Kallmeyer/Maier Sommersemester 2009 Kallmeyer/Maier Sommersemester 2009 Elimination of useless rules (1) Boullier (1998) A useless rule is a rule which cannot be used in a derivation Parsing beyond context-free grammar: ∗ S ( � 0 , n � ) ⇒ ε for any w ∈ T ∗ . Elimination (similar as in the CFG case) in two steps: Simple RCG: 1. Compute the set N T of all symbols A ∈ N that can lead to a Simplifying the Grammar tuple of terminal strings: A ( � α ) → ε ∈ P Laura Kallmeyer, Wolfgang Maier [ A ] Sommersemester 2009 [ A 1 ] , . . ., [ A m ] α ) → A 1 ( � α m ) ∈ P A ( � α 1 ) . . . A m ( � [ A ] Rules that contain non-terminals not in N T are eliminated. Parsing beyond CFG 1 LCFRS/MCFG/sRCG III Parsing beyond CFG 3 LCFRS/MCFG/sRCG III Kallmeyer/Maier Sommersemester 2009 Kallmeyer/Maier Sommersemester 2009 Elimination of useless rules (2) 2. In the resulting simple RCG, eliminate unreachable rules: Compute the set N S of non-terminals reachable from S : Overview [ A ] 1. Elimination of useless rules A ( � α ) → A 1 ( � α 1 ) . . .A m ( � α m ) ∈ P [ S ] [ A 1 ] , . . ., [ A m ] 2. Elimination of ε -Rules Rules that contain non-terminals not in N S are eliminated. 3. Ordering 4. Binarization 5. Conclusion Parsing beyond CFG 2 LCFRS/MCFG/sRCG III Parsing beyond CFG 4 LCFRS/MCFG/sRCG III
Kallmeyer/Maier Sommersemester 2009 Kallmeyer/Maier Sommersemester 2009 Elimination of ε -Rules (1) Elimination of ε -Rules (3) An ε -rule is a rule with one of the lhs arguments being ε . The set N ǫ is constructed recursively with initial value ∅ : 1. For every A ( x 1 , . . ., x k ) → ǫ ∈ P : A simple RCG is ǫ -free if ι ) to N ǫ with for all 1 ≤ i ≤ k : � Add ( A,� ι ( i ) = 0 if x i = ε , else • it either contains no ǫ -rules ι ( i ) = 1 for all 1 ≤ i ≤ k . � • or there is exactly one rule S ( ǫ ) → ǫ and S does not appear in 2. Repeat until N ǫ does not change any more: any rhs of a rule in G For every A ( x 1 , . . ., x k ) → A 1 ( α 1 ) . . .A m ( α m ) and all ( A 1 ,� ι 1 ), . . . , ( A m ,� ι m ) ∈ N ǫ : • Calculate a vector ( x ′ 1 , . . ., x ′ k ) from ( x 1 , . . ., x k ) by removing every variable that is the j th variable of A i in the righthand side such that � ι i ( j ) = 0 with ε . • Then add ( A,� ι ) to N ǫ with for all 1 ≤ i ≤ k : � ι ( i ) = 0 if x ′ i = ǫ , else � ι ( i ) = 1. Parsing beyond CFG 5 LCFRS/MCFG/sRCG III Parsing beyond CFG 7 LCFRS/MCFG/sRCG III Kallmeyer/Maier Sommersemester 2009 Kallmeyer/Maier Sommersemester 2009 Elimination of ε -Rules (2) Elimination of ε -Rules (4) Now we can compute the new set of rules: First, compute for all A ∈ N , all possibilities to have ε -components in their yields: For every clause A ( x 1 , . . ., x k ) → A 1 ( x (1) 1 , . . ., x (1) k 1 ) . . .A m ( x ( m ) , . . ., x ( m ) k m ) and all ι ∈ { 0 , 1 } dim ( A ) , and • We introduce vectors � 1 ( A 1 ,� ι 1 ), . . . , ( A m ,� ι m ) ∈ N ǫ , we compute a clause for the new • we compute the set N ε of pairs ( A,� ι ) where � ι signifies that it is grammar: possible for A to have a tuple τ in its yield with τ ( i ) = ε if • In the rhs, we replace A i with ( A i ,� ι i ) for all 1 ≤ i ≤ m ; � ι ( i ) = 0 and τ ( i ) � = ε if � ι ( i ) � = 0. • in both, lhs and rhs, we delete all variables x ( i ) for 1 ≤ i ≤ m j and 1 ≤ j ≤ k i with � ι i ( j ) = 0; • in the lhs, we replace A with ( A,� ι ) where � ι ( l ) = 0 iff the i th argument is ε ; • finally delete all ε -arguments in lhs and rhs. Furthermore, we add a new start symbol ( S ′ ) with S ′ ( X ) → S 1 ( X ) and, if ε in the language, S ′ ( ε ) → ε . Parsing beyond CFG 6 LCFRS/MCFG/sRCG III Parsing beyond CFG 8 LCFRS/MCFG/sRCG III
Kallmeyer/Maier Sommersemester 2009 Kallmeyer/Maier Sommersemester 2009 Elimination of ε -Rules (5) Ordered simple RCG (2) Transformation into an ordered simple RCG: Example: Original simple RCG rules: P ′ := P ; S ( XY ) → A ( X, Y ), A ( a, ε ) → ε , A ( ε, a ) → ε , A ( a, b ) → ε repeat until P ′ does not change any more: Set of pairs characterizing possibilities for ε -components: for all rules r = A ( � α ) → A 1 ( � α 1 ) . . . A k ( � α k ) in P ′ : for all i , 1 ≤ i ≤ k : N ε = { ( S, 1) , ( A, 10) , ( A, 01) , ( A, 11) } if A i ( � α i ) = A i ( Y 1 , . . . , Y dim ( A i ) ) and Rules after ε -elimination: the order of the Y 1 , . . . , Y dim ( A i ) in � α is S ′ ( X ) → S 1 ( X ), p ( Y 1 , . . . , Y dim ( A i ) ) where p is not the identity S 1 ( X ) → A 10 ( X ), A 10 ( a ) → ε , then α i ) in r with A p replace A i ( � i ( p ( � α i )) ; S 1 ( X ) → A 01 ( X ), A 01 ( b ) → ε , for every A i -rule A i ( � γ ) → Γ ∈ P ′ : S 1 ( XY ) → A 11 ( X, Y ), A 11 ( a, b ) → ε add a new rule A p i ( p ( � γ )) → Γ to P ′ (if not yet in P ′ ) Parsing beyond CFG 9 LCFRS/MCFG/sRCG III Parsing beyond CFG 11 LCFRS/MCFG/sRCG III Kallmeyer/Maier Sommersemester 2009 Kallmeyer/Maier Sommersemester 2009 Ordered simple RCG (1) Ordered simple RCG (3) A simple RCG is ordered if for every rule Example: A ( � α ) → A 1 ( � α 1 ) . . .A k ( � α k ) Original clauses: S ( XY ) → A ( X, Y ), A ( X, Y ) → A ( Y, X ), A ( aX, bY ) → A ( X, Y ), and every A i ( � α i ) = A i ( Y 1 , . . ., Y dim ( A i ) ) (1 ≤ i ≤ k ), the order of A ( a, b ) → ε the components of � α i in � α is Y 1 , . . ., Y dim ( A i ) . Clauses after transformation into ordered RCG: Crucially, in an ordered simple RCG, the order of the components S ( XY ) → A ( X, Y ), A ( X, Y ) → A p ( X, Y ), A ( aX, bY ) → A ( X, Y ), of the lhs predicate of a rule corresponds always to their order in A ( a, b ) → ε , A p ( Y, X ) → A ( Y, X ), A p ( bY, aX ) → A p ( Y, X ), the input. A p ( b, a ) → ε ( p being the permutation that switches the two arguments) For every simple RCG, there exists a weakly equivalent ordered simple RCG. For the transformation, in addition to the A ∈ N , we introduce new predicates A p where p a permutation of � 1 , . . ., dim ( A ) � . Parsing beyond CFG 10 LCFRS/MCFG/sRCG III Parsing beyond CFG 12 LCFRS/MCFG/sRCG III
Kallmeyer/Maier Sommersemester 2009 Kallmeyer/Maier Sommersemester 2009 Binarization (1) Binarization (3) Binarization algorithm: We call the length of the rhs of a rule its rank . The rank of a simple RCG is given by the maximal rank of its rules. α ) → A 0 ( � for all rules r = A ( � α 0 ) . . . A m ( � α m ) in P with m > 1 : remove r from P ; For every simple RCG, there is an equivalent simple RCG of rank 2. R := ∅ ; CNF binarization for CFG: A → BCD replaced with A → BE , pick new predicate names C 1 , . . . , C m − 1 ; E → CD add A ( � α ) → A 0 ( � α 0 ) C 1 ( � γ 1 ) to R where � γ 1 is � α reduced with � α 0 ; A similar binarization algorithm can be applied to LCFRS/simple for all i , 1 ≤ i < m − 1 : RCG: add C i ( � γ i ) → A i ( � α i ) C i +1 ( � γ i +1 ) to R where For every rule with rhs length > 2, we introduce a new predicate γ i +1 is � � γ i reduced with � α i ; that comprises all rhs predicates except the first. This is done add C m − 1 ( γ m − 2 ) → A m − 1 ( α m − 1 ) A m ( � α m ) to R ; � � repeatedly. for every r ′ ∈ R : replace rhs arguments of length > 1 with new variables (in both sides) and add the result to P Parsing beyond CFG 13 LCFRS/MCFG/sRCG III Parsing beyond CFG 15 LCFRS/MCFG/sRCG III Kallmeyer/Maier Sommersemester 2009 Kallmeyer/Maier Sommersemester 2009 Binarization (2) Binarization (4) α 1 ∈ [( T ∪ V ) ∗ ] k 1 by We define the reduction of a � Example: Original simple RCG: � X 1 , . . ., X k 2 � ∈ V k 2 where all X i for 1 ≤ i ≤ k 2 occur in � α 1 as the S ( XY ZUV W ) → A ( X, U ) B ( Y, V ) C ( Z, W ) following vector of variables: A ( aX, aY ) → A ( X, Y ) B ( bX, bY ) → B ( X, Y ) C ( cX, cY ) → C ( X, Y ) We take all variables from � α 1 (in their order) that are not in A ( a, a ) → ε B ( b, b ) → ε C ( c, c ) → ε { X 1 , . . ., X k 2 } while starting a new component whenever a variable is, in � α 1 , in a different component than the preceding variable in For the rule with righthand side of length > 2 we obtain the result or in the same component but not adjacent to it. R = { S ( XY ZUV W ) → A ( X, U ) C 1 ( Y Z, V W ), C 1 ( Y Z, V W ) → B ( Y, V ) C ( Z, W ) } Examples: 1. � aX 1 , X 2 , bX 3 � reduced with � X 2 � yields � X 1 , X 3 � ; Equivalent simple RCG of rank 2: S ( XPUQ ) → A ( X, U ) C 1 ( P, Q ) C 1 ( Y Z, V W ) → B ( Y, V ) C ( Z, W ) 2. � aX 1 X 2 bX 3 � reduced with � X 2 � yields � X 1 , X 3 � . A ( aX, aY ) → A ( X, Y ) B ( bX, bY ) → B ( X, Y ) C ( cX, cY ) → C ( X, Y ) A ( a, a ) → ε B ( b, b ) → ε C ( c, c ) → ε Parsing beyond CFG 14 LCFRS/MCFG/sRCG III Parsing beyond CFG 16 LCFRS/MCFG/sRCG III
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