Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars Karl - PowerPoint PPT Presentation

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

Motivation Previous Work Hardness Result Tree-Adjoining Grammars Initial and auxiliary trees, nodes marked for adjunction A B B a c B A A b B a b c Initial tree Auxiliary tree Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Tree-Adjoining Grammars Adjoining trees A B a c A B B A A B a c B A A = + B a b c b B a b c Initial tree Auxiliary tree b Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Tree-Adjoining Grammars Examples Every CFG { a n b n c n d n | n ∈ N } S S S S a d a d S b c b c (Large parts of) English (XTAG [DEH + 94] ) Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Tree-Adjoining Grammars Examples Every CFG { a n b n c n d n | n ∈ N } (Large parts of) English (XTAG [DEH + 94] ) { aa | a ∈ Σ } S S S S a a S a a Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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 | 2 ω ) using matrix multiplication, ω < 2 . 373 [RY98] Faster algorithms? � Improbable, for | Γ | = Θ( n 12 ) [Sat94] � Now, even for | Γ | = Θ( 1 ) Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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 | 2 ω ) using matrix multiplication, ω < 2 . 373 [RY98] Faster algorithms? � Improbable, for | Γ | = Θ( n 12 ) [Sat94] � Now, even for | Γ | = Θ( 1 ) Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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 | 2 ω ) using matrix multiplication, ω < 2 . 373 [RY98] Faster algorithms? � Improbable, for | Γ | = Θ( n 12 ) [Sat94] � Now, even for | Γ | = Θ( 1 ) Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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 | 2 ω ) using matrix multiplication, ω < 2 . 373 [RY98] Faster algorithms? � Improbable, for | Γ | = Θ( n 12 ) [Sat94] � Now, even for | Γ | = Θ( 1 ) Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Lower Bounds Hard to show lower bounds for natural problems � Use reductions to relate problems Problem P Problem Q Instance I Instance J ( x 1 ∨ x 3 ∨ x 5 ) reduction ∧ ( x 1 ∨ x 3 ∨ x 4 ) r ( n ) time ∧ ( x 1 ∨ x 2 ∨ x 3 ) size s ( n ) size n ⇐ ⇒ I is “yes” instance J is “yes” instance r ( n ) + t ( s ( n )) algorithm ← − t ( n ) algorithm Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Lower Bounds Hard to show lower bounds for natural problems � Use reductions to relate problems Problem P Problem Q Instance I Instance J ( x 1 ∨ x 3 ∨ x 5 ) reduction ∧ ( x 1 ∨ x 3 ∨ x 4 ) r ( n ) time ∧ ( x 1 ∨ x 2 ∨ x 3 ) size s ( n ) size n ⇐ ⇒ I is “yes” instance J is “yes” instance r ( n ) + t ( s ( n )) algorithm ← − t ( n ) algorithm Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Lower Bounds Hard to show lower bounds for natural problems � Use reductions to relate problems Problem P Problem Q Instance I Instance J ( x 1 ∨ x 3 ∨ x 5 ) reduction ∧ ( x 1 ∨ x 3 ∨ x 4 ) r ( n ) time ∧ ( x 1 ∨ x 2 ∨ x 3 ) size s ( n ) size n ⇐ ⇒ I is “yes” instance J is “yes” instance No r ( n ) + t ( s ( n )) algorithm − → No t ( n ) algorithm Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Lower Bounds Hard to show lower bounds for natural problems � Use reductions to relate problems Problem P Problem Q Instance I Instance J ( x 1 ∨ x 3 ∨ x 5 ) reduction ∧ ( x 1 ∨ x 3 ∨ x 4 ) r ( n ) time ∧ ( x 1 ∨ x 2 ∨ x 3 ) size s ( n ) size n I is “yes” instance ⇐ ⇒ J is “yes” instance No r ( n ) + t ( s ( n )) algorithm − → No t ( n ) algorithm � Need hard problems Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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 ( n k ) algorithm ω k 3 ) for 3 | k , using matrix multiplication, ω < 2 . 373 [NP85] O ( n Let us believe that these algorithms are optimal. Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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 ( n k ) algorithm ω k 3 ) for 3 | k , using matrix multiplication, ω < 2 . 373 [NP85] O ( n Let us believe that these algorithms are optimal. Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Hardness for Context-Free Grammars 3 k -Clique Parsing CFG G = ( V , E ) , k CFG Γ , string s Γ = ( T , NT , P , S ) , reduction graphs w/ 3 k -Cliques n k + 1 time s , encoding of G size s ( n ) = | s | = n k + 1 size n = | V | ⇐ ⇒ G contains 3 k -Clique s ∈ L (Γ) No n k + 1 + t ( n k + 1 ) algorithm − → No t ( n ) algorithm Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Hardness for Context-Free Grammars 3 k -Clique Parsing CFG G = ( V , E ) , k CFG Γ , string s Γ = ( T , NT , P , S ) , reduction graphs w/ 3 k -Cliques n k + 1 time s , encoding of G size s ( n ) = | s | = n k + 1 size n = | V | ⇐ ⇒ G contains 3 k -Clique s ∈ L (Γ) No n 3 − ε ′ algorithm No n 3 k ( 1 − ε ) algorithm − → No n ω − ε ′ algorithm ω k 3 ( 1 − ε ) algorithm No n Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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

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

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 C 1 C 2 C 3 · · · · · · (+ | +) 2 (+ | +) 3 (+ | +) 1 ( −| +) 2 ( −| +) 1 ( −| +) 3 ( −|− ) 1 ( −|− ) 3 ( −|− ) 2 + : List clique’s nodes − : List for every node all neighbors Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

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 C 1 C 2 C 3 · · · (+ | +) 1 · · · ( −| +) 2 · · · ( −|− ) 3 · · · + : List clique’s nodes − : List for every node all neighbors Matching + and − form a 2 k -clique. Γ will look like: S → ∗ · · · (+ | S αβ | S βγ |− ) · · · S αβ → ∗ +) · · · ( − S βγ → ∗ +) · · · ( − Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

Motivation Previous Work Hardness Result Hardness for Context-Free Grammars Defining suitable gadgets: Node gadget NG ( v ) := $ binary ( v ) $ List gadget LG ( v ) := � NG ( u ) u ∈ N ( v ) (# NG ( v )#) k Clique node gadget ( + ) CNG ( C ) := � v ∈ C � k � Clique list gadget CLG ( C ) := � # LG ( v )# v ∈ C � Use CLG ( C ) R for − gadgets CNG ( C 1 ) ⊆ CLG ( C 2 ) ⇒ C 1 ∪ C 2 is a 2 k -clique � CFGs can generate aa R Karl Bringmann and Philip Wellnitz Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars