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<Paper uid="E93-1053">
  <Title>Localising Barriers Theory</Title>
  <Section position="2" start_page="0" end_page="443" type="metho">
    <SectionTitle>
2 Dependencies Between Nodes
</SectionTitle>
    <Paragraph position="0"> We take a tree T to be a structure (N,&gt;), where N is a set of nodes and &gt; stands for dominance, a binary relation on N. We say that nodes a and b are connected iff a &gt; b V b &gt; a V a = b. We define the relation of immediate dominance ~- between two nodes a and b as a &gt; b A ~3c : a &gt; c A c &gt; 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.</Paragraph>
    <Paragraph position="1"> (1) z&gt;z A y&gt;z --* x connected with y (2) ~xVy : x &gt; y (3) x&gt;z ~ 3y : x~y A y&gt;z  Chomsky \[1986, 9,30\] discusses several definitions for constraints on unbounded dependencies.</Paragraph>
    <Paragraph position="2"> (13) a c-commands/~ iff a does not dominate/~ \[and/~ does not dominate or equal a\] and every 7 that dominates a dominates/~.</Paragraph>
    <Paragraph position="3"> Where 7 is restricted to maximal projections we will say that a m-commands/?.</Paragraph>
    <Paragraph position="4"> (18) a governs/~ iff a m-commands/~ and there is no 7, 7 a harrier for/~/, such that 7 excludes a.</Paragraph>
    <Paragraph position="5"> (59)/~ is n-subjacent to a iff there are fewer than n+l barriers for/~ that exclude a.</Paragraph>
    <Paragraph position="6"> All of these can be moulded into the general format introduced in (4): Two nodes can only stand in a relation 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 barrierhood entail dominance. We call relations that satisfy axiom (4) definable with barrier concepts, for short BC-definable.</Paragraph>
    <Paragraph position="7">  (4) aRb ~-* a, b unconnected ^</Paragraph>
    <Paragraph position="9"> Balanced relations like government require a definition in terms of two BC-definable relations: Rl(a, b) and R2(b, a).</Paragraph>
    <Paragraph position="10"> (5) B(c,b) ~c&gt;b We can show several properties of BC-definable relations. The nodes are unconnected.</Paragraph>
    <Paragraph position="11"> (6) aRb ---* a, b unconnected In order to investigate BC-definable relations it suffices to investigate the ancestor lines of their second argument b (that is {y J y &gt;_ b}).</Paragraph>
    <Paragraph position="12">  (7) x~-y A z&gt;al A &amp;quot;,y&gt;__al A x&gt;a2 A -w&gt;_a~ A y&gt;b --* (alRb ~ a2Rb) (7) gives rise to equivalence classes for the first argument of R. For a particular pair (a,b) we can always find a y as defined in (8).</Paragraph>
    <Paragraph position="13"> (s) a* ^ x&gt;a ^ y&gt;a ^ y&gt;b Definable relations are never empty. Barriers are preserved in the upward direction of the ancestor line: (9) \[y\]Ry (10) x&gt;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.</Paragraph>
    <Paragraph position="14"> *\[cP=~ \[vP=~ Hei sees Mary \] and John/is happy\].</Paragraph>
  </Section>
  <Section position="3" start_page="443" end_page="445" type="metho">
    <SectionTitle>
3 Barrier Definitions
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="443" end_page="443" type="sub_section">
      <SectionTitle>
3.1 Adjunction
</SectionTitle>
      <Paragraph position="0"> Adjunction rules raise a problem for algebraic investigations of barriers theory (e.g. \[Kracht, 1992\]): They insert material into a tree but do not create 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 &gt; bi --* a &gt; b2 Chomsky \[1986\] defines projection dominance so that dominates ~ only if every segment of a dominates (every segment of) f/. In case this definition is not empty, (1) guarantees a unique minimal segment 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.</Paragraph>
      <Paragraph position="1">  (11) dominate(a,/3) *-+ a e a A b */3 A minimal segment(a) A a &gt; b Likewise, Chomsky's definition of exclusion, viz that a excludes j3 if no segment of a dominates (any segment 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.</Paragraph>
      <Paragraph position="2"> (12) exclude(aft) ~ a E a A b e fl A maximal segment(a) A --a &gt; b  This way, we reduce projection dominance to segment dominance. In (13--15), conditions of segment minimality or maximality are included where they are appropriate by (11) and (12).</Paragraph>
    </Section>
    <Section position="2" start_page="443" end_page="444" type="sub_section">
      <SectionTitle>
3.2 Chomsky's Theory
</SectionTitle>
      <Paragraph position="0"> Chomsky \[1986, 14\] gives the following two core definitions for barriers. We are not concerned about the exact formulation of L-marking (for a definition see \[Chomsky, 1986, 24\]).</Paragraph>
      <Paragraph position="1">  (25) 7 is a blocking category for fl iff 7 is not L-marked and 7 dominates/3.</Paragraph>
      <Paragraph position="2"> (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 &amp;quot;immediately dominate&amp;quot; in (26a) to be a relation between maximal projections (so that 7 immediately dominates 5 in this sense even if a nonmaximal projection intervenes). null 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 definitions is held implicit: Barrier concepts are true iff they comply with a manifest definition (see also 22  and 23).</Paragraph>
      <Paragraph position="3"> (13) blocking category(c,b) C/:: maximal projection(c) A  -,IP(c).</Paragraph>
      <Paragraph position="4"> We regard unary predicates as local conditions (L) and binary predicates as global concepts (B for &amp;quot;barrier concept&amp;quot;). Abstracting over the particular predicates involved we end up with the following definition schemes (16 for 13 and 15, 17 for 14).</Paragraph>
      <Paragraph position="6"> We call the existential subformula of (17) an inheritance clause I. The only global conditions in our system are inheritance clauses and c&gt; 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.</Paragraph>
      <Paragraph position="8"/>
    </Section>
    <Section position="3" start_page="444" end_page="445" type="sub_section">
      <SectionTitle>
3.3 Negative Inheritance Clauses
</SectionTitle>
      <Paragraph position="0"> It has interesting repercussions to incorporate a scheme with a negated inheritance clause, viz. (18).</Paragraph>
      <Paragraph position="2"> For illustration we discuss several applications for negative inheritance clauses.</Paragraph>
      <Paragraph position="3"> Chomsky \[1986, 37\] talks about IPs as inherent barriers, 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.</Paragraph>
      <Paragraph position="5"> A feature of negative inheritance clauses that is desirable 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 resort 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 standard 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 example. null dab \[cP \[tP PRO \[vp \[NP der Wagen\] zu reparieren\]\]\] \[v versucht\] wurde In this example V governs NP but not &amp;quot;PRO&amp;quot; even though &amp;quot;PRO&amp;quot; intervenes between V and NP. CP might be called a phantom barrier. Generally, a phantom (like CP, IP above) is a barrier just in case it does not dominate a non-phantom (VP above). Thus CP shields &amp;quot;PRO&amp;quot; but remains open for government of NP. This state of affairs can be caught in the present framework by a negative inheritance clause.</Paragraph>
      <Paragraph position="7"> Similar cases arise with negation. Again, the literature 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.</Paragraph>
      <Paragraph position="8"> *Either he~ owns a Porsche or John/ hides it.</Paragraph>
      <Paragraph position="9"> Either he/does not own a Porsche or John/  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 amalgamation process triggered by coindexing of this sort. CP is no barrier anymore for LF-amalgamated elements since they become wh-movable. We might model amalgamation with the &amp;quot;nonphantom&amp;quot; clause of (21). Then, this clause would have to hold true for inherently wh-movable elements (bare quantifiers in Cinque's analysis) as well.</Paragraph>
      <Paragraph position="10"> *Molti amici, \[cP ha invitato t, che io sappin. null Molti amici, \[cP \[NegP non ha invitato t, che io sappia.</Paragraph>
    </Section>
    <Section position="4" start_page="445" end_page="445" type="sub_section">
      <SectionTitle>
3.4 Properties of the Definition Schemes
</SectionTitle>
      <Paragraph position="0"> In this paragraph we further investigate properties of the three definition schemes we are dealing with.</Paragraph>
      <Paragraph position="1"> We summarize scheme (16) in (22). def is a variable ranging over the given definitions.</Paragraph>
      <Paragraph position="2"> (22) B(c,b) ~ Bdef: Ldef(c ) A c&gt;b We can collapse all definitions de/into a single definition with local condition K(c) ~ Vd4Ld4(c). In order to summarize the schemes (16--17) we introduce vectors of definitions def&amp;quot; of length n and corresponding sequences of nodes Z of length n + 1. xl is fixed to c and Xn+l to b.</Paragraph>
      <Paragraph position="4"> 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.</Paragraph>
      <Paragraph position="5"> (24) x ~- y A aRx A ~aRy A x &gt; b --* --,aRb In a full-fledged definition scheme where (16--18) are available (24) ceases to hold. In the example discussed above a does not govern y but does govern b. a \[cP=, \[vP=y b In pre-Barriers GB theory and most current computational approaches only inherent barriers are allowed (scheme 16) and the violating number of barriers in axiom (4) is set to null. Note that under these provisos, barriers theory shrinks to command theory: (4') aRb ~ a, b unconnected A</Paragraph>
      <Paragraph position="7"> The following constraint holds in this configuration: A barrier as in (24) is not affected by the triggering first argument.</Paragraph>
      <Paragraph position="8"> (25) x ~-y A Ba : \[aRx A --,aRy\] A bRx ---. --,bRy Chomsky \[1986, 11\] discusses (25) at some length. In his example (see below) &amp;quot;decide&amp;quot; =a does not govern &amp;quot;PRO&amp;quot;, but &amp;quot;e&amp;quot; =b would. He shows that if either of the mentioned requirements (n=O and intrinsic barriers) is not met the theorem is refuted.</Paragraph>
      <Paragraph position="9"> (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.</Paragraph>
      <Paragraph position="10"> (26) a, bl unconnected A a, b2 unconnected A by N- bl A by N- b2 ~ (aP0bl ~ aRb2)</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="445" end_page="446" type="metho">
    <SectionTitle>
4 Localising the Global Constraints
</SectionTitle>
    <Paragraph position="0"> The next step is to localise the definitions (16-18). For ease of reference we repeat the definition schemes.</Paragraph>
    <Paragraph position="2"> 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 between 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.</Paragraph>
    <Paragraph position="3">  (28) xNy A y&gt;_b A &amp;quot;-,y&gt;_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 corresponding node n2 is found (that can supply the missing 0-role or absorb a superfluous 0-role). When searching, ancestor lines are either ascended or descended. Accordingly we have to make a distinction between the upward and downward state of dependency information. null</Paragraph>
    <Section position="1" start_page="446" end_page="446" type="sub_section">
      <SectionTitle>
4.1 Upward States
</SectionTitle>
      <Paragraph position="0"> 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) associated with some node c and some dependency coming from b.</Paragraph>
      <Paragraph position="1"> {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).</Paragraph>
      <Paragraph position="2"> 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.</Paragraph>
      <Paragraph position="3"> Theorem (29) stands to express that as soon as we have found a node y violating the definitions upward search becomes obsolete.</Paragraph>
      <Paragraph position="4"> (29)</Paragraph>
    </Section>
    <Section position="2" start_page="446" end_page="446" type="sub_section">
      <SectionTitle>
4.2 Downward States
</SectionTitle>
      <Paragraph position="0"> Downward states encode information about barrier nodes encountered on the ancestor line above. They are computed when the second argument b of a relation 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.</Paragraph>
      <Paragraph position="1"> 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 formula in the upper downward state can be transformed into a formula holding for the lower node c. False formulae are discarded, while true formulae increase the counter.</Paragraph>
      <Paragraph position="2"> 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 candidate 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).</Paragraph>
    </Section>
    <Section position="3" start_page="446" end_page="446" type="sub_section">
      <SectionTitle>
4.3 Example
</SectionTitle>
      <Paragraph position="0"> Consider the chain of &amp;quot;how&amp;quot; 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 &amp;quot;why&amp;quot; excludes a trace in SpecCP, the BC IP occurs between CP and the next trace. Due to the d-role of &amp;quot;how&amp;quot;, 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.</Paragraph>
    </Section>
  </Section>
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