Localising Barriers Theory 
Michael Schiehlen* 
Institute for Computational Linguistics, University of Stuttgart, 
Azenbergstr. 12, W-7000 Stuttgart 1 
E-mail: mike@adler.ims.uni-stuttgart.de 
1 Introduction 
Government-Binding Parsing has become attractive 
in the last few years. A variety of systems have been 
designed in view of a correspondence as direct as pos- 
sible with linguistic theory (\[Johnson, 1989\], \[Pollard 
and Sag, 1991\], \[Kroch, 1989\]). These approaches 
can be classified by their method of handling global 
constraints. Global constraints are syntactic in na- 
ture: They cover more than one projection. In con- 
trast, local constraints can be checked inside a pro- 
jection and, thus, lend themselves to a treatment in 
the lexicon. Conditions on features have been the 
subject of intensive study and viable logics have 
been proposed for them (see e.g. the CUF formalism 
\[Dhrre and Eisele, 1991\], \[Dorna, 1992\]). In this pa- 
per, we assume such a unification-based mechanism 
to take care of local conditions and focus on global 
constraints. One class of approaches to principle- 
based parsing (see \[Pollard and Sag, 1991\] for HPSG, 
\[Kroch, 1989\] for TAG) attempts to reduce global 
conditions to local constraints and thus to make 
them accessible to treatment in a feature framework. 
This strategy has been pursued only at the expense 
of sacrificing the precise formulation of the theory 
and the definitory power stemming from it. The re- 
sult has been a shift from the structural perspec- 
tive assumed by GB theory to the object-oriented 
view taken by unification formalisms. The other class 
of approaches (\[Johnson, 1989\]) has allowed the full 
range of possible restrictions on trees and has in- 
curred potential undecidability for its parsers. We 
take up a middle stance on the matter in that we 
propose a separate logic for global constraints and 
posit that global constraints only work on ancestor 
lines (see 7). 
We assume "movement" to be encoded by the kind of 
gap-threading technique familiar from HPSG, LFG. 
In order to integrate global constraints a "state" (in- 
formation that serves to express barrier configura- 
tions in the part of the tree which has already been 
built up) is associated with each "chain" (informa- 
tion about a moved element). Following H PSG, LFG, 
we have in mind a rule-based parser. Thus, states are 
manipulated when rules are chained. We need a cal- 
culus that is able to derive global constraints working 
on a local basis. We begin by developing this calculus 
hand in hand with an analysis of Chomsky's frame- 
*I wish to thank Robin Cooper, Mark Johnson and 
Esther KSnig-Baumer for comments on earlier versions 
of this paper. 
work. We then go on to show that many approaches 
to barriers theory and a variety of diverse phenom- 
ena can be moulded into our format and conclude 
with an indication of ways to use the system on-line 
during parsing. 
2 Dependencies Between Nodes 
We take a tree T to be a structure (N,>), where 
N is a set of nodes and > stands for dominance, a 
binary relation on N. We say that nodes a and b 
are connected iff a > b V b > a V a = b. We define 
the relation of immediate dominance ~- between two 
nodes a and b as a > b A ~3c : a > c A c > b. Dominance 
is an irreflexive partial order relation satisfying the 
axioms (1--3). Ancestors of a node are connected (1), 
there exists a (single) root (2), dominance reduces to 
immediate dominance (3). Variables are universally 
quantified unless specified otherwise. 
(1) z>z A y>z --* x connected with y 
(2) ~xVy : x > y 
(3) x>z ~ 3y : x~y A y>z 
Chomsky \[1986, 9,30\] discusses several definitions for 
constraints on unbounded dependencies. 
(13) a c-commands/~ iff a does not domi- 
nate/~ \[and/~ does not dominate or equal a\] 
and every 7 that dominates a dominates/~. 
Where 7 is restricted to maximal projec- 
tions we will say that a m-commands/?. 
(18) a governs/~ iff a m-commands/~ and 
there is no 7, 7 a harrier for/~/, such that 7 
excludes a. 
(59)/~ is n-subjacent to a iff there are fewer 
than n+l barriers for/~ that exclude a. 
All of these can be moulded into the general format 
introduced in (4): Two nodes can only stand in a re- 
lation R if they are unconnected and, furthermore, at 
most n barriers for the second node do not dominate 
the first one. The notion of a barrier B remains to 
be specified. For now, we only demand that barrier- 
hood entail dominance. We call relations that satisfy 
axiom (4) definable with barrier concepts, for short 
BC-definable. 
443 
(4) aRb ~-* a, b unconnected ^ 
I{c I B(c,b) ^ -,e>a}l < n 
Balanced relations like government require a defini- 
tion in terms of two BC-definable relations: Rl(a, b) 
and R2(b, a). 
(5) B(c,b) ~c>b 
We can show several properties of BC-definable re- 
lations. The nodes are unconnected. 
(6) aRb ---* a, b unconnected 
In order to investigate BC-definable relations it suf- 
fices to investigate the ancestor lines of their second 
argument b (that is {y J y >_ b}). 
(7) x~-y A z>al A ",y>__al A x>a2 A -w>_a~ 
A y>b --* (alRb ~ a2Rb) 
(7) gives rise to equivalence classes for the first argu- 
ment of R. For a particular pair (a,b) we can always 
find a y as defined in (8). 
(s) a• ^ x>a ^  y>a ^ y>b 
Definable relations are never empty. Barriers are pre- 
served in the upward direction of the ancestor line: 
(9) \[y\]Ry 
(10) x>y ^  \[xlP  
(10) is less innocent than it looks. I give a revealing 
binding example from Kamp and Reyle \[1993\]. 
If \[cP=~ \[cP=y hei sees Mary \] and she 
smiles\] John/ is happy. 
*\[cP=~ \[vP=~ Hei sees Mary \] and John/is 
happy\]. 
3 Barrier Definitions 
3.1 Adjunction 
Adjunction rules raise a problem for algebraic in- 
vestigations of barriers theory (e.g. \[Kracht, 1992\]): 
They insert material into a tree but do not cre- 
ate new projections. Thus, adjunction rules imply 
a distinction between projections and segment nodes 
that correspond to graph-theoretical nodes. We shall 
use Greek letters to refer to projection nodes and 
Latin letters for segment nodes. The only way to 
create projections covering more than one segment 
is through adjunction. Since adjunction rules have 
equivalent mother and daughter nodes, projections 
are coherent in the sense that: 
Va ~ fl Vbi, b2 • f~ : a > bi --* a > b2 
Chomsky \[1986\] defines projection dominance so that 
dominates ~ only if every segment of a domi- 
nates (every segment of) f/. In case this definition 
is not empty, (1) guarantees a unique minimal seg- 
ment a,~in of a. Thus, we can rephrase Chomsky's 
definition in terms of segment nodes and get that a 
dominates fl just in case the minimal segment of a 
dominates some segment of 3. 
(11) dominate(a,/3) *-+ a e a A b •/3 A 
minimal segment(a) A a > b 
Likewise, Chomsky's definition of exclusion, viz that 
a excludes j3 if no segment of a dominates (any seg- 
ment of) /3, can be transformed to the equivalent 
condition that a excludes/3 if the maximal segment 
of a does not dominate a segment of 3. 
(12) exclude(aft) ~ a E a A b e fl A 
maximal segment(a) A --a > b 
This way, we reduce projection dominance to seg- 
ment dominance. In (13--15), conditions of segment 
minimality or maximality are included where they 
are appropriate by (11) and (12). 
3.2 Chomsky's Theory 
Chomsky \[1986, 14\] gives the following two core def- 
initions for barriers. We are not concerned about the 
exact formulation of L-marking (for a definition see 
\[Chomsky, 1986, 24\]). 
(25) 7 is a blocking category for fl iff 
7 is not L-marked and 7 dominates/3. 
(26) 7 is a barrier for ~ iff (a) or (b): 
a. 7 immediately dominates 6, 
a blocking category for 3; 
b. 7 is a blocking category for 3, 7 ~ IP. 
We understand 7 in (25) and (26) to be 
a maximal projection, and we understand 
"immediately dominate" in (26a) to be a 
relation between maximal projections (so 
that 7 immediately dominates 5 in this 
sense even if a nonmaximal projection in- 
tervenes). 
Formulation of these definitions in first order logic 
yields (13--15). In order to obtain an open-ended 
definition scheme the equivalence of the above defi- 
nitions is held implicit: Barrier concepts are true iff 
they comply with a manifest definition (see also 22 
and 23). 
(13) blocking category(c,b) ¢:: 
maximal projection(c) A 
444 
-, L-marked(c) A 
minimal segment(c) A 
c>b. 
(14) barrier(c,b) 
maximal projection(c) A 
minimal segment(c) A 
3d : blocking category(d,b) A 
c>dA 
Ve:c>e>d--+ 
-, ( maximal projection(e) A 
minimal segment(e) ). 
(15) barrier(c,b) ¢= 
blocking category(c,b) A 
-,IP(c). 
We regard unary predicates as local conditions (L) 
and binary predicates as global concepts (B for "bar- 
rier concept"). Abstracting over the particular predi- 
cates involved we end up with the following definition 
schemes (16 for 13 and 15, 17 for 14). 
(16) B(c, b) ¢= 
L(e) A 
c>b. 
(17) S(e, b) 
L(e) A 
3d : B(d, b) A 
e>dA 
Ve : e>e >d ~ ",L(e). 
We call the existential subformula of (17) an inher- 
itance clause I. The only global conditions in our 
system are inheritance clauses and c> b, a condition 
that always holds for barrier concepts (see 5). We will 
discuss in detail a way to derive inheritance clauses 
on a rule to rule basis. For the sake of conciseness 
we adopt the following abbreviation for inheritance 
clauses. 
35 : B(d, b) A e > d A Ve : c > e > d --* -,L(e) 
,: y 
I(c,b,B,L) 
3.3 Negative Inheritance Clauses 
It has interesting repercussions to incorporate a 
scheme with a negated inheritance clause, viz. (18). 
(18) B(e, b) 
L(c) A 
c>bA 
-,3d : B(d, b) A 
c>dA 
Ve : c>e>d-* -,L(e). 
For illustration we discuss several applications for 
negative inheritance clauses. 
Chomsky \[1986, 37\] talks about IPs as inherent bar- 
riers, this effect being restricted to the most deeply 
embedded tensed IP. To capture this concept we once 
again need a negative inheritance clause: An IP is 
most deeply embedded if it does not dominate any 
other IP. 
(20) barrier(Tfl) ¢= 
tensed IP(7) A 
7>8A 
--,36 : IP(6,8) A 
7>6. 
IP(7,3) ~ IP(7) A 7>8. 
A feature of negative inheritance clauses that is de- 
sirable in many cases is that they allow to cancel 
barriers higher up in the tree. They can be used to 
circumvent (24). Classical GB theory has had to re- 
sort to a variety of tricks to account for discontinuous 
domains. A case in point is the coherent infinitive 
construction found in German and Dutch ~. A stan- 
dard account is to reanalyse 0-structure into another 
structure that lacks the annoying barrier-generating 
nodes. Different submodules of the theory will work 
on different structures. Consider the following exam- 
ple. 
dab \[cP \[tP PRO \[vp \[NP der Wagen\] zu reparieren\]\]\] \[v versucht\] wurde 
In this example V governs NP but not "PRO" even 
though "PRO" intervenes between V and NP. CP 
might be called a phantom barrier. Generally, a phan- 
tom (like CP, IP above) is a barrier just in case it 
does not dominate a non-phantom (VP above). Thus 
CP shields "PRO" but remains open for government 
of NP. This state of affairs can be caught in the 
present framework by a negative inheritance clause. 
(21) barrier(7,#) ¢= 
phantom(7) A 
7>#A 
"~q# : nonphantom(~,3) A 
7>8. 
nonphantom(7,8 ) ¢= nonphantom(7) A 7 > 8. 
Similar cases arise with negation. Again, the litera- 
ture adopts different lines of argument to account for 
the phenomenon. Kamp and Reyle \[1993\] handle the 
binding case below with a rule of double negation 
elimination, an operation that deletes structure. 
*Either he~ owns a Porsche or John/ hides 
it. 
Either he/does not own a Porsche or John/ 
hides it. 
1Mfiller and Sternefeld \[1991\] propose to treat this 
construction within the framework of barrier theory. 
445 
The examples below are drawn from Cinque \[1990, 
83\]. He uses a superscription convention to annotate 
the scope of the negation and assumes an LF amalga- 
mation process triggered by coindexing of this sort. 
CP is no barrier anymore for LF-amalgamated el- 
ements since they become wh-movable. We might 
model amalgamation with the "nonphantom" clause 
of (21). Then, this clause would have to hold true for 
inherently wh-movable elements (bare quantifiers in 
Cinque's analysis) as well. 
*Molti amici, \[cP ha invitato t, che io sap- 
pin. 
Molti amici, \[cP \[NegP non ha invitato t, 
che io sappia. 
3.4 Properties of the Definition Schemes 
In this paragraph we further investigate properties 
of the three definition schemes we are dealing with. 
We summarize scheme (16) in (22). def is a variable 
ranging over the given definitions. 
(22) B(c,b) ~ Bdef: Ldef(c ) A c>b 
We can collapse all definitions de/into a single defi- 
nition with local condition K(c) ~ Vd4Ld4(c). In 
order to summarize the schemes (16--17) we intro- 
duce vectors of definitions def" of length n and corre- 
sponding sequences of nodes Z of length n + 1. xl is 
fixed to c and Xn+l to b. 
(23) B(c,b) *-* B def, Z:Vi • {1,...,n}: 
Ldef(i)(xi) A xi > xi+l. 
For definitions conforming to type (16--17) we can 
show the following property: If we have found a son 
y violating the relation R all descendants b of the 
father x will be inaccessible to R. 
(24) x ~- y A aRx A ~aRy A x > b --* --,aRb 
In a full-fledged definition scheme where (16--18) 
are available (24) ceases to hold. In the example dis- 
cussed above a does not govern y but does govern b. 
a \[cP=, \[vP=y b 
In pre-Barriers GB theory and most current com- 
putational approaches only inherent barriers are al- 
lowed (scheme 16) and the violating number of barri- 
ers in axiom (4) is set to null. Note that under these 
provisos, barriers theory shrinks to command theory: 
(4') aRb ~ a, b unconnected A 
Vc :K(c) A c>b---*c>a 
The following constraint holds in this configuration: 
A barrier as in (24) is not affected by the triggering 
first argument. 
(25) x ~-y A Ba : \[aRx A --,aRy\] A bRx ---. --,bRy 
Chomsky \[1986, 11\] discusses (25) at some length. In 
his example (see below) "decide" =a does not govern 
"PRO", but "e" =b would. He shows that if either of 
the mentioned requirements (n=O and intrinsic bar- 
riers) is not met the theorem is refuted. 
(21) John decided \[cP e \[xP PRO to \[re 
see the movie \]\]\] 
If (16--18) are given then we can show the following 
theorem: Brothers are equivalent when occurring as 
a second argument of a BC-definable relation. 
(26) a, bl unconnected A a, b2 unconnected A 
by N- bl A by N- b2 ~ (aP0bl ~ aRb2) 
4 Localising the Global Constraints 
The next step is to localise the definitions (16-- 
18). For ease of reference we repeat the definition 
schemes. 
(27) B(c,b) ~ 3def: \[Ll(C) A c>b\] V 
ILl(c) A I(c,b,B, L2)\] V 
\[Ll(c) A c>b A -,I(c,b,B, L2)\] 
We only take into account nodes c that separate a 
from b in the sense that they sit on the ancestor line 
of b but not on that of a (see also the restrictions 
of 4 and 5). Theorem (28) specifies a connection be- 
tween the inheritance clauses valid on a father node z 
and those valid on the son y. Recall that inheritance 
clauses are the only global conditions we consider. 
(28) xNy A y>_b A "-,y>_a ---* 
(B(y, b) V (I(y, b, B, L) A -~L(y)) 
*-* I(x, b, B, L)) 
In parsing, an unbounded dependency (formally, a 
relation R) is triggered by a node nl (e.g. because it 
lacks a 0-role or cannot take up a 0-role assigned to 
it) and successfully terminates when a correspond- 
ing node n2 is found (that can supply the missing 
0-role or absorb a superfluous 0-role). When search- 
ing, ancestor lines are either ascended or descended. 
Accordingly we have to make a distinction between 
the upward and downward state of dependency in- 
formation. 
446 
4.1 Upward States 
Upward states supply information about barrier 
nodes encountered on the ancestor line below. They 
are constructed when the second argument b of a 
relation R has been found and the tree is being 
searched for the first argument a. Formally, upward 
states are sets (standing for conjunctions) associ- 
ated with some node c and some dependency coming 
from b. 
{B,L) e UState(c,b) ~ I(c,b,B,L) 
Any inheritance clause that can be derived at c on 
the basis of the lower upward state and the rule 
schemes (27--28) is included in c's upward state. If 
a clause is not in the state, it cannot be inferred by 
(16--18). Consequently, the negation of a missing 
clause must hold. We assume a counter for c and b 
to be increased and checked as defined by the theory 
(computing the number n of passed barriers, see 4). 
IncreaseCounter(c,b) ~ B(c,b) 
We use the upward state to break off search as soon 
as we can infer from the theory that an element 
a cannot possibly be found in the rest of the tree. 
Theorem (29) stands to express that as soon as we 
have found a node y violating the definitions upward 
search becomes obsolete. 
(29) 
4.2 Downward States 
Downward states encode information about barrier 
nodes encountered on the ancestor line above. They 
are computed when the second argument b of a re- 
lation tt is being expected because a first argument 
a has been discovered. Formally, downward states 
are first order formulae associated with some node 
c, some ancestor node ct of c, and some dependency 
leading to b. Atomic formulae of DState(c,cl,b) are 
inheritance clauses I with respect to c and b. 
formula E DState(c,ct,b) 
formula(c,b) ~ IncreaseCounter(cl ,b) 
The rule schemes (27--28) supply all sufficient 
and necessary conditions for transfer of inheritance 
clauses between nodes. Accordingly an atomic for- 
mula in the upper downward state can be trans- 
formed into a formula holding for the lower node c. 
False formulae are discarded, while true formulae in- 
crease the counter. 
We use downward states to restrict the search space. 
By (24) we can sometimes infer that search into 
a subtree will be pointless. Negative inheritance 
clauses, however, can only be checked when a can- 
didate for b has been encountered. When the parser 
descends paths while searching, it always assumes 
that the current path will dominate b. For upward 
states, in contrast, the ancestor line of b is fixed. 
Only downward states scan trees. (26) shows that a 
state will not change for brother nodes. So we only 
have to store one downward state per rule (e.g. under 
its mother node). 
4.3 Example 
Consider the chain of "how" in the following example 
how do \[zp. you \[vP, t \[vP remember 
\[cp t/*why lip Bill t behaved t \]\]\]\]\] 
In a left-to-right top-down parse, the first barrier to 
be encountered would be IP* if it dominated either 
a blocking category (BC) or no other tensed IP. VP* 
is no BC or barrier since it does not dominate the 
intermediate trace (it is not the minimal segment of 
the VP node). CP is L-marked and hence a barrier 
only if it dominates a BC. If "why" excludes a trace 
in SpecCP, the BC IP occurs between CP and the 
next trace. Due to the d-role of "how", government is 
violated leading to an ungrammatical sentence. If an 
intermediate trace is allowed, a new chain is started 
and no BC occurs. IP refutes the hypothesis that IP* 
is the deepest embedded tensed IP, and it turns out 
to be this IP as soon as the variable is found. So 
only one subjacency barrier occurs: The sentence is 
grammatical. 
5 Conclusion 
We have described a mechanism that handles global 
constraints on long movement from a local basis. The 
device has been derived from a logical formulation of 
Chomsky's \[1986\] theory so that equivalence to this 
theory is easily proved. We have sketched methods to 
use the logic for early determination of ungrammat- 
ical readings in a parser. In my thesis (\[Schiehlen, 
1992\]) the technique has been implemented in an 
Earley parser that generates all readings in paral- 
lel. In this system local conditions are couched into 
feature terms. Feature clashes lead to creation and 
abolition of dependencies modelling the GB notion 
of failed feature assignment and last resource. The 
barriers logic restricts rule choice for the predictor 
(descending ancestor lines) and discards analyses in 
the completer (ascending ancestor lines). Ongoing 
work is centred around an application of the bar- 
riers framework to the generation of semantic struc- 
ture (Discourse Representation Structure). Kraeht's 
\[1992\] approach to analysing barriers theory is re- 
lated to the one presented here. However, Kracht's 
emphasis is not so much on parsing. 
447 
A Proofs 
Proof of (6) is trivial. 
The theorem (7) is symmetric for al and a2. Suppose 
alRb A "~a2Rb. a2 and b are unconnected. So there 
exist kl barriers not dominating al (kl < n) and k2 
barriers not dominating a2 (k2 > n). Suppose c is a 
barrier not dominating a2 but dominating al (there 
are at least k2-kl > 1 such barriers), c>b and y>b, 
hence c and y are connected. But y>_c entails y>al. 
Ifc>y then either x>c>y or c>x. But c>x 
implies c > a2. 
To prove (9) note that all barriers for y dominate y 
by (5). Hence they also dominate a e \[y\]. 
We now turn to (10). Take al E \[x\] and a2 E \[y\]. 
a2 and y are not connected. We show that if --c > a2 
and c > b then -~c > al. Assume c > b and c > ax. 
Then x and c are connected both dominating b. We 
know that -~x _> c > ax. Hence c > x > y. Suppose 
y! is y's father. Then c > x >_ y! ~.- y and equally 
c> x > y! ~- a2. We obtain that {c I B(c,b) A -~c> 
el} D {c I B(c,b) ^ -~c>a2}. Hence -~\[x\]Rb. 
We prove (24). Suppose c is a barrier for x. Then 
by (23) there is a sequence of nodes xl = c and 
xn > xn+l = x. But xn > x > b, so c is a barrier for b as 
well. a and y are unconnected. Suppose c is a barrier 
for y but not x. Then xl = c and xn>x~+l = y. xn 
and x are connected both dominating y. We know 
that -~x > xn > y and ~xn > x else c would be a 
barrier for x. Hence Xn = x and we get x, = x > b. 
There are at least as many barriers for b as there are 
for y, so -~aRb. 
To prove (25) we adopt the argumentation of the 
foregoing proof and infer that x is a barrier for y. 
bILz shows that b, x are unconnected, hence -~x > b 
and -~bRy. 
(26) follows if we prove B(c, bl) ~ B(c, b2) by in- 
duction. The theorem is symmetric. Assume a c 
such that B(c, bl). Then either scheme (16) holds: 
L(c) A c>bx hence c>b2. Or (17) and L(c) A 3d : 
B(d, bl) A c> d A Ve : c> e > d ---* ~L(e). By 
induction B(d, b2) as well. For the negative scheme 
(18) we use symmetry to extend the implication 
I(c, bx, B, L) ---, I(c, b2, B, L) to an equivalence. 
For (28) we give a proof by cases. Either B(y, b) --. 
I(z, b, B, L). y is the barrier node d referred to in the 
consequent. Or I(y, b, B, L) A -~L(y) --* I(x, b, B, L). 
We set the barrier node d of the first inheritance 
clause equal to the one of the second. Does a node 
e between x and d satisfy L? y does not, nor do 
the nodes between y and d, and there is no node 
between x and y. But y and e must be connected, 
both dominating d. We show I(x, b, B, L) --* B(y, b) V 
I(y, b, B, L). The barrier node d of the antecedent 
clause and y are connected, both dominating b (see 
5). d cannot sit between x and y. If d - y the first 
disjunct holds. If y > d we set d equal to the barrier 
node of the second disjunct. No e between y and d 
satisfies L. 
We reduce (29) to (10). If a > y > b we make use of (6). 
Otherwise let x! be the smallest node that dominates 
both y and a and let x be such that x! ~- x >__ y. Then 
by (10) "~\[x\] Rb, meaning --~aRb (see 8). 
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448 
