n ..... pty) (which corresponds to a (T q) --~ annotation) and C/I*(q) has no a- attribute. Hence, the pure semantic constraints which cannot be tested in a particular step, the pure functional constraints, and the inter-modular constraints of the rule's constraint set have to be collected in C C/ . The (constraint-driven) generability relation has to then be defined as follows.</Paragraph>
      <Paragraph position="6">  DEFINITION 3.2 A terminal string s is generable from an input structure ~C~n under a constraint set C eel' (F~(Z~,, s, C~e~')) iff there is a sequence wQ ...w. such that o w0 = (as, ~t, cs, ~,,, h~, C$) and  deg for all wi = (c,@,dp, Ein,ha,CC/), w,+l = (d,q2',~b',~in,ha~,C C/') (0 &lt; i &lt; u) there is a generator rule V ~ (er,qlr,C/&amp;quot;,Ch and - V is a label of a leaf p of c o the substructure rooted by h~(~(C/t~,o')) is (apart from the +labels) an extension of li\]r (the substructure of @, rooted by ~,(q~, a)) o for all pure semantic constraints (j a p) =c (i a q) etc. in C~r for which ~f(~b,4, a) and 6 (C/v.i, a) is defined, 8&amp;quot;, (h a (~ (5,4, a)), p) ~t.(h~(~(C/~.,, a)), q) etc.</Paragraph>
      <Paragraph position="7">  - e' is the result of expanding p in e by cr - ~' is the minimal extension of * which results from ~ by unifying C/2. with the substructure rooted by eu o VpeL*~((q~o,p.a)eDom(6 '*) -, :qqeL*(tV*(q~o,o'.q) = 8'*(q~,p.o'))) - if v is a node of e, whose ~value is/f* (q0, P) in ~/, then its C/~-value in @' is tf~*(q~,p).</Paragraph>
      <Paragraph position="8"> if/~.j is a node of d, not contained in c, the value of qt~ is 6 r (qD, q) aud the valse of ~b for ~ is *5*(qo,p), then the value of ~# for p.j is 5'*(q~o,p.q) in ~2'.</Paragraph>
      <Paragraph position="9"> o C C/* containes besides C C/ the pure semantic constraints which cau.ot be tested in this step, the pure functional and the inter-modular constraints of C~ with p attached as a prefix to the node indices of the constraints o Vp~((q'0, .v)~Do.,(~&amp;quot;) -~ h~'(~'*(q'o, ~p)) = ~'. (#, p)) o each node of the substructure of ~. rooted by 8.(q~, a) (the semantic structurt 0 is +-labeled o the substructure of C/2n rooted hy 6~(q~,a) is isomorphic to ~i. (h~ is an isomorphism) - ~. is f-complete and f-coherent - q~n satisfies all constraints in C. C/ o ~. satislies all constraints in C C/$~ s is the terminal string of the c-structure of wn.</Paragraph>
      <Paragraph position="10"> If we define the (unconstrained) generability relation ra as we Call prove I,EMMA 2 V~eDAG Vs*VT~(P.(~ , s) *-* A~(s, E)) The proof from left to right is carried out in analogy to Lemma 1. The right-to-left half follows from the fact that the order of the rule application during the derivation of a sentence with a connected semantic structure can always be rearrauged in such a way that in each step of the derivation the partial semantic structure is commcted. Otherwise the derived sentence would not haw. ~ a connected semantic structure.</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>