A COMPLETE AND RECURSIVE FEATURE THEORY* 
Rolf Backofen and Gert Smolka 
German Research Center for Artificial Intelligence (DFKI) 
W-6600 Saarbr/icken, Germany 
{backofen,smolka} @dfki.uni-sb.de 
Abstract 
Various feature descriptions are being employed in 
constrained-based grammar formalisms. The com- 
mon notational primitive of these descriptions are 
functional attributes called features. The descrip- 
tions considered in this paper are the possibly quan- 
tified first-order formulae obtained from a signature 
of features and sorts. We establish a complete first- 
order theory FT by means of three axiom schemes 
and construct three elementarily equivalent models. 
One of the models consists of so-called feature 
graphs, a data structure common in computational 
linguistics. The other two models consist of so-called 
feature trees, a record-like data structure generaliz- 
ing the trees corresponding to first-order terms. 
Our completeness proof exhibits a terminating 
simplification system deciding validity and satisfia- 
bility of possibly quantified feature descriptions. 
1 Introduction 
Feature descriptions provide for the typically partial 
description of abstract objects by means of functional 
attributes called features. They originated in the late 
seventies with so-called unification grammars \[14\], a 
by now popular family of declarative grammar for- 
malisms for the description and processing of natu- 
ral language. More recently, the use of feature de- 
scriptions in logic programming has been advocated 
and studied \[2, 3, 4, 17, 16\]. Essentially, feature de- 
scriptions provide a logical version of records, a data 
structure found in many programming languages. 
Feature descriptions have been proposed in vari- 
ous forms with various formalizations \[1, 13, 9, 15, 
5, 10\]. We will follow the logical approach pioneered 
by \[15\], which accommodates feature descriptions 
as standard first-order formulae interpreted in first- 
order structures. In this approach, a semantics for 
*We appreciate discussions with Joachim Niehren and 
Ralf Treinen who read a draft version of this paper. The 
research reported in this paper has been supported by 
the Bundesminister ffir Forschung und Technologie under 
contracts ITW 90002 0 (DISCO) and ITW 9105 (Hydra). 
For space limitations proofs are omitted; they can be 
found in the complete paper \[6\]. 
feature descriptions can be given by means of a fea- 
ture theory (i.e., a set of closed feature descriptions 
having at least one model). There are two comple- 
mentary ways of specifying a feature theory: either 
by explicitly constructing a standard model and tak- 
ing all sentences valid in it, or by stating axioms 
and proving their consistency. Both possibilities are 
exemplified in \[15\]: the feature graph algebra ~" is 
given as a standard model, and the class of feature 
algebras is obtained by means of an axiomatization. 
Both approaches to fixing a feature theory have 
their advantages. The construction of a standard 
model provides for a clear intuition and yields a com- 
plete feature theory (i.e., if ¢ is a closed feature de- 
scription, then either ¢ or -~¢ is valid). The presenta- 
tion of a recursively enumerable axiomatization has 
the advantage that we inherit from predicate logic a 
sound and complete deduction system for valid fea- 
ture descriptions. 
The ideal case then is to specify a feature theory 
by both a standard model and a corresponding re- 
cursively enumerable axiomatization. The existence 
of such a double characterization, however, is by no 
means obvious since it implies that the feature theory 
is decidable. In fact, so far no decidable, consistent 
and complete feature theory has been known. 
In this paper we will establish a complete and de- 
cidable feature theory FT by means of three axiom 
schemes. We will also construct three models of FT, 
two consisting of so-called feature trees, and one con- 
sisting of so-called feature graphs. Since FT is com- 
plete, all three models are elementarily equivalent 
(i.e., satisfy exactly the same first-order formulae). 
While the feature graph model captures intuitions 
common in linguistically motivated investigations, 
the feature tree model provides the connection to 
the tree constraint systems \[8, 11, 12\] employed in 
logic programming. 
Our proof of FT's completeness will exhibit a sim- 
plification algorithm that computes for every feature 
description an equivalent solved form from which the 
solutions of the description can be read of easily. For 
a closed feature description the solved form is either 
T (which means that the description is valid) or _L 
(which means that the description is invalid). For 
193 
a feature description with free variables the solved 
form is .L if and only if the description is unsatisfi- 
able. 
1.1 Feature Descriptions 
Feature descriptions are first-order formulae built 
over an alphabet of binary predicate symbols, called features, 
and an alphabet of unary predicate sym- 
bols, called sorts. There are no function symbols. 
In admissible interpretations features must be func- 
tional relations, and distinct sorts must be disjoint 
sets. This is stated by the first and second axiom 
scheme of FT'. 
(Axl) VxVyVz(f(x, y) A I(x, z) --~ y - z) (for 
every feature jr) (Ax2) 
W(A(=) ^ B(.) -~ ±) (for every two dis- 
tinct sorts A and B). 
A typical feature description written in matrix no- 
tation is 
X : 
woman 
father 3y 
husband 
engineer \] 
: age :y 
: \[ painter 
age:y \] 
It may be read as saying that x is a woman whose 
father is an engineer, whose husband is a painter, 
and whose father and husband are both of the same 
age. Written in plain first-order syntax we obtain 
the less suggestive formula 
3y, F, H (woman(X) A 
father(x, F) A engineer(V) A age(V, y) A 
husband(x, H) A painter(H) A age(H, y) ). 
The axiom schemes (Axl) and (Ax2) still ad- 
mit trivial models where all features and sorts are 
empty. The third and final axiom scheme of FT 
states that certain "consistent" descriptions have so- 
lutions. Three Examples of instances of FT's third 
axiom scheme are 
3x, y, z (f(x, y) A A(y) A g(x, z) A B(z)) 
Vu, z 3x, y (f(x, y) A g(y, u) A h(y, z) A YfT ) 
Vz 3x, y (f(x, y) A g(y, x) A h(y, z) A yfT), 
where yfT abbreviates -~3z(f(y, z)). Note that the 
third description 
f(=, y) ^ g(y, =) ^ h(y, z) A f~T 
is "cyclic" with respect to the variables x and y. 
1.2 Feature Trees 
A feature tree (examples are shown in Figure 1) is 
a tree whose edges are labeled with features, and 
whose nodes are labeled with sorts. As one would 
expect, the labeling with features must be determin- 
istic, that is, the direct subtrees of a feature tree 
must be uniquely identified by the features of the 
194 
point 
xval~:val 
point 
xval~lor 
2 3 red 
circle 
xval~yval 
n"s ) 
Figure 1: Examples of Feature Trees. 
edges leading to them. Feature trees can be seen as a 
mathematical model of records in programming lan- 
guages. Feature trees without subtrees model atomic 
values (e.g., numbers). Feature trees may be finite or 
infinite, where infinite feature trees provide for the 
convenient representation of cyclic data structures. 
The last example in Figure 1 gives a finite graph 
representation of an infinite feature tree, which may 
arise as the representation of the recursive type equa- 
tion nat = 0 + s(nat). 
Feature descriptions are interpreted over feature 
trees as one would expect: 
• Every sort symbol A is taken as a unary predi- 
cate, where a sort constraint A(x) holds if and 
only if the root of the tree x is labeled with A. 
• Every feature symbol f is taken as a binary 
predicate, where a feature constraint f(x,y) 
holds if and only if the tree x has the direct 
subtree y at feature f. 
The theory of the corresponding first-order structure 
(i.e., the set of all closed formulae valid in this struc- 
ture) is called FT. We will show that FT is in fact ex- 
actly the theory specified by the three axiom schemes 
outlined above, provided the alphabets of sorts and 
features are both taken to be infinite. Hence FT is 
complete (since it is the theory of the feature tree 
structure) and decidable (since it is complete and 
specified by a recursive set of axioms). 
Another, elementarily equivalent, model of FT is 
the substructure of the feature tree structure ob- 
tained by admitting only rational feature trees (i.e., 
finitely branching trees having only finitely many 
subtrees). Yet another model of FT can be obtained 
from so-called feature graphs, which are finite, di- 
rected, possibly cyclic graphs labelled with sorts and 
features similar to feature trees. In contrast to fea- 
ture trees, nodes of feature graphs may or may not 
be labelled with sorts. Feature graphs correspond to 
the so-called feature structures commonly found in 
linguistically motivated investigations \[14, 7\]. 
1.3 Organization of the Paper 
Section 2 recalls the necessary notions and nota- 
tions from Predicate Logic. Section 3 defines the 
theory FT by means of three axiom schemes. Sec- 
tion 4 establishes the overall structure of the com- 
pleteness proof by means of a lemma. Section 5 
studies quantifier-free conjunctive formulae, gives a 
solved form, and introduces path constraints. Sec- 
tion 6 defines feature trees and graphs and estab- 
lishes the respective models of FT. Section 7 studies 
the properties of so-called prime formulae, which are 
the basic building stones of the solved form for gen- 
eral feature constraints. Section 8 presents the quan- 
tifier elimination lemmas and completes the com- 
pleteness proof. 
2 Preliminaries 
Throughout this paper we assume a signature SOR~ 
FEA consisting of an infinite set SOR of unary pred- 
icate symbols called sorts and an infinite set FEA 
of binary predicate symbols called features. For 
the completeness of our axiomatization it is essen- 
tial that there are both infinitely many sorts and 
infinitely many features.The letters A, B, C will al- 
ways denote sorts, and the letters f, g, h will always 
denote features. 
A path is a word (i.e., a finite, possibly empty 
sequence) over the set of all features. The symbol c 
denotes the empty path, which satisfies cp = p = pc 
for every path p. A path p is called a prefix of a 
path q, if there exists a path p' such that pp' = q. 
We also assume an infinite alphabet of variables 
and adopt the convention that x, y, z always de- 
note variables, and X, Y always denote finite, pos- 
sibly empty sets of variables. Under our signa- 
ture SOR ~ FEA, every term is a variable, and an 
atomic formula is either a feature constraint xfy 
(f(x,y) in standard notation), a sort constraint 
Ax (A(x) in standard notation), an equation x - y, 
_L ("false"), or T ("true"). Compound formulae are 
obtained as usual with the connectives A, V, --+, ~-+, 
-~ and the quantifiers 3 and V. We use 3¢ \[V¢\] to de- 
note the existential \[universal\] closure of a formula 
¢. Moreover, 1)(¢) is taken to denote the set of all 
variables that occur free in a formula ¢. The letters 
¢ and ¢ will always denote formulae. 
We assume that the conjunction of formulae is an 
associative and commutative operation that has T 
as neutral element. This means that we identify 
eA(¢A0) withOA(¢A¢),andeATwith¢(but 
not, for example, xfy A xfy with xfy). A conjunc- 
tion of atomic formulae can thus be seen as the finite 
multiset of these formulae, where conjunction is mul- 
tiset union, and T (the "empty conjunction") is the 
empty multiset. We will write ¢ C ¢ (or ¢ E ¢, if 
¢ is an atomic formula) if there exists a formula ¢~ 
such that ¢ A ¢1 = ¢. 
Moreover, we identify 3x3y¢ with 3y3x¢. If X = 
{xl,...,xn}, we write 3X¢ for 3xl ...3xn¢. IfX = 
0, then 3X¢ stands for ¢. 
Structures and satisfaction of formulae are defined 
as usual. A valuation into a structure `4 is a total 
function from the set of all variables into the universe 
1`4\] of`4. A valuation ~' into,4 is called an x-update 
\[X-update\] of a valuation a into ,4 if (~' and a a~ree 
everywhere but possibly on x \[X\]. We use ¢~ to 
denote the set of all valuations c~ such that ,4, c~ ~ ¢. 
We write ¢ ~ ¢ ("¢ entails ¢") if CA C ¢ A for all 
structures ,4, and ¢ ~ ¢ ("¢ is equivalent to ¢") if 
¢.4 = cA for all structures ,4. 
A theory is a set of closed formulae. A model of 
a theory is a structure that satisfies every formulae 
of the theory. A formula ¢ is a consequence of 
a theory T (T ~ ¢) if V¢ is valid in every model 
of T. A formula ¢ entails a formula ¢ in a theory 
T (¢ ~T ¢) if ¢'4 C_ ¢.4 for every model ,4 of T. 
Two formulae ¢, ¢ are equivalent in a theory T 
(¢ ~T ¢) if cA = ¢.4 for every model ,4 of T. 
A theory T is complete if for every closed formula 
either ¢ or -,¢ is a consequence of T. A theory is 
decidable if the set of its consequences is decidable. 
Since the consequences of a recursively enumerable 
theory are recursively enumerable (completeness of 
first-order deduction), a complete theory is decidable 
if and only if it is recursively enumerable. 
Two first-order structures ,4, B are elementarily 
equivalent if, for every first-order formula ¢, ¢ is 
valid in ,4 if and only if ¢ is valid in B. Note that all 
models of a complete theory are elementarily equiv- 
alent. 
3 The Axioms 
The first axiom scheme says that features are func- 
tional: 
(Ax1) VxVyVz(xfy A z.fz ---* y -- z) (for every 
feature f). 
The second scheme says that sorts are mutually dis- 
joint: 
(Ax2) Vx(Ax A Bx --* 1) (for every two distinct 
sorts A and B). 
The third and final axiom scheme will say that cer- 
tain "consistent feature descriptions" are satisfiable. 
For its formulation we need the important notion of 
a solved clause. 
An exclusion constraint is an additional atomic 
formula of the form zfI ("f undefined on x") taken 
to be equivalent to -~3y (xfy) (for some variable y # 
z). 
A solved clause is a possibly empty conjunction 
¢ of atomic formulae of the form xfy, Ax and xf~ 
such that the following conditions are satisfied: 
1. no atomic formula occurs twice in ¢ 
2. ifAxEeandBxE¢,thenA=B 
3. ifxfyEeandxfzE¢,theny=z 
4. if xfy E ¢, then xfT q~ ¢. 
Figure 2 gives a graph representation of the solved 
clause 
xfu A xgv A zh~ A 
195 
f-~x hT 
=~ B gT 
Figure 2: A graph representation of a solved clause. 
Cu A uhx A ugy A u f z A 
Av ^ vgz ^ vhw ^ vfT A 
Bw A wIT A wg T . 
As in the example, a solved clause can always be seen 
as the graph whose nodes are the variables appearing 
in the clause and whose arcs are given by the feature 
constraints xfy. The constraints Ax, xfT appear as 
labels of the node x. 
A variable x is constrained in a solved clause ¢ 
if ¢ contains a constraint of the form Ax, xfy or 
xfT. We use CV(¢) to denote the set of all variables 
that are constrained in ¢. The variables in V(¢) - 
CV(¢) are called the parameters of a solved clause 
¢. In the graph representation of a solved clause the 
parameters appear as leaves that are not not labeled 
with a sort or a feature exclusion. The parameters 
of the solved clause in Figure 2 are y and z. 
We can now state the third axiom scheme. It says 
that the constrained variables of a solved clause have 
solutions for all values of the parameters: 
(Ax3) ~/qx¢ (for every solved clause ¢ and X = 
cv(¢)). 
The theory FT is the set of all sentences that can 
be obtained as instances of the axiom schemes (Axl), 
(Ax2) and (Ax3). The theory FTo is the set of all 
sentences that can be obtained as instances of the 
first two axiom schemes. 
As the main result of this paper we will show that 
FT is a complete and decidable theory. 
By using an adaption of the proof of Theorem 8.3 
in \[15\] one can show that FTo is undecidable. 
4 Outline of the Completeness Proof 
The completeness of FT will be shown by exhibit- 
ing a simplification algorithm for FT. The following 
lemma gives the overall structure of the algorithm, 
which is the same as in Maher's \[12\] completeness 
proof for the theory of constructor trees. 
Lemma 4.1 Suppose there exists a set of so-called 
prime formulae such that: 
1. every sort constraint Ax, every feature con- 
straint xfy, and every equation x - y such that 
= ~ y is a prime formula 
2. T is a prime formula, and there is no other 
closed prime formula 
3. for every two prime formulae fl and fl' one can 
compute a formula 6 that is either prime or .1_ 
and satisfies 
flAi'MFT6 and )2(6)C_V(flAff) 
4. for every prime formula fl and every variable x 
one can compute a prime formula i' such that 
3xi MFT fl' and Y(t') C_ Y(3xfl) 
5. if i, ill,''' ,fin are prime formulae, then 
ft 
^ 3=(t ^ 
i=1 i----1 
6. for every two prime formulae fl, fl I and every 
variable x one can compute a Boolean combina- 
tion 6 of prime formulae such that 
3~(fl^-,¢) I~FT 6 and Vff) C VO=(fl^-~l')). 
Then one can compute for every formula ¢ a Boolean 
combination ~ of prime formulae such that ¢ MET ~ 
and V(O C_ V(¢). 
Proof. Suppose a set of prime formulae as required 
exists. Let ¢ be a formula. We show by induction on 
the structure of ¢ how to compute a Boolean combi- 
nation df of prime formulae such that ¢ MET 6 and 
V(O C_ V(¢). 
If ¢ is an atomic formula Ax, xfy or x - y, then 
¢ is either a prime formula, or ¢ is a trivial equation 
z - z, in which case it is equivalent to the prime 
formula T. 
If ¢ is -~¢, ¢ ^ ¢' or ¢ V ¢', then the claim follows 
immediately with the induction hypothesis. 
It remains to show the claim for ¢ = 3=¢. By the 
induction hypothesis we know that we can compute a 
Boolean combination df of prime formulae such that 
6 MFT ~) and V(6) C_ V(¢). Now ~ can be trans- 
formed to a disjunctive normal form where prime 
formulae play the role of atomic formulae; that is, 6 
is equivalent to 6'1 V... V ¢,, where every "clause" 
qi is a conjunction of prime and negated prime for- 
mulae. Hence 
3=¢ 14 3=(o-~ v... v ,..) I=13=o-~ v... v 3=o-., 
where all three formulae have exactly the same free 
variables. It remains to show that one can compute 
for every clause ~r a Boolean combination 6 of prime 
formulae such that =1=o- MET 6 and Y(6) C_ V(3xa). 
We distinguish the following cases. 
(i) a = fl for some basic formula i. Then the claim 
follows by assumption (4). Oi) o" i^" 
~ , = Ai=I ti n > 0. Then the claim follows 
with assumptions (5) and (6). n T n 
Oil) tr = Ai=I -~ii, n > 0. Then a MET AA/=I -~fli 
and the claim follows with case (it) since T is a prime 
formula by assumption (2). 
(iv) ~ =ill ^...^tk ^-,ill ^... h t', k > 1, n ___ 0. 
Then we know by assumption (3) that either fll A...A 
flk MFT .L or fll A ... A flk MET fl for some prime 
formula ft. In the former case we choose 8 = -,T, 
and in the latter case the claim follows with case (i) 
or (ii). \[\] 
196 
Note that, provided a set of prime formulae with 
the required properties exists, the preceding lemma 
yields the completeness of FT since every closed for- 
mula can be simplified to T or -~T (since T is the 
only closed prime formula). 
In the following we will establish a set of prime 
formula as required. 
5 Solved Formulae 
In this section we introduce a solved form for con- 
junctions of atomic formulae. 
A basic formula is either 3- or a possibly empty 
conjunction of atomic formulae of the form Ax, xfy, 
and x - y. Note that T is a basic formula since T is 
the empty conjunction. 
Every basic formula ¢ ~ 3- has a unique decom- 
position ¢ = CN ACG into a possibly empty conjunc- 
tion CN of equations "x -- y" and a possibly empty 
conjunction CG of sort constraints "Ax" and feature 
constraints "xfy". We call CN the normalizer and 
and ¢G the graph of ¢. 
We say that a basic formula ¢ binds x to y if 
x - y E ¢ and x occurs only once in ¢. Here it 
is important to note that we consider equations as 
directed, that is, assume that x - y is different from 
y ~ x ifx ~ y. We say that ¢ eliminatesx if¢ 
binds x to some variable y. 
A solved formula is a basic formula 7 ~ 3- such 
that the following conditions are satisfied: 
1. an equation x - y appears in 7 if and only if 7 
eliminates x 
2. the graph of 7 is a solved clause. 
Note that a solved clause not containing exclusion 
constraints is a solved formula, and that a solved 
formula not containing equations is a solved clause. 
The letter 7 will always denote a solved formula. 
We will see that every basic formula is equivalent 
in FT0 to either 3- or a solved formula. 
Figure 3 shows the so-called basic simplification 
rules. With ¢\[x ~-- y\] we denote the formula that 
is obtained from ¢ by replacing every occurrence of 
x with y. We say that a formula ¢ simplifies to a 
formula ¢ by a simplification rule p if ~ is an instance 
of p. We say that a basic formula ¢ simplifies to a 
basic formula ¢ if either ¢ = ¢ or ¢ simplifies to ¢ 
in finitely many steps each licensed by one of basic 
simplification rules in Figure 3. 
Note that the basic simplification rules (1) and (2) 
correspond to the first and second axiom scheme, re- 
spectively. Thus they are equivalence transformation 
with respect to FTo. The remaining three simplifica- 
tion rules are equivalence transformations in general. 
Proposition 5.1 The basic simplification rules are 
terminating and perform equivalence transforma- 
tions with respect to FT0. Moreover, a basic formula 
¢ ~ 3_ is solved if and only if no basic simplification 
rule applies to it. 
Proposition 5.2 Let ¢ be a formula built from 
atomic formulae with conjunction. Then one can 
1. xfy A xfz A ¢ 
xfzAy--zA¢ 
AxABxA¢ 2. 3- A# B 
Ax A Ax A ¢ 3. 
AxA¢ 
x--yA¢ 4. z E 13(¢) and x ~ y 
~- y^¢\[~,-- y\] 
z--xA¢ 5. 
¢ 
Figure 3: The basic simplification rules. 
compute a formula 6 that is either solved or 3_ such 
that ¢ ~FTo 6 and r(6) C_ l;(¢). 
In the quantifier elimination proofs to come it 
will be convenient to use so-called path constraints, 
which provide a flexible syntax for atomic formulae 
closed under conjunction and existential quantifica- 
tion. We start by defining the denotation of a path. 
The interpretations fit, g~ of two features f, g 
in a structure .4 are binary relations on the universe 
1"41 of .4; hence their composition fA o g.a is again a 
binary relation on 1-41 satisfying 
a(f A o gA)b ¢=:¢, 3c ~ 1"41: af Ac A cfAb 
for all a, b E 1"41. Consequently we define the deno- 
tation p~t of a path p = fl "'" .In in a structure .4 
as the composition 
(fl...fn) A :---- f:o...ofn A, 
where the empty path ~ is taken to denote the iden- 
tity relation. If.4 is a model of the theory FTo, then 
every paths denotes a unary partial function on the 
universe of .4. Given an element a E \[.41, p~t is thus 
either undefined on a or leads from a to exactly one 
b ~ 1.41. 
Let p, q be paths, x, y be variables, and A be a 
sort. Then path constraints are defined as follows: 
.4, a ~ zpy :¢:~ o~(x) pA a(y) 
.4, a ~ xp.~yq :¢:=~ 3a E 1.41: °t(x)pa aAa(y)q A a 
.4, a~Azp :~=~3ael.41: a(z)p'4a^aeA "~. 
Note that path constraints xpy generalize feature 
constraints x fy. 
A proper path constraint is a path constraint 
of the form "Axp" or "xp ~. yq". 
Every path constraint can be expressed with the 
already existing formulae, as can be seen from the 
following equivalences: 
x~y ~ x - y 
xfpy ~ 3z(xfz A zpy) (z ~£ x,y) 
xpl yq N 3z(xpz ^ uqz) (z # ~, ~) 
mxp ~ 3y(xpy A my) (y   x). 
197 
The closure \[3`\] of a solved formula 3` is the 
closure of the atomic formulae occurring in 7 with 
respect to the following deduction rules: 
x-y xpy yfz xpz yqz Ay xpy 
xEx xey zpf z xp I Yq Axp 
Recall that we assume that equations x - y are di- 
rected, that is, are ordered pairs of variables. Hence, 
xey E \[71 and yex ~ \[71 if x - y E 7. 
The closure of a solved clause 6 is defined anal- 
ogously. 
Proposition 5.3 Let 7 be a solved formula. Then: 
I. if ~v E \[7\], then 7 ~ ~r 
2. xeyE\[7\] iff x=yorx--yE7 
3. xfy E \[7\] iff zfy E 3` or 3z: z "-- z E 
7 and zfy E 7 
4. xpfy e \[7\] iff 3z: xpz e \[7\] and zfy e 3` 
5. if p 7 £ e and xpy, xpz E \[3`\], then y = z 
5. it is decidable whether a path constraint is in \[3'\]. 
6 Feature Trees and Feature Graphs 
In this section we establish three models of FT con- 
sisting of either feature trees or feature graphs. Since 
we will show that FT is a complete theory, all three 
models are in fact elementarily equivalent. 
A tree domain is a nonempty set D _C FEA* of 
paths that is prefix-closed, that is, if pq E D, then 
p E D. Note that every tree domain contains the 
empty path. 
A feature tree is a partial function a: FEA* --+ 
SOR whose domain is a tree domain. The paths in 
the domain of a feature tree represent the nodes of 
the tree; the empty path represents its root. We 
use D~ to denote the domain of a feature tree ~. A 
feature tree is called finite \[infinite I if its domain 
is finite \[infinite\]. The letters a and 7. will always 
denote feature trees. 
The subtree pa of a feature tree a at a path 
p E Da is the feature tree defined by (in relational 
notation) 
pa := {(q,A) l(pq, A) Ea}. 
A feature tree a is called a subtree of a feature tree 
7- if ~r is a subtree of 7- at some path p E Dr, and a 
direct subtree if p = f for some feature f. 
A feature tree a is called rational if (1) cr has only 
finitely many subtrees and (2) a is finitely branching 
(i.e., for every p E D~, the set {pf E D~ \[ f E FEA} 
is finite). Note that for every rational feature tree 
a there exist finitely many features fl,...,In such 
that Do C_ {fl,..-,fn}*. 
The feature tree structure'T is the SOR~FEA- 
structure defined as follows: 
* the universe of 7- is the set of all feature trees 
• (r E A 7- iff a(c) = A (i.e., a's root is labeled 
with A) 
• (~,7-) EfT" iff f E Da and 7- = fa (i.e., r is the 
subtree of a at f). 
The rational feature tree structure 7~ is the sub- 
structure of T consisting only of the rational feature 
trees. 
Theorem 6.1 The feature tree structures T and 7~ 
are models of the theory FT. 
A feature pregraph is a pair (x, 7) consisting of 
a variable x (called the root) and a solved clause 
7 not containing exclusion constraints such that, for 
every variable y occurring in 7, there exists a path 
p satisfying xpy E \[7\]- If one deletes the exclusion 
constraints in Figure 2, one obtains the graphical 
representation of a feature pregraph whose root is x. 
A feature pregraph (x, 7) is called a subpregraph 
of a feature pregraph (y,~) if 7 _C 6 and x -- y or 
x E \]2(~). Note that a feature pregraph has only 
finitely many subpregraphs. 
We say that two feature pregraphs are equivalent 
if they are equal up to consistent variable renaming. 
For instance, (x, xfy A ygx) and (u, ufx A xgu) are 
equivalent feature pregraphs. 
A feature graph is an element of the quotient 
of the set of all feature pregraphs with respect to 
equivalence as defined above. We use (x, 7) to denote 
the feature graph obtained as the equivalence class 
of the feature pregraph (x, 7). 
In contrast to feature trees, not every node of a 
feature graph must carry a sort. 
The feature graph structure ~ is the SOR 
FEA-structure defined as follows: 
• the universe of ~ is the set of all feature graphs 
• (x,7) EA ~iffAxE7 
• ((x, 7), a) E f6 iff there exists a maximal fea- 
ture subpregraph (y, ~) of (x, 7) such that xfy E 
7 and ~r -- (y, 6). 
Theorem 6.2 The feature graph structure ~ is a 
model of the theory FT. 
Let ~" be the structure whose domain consists of 
all feature pregraphs and that is otherwise defined 
analogous to G. Note that G is in fact the quotient 
of jc with respect to equivalence of feature pregraphs. 
Proposition 6.3 The feature pregraph structure yr 
is a model of FTo but not of FT. 
7 Prime Formulae 
We now define a class of prime formulae having the 
properties required by Lemma 4.1. The prime for- 
mulae will turn out to be solved forms for formulae 
built from atomic formulae with conjunction and ex- 
istential quantification. 
A prime formula is a formula 3X7 such that 
1. 7 is a solved formula 
2. X has no variable in common with the normal- 
izer of 3' 
3. every x E X can be reached from a free variable, 
that is, there exists a path constraint ypx E \[7\] 
such that y ~t X. 
198 
The letter/3 will always denote a prime formula. 
Note that T is the only closed prime formula, and 
that 3X 7 is a prime formula if 3x3X 7 is a prime 
formula. Moreover, every solved formula is a prime 
formula, and every quantifier-free prime formula is a 
solved formula. 
The definition of prime formulae certainly fulfills 
the requirements (1) and (2) of Lemma 4.1. The 
fulfillment of the requirements (3) and (4) will be 
shown in this section, and the fulfillment of the re- 
quirements (5) and (6) will be shown in the next 
section. 
Proposition 7.1 Let 3X 7 be a prime formula, .A 
be a model of FT, and ,4, a ~ 3X7. Then there 
exists one and only one X-update (~' of ~ such that 
A,a' ~7. 
The next proposition establishes that prime formu- 
lae are closed under existential quantification (prop- 
erty (4) of Lemma 4.1). 
Proposition 7.2 For every prime formula /3 and 
every variable x one can compute a prime formula 
/3' such thai 3x/3 ~:~FT /3' and Y(/3') C Y(3x/3). 
Proposition 7.3 If /3 is a prime formula, then 
FT p i/3. 
The next proposition establishes that prime formu- 
lae are closed under consistent conjunction (property 
(3) of Lemma 4.1). 
Proposition 7.4 For every two prime formulae /3 
and/3' one can compute a formula 8 that is either 
prime or _L and satisfies 
/3 A/3' ~FT 8 and 1)(6) C 1)(/3 A/3'). 
Proposition 7.5 Let ¢ be a formula that is built 
from atomic formulae with conjunction and existen- 
tial quantification. Then one can compute a formula 
6 that is either prime or I such that ¢ ~FT 8 and 
Vff) _C V(¢). 
The closure of a prime formula 3X7 is defined 
as follows: 
\[3xv\] := { ~ e \[7\] I v(~) n x = ~ or ~ = xc~ 
or ~ = =¢ 1=~ }- 
The proper closure of a prime formula/3 is de- 
fined as follows: 
\[/3\]* := {Tr • \[/3\] I r is a proper path constraint}. 
Proposition 7.6 If/3 is a prime formula and r • 
\[/3\], then/3 p ~ (and hence --,,~ p --,/3). 
We now know that the closure \[ill, taken as an 
infinite conjunction, is entailed by/3. We are going to 
show that, conversely,/3 is entailed by certain finite 
subsets of its closure \[/3\]. 
An access function for a prime formula/3 = 3X 7 
is a function that maps every x • 1)(7 ) - X to the 
rooted path x¢, and every x E X to a rooted path 
x'p such that x'px • \[7\] and x' ~ X. Note that 
every prime formula has at least one access function, 
and that the access function of a prime formula is 
injective on 1)(3') (follows from Proposition 5.3 (5)). 
The projection of a prime formula/3 = 3X7 with 
respect to an access function @ for/3 is the conjunc- 
tion of the following proper path constraints: 
{Ax'p I Ax E 7, x'p = @x} U 
{='pf~y'q \[ xfy E 7, x'p = @x, y'q = @y}. 
Obviously, one can compute for every prime formula 
an access function and hence a projection. Further- 
more, if )~ is a projection of a prime formula/3, then 
)~ taken as a set is a finite subset of the closure \[/3\]. 
Proposition 7.7 Let )~ be a projection of a prime 
formula/3. Then )t C \[/3\]* and )t ~=~FT /3" 
As a consequence of this proposition one can 
compute for every prime formula an equivalent 
quantifier-free conjunction of proper path con- 
straints. 
We close this section with a few propositions stat- 
ing interesting properties of closures of prime formu- 
lae. These propositions will not be used in the proofs 
to come. 
Proposition 7.8 If fl is a prime formula, then 
/3 ~FT \[/3\]*. 
Proposition 7.9 If/3 is a prime formula, and r is 
a proper path constraint, then 
~e\[Z\]* ¢=~ /3Prr~- 
Proposition 7.10 Let /3, /3' be prime formulae. 
Then/3 ~FT fl' ¢=~ ~\]* _D \[/3'\]*. 
Proposition 7.11 Let/3,/3' be prime formulae, and 
let )d be a projection of/3'. Then \]3 ~FT /3t \[#\]* _~ k'. 
Proposition 7.11 gives us a decision procedure for 
"/3 ~FT /3" since membership in \[/3\]* is decidable, 
k' is finite, and ,V can be computed from/3'. 
8 Quantifier Elimination 
In this section we show that our prime formulae sat- 
isfy the requirements (5) and (6) of Lemma 4.1 and 
thus obtain the completeness of FT. We start with 
the definition of the central notion of a joker. 
A rooted path xp consists of a variable x and a 
path p. A rooted path xp is called unfree in a prime 
formula 13 if 
3 prefix p' of p 3 yq: x 5£ y and xp' I Yq E \[/3\]. 
A rooted path is called free in a prime formula/3 if 
it is not unfree in/3. 
Proposition 8.1 Let/3 = 3X 7 be a prime formula. 
Then: 
1. if xp is free in/3, then x does not occur in the 
normalizer of 7 
2. if x ~ 1)(/3), then xp is free in/3 for every path 
p. 
199 
A proper path constraint 7r is called an z-joker for 
a prime formula/3 if r ~ \[/3\] and one of the following 
conditions is satisfied: 
1. 7r = Axp and xp is free in fl 
2. 7r = xp ~ yq and xp is free in/3 
3. 7r = yp ~ xq and xq is free in/3. 
Proposition 8.2 It is decidable whether a rooted 
path is free in a prime formula, and whether a path 
constraint is an x-joker for a prime formula. 
Lemma 8.3 Let/3 be a prime formula, x be a vari- 
able, 7r be a proper path constraint that is not an 
x-joker for /3, A be a model of FT, .A,c~ ~ fl, 
.4,~' ~ /3, and t~' be an z-update of c~. Then 
A, c~ ~ 7r if and only if.A, a' ~ 7r. 
Lemma 8.4 Let /3 be a prime formula and 
7q,..., rn be x-jokers for/3. Then 
3x/3 ~FT 3Z(/3A Z"nffi )" 
i=1 
The proof of this lemma uses the third axiom 
scheme, the existence of infinitely many features, and 
the existence of infinitely many sorts. 
Lemma 8.5 Let/3, /3' be prime formulae and a be 
a valuation into a model A of FT such that 
,4, ~ p 3x(/3 A/3') and .4, ~ p 3x(/3 A -,/3'). 
Then every projection of/3' contains an z-joker for 
/3. 
Lemma 8.6 If/3, /31,...,/3n are prime formulae, 
then 
::lz(fl A Z "~/3`) ~::~FT Z 3z(fl A "-,fl,). 
i=1 i=l 
Lemma 8.7 For every two prime formulae /3, /3' 
and every variable x one can compute a Boolean com- 
bination 6 of prime formulae such that 
3x(/j A-,/3') ~FT 6 and 12(6) C 12(qx(fl A ~/3')). 
Theorem 8.8 For every formula ~b one can compute 
a Boolean combination 6 of prime formulae such that 
MFT 6 and V(6) C_ V(/3) 
Corollary 8.9 FT is a complete and decidable the- 
ory. 

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