Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions C OMPLEXITY OF NATURAL LANGUAGE QUANTIFIERS C OMPUTATIONAL DICHOTOMY BETWEEN RECIPROCALS Jakub Szymanik Institute for Logic, Language and Computation Universiteit van Amsterdam GLLC 14 1 2 September 28, 2007 Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions A BSTRACT Study reciprocals, like “each other”. Define them as lifts over monadic GQs. Show computational dichotomy: — Strong r.l. over proportional quantifiers are NP-complete. — PTIME quantifiers are closed on intermediate and weak r.l. R.l. are frequent NP-complete constructions. Ask some general mathematical questions about r.l. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions O UTLINE 1 M OTIVATIONS 2 P REVIOUS WORKS 3 R ECIPROCITY IN LANGUAGE 4 R ECIPROCALS AS LIFTS OVER GQ S 5 C OMPLEXITY OF RECIPROCAL LIFTS Strong reciprocity Intermediate and weak reciprocity 6 F URTHER QUESTIONS Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions O UTLINE 1 M OTIVATIONS 2 P REVIOUS WORKS 3 R ECIPROCITY IN LANGUAGE 4 R ECIPROCALS AS LIFTS OVER GQ S 5 C OMPLEXITY OF RECIPROCAL LIFTS Strong reciprocity Intermediate and weak reciprocity 6 F URTHER QUESTIONS Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions Link semantics and computational complexity. Evaluate complexity of semantic constructions in order to: — better understand our linguistic competence. — investigate into robustness of linguistic distinctions. Classify semantic constructions by their complexity. It will be valuable for cognitive science. Clarify concept of “meaning”. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions O UTLINE 1 M OTIVATIONS 2 P REVIOUS WORKS 3 R ECIPROCITY IN LANGUAGE 4 R ECIPROCALS AS LIFTS OVER GQ S 5 C OMPLEXITY OF RECIPROCAL LIFTS Strong reciprocity Intermediate and weak reciprocity 6 F URTHER QUESTIONS Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions GQ S — A SHORT REMINDER D EFINITION A generalized quantifier Q of type ( n 1 , . . . , n k ) is a class of structures of the form M = ( U , R 1 , . . . , R k ) , where R i is a subset of U n i . Additionally, Q is closed under isomorphism. E XAMPLE MOST = { ( U , A M , B M ) : card ( A M ∩ B M ) > card ( A M − B M ) } . Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions Q UANTIFIERS AND COMPLEXITY D EFINITION Let Q be of type ( n 1 , . . . , n k ) . By complexity of Q we mean computational complexity of the corresponding class K Q . Our computational problem is to decide whether M ∈ K Q . Equivalently, does M | = Q [ R 1 , . . . R k ]? D EFINITION We say that Q is NP-hard if K Q is NP-hard. Q is mighty if K Q is NP and K Q is NP-hard. It was Blass and Gurevich 1986 who first studied those notions. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions M IGHTY QUANTIFIERS — FIRST EXAMPLE Let us consider models of the form M = ( U , E M ) , where E M is an equivalence relation. D EFINITION M | = R e xy ϕ ( x , y ) means that there is a set A ⊆ U such that ∀ a ∈ U ∃ b ∈ A E ( a , b ) and for each a , b ∈ A M | = ϕ ( a , b ) . T HEOREM (M OSTOWSKI , W OJTYNIAK 2004) R e is mighty. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions M IGHTY QUANTIFIERS — SECOND EXAMPLE Let us consider models of the form M = ( U , V M , T M ) , where V M , T M are subsets of U . D EFINITION M | = BMost xy ϕ ( x , y ) means that there are sets A ⊆ U and B ⊆ U such that: MOST x ( V ( x ) , A ( x )) ∧ MOST y ( T ( y ) , B ( y )) ∧ ∧∀ x ∀ y ( A ( x ) ∧ B ( y ) ⇒ ϕ ( x , y )) . T HEOREM (S EVENSTER 2006) BMost is mighty. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions M OTIVATION FOR PREVIOUS RESULTS Under branching interpretation the following sentences are NP-complete: (1.) Some relative of each villager and some relative of each townsman hate each other. (2.) Most villagers and most townsmen hate each other. However, all these sentences are ambiguous and can be hardly found in the corpus of language. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions O UTLINE 1 M OTIVATIONS 2 P REVIOUS WORKS 3 R ECIPROCITY IN LANGUAGE 4 R ECIPROCALS AS LIFTS OVER GQ S 5 C OMPLEXITY OF RECIPROCAL LIFTS Strong reciprocity Intermediate and weak reciprocity 6 F URTHER QUESTIONS Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions R ECIPROCAL EXPRESSIONS ARE COMMON IN E NGLISH (1.) Andi, Jarmo and Jakub laughed at one another. (2.) 15 men are hitting one another. (3.) Even number of the PMs refer to each other. (4.) Most Boston pitchers sat alongside each other. (5.) Some pirates were staring at each other in surprise. In BNC there are 10351 occurrences of “each other”. Many sentences contain quantifiers in antecedents. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions V ARIOUS INTERPRETATIONS Dalrymple et al. 1998 classifies possible readings. They explain variations in the meaning by: S TRONG M EANING H YPOTHESIS Reading associated with the reciprocal in a given sentence is the strongest available reading which is consistent with relevant information supplied by the context. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions S TRONG READING (3.) Even number of the PMs refer to each other. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions I NTERMEDIATE READING (4.) Most Boston pitchers sat alongside each other. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions W EAK READING (5.) Some pirates were staring at each other in surprise. Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions O UTLINE 1 M OTIVATIONS 2 P REVIOUS WORKS 3 R ECIPROCITY IN LANGUAGE 4 R ECIPROCALS AS LIFTS OVER GQ S 5 C OMPLEXITY OF RECIPROCAL LIFTS Strong reciprocity Intermediate and weak reciprocity 6 F URTHER QUESTIONS Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions S TRONG RECIPROCAL LIFT Let Q be a monadic monotone increasing quantifier. D EFINITION Ram S ( Q ) AR ⇐ ⇒ ∃ X ⊆ A [ Q ( X ) ∧∀ x , y ∈ X ( x � = y ⇒ R ( x , y ))] . E XAMPLE (3.) Even number of the PMs refer to each other indirectly. (3’.) Ram S ( EVEN ) MP Refer . Jakub Szymanik Computational dichotomy between reciprocals
Motivations Previous works Reciprocity in language Reciprocals as lifts over GQs Complexity of reciprocal lifts Further questions I NTERMEDIATE RECIPROCAL LIFT D EFINITION Ram I ( Q ) AR ⇐ ⇒ ∃ X ⊆ A [ Q ( X ) ∧ ∀ x , y ∈ X ( x � = y ⇒ ∃ sequence z 1 , . . . , z ℓ ∈ X such that ( z 1 = x ∧ R ( z 1 , z 2 ) ∧ . . . ∧ R ( z ℓ − 1 , z ℓ ) ∧ z ℓ = y )] . E XAMPLE (4.) Most Boston pitchers sat alongside each other. (4’.) Ram I ( MOST ) Pitcher Sit . Jakub Szymanik Computational dichotomy between reciprocals
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