Some Remarks on the Decidability of the Generation Problem in 
LFG- and PATR-Style Unification Grammars 
Jiirgen Wedekind 
Institute for Natural Language Processing 
University of Stuttgart 
Azenbergstr. 12 
D-70174 Stuttgart, FRG 
juergen@ims.uni-stuttgart.de 
Abstract 
In this paper, we prove the decidability of 
the generation problem for those unifica- 
tion grammars which are based on context- 
free phrase structure rule skeletons, like 
e.g. LFG and PATR-II. The result shows 
a perhaps unexpected asymmetry, since it 
is valid also for those unification grammars 
whose parsing problem is undecidable, e.g. 
grammars which do not satisfy the off-line 
parsability constraint. The general proof is 
achieved by showing that the space of the 
derivations which have to be considered in 
order to decide the problem for a given in- 
put is always restricted to derivations whose 
length is limited by some fixed upper bound 
which is determined relative to the "size" of 
the input. 
1 Introduction 
Unification Grammars with a context-free skeleton, 
like Lexical Fhnctional Grammar (LFG) and PATR- 
II (cf. e.g. Kaplan and Bresnan 1982, Shieber et 
al. 1983) assign to a sentence not only a constituent 
structure (c-structure), but also an additional lin- 
guistic entity. In the rather restricted grammars of 
the early stage this entity is identified with a special 
graph structure, commonly called feature structure. 
Since a string is regarded as well-formed only if a 
(well-formed) feature structure is assigned to it by 
the grammar, two inverse decidability problems arise 
which had to be solved in order to know whether we 
can formulate terminating parsing and generation al- 
gorithms. If we retain the terminology of the early 
stages then an adequate parsing algorithm requires 
that we can decide for a given grammar and a given 
string whether there exists a feature structure as- 
signed to it by the grammar (parsing problem) and 
an adequate generation algorithm requires that we 
can decide for a given grammar and a given feature 
structure whether there exists a sentence to which 
this structure is assigned by the grammar (genera- 
tion problem). 
While we already know for a long time that the 
parsing problem is undecidable (cf. Kaplan and Bres- 
nan 1982, Johnson 1988), we want to show in this pa- 
per that the generation problem is decidable even for 
unrestricted (not off-line parsable) unification gram- 
mars. For the proof we first introduce in section 2 the 
type of grammar we want to consider. In section 3 
we then define the generation problem and show its 
decidability in two steps. 
2 Preliminaries 
The unification grammars we want to consider con- 
sist of rules with a context-free skeleton and a set of 
annotations associated with the constituents men- 
tioned in the rules. Typical examples taken from 
LFG and PATR-II are given in figure 1. For the for- 
S --~ NP VP S --+ NP VP 
(t SUB J) ----$ j'=$ (VP AGR) = (NP AGR) 
NP -+ John NP -4 Uther 
(1" PRED) = JOHN (NP AGR NUM) = SG 
(NP AGR PER) ---- 3RD 
Figure 1 
Examples of rules in LFG (left) and PATR-II 
format (right). 
mal definition of those grammars we reconstruct the 
annotations as formulas of a quantifier-free sublan- 
guage of a classical first-order language with equality 
whose (nonlogical) symbols are given by a finite set of 
unary partial function symbols and a finite set of con- 
stants. For the translation of LFG and PATR-II an- 
notations we regard the attributes (in figure 1: SUB J, 
PRED, AGR, NUM, PER) 58 unary partial function 
symbols and the atomic values (in figure 1: JOHN, 
45 
SG, 3RD) as individual constants. Furthermore, we 
assume for a context-free rule of the form A ---> w 
(w e (VN U VT)*) that the variable x0 is associated 
with A and that for each occurrence wi in w there 
is a variable xi which is associated with wi. For the 
formal reconstruction of LFG's we assume that each 
occurrence of $ in the annotation of w~ corresponds 
to an occurrence of xi and that each occurrence of 
1" corresponds to an occurrence of x0. For grammars 
in PATR-II format we suppose that occurrences of 
categories in the annotations correspond to the asso- 
ciated variables. 
Before we give the definition of the grammars we 
want to investigate, we introduce the following nota- 
tion. In the following we use S\[xl, .., x~\] to indicate 
that the variables occurring in the set of formulas S 
are included in {xl, .., Xn} and S(Xl, .., xn) if the set 
of variables occurring in S is exactly {Xl,.., xn}. 
1. DEFINITION. A unification grammar is a tuple 
(VN , VT , S, F1, V, V, R>, consisting of a finite nonter- 
minal vocabulary VN, a finite terminal vocabulary 
VT, a start symbol S E VN and a feature-description 
language L determined by a finite set of unary par- 
tial function symbols F~, a finite set of atomic values 
V and a denumerable set of variables 1 
V= {x~ I a e N*} with x~ #x,, for a # a'. 
All vocabularies are pairwise disjoint. R is a finite 
set of rules of the form r = ((A,w),S~\[xo,..,xl~l\] } 
(zi E 1;), with (A, w) e VN x (VN U VT)* (a context- 
free phrase structure rule) and S~\[x0, .., xl~l\] a finite 
set of (quantifier-free) literals of L. 2 
According to our definition the LFG rules in figure 1 
are now expressed as depicted in (la) and the PATR- 
II rules as given in (lb). Note that the structure of 
the terms is now "mirror imaged", since we assume 
the attributes to be unary partial function symbols. 
(1) (a) (S -+ NP VB, {SUBJ Xo ..~ xl,xo ~ x2}) 
(NP --~ John, {PRED XO ~ JOHN}) 
(b) (S --> NP VP, {AOa z2 ~ AGR xl}) 
SNUM AGR XO ~ SO,1 \ (NP --+ Uther, \].PEg AOR Xo ~ 3RD~/ 
For the definition of the sentences derivable by a 
unification grammar we have to specify first what 
derivations are. 
2. DEFINITION. A sequence of pairs ~r0...~rn with 
7to = (Be, 01 (B 6 VN) is called derivation of length 
n iff for each 7ri = (B \[..A~..\]~, S) (0 < i < n) there is 
a rule r (A -+ w~ k = ..win, S~) such that 
= .., ,~\]~..\]~,S~). 
In the definition we assume that the order of the 
arcs of a tree is encoded by numbering the arcs and 
that each node is identified with the sequence of in- 
tegers numbering the arcs along the path from the 
1The syntax and semantics of feature-description lan- 
guages is given in the appendix. 
2A literal is an atomic formula or the negation of an 
atomic formula. 
root (O) to that node. In our bracket notation we 
add to a constituent its root node as the right and 
its root node label as the left index. In order to be 
able to refer to the c-structure derivation and to the 
sequence of feature descriptions and to have access 
to the nodes which are substituted in each step of a 
derivation, we define for a derivation 7r three other 
sequences. 
3. DEFINITION. Let ~r be a derivation of length 
n. We then define two sequences w and ")' for 
each i=O,..,n with lh=(Tc, S) by wi=Tc and 
7i = S and a sequence w for each i-- 1,..,n with 
w~-i = B\[..A~..\]O and Tc = B\[..A\[W~.I,..,wk.m\]~..\]0 
by wi = #. 
Let S be a set of literals and 0 a unary partial map- 
ping over the set of terms. Then the expression S\[0\] 
denotes the set of expressions obtained from S by 
simultaneously replacing each occurrence of a term 
~- in each formula in S by 0(T). The feature descrip- 
tion derived by zr is then defined by means of the 
following operation. 
4. DEFINITION. If ~r is a derivation of length n then 
the feature description derived by 7r from h to k 
(0 <_ h < k <_ n) is given by 
k 
S;-~k = U 7i\[{(xj,xw, j) I xj occurs in 3'i}\]. 
i----h 
EXAMPLE 1. If we start a derivation zr from (So, 0) 
and apply the S-rule in (la) and the following VP- 
rule 
(VP -+ V VP', {xo ~ Xl,XCOMP ~0 ~ X2}) 
we end up with the following sequence. 
~o = (So, 0) 
7rl = (s\[NP1, VP2\]o,{SUBJ xo ~ Xl,XO ~ x2}) 
zr2 ---- (s\[NPI,vp \[V2.1 ,VP'2.21210,{xo ~. Xl,XCOMP Xo ~ X2}) 
For the steps depicted above the sequence w is given 
by wl = 0 and w2 = 2 and the feature description 
derived by 7r from 0 to 2 (S~_~2) is 
{SUBJ X0 ~ Xl,X0 ~ X2,X2 ~ x2.1,XCOMP x2 ~ x2.2}. 
Sentences are then defined as follows. 
5. DEFINITION. A terminal string w (w E V~) is 
a sentence iff there is a derivation (So, 0) = r0..Trn 
with Wn = S\[w\]0 and 3x~1 ..x,~ A S~-"~n(X~tl' "" Z~tm) 
satisfiable. 3 
In the following we write S" for S~_+n if the inter- 
val covers the whole derivation, i.e. if ~r is of length 
n. 
Since a specific reduction algorithm and a few 
model-theoretic facts required in the proofs later on 
can be introduced by showing how satisfiability of 
such existential prenex formulas can be decided, we 
will continue with a short excursion on satisfiability. 
3We use s\[w\]o to denote an S-rooted c-structure with 
yield w. 
46 
2.1 Satisfiability 
In order to test whether for a given finite set of lit- 
erals S of a feature-description language (2) 
(2) 3z~..zt A S(x~,.., zl) 
is satisfiable, we can exploit by skolemization well- 
known test procedures available for quantifier- and 
variable-free sets of such literals. Let C be a set of 
Skolem-constants (\[{xl, ..,xz}\[ = ICl) and 0 be a bi- 
jective function from {Xl, .., xt} to C, then (2) can be 
tested by testing the set of literals (3) over L(C) 4 
(3) S\[0\], 
since (2) and (3) are equi-satisfiable. In the follow- 
ing we complete the procedure by introducing a re- 
duction algorithm that reduces a set of literals (3) 
according to a measure in a sequence of measure 
decreasing rewrite steps to a deductively equivalent 
set (4) (in reduced form) 
(4) (S\[e\])p, 
which is satisfiable iff the terms 7- of all inequalities 
T ~ 7- of (4) do not occur as subterms in equations 
of (4).5 
For the proof we first introduce a few definitions 
and some notation. Let 7- be the set of terms 
of a variable-free feature-description language L(C). 
Then an injective function m • \[7- ~ ~l*\] is a mea- 
sure iff it satisfies the following conditions for all 
T, T' • 7" and a • FI*: 
(i) if \[7-\[ < \[7-'\[, then re(w) < m(7-'), 
(ii) if re(r) < m(7-'), then m(aT) <_ m(a'c'). 
For literals and sets of literals S we extend a mea- 
sure m as usual by m((.~)7- ~ 7-')= m(7-)+ m(7-') 
and re(S) = Era(C). 
Ces 
In the following we use 7- ~7-' iff m(7-) > m(7-') 
and 7-~7-' to denote ambiguously 7- ~ 7-' or 7-~ ~ 7-. 
Let S be a set of literals then E denotes the set of 
all equations in S, 7-s the set of terms occurring in 
the formulas of S (7-s = {~-, 7-' \[ ("~)7- ~ 7-' • S}) and 
SUB(Ts) the set of all subterms of the terms in 7~ 
SUB(7~) = {7-\[a7- • 7~, with a • FI*}. 
For the construction of a reduced form we need a 
specific partial choice function p which satisfies 
p(S) • {7- ~7-' • SIT • SVS(Ts\{r~.,.,})} 
if the specified set is nonempty and undefined other- 
wise. 
6. DEFINITION. For a given finite set of literals S 
and a choice function p we define a sequence of sets 
Sp, (i > O) by induction: 
Spo =S 
f( Spi\{ 7-~-.7-t} )\[r/v' \] U {7-~T t} if p( Sp, ) = 7- ~7-' 
Sp,+, = \[So, if p(S m) undef. 
aThe feature-description language which in addition 
to L provides a distinct set of Skolem-constants C'. Cf. 
the appendix for more details. 
~The algorithm is adapted from Statman 1977 and 
Knuth and Bendix 1970 and first applied to feature- 
description languages by Beierle and Pletat (1988). 
Since m(Sm) > m(Sp,+l ) ifp is defined for Sin, the 
construction terminates with a finite set of literals. 
If we set 
Sp = Spt ; with t = min{i \[ Sp, = Sin+ ~ } 
the following lemma can easily be proven by induc- 
tion on the construction of Sp. 6 
7. LEMMA. For Sp it holds that: 
(i) S ~F S o, 
(ii) if T~T' C S o then T ¢ SUB(Tsp\{r~r,}). 
Since Sp is obviously not satisfiable if it contains 
an inequality T ~ 7 and 7 occurs as a subterm in Ep, 
the whole proof is completed by showing that we can 
construct a canonical model satisfying Sp if Sp does 
not contain such an inequality. 
For the model construction we need the set 
T~p = {r e SUB(TE,) \[ -~3T'(T ~T' e Ep)} 
and the function h c E \[SUB(7-Ep) ~ 7-~,\] which is de- 
fined for each 7- e SUB(TE,) by 
f ,T'(7-~T' Ep) if 7- E, h e (T) = • f\[ T'c 
\[7- otherwise. 
That h e is well-defined results of course from 7(ii). 
8. DEFINITION. For a set of literals S o the canoni- 
cal term model is given by the pair Mp = (Hp, .~p), 
consisting of the universe 
Ltp=\[7~, ifE, ~ 0 
\[.{O} otherwise 
and the interpretation function ~p, which is defined 
forc•VUC, f•/'l and 7-•Hpby: 
\[M(c) if c • SUB(TE,) 
~p(e) = I.undefined otherwise 
~ h~(fT-) if fT- • SUB(TE,) 
"~P(f)(7-) = \[undefined otherwise. 
For Mp which is well-defined the following lemma 
holds: 
9. LEMMA. If 7- is a subterm of Ts, then 
(i) ~p(7-) = he(7-), if 7- • SUB(TE~), 
(ii) 7- • SUB(T~), if T • Dom(.~o). 
PROOF. (By induction on the length of 7-.) The 
lemma is trivial for constants. By showing (i) be- 
fore (ii) we get the induction step for a subterm fT- 
of Ts, in both cases according to 
~p(fT) = ~p(f)(-~p(7-)) = ~,(f)(hC(7-)) = ~p(f)(7-). 
We get .~p(7-) = hC(T) by inductive hypothesis and 
M(7-) = % since 7- ¢ Hp would imply the existence of 
6In order to verify 7(i) cf. e.g. Wedekind 1991 
and 1994. 
47 
T ~ r' • E o and fT could not be a subterm of 7~p 
according to lemma 7(ii). Now, if (i) fT • SUB(TEp) 
then ~p(f)(T) is defined and equal to h~(fr) and 
(ii) if fr • SUB(Ts,) and .~o(fT) is defined then 
fr • SUB(TE~). \[\] 
On the basis of lemma 9 it is now easy to prove: 
10. LEMMA. VT ~ T • So(7" ¢ SVB("fEp)) --~PMp S O. 
PROOF. (If the condition is satisfied ~M, ¢ holds for 
every ¢ • So. ) If ¢ = ~'~T' • S o with m(T') < m(r), 
then v' • T~o by 7(ii) and hence hC(T ') = T'. We get 
then h~(~ -) = T' for m(T') = m(T) by T' = T and for 
m(~-') < m(~-) by the definition of h ~, since r ¢f T~. 
Thus ~p(T) = ~p(T') by 9(i) and hence ~Mo ¢. 
Assume ¢=TCT'. If T~7' were satis- 
fied by Mp, we would get ~p(T)= ~p(T') and 
by 9(ii) T,T'•SUB(TE,). Since 7(ii) ensures 
he(r) = h~(~ -') = v = ~-', we would have ~- ¢ r • Sp 
with T • SUB(TEo). \[\] 
Finally it should be mentioned that Mp is a unique 
(up to isomorphism) minimal model for Sp, i.e. if M 
is a model for So, homomorphic to Mp, then every 
minimal submodel of M that satisfies S o is isomor- 
phic to Mp. 
3 The Generation Problem and its 
Decidability 
Although it was not necessary for the definition of 
the sentences derivable by a unification grammar, we 
now have to make explicit that also a feature descrip- 
tion is assigned to a sentence. 
11. DEFINITION. A terminal string w (w • V~) is 
derivable with feature description 3Xl..Xl¢(Xl,.., xt) 
iff the feature description is satisfiable and there 
is a derivation (S~, O) = ~r0..Ir,~ with w~ = s\[w\]~ and 
¢=AS ~. 
Since deductively equivalent consistent feature de- 
scriptions are assumed to describe the same set of 
feature structures (models), the assignment of en- 
tities to terminal strings determined by a unifica- 
tion grammar is then formally given by a binary 
relation A between terminal strings and sets of 
classes of deductively equivalent feature descriptions 
\[?Xl ..Xl ~)( X l , .., Xl ) \]'-tF .7 
12. DEFINITION. For each terminal string w • V~ 
and each class \[Sxl..xl¢(Xl,.., xl)\]: 
A(w, \[3xl ..xl¢(xl,.., xl)\]) iff w is derivable with 
3zl..zl¢(zl, .., xt). 
Definition 12 now brings us closer to the problem, 
since we can for any unification grammar in rather 
abstract terms specify what parsers and generators 
are: a parser is a procedure which recursively enu- 
merates for any given string w the set 
{\[~Xl..XI~)(Xl,.-, Xl)\] \[ A(W, \[3X 1..xI¢(xl,.., Xl)\])} 
7We omit the index of the equivalence classes in the 
following. 
and a generator is a procedure which recursively enu- 
merates for any given class \[3Xl..Xl¢(xl, .., xl)\]: s 
{w • y~ l A(w, \[3Xl..Z,C(Xl, ..,z,)\])}. 
Whether adequate algorithms (effective proce- 
dures) can be formulated depends on the decidability 
of the corresponding parsing and generation problem. 
In our case (generation), it is the problem whether 
3w • y~(zx(~, \[3Xl..X~¢(xl, .., x~)\])) 
is decidable for any given class \[3xl..xl¢(xl, ..,xl)\]. 
The decidability of the generation problem alone en- 
sures the existence of algorithms which terminate in 
any case with an output, although they might (of 
course) not be able to produce all possible solutions. 
Despite decidability, inputs can still be infinitely am- 
biguous (\[{w • V~ \[A(w, \[3xl..xl¢(Xl, ..,xl)\])}\] infi- 
nite). 
In order to prove the decidability of the generation 
problem (theorem 13), we proceed in two steps. 
13. THEOREM. It is decidable for each feature 
description 3yl..Yk¢(Yl,..,yk) whether there is a 
terminal string w • V~ which is derivable with 
3Xl..Xl¢(xl,.., Xl) and 
3yl ..Yk¢(Yl,.., Yk) qF- 3Xl ..Xt¢(Xl, .., Xl). 
In the first step we show that we can always shorten 
a derivation of a sentence w with (consistent) fea- 
ture description ¢ to a derivation of a sentence w' 
with feature description ¢' and ¢ -t~- ¢' whose length 
is bounded by the "size" of ¢. By showing in the 
second step that two deductively equivalent consis- 
tent feature descriptions have the same "size" the- 
orem 13 follows, since only a finite set of deriva- 
tions (those whose length does not exceed this up- 
per bound) have to be inspected in order to decide 
3w • V~ (A(w, \[¢\])) for an arbitrary consistent input 
¢. 
3.1 Redundant Recursions and Pumping 
For the proof that for a derivation of a sentence w 
with (consistent) feature description ¢ there always 
exists a short derivation of a sentence w' with fea- 
ture description ¢' and ¢ -tt- ¢' we exploit the fact 
that a c-structure may contain recursions of the form 
depicted in figure 2 whose corresponding subderiva- 
tions in ~r are eliminable. Such recursions are called 
redundant. 
14. DEFINITION. Let rr be a derivation of a sentence 
uvzxy of length m + k + 1 whose c-structure deriva- 
tion has the form ~O0..t.dm..O.~rn-t-k..Wm-l-k-bl -~ OQn with 
wm= S\[U, A u, Y\]0 and aJm+ k : s\[U, AIr, Au.~, x\],, Y\]0 
(reorder if necessary). If 7r' is a derivation of 
uzy of length m+l which is defined for each j 
(O < j < rn+l) by 
{r~j if j <_ m 
7r~ = (s\[ury\]~, S) if j > m and 7rj+k = (s\[uvrxy\]~,S) 
SWe assume here strong reversibility, since a generator 
is for a given input y simply a parser which operates on 
A-I: it recursively enumerates instead of {x I A(x ,y)} 
the set {x I A-I(x,Y)} • 
48 
\ 
u v z x y 
Figure 2 
A c-structure with recursion A\[V, A~.~, x\]t, (1~1 > 0). 
and 
, {;rj ifj<_m 
wJ = .t if j>m and wj+k = #.~.~ 
then 7rm+l...~m+ k is a redundant recursion iff 
3x,..x~ h S~(xt, .., x~) ~ 3xl "x'k h S~' (xl, .-, x~)- 
If we assume that a given derivation of a sentence 
is already shortened to a derivation without redun- 
dant recursions it remains to show that the length of 
such a derivation could not exceed the upper bound 
determined by the "size" of the derived feature de- 
scription. 
The "size" of a consistent feature description is on 
the one hand determined by the size of its minimal 
model, and on the other hand determined by a nor- 
mal form into which every feature description can be 
converted. The conversion is performed in two steps. 
In the first step, we eliminate as many variables as 
possible by substitution. 
15. DEFINITION. If S(Xl,..,Xl) is a set of lit- 
erals, then xi is eliminable in S(xl,..,xl) iff 
there is a term T not containing xi such that 
~- 3zl..x~(A S(Xl, .., ~) ~ z~ = r). 
16. NOTATION. In the following we write S\[x~, .., xz\] 
iff each xi is not eliminable in S. 
17. DEFINITION. We assign to a set of liter- 
als S'(x~,..,x~,x~,..,X~k) a set R(S') which con- 
tains a set S\[x~,..,xt\] iff there is a substitution 
O E \[{x~, ..,x~} ~-~ T(x~, ..,x,)\] such that 
~..~,~..x~(A s' ~ ~ ~ o(~)) 
for all ~ (1 < i < k) and Sfz~, ..,~\] = S'\[O\]? 
By the substitutivity theorem we get: 
18. LEMMA. If SfXl, .., xl\] E R(S'(Xl,.., xl,X~l,.., x~k)) 
then 2x~ ..x~k(A S - A S'). 
In the second step, we make the set of literals in- 
dependent, i.e. we remove those literals which are 
implied by the remaining subset. 
~T(x~,..,xt) denotes the set of terms over V, 
{x~, .., x~} and F~. 
19. DEFINITION. A set of literals S(xl,..x~) is in- 
dependent iff there is no formula ¢ E S for which 
F- ~xl..xl(A(S\{¢}) D ¢) holds. 
Normal forms are then defined as follows. 
20. DEFINITION. A consistent feature description 
~xl..xl A Six1,.., xt\] is in normal form (in the fol- 
lowing indicated by a v index) iff S\[xl, .., xl\] is in- 
dependent. 
Furthermore, we call ~xl ..xl A S~ Ix1, .., xl\] a normal 
form of ~xl..xtx'~..x~ h S'(xl, .., x~) iff S~ Ix1, .., xt\] 
is an independent subset of S'\[O\] E R(S') and 
~Z1..Xl(ASv\[Xl,..,Xl\] ~ ASt\[0\]) • 
Lemma 18 and the condition in definition 20 ensure 
that a consistent feature description and its normal 
forms are deductively equivalent. 
In order to be able to show the existence of 
a redundant recursion, we exploit the simple fact 
that the information which contributes a literal 
in a normal form with a minimal model (Up, ~p} 
can be specified by an equation ~-~ 1 -s where 
IT\[ + \[rq < \[Up\[ + 2. A literal r ~ O'a'T 't (\[a I > 0) 
whose terms are longer must always be reducible 
by a loop 5rtT tl ,-~ T II to a shorter equation. Since 
the construction of such an information piece can be 
done with a subderivation of some fixed length, there 
must be a redundant recursion if the length of whole 
derivation exeeds a fixed value which is dependent 
on \[Hp\[ and \[S~\[ and exactly specified in lemma 21.1° 
21. LEMMA. Suppose that w E V~ is derivable with 
¢ = ~xl ..xtx'l..x'k A S~(xl, .., x'k) over ~r of length n, 
that ~xl..xl A S~ Ix1,.., xl\] is a normal form of ¢ and 
that M o = (Up, ~o) is a minimal model of ¢. If ~r 
has no redundant recursions then each path of wn is 
shorter or equal to IVN\[ . (3 \[Hol + 1)- (IS~\[ + 1). 
PROOF. Suppose, one path of wn were longer 
than IVN\[. (3 \[Up\[ + 1). (\[S~\[ + 1), then more than 
(3 I/~p\] + 1). (\[S~ I + 1) different nodes on that path 
had to be labelled by the same A E VN. With- 
out loss of generality we can assume that lr is a 
derivation whose c-structure derivation w has the 
form Wo..wm..wn with wm= s\[u, A,, y\]~ for each node 
# on that path which is labelled by A (reorder if 
necessary). In order to exclude that complex in- 
ferences are used to build up 0, we assume fur- 
thermore that 0 is non-deterministicaUy constructed 
from S ~ by recursive variable substitution, i.e. we 
require for each (X, aT) E 0 either x'~aT E S ~ or 
~x~ay E S'((y,T) E 0). Finally, let S C S ~ with 
S~ = S\[O\]. In order to identify the redundant re- 
cursion we have to consider the following cases. 
1. Suppose there are more than \[Sv\[ + 1 A-labelled 
nodes # such that x i, does not occur in S ", then there 
must be more than \[Su\[ non-overlapping recursions. 
For at least one of those recursions ~rm+~...Trm+k it 
must hold that 
(S \[~l S~+l_+rn_l_k) C (s~r.+rn \[.J S~n+k..kl_~n). 
1°The given factor \[VN\[. (3\[//p\[ + 1). (\[S~\[ + 1) de- 
creases for more restricted grammars, like e.g. grammars 
which allow only feature descriptions with single-rooted 
and/or acyclic minimal models. 
49 
But then 
Sn ~r S c_ ( 0-~m u Sm+,+~) 
and ~rm+~...7rrn+k must be redundant. 
2. If case 1 does not apply there must be more than 
3 I/4p\[ • (\]S~ I + 1) distinct A-labelled nodes # on that 
path such that x, occurrs in S ~ and for more than 
31S-I + 1 of these nodes must pairwise hold 
~.-z~ (h s ~ ~ x, ~ ~.~). 
But then there must be at least three recursions such 
that 
and 
(S n s~+~_~+~+~+z) G ($8~ ~ s~\~+.+~+~_~.). 
We can then assign to each recursion 7rm+l...Trrn+k 
(m=i,k=l; m= i +l,k=v or m=i+l +v,k= z) a 
type which corresponds to the strongest of the fol- 
lowing conditions the recursion satisfies. 
(a) ~,~+~...~m+~ satisfies 
(b) It holds only 
~- ~..x~(A S~+~+~ ~ z,.~ ~ ax,) 
with lal > 0 and aO(x,)..~ O(x,) is implied by 
~-Sxl..x'k(ASo'_,m+~ A S~) where S a is the set of 
ground literals of S. 
(c) Or it holds 
t 7r 
with \]a' I > 0 and a'O(x,.~) ~ O(x,.~) is implied by 
~- ~z~..z~(A S~+l~ A S~). 
(d) If a recursion which satisfies 
(\[a I > O) is not of type (b) then there must be a 
ground term T which is not reducible in terms of x~, 
i.e. t? can not satisfy T = a'O(x,) for some non-empty 
prefix a', and 
(e) For a recursion with 
which is not of type (c) we get for x, the same prop- 
erty as for x,.~ in (d). 
(f) If the previous cases do not apply, the recursion 
might satisfy 
with lal > 0 and Io'1 > 0. Since x~, is not eliminable 
in terms of x,., and vice versa, there must be ground 
terms T, y' such that 
~..~%(A s ~ 3 ~. ~ ~ A x..~ ~ ~' A ~ ~ ~'). 
(g) If a recursion is not of type (a-f) then 
~- 3Xl..Xk( A Sm+l_~m+k D ax ~ x, A a'y ~ x~.~). 
But then x, and x~.~ must be ground eliminable as 
in (f). 
Since a recursion of type (a-c) is not redundant if it 
contains terms T or T' such that T is not reducible in 
terms of x, and r' is not reducible in terms of x~.~ 
and ~- or T ~ are used to eliminate x, and x, ~, there 
must be at least one recursion 7r,~+l...Trm+k such that 
and 0 still follows either by ground inferences or due 
to the properties of (b) and (c). \[\] 
If lmax = max{Iw\[ l ((A,w),Srl e R} then the fol- 
lowing pumping lemma follows immediately as a 
corollary. 
22. COROLLARY. Suppose that w E V~ is deriv- 
lr X I able with ¢= ~Xl..X~X~l..X~kAS (x,..,Xk) over 
of length n, that 3Xl..xl A Sv\[xl,..,xl~ is a nor- 
mal form of ¢ and that Mp=(Hp,~p) is a mini- 
I ~ Iwl "- l IV~F(21U;l+l) then mal model of ¢. j j l max w has 
the form uvzxy with vx >0 and for all i> 1: 
! {~v~zx~y, \[~x~ ..z~ ~'~ .. ~'k/\ ^ S" (~, .., ~k)\]) e A. 
PROOF. If Iwl > lWalx (~IupI+I) then at least one path 
ofwn is longer than \[VNI" (2 \]Hal + 1) and more than 
2\[H;\[ + 1 different nodes on that path are labelled 
by the same A E VN. Without loss of generality 
we assume again that 7r is a derivation whose c- 
structure derivation w has the form w0..w,~..w,~ with 
Wm = s\[u, A,, Y\]0 for each node p on that path which 
is labelled by A, and that 0 is non-deterministically 
constructed from S ~ by recursive variable substi- 
tution, i.e. we require for each (x, ar)E 0 either 
x~ffT e S ~r or 3x~ay • SW((y,T) • 0). Suppose fur- 
thermore that S C S ~ with S~ = S\[0\]. In order to 
isolate the recursion which allows pumping we have 
to distinguish the following cases. 
1. If 7r contains a recursion 7~m~l...7rm+ k with 
wm = s\[u,A~,y\]0, 03mWk = s\[u, Mv, A~.~,x\],,Y\]V 
and Ivx\[ > 0 and x~ and x,.~ do not occur in S ~, 
we take 71"mW1...Tl'rn+k. 
2. If 7r does not contain such a recursion there must 
be at least three distinct A-labelled nodes ~, ~.A, ~.A.v 
on that path such that I' 
wi = S u ,A~,y%, 
fv I A X I~ I1 w~+~=stu,At , ~.X, b,YJ0, 
~+~+~ = s\[~', Ale, Air, A,.~.,, ~\],.~, ~'1~, V'\]0 
with Iv'x'l > O, Irsl > 0 and 
~- ~z~..z'k(A S '~ ~ z~ ~ z~.~, ,~ z~.x.,). 
2.1 Suppose there is a recursion of type (a-c) (cf. 
proof of lemma 21) we choose this one. 
2.2 If 7r does not contain such a recursion each of 
the recursions must be of type (d), (e), (f) or (g). 
But then there must be one recursion "ffrn+l...'ffrn+k 
(m=i,k=Iorm=i+l,k:v) with 
I ff 
50 
for some ground terms T, T'. This recursion is 
choosen for the proof. 
On the basis of the recursion 7rm+i...TrmWk we can 
now define derivations r i as follows. We set 7r i = r 
and define ~r i+i on the basis of 7r ~ by 
I(:\[U,A\[virxi\]~.~',Y\]o,S) ifj>m+ikand ~T~+I 
i = . ~J-~ = (s\[u,A\[vi-1,':~i-1\],.,.,~,-,,y\]~,S) 
~rj if j <_m+ik 
and 
zv~ if j <_ m + ik 
\[#.ai.~ if j > m + ik and i ~j--k ---- ~.~i--l.t. 
By induction on i it can then be shown for all possible 
cases that k 3x~..x~..(A S" - A S'). \[\] 
3.2 Invariance of the Parameters under 
Deductive Equivalence 
Since the universes of the minimal models of two de- 
ductively equivalent consistent feature descriptions 
must have the same cardinality, for the completion 
of the proof of theorem 13 it remains to be shown 
that two deductively equivalent consistent feature 
descriptions have the same "information content", 
i.e. that the sets of literals of their normal forms have 
the same cardinality: 
23. LEMMA. Suppose that 3Xl..Xi A S~rxi,..,xl\] 
and 3Yl..Yk A S~ \[Yi , .., Yk \] are deductively equivalent 
consistent feature descriptions in normal form then 
We proof lemma 23 in two steps. First, we 
show that we can convert s'~rYl,..,Yk\] into a set 
S~\[xi,..,xl\] with the same cardinality such that 
3xl..xl (A S~ rxl, .., Xl\] -~- A S~' rxl, .., xl\] ) holds. 
24. LEMMA. Assume that 3xi..xl A S~rxl,..,xl\] 
and qyi..Yk A S~ \[Yl, .-, Yk \] are deductively equivalent 
consistent feature descriptions in normal form. Then 
H l ---- k and there is a set S~ rXl, ..,xl\] with 
H (i) IS,, rxi,.., :clll = is,, ryl,.., y,ql 
and 
(ii) I- 3x,..xl (A S,, rxl, .., xl\] _= A s~,, \[xi, .., xl\]). 
PROOF. Suppose that {xl,..,xl} N {Yl,..,Yk} = (~ 
(rename if necessary), that S~\[yl,..,yk\] is in re- 
duced form (the reduction of an independent set does 
not change the cardinality) and that M = (~, ~) 
is an arbitrary model of 3xi..Xl A S.rxi,..,xl\] 
and ~yl..ykAS'~ryl,..,yk\]p. Let a and a' 
be assignments such that a ~M Su\[Xl,..,xl\] and 
a' ~M S~\[Yl,..,Yk\];. 
We show first that there is a bijective function 
1) • \[{Xl,..,Xl} ~ {yl,..,Yk}\] such that for all xi 
there is aai • F~ (i = 1, ..,l) and a variable 1)(xi) 
occurring in S' with a(xi) = ~(aiO(xi))(a'). First 
of all ~ is left-total, since a(xi)= ~(T) with T 
variable-free would imply that xi is eliminable 
in S. In order to show that ~ is a func- 
tion, assume a(xi) = ~(ajyj)(a*) = (~(ahYh)(a') for 
Yj,Yh occurring in S' with yj ¢ Yh. Since yj 
and Yh are not eliminable in S' there must 
be terms Tj, Th such that a'(yj)-=~(Tj)(a), 
a'(Yh) = ~(Th)(OL), ~(O'jTj)(OI) ~- ~(qhrh)(O 0 and 
there is no a • Fi* such that ~(aTj)(a) = ~(rh)(a) 
or ~(aVh)(a) = ~(Tj)(a). Thus, vj and Vh must be 
terms in T({xi,..,Xl}\{xi}) and xi would be elim- 
inable in S. Suppose now that yj is not in the 
range of 1). Then there must be a term r with 
.~(T)(a) ---- a'(yj). Since yj is not eliminable in S', 
T must be of the form axi and there must be a 
term T' with a(xi) = .~(T')(a'). If T' is a term in 
T({yi,..,yk}\{Yj}), yj would be eliminable. Oth- 
erwise r' is of the form a'yj and we would get 
a(xi) = c3(a'yj)(a'). Hence 1) is onto. Assume fi- 
nally a(xi) = ~(aiYh)(a') and a(xj) = ~(ajYh)(a') 
with xi ~ xj. Then there must be a term r with 
a'(yh) = ~(T)(a). Since xi and xj would be elim- 
inable in S if T is a term in T({xi, ..,xl}\{xi,xj}), 
T is of the form axi or axj. But then either 
a(z~) = .~(a~x~)(a) or a(x~) = ~(m~xA(~). Thus, 
1) is bijective and l = k. 
On the basis of 1) we then define a sequence of new 
sets S~ (0 < i _< l) by induction as follows (within the 
induction we assume 1)(xi) -= y): 
s~ = s" ryl, .., yl\] p 
, \[s~_,\[y/~,\] if a(x,) = ~'(y) 
Si = ((S~_l \{y~o'o"y} )\[Y/o.x, \] U {x i ~ oJoxi} if (A), 
where (A) means a(xi) ~ a'(y), y~aa'y • S~_ 1 and 
a(xi) = .~(a'y)(a'). In the case where the variables 
refer to different nodes on a loop (a(xi) ~ a'(y)) the 
definition is well-formed, since S~\[yi,..,yl\]p is re- 
duced and normalized and thus there must be ex- 
actly one equation y~aa'y in S~_ 1 describing the 
loop with the node to which xi refers. For S" = S\[, 
IS~ rxl, .., xl\]l ~- Is~ ryl,.., Yk\]l follows immediately 
by induction on the construction of S'. 
Finally we get (ii), since 
3xi..xi(3xi+l..x~ A S~ \[xi, .., xl\] - 
31)(x~+~ )..1)(xl) h s~ rx~, .., :~, 1)(x~+1 ),.., 1)(xl)\]) 
can easily be verified by induction on the construc- 
tion of S". \[\] 
Since two deductively equivalent independent and 
consistent sets of (variable-free) literals reduce to 
the same set of literals in reduced form, lemma 25 
follows by skolemization and completes the proof of 
lemma 23. 
25. LEMMA. If Bxi..xl A Sv \[xi, .., xl\] is a consistent 
feature description m normal form 
andt- 3xi..xl(A S~rxi,..,xl \] ---- A S'~'rxi,..,xl\]) then 
I&rzx,..,zlql --- 
Appendix: Syntax and Semantics of 
Feature-Description Languages 
A feature-descriptionlanguage L(C) consists of the 
logical connectives -.~ (negation), D (implication), 
the equality symbol ~, the existential quantifier 3 
and the parentheses (,). The nonlogical vocabulary 
is given by a finite set of constants V (atomic values), 
51 
a possibly empty finite set of constants C (Skolem- 
constants) and a finite set of unary partial \]unction 
symbols F1 (V, C, F1 pairwise disjoint). The class of 
terms and formulas of L(C) are recursively defined as 
usual. Feature descriptions of L(C) are expressions 
of the form 3xl..xl A Six1, ..,xt\], where S is a finite 
set of (quantifier-free) literals. (We assume that the 
connectives v (disjunction), A (conjunction) and 
(equivalence) are introduced by their usual defini- 
tions.) 
A model for L(C) consists of a nonempty universe 
b/and an interpretation function ~. Since not every 
term denotes an element in/d if the function symbols 
are interpreted as unary partial functions, we gener- 
alize the partiality of the denotation by assuming 
that ~ itself is a partial function. It is only required 
that all Skolem-constants denote. Suppose IX ~-~ Y\] 
designates the set of all partial functions from X to 
Y and IX ~-~ Y\] the set of all total functions from X 
to Y, then a model is defined as follows: 11 
DEFINITION. A model for L(C) is a pair M = (b/, ~), 
consisting of a nonempty set b/and an interpretation 
function ~ = ~v U -~c U ~F1, such that 
(ii) ~c • \[C ~-+/d\], 
(iii) ~F, • IF1 ~ \[U ~/d\]\], 
(iv) Vf • Fl(f • Dom(~) ~ ~(f) ¢O). 
If we extend the denotation function to terms and 
variable assignments c~, the definition of the satisfac- 
tion relation differs only in the clause for the equa- 
tions from the usual one: 
O/ ~M T ,~ T' iff ~(T)(C~) and ~(T')(C~) are defined 
and ~(T)(a) = .~(T')(~). 
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