Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Residuated lattices in syntactic description Alexander Clark Department of Computer Science Royal Holloway, University of London June 2011 QMUL
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Most important problem of linguistics Chomsky’s questions What constitutes knowledge of a language? 1 How is this knowledge acquired by its speakers? 2
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Tension Chomsky, 1986 To achieve descriptive adequacy it often seems necessary to enrich the system of available devices, whereas to solve our case of Plato’s problem we must restrict the system of available devices so that only a few languages or just one are determined by the given data. It is the tension between these two tasks that makes the field an interesting one, in my view.
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Tension Chomsky, 1986 To achieve descriptive adequacy it often seems necessary to enrich the system of available devices, whereas to solve our case of Plato’s problem we must restrict the system of available devices so that only a few languages or just one are determined by the given data. It is the tension between these two tasks that makes the field an interesting one, in my view. Boeckx and Piattelli-Palmarini (2005) "the primary contribution of P&P , in the present connection, was to divorce questions of learning entirely from the question of the “format for grammar”"
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Distributional lattice grammars A recently developed grammatical formalism that tries to resolve this tension: efficient learnability cubic parsing algorithm slightly context sensitive based on a residuated lattice (the syntactic concept lattice)
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Empiricist models Slogan The structure of the representation should be based on the structure of the language, not something arbitrarily imposed on it from outside. Congruence based approaches: DFAs based on the Myhill-Nerode congruence (Angluin, 1982,1987) CFGs based on the syntactic congruence (Clark and Eyraud, 2007, Clark, 2010) MCFGs based on congruence of tuples (Yoshinaka, 2009) Lattice based approaches based on the syntactic concept lattice.
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Recall (or otherwise) Monoid � M , ◦ , 1 � , ◦ is associative and 1 ◦ u = u = u ◦ 1 Example: strings u ◦ v = uv , 1 is empty string Bounded lattice � M , ∧ , ∨ , ⊤ , ⊥� Example: powerset lattice 2 X , ∨ = ∪ , ∧ = ∩ , ⊥ = ∅ Lattice ordered monoid M is a lattice and a monoid such that X ≤ Y , P ≤ Q means X ◦ P ≤ Y ◦ Q
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Slightly stronger condition Residuation operations X ◦ Y ≤ Z iff X ≤ Z / Y iff Y ≤ X \ Z Z / Y = max { X | X ◦ Y ≤ Z } Example Set of all subsets of a monoid X ◦ Y = { xy | x ∈ X , y ∈ Y } Specifically if monoid is Σ ∗ we have the lattice of all languages over Σ , 1 = λ , ⊤ = Σ ∗ , ⊥ = ∅ Denote this by 2 Σ ∗
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Residuated lattices Appear twice Algebraic underpinning for DLGs Models for substructural logics and the Lambek calculus Questions in this talk Is this a coincidence? What is the relationship? How does this relate to the proof theory/model theory argument?
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Rules of inference Correspondence Inferences in the associative Lambek calculus AL are theorems about residuated lattices: Lambek calculus is sound w.r.t residuated lattices. Lambek calculus residuated lattices x ( yz ) → ( xy ) z ( X ◦ Y ) ◦ Z = X ◦ ( Y ◦ Z ) ( x / y ) y → x ( X / Y ) ◦ Y ≤ X x → y / ( x \ y ) X ≤ ( Y / ( X \ Y )) ( x / y )( y / z ) → x / z ( X / Y ) ◦ ( Y / Z ) ≤ ( X / Z )
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Rules of inference Correspondence Inferences in the associative Lambek calculus AL are theorems about residuated lattices: Lambek calculus is sound w.r.t residuated lattices. Lambek calculus residuated lattices x ( yz ) → ( xy ) z ( X ◦ Y ) ◦ Z = X ◦ ( Y ◦ Z ) ( x / y ) y → x ( X / Y ) ◦ Y ≤ X x → y / ( x \ y ) X ≤ ( Y / ( X \ Y )) ( x / y )( y / z ) → x / z ( X / Y ) ◦ ( Y / Z ) ≤ ( X / Z ) X ≤ X ◦ ( Y / Y ) ? X ∧ Y ≤ X
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Distributions The substring context relation Context (or environment ) A context is just a pair of strings ( l , r ) ∈ Σ ∗ × Σ ∗ . ( l , r ) ⊙ u = lur ( l , r ) ⊙ ( x , y ) = ( lx , yr ) Distribution of a string in a language ( l , r ) ∼ L u iff lur ∈ L C L ( u ) = { ( l , r ) | lur ∈ L } = { f | f ⊙ u ∈ L } ( λ, λ ) ∈ C L ( u ) iff u ∈ L
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Syntactic Concept Lattice Galois connection of the substring context relation S is a set of strings, and C is a set of contexts. Polar maps S ′ = { ( l , r ) : ∀ w ∈ S lwr ∈ L } C ′ = { w : ∀ ( l , r ) ∈ C lwr ∈ L } L = { ( λ, λ ) } ′ Closure operator S ′′ ⊇ S If S ′′ = S then S is a closed set of strings Always true that S ′′′ = S ′
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Concepts Formal Concept Analysis Concept A syntactic concept is an ordered pair � S , C � . where C ′ = S and S ′ = C . Alternatively: maximal sets such that C ⊙ S ⊆ L . Defined equally by closed sets of strings.
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Basic properties Partial order � S 1 , C 1 � ≤ � S 2 , C 2 � iff S 1 ⊆ S 2 iff C 1 ⊇ C 2 Lattice The set of concepts of a language form a complete lattice � S x , C x � ∧ � S y , C y � = � S x ∩ S y , ( S x ∩ S y ) ′ � Finite iff L is regular Typical concepts C ( w ) = �{ w } ′′ , { w } ′ � Language L = � L , { ( λ, λ ) } ′′ � = C ( L ) = C (( λ, λ )) Top ⊤ = � Σ ∗ , ∅� Bottom ⊥ = �∅ , Σ ∗ × Σ ∗ � Unit 1 = C ( λ )
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis L = ( ab ) ∗ ⊤ = � Σ ∗ , ∅� � [ a ] , [ λ, b ] , � � [ b ] , [ a, λ ] � L = � [ ab ] ∪ [ λ ] , [ λ, λ ] � � [ ba ] ∪ [ λ ] , [ a, b ] � 1 = � [ λ ] , [ a, b ] ∪ [ λ, λ ] � ⊥ = �∅ , Σ ∗ × Σ ∗ �
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Concatenation Definition � S x , C x � ◦ � S y , C y � = � ( S x S y ) ′′ , ( S x S y ) ′ � = C ( S x S y ) The smallest concept that contains the concatenation of the sets S x and S y . Observation w = a 1 . . . a n is in L iff C ( a 1 ) ◦ · · · ◦ C ( a n ) ≤ L
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Difference L = { ab , c } Free RL 2 Σ ∗ { a } ◦ { b } = { ab } Syntactic Concept Lattice { a } ◦ { b } = { ab , c }
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Residuated lattice This is a complete residuated lattice; written B ( L ) . Concatenation is a monoid: associative and with unit C ( { λ } ) . Suppose X = � S x , C x � and Y = � S y , C y � are concepts. X / Y = C ( C x ⊙ ( λ, S y )) Y \ X = C ( C x ⊙ ( S y , λ ))
Introduction Syntactic concept lattice Distributional lattice grammars Categorial grammars Synthesis Residuated lattice This is a complete residuated lattice; written B ( L ) . Concatenation is a monoid: associative and with unit C ( { λ } ) . Suppose X = � S x , C x � and Y = � S y , C y � are concepts. X / Y = C ( C x ⊙ ( λ, S y )) Y \ X = C ( C x ⊙ ( S y , λ )) Residuation Suppose all elements of X occur in a context ( l , r ) ; Suppose u is in X / Y and v is in Y So uv occurs in ( l , r ) – luvr ∈ L u must occur in context ( l , vr ) which is ( l , r ) ⊙ ( λ, v )
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