Features and Formulae 
Mark Johnson" 
Brown University 
Feature structures are a representational device used in several current linguistic theories. This 
paper shows how these structures can be axiomatized in a decidable class of first-order logic, 
which can also be used to express constraints on these structures. Desirable properties, such 
as compactness and decidability, follow directly. Moreover, additional types of feature values, 
such as "set-valued" features, can be incorporated into the system simply by axiomatizing their 
properties. 
1. Introduction 
Many modern linguistic theories, such as Lexical-Functional Grammar (Bresnan 1982), 
Functional Unification Grammar (Kay 1985), Generalized Phrase Structure Grammar 
(Gazdar et al. 1985), Unification Categorial Grammar (Haddock et al. 1987), (Uszkoreit 
1986), and Head-Driven Phrase Structure Grammar (Pollard and Sag 1987), 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 spec- 
ified indirectly 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 parent node and its immediate descendants. 
That is, lexical entries and syntactic rules used to construct a syntactic phrase 
structure tree all contribute constraints on the feature structures that appear as the 
labels on nodes in the syntactic tree. 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 precise 
nature of the satisfying feature structures (of which there may be infinitely many) is 
of secondary importance. 1 
A variety of different types of feature structures have been proposed in the liter- 
ature, but most work on unification-based grammar has centered on a certain type of 
feature structure known as 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 values may be any other ele- 
ment in the structure, including a complex element. An element can be the value of 
zero, one or several attributes. Attributes are partial: it need not be the case that every 
attribute is defined for every complex element. 
* Department of Cognitive and Linguistic Sciences, Providence, RI 02912 USA 1 The validity problem is also of interest, since it provides a way of "extracting information" about all of 
the satisfying feature structures. In the framework developed below, if ~ is a formula representing a 
system of constraints and ~b --~ 0 is valid, then 0 is a true description of every feature structure 
satisfying ~b. 
(~ 1991 Association for Computational Linguistics 
Computational Linguistics Volume 17, Number 2 
"Mary seems to like John" 
eo 
-pred = 
subj = 
e3 
comp : 
seem 
- 11 I num = sg i agre5 ~pers 3rd pred = mary 
pred = like 
I pred = john 
obj= ~ ~num= 
e2 ~agr e4~pers = 
tense = none 
subj = e 3 
tense = pres 
3;:11 
Figure 1 
An attribute-value structure for Mary seems to like John 
Example 1 
Figure 1 depicts an attribute-value structure. The attribute-value element labeled eo 
in Figure 1 might be associated with the sentence Mary seems to like John. 
The attribute-value structure depicted in Figure 1 contains six complex elements 
eo,..., e5 and eight constant elements seem, like, john, sg, 3rd, mary, none, and pres. The 
element eo is a complex attribute-value element with four attributes: pred, subj, comp, 
and tense: the order in which the attributes appear in the diagram is irrelevant. The 
value of its pred attribute is the constant seem (which abbreviates the relation denoted 
by the verb seem), and the value of its tense attribute is the constant element pres 
(which indicates that the clause is in the present tense). The values of the subj and 
comp attributes are the complex elements e3 and el (which represent the subject and the 
complement of the verb seem, respectively). The element e3 also appears as the subject 
of el, indicating that Mary is also the (understood) subject of the verb likes as well. 
The element el is a complex attribute-value element with four attributes pred, obj, 
subj, and tense. The value of its pred attribute is the constant element like (which abbre- 
viates the relation denoted by the verb like) and the value of its tense attribute is the 
constant element none (which indicates that the clause is untensed). The values of the 
attributes obj and subj of el are the complex elements e2 and e3, respectively (which 
represent the subject and object of the clause). Both e4 and e5 have the same attributes 
hum and pets, and the values of these attributes of e4 are identical to the corresponding 
values of these attributes of e5. Nevertheless, e4 and e5 are distinct elements. 
An operation called unification plays an important role in most accounts of feature 
structures (Kay 1985; Shieber 1986). The unification operation "combines" or "merges" 
two elements into a single element that agrees with both of the original elements on 
the values of all of their defined sequences of attributes, so the unification of two 
complex elements requires the unification of the values of any attributes they have 
in common. The unification operation fails if it requires the unification of distinct 
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Johnson Features and Formulae 
salmon 
swims 
e' 
-pred = salmon 
agr= ~pers= 3rdlp ~ ~0 L J, 
pred = swim 
~num = 
subj = agr f =, \[_pers -- e" 2 
" tense = pres 
3rd 
Figure 2 
Lexical entries for salmon and swim 
"The salmon swims" 
pred = 
• subj = 
e' 
e" 
tense = 
swim s !l 
/ ;Lpers-- 
~pred = salmon 
pres 
Figure 3 
An example of attribute-value unification 
constant elements (a constant-constant clash) or the unification of a constant element 
and a complex element (a constant-complex clash). 
Example 2 
A grammar might assign the attribute-value structures in Figure 2 to the NP the salmon 
and the VP swims, respectively. Note that e' does not have a num attribute, since the 
salmon can be either singular or plural and that e" does not have a pred attribute. 
The attribute-value structure for the sentence (the) salmon swims is obtained by 
unifying e' and e', which corresponds to identifying salmon as the subject of swims. 
The resulting element inherits the value of the pred attribute from e' and the value of 
the num attribute from e'. The unification of e' and e" requires the unification of f~ and 
f" as well. 
Although it might not be obvious from this simple example, a large number of 
syntactic constructions from a variety of natural languages can be described in such 
a unification-based framework (many of the analyses presented in Bresnan 1982 can 
be expressed in such a "pure" unification-based framework). Nevertheless, it is often 
convenient and sometimes necessary to extend the basic unification framework to 
include a wider variety of feature structures. 
For example, "negative values" and "disjunctive values" allow grammars and 
lexical entries to be written much more succinctly, as the following examples show 
(based on Karttunen 1984). 
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Computational Linguistics Volume 17, Number 2 
d/e 
__ m 
cat = determiner 
/~number: singular-~ \ 
agr= ~_gender: femininel~ 
\ ~number: plural ~ / 
nom case= (acc / 
Figure 4 
Disjunction in the lexical entry for die 
swim I 
pred = swim ljl 
I ~num= sg subj = agr = --, 
tense e ~ pre s f~eers = 3rd 
g 
Figure 5 
Negation in the lexical entry for swim 
Example 3 
In German the determiner die must have accusative or nominative case, and agrees 
with either feminine singular nouns or plural nouns of any gender. In a framework 
with disjunctive values only one lexical entry for die is required. 2 
Example 4 
In the basic unification framework described above the tensed verb swim would require 
multiple lexical entries, since it agrees with first person, second person, and plural third 
person subjects; i.e., a subject with any agreement features other than third person 
singular. In a framework with "negative values" it requires only the single lexical 
entry in Figure 5, where "-~" identifies a "negative value." 
As mentioned earlier, other kinds of feature structures besides attribute-value 
structures have been proposed in the literature. Johnson and Klein (1986) and Johnson 
and Kay (1990) show how "set-valued" features can be used to express Discourse Rep- 
resentation Theory (Kamp 1981) in a complex-feature based grammar formalism. The 
highly simplified example below is meant solely to show one way in which set-valued 
features can be used--no claims are made for its linguistic correctness. 
2 "Disjunctive" features are depicted using angle brackets, since curly brackets are used in this paper to 
depict "set-valued" features. Below we reinterpret the "disjunctive" and "negative" features depicted 
in this example and the next as disjunctions and negations of constraints. 
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Johnson Features and Formulae 
~ at = NP 
lindex ~ s J 
she -- I refs-in = s | woman 
u ~efs-out = s~ 
I cat = N 
r index = i' J 
Jj refs-in = s' 
i refs-out = s' u { i' } v\[_ 
Figure 6 
Set-values in the lexical entries for she and woman 
~ at = N , 
I index = i 
I refs-in = { } 
v ~efs-out = { i'} 
Figure 7 
The result of unifying s' in Figure 6 with the empty set 
Example 5 
A naive theory of anaphoric dependencies between indefinite NPs and anaphoric pro- 
nouns can be constructed as follows. Each NP has an index attribute whose value is 
a "reference marker," and two NPs are coreferential iff they share the same reference 
marker. 3 Every feature structure associated with a node in the syntactic tree has at- 
tributes refs-in and refs-out, whose values are the sets of discourse entities available 
preceding and following this node, respectively. The grammar constrains the value 
of the refs-out attribute of an indefinite NP to be the union of its refs-in attribute and 
the singleton set containing the value of the NP's index attribute; this adds the NP's 
index to the set of available indices. Similarly, the grammar requires the values of a 
pronoun's refs-in and refs-out attributes to be identical, and that its index attribute be a 
member of the value of its refs-in attribute. This requires that the pronoun refer to an 
entity previously introduced into the discourse. In a framework with set values the 
lexical entries for (a) woman and she could be as seen in Figure 6. 
Unifying the value s' of the refs-in attribute of the lexical entry for woman with the 
empty set (which corresponds to the empty discourse context) produces the feature 
structure depicted in Figure 7. 
Further, the unification of the value of the refs-out attribute in Figure 7 with the 
value of the refs-in attribute of u in Figure 6 (the lexical entry for she), which corre- 
sponds to interpreting the pronoun as an anaphor within the context established by 
the single NP a woman produces the feature structure depicted in Figure 8. 
Extending the possible feature structures beyond the basic attribute-value features 
complicates the basic unification operation, however. For example, Moshier and Rounds 
(1987) and Pereira (1987) point out that it is not obvious how to extend unification 
3 Reference markers in DRT correspond approximately to the referential indices associated with NPs in GB theory. 
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Computational Linguistics Volume 17, Number 2 
cat = NP \[ 
\] 
index = i' 
I refs-in = { i'} 
u ~efs-out = { i'} 
Figure 8 
The result of unifying the value of the refs-out attribute of Figure 7 with s in Figure 6 
this > def= Tnum= sg 
u agrv L pers = 3rd 
these .def = + num = pl 
~agr-- LPers= 3rd U' V' 
salmon > IP red= salm°n3rd\] 1 
e, Lagr =f\[Pe rs= 
swim rpre i pers nUm sgll 
agr= 3rd subj =e 
g ~ tense = pres 
Figure 9 
Feature structures demonstrating interaction of negative values and unification 
to negative feature values; specifically, some apparently plausible extensions lose the 
associativity property of unification. 
Example 6 
Consider the feature structures in Figure 9, which might be assigned to the singular 
determiner this, the plural determiner these, the noun salmon, and the verb swim (the 
latter two structures are the same as those depicted in Figures 2 and 5). These can be 
used to analyze utterances such as these salmon swim and (the ill-formed utterance) this 
salmon swim, which involve the unification of u, e', and e or u', e', and e, respectively. 
Suppose a negative value is interpreted as a constraint that a feature structure 
either satisfies or does not satisfy, and suppose further that in Figure 9 the negative 
feature constraint f is satisfied by the value f'. Then e' and e in Figure 9 unify, and 
moreover further unification of e ~ with either u ~ or u succeeds, undesirably in the latter 
case. (Reinterpreting the negative constraint f so that f' fails to satisfy it does not help, 
since the unification of e, e', and u ~ should succeed). On the other hand, we obtain the 
results we desire if e' is unified with u or u ~ before being unified with e. If e' is first 
unified with u, then f~ is unified with v, and further unification of e' with e fails, since 
v does not satisfy f. If e' is first unified with u' then f' is unified with v' and further 
unification of e' with e succeeds, since v' does satisfy f. Thus under this interpretation 
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Johnson Features and Formulae 
of negation and unification, the success or failure of a sequence of unifications depends 
on the order in which they are performed. 4 
2. Feature Structures and Function-free Formulae 
These problems have generated a considerable body of work on the mathematical 
properties of feature structures and the constraints and operations that apply to them. 
Following Kaplan and Bresnan (1982), Pereira and Shieber (1984), Kasper and Rounds 
(1986, 1990), and Johnson (1988, 1990a) the constraints that determine the feature struc- 
tures are regarded as formulae from a language for describing feature structures, rather 
than as feature structures themselves. 
Disjunction and negation appear only in expressions from the description lan- 
guage, rather than as components of the feature structures that these expressions de- 
scribe. Thus the lexical entries in the examples above will be interpreted as formulae 
that constrain the feature structures that can be associated with these lexical items in 
a syntactic tree, rather than the feature structures themselves. For example, the feature 
matrices depicted in Figures 2, 4-6, and 9 will be interpreted as graphical depictions of 
formulae expressing constraints on linguistic objects, rather than the linguistic objects 
that satisfy these constraints. This avoids any need to refer to "negative" or "disjunc- 
tive" objects as entities appearing in a feature structure. 
As explained below, the familiar attribute-value "unification algorithm" can be 
interpreted as computing the atomic consequences of a purely conjunctive formula 
(where the graphs it operates on are data structures efficiently representing such for- 
mulae), and unification failure corresponds to the unsatisfiability of that conjunction 
(Kasper and Rounds 1990; Johnson 1988, 1990a; Pereira 1987). 
The most widely known model of feature structures and constraint language is 
the one developed to explain disjunctive feature values by Kasper and Rounds (1986, 
1990) and Kasper (1986, 1987). The Kasper-Rounds treatment resolves the difficulties 
in interpreting disjunctive values by developing a specialized language for expressing 
these constraints. Various proposals to extend the Kasper-Rounds approach to deal 
with negative feature values are described by Moshier and Rounds (1987), Moshier 
(1988), Kasper (1988), Dawar and Vijayashanker (1989, 1990), Langholm (1989); other 
extensions to this framework are discussed by D6rre and Rounds (1989), Smolka (1988, 
1989), and Nebel and Smolka (1989); and Shieber (1989) discusses the integration of 
such feature systems into a variety of parsing algorithms. 
One difficulty with this approach is that the constraint language is "custom built," 
so important properties, such as compactness and decidability, must be investigated 
from scratch. Moreover, it is often unclear if the treatment can be extended to 
handle other types of feature structures as well. Rounds (1988) proposes a model 
for set-valued features, but he does not provide a language for expressing constraints 
on such set-valued entities, or investigate the computational complexity of systems of 
such constraints. 
This paper follows an alternative strategy suggested in Johnson (1990a): axiomatize 
the relevant properties of feature structures in some well-understood language (here 
first-order logic) and translate constraints on these structures into the same language. 
4 It is possible to avoid these problems by augmenting feature structures with "inequality arcs," as was 
first proposed (to my knowledge) by Karttunen (1984) and discussed in Johnson (1990a), Johnson (in 
press) and pages 67-72 of Johnson (1988). However, it is hard to justify the existence of such arcs if feature structures are supposed to be linguistic objects (rather than data structures that represent 
formulae manipulated during the parsing process). 
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Computational Linguistics Volume 17, Number 2 
Thus the satisfiability problem for a set of constraints on feature structures is reduced 
to the satisfiability problem for the axioms conjoined with the translation of these 
constraints in the target language. Importantly, techniques used to determine satisfia- 
bility in the target language can be used to determine the satisfiability of the feature 
constraints as well. In this paper the properties of attribute-value structures and con- 
straints on them are expressed in a decidable class of first-order formulae: this means 
that the satisfiability problem for such formulae, and hence the feature constraints that 
they express, is always decidable. 
Of course, some linguistic analyses make use of feature structure constraint sys- 
tems that can encode undecidable problems. For example, subsumption constraints, 
which are useful in the description of agreement phenomena in coordination con- 
structions (Shieber 1989) can be used to encode undecidable problems, as D6rre and 
Rounds (1989) have shown. Clearly such constraints cannot be expressed in a decidable 
class, but often they can be axiomatized in other standard logics. Johnson (1991) shows 
how (positively occurring) subsumption constraints can be axiomatized in first-order 
logic, and sketches treatments of sort constraints and nonmonotonic devices such as 
ANY values (Kay 1985) and 'constraint equations' (Kaplan and Bresnan 1982) can be 
formalized in second-order logic using circumscription. 
2.1 Axiomatizing Feature Structures with Function-Free Formulae 
This section shows how the important properties of feature structures can be axioma- 
tized using formulae from the Sch6nfinkel-Bernays class, which is the class of first-order 
formulae of the form 
3X1. . . Xn~yl . . . yng~ 
where ~ contains no function symbols or quantifiers. (Thus no existential quantifier 
can appear in the scope of a universal quantifier.) This class of formulae was chosen 
because it is both decidable (see e.g. Lewis and Papadimitriou 1981) and can express 
the quantification needed to describe the particular set operations proposed here, as 
well as a variety of other interesting types of feature structures and constraints. (For 
a general discussion of decidable classes see Gurevich 1976 and Dreben and Goldfarb 
1979.) The next section shows how the various kinds of constraints on feature struc- 
tures described above can be translated into this class of formulae, so any system of 
such feature constraints is decidable as well. 
The elements of a feature structure, both complex and constant, constitute the 
domain of individuals in the intended interpretation. The attributes are binary 
relations over this domain, s We proceed by axiomatizing the conditions under which 
an interpretation corresponds to a well-formed feature structure, formulating them in 
essentially the same way as Smolka (1988, 1989) does. 
The axiomatization begins by describing the properties of the constant elements of 
attribute-value structures. The attribute-value constants are the denotation of certain 
constant symbols of the language of first-order logic, but not all constants (of the first- 
order language) will denote attribute-value constants since it is convenient to have 
constants that denote other entities as well. The following axiom schemata express the 
requirement that attribute-value constants have no attributes and that all attribute- 
5 This differs from earlier work (Johnson 1988) in which values and attributes were both conceptualized 
as individuals. In fact, research in progress indicates that it is advantageous to conceptualize of 
attributes as individuals and attribute relations in terms of a 3-place relation arc, where arc(x, a, y) is 
true iff the value of x's attribute a is y. This permits the quantification over attributes needed to define 
both simple and parameterized sorts to be expressed. 
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Johnson Features and Formulae 
value constants are distinct; i.e., that distinct attribute-value constants denote different 
entities. 
1. For all attribute-value constants c and attributes a, V x -~a(G x). 
2. For all distinct pairs of attribute-value constants Cl, c2, Cl # c2. 
The next axiom schema requires attributes to be single-valued. 
3. For all attributes a, V xyz a(x,y) Aa(x,z) ~ y = z. 
This completes the axiomatization of attribute-value feature structures. 6 The claim is 
that any interpretation that satisfies these axioms is an attribute-value structure, i.e. 
1-3 constitute a definition of attribute-value structures. Such interpretations can be 
viewed as (possibly infinite and disconnected) directed graphs, where the individuals 
constitute the graph's nodes and the attribute relations the arcs between those nodes. 
Thus these axioms admit a much wider class of models than do most other treat- 
ments of feature structures (e.g., Kasper and Rounds (1990) require feature structures 
to be a certain type of finite automata). In fact it is easy to add axioms requiring 
attribute-value structures to have additional properties such as acyclicity. But since 
the axioms that define attribute-value structures are in effect assumptions that stipulate 
the nature of linguistic entities, we obtain a more general theory the weaker these 
axioms are. Thus 1-3 are intended to stipulate only the properties of attribute-value 
structures that are required by linguistic analyses. 
Note that the partiality of attributes is of crucial importance: if attributes were 
required to be total rather than partial functions, we could not axiomatize them with 
formulae from the Sch6nfinkel-Bernays class. (An axiom schema requiring attributes 
to be total functions would have instances of the form Vx 3y a(x,y), which do not 
belong to the Sch6nfinkel-Bernays class). 
Example 7 
The interpretation corresponding to the attribute-value structure depicted in Figure 1 
has as its domain the set D = {seem, like, john, sg, 3rd, mary, pres, none} U {e0,..., es}. 
The attributes denote relations on D x D. For example, pred denotes the relation 
{leo~seem), (el, like), le2,johnl~ (e3~ maryl}. It is straightforward to check that all of the 
axioms hold in this interpretation. 
Instead of providing entities in the interpretation that serve as the denotation for "dis- 
junctive" or "negative" features, we follow Kasper and Rounds (1986, 1990), Moshier 
and Rounds (1987), and Johnson (1988, 1990) in permitting disjunction and negation 
only in the constraint language. Since the classical semantics of disjunction and nega- 
tion for first-order languages is consistent and monotonic, a consistent, monotonic 
semantics for negative and disjunctive feature constraints follows directly. (An exam- 
ple is presented below; see Johnson (1990) and especially Section 2.10 of Johnson (1988) 
for further discussion). 
We turn now to the set-valued features. The most straightforward way of introduc- 
ing set-valued features would be to combine some standard axiomatization of set the- 
ory with the axiomatization of attribute-value structures just presented. Unfortunately, 
6 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. 
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Computational Linguistics Volume 17, Number 2 
all of the formulations of set-theory I am aware of, such as Zermelo-Fraenkel set- 
theory, are expressed in languages whose satisfiability problem is undecidable. While 
this does not imply that the satisfiability problem for set-valued feature-structure con- 
straints is also undecidable (since the feature constraint language may have restricted 
expressiveness), it does mean that its decidability cannot be shown by noting that a 
translation into a decidable class of formulae exists. 
Also, as an anonymous reviewer points out, since the intended linguistic applica- 
tions only require finite sets and operations such as union and intersection, standard 
theories of sets (such as Zermelo-Fraenkel set-theory) are much more powerful than 
needed. 
Instead, we axiomatize just those properties of set-valued features that our feature 
constraints require using formulae from the Sch6nfinkel-Bernays class. We interpret 
the two-place relation in as the membership relation; in(x, y) is true in a model iff x is a 
member of y. We place no restrictions on this relation, but in other formulations axioms 
of foundation and extensionality could be used to ensure that the in relation can be 
interpreted as the set-membership relation of Zermelo-Fraenkel set theory. Thus this 
axiomatization presented here will admit models in which the values of set-valued fea- 
tures do not have these properties. 7 These additional properties of the set-membership 
relation don't seem to be needed in linguistic analyses, so such stipulations are not 
made here. 
The axiom of foundation requires that all sets are well founded; i.e., that the tran- 
sitive closure of the set-membership relation is irreflexive, or more informally, that no 
set directly or indirectly contains itself as a member. Versions of set-theory that relax 
this restriction have been proposed by, e.g., Aczel (1988), and Rounds (1988) argues 
that non-well founded sets may be appropriate models of set values in feature struc- 
tures. The paradoxes associated with non-well founded set theories are avoided here 
because the axiom of comprehension that asserts the existence of paradoxical sets is 
not included in this axiomatization; i.e., the only way of defining a set is either by 
explicitly listing its members or by means of union and intersection operations. 
The axiom of extensionality requires that if sets $1 and $2 contain exactly the same 
members then $1 = $2; without extensionality it is possible for two different sets to 
contain exactly the same members. Admittedly the primary reason for omitting an 
extensionality axiom is that it does not appear to be axiomatizable using Sch6nfinkel- 
Bernays' formulae, but three other reasons motivate this decision. 
First, as noted in Shieber (1986) and in Example 1 above, feature structures in 
general are not extensional (e.g., two distinct attribute-value elements can have ex- 
actly the same attributes and values), and it seems reasonable to treat set-values in a 
nonextensional fashion as well. 
Second, extensionality could produce undesirable interactions with the attribute- 
value component of feature structures. Since set-valued features can also have at- 
tributes (for example, in LFG (Kaplan and Bresnan 1982)) a conjunction is associated 
with a set-value that also has attributes), extensionality would prohibit there being 
7 In fact there are Sch6nfinkel-Bernays axioms that require the in relation to be acyclic. Define a new 
relation, say in + , by the axioms 
V es in(e,s) ---* in+(e,s) 
V ess' in+(e,s) A in+(s,s ') ~ in+(e,J). 
Then in any model in + denotes a superset of the transitive closure of the in relation. The following 
axiom requires that this transitive closure is irreflexive, i.e. that no set is contained in itself. 
v s ~in + (s, s). 
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Johnson Features and Formulae 
two set-valued features that contain exactly the same elements but that differ on the 
value of some attribute, something a linguistic analysis might reasonably require. 
Third, as far as I am aware, no linguistic analysis requires sets to be extensional. 
Appealing to the same general considerations that were used to justify the attribute- 
value axioms, since the assumption that sets are extensional is not required, the stip- 
ulation is not made here. 
It is necessary to define some predicates that describe set-values. We begin by pre- 
senting a general first-order axiomatization of these predicates, and then approximate 
these with formulae from the Sch6nfinkel-Bernays class. 
Most of the definitions are straightforward, and are given without explanation, s 
The unary predicate null is true of an element iff that element has no members. 
4. Vx null(x) ~ -~y in(y,x) 
The ternary relation union(x, y, z) is true only if every element in z is in x or y. 
5. Vxyz union(x,y,z) ~ Vu in(u,z) ~ in(u,x) V in(u,y) 
The ternary relation intersection(x, y, z) is true only if every element in z is in x and in y. 
6. Vxyz intersection(x,y,z) ~ Vu in(u,z) *--* in(u,x) A in(u,y) 
The binary relation singleton(u, x) is true if and only if u is the only member of x. 
7. Vux singleton(u, x) ~ in(u, x) A Vv in(v, x) ~ u = v 
Unfortunately the axioms 4-7 do not belong to the Sch6nfinkel-Bernays class, so we 
cannot guarantee the decidability of systems of constraints defined using them simply 
by noting a translation into this class exists. However, in all of the linguistic applica- 
tions I am aware of these predicates always appear positively, and in this case these 
axioms can be replaced by the corresponding "one-sided" axioms given below. (The 
predicate null is an exception, since some HPSG analyses (Pollard and Sag 1987) re- 
quire the set of unsaturated arguments of some phrases to be non-null. However, it 
is possible to require that a set s is nonempty by introducing a new constant u and 
require that in(u, s).) 
4'. Vxy null(x) --* -,in(y, x) 
5'. Vuxyz union(x,y,z) --. (in(u,z) ~ in(u,x) V in(u,y)) 
6'. Vuxyz intersection(x,y,z) --. (in(u,z) ~ in(u,x) A in(u,y)) 
7'. Vuxv singleton(u, x) --. (in(u, x) A in(v, x) --* u = v) 
These one-sided definitions are incorrect when these predicates appear negatively (i.e., 
in the scope of an odd number of negation symbols after all other proposition con- 
nectives have been expressed in terms of A, V, and -,). For example, an interpretation 
8 In the following axioms all of the connectives are to be interpreted as right-associative. 
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Computational Linguistics Volume 17, Number 2 
with an empty in relation can satisfy =null(x). As Johan van Benthem and the anony- 
mous reviewer independently pointed out to me, it is possible to prove that so long as 
the relations null, union, intersection, and singleton appear only positively in linguistic 
constraints, any model satisfying 4'-7' differs from a :model satisfying 4-7 at most in 
the denotation of these relations; other relations, in particular the attribute relations or 
even the in relation, are not affected by the one-sided approximation. The following 
proposition expresses this. 
Proposition 
Let x be a tuple of variables, A be any relation symbol, ~(A) be any formula in which 
A appears only positively, and ~(x) be a formula in which A does not appear. Then 
(i) .M ~ @(A) A Vx A(x) *-+ ~(x) 
if and only if there is a model M' differing from A4 only on the denotation it assigns 
to A such that 
(ii) A4' ~ @(A) A Vx A(x) -+ ~(x). 
Proof 
The left to right direction is obvious. The proposition follows from right to left as 
follows. Let A,/' be any model that satisfies (ii). A model A4 that satisfies (i) can be 
constructed as follows. Let A4 be the model that agrees with A4' except possibly on A, 
where IAIM = IAI~, U ~Ax~(x)~,. Now we check that M satisfies (i). Since ~A~ 2 
~A~, and A appears only positively in @(A), A4 ~ ~(A). Further dr4 ~ VxA(x) *-- ~(x) 
by construction. Since A does not appear in ~(x), ~Ax~(x)~ = ~Ax~(x)~,, and since 
~A~M D \[A~,, M ~ VxA(x) --+ ~(x) as well. Thus M satisfies (i) as required. In 
fact we have shown something stronger; the denotation of A in A4' is a subset of the 
denotation of A in d~4. • 
2.2 Expressing Constraints 
A feature structure is specified implicitly by means of the constraints that it must sat- 
isfy. This section shows how such constraints can be translated into quantifies-free and 
function-free prenex formulae. There is a plethora of different notations for expressing 
these 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 Ix a / -- y in PATR-II (Shieber 
1986), as (x a) = y in GFG (Kaplan and Bresnan 1982), and as x(a) ~ y in Johnson 
(1988), for example. Here we express attribute-value constraints using the attribute 
relations a, so this constraint would be expressed as a(x, y). Set-valued constraints are 
expressed using the relations in, null, union, and singleton defined in the previous sec- 
tion. The propositional connectives are used to express negative and disjunctive feature 
constraints. This section shows how constraints on feature bundles can be specified 
using equality, the attribute relations, and the set predicates axiomatized in the last 
section. (In fact as far as the theoretical results of this paper are concerned all that is 
important is that the constraints are taken to mean the same thing as these formulae, 
irrespective of the notation in which they are expressed.) 
142 
Johnson Features and Formulae 
Example 2 (continued) 
The lexical entries for salmon and swims in Figure 2 are the following formulae, where 
e ~, e', f~, f" and g" are constants of the first-order language that are not attribute-value 
constants. 9 
8a. pred(e'~ salmon) A agr(e', f') A pers(f', 3rd) 
8b. pred(g", swim)A tense(g", pres)A subj(g", e")A agr(e',f")A 
num(f", sg)A pers(f", 3rd). 
Example 3 (continued) 
The lexical entry for the determiner die of Figure 4 is the following formula, where x is 
a (non-attribute-value) constant that denotes the feature structure of the determiner, 
and y and z are constants that are not attribute-value constants. 
9. cat(x, determiner) A agr( x, y ) A (case(x, nom ) V case(x, acc ) ) A 
(number(y, plural) V (number(y, singular)A gender(y, feminine) ) ) 
Example 4 (continued) 
The lexical entry for the verb swim of Figure 5 is the following formula, where g is a 
constant that denotes the feature structure of the verb, and e and f are constants that 
are not attribute-value constants. 1° 
10. pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e,f)A 
-~(num(f, sg)A pers(f , 3rd) ) 
The lexical entries for the determiners this and these of Figure 9 are the following 
formulae, where u, v, u ~ and v ~ are constants that are not attribute-value constants, 
and u denotes the feature structure of this and u ~ denotes the feature structure of these. 
11. def(u, +) A agr(u, v) A num(v, sg) A pers(v, 3rd) 
12. def(u', +) A agr(u', v') A num(v',pl) A pers(v',3rd) 
9 Instead of naming all of the nonroot attribute-value elements with constants as is done here, it is 
possible to merely assert their existence using an existential quantification. For example, the lexical 
entry for salmon could be the formula 
3f'pred(e', salmon) A agr(e', f I) A pers(f , 3rd) 
where fr is an existentially quantified variable. This formulation has the advantage that no 'renaming' 
is needed when determining subsumption of systems of attribute-value constraints. (The subsumption 
relation between systems of constraints is used in certain types of 'unification based' parsers (Shieber 
1989).) That is, a system of constraints represented by a formula ~ subsumes another system of 
constraints represented by 8 iff A ~ 0 ~ ~, where A is the conjunction of the axioms defining the 
relevant types of feature structures. 
10 The formulation (10) of the negative constraint depicted in Figure 5 does not imply that f has either a 
num or pers attribute. Conceivably, one might want to interpret such a negative constraint as requiring 
f to have both num and pers attributes with values differing from either sg or 3rd, respectively. The 
formula below expresses this interpretation. 
pred(g, swim) A tense(g, pres) A subj(g, e) A agr(G f) A 
hum(f+ u) A pers(f , v) A -~(u = sg A v = 3rd) 
143 
Computational Linguistics Volume 17, Number 2 
Example 5 (continued) 
The lexical entries for she and woman of Figure 6 are the formulae (13) and (14), where 
u denotes the feature structure of the pronoun, v denotes the feature structure of the 
noun, and w, s, s', s', i, and i' are constants that are not attribute-value constants. 
13. cat(u, np) A refs-in(u, s) A refs-out(u, s) A index(u, i) A in(i, s) 
14. cat(v, n)A index(v, i')A refs-in(v, s')A refs-out(v, s")A singleton(i', w)A 
union(s', w, s") 
In general then, a system of feature structure constraints can be viewed as a function- 
free and quantifier-free formula. These constraints are satisfiable if and only if there is 
an interpretation that simultaneously satisfies the corresponding formula and the ax- 
ioms presented in the previous section, or equivalently, the conjunction of this formula 
and the relevant axioms from the axiomatization. This conjunction is itself a formula 
from the Sch6nfinkel-Bernays class, and so the satisfiability problem for systems of 
feature structure constraints is decidable. 
Further, we can apply results on the computational complexity of the satisfiability 
problem for the Sch6nfinkel-Bernays class to determine the computational complexity 
of the satisfiability problem for systems of such feature constraints. Since (universal) 
quantifiers appear only in the axiomatization of feature structures and not in the 
feature constraints themselves, the number of quantifiers appearing in the conjunction 
of the feature constraints and the axiomatization is a constant, and does not vary with 
the size of the system of feature constraints. By Proposition 3.2 of Lewis (1980), the 
satisfiability problem for a formula F with u universal quantifiers in the Sch6nfinkel- 
Bernays class requires nondeterministic time polynomial in IFI u, so the problem is in 
NP. The reductions presented in Kasper and Rounds (1986) and Johnson (1988) can 
be used to show that the problem is NP-hard, so the satisfiability problem for feature 
constraints with set-values (as defined above) is NP-complete. 
2.3 Unification and Satisfaction 
This section discusses the relationship between unification and the axiomatization 
presented above. 
Unification identifies or merges exactly the elements that the axiomatization im- 
plies are equal. The unification of two complex elements e and e' causes the unification 
of the values of all attributes a that are defined on both e and e'. Similarly, the con- 
junction of the formulae e = e', a(6 f), a(e', f') and the axioms given above implies that 
f = f, since axiom schema (3) requires that attributes are single valued. 
Similarly, the unification of two attribute-value structures fails either when two 
distinct constant elements are unified (a constant-constant clash) or when a constant 
and a complex element are unified. The formula x = x ~ is unsatisfiable under exactly 
the same circumstances in the theory axiomatized above. The formula x = x' conjoined 
with x = c and x' = c' for distinct attribute-value constants c, c' is unsatisfiable, since 
c ¢ c' by axiom schema (2). Also, x = x' is unsatisfiable when conjoined with a(x,y) 
for any y and x' = c, since ~a(c,y) by axiom schema (1). 
If attention is restricted to purely conjunctive attribute-value systems, the corre- 
sponding formulae can be represented as a directed graph, where nodes represent 
(first-order) constants, and an arc labeled a from x to y encodes the atom a(x,y). 
Then the standard attribute-value 'unification algorithm' can be used as a specialized 
inference procedure that takes as input such a graph encoding of a conjunction of 
144 
Johnson Features and Formulae 
attribute-value relations and returns (the graph encoding of) the conjunction of all of 
their atomic consequences. 
As Kasper (1986, 1987) noted in a different setting, the steps of the attribute-value 
unification algorithm are just applications of the axioms 1-3. It 'forward chains' using 
axiom schema (3) (for which the graph representation provides efficient indexing), 
and checks at each step that 1 and 2 are not falsified; if they are falsified the unifi- 
cation algorithm halts and reports a unification failure. Atomic equalities x = y are 
represented by a 'forwarding pointer' from x to y (as in the UNION-HND algorithm 
(Gallier 1986; Nelson and Oppen 1980; Johnson in press)). 
Example 2 (continued) 
The unification of e ~ and e" in Figures 2 and 3 corresponds to conjoining the formula 
e ~ = e" to the conjunction of 8a and 8b, resulting in the formula 15a. 
15a. e'= e"A pred(e', salmon) A agr(e', f') A pers(f', 3rd)A pred(g", swim)A 
tense(g", pres)A subj(g", e")A agr(e", f")A num~f", sg )A pers(~", 3rd). 
This formula can be simplified by substituting e r for e" to yield 15b (this substitution 
corresponds exactly to the first step of the unification algorithm, viz. redirecting e" 
to e0. The affected subformulae are in boldface below. 
15b. e' = e"A pred(e'~ salmon)A agr(e',f')A pers~f', 3rd)A pred(g"~ swim)A 
tense(g", pres)A subj(g"~ e')A agr(e'~f")A num(f", sg)A pers(f", 3rd). 
Since 15b contains the conjunction of agr(e',f') and agr(e',f"), axiom schema (3) re- 
quires that f' = f", so 15b can be further simplified by substituting f' for f" to 
yield 15c. 
15c. e' = e"A f' = f"A pred(e', salmon)A agr(e',f')A pers(f', 3rd)A 
pred(g", swim)A tense(g"~ pres)A subj(g", e')A agr(e'~ f')A num~f'~ sg)A 
pers(f'~ 3rd). 
The duplicate occurrences of agr(d,f') and pers(fr,3rd) can be deleted, yielding 15d 
(these last two steps correspond exactly to the unification of f~ and f" in Figure 3). 
15d. e' = e"A f'= f"A pred(e', salmon)A agr(e',f')A pers(f',3rd)A 
pred(g", swim )A tense(g", pres )A subj(g", e')A num(f', sg). 
No further simplifications are possible, and 15d is satisfiable. In fact 15d describes the 
structure depicted in Figure 3, as expected. 
The standard unification algorithm is unable to handle negative constraints correctly, 
as noted above. However, because negation is interpreted declaratively (in fact, clas- 
sically) in the first-order language used to express constraints here, its treatment is 
straightforward and unproblematic, and suggests ways of extending the unification 
algorithm to cover these cases (Johnson 1990b, to appear). 
145 
Computational Linguistics Volume 17, Number 2 
- pred = swim 
I pred = salmon 
subj = \[ agr = -! num= ~sg 1 
~'L 3fLpers= 3rd 
tense = pres 
Figure 10 
A graphical depiction of the formula 16b 
Example 4 (continued) 
The unification of e ~ and e (i.e. the lexical entries for salmon and swim) of Figure 9 
corresponds to the conjunction of the formula e = e I to the conjunction of 8a and 10, 
which is the formula 16a. 
16a. e = e'A pred(e', salmon)A agr(e',f)A pers(f', 3rd)A pred(g, swim)A 
tense(g, pres)A subj(g, e)A agr( G f)A -~(num(f , sg)A pers(f , 3rd) ) 
This can be simplified by straightforward applications of axiom schema (3), equality 
substitution, and propositional equivalences to obtain 16b. 
16b. e = e'A f = f'A pred(G salmon)A pers(f, 3rd)A pred(g, swim)A 
tense(g, pres)A subj(g, e)A agr(G f)A -~num(f , sg). 
This formula could be depicted as in Figure 10, where such matrices are now to be 
understood as graphical depictions of formulae. The further unification of e' with u, 
the lexical entry for this, corresponds to the conjunction of e ~ = u to the conjunction of 
the formulae 16b and 11, which is the formula 16c. 
16c. e = e'A f = f'A e' = uA pred(G salmon)A pers(f, 3rd)A pred(g, swim)A 
tense(g, pres)A subj(g, e)A agr(e, f)A -~num(f , sg)A def(u, +)A agr(u, v)A 
num(v, sg)A pers(v, 3rd). 
By substituting e for both e I and u in 16c, we obtain 16d. 
16d. e = e'A f = f'A e = uA pred(e, salmon)A pers(f, 3rd)A pred(g, swim)A 
tense(g, pres)A subj(g, e)A agr(e,f)A -~num(f, sg)A def(e~ +)A agr(e~ v)A 
num(v, sg)A pers(v, 3rd). 
Again, since 16d contains the conjunction of agr(e,f) and agr(G v), axiom schema (3) 
requires that f = v, so 16d can be further simplified by substituting f for v, yielding 16e. 
16e. e = e'A f = f'A e = uA f = vA pred(e, salmon)A pers(f, 3rd)A pred(g, swim)A tense(g, pres)A subj(g, 
e)A agr( G f)A -~num(~, sg)A 
def(G +)A agr(e,f)A num(f, sg)A pers(f,3rd). 
146 
Johnson Features and Formulae 
- pred = 
subj 
e, 
e, u 
tense = 
swim 
- pred = salmon 
def = + 
agr= Inum=pl 1 !,~ 
pers = 3rd 
)res 
Figure 11 
A graphical depiction of the formula 16f' 
The formula 16e is unsatisfiable, since it contains conjunction of both num(f, sg) and 
its negation -~num(f, sg). This is the desired result, since the utterance this salmon swim 
is ill formed. 
On the other hand, the unification e' in 16b (c.f. Figure 10) is with u', the lexical 
entry for these, corresponds to the conjunction of e' = u', 16b and 12, which is the 
formula 16c'. 
16c'. e = e'A f = f'A e' = u'A pred(e, salmon)A pers(f, 3rd)A pred(g, swim)A 
tense(g, pres)A subj(g, e)A agr(e, d)A -~numff , sg)A def(u', +)A agr(u', v')A 
num(v', pl)A pers(v', 3rd). 
By following the same steps as were used to simplify 16c to 16e, 16c' can be simplified 
to 16e'. 
16e'. e = e'A f =f'A e = u'A f = v'A pred(e, salmon)A pers(f, ard)A 
pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e, f)A -~nurn(f , sg)A 
def(e, +)A agr(e,f)A num(f , pl)A pers(f , 3rd). 
One of duplicate conjuncts agr(e,f) can be deleted, and since num(f, pl) implies 
-~num(f, sg) (by instances sg   pl of (2) and Vxyz num(x,y) A num(x,z) --+ y = z of 
(3)), 16e' can be further simplified to 16f', where -~num(f, sg) has also been deleted. 
16f'. e = e'A f = f'A e = u'A f = v'A pred(e, salmon)A pers(f, 3rd)A 
pred(g, swim)A tense(g, pres)A subj(g, e)A agr(e, f)A def(e, +)A num(f , pl)A 
pers(f, 3rd). 
This formula is satisfiable, as desired, since the utterance these salmon swim is well 
formed. This formula could be depicted as in Figure 11, where again the matrix is to 
be understood as a graphical depiction of the formula 16f'. 
The set-valued examples are somewhat more complicated because they involve quan- 
tification. 
147 
Computational Linguistics Volume 17, Number 2 
Example 5 (continued) 
The unification of s' with the empty set in Figure 6 corresponds to the conjunction of 
14 with the formula null(s'), as given in 17a. 
17a. cat(v, n)A index(v, i')A refs-in(v, s')A refs-out(v~ s")A singleton(i', w)A 
union(d, w, s")A null(s'). 
Now singleton(i', w) A union(d, w, s") implies by axioms (5) and (7) that Vu in(u~ s') *-* 
u = i' V in(u, s'). Further, since null(s') implies by axiom (4) that Vu-~in(u, s'), it follows 
that 17a is equivalent to 17b. 
17b. cat(v, n)A index(v, i')A refs-in(v, s')A refs-out(v, s')A singleton(i', w)A 
union(s',w,s")A null(s')A Vu (in(u,s") ~ u = i'). 
Unifying the value of the refs-out attribute of Figure 7 with the value of the refs-in 
attribute of u in Figure 6 corresponds to conjoining s = s" with the conjunction of 17b 
and 13, yielding 17c. 
17c. s = s"A cat(u, np)A refs-in(u,s)A refs-out(u,s)A index(u, i)A in(i,s)A 
cat(v, n)A index(v, i')A refs-in(v, s')A refs-out(v, s')A singleton( i', w)A 
union(s',w,s")A null(s')A Vu (in(u,s') ~ u = i'). 
This can be simplified by substituting s for s" and noting that Vu (in(u, s) ~ u = i') 
and in(i, s) implies that i = i', as required. 
17d. s = s'A cat(u, np)A refs-in(u, s)A refs-out(u, s)A index(u, i)A in(i, s)A 
cat(v, n)A index(v, i)A refs-in(v, s')A refs-out(v, s)A singleton(i, w)A 
union(d, w, s)A null(s')A Vu (in(u, s) ~ u = i). 
3. Conclusion 
The general approach adopted here of separating the feature structures and the con- 
straints that they must satisfy is used in most accounts of feature structures. The novel 
aspect of this work is that feature structures are axiomatized in and the constraints on 
feature structures are expressed in a decidable class of first-order logic, so important 
results such as decidability and compactness follow directly. The Sch6nfinkel-Bernays 
class of formulae used in this paper are sufficiently expressive so that "set-valued" 
features can be axiomatized quite directly. 
We conclude with some tentative remarks about the implementation of the system 
described here. Although a general-purpose first-order logic theorem prover could 
be used to determine the satisfiability of Sch6nfinkel-Bernays formulae, it should be 
possible to take advantage of the syntactic restrictions these formulae satisfy to obtain 
a more efficient implementation. One way in which this might be done is as follows. 
First, the axioms should be expressed in clausal form, i.e. in the form 
3xl ... xnV yl ... yn A1 A ... A Am --~ B1 V ... V Bn 
where the Ai and Bj are atoms. These can be used in a 'forward chaining' inference pro- 
cedure using 'semi-naive evaluation' (see Genesereth and Nilsson (1987) for details). 
148 
Johnson Features and Formulae 
For example, the clausal form expansion of axiom (5') for union is 
18a. V xyzu union(x,y,z) A in(u,z) ~ in(u,x) V in(u,y) 
18b. V xyzu union(x,y,z) A in(u,x) --* in(u,z) 
18c. V xyzu union(x,y,z) A in(u,y) ~ in(u,z). 
Second, if efficiency comparable to the standard (purely conjunctive) unification algo- 
rithm is to be achieved, it is necessary to efficiently index atoms on their arguments 
(both from the original constraints and those produced as consequences during the in- 
ference process just described). If we were dealing with only purely conjunctive formu- 
lae we could use a graph-based representation similar to the one used in the standard 
attribute-value unification algorithm, but since axioms such as 18a have disjunctive 
consequents we need a data structure that can represent nonconjunctive formulae, 
even if all of the linguistic constraints associated with lexical entries and syntactic 
rules are purely conjunctive. This problem is an instance of the general problem of 
disjunction, and it seems that some of the techniques proposed in the feature-structure 
literature to deal with disjunction (e.g. Eisele and D6rre 1988; D6rre and Eisele 1990; 
Maxwell and Kaplan 1989a, 1989b) can be applied here too. 
Acknowledgments 
I would like to thank Nick Asher, Jochen 
D6rre, Andreas Eisele, Martin Emele, 
Martin Kay, Ron Kaplan, Lauri Karttunen, 
Harry Lewis, John Maxwell, Bill Rounds, 
Gert Smolka, Stuart Shieber, Rich 
Thomason, Johan van Benthem, and J6rgen 
Wedekind as well as an anonymous 
reviewer and audiences at the CLIN Dag in 
Utrecht and at the DFKI in Saarbrficken for 
their important contributions to the material 
presented in this paper. The idea of 
translating feature constraints into a 
specialized language with desirable 
computational properties arose in 
conversation with J/irgen Wedekind. Harry 
Lewis noted that attribute-value structures 
could be axiomatized using formulae from 
the Sch6nfinkel-Bernays class, and guided 
me to the relevant results. Naturally, all 
errors remain my own. The final revision of 
this paper was completed at the Institut f/Jr 
maschinelle Sprachverarbeitung, Universitat 
Stuttgart, which I thank for providing a 
congenial research environment. 
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