An Attributive Logic of Set Descriptions 
Set Operations 
Suresh Manandhar 
HCRC Language Technology Group 
The University of Edinburgh 
2 Buccleuch Place 
Edinburgh EH8 9LW, UK 
Internet: Suresh. Manandhar@ed. ac. uk 
and 
Abstract 
This paper provides a model theoretic semantics to fea- 
ture terms augmented with set descriptions. We pro- 
vide constraints to specify HPSG style set descriptions, 
fixed cardinality set descriptions, set-membership con- 
straints, restricted universal role quantifications, set 
union, intersection, subset and disjointness. A sound, 
complete and terminating consistency checking proce- 
dure is provided to determine the consistency of any 
given term in the logic. It is shown that determining 
consistency of terms is a NP-complete problem. 
Subject Areas: feature logic, constraint-based gram- 
mars, HPSG 
1 Introduction 
Grammatical formalisms such as HPSG 
\[Pollard and Sag, 1987\] \[Pollard and Sag, 1992\] and 
LFG \[Kaplan and Bresnan, 1982\] employ feature de- 
scriptions \[Kasper and Rounds, 1986\] \[Smolka, 1992\] 
as the primary means for stating linguistic theories. 
However the descriptive machinery employed by these 
formalisms easily exceed the descriptive machinery 
available in feature logic \[Smolka, 1992\]. Furthermore 
the descriptive machinery employed by both HPSG 
and LFG is difficult (if not impossible) to state in fea- 
ture based formalisms such as ALE \[Carpenter, 1993\], 
TFS \[Zajac, 1992\] and CUF \[D6rre and Dorna, 1993\] 
which augment feature logic with a type system. 
One such expressive device employed both within 
LFG \[Kaplan and Bresnan, 1982\] and HPSG but is 
unavailable in feature logic is that of set descriptions. 
Although various researchers have studied set de- 
scriptions (with different semantics) \[Rounds, 1988\] 
\[Pollard and Moshier, 1990\] two issues remain unad- 
dressed. Firstly there has not been any work on consi- 
stency checking techniques for feature terms augmen- 
ted with set descriptions. Secondly, for applications 
within grammatical theories such as the HPSG forma- 
lism, set descriptions alone are not enough since de- 
scriptions involving set union are also needed. Thus 
to adequately address the knowledge representation 
needs of current linguistic theories one needs to provide 
set descriptions as well as mechanisms to manipulate 
these. 
In the HPSG grammar forma- 
lism \[Pollard and Sag, 1987\], set descriptions are em- 
ployed for the modelling of so called semantic indices 
(\[Pollard and Sag, 1987\] pp. 104). The attribute INDS 
in the example in (1) is a multi-valued attribute whose 
value models a set consisting of (at most) 2 objects. 
However multi-valued attributes cannot be descri- 
bed within feature logic \[Kasper and Rounds, 1986\] 
\[Smolka, 1992\]. 
(1) 
Io DREL --4 °~TIs~R\[\] / 
Ls'~E~ w J 
\[NDS IRESTINAME ~andy \]\['IRESTINAME kim II ¢ 
L L N*M" D JIL L JJJ 
A further complication arises since to be able to deal 
with anaphoric dependencies we think that set mem- 
berships will be needed to resolve pronoun dependen- 
cies. Equally, set unions may be called for to incremen- 
tally construct discourse referents. Thus set-valued 
extension to feature logic is insufficient on its own. 
Similarly, set valued subcategorisation frames (see (2)) 
has been considered as a possibility within the HPSG 
formalism. (2) 
believes= IYNILOCISUBCAT~ \[\[SYN~LOOIHEADICAT v\] 
But once set valued subeategorisation frames are em- 
ployed, a set valued analog of the HPSG subcategorisa- 
tion principle too is needed. In section 2 we show that 
the set valued analog of the subcategorisation principle 
can be adequately described by employing a disjoint 
union operation over set descriptions as available wit- 
hin the logic described in this paper. 
2 The logic of Set descriptions 
In this section we provide the semantics of feature 
terms augmented with set descriptions and various 
constraints over set descriptions. We assume an al- 
phabet consisting of x, y, z,... 6 )2 the set of variables; 
f,g,... E Y: the set of relation symbols; el, c2,... E C 
the set of constant symbols; A,B,C,... 6 7 ) the set 
of primitive concept symbols and a,b,... 6 .At the 
set of atomic symbols. Furthermore, we require that 
/,T E T'. 
255 
The syntax of our term language defined by the follo- 
wing BNF definition: 
P > x I a t c I C \[ -~x I -~a \[ -~c \[ -~C 
S,T- > 
P 
f : T feature term 
Sf : T existential role quantification 
Vf : P universal role quantification 
f: {T1,...,Tn} set description 
f {T1,.., Tn}= fixed cardinality set description 
f : g(x) U h(y) union 
f: g(x) rq h(y) intersection 
f :~ g(x) subset 
f(x) # g(y) disjointness 
S Iq T conjunction 
where S, T, T1,..., Tn are terms; a is an atom; c is a 
constant; C is a primitive concept and f is a relation 
symbol. 
The interpretation of relation symbols and atoms is 
provided by an interpretation Z =</4I I > where/41 
is an arbitrary non-empty set and I is an interpretation 
function that maps : 
1. every relation symbol f • ~" to a binary relation 
fl C_/4I x/4I 
2. every atom a • .At to an element a I • bl x 
Notation: 
• Let if(e) denote the set {e'\[ (e,e') • if} 
• Let fI(e) T mean fl(e) = 0 
Z is required to satisfy the following properties : 
1. if al ~ a2 then all # hi2 (distinctness) 
2. for any atom a • At and for any relation f • ~" there 
exists no e • U 1 such that (a, e) • fl (atomicity) 
For a given interpretation Z an Z-assignment a is a 
function that maps : 
1. every variable x • \]2 to an element a(x) • 141 
2. every constant c • C to an element a(c) •/41 such 
that for distinct constants Cl, c2 : a(cl) # a(c2) 
3. every primitive concept C • 7 ) to a subset a(C) C 
/41 such that: 
• ~(_L) = 0 
• a(T) =/41 
The interpretation of terms is provided by a denotation 
function \[\[.\]\]z,a that given an interpretation Z and an 
Z-assignment a maps terms to subsets of/41. 
The function \[.\]\]z,a is defined as follows : 
~x~z," = {,~(x)} 
\[\[a\]\]Z, ~ = {a I} 
\[cK'" = {a(e)} 
Iv\] z,~ = ~(c) 
If: T\] z'" = {e •/411 he' •/4i: fZ(e ) 
= {e'} A e' • ~T\] z'e} 
\[3f : T~ :r'a = 
{e•/4 llqe'•/4(l:(e,e') •f! A e' • IT\] z'"} 
IV f: T\]\] z'~ = 
{e • W' lye' •/41: (e, e') • f1 =~ e' • IfT\] z'"} 
U: {T,,...,T~}K," = 
{e E U I \[ 9el,...,ge~ e U I : 
f1(e) = {el,...,e,}^ 
el e IT1\] z'a A... A e,~ • \[T,~\] z'~} 
If: {T1,..., Tn}=\] z'a = 
{e •/4I I 9el,...,ge~ •/4I : 
Ifl(e) l = n A fI(e) = {el,... ,en}A 
el • \[Tx\]Z'a A... A e~ • \[T,\] z'"} 
If: g(x) U h(y)\]\] z'a = 
{e • LI I I fl(e) = gl(a(x)) U hI(a(y))} 
If: g(x) N h(y)\] z'a = 
{e •/41 \[ fi (e ) = gi (c~(x) ) rq hl (c~(y) ) } 
If :~_ g(x)lz, ~' = 
{e • u ~ I f(e) ~ g1(~(x))} 
if(x ) # g(y)\]\]z,c~ = 
• 0 if fl(a(x)) n gl(a(y)) # O 
• U I if f1(a(x)) A g1(a(y)) = 0 
IS rl T\]\] z,a = \[\[S\]\] z,a N \[T\]\] z,a 
\[-~T~ ~," = U' - \[T~ z," 
The above definitions fix the syntax and semantics of 
every term. 
It follows from the above definitions that: 
I:T - /:{T} -I:{T}= 
Figure 1 
Although disjoint union is not a primitive in the logic 
it can easily be defined by employing set disjointness 
and set union operations: 
f: g(x) eJ h(y) =de/ g(x) # h(y) ~q f: g(x) U h(y) 
Thus disjoint set union is exactly like set union except 
that it additionally requires the sets denoted by g(x) 
and h(y) to be disjoint. 
The set-valued description of the subcategorisation 
principle can now be stated as given in example (3). 
(3) Subcategorisation Principle 
SYN,LOC Y \]\] 
TRS X n \[HL-DTR\[SYN\[LOC\[SUBCAT c-dtrs(X) ~ subcat(Y) 
The description in (3) simply states that the subcat 
value of the H-DTR is the disjoint union of the subcat 
value of the mother and the values of C-DTRS. Note 
that the disjoint union operation is the right operation 
to be specified to split the set into two disjoint subsets. 
Employing just union operation would not work since 
256 
Decomposition rules 
x=F:TAC~ (DFeat) x=F:yAy=TACs 
if y is new and T is not a variable and F ranges over Sf, f 
x = Vf : ~ A C~ (DForall) x=Vf:yAy=~ACs 
if y is new and ~ ranges over a, c. 
(DSet) x = f: {Ti,...,T~} A C~ 
x = I: {xl,...,x~}^xl =T1 ^...ix~ =T~ACs 
if xi,..., xn are new and at least one of Ti : 1 < i < n is not a variable 
x= f : {Ti,...,T,}=A Cs 
(DSetF) x = f : {Xl,... , xn} A X = f: {Xl,... , Xn}= A X 1 = T 1 ^... i x n = T n i C s 
if xi,..., x~ are new and at least one of Ti : 1 < i < n is not a variable 
x=SNTAC,~ (DConj) x = S i x = T A gs 
Figure 2: Decomposition rules 
it would permit repetition between members of the 
SUBCAT attribute and C-DTRS attribute. 
Alternatively, we can assume that N is the only multi- 
valued relation symbol while both SUBCAT and C-DTRS 
are single-valued and then employ the intuitively ap- 
pealing subcategorisation principle given in (4). 
(4) Subcategorisation Principle 
TRS \[H-DTRISYNILOCISUBCATIN N(X) ~ N(Y) C-DTRS X 
With the availability of set operations, multi-valued 
structures can be incrementally built. For instance, by 
employing union operations, semantic indices can be 
incrementally constructed and by employing members- 
hip constraints on the set of semantic indices pronoun 
resolution may be carried out. 
The set difference operation f : g(y) - h(z) is not avai- 
lable from the constructs described so far. However, 
assume that we are given the term x R f : g(y) - h(z) 
and it is known that hZ(~(z)) C_ gZ(a(y)) for every in- 
terpretation 27, (~ such that \[x R f : g(y)- h(z)~ z,~ ¢ 0. 
Then the term x N f : g(y) - h(z) (assuming the ob- 
vious interpretation for the set difference operation) is 
consistent iff the term y \[\] g : f(x) t~ h(z) is consistent. 
This is so since for setsG, F,H:G-F=HAFCG 
i\]:f G = F W H. See figure 1 for verification. 
3 Consistency checking 
To employ a term language for knowledge representa- 
tion tasks or in constraint programming languages the 
minimal operation that needs to be supported is that 
of consistency checking of terms. 
A term T is consistent if there exists an interpreta- 
tion 2: and an/:-assignment (~ such that \[T\] z'a ~ 0. 
In order to develop constraint solving algorithms for 
consistency testing of terms we follow the approaches 
in \[Smolka, 1992\] \[Hollunder and Nutt, 1990\]. 
A containment constraint is a constraint of the 
form x = T where x is a variable and T is an term. 
Constraint simplification rules - I 
x=yACs (SEquals) x = y A \[x/y\]Cs 
if x ~ y and x occurs in Cs 
(SConst) x=~Ay=~ACs x=yAx=~ACs 
where ~ ranges over a, c. 
(SFeat) x= f :yAx= F :zZACs x=/:yAy= ACs 
where F ranges over f, 3f, Vf 
(SExists) x=gf:yAx=Vf:zAC~ x= f :yAy=zACs 
(SForallE) x = V__\] : C A x = 9f : y A C~_ 
x =V/: CAx = 3/: yAy = CAC~ 
if C ranges over C, -~C, -~a, --c, -~z and 
Cs Vy =C. 
Figure 3: Constraint simplification rules - I 
In addition, for the purposes of consistency checking 
we need to introduce disjunctive constraints which 
are of the form x -- Xl U ... U x,~. 
We say that an interpretation Z and an/-assignment 
a satisfies a constraint K written 27, a ~ K if. 
• Z,a~x=Tv=~a(x) E\[T~ z'a 
• Z,a~x=xlU...Uxn.: ~.a(x)=a(xi)forsome 
xi:l <i<n. 
A constraint system Cs is a conjunction of con- 
straints. 
We say that an interpretation 27 and an Z-assignment 
a satisfy a constraint system Cs iffZ, a satisfies every 
constraint in Cs. 
The following lemma demonstrates the usefulness of 
constraint systems for the purposes of consistency 
checking. 
Lemma 1 An term T is consistent iff there exists a 
variable x, an interpretation Z and an Z-assignment a 
such that Z, a satisfies the constraint system x = T. 
Now we are ready to turn our attention to constraint 
solving rules that will allow us to determine the con- 
sistency of a given constraint system. 
257 
Constraint simplification rules - II 
(SSetF) x=F:yAx=f:{Xl,...,xn}AC8 
x= f :yAy=xlA...Ay=xnACs 
where F ranges over f, Vf 
(SSet) x = f: {y} A C8 
x= f :yAC8 
(SDup) x=f:{Xl,...,xi,...,xj,...,x,~}AC8 
x = f : {Zl,...,x,...,...,x,} ^ C8 
if xi -- x i 
(SForaU) x = Vf : CA x = f : {xl,...,xn} A C8 
x =f: =-C^C8 
if C ranges over C, -~C, -~a, -~c, -~z and 
there exists xi : 1 < i < n such that Cs ~1 xi = C. 
x = Bf : yAx = f : {Xl,...,x,~} A C8 
(SSetE) x=f:{Xl,...,x,~}Ay=xlU...UxnAC8 
(SSetSet) X=f:{Xl,...,Xn}AX=f:{yl,...,ym}AC8 
x = I: 
Xl = Yl II... II Ym ^ • .. ^ Xn = Yl II... II ymA 
Yl ---- xz \[J .. • II xn A... A Ym = Xl II... II xn A 68 
where n _< m 
x= x I II...Uxn ACs 
(SDis) x = Xl M... IJ x~ A x = xi A C8 
ifl <i<nand 
there is no x j, 1 < j < n such that C8 F x = x: 
Figure 4: Constraint 
We say that a constraint system C8 is basic if none of 
the decomposition rules (see figure 2) are applicable to c8. 
The purpose of the decomposition rules is to break 
down a complex constraint into possibly a number of 
simpler constraints upon which the constraint simpli- 
fication rules (see figures 3, 4 and 5 ) can apply by 
possibly introducing new variables. 
The first phase of consistency checking of a term T 
consists of exhaustively applying the decomposition 
rules to an initial constraint of the form x = T (where 
x does not occur in T) until no rules are applicable. 
This transforms any given constraint system into basic 
form. 
The constraint simplification rules (see figures 3, 4 and 
5 ) either eliminate variable equalities of the form x = 
y or generate them from existing constraints. However, 
they do not introduce new variables. 
The constraint simplification rules given in figure 3 are 
the analog of the feature simplification rules provided 
in \[Smolka, 1991\]. The main difference being that our 
simplification rules have been modified to deal with 
relation symbols as opposed to just feature symbols. 
The constraint simplification rules given in figure 4 
simplify constraints involving set descriptions when 
they interact with other constraints such as feature 
constraints - rule (SSetF), singleton sets - rule (SSet), 
duplicate elements in a set - rule (SDup), universally 
quantified constraint - rule (SForall), another set de- 
scription - rule (SSetSet). Rule (SDis) on the other 
hand simplifies disjunctive constraints. Amongst all 
simplification rules - II 
the constraint simplification rules in figures 3 and 4 
only rule (SDis) is non-deterministic and creates a n- 
ary choice point. 
Rules (SSet) and (SDup) are redundant as comple- 
teness (see section below) is not affected by these rules. 
However these rules result in a simpler normal form. 
The following syntactic notion of entailment is em- 
ployed to render a slightly compact presentation of the 
constraint solving rules for dealing with set operations 
given in figure 5. 
A constraint system Cs syntactically entails the (con- 
junction of) constraint(s) ¢ if Cs F ¢ is derivable from 
the following deduction rules: 
1. ¢AC8 F¢ 
2. C~Fx=x 
3. CsFx=y >CsFy=x 
4. CsFx=yACsFy=z >CsFx=z 
5. Cs F x = -~y > C~ F y = -~x 
6. CsFx=f:y >CsFx=3f:y 
7. CsFx=f:y >CsFx=Vf:y 
8. CsFx=I:{...,xi,...} >C~Fz=3I:zi 
Note that the above definitions are an incomplete list 
of deduction rules. However C~ I- ¢ implies C~ ~ ¢ 
where ~ is the semantic entailment relation defined as 
for predicate logic. 
We write C8 t/¢ if it is not the case that C~ I- ¢. 
The constraint simplification rules given in figure 5 
deal with constraints involving set operations. Rule 
(C_) propagates g-values of y into I-values of x in 
the presence of the constraint x = f :_D g(y). Rule 
258 
Extended 
(c_) x = 
if: 
(ULeft) x= 
if Cs 
Constraint simplification rules 
x = f :D g(y) A C~ 
f :D g(y) A z = 3f : Yi A Cs 
F/x = 3f : yi and 
F y = 3g : yi 
x = I: g(y) u h(z) A 
f: g(y) W h(z) A x = f :D g(y) A Cs 
~/ x = f :D g(y) 
(URight) x = f: g(y) U h(z) A Cs 
x = f: g(y) U h(z) A x = f :D h(z) A Cs 
if Cs V z = f :__D h(z) 
(UDown) 
x = f: g(y) U h(z) A Cs 
x = f : g(y) U h(z) A y = 3g : xi I z = 3h : xi A Cs 
if: 
• C~/y=3g:xiand 
• Cst/z=3h:xiand 
• C~l-x=3f:xi 
( nDown ) 
= f: g(y) n h(z) A 
x = f : g(y) n h(z) A y = 3g : xi A z = 3h : xi A C 
if: 
• (Cs\[/y=3g:xiorCsVz=3h:xi) and 
• C~Fx=3f:x~ 
x = f: g(y) n h(z) A Cs 
(nUp) x = f : g(y) n h(z) A x = 3f : xi A Cs 
if: 
• Cs ~x=3f:xi and 
• CsFy=3g:xiand 
• C~Fz=3h:xi 
Figure 5: Constraint solving with set operations 
(ULeft) (correspondingly Rule (URight)) adds the 
constraint x = f :_D g(y) (correspondingly x = f :D 
h(z)) in the presence of the constraint x = f : g(y) U 
h(z). Also in the presence of x = f : g(y) U h(z) rule 
(UDown) non-deterministically propagates an I-value 
of x to either an g-value of y or an h-value of z (if 
neither already holds). The notation y = 3g : xi \] z = 
3h : xi denotes a non-deterministic choice between 
y = 3g : x~ and z = 3h : xi. Rule (nDown) propaga- 
tes an f-value of x both as a g-value of y and h-value of 
z in the presence of the constraint x = f : g(y) n h(z). 
Finally, rule (nUp) propagates a common g-value of y 
and h-value of z as an f-value of x in the presence of 
the constraint x = f : g(y) n h(z). 
4 Invariance, Completeness and 
Termination 
In this section we establish the main results of this 
paper - namely that our consistency checking proce- 
dure for set descriptions and set operations is invari- 
ant, complete and terminating. In other words, we 
have a decision procedure for determining the consi- 
stency of terms in our extended feature logic. 
For the purpose of showing invariance of our ru- 
les we distinguish between deterministic and non- 
deterministic rules. Amongst all our rules only rule 
(SDis) given in figure 4 and rule (UDown) are non- 
deterministic while all the other rules are determini- 
stic. 
Theorem 2 (Invariance) 1. If a decomposition rule 
transforms Cs to C~s then Cs is consistent iff C~ is 
consistent. 
2. Let Z,a be any interpretation, assignment pair and 
let Cs be any constraint system. 
• If a deterministic simplification rule transforms 
Cs to C' s then: 
iff p c" 
• If a non-deterministic simplification rule applies 
to Cs then there is at least one non-deterministic 
choice which transforms Cs to C' s such that: 
z,a p iffz, apc; 
A constraint system Cs is in normal form if no rules 
are applicable to Cs. 
Let succ(x, f) denote the set: 
succ(x, f) = {y I c8 x = 3f : y} 
A constraint system Cs in normal form contains a 
clash if there exists a variable x in C8 such that any 
of the following conditions are satisfied : 
1. C~Fx=al andC~Fx=a2suchthatal ~a2 
2. Cs F x = cl and Cs F x = c2 such thatcl ~c2 
3. Cs F x = S and Cs F x = -,S 
where S ranges over x, a, c, C. 
4. CsFx=3f:yandCsFx=a 
5. C~ F f(x) ¢ g(y) and succ(x, f) n succ(y, g) 7~ 
6. C~ F x = f: {xz,...,xn}= and Isucc(x,f) I < n 
If Cs does not contain a clash then C~ is called clash- 
free. 
The constraint solving process can terminate as soon 
as a clash-free constraint system in normal form is fo- 
und or alternatively all the choice points are exhau- 
sted. 
The purpose of the clash definition is highlighted in 
the completeness theorem given below. 
For a constraint system Cs in normal form an equiva- 
lence relation ~_ on variables occurring in Cs is defined 
as follows: 
x-~ y ifC~ F x = y 
For a variable x we represent its equivalence class by 
Theorem 3 (Completeness) A constraint system 
Cs in normal form is consistent iff Cs is clash-free. 
Proof Sketch: For the first part, let C~ be a constraint 
system containing a clash then it is clear from the de- 
finition of clash that there is no interpretation Z and 
Z-assignment a which satisfies Cs. 
Let C~ be a clash-free constraint system in normal 
form. 
We shall construct an interpretation 7~ =< L/R, .R > 
259 
and a variable assignment a such that T~, a ~ Cs. 
Let U R = V U ,4t UC. 
The assignment function a is defined as follows: 
1. For every variable x in )2 
(a) if C8 }- x = a then ~(x) = a 
(b) if the previous condition does not apply then 
~(x) = choose(Ix\]) where choose(\[x\]) denotes a 
unique representative (chosen arbitrarily) from 
the equivalence class \[x\]. 
2. For every constant c in C: 
(a) if Cs F x = c then a(c) = (~(x) 
(b) if c is a constant such that the previous condition 
does not apply then (~(c) -- c 
3. For every primitive concept C in P: 
= I C8 x = 
The interpretation function .n is defined as follows: 
• fR(x) = succ( , f) • aR=a 
It can be shown by a case by case analysis that for 
every constraint K in C~: 
7~,a~ K. 
Hence we have the theorem. 
Theorem 4 (Termination) 
The consistency checking procedure terminates in a fi- 
nite number of steps. 
Proof Sketch: Termination is obvious if we observe the 
following properties: 
1. Since decomposition rules breakdown terms into 
smaller ones these rules must terminate. 
2. None of the simplification rules introduce new va- 
riables and hence there is an upper bound on the 
number of variables. 
3. Every simplification rule does either of the following: 
(a) reduces the 'effective' number of variables. 
A variable x is considered to be ineffective if it 
occurs only once in Cs within the constraint x = 
y such that rule (SEquals) does not apply. A 
variable that is not ineffective is considered to be 
effective. 
(b) adds a constraint of the form x = C where C 
ranges over y, a, c, C, -~y, -~a, -~c, -~C which means 
there is an upper bound on the number of con- 
straints of the form x = C that the simplification 
rules can add. This is so since the number of va- 
riables, atoms, constants and primitive concepts 
are bounded for every constraint system in basic 
form. 
(c) increases the size of succ(x,f). But the size of 
succ(x, f) is bounded by the number of variables 
in Cs which remains constant during the applica- 
tion of the simplification rules. Hence our con- 
straint solving rules cannot indefinitely increase 
the size of succ(x, f). 
5 NP-completeness 
In this section, we show that consistency checking 
of terms within the logic described in this paper is 
NP-complete. This result holds even if the terms 
involving set operations are excluded. We prove 
this result by providing a polynomial time transla- 
tion of the well-known NP-complete problem of de- 
termining the satisfiability of propositional formulas 
\[Garey and Johnson, 1979\]. 
Theorem 5 (NP-Completeness) Determining 
consistency of terms is NP-Complete. 
Proof: Let ¢ be any given propositional formula for 
which consistency is to be determined. We split our 
translation into two intuitive parts : truth assignment 
denoted by A(¢) and evaluation denoted by r(¢). 
Let a, b,... be the set of propositional variables occur- 
ring in ¢. We translate every propositional variable a 
by a variable xa in our logic. Let f be some relation 
symbol. Let true, false be two atoms. 
Furthermore, let xl, x2,.., be a finite set of variables 
distinct from the ones introduced above. 
We define the translation function A(¢) by: 
A(¢) = f: {true, false}n 
3f :xa nSf :xbn...n 
3f : xl n 3f : x2 n ... 
The above description forces each of the variable 
Xa,Xb,... and each of the variables xl,x2,.., to be 
either equivalent to true or false. 
We define the evaluation function T(¢) by: 
= xo 
T(S&T) = T(S) n r(T) 
T(SVT) = xi n 3f : (\]: {~(S),r(T)} n 3f: xi) 
where xi 6 {xl,x2,...} is a new variable 
r(~S) = xi n 3f : (r(S) n ~z~) 
where xi 6 {xl,x2,...} is a new variable 
Intuitively speaking T can be understood as follows. 
Evaluation of a propositional variable is just its value; 
evaluating a conjunction amounts to evaluating each 
of the conjuncts; evaluating a disjunction amounts to 
evaluating either of the disjuncts and finally evaluating 
a negation involves choosing something other than the 
value of the term. 
Determining satisfiability of ¢ then amounts to deter- 
mining the consistency of the following term: 
3f : A(¢) n 3f: (true n r(¢)) 
Note that the term truenT(¢) forces the value of T(¢) 
to be true. This translation demonstrates that deter- 
mining consistency of terms is NP-hard. 
On the other hand, every deterministic completion of 
our constraint solving rules terminate in polynomial 
time since they do not generate new variables and the 
number of new constraints are polynomially bounded. 
This means determining consistency of terms is NP- 
easy. Hence, we conclude that determining consistency 
of terms is NP-complete. 
6 Translation to Sch6nfinkel-Bernays 
class 
The Schhnfinkel-Bernays class (see \[Lewis, 1980\]) con- 
sists of function-free first-order formulae which have 
260 
the form: 
3xt ... xnVyl • .. ym6 
In this section we show that the attributive logic 
developed in this paper can be encoded within the 
SchSnfinkel-Bernays subclass of first-order formulae by 
extending the approach developed in \[Johnson, 1991\]. 
However formulae such as V f : (3 f : (Vf : T)) which 
involve an embedded existential quantification cannot 
be translated into the SchSnfinkel-Bernays class. This 
means that an unrestricted variant of our logic which 
does not restrict the universal role quantification can- 
not be expressed within the SchSnfinkel-Bernays class. 
In order to put things more concretely, we provide 
a translation of every construct in our logic into the 
SchSnfinkel-Bernays class. 
Let T be any extended feature term. Let x be a va- 
riable free in T. Then T is consistent iff the formula 
(x = T) 6 is consistent where 6 is a translation function 
from our extended feature logic into the SchSnfinkel- 
Bernays class. Here we provide only the essential de- 
finitions of 6: 
• 
• =x#a 
• (x = f : T) ~ = 
f(x, y) & (y = T) ~ ~ Vy'(f(x, y') -+ y = y') 
where y is a new variable 
• (x=qf:T) ~=f(x,y) & (y=T) '~ 
where y is a new variable 
• (x = V f: a) ~ = Vy(f(x,y) --+ y = a) 
• (x = V f: ~a) ~ = Vy(f(x,y) .-+ y # a) 
• (x = f: {T1,...,Tn}) ~ -- 
f(X, Xl) & ... ~ f(X, Xn),~ 
Vy(f(x,y) --~ y = Xl V ... V y = xn)& 
(xl = T1) & ... & (zl = 
where Xl ,..., Xn are new variables 
• (x = f: g(y) U h(z)) ~ = Vxi(f(x, 
xi) -'+ g(y, xi) V h(z, xi)) ~: 
Vy,(g(y, Yi) -4 f(x, Yi)) & 
Vzi(h(z, zi) -+ f(x, zi)) 
• (x = f: (y) # g(z)) ~ = 
Vyizj(f(y, yi) & g(z, zi) --+ Yi # zi) 
• (x=SlqT) '~=(x=S) ~ & (x=T) ~ 
These translation rules essentially mimic the decom- 
position rules given in figure 2. 
Furthermore for every atom a and every feature f in 
T we need the following axiom: 
• Vax(-~f(a, x)) 
For every distinct atoms a, b in T we need the axiom: 
•a#b 
Taking into account the NP-completeness result 
established earlier this translation identifies a NP- 
complete subclass of formulae within the SchSnfinkel- 
Bernays class which is suited for NL applications. 
7 Related Work 
Feature logics and concept languages suchas 
KL-ONE are closely related family of languages 
\[Nebel and Smolka, 1991\]. The principal difference 
being that feature logics interpret attributive labels 
as functional binary relations while concept langua- 
ges interpret them as just binary relations. However 
the integration of concept languages with feature lo- 
gics has been problematic due to the fact the while 
path equations do not lead to increased computatio- 
nal complexity in feature logic the addition of role- 
value-maps (which are the relational analog of path 
equations) in concept languages causes undecidabi- 
lity \[Schmidt-Schant3, 1989\]. This blocks a straight- 
forward integration of a variable-free concept language 
such as ALC \[Schmidt-SchanB and Smolka, 1991\] with 
a variable-free feature logic \[Smolka, 1991\]. 
In \[Manandhax, 1993\] the addition of variables, fea- 
ture symbols and set descriptions to ALC is investi- 
gated providing an alternative method for integrating 
concept languages and feature logics. It is shown that 
set descriptions can be translated into the so called 
"number restrictions" available within concept langu- 
ages such as BACK \[yon Luck et al., 1987\]. However, 
the propositionally complete languages ALV and ALS 
investigated in \[Manandhar, 1993\] are PSPACE-hard 
languages which do not support set operations. 
The work described in this paper describes yet another 
unexplored dimension for concept languages - that of 
a restricted concept language with variables, feature 
symbols, set descriptions and set operations for which 
the consistency checking problem is within the com- 
plexity class NP. 
8 Summary and Conclusions 
In this paper we have provided an extended feature lo- 
gic (excluding disjunctions and negations) with a range 
of constraints involving set descriptions. These con- 
straints are set descriptions, fixed cardinality "set de- 
scriptions, set-membership constraints, restricted uni- 
versal role quantifications, set union, set intersection, 
subset and disjointness. We have given a model theo- 
retic semantics to our extended logic which shows that 
a simple and elegant formalisation of set descriptions 
is possible if we add relational attributes to our logic 
as opposed to just functional attributes available in 
feature logic. 
For realistic implementation of the logic described in 
this paper, further investigation is needed to develop 
concrete algorithms that are reasonably efficient in the 
average case. The consistency checking procedure de- 
scribed in this paper abstracts away from algorithmic 
considerations and clearly modest improvements to the 
basic algorithm suggested in this paper are feasible. 
However, a report on such improvements is beyond 
the scope of this paper. 
For applications within constraint based grammar 
formalisms such as HPSG, minimally a type sy- 
stem \[Carpenter, 1992\] and/or a Horn-like extension 
\[HShfeld and Smolka, 1988\] will be necessary. 
We believe that the logic described in this paper pro- 
vides both a better picture of the formal aspects of 
261 
current constraint based grammar formalisms which 
employ set descriptions and at the same time gives 
a basis for building knowledge representation tools in 
order to support grammar development within these 
formalisms. 
9 Acknowledgments 
The work described here has been carried out as part 
of the EC-funded project LRE-61-061 RGR (Reusa- 
bility of Grammatical Resources). A longer version 
of the paper is available in \[Erbach et al., 1993\]. The 
work described is a further development of the aut- 
hor's PhD thesis carried out at the Department of Ar- 
tificial Intelligence, University of Edinburgh. I thank 
my supervisors Chris Mellish and Alan Smaill for their 
guidance. I have also benefited from comments by an 
anonymous reviewer and discussions with Chris Brew, 
Bob Carpenter, Jochen DSrre and Herbert Ruessink. 
The Human Communication Research Centre (HCRC) 
is supported by the Economic and Social Research 
Council (UK). 

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