Complexity of the Lambek Calculus and Its Fragments Mati Pentus - PowerPoint PPT Presentation

Complexity of the Lambek Calculus and Its Fragments Mati Pentus http://lpcs.math.msu.su/~pentus/ • The Lambek calculus (denoted L) is a mathematical tool for formal language specification. It generates the class of all context-free languages without the empty word. • The Lambek calculus with empty antecedents (denoted L ∗ ) generates the class of all context-free languages. • Proof nets provide a convenient criterion for derivability in L ∗ . • The derivability problems for L ∗ ( \ , / ) and L( \ , / ) are NP-complete . • The derivability problems for L ∗ ( \ ) and L( \ ) are decidable in deterministic polynomial time . 1

1 “Hilbert style” Lambek calculus J. Lambek, The mathematics of sentence structure , American Mathematical Monthly 65 (1958), no. 3, 154–170. Definition 1. The set of all types is defined as the minimal set Tp such that • { p 0 , p 1 , p 2 , . . . } ⊂ Tp • If A ∈ Tp and B ∈ Tp, then ( A · B ) ∈ Tp, ( A \ B ) ∈ Tp, and ( A/B ) ∈ Tp. Example 1. ( p 1 · ( p 1 \ p 2 )) ∈ Tp. Below, we shall use q , r , s , and t instead of p 0 , p 1 , p 2 , and p 3 . Derivable objects of L H are A → B , where A ∈ Tp and B ∈ Tp. Example 2. L H ⊢ ( s/q ) · q → s , but L H � q · ( s/q ) → s . 2

Axioms and rules of L H (the “Hilbert style” Lambek calculus) A → A ( A · B ) · C → A · ( B · C ) A · ( B · C ) → ( A · B ) · C A · B → C A · B → C A → B B → C A → C/B B → A \ C A → C A → C/B B → A \ C A · B → C A · B → C We write L H ⊢ A → B for “ A → B is derivable in the calculus L H ”. Example 3. L H ⊢ q → s/ ( q \ s ). q \ s → q \ s q · ( q \ s ) → s q → s/ ( q \ s ) Remark. L H � s/ ( q \ s ) → q . 3

Example 4. L H ⊢ A · ( A \ B ) → B, L H ⊢ ( B/A ) · A → B, L H ⊢ ( A \ B ) · ( B \ C ) → A \ C, L H ⊢ ( C/B ) · ( B/A ) → C/A, L H ⊢ A → B/ ( A \ B ) , L H ⊢ A → ( B/A ) \ B, L H ⊢ ( A \ B ) /C → A \ ( B/C ) , L H ⊢ A \ ( B/C ) → ( A \ B ) /C. Definition 2. A ↔ B iff L H ⊢ A → B and L H ⊢ B → A . Example 5. ( A \ B ) /C ↔ A \ ( B/C ) , A/ ( B · C ) ↔ ( A/C ) /B, A · ( A \ ( A · B )) ↔ A · B, A \ ( A · ( A \ B )) ↔ A \ B. Example 6. L H ⊢ (( r/q ) \ s ) \ t → ( r \ s ) \ ( q \ t ) , L H � (( q \ r ) \ s ) \ t → s \ (( r \ q ) \ t ) , L H ⊢ (( q \ r ) \ ( q \ q )) \ t → ( q \ q ) \ (( r \ q ) \ t ) . 4

2 Gentzen style Lambek calculus Tp ∗ denotes the set of all finite sequences of types. Tp + denotes the set of all non-empty finite sequences of types. Derivable objects of the calculus L (the Gentzen style Lambek calculus) are sequents Γ → A , where A ∈ Tp and Γ ∈ Tp + . Axioms and rules of L Φ → B Γ B ∆ → A (cut) A → A Γ Φ ∆ → A A Π → B Φ → A Γ B ∆ → C Π → A \ B ( → \ ) (Π is non-empty) ( \ → ) Γ Φ ( A \ B ) ∆ → C Π A → B Φ → A Γ B ∆ → C Π → B/A ( → / ) (Π is non-empty) ( / → ) Γ ( B/A ) Φ ∆ → C Γ A B ∆ → C Γ → A ∆ → B Γ ( A · B ) ∆ → C ( · → ) ( → · ) Γ ∆ → A · B Here A, B, C ∈ Tp and Γ , ∆ , Φ , Π ∈ Tp ∗ . 5

Example 7. L ⊢ A · ( B/C ) → ( A · B ) /C C → C B → B ( / → ) A → A ( B/C ) C → B ( → · ) A ( B/C ) C → ( A · B ) A ( B/C ) → ( A · B ) /C ( → / ) A · ( B/C ) → ( A · B ) /C ( · → ) Theorem 1 (J. Lambek, 1958) . L ⊢ A 1 . . . A n → B if and only if L H ⊢ A 1 · . . . · A n → B . Theorem 2 (cut-elimination, J. Lambek, 1958) . A sequent is derivable in L if and only if it is derivable in L without (cut) . Corollary. The derivability problem for L (and for L H ) is decidable in nondeterministic polynomial time. Remark. L � ( A · B ) /C → A · ( B/C ). 6

3 Grammars The purpose of a Lambek categorial grammar is to provide an algorithm for distinguishing sentences from nonsentences in a fragment of a natural language. Example 8. Mary np John np smiles np \ s np = p 1 s = p 2 sees ( np \ s ) /np charmingly ( np \ s ) \ ( np \ s ) np → np s → s ( \ → ) np → np np ( np \ s ) → s → s ( / → ) np (( np \ s ) /np ) np Mary sees John np → np s → s ( \ → ) ( np \ s ) → ( np \ s ) np ( np \ s ) → s → s ( \ → ) np ( np \ s ) (( np \ s ) \ ( np \ s )) Mary smiles charmingly 7

4 Lambek calculus with empty antecedents Derivable objects of the calculus L ∗ are sequents Γ → A , where A ∈ Tp and Γ ∈ Tp ∗ (Tp ∗ denotes the set of all finite sequences of types). Axioms and rules of L ∗ Φ → B Γ B ∆ → A (cut) A → A Γ Φ ∆ → A A Π → B Φ → A Γ B ∆ → C Π → A \ B ( → \ ) ( \ → ) Γ Φ ( A \ B ) ∆ → C Π A → B Φ → A Γ B ∆ → C Π → B/A ( → / ) ( / → ) Γ ( B/A ) Φ ∆ → C Γ A B ∆ → C Γ → A ∆ → B Γ ( A · B ) ∆ → C ( · → ) ( → · ) Γ ∆ → A · B Example 9. B → B → B \ B ( → \ ) A → A ( / → ) A/ ( B \ B ) → A Cut-elimination theorem. We may drop (cut) from L ∗ . 8

5 Interpretation in the free group If a sequent is derivable in L ∗ , then in the free group its translation is equal to the unit. Here A/B = A · B − 1 and A \ B = A − 1 · B . Example 10. L ∗ ⊢ q → ( s/q ) \ s . q \ (( s/q ) \ s ) = q − 1 · (( s · q − 1 ) − 1 · s ) = q − 1 · (( q · s − 1 ) · s ) = q − 1 · q · s − 1 · s = 1 9

6 Cyclic linear logic Noncommutative linear logic was suggested by J.-Y. Girard in 1987 and expounded by D. N. Yetter. D. N. Yetter, Quantales and noncommutative linear logic , Journal of Symbolic Logic, 55 (1990), no. 1, pp. 41–64. Definition 3. Let At ⇌ { p 0 , p 1 , p 2 , . . . } ∪ { p 0 , p 1 , p 2 , . . . } . Linear formulas are the elements of the minimal set Fm such that • At ⊂ Fm, • if A ∈ Fm and B ∈ Fm, then ( A ⊗ B ) ∈ Fm and ( A � B ) ∈ Fm. ( p i ) ⊥ ⇌ p i ( p i ) ⊥ ⇌ p i ( A � B ) ⊥ ⇌ ( B ) ⊥ ⊗ ( A ) ⊥ ( A ⊗ B ) ⊥ ⇌ ( B ) ⊥ � ( A ) ⊥ 10

A embeds L ∗ into cyclic linear logic. The mapping A �→ � p = p � � A/B = � A � ( � B ) ⊥ � A \ B = ( � A ) ⊥ � � B � A · B = � A ⊗ � B � � Example 11. q/ ( s · r ) = q � ( r � s ), and ( q/r ) /s = ( q � r ) � s . Derivable objects of cyclic linear logic are sequents → A 1 . . . A n , where A i ∈ Fm. The intended meaning of → A 1 . . . A n is A 1 � . . . � A n . Axioms and rules of CMLL → p i p i → p i p i → Γ A B ∆ → Γ A → Φ B ∆ → Γ A Π → B ∆ → Γ ( A � B ) ∆ → Φ Γ ( A ⊗ B ) ∆ → Γ ( A ⊗ B ) ∆ Π 11

Example 12. CMLL ⊢ → ( p ⊗ q ) ( q ⊗ r ) ( r � p ). → p p → q q → ( p ⊗ q ) q p → r r → ( p ⊗ q ) ( q ⊗ r ) r p → ( p ⊗ q ) ( q ⊗ r ) ( r � p ) Example 13. CMLL ⊢ → ( r ⊗ r ) ( r ⊗ r ) ( r � r ) ⊥ . . . � ⊥ � Remark. L ∗ ⊢ A 1 . . . A n → B if and only if CMLL ⊢ → � A n A 1 B . Example 14. L ∗ ⊢ (( q \ r ) · s ) → ( q \ ( r · s )) and CMLL ⊢ → ( s � ( r ⊗ q )) ( q � ( r ⊗ s )). → r r → s s → s r ( r ⊗ s ) → q q → s ( r ⊗ q ) q ( r ⊗ s ) → s ( r ⊗ q ) ( q � ( r ⊗ s )) → ( s � ( r ⊗ q )) ( q � ( r ⊗ s )) 12

Region proof nets for CMLL and L ∗ 7 M. Pentus, Free monoid completeness of the Lambek calculus allowing empty premises , Proceedings of LC 1996, pp. 171–209. For each sequent we build a tree. q → (( s/q ) \ s ) q \ (( s/q ) \ s ) � � q � s � q � s q � (( q ⊗ s ) � s ) � � q s ⊗ s � q � � 13

A region proof net consists of the tree, nonintersecting axiom links (green), and arcs leading from each ⊗ to a � in the same region. The oriented graph consisting of black arcs must be acyclic. Example 15. L ∗ ⊢ q → ( s/q ) \ s . q s ⊗ s � q � � Example 16. L ∗ � ( s/q ) \ s → q . (( s/q ) \ s ) \ q s � q � s � q ( s ⊗ ( s � q )) � q � � s q � s ⊗ q � � 14

Example 17. L ∗ ⊢ ( s \ p ) \ t → ( r \ p ) \ (( s \ r ) \ t ). � � � � ( s \ p ) \ t \ ( r \ p ) \ (( s \ r ) \ t ) s � p � t � r � p � r � p � t � � ( t ⊗ ( s � p )) � ( p ⊗ r ) � ( r ⊗ s ) � t s r ⊗ p p r s t ⊗ � � t ⊗ � � � 15

Example 17. L ∗ ⊢ ( s \ p ) \ t → ( r \ p ) \ (( s \ r ) \ t ). � � � � ( s \ p ) \ t \ ( r \ p ) \ (( s \ r ) \ t ) s � p � t � r � p � r � p � t � � ( t ⊗ ( s � p )) � ( p ⊗ r ) � ( r ⊗ s ) � t s r ⊗ p p r s t ⊗ � � t ⊗ � � � 16

Example 17. L ∗ ⊢ ( s \ p ) \ t → ( r \ p ) \ (( s \ r ) \ t ). � � � � ( s \ p ) \ t \ ( r \ p ) \ (( s \ r ) \ t ) s � p � t � r � p � r � p � t � � ( t ⊗ ( s � p )) � ( p ⊗ r ) � ( r ⊗ s ) � t s r ⊗ p p r s t ⊗ � � t ⊗ � � � Theorem 3. A sequent is derivable in CMLL if and only if there exists a region proof net for it. 17