A Model-Theoretic Framework for Theories of Syntax 
James Rogers 
Institute for Research in Cognitive Science 
University of Pennsylvania 
Suite 400C, 3401 Walnut Street 
Philadelphia, PA 19104 
j rogers©linc, cis. upenn, edu 
Abstract 
A natural next step in the evolution of 
constraint-based grammar formalisms from 
rewriting formalisms is to abstract fully 
away from the details of the grammar 
mechanism--to express syntactic theories 
purely in terms of the properties of the 
class of structures they license. By fo- 
cusing on the structural properties of lan- 
guages rather than on mechanisms for gen- 
erating or checking structures that exhibit 
those properties, this model-theoretic ap- 
proach can offer simpler and significantly 
clearer expression of theories and can po- 
tentially provide a uniform formalization, 
allowing disparate theories to be compared 
on the basis of those properties. We dis- 
cuss L2,p, a monadic second-order logical 
framework for such an approach to syn- 
tax that has the distinctive virtue of be- 
ing superficially expressive--supporting di- 
rect statement of most linguistically sig- 
nificant syntactic properties--but having 
well-defined strong generative capacity-- 
languages are definable in L2K,p iff they are 
strongly context-free. We draw examples 
from the realms of GPSG and GB. 
1 Introduction 
Generative grammar and formal language theory 
share a common origin in a procedural notion of 
grammars: the grammar formalism provides a gen- 
eral mechanism for recognizing or generating lan- 
guages while the grammar itself specializes that 
mechanism for a specific language. At least ini- 
tially there was hope that this relationship would 
be informative for linguistics, that by character- 
izing the natural languages in terms of language- 
theoretic complexity one would gain insight into the 
structural regularities of those languages. More- 
over, the fact that language-theoretic complexity 
classes have dual automata-theoretic characteriza- 
tions offered the prospect that such results might 
provide abstract models of the human language fac- 
ulty, thereby not just identifying these regularities, 
but actually accounting for them. 
Over time, the two disciplines have gradually be- 
come estranged, principally due to a realization that 
the structural properties of languages that charac- 
terize natural languages may well not be those that 
can be distinguished by existing language-theoretic 
complexity classes. Thus the insights offered by for- 
mal language theory might actually be misleading 
in guiding theories of syntax. As a result, the em- 
phasis in generative grammar has turned from for- 
malisms with restricted generative capacity to those 
that support more natural expression of the observed 
regularities of languages. While a variety of dis- 
tinct approaches have developed, most of them can 
be characterized as constrain~ based--the formalism 
(or formal framework) provides a class of structures 
and a means of precisely stating constraints on their 
form, the linguistic theory is then expressed as a sys- 
tem of constraints (or principles) that characterize 
the class of well-formed analyses of the strings in the 
language. 1 
As the study of the formal properties of classes of 
structures defined in such a way falls within domain 
of Model Theory, it's not surprising that treatments 
of the meaning of these systems of constraints are 
typically couched in terms of formal logic (Kasper 
and Rounds, 1986; Moshier and Rounds, 1987; 
Kasper and Rounds, 1990; Gazdar et al., 1988; John- 
son, 1988; Smolka, 1989; Dawar and Vijay-Shanker, 
1990; Carpenter, 1992; Keller, 1993; Rogers and 
Vijay-Shanker, 1994). 
While this provides a model-theoretic interpre- 
tation of the systems of constraints produced 
by these formalisms, those systems are typi- 
cally built by derivational processes that employ 
extra-logical mechanisms to combine constraints. 
More recently, it has become clear that in many 
cases these mechanisms can be replaced with or- 
dinary logical operations. (See, for instance: 
1This notion of constraint-based includes not only the 
obvious formalisms, but the formal framework of GB as 
well. 
10 
Johnson (1989), Stabler, Jr. (1992), Cornell (1992), 
Blackburn, Gardent, and Meyer-Viol (1993), 
Blackburn and Meyer-Viol (1994), Keller (1993), 
Rogers (1994), Kracht (1995), and, anticipating all 
of these, Johnson and Postal (1980).) This ap- 
proach abandons the notions of grammar mecha- 
nism and derivation in favor of defining languages as 
classes of more or less ordinary mathematical struc- 
tures axiomatized by sets of more or less ordinary 
logical formulae. A grammatical theory expressed 
within such a framework is just the set of logical con- 
sequences of those axioms. This step completes the 
detachment of generative grammar from its proce- 
dural roots. Grammars, in this approach, are purely 
declarative definitions of a class of structures, com- 
pletely independent of mechanisms to generate or 
check them. While it is unlikely that every theory 
of syntax with an explicit derivational component 
can be captured in this way, ~ for those that can the 
logical re-interpretation frequently offers a simpli- 
fied statement of the theory and clarifies its conse- 
quences. 
But the accompanying loss of language-theoretic 
complexity results is unfortunate. While such results 
may not be useful in guiding syntactic theory, they 
are not irrelevant. The nature of language-theoretic 
complexity hierarchies is to classify languages on the 
basis of their structural properties. The languages 
in a class, for instance, will typically exhibit cer- 
tain closure properties (e.g., pumping lemmas) and 
the classes themselves admit normal forms (e.g., rep- 
resentation theorems). While the linguistic signifi- 
cance of individual results of this sort is open to de- 
bate, they at least loosely parallel typical linguistic 
concerns: closure properties state regularities that 
are exhibited by the languages in a class, normal 
forms express generalizations about their structure. 
So while these may not be the right results, they 
are not entirely the wrong kind of results. More- 
over, since these classifications are based on struc- 
tural properties and the structural properties of nat- 
ural language can be studied more or less directly, 
there is a reasonable expectation of finding empiri- 
cal evidence falsifying a hypothesis about language- 
theoretic complexity of natural languages if such ev- 
idence exists. 
Finally, the fact that these complexity classes have 
automata-theoretic characterizations means that re- 
sults concerning the complexity of natural languages 
will have implications for the nature of the human 
language faculty. These automata-theoretic charac- 
terizations determine, along one axis, the types of 
resources required to generate or recognize the lan- 
2Whether there are theories that cannot be captured, 
at least without explicitly encoding the derivations, is 
an open question of considerable theoretical interest, as 
is the question of what empirical consequences such an 
essential dynamic character might have. 
11 
guages in a class. The regular languages, for in- 
stance, can be characterized by finite-state (string) 
automata--these languages can be processed using 
a fixed amount of memory. The context-sensitive 
languages, on the other had, can be characterized 
by linear-bounded automata--they can be processed 
using an amount of memory proportional to the 
length of the input. The context-free languages 
are probably best characterized by finite-state tree 
automata--these correspond to recognition by a col- 
lection of processes, each with a fixed amount of 
memory, where the number of processes is linear in 
the length of the input and all communication be- 
tween processes is completed at the time they are 
spawned. As a result, while these results do not 
necessarily offer abstract models of the human lan- 
guage faculty (since the complexity results do not 
claim to characterize the human languages, just to 
classify them), they do offer lower bounds on cer- 
tain abstract properties of that faculty. In this way, 
generative grammar in concert with formal language 
theory offers insight into a deep aspect of human 
cognition--syntactic processing--on the basis of ob- 
servable behavior--the structural properties of hu- 
man languages. 
In this paper we discuss an approach to defining 
theories of syntax based on L 2 (Rogers, 1994), a K,P 
monadic second-order language that has well-defined 
generative capacity: sets of finite trees are defin- 
able within L 2 iff they are strongly context-free K,P 
in a particular sense. While originally introduced 
as a means of establishing language-theoretic com- 
plexity results for constraint-based theories, this lan- 
guage has much to recommend it as a general frame- 
work for theories of syntax in its own right. Be- 
ing a monadic second-order language it can capture 
the (pure) modal languages of much of the exist- 
ing model-theoretic syntax literature directly; hav- 
ing a signature based on the traditional linguistic 
relations of domination, immediate domination, lin- 
ear precedence, etc. it can express most linguistic 
principles transparently; and having a clear charac- 
terization in terms of generative capacity, it serves 
to re-establish the close connection between genera- 
tive grammar and formal language theory that was 
lost in the move away from phrase-structure gram- 
mars. Thus, with this framework we get both the 
advantages of the model-theoretic approach with re- 
spect to naturalness and clarity in expressing linguis- 
tic principles and the advantages of the grammar- 
based approach with respect to language-theoretic 
complexity results. 
We look, in particular, at the definitions of a single 
aspect of each of GPSG and GB. The first of these, 
Feature Specification Defaults in GPSG, are widely 
assumed to have an inherently dynamic character. 
In addition to being purely declarative, our reformal- 
ization is considerably simplified wrt the definition 
in Gasdar et al. (1985), 3 and does not share its mis- 
leading dynamic flavor. 4 We offer this as an example 
of how re-interpretations of this sort can inform the 
original theory. In the second example we sketch a 
definition of chains in GB. This, again, captures a 
presumably dynamic aspect of the original theory in 
a static way. Here, though, the main significance of 
the definition is that it forms a component of a full- 
scale treatment of a GB theory of English S- and 
D-Structure within L 2 This full definition estab- K,P" 
lishes that the theory we capture licenses a strongly 
context-free language. More importantly, by exam- 
ining the limitations of this definition of chains, and 
in particular the way it fails for examples of non- 
context-free constructions, we develop a character- 
ization of the context-free languages that is quite 
natural in the realm of GB. This suggests that the 
apparent mismatch between formal language theory 
and natural languages may well have more to do with 
the unnaturalness of the traditional diagnostics than 
a lack of relevance of the underlying structural prop- 
erties. 
Finally, while GB and GPSG are fundamentally 
distinct, even antagonistic, approaches to syntax, 
their translation into the model-theoretic terms of 
L 2 allows us to explore the similarities between K,P 
the theories they express as well as to delineate ac- 
tual distinctions between them. We look briefly at 
two of these issues. 
Together these examples are chosen to illustrate 
the main strengths of the model-theoretic approach, 
at least as embodied in L2K,p, as a framework for 
studying theories of syntax: a focus on structural 
properties themselves, rather than on mechanisms 
for specifying them or for generating or checking 
structures that exhibit them, and a language that 
is expressive enough to state most linguistically sig- 
nificant properties in a natural way, but which is 
restricted enough to have well-defined strong gener- 
ative capacity. 
2 L~,p--The Monadic Second-Order 
Language of Trees 
L2K,p is the monadic second-order language over 
the signature including a set of individual constants 
(K), a set of monadic predicates (P), and binary 
predicates for immediate domination (,~), domina- 
tion (,~*), linear precedence (-~) and equality (..~). 
The predicates in P can be understood both as 
picking out particular subsets of the tree and as 
(non-exclusive) labels or features decorating the 
tree. Models for the language are labeled tree do- 
3We will refer to Gazdar et al. (1985) as GKP&S 
4We should note that the definition of FSDs in 
GKP&S is, in fact, declarative although this is obscured 
by the fact that it is couched in terms of an algorithm 
for checking models. 
mains (Gorn, 1967) with the natural interpretation 
of the binary predicates. In Rogers (1994) we have 
shown that this language is equivalent in descrip- 
tive power to SwS--the monadic second-order the- 
ory of the complete infinitely branching tree--in the 
sense that sets of trees are definable in SwS iff they 
are definable in L 2 This places it within a hi- K,P" 
erarchy of results relating language-theoretic com- 
plexity classes to the descriptive complexity of their 
models: the sets of strings definable in S1S are ex- 
actly the regular sets (Biichi, 1960), the sets of fi- 
nite trees definable in SnS, for finite n, are the rec- 
ognizable sets (roughly the sets of derivation trees 
of CFGs) (Doner, 1970), and, it can be shown, the 
sets of finite trees definable in SwS are those gener- 
ated by generalized CFGs in which regular ,expres- 
sions may occur on the rhs of rewrite rules (Rogers, 
1996b). 5 Consequently, languages are definable in 
L2K,p iff they are strongly context-free in the mildly 
generalized sense of GPSG grammars. 
In restricting ourselves to the language of L 2 K,P 
we are restricting ourselves to reasoning in terms of 
just the predicates of its signature. We can expand 
this by defining new predicates, even higher-order 
predicates that express, for instance, properties of 
or relations between sets, and in doing so we can use 
monadic predicates and individual constants freely 
since we can interpret these as existentially bound 
variables. But the fundamental restriction of L 2 K,P 
is that all predicates other than monadic first-order 
predicates must be explicitly defined, that is, their 
definitions must resolve, via syntactic substitution, 
2 into formulae involving only the signature of LK, P. 
3 Feature Specification Defaults in 
GPSG 
We now turn to our first application--the def- 
inition of Feature Specification Defaults (FSDs) 
in GPSG. 6 Since GPSG is presumed to license 
(roughly) context-free languages, we are not con- 
cerned here with establishing language-theoretic 
complexity but rather with clarifying the linguis- 
tic theory expressed by GPSG. FSDs specify con- 
ditions on feature values that must hold at a node 
in a licensed tree unless they are overridden by some 
other component of the grammar; in particular, un- 
less they are incompatible with either a feature spec- 
ified by the ID rule licensing the node (inherited fea- 
tures) or a feature required by one of the agreement 
principles--the Foot Feature Principle (FFP), Head 
Feature Convention (HFC), or Control Agreement 
Principle (CAP). It is the fact that the default holds 
5There is reason to believe that this hierarchy can 
be extended to encompass, at least, a variety of mildly 
context-sensitive languages as well. 
6A more complete treatment of GPSG in L 2 I¢.,P can 
be found in Rogers (1996c). 
12 
just in case it is incompatible with these other com- 
ponents that gives FSDs their dynamic flavor. Note, 
though, in contrast to typical applications of default 
logics, a GPSG grammar is not an evolving theory. 
The exceptions to the defaults are fully determined 
when the grammar is written. If we ignore for the 
moment the effect of the agreement principles, the 
defaults are roughly the converse of the ID rules: a 
non-default feature occurs iff it is licensed by an ID 
rule. 
It is easy to capture ID rules in L 2 For instance K,P" 
the rule: 
VP , HI5\], NP, NP 
can be expressed: 
IDh(x, yl, Y2, Y3) -= 
Children(x, Yl, Y2, Y3) A VP(x)A 
H(yl) A (SUBCAT, 5)(Yl) A NP(y2) A NP(y3), 
where Children(z, Yl, Y~, Y3) holds iff the set of nodes 
that are children of x are just the Yi and VP, 
(SUBCAT, 5), etc. are all members of p.7 A se- 
quence of nodes will satisfy ID5 iff they form a local 
tree that, in the terminology of GKP&S, is induced 
by the corresponding ID rule. Using such encodings 
we can define a predicate Free/(x) which is true at 
a node x iff the feature f is compatible with the 
inherited features of x. 
The agreement principles require pairs of nodes 
occurring in certain configurations in local trees to 
agree on certain classes of features. Thus these prin- 
ciples do not introduce features into the trees, but 
rather propagate features from one node to another, 
possibly in many steps. Consequently, these prin- 
ciples cannot override FSDs by themselves; rather 
every violation of a default must be licensed by an 
inherited feature somewhere in the tree. In order 
to account for this propagation of features, the def- 
inition of FSDs in GKP&S is based on identifying 
pairs of nodes that co-vary wrt the relevant features 
in all possible extensions of the given tree. As a re- 
suit, although the treatment in GKP&S is actually 
declarative, this fact is far from obvious. 
Again, it is not difficult to define the configura- 
tions of local trees in which nodes are required to 
agree by FFP, CAP, or HFC in L 2 Let the predi- K,P" 
cate Propagatey(z, y) hold for a pair of nodes z and 
y iff they are required to agree on f by one of these 
principles (and are, thus, in the same local tree). 
Note that Propagate is symmetric. Following the 
terminology of GKP&S, we can identify the set of 
nodes that are prohibited from taking feature f by 
the combination of the ID rules, FFP, CAP, and 
HFC as the set of nodes that are privileged wrt f. 
This includes all nodes that are not Free for f as well 
7We will not elaborate here on the encoding of cat- 
egories in L 2 K,P, nor on non-finite ID schema like the 
iterating co-ordination schema. These present no signif- 
icant problems. 
as any node connected to such a node by a sequence 
of Propagate/ links. We, in essence, define this in- 
ductively. P' (X) is true of a set iff it includes all \] 
nodes not Free for f and is closed wrt Propagate/. 
PrivSet\] (X) is true of the smallest such set. 
P; (x) - 
(Vx)\[- Frees (x) X(x)\] ^ 
(Vx)\[(3y)\[X(y) A Propagate\] (x, y)\] ---* X(x)\] 
PrivSetl(X) = P)(X) A 
(VY)\[P) (Y) --~ Subset(X, Y)\]. 
There are two things to note about this definition. 
First, in any tree there is a unique set satisfying 
PrivSet/(X) and this contains exactly those nodes 
not Free for f or connected to such a node by 
Propagate\]. Second, while this is a first-order in- 
ductive property, the definition is a second-order ex- 
plicit definition. In fact, the second-order quantifi- 
cation of L 2 allows us to capture any monadic K,P 
first-order inductively or implicitly definable prop- 
erty explicitly. 
Armed with this definition, we can identify indi- 
viduals that are privileged wrt f simply as the mem- 
bers of PrivSetl.s 
Privileged\] (x) = (3X)\[PrivSety (X) A X(z)\]. 
One can define Privileged_,/(x) which holds when- 
ever x is required to take the feature f along similar 
lines. 
These, then, let us capture FSDs. For the default 
\[-INV\], for instance, we get: 
(¥x)\[-~Privileged\[_ INV\](X) ""+ \[-- INV\](x)\]. 
For \[BAR0\] D,,~ \[PAS\] (which says that \[Bar 0\] 
nodes are, by default, not marked passive), we get: 
(Vz)\[ (\[BAR 0\](x) A ~Privileged_,\[pAs\](X)) 
-~\[PAS\](x)\]. 
The key thing to note about this treatment of 
FSDs is its simplicity relative to the treatment of 
GKP&S. The second-order quantification allows us 
to reason directly in terms of the sequence of nodes 
extending from the privileged node to the local tree 
that actually licenses the privilege. The immediate 
benefit is the fact that it is clear that the property of 
satisfying a set of FSDs is a static property of labeled 
trees and does not depend on the particular strategy 
employed in checking the tree for compliance. 
SWe could, of course, skip the definition of PrivSet/ 
and define Privilegedy(x) as (VX)\[P'(X) ---* Z(x)\], but 
we prefer to emphasize the inductive nature of the 
definition. 
13 
4 Chains in GB 
The key issue in capturing GB theories within L 2 K,P 
is the fact that the mechanism of free-indexation is 
provably non-definable. Thus definitions of prin- 
ciples that necessarily employ free-indexation have 
no direct interpretation in L 2 (hardly surprising, K,P 
as we expect GB to be capable of expressing non- 
context-free languages). In many cases, though, ref- 
erences to indices can be eliminated in favor of the 
underlying structural relationships they express. 9 
The most prominent example is the definition of 
the chains formed by move-a. The fundamental 
problem here is identifying each trace with its an- 
tecedent without referencing their index. Accounts 
of the licensing of traces that, in many cases of 
movement, replace co-indexation with government 
relations have been offered by both Rizzi (1990) 
and Manzini (1992). The key element of these ac- 
counts, from our point of view, is that the antecedent 
of a trace must be the closest antecedent-governor of 
the appropriate type. These relationships are easy 
to capture in L 2 For A-movement, for instance, K,P" 
we have: 
A-Antecedent-Governs(x, y) 
-~A-pos(x) A C-Commands(x, y) A F.Eq(x, y) A 
--x is a potential antecedent in an 
A-position 
-~(3z)\[Intervening-Barrier(z, x, y)\] A 
--no barrier intervenes 
-~(Bz)\[Spec(z) A-~A-pos(z) A 
C-Commands(z, x) A Intervenes(z, x, y)\] 
--minimality is respected 
where F.Eq(x, y) is a conjunction of biconditionals 
that assures that x and y agree on the appropriate 
features and the other predicates are are standard 
GB notions that are definable in L 2 K,P" 
Antecedent-government, in Rizzi's and Manzini's 
accounts, is the key relationship between adjacent 
members of chains which are identified by non- 
referential indices, but plays no role in the definition 
of chains which are assigned a referential index3 ° 
Manzini argues, however, that referential chains can- 
not overlap, and thus we will never need to distin- 
guish multiple referential chains in any single con- 
text. Since we can interpret any bounded number of 
indices simply as distinct labels, there is no difficulty 
in identifying the members of referential chains in 
L 2 On these and similar grounds we can extend K,P" 
these accounts to identify adjacent members of ref- 
erential chains, and, at least in the case of English, 
9More detailed expositions of the interpretation 
2 of GB in LK,p can be found in Rogers (1996a), 
Rogers (1995), and Rogers (1994). 
1°This accounts for subject/object asymmetries. 
of chains of head movement and of rightward move- 
ment. This gives us five mutually exclusive relations 
which we can combine into a single link relation that 
must hold between every trace and its antecedent: 
Link(x,y) - A-Link(z, y) V A-Ref-Link(x, y) V 
A---Ref-Link(x, y) V X°-Link(x, y) V 
Right-Link(x, y). 
The idea now is to define chains as sequences of 
nodes that are linearly ordered by Link, but before 
we can do this there is still one issue to resolve. 
While minimality ensures that every trace must have 
a unique antecedent, we may yet admit a single an- 
tecedent that licenses multiple traces. To rule out 
this possibility, we require chains to be closed wrt 
the link relation, i.e., every chain must include every 
node that is related by Link to any node already in 
the chain. Our definition, then, is in essence the def- 
inition, in GB terms, of a discrete linear order with 
endpoints, augmented with this closure property. 
Chain(X) -- 
(3!x)\[X(x) A Target(x)\] A 
--X contains exactly one Target 
(3!x)\[X(x) A Base(x)\] A 
--and one Base 
(Vx)\[X(x) A -~Warget(x) ---* 
(3!y)\[Z(y) A Link(y,x)\]\] A 
--All non-Target have a unique an- 
tecedent in X 
(Vx)\[X(x) A-~Base(x) --~ 
(3!y)\[X(y) A Link(x, y)\]\] A 
--All non-Base have a unique suc- 
cessor in X 
(Vx, y)\[X(x) A (Link(x, y) V Link(y, x)) ---* 
X(y)\] 
--X is closed wrt the Link relation 
Note that every node will be a member of exactly 
one (possibly trivial) chain. 
The requirement that chains be closed wrt Link 
means that chains cannot overlap unless they are of 
distinct types. This definition works for English be- 
cause it is possible, in English, to resolve chains into 
boundedly many types in such a way that no two 
chains of the same type ever overlap. In fact, it fails 
only in cases, like head-raising in Dutch, where there 
are potentially unboundedly many chains that may 
overlap a single point in the tree. Thus, this gives us 
a property separating GB theories of movement that 
license strongly context-free languages from those 
that potentially don't--if we can establish a fixed 
bound on the number of chains that can overlap, 
then the definition we sketch here will suffice to 
capture the theory in L 2 and, consequently, the K,P 
theory licenses only strongly context-free languages. 
14 
This is a reasonably natural diagnostic for context- 
freeness in GB and is close to common intuitions 
of what is difficult about head-raising constructions; 
it gives those intuitions theoretical substance and 
provides a reasonably clear strategy for establishing 
context-freeness. 
this distinction is; one particularly interesting ques- 
tion is whether it has empirical consequences. It is 
only from the model-theoretic perspective that the 
question even arises. 
6 Conclusion 
5 A Comparison and a Contrast 
Having interpretations both of GPSG and of a 
GB account of English in L 2 provides a certain K,P 
amount of insight into the distinctions between these 
approaches. For example, while the explanations of 
filler-gap relationships in GB and GPSG are quite 
dramatically dissimilar, when one focuses on the 
structures these accounts license one finds some sur- 
prising parallels. In the light of our interpretation of 
antecedent-government, one can understand the role 
of minimality in l~izzi's and Manzini's accounts as 
eliminating ambiguity from the sequence of relations 
connecting the gap with its filler. In GPSG this con- 
nection is made by the sequence of agreement rela- 
tionships dictated by the Foot Feature Principle. So 
while both theories accomplish agreement between 
filler and gap through marking a sequence of ele- 
ments falling between them, the GB account marks 
as few as possible while the GPSG account marks 
every node bf the spine of the tree spanning them. 
In both cases, the complexity of the set of licensed 
structures can be limited to be strongly context-free 
iff the number of relationships that must be distin- 
guished in a given context can be bounded. 
One finds a strong contrast, on the other hand, in 
the way in which GB and GPSG encode language 
universals. In GB it is presumed that all princi- 
ples are universal with the theory being specialized 
to specific languages by a small set of finitely vary- 
ing parameters. These principles are simply prop- 
erties of trees. In terms of models, one can un- 
derstand GB to define a universal language--the 
set of all analyses that can occur in human lan- 
guages. The principles then distinguish particular 
sub-languages--the head-final or the pro-drop lan- 
guages, for instance. Each realized human language 
is just the intersection of the languages selected by 
the settings of its parameters. In GPSG, in contrast, 
many universals are, in essence, closure properties 
that must be exhibited by human languages--if the 
language includes trees in which a particular config- 
uration occurs then it includes variants of those trees 
in which certain related configurations occur. Both 
the ECPO principle and the metarules can be under- 
stood in this way. Thus while universals in GB are 
properties of trees, in GPSG they tend to be proper- 
ties of sets of trees. This makes a significant differ- 
ence in capturing these theories model-theoretically; 
in the GB case one is defining sets of models, in the 
GPSG case one is defining sets of sets of models. It 
is not at all clear what the linguistic significance of 
We have illustrated a general formal framework for 
expressing theories of syntax based on axiomatiz- 
ing classes of models in L 2 This approach has a K,P* 
number of strengths. First, as should be clear from 
our brief explorations of aspects of GPSG and GB~ 
re-formalizations of existing theories within L 2 K,P 
can offer a clarifying perspective on those theories, 
and, in particular, on the consequences of individ- 
ual components of those theories. Secondly, the 
framework is purely declarative and focuses on those 
aspects of language that are more or less directly 
observable--their structural properties. It allows us 
to reason about the consequences of a theory with- 
out hypothesizing a specific mechanism implement- 
ing it. The abstract properties of the mechanisms 
that might implement those theories, however, are 
not beyond our reach. The key virtue of descrip- 
tive complexity results like the characterizations of 
language-theoretic complexity classes discussed here 
and the more typical characterizations of computa- 
tional complexity classes (Gurevich, 1988; Immer- 
man, 1989) is that they allow us to determine the 
complexity of checking properties independently of 
how that checking is implemented. Thus we can use 
such descriptive complexity results to draw conclu- 
sions about those abstract properties of such mech- 
anisms that are actually inferable from their observ- 
able behavior. Finally, by providing a uniform repre- 
sentation for a variety of linguistic theories, it offers 
a framework for comparing their consequences. Ul- 
timately it has the potential to reduce distinctions 
between the mechanisms underlying those theories 
to distinctions between the properties of the sets of 
structures they license. In this way one might hope 
to illuminate the empirical consequences of these dis- 
tinctions, should any, in fact, exist. 

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