Markus Egg, Alexander Koller, Joachim Niehren The Constraint Language for Lambda Structures Ventsislav Zhechev SfS, Universität Tübingen e-mail: [email protected]@sfs.uni-tuebingen.de

Markus Egg, Alexander Koller, Joachim Niehren

The Constraint Language for Lambda Structures

Markus Egg, Alexander Koller, Joachim Niehren

The Constraint Language for Lambda Structures

Ventsislav ZhechevSfS, Universität Tübingene-mail: [email protected]

Ventsislav ZhechevSfS, Universität Tübingene-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

19.01.2005 2

AgendaAgenda

• Introduction• Motivation• Basic Terms

• Elements of CLLS• -terms• -structures• Discussed Phenomena







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Agenda (continued)Agenda (continued)

• Syntax and Semantics of CLLS• Tree Structures• -structures• Dominance and Parallelism• The CLLS• Constraint Graphs






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• Interaction of Quantifiers, Anaphora and Ellipsis• Quantifier Parallelism• Strict/Sloppy Ambiguities• Nested Ellipses• A Complex Interaction• Antecedent-Contained Ellipsis






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• The Syntax-Semantics Interface• Grammar• Semantic Construction• An Example

• Computational Aspects• Conclusion







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• Linguistic Phenomena• Scope Ambiguities• Anaphora• VP Ellipsis

• Underspecification

• Linguistic Phenomena• Scope Ambiguities• Anaphora• VP Ellipsis

• Underspecification



IntroductionIntroduction

MotivationMotivation

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• Trees• Underspecification• Constraint Language for Lambda Structures:

A Combination of Constraints• Dominance Constraints• Anaphoric Binding Constraints• Parallelism Constraints• -binding Constraints

• Trees• Underspecification• Constraint Language for Lambda Structures:

A Combination of Constraints• Dominance Constraints• Anaphoric Binding Constraints• Parallelism Constraints• -binding Constraints





Basic TermsBasic Terms

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• Every linguist attends a workshop.• (a workshop)(x

(every linguist)(y(attend x) y))

• Types• e individuals• t truth values (0 or 1 / true or false)• <e,t> one-place predicates• <e,<e,t>> two-place predicates• etc.

• Every linguist attends a workshop.• (a workshop)(x


• Types• e individuals• t truth values (0 or 1 / true or false)• <e,t> one-place predicates• <e,<e,t>> two-place predicates• etc.







Elements of CLLSElements of CLLS

-terms-terms

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• (a workshop)(x(every linguist)(y

(attend x) y))•

• lam -abstraction• @ functional application• var bound variable• variable binding

• (a workshop)(x(every linguist)(y

(attend x) y))•

• lam -abstraction• @ functional application• var bound variable• variable binding







-structures-structuresattend

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• Scope Ambiguity• Every linguist attends a workshop.• (a workshop)(x


• (every linguist)(y (a workshop)(x

(attend x) y))•

• dominance

• Scope Ambiguity• Every linguist attends a workshop.• (a workshop)(x


• (every linguist)(y (a workshop)(x

(attend x) y))•

• dominance







Discussed PhenomenaDiscussed Phenomena

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• VP Ellipsis• Every man sleeps, and so does Mary.•

• Parallelism Constraint:X1/X2~Y1/Y2

•

• VP Ellipsis• Every man sleeps, and so does Mary.•

• Parallelism Constraint:X1/X2~Y1/Y2

•



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• Anaphora• Johni said hei

j liked hisj mother.

•

• ana anaphora• anaphoric link

• Anaphora• Johni said hei

j liked hisj mother.

•

• ana anaphora• anaphoric link



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• The Capturing Problem• Variable binding in -terms is usually indicated by

using variable names, i.e. x binds all occurrences of x in its scope

• Possible Problems:• -calculus has to exclude the capturing of free

variables by unintended binders• Problems with constraints used for scope ambiguities

• Problems in the presence of parallelism constraints

• The Capturing Problem• Variable binding in -terms is usually indicated by

using variable names, i.e. x binds all occurrences of x in its scope

• Possible Problems:• -calculus has to exclude the capturing of free

variables by unintended binders• Problems with constraints used for scope ambiguities

• Problems in the presence of parallelism constraints



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• Trees and Tree Structures• The Algebra of Trees

• Let be a set of function symbols, f, g, a, b• Each function symbol f has fixed arity• We write fk for a function symbol f with arity k ≥ 0• We define tree as a ground term built from a set of

function symbols• We define path as a word over ℕ (the natural numbers)

• We identify each node in a tree with the path from the root to this node

• The empty word, , identifies the root• Concatenation is written as • A word is a prefix of , iff there is a word 1

such that =1

• Trees and Tree Structures• The Algebra of Trees

• Let be a set of function symbols, f, g, a, b• Each function symbol f has fixed arity• We write fk for a function symbol f with arity k ≥ 0• We define tree as a ground term built from a set of

function symbols• We define path as a word over ℕ (the natural numbers)

• We identify each node in a tree with the path from the root to this node

• The empty word, , identifies the root• Concatenation is written as • A word is a prefix of , iff there is a word 1

such that =1







Syntax and Semantics of CLLS

Syntax and Semantics of CLLS

Tree StructuresTree Structures

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• Tree Structures• tree domain is a finite nonempty set of nodes,

which is prefix closed ( ) and closed under left siblings (i j for all 1≤j<i)

• tree structure is defined as follows:

• Given nodes 0,..., n, we write 0:f(1,..., n) for(0, 1,..., n) :f

• Tree Structures• tree domain is a finite nonempty set of nodes,

which is prefix closed ( ) and closed under left siblings (i j for all 1≤j<i)

• tree structure is defined as follows:

• Given nodes 0,..., n, we write 0:f(1,..., n) for(0, 1,..., n) :f



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• Tree Structures and -structures• Formalization

• For -structures we assume:{var0, ana0, lam1, @2}

• We define -structures as follows:

• We draw -structures as tree-like graphs

• Tree Structures and -structures• Formalization

• For -structures we assume:{var0, ana0, lam1, @2}

• We define -structures as follows:

• We draw -structures as tree-like graphs





-structures-structures



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• Dominance• Let be a -structure and , two of its nodes.

We say that dominates (⊲*), if lies above ,i.e. is a prefix of

• Dominance is a partial order on the domain of and it is reflexive, transitive and antisymmetric

• Parallelism• We call any pair / of nodes , in with ⊲* a

segment of , where is called the root and the hole of the segment

• We define:

• Dominance• Let be a -structure and , two of its nodes.

We say that dominates (⊲*), if lies above ,i.e. is a prefix of

• Dominance is a partial order on the domain of and it is reflexive, transitive and antisymmetric

• Parallelism• We call any pair / of nodes , in with ⊲* a

segment of , where is called the root and the hole of the segment

• We define:





Dominance and ParallelismDominance and Parallelism



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• Correspondence Functions between segments:

• Parallelism Relation:

• Correspondence Functions between segments:

• Parallelism Relation:



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• We assume an infinite set of node variables, ranged over X, Xi, Y, etc.

• We pick relation symbols for all relations defined so far

• Finally we define CLLS with the following abstract syntax:

• The Semantics of CLLS is defined by interpretation of constraints over the class of-structures:• A pair of a -structure and a variable assignment

into the domain of satisfies a constraint , iff it satisfies each atomic conjunct of it

• We call (, ) a solution of in this case

• We assume an infinite set of node variables, ranged over X, Xi, Y, etc.

• We pick relation symbols for all relations defined so far

• Finally we define CLLS with the following abstract syntax:

• The Semantics of CLLS is defined by interpretation of constraints over the class of-structures:• A pair of a -structure and a variable assignment

into the domain of satisfies a constraint , iff it satisfies each atomic conjunct of it

• We call (, ) a solution of in this case





The CLLSThe CLLS

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• CLLS Constraints are usually hard to read in the standard syntax. That is why we will use constraint graphs for presenting the constraints

• For Example:

• CLLS Constraints are usually hard to read in the standard syntax. That is why we will use constraint graphs for presenting the constraints

• For Example:





Constraint GraphsConstraint Graphs



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• Throughout the chapter we assume a fixed signature:={@2, lam1, var0, ana0, before2, mary0, read0, ...}

• We follow the convention that proper nouns are always analyzed as constants of type e, except as contrasting elements in ellipses where the other contrasting element is a quantifier

• Throughout the chapter we assume a fixed signature:={@2, lam1, var0, ana0, before2, mary0, read0, ...}

• We follow the convention that proper nouns are always analyzed as constants of type e, except as contrasting elements in ellipses where the other contrasting element is a quantifier

Interaction of Quantifiers, Anaphora and Ellipsis

Interaction of Quantifiers, Anaphora and Ellipsis



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• Every linguist attends a workshop.• Every computer scientist does, too.• The pair of sentences has three possible

readings, although it may seem that there are four

• The CLLS constraint for the two sentences looks like this:

• Every linguist attends a workshop.• Every computer scientist does, too.• The pair of sentences has three possible

readings, although it may seem that there are four

• The CLLS constraint for the two sentences looks like this:



Quantifier ParallelismQuantifier Parallelism



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• John likes his mother, and Bill does too.• The sentence has two readings: strict (Bill

likes John’s mother) and sloppy (Bill likes Bill’s mother)

• We describe the meaning of the sentence using parallelism and anaphoric linking constraints:

• John likes his mother, and Bill does too.• The sentence has two readings: strict (Bill

likes John’s mother) and sloppy (Bill likes Bill’s mother)

• We describe the meaning of the sentence using parallelism and anaphoric linking constraints:



Strict/Sloppy AmbiguitiesStrict/Sloppy Ambiguities



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• According to the parallelism constraint the tree part of the -structure below Xt is the same as the one below Xs, except for the contrasting elements, as follows:

• This is yet not a complete -structure, because the anaphor at Xa’ doesn’t have an antecedent

• According to the parallelism constraint the tree part of the -structure below Xt is the same as the one below Xs, except for the contrasting elements, as follows:

• This is yet not a complete -structure, because the anaphor at Xa’ doesn’t have an antecedent

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• John revised his paper before the teacher did, and so did Bill.

• This sentence comprises nested ellipsis: the source clause of the ellipsis is elliptical itself

• The sentence is further complicated by the presence of the anaphor, which induces a complex strict/sloppy ambiguity

• We follow Dalrymple et al. (1991) in assuming five readings for the sentence

• John revised his paper before the teacher did, and so did Bill.

• This sentence comprises nested ellipsis: the source clause of the ellipsis is elliptical itself

• The sentence is further complicated by the presence of the anaphor, which induces a complex strict/sloppy ambiguity

• We follow Dalrymple et al. (1991) in assuming five readings for the sentence



Nested EllipsesNested Ellipses



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• All five readings are represented by the following constraint:

• All five readings are represented by the following constraint:

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• Mary read a book she liked before Sue did.• The sentence has three readings

• In the first reading the indefinite NP a book she liked outscopes both clauses

• The second and the third reading arise from a strict/sloppy ambiguity that occurs if the operator before outscopes the indefinite

• Here is a constraint describing the readings:

• Mary read a book she liked before Sue did.• The sentence has three readings

• In the first reading the indefinite NP a book she liked outscopes both clauses

• The second and the third reading arise from a strict/sloppy ambiguity that occurs if the operator before outscopes the indefinite

• Here is a constraint describing the readings:



A Complex InteractionA Complex Interaction



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• Schematic representations of the solutions:• First reading:

• Second reading:

• Third reading:

• Schematic representations of the solutions:• First reading:

• Second reading:

• Third reading:







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• John greeted every person that Max did.• The problem is that the ellipsis is contained in

the VP it refers to• In CLLS the meaning of the sentence is

described as follows:

• John greeted every person that Max did.• The problem is that the ellipsis is contained in

the VP it refers to• In CLLS the meaning of the sentence is

described as follows:



Antecedent-Contained EllipsisAntecedent-Contained Ellipsis



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• There is one problem with this analysis:the notion of binding equivalence as defined is too strong a restriction for ACD

• The redefinition is given as:

• There is one problem with this analysis:the notion of binding equivalence as defined is too strong a restriction for ACD

• The redefinition is given as:

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• The preconditions for the two branches of the definitions are given here as (a) and (b) respectively:

• This analysis also accounts for the difference between the following two sentences (the first one lacking one of the two readings of the second sentence):• John wants Bill to read everything that Max does.• John wants Bill to read everything Max wants him to

read.

• The preconditions for the two branches of the definitions are given here as (a) and (b) respectively:

• This analysis also accounts for the difference between the following two sentences (the first one lacking one of the two readings of the second sentence):• John wants Bill to read everything that Max does.• John wants Bill to read everything Max wants him to

read.

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• Phrase Structure Rules:

• The Lexicon is defined by a relation Lex, which relates words W and lexical categories{Det, N, IV, TV, SV, RP, ...}. Terminal productions (a13) expand lexical categories to words of this category

• Phrase Structure Rules:

• The Lexicon is defined by a relation Lex, which relates words W and lexical categories{Det, N, IV, TV, SV, RP, ...}. Terminal productions (a13) expand lexical categories to words of this category

The Syntax-Semantics InterfaceThe Syntax-Semantics Interface

GrammarGrammar





• Computational Aspects• Conclusion• Computational Aspects• Conclusion

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• The Syntax-Semantics Interface should factor out as much of the constraint construction as possible into the interface rules

• Most of the lexical entries introduce just one labeling constraint

• For each node in the syntax tree a constraint is generated; the constraint of the whole tree is the conjunction of these subconstraints

• Each node ℕ* in the syntax tree is associated with two variables, X

s (the local scope domain of ) and X

r (the root of the subconstraint for )

• The Syntax-Semantics Interface should factor out as much of the constraint construction as possible into the interface rules

• Most of the lexical entries introduce just one labeling constraint

• For each node in the syntax tree a constraint is generated; the constraint of the whole tree is the conjunction of these subconstraints

• Each node ℕ* in the syntax tree is associated with two variables, X

s (the local scope domain of ) and X

r (the root of the subconstraint for )

Semantic ConstructionSemantic Construction






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• We add a constraint Xs⊲*X

r for each determiner which is not an indefinite. We also add this constraint whenever is a verb

• We associate with each NP an index i that is used in the syntactic tree for coindexation with a variable Xi

• The variables associated with syntactic nodes are related by the following rules:

• We add a constraint Xs⊲*X

r for each determiner which is not an indefinite. We also add this constraint whenever is a verb

• We associate with each NP an index i that is used in the syntactic tree for coindexation with a variable Xi

• The variables associated with syntactic nodes are related by the following rules:


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• The complete constraint which the interface produces is the conjunction of all the local constraints we just mentioned, plus the labeling constraints for the lexical entries, of the typeX

r:sleep• Exceptions to this rule:

• The elliptic does (too) does not add a labeling constraint; its semantics is determined via a parallelism constraint

• Whenever coindexation signifies a relation between an anaphor ’ and its antecedent , we add the constraintX

r=Xi, when we process and the constraint X’r:ana

ante(X’r)=Xi when we process ’

• The complete constraint which the interface produces is the conjunction of all the local constraints we just mentioned, plus the labeling constraints for the lexical entries, of the typeX

r:sleep• Exceptions to this rule:

• The elliptic does (too) does not add a labeling constraint; its semantics is determined via a parallelism constraint

• Whenever coindexation signifies a relation between an anaphor ’ and its antecedent , we add the constraintX

r=Xi, when we process and the constraint X’r:ana

ante(X’r)=Xi when we process ’


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• The constraint for a relative pronoun with index i at isX

r=Xi Xi:var; and the constraint for the corresponding trace (say, at ’) is X’

r=Xi. This, together with rule (b11), enforces correct binding of the trace

• The constraints for possessive pronouns, such as his, are as follows:

• The constraint for a relative pronoun with index i at isX

r=Xi Xi:var; and the constraint for the corresponding trace (say, at ’) is X’

r=Xi. This, together with rule (b11), enforces correct binding of the trace

• The constraints for possessive pronouns, such as his, are as follows:


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• Every linguist attends a workshop.•

• First the lexical elements introduce several labeling constraints:X11

r:every, X121r:linguist, X21

r:attend,X221

r:a, X2221r:workshop

• Every linguist attends a workshop.•

• First the lexical elements introduce several labeling constraints:X11

r:every, X121r:linguist, X21

r:attend,X221

r:a, X2221r:workshop

An ExampleAn Example






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• The constraints for the NPs are built from the above by the rules (b8) and (b9):•

• A1:@(XA2r, XL1

r) A2:@(X11r, X121

r) L1:lam(XL2r)

X1r:var XL2

r⊲*X1r (X1

r)=XL1r XL1

r≠X1r X1

s=X11s=X121

s

• A3:@(XA4r, XL3

r) A4:@(X221r, X2221

r) L3:lam(XL4r)

X22r:var XL4

2⊲*X22r (X22

r)=XL3r XL3

r≠X22r

X22s=X221

s=X2221s

• Then rule (b3) combines the transitive verb and its object• X2

r:@(X21r, X22

r) X2s=X21

s=X22s

• The constraints for the NPs are built from the above by the rules (b8) and (b9):•

• A1:@(XA2r, XL1

r) A2:@(X11r, X121

r) L1:lam(XL2r)

X1r:var XL2

r⊲*X1r (X1

r)=XL1r XL1

r≠X1r X1

s=X11s=X121

s

• A3:@(XA4r, XL3

r) A4:@(X221r, X2221

r) L3:lam(XL4r)

X22r:var XL4

2⊲*X22r (X22

r)=XL3r XL3

r≠X22r

X22s=X221

s=X2221s

• Then rule (b3) combines the transitive verb and its object• X2

r:@(X21r, X22

r) X2s=X21

s=X22s


A2

A1

L1

L2

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• Rule (b1) analogously combines the subject and the VP• X

r:@(X2r, X1

r) Xs=X2

s=X1s

• So far we have the following constraint:•

• X11r:every X121

r:linguist X21r:attend X221

r:a X2221

r:workshop A1:@(XA2r, XL1

r) A2:@(X11r, X121

r) L1:lam(XL2

r) X1r:var XL2

r⊲*X1r (X1

r)=XL1r XL1

r≠X1r

A3:@(XA4r, XL3

r) A4:@(X221r, X2221

r) L3:lam(XL4r) X22

r:var XL4

2⊲*X22r (X22

r)=XL3r XL3

r≠X22r X2

r:@(X21r, X22

r) X

r:@(X2r, X1

r) X1s=X11

s=X121s=X22

s=X221s=X2221

s=X21s=X

s=X2s

• Rule (b1) analogously combines the subject and the VP• X

r:@(X2r, X1

r) Xs=X2

s=X1s

• So far we have the following constraint:•

• X11r:every X121


r:a X2221


r) A2:@(X11r, X121

r) L1:lam(XL2

r) X1r:var XL2

r⊲*X1r (X1

r)=XL1r XL1

r≠X1r

A3:@(XA4r, XL3

r) A4:@(X221r, X2221

r) L3:lam(XL4r) X22

r:var XL4

2⊲*X22r (X22

r)=XL3r XL3

r≠X22r X2

r:@(X21r, X22

r) X

r:@(X2r, X1

r) X1s=X11

s=X121s=X22

s=X221s=X2221

s=X21s=X

s=X2s


A4

A3A2

A1

L4

L3

L2

L1

Xr

X2r

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• Finally we add the relevant scope island constraints:• The complete sentence is associated with the

variable Xs, and all other Xs variables are forced to

be equal to this one by other constraints• Node 11 is a determiner and node 21 is a verb; so

we add the constraint X11s ⊲* X11

r X21s ⊲* X21

r

• Finally we add the relevant scope island constraints:• The complete sentence is associated with the

variable Xs, and all other Xs variables are forced to

be equal to this one by other constraints• Node 11 is a determiner and node 21 is a verb; so

we add the constraint X11s ⊲* X11

r X21s ⊲* X21

r


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• The final constraint we get is the following: •

• X11r:every X121


r:a X2221


r) A2:@(X11r, X121

r) L1:lam(XL2

r) X1r:var XL2

r⊲*X1r (X1

r)=XL1r XL1

r≠X1r

A3:@(XA4r, XL3

r) A4:@(X221r, X2221

r) L3:lam(XL4r) X22

r:var XL4

2⊲*X22r (X22

r)=XL3r XL3

r≠X22r X2

r:@(X21r, X22

r) X

r:@(X2r, X1

r) X1s=X11

s=X121s=X22

s=X221s=X2221

s=X21s=X

s=X2s

Xs ⊲* X11

r Xs ⊲* X21

r

• The final constraint we get is the following: •

• X11r:every X121


r:a X2221


r) A2:@(X11r, X121

r) L1:lam(XL2

r) X1r:var XL2

r⊲*X1r (X1

r)=XL1r XL1

r≠X1r

A3:@(XA4r, XL3

r) A4:@(X221r, X2221

r) L3:lam(XL4r) X22

r:var XL4

2⊲*X22r (X22

r)=XL3r XL3

r≠X22r X2

r:@(X21r, X22

r) X

r:@(X2r, X1

r) X1s=X11

s=X121s=X22

s=X221s=X2221

s=X21s=X

s=X2s

Xs ⊲* X11

r Xs ⊲* X21

r


A4

A3

A2

A1

L4

L3

L2

L1

Xr

X2r

X221r X2221

rX121

r

X22r

X1r

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• Disambiguation of arbitrary CLLS description is very complex: it has been shown that even CLLS without binding is equivalent to context unification, whose decidability is an open problem in theoretical computer science

• There are, however, semi-decision procedures which will eventually enumerate all solved forms of a constraint

• For the sublanguage of dominance constraints it was shown, that the satisfiability problem is decidable, but NP-complete

• Disambiguation of arbitrary CLLS description is very complex: it has been shown that even CLLS without binding is equivalent to context unification, whose decidability is an open problem in theoretical computer science

• There are, however, semi-decision procedures which will eventually enumerate all solved forms of a constraint

• For the sublanguage of dominance constraints it was shown, that the satisfiability problem is decidable, but NP-complete

Computational AspectsComputational Aspects

• Computational Aspects• Conclusion• Computational Aspects• Conclusion• Conclusion• Conclusion• Computational Aspects• Computational Aspects

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• An implementation of a solver for dominance constraints can be obtained by employing constraint programming with finite sets. The constraints can be solved by always performing deterministic propagation steps to eliminate hopeless choices before making case distinctions.

• It can be shown that all dominance constraints that are needed for the linguistic application belong to a fragment called normal dominance constraints. Satisfiability of a normal constraint can be checked by a graph algorithm of polynomial runtime; each reading can be enumerated in polynomial time as well. However the graph algorithm is not a complete solver for all dominance constraints.

• An implementation of a solver for dominance constraints can be obtained by employing constraint programming with finite sets. The constraints can be solved by always performing deterministic propagation steps to eliminate hopeless choices before making case distinctions.

• It can be shown that all dominance constraints that are needed for the linguistic application belong to a fragment called normal dominance constraints. Satisfiability of a normal constraint can be checked by a graph algorithm of polynomial runtime; each reading can be enumerated in polynomial time as well. However the graph algorithm is not a complete solver for all dominance constraints.

• Conclusion• Conclusion• Computational Aspects• Computational Aspects

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• Computational Aspects• Computational Aspects

• CLLS allows the representation of scope ambiguities, anaphora and ellipsis in simple underspecified structures that are transparent and suitable for processing.

• We have shown that CLLS correctly represents many notorious problems from the literature involving scope, anaphora, ellipses and their interactions.

• Furthermore CLLS can be used to model reinterpretation (meaning shift) of aspect and NPs in an underspecified way.

• Nevertheless the linguistic coverage of CLLS still has to be extended.

• Various more formal aspects can also be pursued in the future.

• CLLS allows the representation of scope ambiguities, anaphora and ellipsis in simple underspecified structures that are transparent and suitable for processing.

• We have shown that CLLS correctly represents many notorious problems from the literature involving scope, anaphora, ellipses and their interactions.

• Furthermore CLLS can be used to model reinterpretation (meaning shift) of aspect and NPs in an underspecified way.

• Nevertheless the linguistic coverage of CLLS still has to be extended.

• Various more formal aspects can also be pursued in the future.

ConclusionConclusion

• Conclusion• Conclusion• Conclusion• Conclusion

19.01.2005 49

Thank you!Thank you!

• Conclusion• Conclusion

Documents

Markus Egg, Alexander Koller, Joachim Niehren The Constraint Language for Lambda Structures Ventsislav Zhechev SfS, Universität Tübingen e-mail: [email protected]@sfs.uni-tuebingen.de