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Kallmeyer/Maier ESSLLI 2008 Kallmeyer/Maier ESSLLI 2008 Range Concatenation Grammar The idea behind range concatenation grammar (RCG) is comparable to the idea behind MCFG. Predicate-rewriting clauses describe ranges which are not Parsing

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Kallmeyer/Maier ESSLLI 2008

Parsing beyond context-free grammar: Range Concatenation Grammar Parsing

Laura Kallmeyer, Wolfgang Maier University of T¨ ubingen ESSLLI Course 2008

Parsing beyond CFG 1 RCG Parsing Kallmeyer/Maier ESSLLI 2008

Overview

1. Range Concatenation Grammars (RCG)
2. Parsing RCG

(a) Directional top-down parsing (b) Earley-style parsing

3. Uses of RCG

Parsing beyond CFG 2 RCG Parsing Kallmeyer/Maier ESSLLI 2008

Range Concatenation Grammar The idea behind range concatenation grammar (RCG) is comparable to the idea behind MCFG.

Predicate-rewriting clauses describe ranges which are not

necessarily adjacent.

One predicate can be true or false for a certain string.
Some string w is in the language of an RCG if the start

predicate is true for w.

While in MCFG, a string is generated, in RCG, a string is

reduced to ǫ.

Parsing beyond CFG 3 RCG Parsing Kallmeyer/Maier ESSLLI 2008

Expressivity of RCG

RCG exactly covers the class of PTIME recognizable languages

(Bertsch&Nederhof, 2001).

Simple RCG (basically non-deleting non-copying RCG) is

equivalent to MCFG

RCG can represent languages beyond mild context-sensitivity

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Definition of RCGs: Grammar Definition A RCG is a tuple G = N, T, V, P, S such that

N is a finite set of predicates, each with a fixed arity,
T and V are disjoint finite sets of terminals and variables,
S ∈ N is the start predicate of arity 1, and
P is a finite set of clauses of the form

A0(x01, . . ., x0a0) → ǫ

A0(x01, . . ., x0a0) → A1(x11, . . ., x1a1) . . .An(xn1, . . ., xnan) with n ≥ 1 and Ai ∈ N, xij ∈ (T ∪ V )∗ and ai being the arity of

Ai. A predicate An(xn1, . . ., xnan) can be written as An(

xn)

Parsing beyond CFG 5 RCG Parsing Kallmeyer/Maier ESSLLI 2008

Definition of RCGs: Instantiation A given clause C is instantiated with respect to a string w if variables and arguments are consistently replaced by ranges of w. Example:

A(i . . .j) → B(i + 1 . . .j)

is an instantiation of the clause

A(aX1) → B(X1)

if wi+1 = a.

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Definition of RCGs: Derivation Relation, Language

The derivation relation is defined as follows:

For a predicate A of arity k, a clause A(. . .) → . . ., and ranges i1, j1, . . ., ik, jk with respect to a given w: if there is an instantiation of this clause with LHS A(i1, j1, . . ., ik, jk), then A(i1, j1, . . ., ii, jk) can be replaced with the RHS of this instantiation.

The language of an RCG G is the set of strings that can be

reduced to the empty word: L(G) = {w | S(0, |w|)

∗

⇒ ǫ with respect to w}.

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A sample RCG (1) Sample RCG G for the string language {anbkan | k, n ∈ IN}: An RCG with N = {S, A, B}, T = {a, b}, V = {X, Y, Z}, start predicate S and clauses

S(X Y Z) → A(X, Z) B(Y ),
A(a X, a Y ) → A(X, Y ),
B(b X) → B(X),
A(ǫ, ǫ) → ǫ,
B(ǫ) → ǫ

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A sample RCG (2) As an example consider the reduction of w = aabaa: S(X Y Z) → A(X, Z) B(Y ) w0,2 w2,3 w3,5 w0,2 w3,5 w2,3 aa b aa aa aa b With this instantiation, S(w0,5) ⇒ A(w0,2, w3,5)B(w2,3). Then

Parsing beyond CFG 9 RCG Parsing Kallmeyer/Maier ESSLLI 2008

A sample RCG (3) B(b X) → B(X) w2,3 w3,3 w3,3 b ǫ ǫ and B(ǫ) → ǫ lead to A(w0,2, w3,5)B(w2,3) ⇒ A(w0,2, w3,5)B(w3,3) ⇒ A(w0,2, w3,5).

Parsing beyond CFG 10 RCG Parsing Kallmeyer/Maier ESSLLI 2008

A sample RCG (4) A(a X, a Y ) → A(X, Y ) w0,1 w1,2 w3,4 w4,5 w1,2 w4,5 a a a a a a leads to A(w0,2, w3,5) ⇒ A(w1,2, w4,5). Then

Parsing beyond CFG 11 RCG Parsing Kallmeyer/Maier ESSLLI 2008

A sample RCG (5) A(a X, a Y ) → A(X, Y ) w1,2 w2,2 w4,5 w5,5 w2,2 w5,5 a ǫ a ǫ ǫ ǫ and A(ǫ, ǫ) → ǫ lead to A(w1,2, w4,5) ⇒ A(w2,2, w5,5) ⇒ ǫ

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Definition of RCGs: Other properties (1)

An RCG with maximal predicate arity k is called an RCG of

arity k (also called a k-RCG).

An RCG is called non-combinatorial if each of the arguments

in the right-hand sides of the productions are single variables.

An RCG is called linear if no variable appears more than once

in the left-hand sides of the productions and no variable appears more than once in the right-hand side of the productions.

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Definition of RCGs: Other properties (2)

An RCG is called non-erasing if for each production, each

variable occurring in the left-hand side occurs also in the right-hand side and vice versa.

An RCG is called simple if it is non-combinatorial, linear and

non-erasing.

A simple RCG is called ordered simple if the range variables

are ordered the same way in the RHS and the LHS predicates. Ordered simple RCG is equivalent to simple RCG.

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RCG parsing: Treatment of terminals Without loss of generality, we presuppose that all non-ǫ clauses contain no terminals in their arguments. For each t ∈ T, we introduce a new clause Tt(t) → ǫ and for each clause C ∈ P,

we replace each occurrence t′ of t in all arguments of all

predicates with a variable Vt′,

for each Vt′, we add the predicate Tt(Vt′) to the RHS of C.

Furthermore, for all clauses we assume that its variables are continuously numbered from 1 to some j.

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RCG parsing: Range vectors We will use range vectors similar to those used for MCFG parsing. Range vectors are used to describe variable bindings.

φ = (x1, y1, . . ., xk, yk) is a range vector in w if all xi, yi

are ranges in w for 1 ≤ i ≤ k.

φ = (x1, y1, . . ., xk, yk) is a range constraint vector if it

contains pairs x, y where x, y ∈ Pos(w) ∪ Vr (Vr is a set {r1, r2, . . .} of range boundary variables) such that if x, y ∈ Pos(w)2 then it is a range.

k is called the dimension of φ
φ(i).l denotes then the first component and φ(i).r the second

component of the ith element of φ.

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RCG parsing: Variable constraint vectors The variable constraint vector φ of a non-ǫ clause A( x) → Φ is a range constraint vector of dimension j, j being the highest variable index in the clause. It contains only x ∈ Vr × Vr and must be consistent with variable adjacencies in the clause. Formally, the elements of φ are pairs from Vr × Vr such that φ(h).r = φ(i).l iff XhXi occurs as a substring in one of the arguments of the clause.

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Update of range vectors We define an update φ′ of a range constraint vector φ with respect to an identity x = y, x, y ∈ Pos(w) ∪ Vr as follows:

if x = y, then φ′ = φ;
else if x ∈ Vr and the result ψ of replacing all occurrences of x

in φ with y is a range constraint vector, then φ′ = ψ;

else if y ∈ Vr and the result ψ of replacing all occurrences of y

in φ with x is a range constraint vector, then φ′ = ψ;

otherwise, φ′ is undefined.

Parsing beyond CFG 18 RCG Parsing Kallmeyer/Maier ESSLLI 2008

Directional top-down parsing Corresponds to the algorithm presented in Boullier (2000). Item form:

Active items: [A(

X) → Φ • Ψ, φ]

Passive items: [A, ψ, flag]

where

φ is a range vector of dimension j, j being the highest variable

index in the clause,

ψ is a range vector of dimension k, k being the arity of A,
flag= {p, c} indicates if a passive item is predicted or

completed.

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Directional top-down parsing (axiom and goal)

Axiom:

[S, (0, n), p]

The goal item is [S, (0, n), c].

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Directional top-down parsing (predict-rule) We have two predict operations. Predict-rule predicts an active item for a previously introduced passive item. [A, ψ, p] [A( x) → •Ψ, φ] thereby, the variable bindings in φ applied to x yield ψ. Furthermore, φ respects the adjacency constraints imposed by the variable constraint vector of the clause.

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Directional top-down parsing (predict-pred) Predict-pred predicts a passive item. [A(. . .) → Φ • B( x)Ψ, φ] [B, ψ, p] thereby, ψ results from applying φ to x.

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Directional top-down parsing (scan) Scan: [A, (l, r), p] [A, (l, r), c] A(x) → ǫ, l, r(w) = x

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Directional top-down parsing (complete) Complete moves the dot over a predicate in the RHS of an active item if the corresponding passive item has been completed. [B, φB, c], [A(. . .) → Φ • B( x)Ψ, φ] [A(. . .) → ΦB( x) • Ψ, φ] where φB must be the result of applying φ to x.

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Directional top-down parsing (convert) Once the dot has arrived at the right end of the RHS of a clause, we can convert the active item to a passive item. Convert: [A( x) → Φ•, φ] [A, φ, c]

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Earley-style parsing Presented in Kallmeyer&Maier (2009) (in preparation). Item form:

Active items: [A(

x) → Φ • Ψ, φ]

Passive items: [A, ψ, flag]

where

φ is a range constraint vector of dimension j, j being the

highest variable index in the clause,

ψ is a range constraint vector of dimension k, k being the arity
f A,
flag= {p, c} indicates if a passive item is predicted or

completed.

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Earley-style parsing (initialization and goal)

Initialize:

[S, (0, n), p]

The goal item is [S, (0, n), c].

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Earley-style parsing (predict) We have two predict operations. As for the top-down case, predict-rule predicts active items with the dot on the left of the RHS, for a given previously introduced passive item. [A, ψ, p] [A(x1 . . . y1, . . ., xk . . . yk) → •Ψ, φ′] where, starting from the variable constraint vector φ of the clause, we obtain φ′ by updating with the following identities: φ(xi).l = ψ(i).l, φ(yi).r = ψ(i).r for all 1 ≤ i ≤ k. Note the difference to the top-down case: We are now dealing with range constraint vectors, i.e., some variable boundaries remain unspecified.

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Earley-style parsing (predict-pred) Also as for the top-down case, predict-pred predicts a passive item for the predicate following the dot in an active item. [A(. . .) → Φ • B(x1 . . . y1, . . ., xk . . . yk)Ψ, φ] [B, ψ, p] where ψ(i).l = φ(xi).l, ψ(i).r = φ(yi).r for all 1 ≤ i ≤ k.

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Earley-style parsing (scan) Scan: [A, (l, r), p] [A, (l′, r′), c] A(x) → ǫ, l′, r′(w) = x, l, r compatible with l′, r′

Reduce a single terminal to ǫ, recall definition
Here, “compatible with” means that there is a function

f : {l, r} → {l′, r′} such that f(l) = l′, f(r) = r′ and f(x) = x if x ∈ Pos(w).

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Earley-style parsing (complete) Complete moves the dot over a predicate in the RHS of an active item if the corresponding passive item has been completed. [B, φB, c], [A( x) → Φ • B(x1 . . . y1, . . ., xk . . . yk)Ψ, φ] [A( x) → ΦB(x1 . . . y1, . . ., xk . . .yk) • Ψ, φ′] where φ′ is φ updated with all new constraint information coming from φB, i.e., φ′ is an update of φ wrt. the identities φ(xj).l = φB(j).l and φ(yj).r = φB(j).r for all 1 ≤ j ≤ k.

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Earley-style parsing (convert) Convert turns an active item with the dot at the end of the righthand side into a completed passive item [A(x1 . . . y1, . . ., xk . . . yk) → Ψ•, φ] [A, ψ, c] where ψ(i).l = φ(xi).l and ψ(i).r = φ(yi).r for all 1 ≤ i ≤ k.

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RCG as a tool One can use RCG as an intermediary device, resp. a pivot

formalism. We will see two applications:
RCG for TAG parsing
RCG for Syntax-Directed Machine Translation

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TAG → simple RCG A TAG can straightforwardly be converted into an RCG.

Introduce a predicate for each elementary tree
A predicate corresponding to

– an aux tree β has the form β(L, R), where L and R covers the yield of β to the left and the right of the footnode, including all material added to it – an initial tree α have the form α(X), with X covering the yield of α and all trees added to it

A predicate α/β reduces the input by determining which parts
f the string come from the α/β respectively and which parts

come from substituted/adjoined trees

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TAG → simple RCG: Example S SNA SNA ǫ a S b S S∗

a S∗

b Start predicate: S(X) → α(X) α(ǫ) → ǫ α(B1B2) → β1(B1, B2)|β2(B1, B2) β1(aB1, aB2) → β1(B1, B2)|β2(B1, B2) β2(bB1, bB2) → β1(B1, B2)|β2(B1, B2) β1(a, a) → ǫ β2(b, b) → ǫ

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RCG for MT

Binary 2-RCG can be used for efficient syntax-based machine

translation.

Intuitively, the first argument of a clause specifies the source

language input, while the parser determines the destination language via string variables, i.e., variables in the parser input that are instantiated by lexical items in parsing.

Main advantage over previous systems based on synchronous

versions of CFG/TAG/etc.: Higher expressivity through availability of copying/deleting while still in the same complexity class (O(n6)).

Refer to Søgaard (2008) (COLING 2008) for complete

presentation.

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RCG for MT: Example (1) Example grammar:

S(X1X2, Y1Y2) → NP(X1, Y1)VP(X2, Y2) VP(X1X2, Y1Y2Y3) → V (X1, Y1)ObjP(X2, Y2)Part(X1, Y3) VP(X1, Y1Y2) → V (X1, Y1)Part(X1, Y2) ObjP(X1, Y1Y2) → NP(X1, Y2)Prep(X1, Y1) NP(X1X2, Y1Y2) → Art(X1, Y1)N (X2, Y2) NP(he, er) → ǫ V (entered, trat) → ǫ Part(entered, ein) → ǫ Prep(the room, in) → ǫ Art(the, das) → ǫ N (room, Zimmer) → ǫ Parsing beyond CFG 37 RCG Parsing Kallmeyer/Maier ESSLLI 2008

RCG for MT: Example (2)

Call the parser with the input string w =He entered S, where

S is a string variable, and the start predicate S(X1X2, Y1Y2).

The algorithm should infer that S = Y1Y2 = trat ein in order