Parallel Systems as Mildly Context-Sensitive Grammar ... · Parallel Systems as Mildly Context-Sensitive Grammar Formalisms Non-context-free phenomena in natural languages • Structural

Parallel Systems as Mildly Context-Sensitive

Grammar Formalisms

Suna Bensch

Parallel Systems as Mildly Context-Sensitive Grammar Formalisms

Introduction

• Motivation for mildly context-sensitive grammar formalisms

Suna Bensch


Introduction


Sequential mildly context-sensitive grammar formalisms

Suna Bensch


Introduction



Parallel mildly context-sensitive grammar formalisms

Suna Bensch


Introduction




• Motivation for parallel mildly context-sensitive grammar formalisms

Suna Bensch


Introduction





• Notions: ET0L systems, k-uniformly-limited ET0L systems

Suna Bensch


Introduction






• Left-restricted k-uniformly-limited ET0L systems and languages

Suna Bensch


Introduction







• Results

Suna Bensch


Introduction







• Results

Conclusions

Suna Bensch


Non-context-free phenomena in natural languages

• Structural aspects, that cannot be described with context-free grammars mmms[Stuart Shieber, 1985] i.a.

– For example crossed dependencies in Swiss-German:

. . . mer em Hans es huus haend wele halfe aastriiche.

. . . we em Hans the house have wanted help paintDAT ACC“we have wanted to help Hans to paint the house”

• Crossed dependencies from a formal languages point of view:

L = { anbmcndm | n,m ≥ 1 }

Suna Bensch


Mildly context-sensitivity

• [Aravind Joshi, 1985]

• Characterisation of a class of grammar formalisms for an adequate sunaaaadescription of aspects in natural languages

• Certain non-context-free languages (crossed dependencies)

• Constant growth property or semilinearity

• Polynomial parsing

Suna Bensch



• Investigations on some sequential mildly context-sensitive grammar formalisms[Joshi et al., 1991], [Vijay-Shanker et al., 1994], u.a.:

L(CS)6

L(LCFRS) = L(MCTAG)6

L(TAG)= L(HG)= L(LIG)= L(CCG)6

L(CF)

'

&

$

%

Context-free grammars (CF), Tree adjoining grammars (TAG), Linear indexed grammars (LIG),

Head grammars (HG), Combinatory categorial grammars (CCG), Context-sensitive grammmars (CS),

Linear context-free rewriting systems (LCFRS), Multiple component TAGs (MCTAG)

Suna Bensch


Definition [Aravind Joshi, 1985], [Henning Bordihn, 2004]:

A grammar family GF is said to be mildly context-sensitive if the following sunaconditions are satisfied:

• for every context-free language L, there is a grammar G in GF with

L(G) = L; and there are grammars G1, G2, G3 in GF , such that

L(G1) = {anbncn | n ≥ 1},

L(G2) = {anbmcndm | n,m ≥ 1},

L(G3) = {ww | w ∈ (a, b)+}, and

• for every grammar G in GF the language L(G) is semilinear, and

• for GF the membership problem is decidable in deterministic polynomial time. l

Suna Bensch



• Speechprocessing is not an absolutely sequential process [Ray Jackendoff, 1997],[Jerrold Sadock, 1991].

• Are there any parallel grammar formalisms, that are mildly context-sensitive?

• Lindenmayer systems (or L systems) [Aristid Lindenmayer, 1968] were suintroduced to model the development of multicellular organisms.

• The main difference between L systems and grammars in the Chomsky sunahierarchy is the parallelism.

Suna Bensch


Definition

• An ET0L system (extended tabled Lindemayer system without interaction) istis a tuple G = (Σ,∆, H, ω), where

- Σ is the alphabet,

- ∆ ⊆ Σ is the terminal alphabet,

- H is a finite set of finite substitutions, (called tables),

- ω ∈ Σ∗ is the axiom.

Suna Bensch


Example ET0L system:

table 1 table 2

ω = S

S → ABCA → aAB → bBC → cCa → a

b → bc → c

S → SA → aB → bC → ca → a

b → bc → c

Suna Bensch



table 1 table 2

ω = S


b → bc → c


b → bc → c

S

Suna Bensch



table 1 table 2

ω = S


b → bc → c


b → bc → c

S

CBA

Suna Bensch



table 1 table 2

ω = S


b → bc → c


b → bc → c

S

C

Cc

B

Bb

A

Aa

Suna Bensch



table 1 table 2

ω = S


b → bc → c


b → bc → c

S

C

C

Cc

c

c

B

B

Bb

b

b

A

A

Aa

a

a

Suna Bensch



table 1 table 2

ω = S


b → bc → c


b → bc → c

S

C

C

C

c

c

c

c

c

c

B

B

B

b

b

b

b

b

b

A

A

A

a

a

a

a

a

a

Suna Bensch


Parallel grammar formalisms

• Some parallel grammar formalisms, that do not fulfill the conditions for mild mildmild context-sensitivity:

0L

ET0L

ET0L with finite index

Indian parallel grammars

Russian parallel grammars

Suna Bensch


Restricting ET0L systems to uniform k limitation

[Watjen & Unruh 1990]: k-uniformly-limited ET0L systems (kulET0L systems), sk ∈ IN.

In each derivation step k symbols, if existing, are rewritten.

Suna Bensch





Example of a possible derivation, k = 3:

S

C

Cc

B

Bb

A

Aa

Suna Bensch






S

C

Cc

B

Bb

A

Aa ��

Suna Bensch






S

C

C

Cc

c

B

Bb

b

A

A

Aa

a ��

Suna Bensch






S

C

C

Cc

c

B

Bb

b

A

A

Aa

a �� n��

Suna Bensch






S

C

C

Cc

c

B

Bb

b

A

A

Aa

a �� n��

A kulET0L system is propagating (kulEPT0L system), if there are no erasingproductions.

Suna Bensch


Results for kulEPT0L systems and languages

• We have:

- L(1ulET0L) = L(CF).

- The non-context-free languages L1, L2 und L3 can be generated with 2ulEPT0Lsuna2ulEPT0L systems.

- The family of kulEPT0L languages, k ∈ IN, contains only semilinear

languages.

- Normalform for kulEPT0L systems, k ∈ IN.

• We need:

– A recogniser ...

Suna Bensch


Leftrestriction on kulEPT0L systems

• Mathematical point of view:

– Better parsing strategies

• Linguistical point of view:

– Reduction of ambiguity– Avoidence of rewriting terminal symbols– Time sequence of speech

Suna Bensch


Leftrestriction on kulET0L systems

Left-restricted kulET0L systems (LR-kulET0L systems), k ∈ IN:

In each derivation step the k leftmost nonterminal symbols, if existing, arereplaced.

Suna Bensch






S

CBA

Suna Bensch






S

CBA��

Suna Bensch






S

CB

Bb

A

Aa

Suna Bensch






S

CB

Bb

A

Aa ��

Suna Bensch





Example of a possible derivation, k = 2:S

CB

B

Bb

b

A

A

Aa

a

Suna Bensch





Example of a possible derivation, k = 2:S

CB

B

Bb

b

A

A

Aa

a

��

Suna Bensch






S

CB

B

Bb

b

A

A

Aa

a

��

An LR-kulET0L system is propagating (LR-kulEPT0L system), if the are noerasing productions.

Suna Bensch


Results for LR-kulEPT0L systems and languages

• L(LR-1ulEPT0L) = L(CF \ λ).

• The three non-context-free languages L1, L2 and L3 can be generated withLR-2ulEPT0L systems.

• The family of LR-kulEPT0L languages, k ∈ IN, contains only semilinearlanguages.

• We have a recogniser for LR-2ulEPT0L languages, which achieves polynomialtime.

• The family of LR-kulEPT0L languages, k ∈ IN, is a proper subset of the familyof context-sensitive languages.

Suna Bensch


Idea of the semilinearity proof for the LR-kulEPT0L languages

Notations

• For a word w ∈ V ∗ we define it’s Parikhvector ψ(w) as follows

ψ(w) = ( |w|a1, |w|a2

, . . . , |w|an ), where ai ∈ V, 1 ≤ i ≤ n.

• A set of vectors is semilinear, if it is a union of finite linear sets of the followingform

{ v0 +m

∑

i=1

xivi | xi ∈ IN0, vi ∈ INn0 , 0 ≤ i ≤ m }.

• A language is semilinear, if the set ψ(L) = {ψ(w) | w ∈ L } is a semilinear set.

Suna Bensch


Periodical subtrees

A1A2 . . . Ak

@@

@@

@@

��

��

��

v1A1v2A2 . . . vkAkvk+1

Suna Bensch


Periodical subtrees

A1A2 . . . Ak

@@

@@

@@

��

��

��


Let P be the set of all periodical subtrees:

P = {P1: A1A2 . . . Ak

@@

@@

@@

��

��

��


, . . . , Pφ:N1N2 . . . Nk

@@

@@

@@

��

��

��

v1N1v2N2 . . . vkNkvk+1

}.

Suna Bensch


Let A be the set

A = { A1:ω

��

��

@@

@@

a1a2 . . . an

, . . . , Aα:, . . . ,ω

��

��

@@

@@

b1b2 . . . bm

}.

Suna Bensch


Let A be the set

A = { A1:ω

��

��

@@

@@

a1a2 . . . an

, . . . , Aα:, . . . ,ω

��

��

@@

@@

b1b2 . . . bm

}.

Let B be the set

B = {B1: ω

@@

@@

@@

��

��

��

@@

@

��

�

v1v2 . . . vn

, . . . , Bβ:ω

@@

@@

@@

��

��

��

x1x2 . . . xm

@@

@

��

� }.

Suna Bensch


Aw ={Words of the derivation trees in A}

Bv ={Words of the derivation trees in B}

Pu ={Words of the periodical subtrees

associated to the trees in B }

Suna Bensch


Aw ={Words of the derivation trees in A}

Bv ={Words of the derivation trees in B}


associated to the trees in Bζ( )}

Suna Bensch


Aw ={Words of the derivation trees in A}ψ( )

Bv ={Words of the derivation trees in B}ψ( )


associated to the trees in Bζ( )}ψ( )

Suna Bensch


Aw ={Words of the derivation trees in A}ψ( ) =(a1, a2, . . . , aα).

Bv ={Words of the derivation trees in B}ψ( ) =(b1, b2, . . . , bβ).


associated to the trees in Bζ( )}ψ( ) =(c11, c12, . . . , cβµβ).

Suna Bensch


Let M be the set

a1 ∪ a2 ∪ . . . aα ∪

{ b1 +

µ1∑

j=1

xj · c1j| xj ≥ 0, 1 ≤ j ≤ µ1 } ∪

{ b2 +

µ2∑

j=1

xj · c2j| xj ≥ 0, 1 ≤ j ≤ µ2 } ∪

...

{ bβ +

µβ∑

j=1

xj · cβj| xj ≥ 0, 1 ≤ j ≤ µβ }.

The set M is semilinear and M = ψ(L), L some LR-kulEPT0L language.

Suna Bensch


Recogniser for LR-2ulEPT0L languages

Based on the Earley Recogniser for context-free languages

Input:

An LR-2ulEPT0L system G = (Σ,M, ω,∆, 2) with L(G) = L.

An input string w = x1x2 . . . xn.

Output:

The parse sets I0, I1, . . . , In, if w ∈ L.

Method:

During the recognition process for some w = x1x2 . . . xn the algorithm constructssuccessively the item sets I0, I1, . . . , In.

Suna Bensch


Items, item sets and the dot • as metasymbol

• Every item set Ij, 0 ≤ j ≤ n consists of items.

• Structure of an item:

[

A1 → α, i

A2 → β, (i′)

]

, 0 ≤ i < i′ ≤ n− 1.

• Structure of a composite-predict item:

[

A1 → α, i

A2 → β, (i′)

]

, 0 ≤ i < i′ ≤ n− 1.

• On each right hand side of an item the • occurs:

[

A→ α • α′, i

B → β

]

∈ Ij.

Suna Bensch


Operations

• Scan[

A→ γ • aγ′

B → β

]

Suna Bensch


Operations

• Scan[

A→ γ • aγ′

B → β

]

⇒

[

A→ γa • γ′

B → β

]

Suna Bensch


• Predict[

A→ •X1X2

B → β

]

Suna Bensch


• Predict[

A→ •X1X2

B → β

]

⇒

[

X1 → •γ1

X2 → γ2

]

Suna Bensch


• Composite Predict[

A1 → •XA2 → a

]

Suna Bensch



A1 → •XA2 → a

]

⇒

s′[

B1 →B2 →

]

A1A2Y

ss

[

A1 →A2 →

]

•Xa

��

6��

6��

Suna Bensch



A1 → •XA2 → a

]

⇒

s′[

B1 →B2 →

]

A1A2Y

ss

[

A1 →A2 →

]

•Xa

��

6��

6��

[

X → •αY → β

]

Suna Bensch


• Complete[

A1 → α1•A2 → α2

]

Suna Bensch


• Complete[

A1 → α1•A2 → α2

]

⇒

[

B1 →B2 →

]

A1 • A2

Suna Bensch


• Backward Search[

X1 → α

X2 → β

]

[

A→B →

]

X1 •X2

Suna Bensch


• Backward Search[

X1 → α•X2 → β

]

[

A→B →

]

X1 •X2��

JJ

JJ

J]��

Suna Bensch


Polynomial time complexity

Suna Bensch


Polynomial time complexity

O(n6)

Suna Bensch


Conclusions

Suna Bensch


Conclusions

• We have:

– L(LR-2ulEPT0L) is mildly context-sensitive.

Suna Bensch


Conclusions

• We have:

– L(LR-2ulEPT0L) is mildly context-sensitive.

• To do:

– Extending the recogniser for LR-kulEPT0L systems, k ∈ IN.– Developing a recogniser for kulEPT0L languages, k ∈ IN.

Suna Bensch


Thanks a lot for your attention!

Suna Bensch

Documents

Parallel Systems as Mildly Context-Sensitive Grammar ... · Parallel Systems as Mildly Context-Sensitive Grammar Formalisms Non-context-free phenomena in natural languages • Structural