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Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars Karl Bringmann and Philip Wellnitz Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrcken, Germany May 8, 2019 Motivation Previous Work Hardness

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Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Karl Bringmann and Philip Wellnitz

Max Planck Institute for Informatics, Saarland Informatics Campus (SIC), Saarbrücken, Germany

May 8, 2019

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Motivation Previous Work Hardness Result

Tree-Adjoining Grammars

Initial and auxiliary trees, nodes marked for adjunction A c B b a Initial tree B A c A b B B a Auxiliary tree

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Tree-Adjoining Grammars

Adjoining trees A c B b a Initial tree B A c A b B B a Auxiliary tree + A c B A c A b B b B a a =

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Tree-Adjoining Grammars

Examples Every CFG {anbncndn | n ∈ N} S d S c b a S d S c S b a (Large parts of) English (XTAG [DEH+94])

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Tree-Adjoining Grammars

Examples Every CFG {anbncndn | n ∈ N} (Large parts of) English (XTAG [DEH+94]) {aa | a ∈ Σ} S S a a S S a S a

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Tree-Adjoining Grammars

Check for s ∈ T ∗ if s ∈ L(Γ), i.e. parse s O(|s|6) algorithm using dynamic programming [VSJ85, SJ88] O(|s|2ω) using matrix multiplication, ω < 2.373 [RY98] Faster algorithms? Improbable, for |Γ| = Θ(n12) [Sat94] Now, even for |Γ| = Θ(1)

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Tree-Adjoining Grammars

Check for s ∈ T ∗ if s ∈ L(Γ), i.e. parse s O(|s|6) algorithm using dynamic programming [VSJ85, SJ88] O(|s|2ω) using matrix multiplication, ω < 2.373 [RY98] Faster algorithms? Improbable, for |Γ| = Θ(n12) [Sat94] Now, even for |Γ| = Θ(1)

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Tree-Adjoining Grammars

Check for s ∈ T ∗ if s ∈ L(Γ), i.e. parse s O(|s|6) algorithm using dynamic programming [VSJ85, SJ88] O(|s|2ω) using matrix multiplication, ω < 2.373 [RY98] Faster algorithms? Improbable, for |Γ| = Θ(n12) [Sat94] Now, even for |Γ| = Θ(1)

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Tree-Adjoining Grammars

Check for s ∈ T ∗ if s ∈ L(Γ), i.e. parse s O(|s|6) algorithm using dynamic programming [VSJ85, SJ88] O(|s|2ω) using matrix multiplication, ω < 2.373 [RY98] Faster algorithms? Improbable, for |Γ| = Θ(n12) [Sat94] Now, even for |Γ| = Θ(1)

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Lower Bounds

Hard to show lower bounds for natural problems Use reductions to relate problems Problem P Instance I size n I is “yes” instance r(n) + t(s(n)) algorithm reduction r(n) time ⇐ ⇒ ← − Problem Q Instance J size s(n) J is “yes” instance t(n) algorithm

(x1 ∨ x3 ∨ x5) ∧ (x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3)

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Lower Bounds

Hard to show lower bounds for natural problems Use reductions to relate problems Problem P Instance I size n I is “yes” instance r(n) + t(s(n)) algorithm reduction r(n) time ⇐ ⇒ ← − Problem Q Instance J size s(n) J is “yes” instance t(n) algorithm

(x1 ∨ x3 ∨ x5) ∧ (x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3)

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Lower Bounds

Hard to show lower bounds for natural problems Use reductions to relate problems Problem P Instance I size n I is “yes” instance No r(n) + t(s(n)) algorithm reduction r(n) time ⇐ ⇒ − → Problem Q Instance J size s(n) J is “yes” instance No t(n) algorithm

(x1 ∨ x3 ∨ x5) ∧ (x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3)

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Lower Bounds

Hard to show lower bounds for natural problems Use reductions to relate problems

Problem P Instance I size n I is “yes” instance No r(n) + t(s(n)) algorithm reduction r(n) time ⇐ ⇒ − → Problem Q Instance J size s(n) J is “yes” instance No t(n) algorithm (x1 ∨ x3 ∨ x5) ∧ (x1 ∨ x3 ∨ x4) ∧ (x1 ∨ x2 ∨ x3)

Need hard problems

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

k-Clique

Given a graph G = (V, E) and k ∈ N, does G contain a clique of size k? Naïve O(nk) algorithm O(n

ωk 3 ) for 3 | k, using matrix multiplication, ω < 2.373 [NP85]

Let us believe that these algorithms are optimal.

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

k-Clique

Given a graph G = (V, E) and k ∈ N, does G contain a clique of size k? Naïve O(nk) algorithm O(n

ωk 3 ) for 3 | k, using matrix multiplication, ω < 2.373 [NP85]

Let us believe that these algorithms are optimal.

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Context-Free Grammars

3k-Clique G = (V, E), k size n = |V| G contains 3k-Clique No nk+1 + t(nk+1) algorithm reduction nk+1 time ⇐ ⇒ − → Parsing CFG CFG Γ, string s size s(n) = |s| = nk+1 s ∈ L(Γ) No t(n) algorithm

Γ = (T, NT, P, S), graphs w/ 3k-Cliques s, encoding of G

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Context-Free Grammars

3k-Clique G = (V, E), k size n = |V| G contains 3k-Clique No n3k(1−ε) algorithm No n

ωk 3 (1−ε) algorithm

reduction nk+1 time ⇐ ⇒ − → Parsing CFG CFG Γ, string s size s(n) = |s| = nk+1 s ∈ L(Γ) No n3−ε′ algorithm No nω−ε′ algorithm

Γ = (T, NT, P, S), graphs w/ 3k-Cliques s, encoding of G

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Context-Free Grammars

String s: list all k-cliques of G in a special way Γ: all graphs where 3 k-cliques form a triangle

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Context-Free Grammars

String s: list all k-cliques of G in a special way Γ: all graphs where 3 k-cliques form a triangle

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Context-Free Grammars

String s: list all k-cliques of G in a special way Γ: all graphs where 3 k-cliques form a triangle C2 C3 C1

(+|+)2 (−|+)2 (−|−)2 (+|+)3 (−|+)3 (−|−)3 (+|+)1 (−|+)1 (−|−)1 · · · · · · +: List clique’s nodes −: List for every node all neighbors

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Context-Free Grammars

String s: list all k-cliques of G in a special way Γ: all graphs where 3 k-cliques form a triangle C2 C3 C1

(+|+)1 (−|+)2 (−|−)3 · · · · · · · · · · · · +: List clique’s nodes −: List for every node all neighbors Matching + and − form a 2k-clique.

Γ will look like: S →∗ · · · (+|Sαβ|Sβγ|−) · · · Sαβ →∗ +) · · · (− Sβγ →∗ +) · · · (−

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Context-Free Grammars

Defining suitable gadgets: Node gadget NG(v) := $ binary(v) $ List gadget LG(v) :=

u∈N(v)

NG(u) Clique node gadget (+) CNG(C) :=

v∈C

(#NG(v)#)k Clique list gadget CLG(C) :=

v∈C

#LG(v)# k Use CLG(C)R for − gadgets CNG(C1) ⊆ CLG(C2) ⇒ C1 ∪ C2 is a 2k-clique CFGs can generate aaR

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Parsing TAG TAG Γ, string s size |s| = nk+1 s ∈ L(Γ) No |s|6−ε algorithm No |s|2ω−ε algorithm reduction nk+1 time ⇐ ⇒ − → 6k-Clique G = (V, E), k size n = |V| G contains 6k-clique No n6k(1−ε′) algorithm No n2ωk(1−ε′) algorithm

Γ = (I, A), graphs w/ 6k-cliques s, encoding of G

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Want to “solve” 6k-clique using TAG parser Need string and a grammar String: list all k-cliques in graph G in a special way Grammar: all graphs where 6 k-cliques form a 6-clique Here: Will show how to generate 4k-cliques

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Want to “solve” 6k-clique using TAG parser Need string and a grammar String: list all k-cliques in graph G in a special way Grammar: all graphs where 6 k-cliques form a 6-clique Here: Will show how to generate 4k-cliques

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Can generate (almost) 4k-cliques P(·, ·, ·, ·) 6k-cliques decompose into 3 of these Need to use P(·, ·, ·, ·) 3 times Similar to CFG hardness C1 C6 C3 C4 C1 C6 C2 C5 C3 C4

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Can generate (almost) 4k-cliques P(·, ·, ·, ·) 6k-cliques decompose into 3 of these Need to use P(·, ·, ·, ·) 3 times Similar to CFG hardness C1 C6 C3 C4 C1 C6 C2 C5 C3 C4

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Can generate (almost) 4k-cliques P(·, ·, ·, ·) 6k-cliques decompose into 3 of these Need to use P(·, ·, ·, ·) 3 times Similar to CFG hardness C1 C6 C3 C4 C1 C6 C2 C5 C3 C4

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

String: list all k-cliques in graph G in a special way Grammar: all graphs where 6 k-cliques form a 6-clique CNG(Ck) list vertices of k-clique Ck CLG(Ck) list neighbors of vertices of k-clique Ck CNG(Ck) ⊂ CLG(C′

k) iff Ck ∪ C′ k is a 2k-clique

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

String: list all k-cliques in graph G in a special way Grammar: all graphs where 6 k-cliques form a 6-clique CNG(Ck) list vertices of k-clique Ck CLG(Ck) list neighbors of vertices of k-clique Ck CNG(Ck) ⊂ CLG(C′

k) iff Ck ∪ C′ k is a 2k-clique

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

CNG(Ck) list vertices of k-clique Ck CLG(Ck) list neighbors of vertices of k-clique Ck CNG(Ck) ⊂ CLG(C′

k) iff Ck ∪ C′ k is a 2k-clique

String (to detect 4k-cliques):

GGk(G) :=

C∈Ck

|CNG(C)§CLG(C)R§CLG(C)§CLG(C)R|

C∈Ck

|CLG(C)§CLG(C)R§CNG(C)§CLG(C)R|

C∈Ck

|CLG(C)§CLG(C)R§CNG(C)§CLG(C)R|

C∈Ck

|CNG(C)§CLG(C)R§CLG(C)§CLG(C)R|

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

CNG(Ck) ⊂ CLG(C′

k) iff Ck ∪ C′ k is a 2k-clique

String (to detect 4k-cliques):

GGk(G) :=

C∈Ck

|CNG(C)§CLG(C)R§CLG(C)§CLG(C)R|

C∈Ck

|CLG(C)§CLG(C)R§CNG(C)§CLG(C)R|

C∈Ck

|CLG(C)§CLG(C)R§CNG(C)§CLG(C)R|

C∈Ck

|CNG(C)§CLG(C)R§CLG(C)§CLG(C)R|

Claim: There is a TAG that generates {GGk(G) | G contains a 4k-clique} ∪ {some strings that are not encodings of graphs}

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Reminder: Tree-Adjoining Grammars

Initial and auxiliary trees, nodes marked for adjunction A c B b a Initial tree B A c A b B B a Auxiliary tree

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Consider sets of special trees Program Easily combinable NIn NOut NIn n2 n1 n3 n4

NIn NOut MOut NOut NIn n2 m2 m1 n1 n3 m3 m4 n4

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Writing characters W(a, b, c, d) W(a, b, c, d)In d W(a, b, c, d)Out c W(a, b, c, d)In b a Generates (a, b, c, d)

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness Result

Testing for equality Eq(Σ) Eq(Σ)In Eq(Σ)Out Eq(Σ)In Eq(Σ)In σ Eq(Σ)In σ Eq(Σ)In σ σ ∀σ ∈ Σ Generates {(s, sR, s, sR) | s ∈ Σ∗}

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Writing anything A(Σ) A(Σ)In A(Σ)Out A(Σ)In A(Σ)In σ A(Σ)In A(Σ)In A(Σ)In A(Σ)In A(Σ)In σ A(Σ)In A(Σ)In A(Σ)In σ A(Σ)In A(Σ)In σ A(Σ)In ∀σ ∈ Σ Generates Σ∗ × Σ∗ × Σ∗ × Σ∗

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Programs

Combine programs to detect claws of cliques CNG CLG CLG CLG Detect 4k-clique as 4 claws of cliques

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

Hardness for Tree-Adjoining Grammars

Detecting claws NC := W(#) · A({0, 1, $}) · W($) · Eq({0, 1}) · W($) · A({0, 1, $}) · W(#) Detecting claws of k-cliques (applying NC k2 times) CCIn NCOut CCIn NCOut NCIn NCOut NCOut CCOut NCOut

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Motivation Previous Work Hardness Result

CC is completely symmetric Can simply use it 4 times: C := A(0, 1, #, §, |) · W(|) · CC · W(§) · CC · W(§) · CC · W(§) · CC · W(|) · A(0, 1, #, §, |)

GGk(G) :=

C∈Ck

|CNG(C)§CLG(C)R§CLG(C)§CLG(C)R|

C∈Ck

|CLG(C)§CLG(C)R§CNG(C)§CLG(C)R|

C∈Ck

|CLG(C)§CLG(C)R§CNG(C)§CLG(C)R|

C∈Ck

|CNG(C)§CLG(C)R§CLG(C)§CLG(C)R|

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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Conclusions

Current TAG parsers are very likely optimal There is a natural problem with the running time – Θ(n2ω) using matrix multiplication – Θ(n6) not using matrix multiplication

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More Details

More Details for TAG-hardness

Overview of the grammar parsing 6k-cliques

S P(1, 3, 4, 6)In P(1, 3, 4, 6)Out P(1, 2, 5, 6)In P(1, 2, 5, 6)Out P(1, 2, 5, 6)In P(2, 3, 4, 5)In P(2, 3, 4, 5)Out P(2, 3, 4, 5)In P(1, 3, 4, 6)Out P(1, 3, 4, 6)In e r3(C3)| . . . . . . | (C3)l3 r2(C2)| . . . . . . | (C2)l2 r1(C1) | . . . . . . |(C1)l1 . . . | (C4)l4 r4(C4) | . . . . . . |(C5)l5 r5(C5) | . . . . . . | (C6)l5 r6(C6)| . . .

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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More Details

More Details for TAG-hardness

Encoding of the graph, the string s

GGk(G) :=

C∈Ck

| CNG(C) § CLG(C)R l1 r1 CLG(C) § CLG(C)R |

C∈Ck

| CNG(C) § CLG(C)R l2 r2 CLG(C) § CLG(C)R |

C∈Ck

| CNG(C) § CLG(C)R l3 r3 CLG(C) § CLG(C)R |

C∈Ck

| CLG(C) § CLG(C)R l4 r4 CNG(C) § CLG(C)R |

C∈Ck

| CLG(C) § CLG(C)R l5 r5 CNG(C) § CLG(C)R |

C∈Ck

| CLG(C) § CLG(C)R l6 r6 CNG(C) § CLG(C)R |

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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More Details

References I

Christy Doran, Dania Egedi, Beth Ann Hockey, Bangalore Srinivas, and Martin Zaidel, XTAG system: a wide coverage grammar for English, 15th Conference on Computational Linguistics, COLING’94, 1994, pp. 922–928. Jaroslav Nešetˇ ril and Svatopluk Poljak, On the complexity of the subgraph problem, Commentationes Mathematicae Universitatis Carolinae 26 (1985), no. 2, 415–419. Sanguthevar Rajasekaran and Shibu Yooseph, TAL recognition in O(M(N2)) time, JCSS 56 (1998), no. 1, 83–89.

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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References II

Giorgio Satta, Tree-adjoining Grammar Parsing and Boolean Matrix Multiplication, Comput. Linguist. 20 (1994), no. 2, 173–191. Yves Schabes and Aravind K. Joshi, An Earley-type parsing algorithm for tree adjoining grammars, 26th Annual Meeting

f the Association for Computational Linguistics, ACL

’88, 1988, pp. 258–269.

K. Vijay-Shankar and Aravind K. Joshi, Some computational

properties of tree adjoining grammars, 23rd Annual Meeting

f the Association for Computational Linguistics, ACL

’85, 1985, pp. 82–93.

Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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