Planning Reference Choices for Argumentative Texts 
Xiaorong Huang 
Fachbereich Informatik, Universit£t des Saarlandes 
Postfach 15 11 50, D-66041 Saarbr/icken, Germany 
Email: huangQcs.uni-sb.de 
Abstract 
This paper deals with the reference choices involved in the 
generation of argumentative text. A piece of argument- 
ative text such as the proof of a mathematical theorem 
conveys a sequence of derivations. For each step of de- 
rivation, the premises (previously conveyed intermediate 
results) and the inference method (such as the applica- 
tion of a particular theorem or definition) must be made 
clear. The appropriateness of these references crucially 
affects the quality of the text produced. 
Although hot restricted to nominal phrases, our refer- 
ence decisions are similar to those concerning nominal 
subsequent referring expressions: they depend on the 
availability of the object referred to within a context and 
are sensitive to its attentional hierarchy. In this paper, 
we show how the current context can be appropriately 
segmented into an attentional hierarchy by viewing text 
generation as a combination of planned and unplanned 
behavior, and how the discourse theory of Reichmann can 
be adapted to handle our special reference problem. 
1 Introduction 
This paper describes how reference decisions are made 
in PROVERB, a system that verbalizes machine-found 
natural deduction (ND) proofs. A piece of argumentative 
text such as the proof of a mathematical theorem can be 
viewed as a sequence of derivations. Each such derivation 
is called a proof communicative act (PCA), following the 
viewpoint that speeches are actions. By reference choices 
we mean the explicitness of the verbalization of certain 
entities in the PCAs. Concretely, such decisions must be 
made for intermediate conclusions used as premises, as 
well as for the inference method. As an example, let us 
look at the PCA with the name Derive below: 
(Derive Derived-Formula: u * Iu = u 
Reasons : (unit(lu, U, *), u 6U) 
Method : Def-Semigroup*unit) 
Here, Derived-Formula is filled by a new intermediate 
conclusion the current PCA aims to convey, which is de- 
rivable by applying the filler of Method, with the filler of 
Reasons as premises. While the new conclusion will usu- 
ally be handed over unchanged for verbalization, there 
are alternatives for referring to both the Reasons and the 
Method. Depending on the discourse history, the follow- 
ing are two of the possible verbalizations: 
1. (inference method omitted): "Since lu is the unit ele- 
ment of U, and u is an element of U, u* 1v = u." 
2. (reasons omitted): "According to the definition of unit 
element, u * 1v= u." 
Note that, an explicit reference to a premise or an in- 
ference method is not restricted to a nominal phrase, as 
opposed to the traditional subsequent references. Despite 
this difference, the choices to be made here have much in 
common with the choices of subsequent references dis- 
cussed in more general frameworks \[Rei85, GS86, Da192\]: 
they depend on the availability of the object to be re- 
ferred to in the context and are sensitive to the segment- 
ation of the current context into an attentional hierarchy. 
Although this observation is widely agreed upon for sub- 
sequent references, no consensus about where the segment 
boundaries lie has been reached. In PROVERB, we at- 
tack this problem by viewing text generation as a com- 
bination of hierarchical planning \[Hov88, Moo89, Reigl, 
Dal92\] and local organization \[Sib90\]. Following \[GS86\], 
moreover, we assume that every posting of a new task by 
the hierarchical planning mechnism creates a new atten- 
tional unit. As a consequence, the attentional hierarchy is 
equivalent to the plan hierarchy. Based on this segment- 
ation of context, PRO VERB makes reference choices ac- 
cording to a discourse theory adapted from that of Reich- 
man \[Rei85, Hua90\]. 
2 The System PROVERB 
PROVERB is a text planner that verbalizes natural de- 
duction (ND) style proofs \[Gen35\]. Several similar at- 
tempts can be found in previous work. The system EX- 
POUND \[Che76\] is an example of direct translation: Al- 
though a sophisticated linearization is applied on the in- 
put ND proofs, the steps are then translated locally in 
a template driven way. ND proofs were tested as inputs 
to an early version of MUMBLE \[McD83\], the main aim 
however, was to show the feasibility of the architecture. A 
more recent attempt can be found in THINKER \[EP93\], 
which implements several interesting but isolated proof 
presentation strategies. PRO VERB therefore can be seen 
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as the first serious attempt to devise a comprehensive 
computational model that produces adequate argument- 
ative texts from ND style proofs. 
Most current NL text planners assume that language 
generation is planned behavior and therefore adopt a 
hierarchical planning approach \[Hov88, Moo89, Da192, 
Rei91\]. Nevertheless there is psychological evidence that 
language has an unplanned, spontaneous aspect as well 
\[Och79\]. Based on this observation, Sibun \[Sib90\] imple- 
mented a system for generating descriptions of objects 
with a strong domain structure, such as houses, chips 
and families. Once a discourse is started, local struc- 
tures suggest the next objects available. From a compu- 
tational point of view, a hierarchical planner elaborates 
recursively on the initial communicative goal until the 
final subgoals can be achieved by applying primitive op- 
erators. Local organization, on the other hand, chooses a 
part of the remaining task and carries it out. 
2.1 The Planning Framework 
PROVERB combines both of these approaches in a uni- 
form planning framework \[Hua94c\]. The hierarchical 
planning is realized by so-called top-down presentation 
operators that split the task of presenting a particular 
proof into subtasks of presenting subproofs. While the 
• overall planning mechanism follows the RST-based plan- 
ning approach \[Hov88, Moo89, Rei91\], the planning oper- 
ators resemble more the schemata in schema-based plan- 
ning \[McK85, Par88\]. Bottom-up presentation operators 
are devised to simulate the unplanned aspect, where the 
next intermediate conclusion to be presented is chosen un- 
der the guidance of the local focus mechanism. It is called 
bottom-up since one new intermediate conclusion or sub- 
proof is chosen and presented, using previously presented 
intermediate conclusions as premises. 
The two kinds of presentation operators are treated dif- 
ferently. Since top-down operators embody explicit com- 
municative norms, they are given a higher priority. Only 
when no top-down presentation operator is applicable, 
will a bottom-up presentation operator be chosen. The 
overall planning framework is realized by a function called 
Present. Taking as input a subproof, Present repeatedly 
executes a basic planning cycle until the input subproof 
is conveyed. Each cycle carries out one presentation op- 
erator, where Present always tries first to choose and 
apply a top-down operator. If impossible, a bottom-up 
operator will be chosen. The function Present is first 
called with the entire proof as the presentation task. The 
execution of a top-down presentation operator may gen- 
erate subtasks by calling it recursively. 
2.2 The Discourse Model and the Atten- 
tional Hierarchy 
The discourse carried out so far is recorded in a discourse 
model. Rather than recording the semantic objects and 
their properties, the intermediate conclusions of a ongoing 
argument or mathematical proof are stored. Therefore, 
our discourse model consists basically of the part of the 
input proof tree which has already been conveyed. The 
segmentation of the discourse is described in section 3. 
The following are some notions useful for the formulation 
of the presentation operators: 
• Task is the subproof in the input proof whose present- 
ation is the current task. 
• Local focus is the intermediate conclusion last presen- 
ted, the semantic objects involved in the local focus are 
called the focal centers. 
2.3 Proof Communicative Acts 
PCAs are the primitive actions planned to achieve com- 
municative goals. When enriched with reference de- 
cisions, they are called preverbal messages (PM). Like 
speech acts, PCAs can be defined in terms of the com- 
municative goals they fulfill as well as their possible verb- 
alizations. Based on analysis on proofs in mathematical 
textbooks, thirteen PCAs are identified and employed in 
PROVERB, see \[Hua94b\] for details. The simplist one 
conveying the derivation of a new intermediate conclu- 
sion is illustrated in the introduction. There are also 
PCAs that update the global attentional structure. These 
PCAs also convey a partial plan for the further present- 
ation. For instance, the PCA 
(Begin-Cases Goal: Formula 
Assumptions: (A B)) 
creates two attentional units with A and B as the as- 
sumptions, and Formula as the goal by producing the 
verbalization: 
"To prove Formula, let us consider the two cases by 
assuming A and B." 
2.4 Top-Down Planning 
Top-down presentation operators express communicative 
norms concerning how a proof to be presented can be split 
into subproofs, as well as how the hierarchically struc- 
tured subproofs can be mapped onto some linear order 
for presentation. Because it is not the main concern 
of this paper, we will look at only one such operator, 
which handles proof segments containing cases. The cor- 
responding schema of such a proof tree is shown in Fig- 
ure 1, where the subproof rooted at ?L4 leads to F V G, 
while subproofs rooted at ?L2 and ?La are the two cases 
proving Q by assuming F or G, respectively. Under two 
circumstances a writer may recognize that he is confron- 
ted with a proof segment containing cases. First, when 
the subproof that has the structure as given above is the 
current presentation task, tested by (task ?L1) 1. Second, 
1 Labels stand for the corresponding nodes 
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F G 
: : : 
7;4: FVa CASE ?L1 :A  O 
Figure 1: Proof Schema Case 
when the disjunction FVG has just been presented in the 
bottom-up mode, tested by (local-focus ?L4). In both cir- 
cumstances, a communication norm motivates the writer 
to first present the part leading to F V G (in the second 
case this subgoal has already been achieved), and then to 
proceed with the two cases. This norm also requires that 
certain PCAs be used to mediate between parts of proofs. 
This procedure is ,captured by the presentation operator 
below. 
Case-Implicit 
• Proof: as given in Figure 1 
• Applicability Condition: ((task ?L1) V 
(local-focus ?L4)) A (not-conveyed (?L2 ?L3)) 
• Acts: 
1. if ?L4 has not been conveyed, then present ?L4 (sub- 
goal 1) 
2. a PCA with the verbalization: "First, let us consider 
the first case by assuming F." 
3. present ?L2 (subgoal 2) 
4. a PCA with the verbalization: "Next, we consider 
the second case by assuming G." 
5. present ?L3 (subgoal 3) 
6. mark ?L1 as conveyed 
• features: (top-down compulsory implicit) 
2.5 Bottom-up Presentation 
The bottom-up presentation process simulates the un- 
planned part of proof presentation. Instead of splitting 
presentation goals into subgoals, it follows the local deriv- 
ation relation to find a proof node to be presented next. 
2.5.1 The Local Focus 
The node to be presented next is suggested by the mech- 
anism of local focus. Although logically any proof node 
having the local focus as a child could be chosen for the 
next step, usually the one with the greatest semantic over- 
lap with the focal centers is preferred. As mentioned 
above, focal centers are semantic objects mentioned in 
the proof node which is the local focus. This is based on 
the observation that if one has proved a property about 
some semantic objects, one will tend to continue to talk 
about these particular objects, before turning to new ob- 
jects. Let us examine the situation when the proof below 
is awaiting presentation. 
\[1\]: P(a,b) \[1\]: P(a,b), \[3\]: S(c) 
\[2\]: Q(a,b)' \[4\]: R(b,c) 
\[5\]: Q(a, b) A R(b, c) 
Assume that node \[1\] is the local focus, {a, b} is the set 
of focal centers, \[3\] is a previously presented node and 
node \[5\] is the current task. \[2\] is chosen as the next 
node to be presented, since it does not (re)introduce any 
new semantic element and its overlapping with the focal 
centers ({a, b}) is larger than the overlap of \[4\] with the 
focal centers ({b}). 
2.5.2 The Bottom-Up Presentation Operators 
Under different circumstances the derivation of the next- 
node is also presented in different ways. The corres- 
ponding presentation knowledge is encoded as bottom-up 
presentation operators. In this paper, we only examine 
the most frequently used bottom-up operator below: 
Derive-Bottom-Up 
* Proof: ?Node1,.. "7 ?Noden ?M ?Noden+l 
* Applicability Condition: ?Noden+l is suggested by the 
focus mechanism as the next node, and ?Node1, ..., 
?Noden are conveyed. 
• Acts: a PCA that conveys the fact that the conclu- 
sion ?Noden+l is derived from the premises ?Node1, 
..., ?Noden by applying the method ?M. 
• Features: (bottom-up general explicit detailed) 
If the conclusion, the premises and the method are in- 
stantiated to a E $1, (a E $2, $1 E $2), and def-subset 
respectively, the following verbalization can be produced: 
"Since a is an element of $1, and S1 is a subset of $2, a 
is an element of $2 according to the definition of subset." 
Currently seven bottom-up operators are integrated in 
PROVERB. 
3 The Attentional Hierarchy 
The distinction between planned and unplanned present- 
ation leads to a very natural segmentation of the discourse 
into an attentional hierarchy, since following the theory 
of Grosz and Sidner \[GS86\], there is a one-to-one cor- 
respondence between the intentional hierarchy and the 
attentional hierarchy. In this section, we illustrate the 
attentional hierarchy with the help of an example, which 
will be used to discuss reference choices. 
The input proof in Figure 2 2 is an ND style proof at the 
assertion level, abstracted from a machine-generated ND 
proof \[Hua94a\], for a theorem taken from a mathematical 
textbook. 
2The first 6 lines axe definitions and theorems use, which are 
omitted 
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7th International Generation Workshop • Kennebunkport, Maine * June 21-24, 1994 
NNo S;D Formula 
7. 7; • group(F,*)Asubgroup(U,F,*)Aunit(F,l,*)Aunit(U, 1v,*) 
8. 7; • UCF 
9. 7; • 1v E U 
10. 7; • 3~x E U 
11. ;11 • u E U 
12. 7;11 • u* 1u = u 
13. 7;11 • u E F 
14. 7;11 • Iu E F 
15. 7;11 • semigroup(F, *) 
16. 7;11 • solution(u, u, 1u, F, *) 
17. 7;11 • u * 1 = u 
18. 7;11 • 1 E F 
19. 7;11 • solution(u, u, 1, F, *) 
20. 7;11 • 1 = IU 
21. 7; • 1 = 1U 
22. ; • group(F, *) A subgroup(U, F, *) A unit(F, 1, *) A 
unit(U, 1u,*) ~ 1 = Iy 
F~eason 
(Hyp) 
(Def-subgroup 7) 
(Def-unit 7) (3 9) 
(Hyp) 
(Def-unlt 7 11) 
(Def-subset 8 11) 
(Defosubset 8 9) 
(Def-group 7) 
(Def-solutionl2 13 14 15) 
(Def-tm.it 7 13) 
(Def-unlt 7) 
(Def-solutionl3 17 18 15) 
(Th-solution 17 16 19) 
(Choice 10 20) 
(Ded 7 21) 
Figure 2: Abstracted Proof about Unit Element of Subgroups 
Theorem: 
Let F be a group and U a subgroup of F. If 1 and lu 
are unit elements of F and U respectively, then 1 = 1v. 
The definitions of semigroup, group, and unit are ob- 
vious, solution(a, b, c, F, *) should be read as "c is a solu- 
tion of the equation a * z = b in F." 
The proof-to be presented is represented in a linearized 
version of ND proofs. In this formalism, every proof is a 
sequence of proof lines, each of them is of the form: 
Label ~ ~- Conclusion (Justification reason-pointers) 
where Justification is either an ND inference rule, a defin- 
ition or theorem, which justifies the derivation of the 
Conclusion using formulas in lines pointed to by reason- 
pointers as the premises. ~ can be ignored for our pur- 
pose. 
The corresponding discourse model after the comple- 
tion of the presentation is a proof tree shown in Figure 
3. Children of nodes are given in the order as they have 
been presented. The circles denote nodes which are first 
derived at this place, and nodes in the form of small boxes 
are copies of some previously derived nodes, which are 
used as premises again. The big boxes represent atten- 
tional units called proof units, created during the present- 
ation process. The naturalness of this segmentation is 
largely due to the naturalness of the top-down presenta- 
tion operators. For example, unit U2 has two subordinate 
units U3 and U4. This reflects a natural shift of atten- 
tion between a subproof that derives a formula of the 
pattern 3~P(x) (node 10, 3~x E U), and the subproof 
that proceeds after assuming a new constant u satisfying 
P(u) (node 11, ul E U). There are also elementary units 
composed of multiple PCAs, such as U5 and U6. They 
produce two important premises required by a theorem 
about the concept solution, which are applied at node 
20. It is interesting to node that elementary attentional 
units that contain multiple PCAs would be impossible, if 
we did not distinguish between hierarchical planning and 
local organization. 
Adapting the theory of Reichman for our purpose 
\[Rei85\], we assume that each proof unit may have one 
of the following status: 
• a unit is said to be. open, if its root is still awaiting to 
be conveyed. 
- The active proof unit is the innermost proof unit con- 
taining the local focus. There is exactly one active 
proof unit at any moment. 
- The controlling proof unit is the innermost proof unit 
containing the active unit. 
- precontrol proof units are proof units containing the 
controlling proof unit. 
• Closed units are proof units containing only conveyed 
proof nodes. 
4 A Classification of Reference 
Forms 
This section presents a classification of the possible forms 
with which mathematicians refer to intermediate conclu- 
sions previously proved (called reasons) or to methods of 
inference. The classification is based on our analysis of 
proofs presented in mathematical textbooks. 
4.1 Reference Forms for Reasons 
Three reference forms have been identified for reasons in 
naturally occurring proofs: 
1. The omit form: where a reason is not mentioned at all. 
2. The explicit form: where a reason is literally stated. 
For instance, if the omit form and the explicit form 
are adopted for the first respectively second reason in 
the PCA in Section 1, the sentence may be produced: 
"Since u is an element in U, u * 1u = u." 
3. The implicit form: By an implicit form we mean that 
although a reason is not verbalized directly, an implicit 
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7th International Generation Workshop • Kennebunkport, Maine • June 21-24, 1994 
us 
U4 
°' C3 io, i 
I 
I 
U2 
UX 
Figure 3: Proof Tree for Satz 1.9 
hint is nevertheless given in other components of the 
PCA. That is, either in the verbalization of the infer- 
ence method, or in that of the conclusion. For example, 
in the verbalization below 
"Since u is an element in U, u* 1v = u by the 
definition of unit." 
the first reason of the PCA in Section 1, "since 1v is 
the unit element of U" is hinted at by the inference 
method which reads "by the definition of unit". 
Note that although omit and implicit forms lead to an 
identical surface structure, the existence of an implicit 
hint in the other part of the verbalization affects a reader's 
understanding. 
4.2 Reference Forms for Methods 
PROVERB must select referring expressions for methods 
of inference in PCAs as well. Below are the three reference 
forms identified, which are analogous to the correspond- 
ing cases for reasons: 
1. the explicit form: this is the case where a writer may 
decide to indicate explicitly which inference rule he is 
using. For instance, explicit translations of domain- 
specific rules could look like: "by the definition of unit 
element", or "by the uniqueness of solution." ND rules 
have usually standard verbalizations. 
2. the omit form: in this case a word such as "thus" or 
"therefore" will be used. 
3. The implicit form: Similar to the implicit form for 
reasons, an implicit hint to a domain-specific inference 
method can be given either in the verbalization of the 
reasons, or in that of the conclusion. 
5 Making Reference Choices for 
Reasons 
Because reasons are intermediate conclusions proved pre- 
viously in context, their reference choices have much in 
common with the problem of choosing anaphoric referring 
expressions in natural language generation in general. A 
number of theories have been put forward to account for 
the pronominalization, which is usually ascribed to the fo- 
cus mechanism. For this purpose, concepts like activated- 
ness, foregroundness and consciousness have been intro- 
duced. More recently, the shift of focus has been further 
investigated in the light of a more structured flow of dis- 
course \[Rei85, GS86, Dal92\]. The issue of salience is also 
studied in a broader framework in \[PC93\]. Apart from 
salience, it is also shown that referring expressions are 
strongly influenced by other aspects of human preference. 
For example, easily perceivable attributes and basic-level 
attributes values are preferred \[DH91, Da192, RD92\]. 
Common to all discourse based theories, the update of 
the focus status is tightly coupled to the factoring of the 
flux of text into segments. As illustrated in section 3, we 
basically follow the approach of Grosz and Sidner \[GS86\] 
in that a direct correspondence between the plan hier- 
archy and the attentional spaces is assumed. 
With the segmentation problem settled, the reference 
choices in our theory largely follow the approach of Reich- 
man. Reichman handles the reference problem in a more 
general framework of her discourse grammar \[Rei85\]. 
Based on empirical data, Reichman argues that the choice 
of referring expressions is constrained both by the status 
of the discourse space and by the object's level of focus 
within this space. In her theory, there are seven status as- 
signments a discourse space may have at any given time. 
Within a discourse space, four levels of focus can be as- 
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7th International Generation Workshop • Kennebunkport, Maine • June 21-24, 1994 
signed to individual objects: high, medium, low, or zero, 
since there are four major ways of referring to an object 
using English, namely, by using a pronoun, by name, by a 
description, or implicitly. Thirteen rules are formulated 
to assign level of focus to objects when they are activ- 
ated, either with the initialization of a discourse unit or 
when they are added to the active unit. Four rules further 
reassign the level of focus on reentrance to a suspended 
discourse space. Based on the status assignment of dis- 
course spaces, as well as the level of focus of individual 
objects, four rules are formulated constraining adequate 
referring expressions. 
In short, Reichman takes into account both the fore- 
ground and background status of discourse spaces as well 
as the level of focus of individual objects. As a simplifica- 
tion for argumentative discourse, the notions of structural 
closeness and textual closeness are introduced. 
The structural closeness of a reason reflects the fore- 
ground and background character of the innermost proof 
unit containing it. Intuitively, reasons that may still re- 
main in the focus of attention at the current point from 
the structural perspective are considered as structurally 
close. Otherwise they are considered as structurally dis- 
tant. If a re,on, for instance, is last mentioned or proved 
in the active proof unit (the unit a writer is currently 
working on), it is more likely that this reason should still 
remain in his focus of attention. On the other hand, if 
the reason is in a closed unit, and is not the root, it is 
very likely that the reason has already been moved out 
of the writer's focus of attention. Although the notion of 
fore- and backgroundness might actually be a continuum, 
our theory only distinguishes between reasons residing in 
proof units which are structurally close or structurally dis- 
rant. Rules assigning this structural status are given as 
following. 
1. Reasons in the active unit are structurally close. 
2. Reasons in the controlling unit are structurally close. 
3. Reasons in closed units: 
(a) reasons that are the root of a closed proof unit imme- 
diate subordinate to the active unit are structurally 
close. 
(b) Other reasons in a closed unit are structurally dis- 
tant. 
4. Reasons in precontrol proof units are structurally dis- 
tant. 
Note that, the rules are specified with respect to the 
innermost proof unit containing a proof node. Rule 3 
means that only the conclusions of closed subordinated 
subproofs still remain in the focus of attention. As a 
special treatment, premises of the entire theorem will be 
defined as both structurally distant and far in distance, 
if they are not repeated at the beginning of the proof. 
The textual closeness is used as an approximation to the 
level of focus of an individual reason. In general, the level 
of focus of an object is established when it is activated, 
and decreases with the flow of discourse. In Reichman's 
theory, although four levels of focus can be established 
upon activation, only one is used in the formulation of the 
four reference rules. In other words, it suffices to track the 
status high alone. Based on the discussion above, we use 
only two values to denote the level of focus of individual 
intermediate conclusions, depending solely on the textual 
distance between the last mentioning of a reason and the 
current sentence where the reason is referred to. 
In summary, we assume that each intermediate conclu- 
sion is put into high focus when it is presented as a newly 
derived result or cited as a reason supporting the deriv- 
ation of a further intermediate result. This level of focus 
decreases, either when a proof unit is moved out of the 
foreground of discussion, or with the increase of textual 
distance. On account of the above, the four reference 
rules used in our computational model are given below. 
Choices for Referring Expressions for Reasons 
1. If a reason is both structurally and textually close, it 
will be omitted. 
2. If a reason is structurally close but textually far, first 
try to find an implicit form, if impossible, use an expli- 
cit form. 
3. If a reason is structurally distant but textually close, 
first try to find an implicit form, if impossible, omit it. 
4. An explicit form will be used for reasons that are both 
structurally and textually far. 
Notice that the result of applying rule 2 and rule 3 
depends on the fact that an implicit form is available, 
which often interacts with the verbalization of the rest 
of the PCA. In particular, it interacts with the reference 
choices for inference methods. In PROVERB as it cur- 
rently stands, the interaction is realized by associating a 
keyword with the verbalization of every predicate, func- 
tion, and assertion. For instance, suppose the verbaliza- 
tion of unit(F, 1, *) as a reason is "since 1 is an unit ele- 
ment of F", and the verbalization of the definition of unit 
element as an inference method is "by the definition of 
the unit element". Both the predicate unit and the defin- 
ition are associated with the same keyword "unit". Based 
on this information, PROVERB assumes that the verb- 
alization of the reason unit(F, 1, *) and the verbalization 
of the definition of unit element hint at each other. Dis- 
tance is currently calculated in an ad hoc way by counting 
the PCAs uttered after the corresponding reason was last 
explicitly mentioned. 
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7th International Generation Workshop • Kennebunkport, Maine • June 21-24, 1994 
6 Making Reference Choices for 
Inference Methods 
Like the reference to a reason, the explicitness or im- 
plicitness of referring to an inference method at a par- 
ticular point depends on whether the particular method 
can be easily called into the foreground of the focus of 
attention. In contrast to references to reasons, this is 
evidently irrelevant to the particular discourse structure 
concerned. Actually it is less concerned with the proof 
context than with the user's familiarity with the partic- 
ular inference method. This referring behavior remains 
• the same throughout a whole discourse, similar to the 
referring behavior relating to the so-called canonical sali- 
ence \[PC93\]. In the case of applications of definitions or 
theorems, it depends on the reader's familiarity with the 
corresponding definition or theorem. This is found to be 
sensitive to the global hierarchy of the mathematical the- 
ories. As it currently stands, PROVERB distinguishes 
only between assertions in the underlying theories and 
assertions belonging to the current theory. The reference 
choice rules for inference methods currently incorporated 
are listed as follows. 
° 
Choices for Referring Expressions for Methods 
1. Reference Choices for ND Inference Rules 
(a) All non-structural ND rules (such as eliminations 
of quantifiers) will be omitted (In the case of PCA 
DERIVE, a word like "thus", "hence", etc. will be 
used), because the readers are supposed to be famil- 
iar with the elementary logic. 
(b) All structural ND rules (such as the one dividing 
proofs into eases) will be explicitly given. Although 
they are also familiar to the readers, they provide 
landmarks for the overall proof structure. 
2. Reference Choices for Applications of Assertions 
Readers are assumed to be familiar with definitions and 
theorems of the "underlying theories" upon which the 
current theory is based. For example, when we are 
reasoning about properties of group theory we assume 
that the users are familiar with basic set theory: 
(a) Applications of definitions and theorems of underly- 
ing theories will be omitted. 
(b) For applications of definitions or theorems of the cur- 
rent theory, try first to find an implicit form. If im- 
possible, an explicit indication will be given. 
7 An Integrated Algorithm for 
Reference Choices 
As illustrated above, reference choices for reasons and for 
methods interact with each other. This section describes 
an algorithm that combines the reference choice rules for 
reason and the reference choice rules for methods, to pro- 
duce preverbal messages (PMs) from PCAs. As such, the 
main task is to utilize the interaction between the two 
sets of reference rules to eliminate the indeterminacy in 
both of the rule sets. The indeterminacy lies in reference 
rule 1 and 2 in Section 5 and in reference rule 2(b) in Sec- 
tion 6, which need information on decisions made by the 
other set of rules. In other words, decisions in one rule 
set may help to narrow the alternatives in the other set. 
PROVERB first makes the reference choice for the infer- 
ence method. While doing so, it looks ahead and takes 
the possible reference choices for reasons into account. If 
still no unique choice can be made, the decision is made 
according to a predetermined order. Concretely the ex- 
plicit form will be chosen for rule 2(b) in Section 6. After 
the reference form of the method has been determined, a 
unique reference form can always be determined for the 
reasons. The concrete algorithm is omitted due to space 
restrictions. 
Now we continue with the subgroup example introduced 
in section 2. The PCA below aims to convey the deriv- 
ation of proof node 9 (1v E U) from a part of node 7 
(unit(Iv, U, *)), justified by the application of the defin- 
ition of the unit element in semigroups. 
(Derive Derived-Formula: iu EU 
Reasons : unit(iU, U, *) 
Method: Def-Semigroup*Unit) 
The current unit is U3. U2 and U1 are the controlling 
and precontrol unit, respectively, see Figure 3. Since node 
7 is in the controlling unit and is mentioned last only two 
steps earlier, it is therefore judged as both structurally 
close and near in distance. Rule 1 in Section 5 is applic- 
able and the omit form is chosen. Since the definition of 
the unit element resides in the current theory, Rule 2(b) 
in Section 6 suggests that its application be referred to 
either implicitly or explicitly. Because the implicit option 
is ruled out by the omit form for reasons, the explicit form 
is chosen. The PM below is therefore generated: 
(Derive Derived-Formula: iu qU 
Method: Def-Semigroup*Unit) 
Next let us jump over some steps and consider the PCA 
below. 
(Derive Derived-Formula: u * IU = u 
Reasons: (unit(1u, U,*), u qU) 
Method: Def-unit) 
This PCA is generated to convey that node 12 (u * 
1u) can be derived from part of node 7 (unit(1u, U,*)) 
and node 11 (u E U) by applying the definition of the 
unit element. U5 is now the current unit, with U4 and 
U2 as controlling and precontrol unit. U3 is the unique 
closed unit. Reason node 7 is now structurally distant 
but still near in distance, and node 11 is in the current 
unit, and is the node last conveyed, therefore, both 7 and 
151 
7th International Generation Workshop • Kennebunkport, Maine • June 21-24, 1994 
11 are omitted. The reference form for the definition of 
unit element is decided upon as above. The PM below is 
generated: 
(Derive Derived-Formula: u • iu ---- u 
Method : Def-Semigroup*Unit) 
Fifteen preverbal messages are generated by PRO VERB 
for our example. One sentence in English is gener- 
ated from each PM by the surface generator TAG-GEN 
\[Kilger94\]. The text as generated follows. 
Let F be a group, U be a subgroup of F, 1 be a unit 
element ofF and 1u be a unit element of U. According 
to the definition of unit element, 1u E U. Therefore 
there is an X, X E U. Now suppose that ul is such an 
X. According to the definition of unit element, ut * lu = 
ul. Since U is a subgroup of F, U C F. Therefore 
1u E F. Similarly ul E F, since ul E U. Since F is a 
group, F is a semigroup. Because ul * 1u = ul, 1u is a 
solution of the equation ul * X = u~. Since 1 is a unit 
element ofF, ul * 1 = ul. Since 1 is a unit element 
ofF, 1 E F. Because ul E F, 1 is a solution of the 
equation ul * X = ul. Since F is a group, 1u = 1 by the 
uniqueness_of solution. This conclusion is independent 
of the choice of the element ul. 
8 Conclusion 
This paper describes the way in which PROVERB refers 
to previously conveyed intermediate results and inference 
methods while verbalizing natural deduction style proofs. 
By distinguishing between the planned and unplanned 
part of NL generation, PRO VERB achieves a natural seg- 
mentation of context into an attentional hierarchy. Based 
on this segmentation, PROVERB makes reference de- 
cisions basically according to a discourse theory adapted 
from Reichmann for this special application. The first 
experience shows that output texts are of good quality. 
Currently, it proves difficult to compare text generated 
by PROVERB with naturally occurring texts, since the 
latter are usually at a still higher level of abstraction then 
the assertion level proofs we can reconstruct \[Hua94a\]. 
Nonetheless it might still be useful to build up a small 
corpus of texts. On the other hand, the naturMness of 
references could also be improved by further experiment- 
ing with different settings of the ad hoc thresholds in 
the system. We are also exploring more flexible lexicon 
choices as well as refinement of text planning process. 
Acknowledgment This work was supported by 
the Deutsche Forschungsgemeinschaft, SFB 314 (D2). 
Thanks are due to Dan Nesmith, who carefully read this 
final version. I am also grateful to the two anonymous 
reviewers for their critical and constructive remarks. 

References 
\[Che76\] D. Chester. The translation of formal proofs into 
English. AL 7, 1976. 
\[Dal92\] R. Dale. Generating Referring Ezpressions. MIT 
Press, 1992. 
\[DH91\] R. Dale and N. Haddock. Content determination 
in the generation of referring expressions. Compu- 
tational Intelligence, 7(4), 1991. 
\[EP93\] A. Edgar and F. J. Pelletier. Natural language ex- 
planation of natural deduction proofs. In Proc. o\] 
the lth Conj. o\] the Pacific Assoc. for Comp. Lin- 
guistics. Simon Fraser University, 1993. 
\[Gen35\] G. Gentzen. Untersuchungen fiber das logische 
Schliet3en I. Math. Zeitschrift, 39, 1935. 
\[GS86\] B.J. Grosz and C. L. Sidner. Attention, intentions, 
and the structure of discourse. Computational Lin- 
guistics, 12(3), 1986. 
\[Hov88\] E. H. Hovy. Generating Natural Language under 
Pragmatic Constraints. Lawrence Erlbaum, 1988. 
\[Huag0\] X Huang. Reference choices in mathematical proofs. 
In Proc. of ECAI.90, Pitman, 1990. 
\[Hua94a\] X. Huang. Reconstructing proofs at the assertion 
level. In Proc. o\] l~th CADE. 1994, forthcoming. 
\[Hua94b\] X. Huang. Human Oriented Proof Presentation: A 
Reconstructive Approach. PhD thesis, University of 
Saarbrficken, 1994, forthcoming. 
\[Hua94c\] X. Huang. Planning argumentative text. In Proc. of 
15th COLING. 1994, forthcoming. 
\[Kilger94\] A. Kflger. Using UTAGs for Incremental and Par- 
allel Generation. Computational Intelligence, 1994, 
forthcoming. 
\[Lev89\] W. J. M. Levelt. Speaking: From Intention to Artic- 
ulation. MIT Press, 1989. 
\[McD83\] David D. McDonald. Natural language generation as 
a computational problem. In Brady/Berwick: Com- 
putational Models o\] Discourse. MIT Press, 1983. 
\[McK85\] K. R. McKeown. Text Generation. Cambridge Uni- 
versity Press, 1985. 
\[Moo89\] J. D. Moore. A Reactive Approach to Explanation 
in Expert and Advice-Giving Systems. PhD thesis, 
Univ. of California, 1989. 
\[Och79\] E. Ochs. Planned and unplanned discourse. Syntax 
and Semantics, 12, 1979. 
\[Par88\] C. Paris. Tailoring object descriptions to a user's 
level of expertise. Computational Linguistics, 14, 
1988. 
\[PC93\] T. Pattabhiraman and N Cercone. Decision-theoretic 
salience interactions in language generation. Proc. of 
I\]CAI-93, Morgan Kanfmann, 1993. 
\[RD92\] E. Reiter and R. Dale. A fast algorithm for the gener- 
ation of referring expressions. In Proc. o\] COLING- 
92, 1992. 
\[Rei85\] R. Reichman. Getting Computer to Talk Like You 
and Me. MIT Press, 1985. 
\[Rei91\] N. Reithinger. Eine parallele Architektur zur inkre- 
menteller Dialogbeitr~ige. PhD thesis, University of 
Saarbrficken, 1991. 
\[Sib90\] P. Sibun. The local organization of text. In Proc. of 
the 5th International Natural Language Generation 
Workshop, 1990. 
