A Model-Theoretic Framework for Grammaticality Judgements Denys Duchier Jean-Philippe Prost Thi-Bich-Hanh Dao LIFO, Universit´ e d’Orl´ eans Formal Grammars, 2009 Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Foreword ungrammatical utterances are an everyday phenomenon some utterances are more ungrammatical than others JP Prost’s PhD thesis [2008] contributions: model-theoretic semantics for property grammars loose models for quasi-expressions scoring functions for comparative judgements of admissibility Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Outline Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Sentences of decreasing acceptability Les employ´ es ont rendu un rapport tr` es complet ` a leur employeur [100%] 1 The employees have sent a report very complete to their employer Les employ´ es ont rendu rapport tr` es complet ` a leur employeur [92.5%] 2 The employees have sent report very complete to their employer 3 Les employ´ es ont rendu un rapport tr` es complet ` a [67.5%] The employees have sent a report very complete to 4 Les employ´ es un rapport tr` es complet ` a leur employeur [32.5%] The employees a report very complete to their employer 5 Les employ´ es un rapport tr` es complet ` a [5%] The employees a report very complete to their employer Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Gradience We are interested in two questions: given an expression or a quasi-expression: what is the best (quasi-)analysis for it? how grammatical is it? Bas Aarts [2007]: intersective gradience (classification) subsective gradience (prototypicality) Examples: bat, pinguin Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Possible models for a quasi-expression *VP S NP V V *PP NP NP N a emprunt´ e NP P N V V *PP has taken Marie a emprunt´ e NP P Marie D AP N pour Marie has taken Marie on D AP pour un Adv A chemin on a path un Adv A N tr` es long a very long tr` es long chemin very long path *PP NP VP P N V V NP pour on Marie a emprunt´ e D AP N Marie has taken un Adv A chemin a path tr` es long very long Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Models for related quasi-expressions S NP VP D N V V *NP PP les employ´ es ont rendu N AP P NP the employees have sent rapport Adv A ` a D N report to tr` es complet leur employeur very complete their employer *Star NP *PP D N NP P les employ´ es D N AP a ` the employees to un rapport Adv A a report tr` es complet very complete Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
(some) Formal options GES/MTS (Pullum&Scholtz [2001], Pullum [2007]) GES: ill-suited MTS: grammar = constraint defined in terms of satisfaction (open to violations) compatible with degrees of ungrammaticality OT (Prince&Smolensky [1993]) grammaticality = optimality cannot distinguish between expressions and quasi-expressions Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Property grammars (Blache [2001]) Property Grammars Property Grammars are the transposition of phrase structure grammars from the GES perspective into the MTS perspective Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Production rules as constraints NP → D N GES: rewrite rule MTS: constraint satisfied in a tree iff satisfied at every node satisfied at a node iff: either the node is not labeled with NP , or it has exactly 2 children, the 1st labeled with D , the 2nd labeled with N Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Model-theoretic semantics for CFG A CFG is a set of production rules (1 per non-terminal; use alternation where necessary) class of models: trees labeled with categories a tree is a model of the grammar iff every rule is satisfied at every node α → β 1 . . . β n is satisfied at a node iff: either the node does not have category α , or it has a sequence of exactly n children labeled respectively β 1 through β n Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Coarse-grained constraints NP → D N For a NP there must be: (1) a D child (2) only one (3) a N child (4) only one (5) nothing else (6) the D child must precede the N child Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Properties obligation A : △ B at least one B child uniqueness A : B ! at most one B child linearity A : B ≺ C a B child precedes a C child requirement A : B ⇒ C if there is a B child, then also a C child exclusion A : B �⇔ C B and C children are mutually exclusive constituency A : S ? the category of any child must be one in S Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Fine-grained constraints NP → D N becomes: (1) NP : △ D (a D child) (2) NP : D ! (only one) (3) NP : △ N (a N child) (4) NP : N ! (only one) (5) NP : { D , N } ? (nothing else) (6) NP : D ≺ N (the D child must precede the N child) these can be independently violated Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Property Grammar for French S (Utterance) AP (Adjective Phrase) PP (Propositional Phrase) obligation : △ VP obligation : △ ( A ∨ V [past part] ) obligation : △ P uniqueness : NP ! uniqueness : P ! uniqueness : A ! : VP ! : NP ! : V [past part] ! linearity : NP ≺ VP linearity : P ≺ NP : Adv ! dependency : NP � VP : P ≺ VP linearity : A ≺ PP requirement : P ⇒ NP : Adv ≺ A dependency : P � NP exclusion : A �⇔ V [past part] NP (Noun Phrase) VP (Verb Phrase) obligation : △ ( N ∨ Pro ) obligation : △ V uniqueness : D ! uniqueness : V [main past part] ! : N ! : NP ! : PP ! : PP ! : Pro ! linearity : V ≺ NP linearity : D ≺ N : V ≺ Adv : D ≺ Pro : V ≺ PP : D ≺ AP requirement : V [past part] ⇒ V [aux] : N ≺ PP exclusion : Pro [acc] �⇔ NP requirement : N ⇒ D : Pro [dat] �⇔ Pro [acc] : AP ⇒ N dependency : V � � Pro � exclusion : N �⇔ Pro type pers pers 1 dependency : N � � D case nom � � � num 2 gend 1 gend 1 pers 1 num num 2 2 num 2 Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Formal definition of property grammars L a finite set of labels, S a finite set of strings P L = { c 0 : c 1 ≺ c 2 , c 0 : △ c 1 , c 0 : c 1 ! , c 0 : c 1 ⇒ c 2 , c 0 : c 1 �⇔ c 2 , c 0 : s 1 ? | ∀ c 0 , c 1 , c 2 ∈ L , ∀ s 1 ⊆ L} Property grammar G = ( P G , L G ) P G ⊆ P L L G ⊆ L × S Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Semantics of PG by interpretation over syntax tree structures syntax tree τ = ( D τ , L τ , R τ ) tree domain D τ labeling function L τ : D τ → L realization function R τ : D τ → S ∗ tree domain a finite subset of N ∗ 0 closed for prefixes and for left-siblings, where N 0 = N \ { 0 } arity A τ ( π ) = max { 0 } ∪ { i ∈ N 0 | π i ∈ D τ } Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Instances of Properties Every property in P G must be checked at every node in D τ and for all possible choices among its children. I τ [ [ c 0 : c 1 ≺ c 2 ] ] = { ( c 0 : c 1 ≺ c 2 )@ � π, π i , π j � | ∀ π, π i , π j ∈ D τ , i � = j } Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Instances of Properties Every property in P G must be checked at every node in D τ and for all possible choices among its children. I τ [ [ G ] ] = ∪{I τ [ [ p ] ] | ∀ p ∈ P G } I τ [ [ c 0 : c 1 ≺ c 2 ] ] = { ( c 0 : c 1 ≺ c 2 )@ � π, π i , π j � | ∀ π, π i , π j ∈ D τ , i � = j } I τ [ [ c 0 : △ c 1 ] ] = { ( c 0 : △ c 1 )@ � π � | ∀ π ∈ D τ } I τ [ [ c 0 : c 1 !] ] = { ( c 0 : c 1 !)@ � π, π i , π j � | ∀ π, π i , π j ∈ D τ , i � = j } I τ [ [ c 0 : c 1 ⇒ s 2 ] ] = { ( c 0 : c 1 ⇒ s 2 )@ � π, π i , π j � | ∀ π, π i , π j ∈ D τ , i � = j } I τ [ [ c 0 : c 1 �⇔ c 2 ] ] = { ( c 0 : c 1 �⇔ c 2 )@ � π, π i , π j � | ∀ π, π i , π j ∈ D τ , i � = j } I τ [ [ c 0 : s 1 ? ] ] = { ( c 0 : s 1 ? )@ � π, π i � | ∀ π, π i ∈ D τ } Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
Pertinence P τ (( c 0 : c 1 ≺ c 2 )@ � π, π i , π j � ) ≡ L τ ( π ) = c 0 ∧ L τ ( π i ) = c 1 ∧ L τ ( π j ) = c 2 Denys Duchier, Jean-Philippe Prost, Thi-Bich-Hanh Dao A Model-Theoretic Framework for Grammaticality Judgements
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