GENERALIZED CHART ALGORITHM: 
AN EFFICIENT PROCEDURE FOR 
COST-BASED ABDUCTION 
Yasuharu Den 
ATR Interpreting Telecommunications Research Laboratories 
2-2 Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, JAPAN 
Tel: +81-7749-5-1328, Fax: +81-7749-5-1308, e-mail: denQitl.atr.co.jp 
Abstract 
We present an efficient procedure for cost-based ab- 
duction, which is based on the idea of using chart 
parsers as proof procedures. We discuss in de- 
tail three features of our algorithm -- goal-driven 
bottom-up derivation, tabulation of the partial re- 
sults, and agenda control mechanism -- and report 
the results of the preliminary experiments, which 
show how these features improve the computational 
efficiency of cost-based abduction. 
Introduction 
Spoken language understanding is one of the most 
challenging research areas in natural language pro- 
cessing. Since spoken language is incomplete in var- 
ious ways, i.e., containing speech errors, ellipsis, 
metonymy, etc., spoken language understanding 
systems should have the ability to process incom- 
plete inputs by hypothesizing the underlying infor- 
mation. The abduction-based approach (Hobbs et 
al., 1988) has provided a simple and elegant way to 
realize such a task. 
Consider the following 3apanese sentence: 
(1) Sfseki kat- ta 
(a famous writer) buy PAST 
This sentence contains two typical phenomena aris- 
ing in spoken language, i.e., metonymy and the el- 
lipsis of a particle. When this sentence is uttered 
under the situation where the speaker reports his 
experience, its natural interpretation is the speaker 
bought a SSseki novel. To derive this interpreta- 
tion, we need to resolve the following problems: 
• The metonymy implied by the noun phrase 
S6seki is expanded to a S6seki novel, based on 
the pragmatic knowledge that the name of a 
writer is sometimes used to refer to his novel. 
• The particle-less thematic relation between the 
verb katta and the noun phrase SSseki is deter- 
mined to be the object case relation, based on the 
semantic knowledge that the object case relation 
between a trading action and a commodity can 
be linguistically expressed as a thematic relation. 
This interpretation is made by abduction. For 
instance, the above semantic knowledge is stated, 
in terms of the predicate logic, as follows: 
(2) sem(e,x) C trade(e) A commodity(x) A obj(e,x) 
Then, the inference process derives the consequent 
sem(e,x) by hypothesizing an antecedent obj(e,x), 
which is never proved from the observed facts. This 
process is called abduction. 
Of course, there may be several other possibili- 
ties that support the thematic relation sem(e,x). 
For instance, the thematic relation being deter- 
mined to be the agent case relation, sentence (1) 
can have another interpretation, i.e., Sfseki bought 
something, which, under some other situations, 
might be more feasible than the first interpretation. 
To cope with feasibility, the abduction-based model 
usually manages the mechanism for evaluating the 
goodness of the interpretation. This is known as 
cost-based abduction (Hobbs et al., 1988). 
In cost-based abduction, each assumption 
bears a certain cost. For instance, the assump- 
tion obj(e,x), introduced by applying rule (2), is 
specified to have a cost of, say, $2. The goodness of 
the interpretation is evaluated by accumulating the 
costs of all the assumptions involved. The whole 
process of interpreting an utterance is depicted in 
the following schema: 
1. Find all possible interpretations, and 
2. Select the one that has the lowest cost. 
In our example, the interpretation that as- 
sumes the thematic relation to be the object case 
relation, with the metonymy being expanded to 
a S6seki novel, is cheaper than the interpretation 
that assumes the thematic relation to be the agent 
case relation; hence, the former is selected. 
An apparent problem here is the high compu- 
tational cost; because abduction allows many pos- 
sibilities, the schema involves very heavy compu- 
tation. Particularly in the spoken language under- 
standing task, we need to consider a great number 
of possibilities when hypothesizing various underly- 
ing information. This makes the abduction process 
218 
computationally demanding, and reduces the prac- 
ticality of abduction-based systems. The existing 
models do not provide any basic solution to this 
problem. Charniak (Charniak and Husain, 1991; 
Charniak and Santos Jr., 1992) dealt with the prob- 
lem, but those solutions are applicable only to the 
propositional case, where the search space is rep- 
resented as a directed graph over ground formulas. 
In other words, they did not provide a way to build 
such graphs from rules, which, in general, contain 
variables and can be recursive. 
This paper provides a basic and practical so- 
lution to the computation problem of cost-based 
abduction. The basic idea comes from the natural 
language parsing literature. As Pereira and War- 
ren (1983) pointed out, there is a strong connec- 
tion between parsing and deduction. They showed 
that parsing of DCG can be seen as a special case 
of deduction of Horn clauses; conversely, deduction 
can be seen as a generalization of parsing. Their 
idea of using chart parsers as deductive-proof pro- 
cedures can easily be extended to the idea of using 
chart parsers as abductive-proof procedures. Be- 
cause chart parsers have many advantages from the 
viewpoint of computational efficiency, chart-based 
abductive-proof procedures are expected to nicely 
solve the computation problem. Our algorithm, 
proposed in this paper, has the following features, 
which considerably enhance the computational ef- 
ficiency of cost-based abduction: 
1. Goal-driven bottom-up derivation, which reduces 
the search space. 
2. Tabulation of the partial results, which avoids the 
recomputation of the same goal. 
3. Agenda control mechanism, which realizes var- 
ious search strategies to find the best solution 
efficiently. 
The rest of the paper is organized as follows. 
First, we explain the basic idea of our algorithm, 
and then present the details of the algorithm along 
with simple examples. Next, we report the results 
of the preliminary experiments, which clearly show 
how the above features of our algorithm improve 
the computational efficiency. Then, we compare 
our algorithm with Pereira and Warren's algorithm, 
and finally conclude the paper. 
Head-driven Derivation 
Pereira and Warren showed that chart parsers 
can be used as proof procedures; they presented the 
Earley deduction proof procedure, that is a gener- 
alization of top-down chart parsers. However, they 
mentioned only top-down chart parsers, which is 
not always very efficient compared to bottom-up 
(left-corner) chart parsers. It seems that using left- 
corner parsers as proof procedures is not so easy, 
...........':"'"'" 
Figure 1: Concept of Head-driven Derivation 
unless the rules given to the provers have a certain 
property. Here, we describe under what conditions 
left-corner parsers can be used as proof procedures. 
Let us begin with the general problems of Horn 
clause deduction with naive top-down and bottom- 
up derivations: 
• Deduction with top-down derivation is affected 
by the frequent backtracking necessitated by the 
inadequate selection of rules to be applied. 
• Deduction with bottom-up derivation is affected 
by the extensive vacuous computation, which 
never contributes to the proof of the initial goal. 
These are similar to the problems that typi- 
cally arise in natural language parsing with naive 
top-down and bottom-up parsers. In natural lan- 
guage parsing, these problems are resolved by intro- 
ducing a more sophisticated derivation mechanism, 
i.e., left-corner parsing. We have attempted to ap- 
ply such a sophisticated mechanism to deduction. 
Suppose that the proof of a goal g(x,y) can 
be represented in the manner in Figure 1; the first 
argument x of the goal g(x,y) is shared by all the 
formulas along the path from the goal g(z,y) to 
the left corner am(z,zm). In such a case, we can 
think of a derivation process that is similar to left- 
corner parsing. We call this derivation head-driven 
derivation, which is depicted as follows: 
Step 1 Find a fact a(w,z) whose first argument 
w unifies with the first argument x of the goal 
g(x,y), and place it on the left corner. 
Step 2 Find a rule am-l(W,Zrn-l) C a(W,Zm)/~ 
BZ ^ ... A Bn whose leftmost antecedent 
a(W,Zm) unifies with the left-corner key a(x,z), 
and introduce the new goals B1, ..., and Bn. If 
all these goals are recursively derived, then cre- 
ate the consequent a,,~_ 1 ( z ,zm_ 1 ), which domi- 
nates a(x,zm), B1, ..., and Bn, and place it on 
the left corner instead of a(x,z). 
Step3 If the consequent am-l(x,zm_l) unifies 
with the goal g(z,y), then finish the pro- 
cess. Otherwise, go back to step2 with 
am- 1 (x,zm_l) being the new left-corner key. 
219 
Left-corner parsing of DCG is just a special 
case of head-driven derivation, in which the in- 
put string is shared along the left border, i.e., the 
path from a nonterminal to the leftmost word in 
the string that is dominated by that nonterminal. 
Also, semantic-head-driven generation (Shieber el 
al., 1989; van Noord, 1990) and head-corner pars- 
ing ivan Noord, 1991; Sikkel and op den Akker, 
1993) can be seen as head-driven derivation, when 
the semantic-head/syntactic-head is moved to the 
leftmost position in the body of each rule and the 
argument representing the semantic-feature/head- 
feature is moved to the first position in the argu- 
ment list of each formula. 
To apply the above procedures, all rules must 
be in chain form arn--l(W,Zrn-~) C arn(W,Zm) A B1 
A ... A Bn; that is, in every rule, the first argu- 
ment of the leftmost antecedent must be equal to 
the first argument of the consequent. This is the 
condition under which left-corner parsers can be 
used as proof procedures. Because this condition is 
overly restrictive, we extend the procedures so that 
they allow non-chain rules, i.e., rules not in chain 
form. Step 1 is replaced by the following: 
Step 1 Find a non-chain rule a(w,z) C B1 A... A 
B~ such that the first argument w of the con- 
sequent a(w,z) unifies with the first argument 
z of the goal g(x,y), and introduce the new 
goals B1, ..., and /3,. A fact is regarded as 
a non-chain rule with an empty antecedent. If 
all these goals are recursively derived, then cre- 
ate the consequent a(z,z), which dominates B1, 
..., and B,, and place it on the left corner. 
Generalized Chart Algorithm 
The idea given in the previous section realizes the 
goal-driven bottom-up derivation, which is the first 
feature of our algorithm. Then, we present a more 
refined algorithm based upon the idea, which real- 
izes the other two features as well as the first one. 
Chart Parsing and its Generalization 
Like left-corner parsing, which has the drawback of 
repeatedly recomputing partial results, head-driven 
derivation will face the same problem when it is 
executed in a depth-first manner with backtrack- 
ing. In the case of left-corner parsing, the prob- 
lem is resolved by using the tabulation method, 
known as chart parsing (Kay, 1980). A recent 
study by Haruno et al. (1993) has shown that 
the same method is applicable to semantic-head- 
driven generation. The method is also applicable 
to head-driven derivation, which is more general 
than semantic-head-driven generation. 
To generalize charts to use in proof procedures, 
m( <\[AJ,\[B\]>,\[A,B\]) ,oO.O*°"°"°O°Oo..,. 
...."" ..... 
/ / ',..., ,.. 
/ / ... k i /. '-.... \ 
/ \\ 
h( <IA\],\[BI> A~> ) I <II.\[BI>~.. m( <\[\],~f~..){\]) 
g-" ..... -%- "A ..... C'" "" ........ ":':~ ~\[A1JBI~(" -- ~/ <\[l \[l>~ 
Z \ u 
".... ..... I (Some labels ..... m(<\[A\],\[_\]..~',tA\]) 
are omitted) .................... .. 
m( <\[A\],IB\]>,\[B,A\]) 
Figure 2: Example of Generalized Charts 
we first define the chart lexicons. In chart pars- 
ing, lexicons are the words in the input string, 
each of which is used as the index for a subset 
of the edges in the chart; each edge incident from 
(the start-point of) lexicon w represents the sub- 
structure dominating the sub-string starting from 
w. In our case, from the-similarity between left- 
corner parsing and head-driven derivation, lexicons 
are the terms that occur in the first argument po- 
sition of any formula; each edge incident from (the 
start-point of) lexicon x represents the substruc- 
ture dominating the successive sequence of the de- 
rived formulas starting from the fact in which z 
occupies the first argument position. For example, 
in the chart representing the proof in Figure 1, all 
the edges corresponding to the formulas on the left 
border, i.e. am(X,Zrn), am--l(Z,Zm--1),..., al(x,zl) 
and g(z,y), are incident from (the start-point of) 
lexicon z, and, hence, x is the index for these edges. 
Following this definition of the chart lexicons, 
there are two major differences between chart 
parsing and proof procedures, which Haruno also 
showed to be the differences between chart parsing 
and semantic-head-driven generation. 
1. In contrast to chart parsing, where lexicons are 
determined immediately upon input, in proof 
procedures lexicons should be incrementally in- 
troduced. 
2. In contrast to chart parsing, where lexicons are 
connected one by one in a linear sequence, in 
proof procedures lexicons should be connected in 
many-to-many fashion. 
In proof procedures, the chart lexicons are not 
determined at the beginning of the proof (because 
220 
we don't know which formulas are actually used in 
the proof), rather they are dynamically extracted 
from the subgoals as the process goes. In addi- 
tion, if the rules are nondeterministic, it sometimes 
happens that there are introduced, from one left- 
corner key, a(x,z), two or more distinct succes- 
sive subgoals, bl(wl,y~), b2(w2,y2), etc., that have 
different first arguments, w 1, w 2, etc. In such a 
case, one lexicon x should be connected to two or 
more distinct lexicons, w 1, w 2, etc. Furthermore, 
it can happen that two or more distinct left-corner 
keys, al(xl,zl), a2(x2,z2), etc., incidentally intro- 
duce the successive subgoals, bl(w,yl), b2(w,y~), 
etc., with the same first argument w. In such a 
case, two or more distinct lexicons, x 1, x 2, etc., 
should be connected to one lexicon w. Therefore, 
the connections among lexicons should be many- 
to-many. Figure 2 shows an example of charts with 
many-to-many connections, where the connections 
are represented by pointers A, B; etc. 
The Algorithm 
We, so far, have considered deduction but not ab- 
duction. Here, we extend our idea to apply to ab- 
duction, and present the definition of the algorithm. 
The extension for abduction is very simple. 
First, we add a new procedure, which introduces 
an assumption G for a given goal G. An assump- 
tion is treated as if it were a fact. This means that 
an assumption, as well as a fact, is represented as a 
passive edge in terms of the chart algorithm. Sec- 
ond, we associate a set S of assumptions with each 
edge e in the chart; S consists of all the assump- 
tions that are contained in the completed part of 
the (partial) proof represented by the edge e. More 
formally, the assumption set 5 associated with an 
edge e is determined as follows: 
1. If e is a passive edge representing an assumption 
A, then S-- {A}. 
2. If e is a passive/active edge introduced from a 
non-chain rule, including fact, then S is empty. 
3. If e is a passive/active edge predicted from a 
chain rule with a passive edge e' being the left- 
corner key, then S is equal to the assumption set 
S' of e'. 
4. If e is a passive/active edge created by combining 
an active edge el and a passive edge e2, then 
,-q = $1 U $2 where 81 and ~q2 are the assumption 
sets of el and e2, respectively. 
Taking these into account, the definition of our 
algorithm is as follows, f is a function that assigns 
a unique vertex to each chart lexicon. The notation 
A:S stands for the label of an edge e, where A is 
the label of e in an ordinary sense and S is the 
assumption set associated with e. 
Initialization Add an active edge \[\[?IG\]-I-:¢ to 
the chart, looping at vertex 0, where G is the 
initial goal. 
Apply the following procedures repeatedly until 
no procedures are applicable. 
Introduction Let e be an active edge labeled 
\[...\[?\]Bj...\]A:S incident from vertex s to t, 
where Bj = bj (zj,yj) is the first open box in e. 
1. If the lexicon xj is never introduced in the 
chart, then introduce it and run a pointer 
from t to f(zj). Then, do the following: 
(a) For every non-chain rule a(w,z)C B1 A 
... A Bn, including fact, such that w uni- 
fies with zi, create an active edge la- 
beled \[\[?\]Bl'"\[?lS,~\]a(xj,z):¢ between ver- 
tex f(xj) and f(zj) + 1. (Create, instead, 
a passive edge labeled a(xj,z):¢ when the 
rule is a fact, i.e. n = 0.) 
(b) Create a passive edge labeled Bj:{Bj} be- 
tween vertex f(xj) and f(zj) + 1. 
2. If the lexicon zj was previously introduced in 
the chart, then run a pointer from t to f(xj). 
In addition, if the passive edge Bj :{Bj } never 
exists in the chart, create it between vertex 
f(rj) and f(xj) + 1. 
Prediction Let e be a passive edge labeled C:S 
incident from vertex s to t. For every chain 
rule A' C A A B1 A ... A Bn such that A 
unifies with C, create an active edge labeled 
\[A\[?\]B1...\[?\]Bn\]A':,~ between vertex s and t. 
(Create, instead, a passive edge labeled A':S 
when A is the single antecedent, i.e., n = 0.) 
Combination Let ez be an active edge labeled \['" "\[?\]Bj\[?\]Bj+I'" 
.\[?\]B,~\]A:$1 incident from ver- 
tex s to t, where Bj is the first open box in ez 
and let e2 be a passive edge labeled C:S~ inci- 
dent from vertex u to v. If Bj and C unify and 
there is a pointer from t to u, then create an ac- 
tive edge labeled \[-..Bj\[?\]Bj+I...\[?\]Bn\]A:S1 US2 
between vertex s and v. (Create, instead, a pas- 
sive edge labeled A:S1 U S: when B 1 is the last 
element, i.e., j = n.) 
Each passive edge T:S represents an answer. 
Examples 
Here, we present a simple example of the appli- 
cation of our algorithm to spoken language un- 
derstanding. Figure 3 provides the rules for spo- 
ken Japanese understanding, with which the sen- 
tence (1) is parsed and interpreted. They include 
the pragmatic, semantic and knowledge rules as 
well as the syntactic and lexical rules. 
The syntactic rules allow the connection be- 
tween a verb and a noun phrase with or with- 
221 
Syntactic Rules 
s(i,k,e)Cvp(i,k,e) 
vp(i,k,e)Cnp(i,j,c,x) A vp(j,k,e) A depend( (c,e,x)d) 
vp( i,k,e)C np( i,j,x) A vp(j,k,e) A depend( (c,e,X)d) 
np(i,k,c,x)Cnp(i,j,x) A p(j,k,c) 
depend( (c,e,x)d)Cprag( (x)p,y) ^ sem( (c,e,y), ) 
Lexical Rules 
np(\[S6seki\]k\],k,x)C soseki(x) $~ 
vp(\[katta\]k\],k,e)C buy( e) *1 
p(\[galk\],k,c)c ga( e) 
p(\[ olk \],k,c)C wo( c) .1 
Pragmatic Rules 
prag((x)p, ) 
prag( (x)p,y)C r ter(x) ^ wr te( ^ novel(y)Sl 
Semantic Rules 
sem( s)C ga( s,e) A ga(e) $3 
sem(s)Cwo(s,e) ^  o(e) .3 
ga( ( c,e,x) 8 ,c)C intend( e ) A person(x) A agt( ( e,x) e ) $2° 
wo( (c,e,x), ,c)C trade(e) A commodity(z) ^ obj( (e,x),) $~ 
Knowledge Rules 
person( x )C soseki( x ) 
w~ter(x)Csoseki(x) 
book(x)Cnovd(x) 
eommodity( ~ )C book(z) 
trade(e)Cbuy(e) 
intend( e)C trade( e) 
Figure 3: Example of Rules 
out a particle, which permit structures like 
\[VP\[NpS6sek2\]\[vpkatla\]\]. Such a structure is evalu- 
ated by the pragmatic and semantic criteria. That 
is, the dependency between a verbal concept e and a 
nominal concept x is supported if there is an entity 
y such that x and y have a pragmatic relation, i.e., 
a metonymy relation, and e and y have a semantic 
relation, i.e., a thematic relation. The metonymy 
relation is defined by the pragmatic rules, based on 
certain knowledge, such as that the name of a writer 
is sometimes used to refer to his novel. Also, the 
thematic relation is defined by the semantic rules, 
based on certain knowledge, such as that the object 
case relation between a trading action and a com- 
modity can be linguistically expressed as a thematic 
relation. 
The subscript $c of a formula A represents 
the cost of assuming formula A. A is easy to as- 
sume when c is small, while A is difficult to as- 
sume when c is large. For instance, the cost of 
interpreting the thematic relation between a trad- 
ing action and a commodity as the object case re- 
lation is low, say $2, while the cost of interpret- 
ing the thematic relation between an intentional 
action and a third person as the agent case rela- 
tion is high, say $20. This assignment of costs is 
suitable for a situation in which the speaker re- 
ports his experience. In spite of the difficulty of 
assigning suitable costs in general, the cost-based 
interpretation is valuable, because it provides a uni- 
form criteria for syntax, semantics and pragmat- 
ics. Hopefully, several techniques, independently 
developed in these areas, e.g., stochastic parsing, 
example-based/corpus-based techniques for word 
sense/structural disambiguation, etc., will be us- 
able for better cost assignment. Probability will 
also be a key technique for the cost assignment 
(Charniak and Shimony, 1990). 
Figure 4 and Table 1 show the chart that is 
created when a sentence (1) is parsed and inter- 
preted using our algorithm. Although the diagram 
seems complicated, it is easy to understand if we 
break down the diagram. Included are the syntac- 
tic parsing of the sentence (indicated by edges 2, 6, 
7, 14, 52 and 53), the pragmatic interpretation of 
the metonymy by S6seki S (indicated by edges 17, 
18, 20 and 24), the semantic interpretation of the 
thematic relation between a buying event B and a 
novel N written by S6seki (indicated by edges 42, 
44, 45, 47, 48 and 50), and so on. In the pragmatic 
interpretation, assumption novel(N) (edge 21) is 
introduced, which is reused in the semantic inter- 
pretation. In other words, a single assumption is 
used more than once. Such a tricky job is naturally 
realized by the nature of the chart algorithm. 
Agenda Control 
Since the aim of cost-based abduction is to find 
out the best solution, not all solutions, it is reason- 
able to consider combining heuristic search strate- 
gies with our algorithm to find the best solution 
efficiently. Our algorithm facilitates such an exten- 
sion by using the agenda control mechanism, which 
is broadly used in advanced chart parsing systems. 
The agenda is a storage for the edges created by 
any of the three procedures of the chart algorithm, 
out of which edges to be added to the chart are 
selected, one by one, by a certain criterion. The 
simplest strategy is to select the edge which has 
the minimal cost at that time, i.e., ordered search. 
Although ordered search guarantees that the 
first solution is the best one, it is not always very ef- 
ficient. We can think of other search strategies, like 
best first search, beam search, etc., which are more 
practical than ordered search. To date, we have not 
investigated any of these practical search strategies. 
However, it is apparent that our chart algorithm, 
together with the agenda control mechanism, will 
provide a good way to realize these practical search 
strategies. 
222 
\[Soseki,katta\] 
i 
I 
I 
i 
I 
I 
I 
i 
I 
I 
I 
I 
l 
39,,,5..4. -.- ~°..°.°-° °°o°° 
g~gllIRmBSml~00 -? --....... 
6"7,8 
% 
\ 
f.l -*-o.~ 
i ............... !. 
i 35.. .... 34,49 " i 
! 
a F <B,S>s i~ 1.~ 
iii ........ <s.v  
I " ..'L_ 
3,4,5 ~. S 
Q.-" 
20 /24 
g 
/ 
k "\ _ X Z:" ..... L 
25 
\[katta\] 
°°°'~°°°'o0 
o 
• '...... 
28,29 
IW-. 
I 
I 
I 
I 
I i D 
! 
P n 
\\ 
13 ~.1-' 
-) 2 
i 
-" I 
" J 
...10,11,1 
passive edge 
active edge 
pointer 
Figure 4: Chart Diagram for SSseki katta 
Preliminary Experiments 
We conducted preliminary experiments to compare 
four methods of cost-based abduction: top-down al- 
gorithm (TD), head-driven algorithm (HD), gener- 
alized chart algorithm with full-search (GCF), and 
generalized chart algorithm with ordered search 
(GCO). The rules used for the experiments are in 
the spoken language understanding task, and they 
are rather small (51 chain rules + 35 non-chain 
rules). The test sentences include one verb and 1-4 
noun phrases, e.g., sentence (1). 
Table 2 shows the results. The performance of 
each method is measured by the number of compu- 
tation steps, i.e., the number of derivation steps 
in TD and HD, and the number of passive and 
active edges in GCF and GCO. The decimals in 
parentheses show the ratio of the performance of 
each method to the performance of TD. The table 
clearly shows how the three features of our algo- 
rithm improve the computational efficiency. The 
improvement from TD to HD is due to the first fea- 
ture, i.e., goal-driven bottom-up derivation, which 
eliminates about 50% of the computation steps; the 
improvement from HD to GCF is due to the sec- 
ond feature, i.e., tabulation of the partial results, 
Table 2: Comp. among TD, HD, GCF, and GCO 
Ns II TDI ttD GCF 
1 215 112 (0.52) 83 (0.39) 
2 432 218 (0.50) 148 (0.34) 
3 654 330 (0.50) 193 (0.30) 
4 876 442 (0.50) 238 (0.27) 
GCO 
75 (0.35) 
113 (0.26) 
160 (0.24) 
203 (0.23) 
which decreases the number of steps another 13%- 
23%; the improvement from GCF to GCO is due to 
the last feature, i.e., the agenda control mechanism, 
which decreases the number of steps another 4%- 
8%. In short, the efficiency is improved, maximally, 
about four times. 
Comparison with Earley Deduction 
We describe, here, some differences between our al- 
gorithm and Earley deduction presented by Pereira 
and Warren. First, as we mentioned before, our al- 
gorithm is mainly based on bottom-up (left-corner) 
derivation rather than top-down derivation, that 
Earley deduction is based on. Our experiments 
showed the superiority of this approach in our par- 
223 
titular, though not farfetched, example. 
Second, our algorithm does not use sub- 
sumption-checking of edges, which causes a serious 
computation problem in Earley deduction. Our al- 
gorithm needs subsumption-checking only when a 
new edge is introduced by the combination proce- 
dure. In the parsing of augmented grammars, even 
when two edges have the same nonterminal symbol, 
they are different in the annotated structures asso- 
ciated with those edges, e.g., feature structures; in 
such a case, we cannot use one edge in place of 
another. Likewise, in our algorithm, edges are al- 
ways annotated by the assumption sets, which, in 
most cases, prevent those edges from being reused. 
Therefore, in this case, subsumption-checking is not 
effective. In our algorithm, reuse of edges only be- 
comes possible when a new edge is introduced by 
the introduction procedure. However, this is done 
only by adding a pointer to the edge to be reused, 
and, to invoke this operation, equality-checking of 
lexicons, not edges, is sufficient. 
Finally, our algorithm has a stronger connec- 
tion with chart parsing than Earley deduction does. 
Pereira and Warren noted that the indexing of for- 
mulas is just an implementation technique to in- 
crease efficiency. However, indexing plays a con- 
siderable role in chart parsing, and how to index 
formulas in the case of proof procedures is not so 
obvious. In our algorithm, from the consideration 
of head-driven derivation, the index of a formula 
is determined to be the first argument of that for- 
mula. All formulas with the same index are derived 
the first time that index is introduced in the chart. 
Pointers among lexicons are also helpful in avoiding 
nonproductive attempts at applying the combina- 
tion procedure. All the devices that were originally 
used in chart parsers in a restricted way are in- 
cluded in the formalism, not in the implementation, 
of our algorithm. 
Concluding Remarks 
In this paper, we provided a basic and practi- 
cal solution to the computation problem of cost- 
based abduction. We explained the basic concept 
of our algorithm and presented the details of the 
algorithm along with simple examples. We also 
showed how our algorithm improves computational 
efficiency on the basis of the results of the prelimi- 
nary experiments. 
We are now developing an abduction-based 
spoken language understanding system using our 
algorithm. The main problem is how to find a good 
search strategy that can be implemented with the 
agenda control mechanism. We are investigating 
this issue using both theoretical and empirical ap- 
proaches. We hope to report good results along 
these lines in the future. 
Acknowledgments 
The author would like to thank Prof. Yuji Matsu- 
moto of Nara Institute of Science and Technology 
and Masahiko Haruno of NTT Communication Sci- 
ence Laboratories for their helpful discussions. 

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