EXPRESSING DISJUNCTIVE AND NEGATIVE FEATURE CONSTRAINTS WITH 
CLASSICAL FIRST-ORDER LOGIC. 
Mark Johnson, 
Cognitive and Linguistic Sciences, Box 1978, 
Brown University, Providence, RI 02912. 
mj@cs.brown.edu 
ABSTRACT 
In contrast to the "designer logic" approach, this 
paper shows how the attribute-value feature 
structures of unification grammar and 
constraints on them can be axiomatized in 
classical first-order logic, which can express 
disjunctive and negative constraints. Because 
only quantifier-free formulae are used in the 
axiomatization, the satisfiability problem is NP- 
complete. 
INTRODUCTION. 
Many modern linguistic theories, such as 
Lexical-Functional Grammar \[1\], Functional 
Unification Grammar \[12\] Generalized Phrase- 
Structure Grammar \[6\], Categorial Unification 
Grammar \[20\] and Head-driven Phrase- 
Structure Grammar \[18\], replace the atomic 
categories of a context-free grammar with a 
"feature structure" that represents the.syntactic 
and semantic properties of the phrase. These 
feature structures are specified in terms of 
constraints that they must satisfy. Lexical 
entries constrain the feature structures that can 
be associated with terminal nodes of the 
syntactic tree, and phrase structure rules 
simultaneously constrain the feature structures 
that can be associated with a parents and its 
immediate descendants. The tree is well-formed 
if and only if all of these constraints are 
simultaneously satisfiable. Thus for the 
purposes of recognition a method for 
determining the satisfiability of such constraints 
is required: the nature of the satisfying feature 
structures is of secondary importance. 
Most work on unification-based 
grammar (including the references cited above) 
has adopted a type of feature structure called an 
attribute-value structure. The elements in an 
attribute-value structure come in two kinds: 
constant elements and complex elements. Constant 
elements are atomic entities with no internal 
structure: i.e. they have no attributes. Complex 
elements have zero or more attributes, whose 
173 
values may be any other element in the structure 
(including a complex element) and ally element 
can be the value of zero, one or several 
attributes. Attributes are partial: it need not be 
the case that every attribute is d ef!ned for every 
complex element. The set of attribute-value 
structures partially ordered by the subsumption 
relation (together with all additional entity T 
that every attribute-value structure subsumes) 
forms a lattice, and the join operation on this 
lattice is called the unification operati(m 119\]. 
Example: (from \[16\]). The attribute-value 
structure (1) has six complex elements labelled 
el ... e6 and two corastant elements, singular and 
third. The complex element el has two 
attributes, subj and pred, the value of which are 
the complex elements e 2 and e 3 respectively. 
(1) 
e2 
e2¢ r 
number 
singular 
el 
subj~pred 
"~e 3 
verb 
agr 
 'e6 
person 
third 
e 7 
(2) ~pred )ubi ""5 e,) 
verb e8 )el() 
agr.. 
ell 
The unification of elements el of(l) and e7 of(2) 
results in the attribute-value structure (3), the 
minimal structure in the subsumption lattice 
which subsumes both (1)and (2). 
¢1 ¢7 
(3) ~pred ~ 
ubj "~e3 e9 
e2 e8 verb 
...~e 5 el0 agr~agr 
number person 
singular third 
If constraints on attribute-value structures are 
restricted to conjunctions of equality constraints 
(i.e. requirements that the value of a path of 
attributes is equal to a constant or the value of 
another path) then the set of satisfying attribute- 
value structures is the principal filter generated 
by the minimal structure that satisfies the 
constraints. The generator of the satisfying 
principal filter of the conjunction of such 
constraints is the unification of the generators of 
the satisfying principal filters of each of the 
conjuncts. Thus the set of attribute-value 
structures that simultaneously satisfy a set of 
such constraints can be characterized by 
computing the unification of the generators of 
the corresponding principal filters, and the 
constraints are satisfiable iff the resulting 
generator is not "T (i.e. -T- represents unification 
failure). Standard t, nification-based parsers use 
unification in exactly this way. 
When disjunctions and negations of 
constraints are permitted, the set of satisfying 
attribute-value structures does not always form 
a principal filter \[11\], so the simple unification- 
based technique for determining the 
satisfiability of feature structure constraints 
must be extended. Kasper and Rounds \[11\] 
provide a formal framework for investigating 
such constraints by reviving a distinction 
originally made (as far as I am aware) by Kaplan 
and Bresnan \[10\] between the language in which 
feature structure constraints are expressed and 
the structures that satisfy these constraints. 
Unification is supplanted by conjunction of 
constraints, and disjunction and negation appear 
only in the constraint language, not in the 
feature structures themselves (an exception is \[3\] 
and \[2\], where feature bundles may contain 
negative arcs). 
Research in this genre usually follows a 
general pattern: an abstract model for feature 
structures and a specialized language for 
expressing constraints on such structures are 
"custom-crafted" to treat some problematic 
feature constraint (such as negative feature 
constraints). Table 1 sketches some of the 
variety of feature structure models and 
constraint types that previous analyses have 
used. 
This paper follows Kasper and Rounds 
and most proposals listed in Table 1 by 
distinguishing the constraint language from 
feature structures, and restricts disjunction and 
negation to the constraint language alone. It 
Table 1: Constraint Languages and Feature Structure Models. 
Author 
Kaplan and Bresnan \[10\] 
Model of Feature Structures 
Partial functions 
Constraint Lanl~ua~e Features 
Disjunction, negation, set- 
values 
Pereira and Shieber \[17\] Information Domain 
F=\[A---)F\]+C 
Kasper and Rounds \[11\] Acyclic finite automata Disjunction 
Moshier and Rounds \[14\] Forcing sets of acyclic finite lntuitionistic negation 
automata 
Dawar and Vijayashankar \[3\] Acyclic finite automata Three truth values, negation 
Gazdar, Pullum, Carpenter, Category structures Based on propositional modal 
Klein, Hukari and Levine \[7\] logic 
Johnson \[9\] "Attribute-value structures" Classical negation, 
disjunction... 
174 
(A1) For all Constants c and attributes a, a(c) = 3-. 
(A2) For all distinct pairs of constants Cl, c2, Cl ~ c2. 
(A3) For all attributes a, a(3-) = ±. 
(A4) For all constants c, c ~ ±. 
(A5) For all terms u, v, U = V ~-~ ( U = V A U # ± ) 
Figure 1: The axiom schemata that define attribute-value structures. 
differs by not proposing a custom-built 
"designer logic" for describing feature 
structures, but instead uses standard first-order 
logic to axiomatize attribute-value structures 
and express constraints on them, including 
disjunctive and negative constraints. The 
resulting system is a simplified version of 
Attribute-Value Logic \[9\] which does not allow 
values to be used as attributes (although it 
would be easy to do this). The soundness and 
completeness proofs in \[9\] and other papers 
listed in Table 1 are not required here because 
these results are well-known properties of first- 
order logic. 
Since both the axiomatizion and the 
constraints are actually expressed in a decidable 
class of first-order formulae, viz. quantifier-free 
formulae with equality, 1 the decidability of 
feature structure constraints follows trivially. In 
fact, because the satisfiability problem for 
quantifier-free formulae is NP-complete \[15\] and 
the relevant portion of the axiomatization and 
translation of constraints can be constructed in 
polynomial time, the satisfiability problem for 
feature constraints (including negation) is also 
NP-complete. 
AXIOMATIZING ATTRIBUTE-VALUE 
STRUCTURES 
This section shows how attribute-value 
structures can be axiomatized using first-order 
quantifier-free formulae with equality. In the 
next section we see that equality and inequality 
constraints on the values of the attributes can 
also be expressed as such formulae, so systems 
of these constraints can be solved using standard 
techniques such as the Congruence Closure 
algorithm \[15\], \[5\]. 
The elements of the attribute-value 
structure, both constant and complex, together 
with an additional element ± constitute the 
domain of individuals of the intended 
interpretation. The attributes are unary partial 
functions over this domain (i.e. mappings from 
elements to elements) which are always 
undefined on constant elements. We capture 
this partiality by the standard technique of 
adding an additional element 3_ to the domain to 
serve as the value 'undefined'. Thus a(x) = 3_ if x 
does not have an attribute a, otherwise a(x) is the 
value of x's attribute a. 
We proceed by specifying the conditions 
an interpretation must satisfy to be an attribute- 
value structure. Modelling attributes with 
functions automatically requires attributes to be 
single-valued, as required. 
Axiom schema (A1)describes the 
properties of constants. It expresses the 
requirement that constants have no attributes. 
Axiom schema (A2) requires that 
distinct constant symbols denote distinct elements 
in any satisfying model. Without (A2) it would 
be possible for two distinct constant elements, 
say singular and plural, to denote the same 
individual. 2 
Axiom schema (A3) and (A4) state the 
properties of the "undefined value" 3-. It has no 
attributes, and it is distinct from all of the 
constants (and from all other elements as well - 
this will be enforced by the translation of 
equality constraints). 
This completes the axiomatization. This 
axiomatization is finite iff the sets of attribute 
symbols and constant symbols are finite: in the 
intended computational and linguistic 
applications this is always the case. The claim is 
that any interpretation which satisfies all of these 
The close relationship between quantifier- 
free formulae and attribute-value constraints 
was first noted in Kaplan and Bresnan \[10\]. 
175 
Such a schema is required because we are 
concerned with satisfiability rather than 
validity (as in e.g. logic programming). 
axioms is an attribute-value structure; i.e. (A1) - 
(A4) constitute a definition of attribute-value 
structures. 
Example (continued): The interpretation 
corresponding to the attribute-value structure 
(1) has as its domain the set D = { el ..... e6, 
singular, third, 3-}. The attributes denote 
functions from D to D. For example, agr denotes 
the function whose value is 3_ except on e2 and 
es, where its values are e4 and e6 respectively. It 
is straight-forward to check that all the axioms 
hold in the three attribute-value structures given 
above. 
In fact, any model for these axioms can be 
regarded as a (possibly infinite and 
disconnected) attribute-value feature structure, 
where the model's individuals are the elements 
or nodes, the unary functions define how 
attributes take their values, the constant symbols 
denote constant elements, and _L is a sink state. 
EXPRESSING CONSTRAINTS AS 
QUANTIFIER-FREE FORMULAE. 
Various notations are currently used to express 
attribute-value constraints: the constraint 
requiring that the value of attribute a of (the 
entity denoted by) x is (the entity denoted by) y 
is written as (x a> = y in PATR-II \[19\], as (x a) = y 
in LFG \[10\], and as x(a) = y in \[9\], for example. 
At the risk of further confusion we use another 
notation here, and write the constraint requiring 
that the value of attribute a of x is y as a(x) = y. 
This notation emphasises the fact that attributes 
are modelled by functions, and simplifies the 
definition of '-'. 
Clearly for an attribute-value structure 
to satisfy the constraint u = v then u and v must 
denote the same element, i.e. u = v. However 
this is not a sufficient condition: num(x) = num(y) 
is not satisfied if num(x) or num(y) is I. Thus it 
is necessary that the arguments of '=' denote 
identical elements distinct from the denotation 
of_L. 
Even though there are infinitely many 
instances of the schema in (A5) (since there are 
infinitely many terms) this is not problematic, 
since u = v can be regarded as an abbreviation for 
U=VAU~:/. 
Thus equality constraints on attribute- 
value structures can be expressed with 
quantifier-free formulae with equality. We use 
classically interpreted boolean connectives to 
express conjunctive, disjunctive and negative 
feature constraints. 
Example (continued): Suppose each variable 
xi denotes the corresponding e i, 1 <_i <_11, of(l) 
and (2). Then subj(xl) ~ x2, 
number(x4) = singular and number(agr(x2 ) ) 
= number(x 4) are true, for example. Since e 4 and 
e5 are distinct elements, x8 = Xll is false and 
hence x8 ~Xll is true. Thus " ~" means "not 
identical to" or "not unified with", rather than 
"not unifiable with". 
Further, since agr(xl ) = J-, 
agr( x l ) = agr(x l ) is false, even though 
agr(xl) = agr(xl) is true. Thus t = t is not a 
theorem because of the possibility that t = J_. 
SATISFACTION AND UNIFICATION 
Given any two formulae ~ and q0, the set of 
models that satisfy both ~) and q0 is exactly the set 
of models that satisfy ~ ^ q). That is, the 
conjunction operation can be used to describe 
the intersection of two sets of models each of 
which is described by a constraint formula, even 
if these satisfying models do not form principal 
filters \[11\] \[9\]. Since conjunction is idempotent, 
associative and commutative, the satisfiability of 
a conjunction of constraint formulae is 
independent of the order in which the conjuncts 
are presented, irrespective of whether they 
contain negation. Thus the evaluation (i.e. 
simplification) of constraints containing 
negation can be freely interleaved with other 
constraints. 
Unification identifies or merges exactly 
the elements that the axiomatization implies are 
equal. The unification of two complex elements 
e and e' causes the unification of elements a(e) 
and a(e') for all attributes a that are defined on 
both e and e'. The constraint x = x' implies 
a(x) : a(x') in exactly the same circumstances; i.e. 
when a(x) and a(x') are both distinct from 3-. 
Unification fails either when two different 
constant elements are to be unified, or when a 
complex element and a constant element are 
unified (i.e. constant-constant clashes and constant- 
complex clashes). The constraint x : x' is 
unsatisfiable under exactly the same 
circumstances, x -~ x' is unsatisfiable when x and 
x' are also required to satisfy x = c and x' = c' for 
distinct constants c, c', since c ~ c' by axiom 
schema (A2). x = x" is also unsatisfiable when x 
and x' are required to satisfy a(x) : t and x' ~ c' 
176 
for any attribute a, term t and constant c', since 
a(c') = _t_ by axiom schema (A3). 
Since unification is a technique for 
determining the satisfiability of conjunctions of 
atomic equality constraints, the result of a 
unification operation is exactly the set of atomic 
consequences of the corresponding constraints. 
Since unification fails precisely when the 
corresponding constraints are unsatisfiable, 
failure of unification occurs exactly when the 
corresponding constraints are equivalent to 
False. 
Example (continued): The sets of satisfying 
models for the formulae (1") and (2') are precisely 
the principal filters generated by (1) and (2) 
above. 
(1') subj(xl) = x2 ^ agr(x2) = x4 ^ 
number(x4) = singular A pred(xl) = x3 A 
verb(x3) = x5 A agr(x 5) ~- X6 ^ 
person(x6) = third 
(2') subj(x7) = x8 ^ agr(x8) = Xll ^ pred(x7) = x9 a 
verb(x9) = Xl0 A agr(xlO) = Xll 
Because the principal filter generated by the 
unification of el and e7 is the intersection of the 
principal filters generated by (1) and (2), it is 
also the set of satisfying models for the 
conjunction of (1') and (2') with the formula 
Xl = x7 (3'). 
(3') subj(xl) = x 2 ^ agr(x 2) = x4 ^ 
nmber(x4) = singular ^ pred(xl) ~- x3 ^ 
verb(x3) = x5 ^ agr(x5) = x6 ^ 
person(x6) -~ third a subj(x7) = x8 ^ 
agr(x8) = Xll A pred(x7) ~- x9 A 
verb(x 9) = Xl0 A agr(xlO) = Xll A X 1 ~ X 7 . 
The satisfiability of a formula like (3') can be 
shown using standard techniques such as the 
Congruence Closure Algorithm \[15\], \[5\]. In 
fact, using the substitutivity and transitivity of 
equality, (3') can be simplified to (3"). It is easy 
to check that (3) is a satisfying model for both 
(3") and the axioms for attribute-value 
structures. 
The treatment of negative and disjunctive 
constraints is straightforward. Since negatiou is 
interpreted classically, the set of satisfying 
models do not ahvays form a filter (i.e. they are 
not always upward closed \[16\]). Nevertheless, 
the quantifier-free language itself is capable of 
characterizing exactly the set of feature 
structures that satisfy any boolean combination 
of constraints, so the failure of upward closure is 
not a fatal flaw of this approach. 
At a methodological level, I claim that 
after the mathematical consequences of two 
different interpretations of feature structure 
constraints have been investigated, such as the 
classical and intuitionistic interpretations of 
negation in feature structure constraints \[14\], it 
is primarily a linguistic question as to which is 
better suited to the description of natural 
language. I have been unable to find any 
linguistic analyses which can yield a set of 
constraints whose satisfiablity varies under the 
classical and intuitionistic interpretations, so the 
choice between classical and intuitionistic 
negation may be moot. 
For reasons of space the following 
example (based on Pereira's example 116\] 
demonstrating a purported problem arising 
from the failure of upward closure with classical 
negation) exhibits only negative constraints. 
Example: The conjunction of the formulae 
number(agr(x) ) = singular 
and 
agr(x) = y A ~ (pers(y) = 3rd A 
number(y) = singular ) 
can be simplified by substitution and transitivity 
of equality and boolean equivalences to 
(4') agr(x) = y A number(y) ~- singular A 
pers(y) ~ 3rd. 
This formula is satisfied by the structure (4) 
when x denotes e and y denotes f. Note the 
failure of upward closure, e.g. (5) does not satisfy 
(4'), even though (4) subsumes (5). 
(3") subj(xl) = x2 A agr(x2) = x4 A 
number(x4) = singular A person(x4) = third A 
pred(xl) = x3 A verb(x 3) = x5 A agr(xs) = X4 A 
Xl = X7 ^ X2 = X5 ^ X3 = X9 AX5 = Xl0 ^ 
X4 ~- X6 A X4 = X11. 
177 
(4) el (5) el 
number number pers 
singular singular 3rd 
However, if (4') is conjoined with 
pers(agr(x) ) ~- 3rd the resulting formula (6)/s 
unsatisfiable since it is equivalent to (6'), and 
3rd ~ 3rd is unsatisfiable. 
(6) agr(x) ~, y ^ number(y) = singular ^ 
pers(y) ~ 3rd ^ pers(agr(x)) = 3rd. 
(6') agr(x) = y a number(y) ~ singular ^ 
pers(y) = 3rd ^ 3rd ~ 3rd. 
CONCLUSION 
This paper has shown how attribute-value 
structures and constraints on them can be 
axiomatized in a decidable class of first-order 
logic. The primary advantage of this approach 
over the "designer logic" approach is that 
important properties of the logic of the feature 
constraint language, such as soundness, 
completeness, decidability and compactness, 
follow immediately, rather than proven from 
scratch. A secondary benefit is that the 
substantial body of work on satisfiability 
algorithms for first-order formulae (such as 
ATMS-based techniques that can efficiently 
evaluate some disjunctive constraints \[13\]) can 
immediately be applied to feature structure 
constraints. 
Further, first-order logic can be used to 
axiomatize other types of feature structures in 
addition to attribute-value structures (such as 
"set-valued" elements) and express a wider 
variety of constraints than equality constraints 
(e.g. subsumption constraints). In general these 
extended systems cannot be axiomatized using 
only quantifier-free formulae, so their 
decidability may not follow directly as it does 
here. However the decision problem for 
sublanguages of first-order logic has been 
intensively investigated \[4\], and there are 
decidable classes of first-order formulae \[8\] that 
appear to be expressive enough to axiomatize an 
interesting variety of feature structures (e.g. 
function-free universally-quantified prenex 
formulae can express linguistically useful 
constraints on "set-valued" elements). 
An objection that might be raised to this 
general approach is that classical first-order 
logic cannot adequately express the inherently 
"partial information" that feature structures 
represent. While the truth value of any formula 
with respect to a model (i.e. an interpretation 
and variable assignment function) is completely 
determined, in general there will be many 
models that satisfy a given formula, i.e. a 
formula only partially identifies a satisfying 
model (i.e. attribute-value structure). The claim 
is that this partiality suffices to describe the 
partiality of feature structures. 

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