FEATURE LOGIC WITH WEAK SUBSUMPTION 
CONSTRAINTS 
Jochen Dbere 
IBM Deutschland OmbH 
Science Center - IKBS 
P.O. Box 80 08 80 
D-7000 Stuttgart 80, Germany 
ABSTRACT 
In the general framework of a constraint-based 
grammar formalism often some sort of feature 
logic serves as the constraint language to de- 
scribe linguistic objects. We investigate the ex- 
tension of basic feature logic with subsumption 
(or matching) constraints, based on a weak no- 
tion of subsumption. This mechanism of one- 
way information flow is generally deemed to be 
necessary to give linguistically satisfactory de- 
scriptions of coordination phenomena in such 
formalisms. We show that the problem whether 
a set of constraints is satisfiable in this logic is 
decidable in polynomial time and give a solution 
algorithm. 
1 Introduction 
Many of the current constralnt-based grammar 
formalisms, as e.g. FUG \[Kay 79, Kay 85\], LFG 
\[Kaplan/Bresnan 82\], HPSG \[Pollard/Sag 87\], 
PATR-II \[Shieber et al. 83\] and its derivates, 
model linguistic knowledge in recursive fea- 
ture structures. Feature (or functional) equa- 
tions, as in LFG, or feature terms, as in FUG 
or STUF \[Bouma et al. 88\], are used as con- 
straints to describe declaratively what proper- 
ties should be assigned to a linguistic entity. 
In the last few years, the study of the for- 
real semantics and formal properties of logics 
involving such constraints has made substan- 
tial progress \[Kasper/Rounds 86, Johnson 87, 
Smolka 88, Smolka 89\], e.g., by making precise 
which sublanguages of predicate logic it corre- 
sponds to. This paves the way not only for reli- 
able implementations of these formalisms, but 
also for extensions of the basic logic with a 
precisely defined meaning. The extension we 
present here, weak subsumption constraints, is 
a mechanism of one-way information flow, often 
proposed for a logical treatment of coordination 
in a feature-based unification grammar. 1 It can 
I Another application would be type inference in a 
grammar formalism (or programming language) that 
be informally described as a device, which en- 
ables us to require that one part of a (solution) 
feature structure has to be subsumed (be an in- 
stance of) another part. 
Consider the following example of a coordina- 
tion with "and", taken from \[Shieber 89\]. 
(1) Pat hired \[tcP a Republican\] and 
\[NP a banker\]. 
(2) *Pat hired \[NP a Republican\] and lAP 
proud of it\]. 
Clearly (2) is ungrammatical since the verb 
"hire" requires a noun phrase as object com- 
plement and this requirement has to be ful- 
filled by both coordinated complements. This 
subcategorization requirement is modeled in 
a unification-based grammar generaUy using 
equations which cause the features of a comple- 
ment (or parts thereof encoding the type) to get 
unified with features encoding the requirements 
of the respective position in the subcategoriza- 
tion frame of the verb. Thus we could assume 
that for a coordination the type-encoding fea- 
tures of each element have to be "unified into" 
the respective position in the subcategorisation 
frame. This entails that the coordinated ele- 
ments are taken to be of one single type, which 
then can be viewed as the type of the whole 
coordination. This approach works fine for the 
verb "hire", but certain verbs, used very fre- 
quently, do not require this strict identity. 
(3) Pat has become \[NP a banker\] and 
\[AP very conservative\]. 
(4) Pat is lAP healthy\] and \[pp of 
sound mind\]. 
The verb "become" may have either noun- 
phrase or adjective-phrase complements, "to 
be" Mlows prepositional and verb phrases in 
addition, and these may appear intermixed in 
a coordination. In order to allow for such 
"polymorphic" type requirements, we want to 
l~e~ a-type discipline with polymorphic types. 
256 
state, that (the types of) coordinated arguments 
each should be an instance of the respective re- 
quirement from the verb. Expressed in a gen- 
eral rule for (constituent) coordination, we want 
the structures of coordinated phrases to be in- 
stances of the structure of the coordination. Us- 
ing subsumption constraints the rule basically 
looks like this: 
E ~ C and D 
E~C 
E~D 
With an encoding of the types like the one pro- 
posed in HPSG we can model the subcatego- 
risation requirements for"to be" and "to be- 
come" as generalizations of all allowed types (cf. 
Fig. 1). 
i n: \] \] NP= v: - AP= v: + 
bar: 2 bar: 2 
VPffi v: + PP= v: - 
bar: 2 bar: 2 
'to be' requires: 'to become' requires: 
Figure 1: Encoding of syntactic type 
A similar treatment of constituent coordina- 
tion has been proposed in \[Kaplan/Maxwell 88\], 
where the coordinated elements are required to 
be in a set of feature structures and where the 
feature structure of the whole set is defined as 
the generalisation (greatest lower bound w.r.t. 
subsumption) of its elements. This entails 
the requirement stated above, namely that the 
structure of the coordination subsumes those of 
its elements. In fact, it seems that especially in 
the context of set-valued feature structures (cf. 
\[Rounds 88\]) we need some method of inheri- 
tance of constraints, since if we want to state 
general combination rules which apply to the 
set-valued objects as well, we would like con- 
straints imposed on them to affect also their 
members in a principled way. 
Now, recently it turned out that a feature logic 
involving subsumption constraints, which are 
based on the generally adopted notion of sub- 
sumption for feature graphs is undecidable (cf. 
\[D rre/Rounds 90\]). In the present paper we 
therefore investigate a weaker notion of sub- 
sumption, which we can roughly characterize as 
257 
relaxing the constraint that an instance of a fea- 
ture graph contains all of its path equivalencies. 
Observe, that path equivalencies play no role in 
the subcategorisation requirements in our ex- 
amples above ...... ~ 
2 Feature Algebras 
In this section we define the basic structures 
which are possible interpretations of feature de- 
scriptions, the expressions of our feature logic. 
Instead of restricting ourselves to a specific in- 
terpretation, like in \[Kasper/Rounds 86\] where 
feature structures are defined as a special kind 
of finite automata, we employ an open-world se- 
mantics as in predicate logic. We adopt most 
of the basic definitions from \[Smolka 89\]. The 
mathematical structures which serve us as in- 
terpretations are called feature algebras. 
We begin by assuming the pairwise disjoint sets 
of symbols L, A and V, called the sets of fea- 
tures (or labels), atoms (or constants) and vari- 
ables, respectively. Generally we use the letters 
/,g, h for features, a, b, c for atoms, and z, ~, z 
for variables. The letters s and t always denote 
variables or atoms. We assume that there are 
infinitely many variables. 
A feature algebra .A is a pair (D ~4, ..4) consisting 
of a nonempty set D ~t (the domain of.4) and an 
interpretation .~ defined on L and A such that 
* a ~4 E D "4 for a E A. (atoms are constants) 
• Ifa ~ b then a "4 ~ b ~4. (unique name as- 
sumption) 
• If f is a feature then/~4 is a unary partial 
function on D ~4. (features are functional) 
• No feature is defined on an atom. 
Notation. We write function symbols on the 
right following the notation for record fields in 
computer languages, so that f(d) is written dr. 
If f is defined at d, we write d.f ~, and other- 
wise d/ T. We use p,q,r to denote strings of 
features, called paths. The interpretation func- 
tion .Jr is straightforwardly extended to paths: 
for the empty path e, ~.4 is the identity on D~4; 
for a path p = fl ... f-, p~4 is the unary partial 
function which is the composition of the filnc- 
tions fi"4.., f.4, where .fl "4 is applied first. 
A feature algebra of special interest is the Fea- 
ture Graph Algebra yr since it is canonical 
in the sense that whenever there exists a solu- 
tion for a formula in basic feature logic in some 
feature algebra then there is also one in the Fea- 
ture Graph Algebra. The same holds if we ex- 
tend our logic to subsumption constraints (see 
~DSrre/Rounds 90\]). A feature graph is a rooted 
and connected directed graph. The nodes are 
either variables or atoms, where atoms may ap- 
pear only as terminal nodes. The edges are la- 
beled with features and for every node no two 
outgoing edges may be labeled with the same 
feature. 
We formalize feature graphs as pairs (s0, E) 
where So E VUA is the root and E C V x 
L x (V U A) is a set of triples, the edges. The 
following conditions hold: 
1. If s0EA, thenE=0. 
2. If (z, f, s) and (z, f, t) are in E, then s : t. 
3. If (z, f, 8) is in E, then E contains edges 
leading from the root s0 to the node z. 
Let G - (z0, E) be a feature graph containing 
an edge (z0, f, s). The subgraph under f of G 
(written G/f) is the maximal graph (s, E') such 
that E t C E. 
Now it is clear how the Feature Graph Algebra 
~" is to be defined. D ~r is the set of all feature 
graphs. The interpretation of an atom a ~r is the 
feature graph (a, ~), and for a feature f we let 
G.f 7~ = G/.f, if this is defined. It is easy to 
verify that ~r is a feature algebra. 
Feature graphs are normally seen as data ob- 
jects containing information. From this view- 
point there exists a natural preorder, called sub- 
sumptlon preorder, that orders feature graphs 
according to their informational content thereby 
abstracting away from variable names. We do 
not introduce subsumption on feature graphs 
here directly, but instead we define a subsump- 
tion order on feature algebras in general. 
Let .A and B be feature algebras. A simulation 
between .A and B is a relation A C D ~4 × D v 
satisfying the following conditions: 
1. if (a ~4, d) E A then d = a B, for each atom 
a, and 
2. for any d E D~,e E D B and f E L: if 
df A ~ and (d,e) E A, then ef B ~ and 
(dr ~4, ef B) E A. 
Notice that the union of two simulations and 
the transitive closure of a simulation are also 
simulations. 
A partial homomorphlsm "y between .A and 
B is a simulation between the two which is a 
partial function. If.A = B we also call T a partial 
endomorphism. 
Definition. Let .A be a feature algebra. The 
(strong) subsumption preorder ff_A and 
258 
the weak subsumption preorder ~4 of ~4 
are defined as follows: 
* d (strongly) subsumes e (written d E ~4 e) 
iff there is an endomorphism "y such that 
= e. 
* d wealcly subsumes e (written d ~4 e) iff 
there is a simulation A such that dAe. 
It can be shown (see \[Smolka 89\]) that the 
subsumption preorder of the feature graph 
algebra coincides with the subsumption or- 
der usually defined on feature graphs, e.g. in 
\[Kasper/Rounds 86\]. 
Example: Consider the feature algebra de- 
picted in Fig. 2, which consists of the elements 
{1, 2, 3, 4, 5, a, b) where a and b shall be (the pic- 
tures of) atoms and f, g, i and j shall be features 
whose interpretations are as indicated. 
i i  
simulation A 
f g 1A3 
2A4 
2A5 
aAa 
bAb 
a a b 
Figure 2: Example of Weak Subsumption 
Now, element 1 does not strongly subsume 3, 
since for 3 it does not hold, that its f-value 
equals its g-value. However, the simulation A 
demonstrates that they stand in the weak sub- 
sumption relation: 1 ~ 3. 
3 Constraints 
To describe feature algebras we use a relational 
language similar to the language of feature de- 
scriptions in LFG or path equations in PATR- 
II. Our syntax of constraints shall allow for the 
forms 
zp "---- ~q, zp "---- a, zp ~ ~q 
where p and q are paths (possibly empty), a E 
A, and z and ~/are variables. A feature clause 
is a finite set of constraints of the above forms. 
As usual we interpret constraints with respect 
to a variable assignment, in order to make sure 
that variables are interpreted uniformly in the 
whole set. An assignment is a mapping ~ of 
variables to the elements of some feature alge- 
bra. A constraint ~ is satisfied in .,4 under as- 
signment a, written (A, a) ~ ~, as follows: 
(.,4, a) ~ zp - vq iff a(z)p A = a(v)q A 
(.4, a) ~ zp -- a aft a(z)p A 
if  (v)qA. 
The solutions of a clause C in a feature alge- 
bra .4 are those assignments which satisfy each 
constraint in C. Two clauses C1 and C2 are 
equivalent iff they have the same set of solu- 
tions in every feature algebra .A. 
The problem we want to consider is the follow- 
ing: 
Given a clause C with symbols from 
V, L and A, does C have a solution in 
some feature algebra? 
We call this problem the weak semiunification 
problem in feature algebras) 
4 An Algorithm 
4.1 Presolved Form 
We give a solution algorithm for feature clauses 
based on normalization, i.e. the goal is to de- 
fine a normal form which exhibits unsatisfiabil- 
ity and rewrite rules which transform each fea- 
ture clause into normal form. The normal form 
we present here actually is only half the way to 
a solution, but we show below that with the use 
of a standard algorithm solutions can be gener- 
ated from it. 
First we introduce the restricted syntax of the 
normal form. Clauses containing only con- 
straints of the following forms are called sim- 
ple: 
zf --y, z--s, z ~ y 
where s is either a variable or an atom. Each 
feature clause can be restated in linear time as 
an equisatisfiable simple feature clause whose 
solutions are extensions of the solutions of the 
original clause, through the introduction of aux- 
iliary variables. This step is trivial. 
A feature clause C is called presolved iff it is 
simple and satisfies the following conditions. 
~The anMogous problem for (strong) subsumption 
constraints is undecidable, even if we restrict ourselves 
to finite feature algebras. Actually, this problem could 
be shown to be equivalent to the semiunification prob- lem for rational trees, i.e. first-order terms which may 
contain cycles. The interested reader is referred to 
\[D~rre/Rounds 90\]. 
C1. If z - ~/is in C, then z occurs exactly once 
in C. 
C2. Ifzf-yandzf-zareinC, theny=z. 
C3. Ifz~vandy~zareinC, thenz~zis 
in C (transitive closure). 
C4. Ifz ~V and z f-- z t and Vf -- V t are in 
C, then z' ~ V' is in C (downward propa- 
gation closure). 
In the first step our algorithm attempts to trans- 
form feature clauses to presolved form, thereby 
solving the equational part. In the simplifica- 
tion rules (cf. Fig. 3) we have adapted some 
of Smolka's rules for feature clauses including 
complements \[Smolka 89\]. In the rules \[z/s\]C 
denotes the clause C where every occurrence of 
z has been replaced with s, and ~ & C denotes 
the feature clause {~} U C provided ~b ~ C. 
Theorem 1 Let C be a simple feature clause. 
Then 
I. if C can be rewritten to 19 using one of 
the rules, then 1) i8 a simple feature clause 
equivalent to C, 
f. for every non-normal simple feature clause 
one of the rewrite rules applies, 
3. there is no infinite chain C --* U1 --* C2 --, 
ProoL 3 The first part can be verified straight- 
forwardly by inspecting the rules. The same 
holds for the second part. To show the termina- 
tion claim first observe that the application of 
the last two rules can safely be postponed until 
no one of the others can apply any more, since 
they only introduce subsumption constraints, 
which cannot feed the other rules. Now, call 
a variable z isolated in a clause C, if C contains 
an equation z - 7/and z occurs exactly once in 
C. The first rule strictly increases the number 
of isolated variables and no rule ever decreases 
it. Application of the second and third rule de- 
crease the number of equational constraints or 
the number of features appearing in C, which 
no other rule increase. Finally, the last two 
rules strictly increase the number of subsump- 
tion constraints for a constant set of variables. 
Hence, no infinite chain of rewriting steps may 
be produced. \[\] 
We will show now, that the presolved form can 
be seen as a nondeterministic finite automaton 
~Part of this proof has been directly adapted from 
\[S molka 89\]. 
259 
z-y&C 
z-z&C 
zf -1/ gr zf - z & C 
zg~ztzC 
--4 z--l/ & \[z/1/\]C, if z occurs in C and z~l/ 
--, C 
--+ z~y&zf "--z'gryf "--yt&zt~y'&C 
if z t ~ ~ ~ C 
(1) 
(2) 
(3) 
(4) 
Ca) 
Figure 3: Rewriting to presolved form 
with e-moves and that we can read off solutions 
from its deterministic equivalent, if that is of 
a special, trivially verifiable, form, called clash- 
bee. 
4.2 The Transition Relation 6c of a 
Presolved Clause C 
The intuition behind this construction is, that 
subsumption constraints basically enfoice that 
information about one variable (and the space 
teachable hom it) has to be inherited by (copied 
to) another variable. For example the con- 
straints z H y and zp - a entail that also 
lip - a has to hold. 4 Now, if we have a con- 
straint z ~ T/, we could think of actually copying 
the information found under z to y, e.g. zf - z ~ 
would be copied to 1/f - 1/t, where 1/I is a new 
variable, and z I would be linked to yl by z p ~ ?/. 
However, this treatment is hard to control in the 
presence of cycles, which always can occur. In- 
stead of actually copying we also can regard a 
constraint z g 7/as a pointer ¢rom ~ back to z 
leading us to the information which is needed to 
construct the local solution of ~. To extend this 
view we regard the whole p~esolved chase C as 
a finite automaton: take variables and atoms 
as nodes, a feature constraint as an arc labeled 
with the feature, constraints z - s and 1/~ z 
as e-moves horn z to s or ~/. We can show then 
that C is unsatisfiable iff there is some z hom 
which we reach atom a via path p such that we 
can also reach b(~ a) via p or there is a path 
starting from z whose proper prefix is p. 
Formally, let NFA Arc of presolved clause C be 
~F~rora this point of view the difference between weak 
and strong subsumption can be captured in the type 
of information they enforce to be inherited. Strong 
subsumption requires path equivalences to be inherited 
(x ~ y and ~p -" zq implies yp - yq), whereas weak 
subsumption does not. 
260 
defined as follows. Its states are the variables 
occurring in C (Vc) plus the atoms plus the 
states qF and the initial state q0. The set of 
final states is Vc U {qp}. The alphabet of Arc is 
vcu z, u A u {e}. 5 
The transition relation is defined as follows: s 
6c := vc} 
o {(a,a,q~)la~ A} 
u I • g c} 
u f, I -" c} 
v • c} 
As usual, let ~c be the extension of 6c to paths. 
Notice that zpa E L(Afc) iff (z,p,a) E ~c. 
The language accepted by this automaton con- 
tains strings of the forms zp or zpa, where a 
string zp indicates that in a solution a the ob- 
ject ol(z)p ~t should be defined and zpa tells us 
further that this object should be a A. 
A set of strings of (V x L*) U (V x L* x A) is 
called clash-free iff it does not contain a string 
zpa together with zpb (where a ~ b) or together 
with zpf. It is clear that the property of a reg- 
ular language L of being dash-free with respect 
to L and A can be read off immediately from 
a DFA D for it: if D contains a state q with 
5(q, a) E F and either 6(q, b) E F (where a ~ b) 
or 6(q, f) E F, then it is not clash-free, other- 
wise it is. 
We now present our centrM theorem. 
Theorem 2 Let Co be a feature clause, C its 
presolved form and Arc the NFA as constructed 
sir L or A are infinite we restrict ourselves to the sets 
of symbols actually occurring in C. 
6Notice that if x - s E C, then either s is an atom or 
occurs only once. Thus it is pointless to have an arc 
fr,)m s to ~, since we either have already the maximum 
of information for s or ~ will not provide any new arcs. 
above. Then the following conditions are equiv- 
alent: 
i. L(Are) is cZash- ,ee 
YL There exists a finite feature algebra .A and 
an assignment c~ such that (.A,c~) ~ Co, 
provided the set of atoms is finite. 
3. There exists a feature algebra .4 and an as- 
8ignraent ol such that (.A, c~) ~ Co. 
Proof. see Appendix A. 
Now the algorithm consists of the following sim- 
ple or well-understood steps: 
1: (a) Solve the equationai constraints of C, 
which can be done using standard uni- 
fication methods, exemplified by rules 
1) to 3). 
(b) Make the set of weak subsumption 
constraints transitively and "down- 
ward" closed (rules 4) and 5)). 
2: The result interpreted as an NFA is made 
deterministic using standard methods and 
tested of being clash-free. 
4.3 Determining Clash-Freeness Di- 
rectly 
For the purpose of proving the algorithm cor- 
rect it was easiest to assume that clash-freeness 
is determined after transforming the NFA of the 
presolved form into a deterministic automaton. 
However, this translation step has a time com- 
plexity which is exponential with the number 
of states in the worst case. In this section\[A we 
consider a technique to determine clash-freeness 
directly from the NFA representation of the pre- 
solved form in polynomial time. We do not go 
into implementational details, though. Instead 
we are concerned to describe the different steps 
more from a logical point of view. It can be 
assumed that there is still room left for opti- 
mizations which improve ef\[iciency. 
In a first step we eliminate all the e-transitions 
from the NFA Arc- We will call the result 
still Arc. For every pair of a variable node 
z and an atom node a let Arc\[z,a\] be the 
(sub-)automaton of all states of Arc reachable 
horn z, but with the atom a being the only final 
state. Thus, Afc\[z,g\] accepts exactly the lan- 
guage of all strings p for which zpg E L(Arc). 
Likewise, let Afc\[z,~\] be the (sub-)automaton 
of all states olaf C reachable from z, but where 
every atom node besides a is in the set of fi- 
nal states as well as every node with an outgo- 
ing feature arc. The set accepted by this ma- 
chine contains every string p such that zpb E 
L(ArC), (b ~ a) or zpf E L(Arc). If and only if 
the intersection of these two machines is empty 
for every z and a, L(Arc) is clash-free. 
4.4 Complexity 
Let us now examine the complexity of the dif- 
ferent steps of the algorithm. 
We know that Part la) can be done (using 
the efficient union/find technique to maintain 
equivalence classes of variables and vectors of 
features for each representative) in nearly lin- 
ear time, the result being smaller or of equal 
size than Co. Part lb) may blow up the clause 
to a size at most quadratic with the number 
of different variables n, since we cannot have 
more subsumption constraints than this. For 
every new subsumption constraint, trying to ap- 
ply ruh 4) might involve at most 2n membership 
test to check whether we are actually adding a 
new constraint, whereas for rule 5) this number 
only depends on the size of L. Hence, we stay 
within cubic time until here. 
Determining whether the presolved form is 
dash-free from the NPA representation is done 
in three steps. The e-free representation of Arc 
does not increase the number of states. If n,a 
and l are the numbers of variables, atoms and 
features resp. in the initial clause, then the 
number of edges is in any case smaller than 
(n + a) ~ • l, since there are only n + a states. 
This computation can be performed in time of 
an order less than o((~z + a)3). 
Second, we have to build the intersections for 
Arc\[z,a\] and Arc\[z,g\] for every z and a. Inter- 
section of two NFAs is done by building a cross- 
product machine, requiring maximally o((~z + 
a) 4 • l) time and space. ¢ The test for emptiness 
of these intersection machines is again trivial 
and can be performed in constant time. 
Hence, we estimate a total time and space com- 
plexity of order n- a. (Tz + a) 4 • I. 
7This is an estimate for the number of edges, since the 
nmuber of states is below (n + a) 2. As usual, we assume 
appropriate data structures where we can neglect the 
order of access times. Probably the space (and time) 
complexity can be reduced hrther, since we actually do 
not need the representations of the intersection machines 
besides for testing, whether they can accept anything. 
261 
5 Conclusion 
We proposed an extension to the basic feature 
logic of variables, features, atoms, and equa- 
tional constraints. This extension provides a 
means for one-way information passing. We 
have given a simple, but nevertheless completely 
formal semantics for the logic and have shown 
that the satisfiability (or unification) problem 
in the logic involving weak subsumption con- 
straints is decidable in polynomial time. Fur- 
thermote, the first part of the algorithm is a sur- 
prisingly simple extension of a standard unifica- 
tion algorithm for feature logic. We have formu- 
lated the second part of the problem as a simple 
property of the regular language which the out- 
come of the first part defines. Hence, we could 
make use of standard techniques from automata 
theory to solve this part of the problem. The 
algorithm has been proved to be correct, com- 
plete, and guaranteed to terminate. There are 
no problems with cycles or with infinite chains 
of subsumption relations as generated by a con- 
straint like z ~ zf. s 
The basic algorithmic requirements to solve the 
problem being understood, the challenge now is 
to find ways how solutions can be found in a 
more incremental way, if we already have solu- 
tions for subsets of a clause. To achieve this we 
plan to amalgamate more closely the two parts 
algorithms, for instance, through implementing 
the check for clash-freeness also with the help 
of (a new form of) constraints. It would be in- 
teresting also from a theoretical point of view 
to find out how much of the complexity of the 
second part is really necessary. 
Acknowledgment 
I am indebted to Bill Romtds for reading a first draft 
of this paper and pointing out to me a way to test 
dash-freeness in polynomial time. Of course, any 
remaining errors are those of the author. I would 
also llke to thank Gert Smolka for giving valuable 
comments on the first draft. 

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