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Computational complexity of polyadic lifts of generalized quantifiers in natural language
, 2010
"... We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multiquantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, ..."
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Cited by 20 (11 self)
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We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multiquantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multiquantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic distinctions between quantified reciprocal sentences. We show a computational dichotomy between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility of revising the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multiquantifier sentences. The paper not only contributes to the field of formal semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science.
Unary Quantifiers on Finite Models
 J. Logic, Language and Information
, 1997
"... In this paper all quantifiers are assumed to be so called simple unary quantifiers, and all models are assumed to be finite. We give a necessary and sufficient condition for a quantifier to be definable in terms of monotone quantifiers. For a monotone quantifier we give a necessary and sufficien ..."
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Cited by 12 (1 self)
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In this paper all quantifiers are assumed to be so called simple unary quantifiers, and all models are assumed to be finite. We give a necessary and sufficient condition for a quantifier to be definable in terms of monotone quantifiers. For a monotone quantifier we give a necessary and sufficient condition for being definable in terms of a given set of bounded monotone quantifiers. Finally, we give a necessary and sufficient condition for a monotone quantifier to be definable in terms of a given monotone quantifier. Our analysis shows that the quantifier "at least one half" and its relatives behave differently than other monotone quantifiers. 1 Introduction A simple unary (or monadic) quantifier, in this paper just a quantifier, is a class Q of structures (A; R), where R ` A, which is closed under isomorphisms. This concept was introduced by Mostowski [9]. A more general concept of a quantifier was introduced by Lindstrom [7] and a vast literature has emerged on the topic. W...
The Hierarchy Theorem for Generalized Quantifiers
, 1999
"... The concept of a generalized quantifier of a given similarity type was defined in [Lin66]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantif ..."
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Cited by 11 (3 self)
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The concept of a generalized quantifier of a given similarity type was defined in [Lin66]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindstrom [Wes] with a counting argument. We extend his method to arbitrary similarity types. 1 Introduction According to Lindstrom [Lin66], generalized quantifiers are simply classes of structures of a fixed similarity type such that the class is closed under isomorphisms. We identify similarity types with finite sequences of positive integers. A structure A of (similarity) type t = (t 1 ; : : : ; t u ) consists of a finite Key words: generalized quantifier, finite model theory, abstract model theory, y Par...
Using Answer Set Programming and Lambda Calculus to Characterize Natural Language Sentences with Normatives and Exceptions ∗
"... One way to solve the knowledge acquisition bottleneck is to have ways to translate natural language sentences and discourses to a formal knowledge representation language, especially ones that are appropriate to express domain knowledge in sciences, such as Biology. While there have been several pro ..."
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Cited by 8 (4 self)
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One way to solve the knowledge acquisition bottleneck is to have ways to translate natural language sentences and discourses to a formal knowledge representation language, especially ones that are appropriate to express domain knowledge in sciences, such as Biology. While there have been several proposals, including by Montague (1970), to give model theoretic semantics for natural language and to translate natural language sentences and discourses to classical logic, none of these approaches use knowledge representation languages that can express domain knowledge involving normative statements and exceptions. In this paper we take a first step to illustrate how one can automatically translate natural language sentences about normative statements and exceptions to representations in the knowledge representation language Answer Set Programming (ASP). To do this, we use λcalculus representation of words and their composition as dictated by a CCG grammar.
A Remark on Collective Quantification
, 2007
"... We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to de ne the meanings of collective determiners, quantifying over collections, using certain typeshifting operations. These typeshifting operations, i.e., ..."
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We consider collective quantification in natural language. For many years the common strategy in formalizing collective quantification has been to de ne the meanings of collective determiners, quantifying over collections, using certain typeshifting operations. These typeshifting operations, i.e., lifts, define the collective interpretations of determiners systematically from the standard meanings of quantifiers. All the lifts considered in the literature turn out to be definable in secondorder logic. We argue that secondorder definable quantifiers are probably not expressive enough to formalize all collective quantification in natural language.
Dependence Logic with Generalized Quantifiers: Axiomatizations
"... Abstract. We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the sense that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q i ..."
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Abstract. We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the sense that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as “there exist uncountably many. ” Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences. 1
Almost All Complex Quantifiers are Simple
"... Abstract. We prove that PTIME generalized quantifiers are closed under Boolean operations, iteration, cumulation and resumption. Key words: generalized quantifiers; computational complexity; polyadic quantifiers; Boolean combinations; iteration; cumulation; resumption 1 ..."
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Abstract. We prove that PTIME generalized quantifiers are closed under Boolean operations, iteration, cumulation and resumption. Key words: generalized quantifiers; computational complexity; polyadic quantifiers; Boolean combinations; iteration; cumulation; resumption 1
A Dichotomy Result for Ramsey Quantifiers?
"... Abstract. Ramsey quantifiers are a natural object of study not only for logic and computer science, but also for formal semantics of natural language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomialtime computable or NPhar ..."
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Abstract. Ramsey quantifiers are a natural object of study not only for logic and computer science, but also for formal semantics of natural language. Restricting attention to finite models leads to the natural question whether all Ramsey quantifiers are either polynomialtime computable or NPhard, and whether we can give a natural characterization of the polynomialtime computable quantifiers. In this paper, we first show that there exist intermediate Ramsey quantifiers and then we prove a dichotomy result for a large and natural class of Ramsey quantifiers, based on a reasonable and widelybelieved complexity assumption. We show that the polynomialtime computable quantifiers in this class are exactly the constantlogbounded Ramsey quantifiers. 1
Enhancing Fixed Point Logic With Cardinality Quantifiers
"... Let Q IFP be any quantifier such that FO(Q IFP ), first order logic enhanced with Q IFP and its vectorizations, equals inductive fixed point logic, IFP in expressive power. It is known that for certain quantifiers Q, the equivalence FO(Q IFP ) j IFP is no longer true if Q is added on both sides [12 ..."
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Let Q IFP be any quantifier such that FO(Q IFP ), first order logic enhanced with Q IFP and its vectorizations, equals inductive fixed point logic, IFP in expressive power. It is known that for certain quantifiers Q, the equivalence FO(Q IFP ) j IFP is no longer true if Q is added on both sides [12, 13]. Rather, we have FO(Q IFP ; Q) ! IFP(Q) in such cases. We extend these results to a great variety of quantifiers, namely all unbounded simple cardinality quantifiers. Our argument also applies to partial fixed point logic, PFP . In order to establish an analogous result for least fixed point logic, LFP , we exhibit a general method to pass from arbitrary quantifiers to monotone quantifiers. Our proof shows that the tree isomorphism problem is not definable in L ! 1! (Q 1 ) ! , infinitary logic extended with all monadic quantifiers and their vectorizations, where a finite bound is imposed to the number of variables as well as to the number of nested quantifiers in Q 1 . This streng...
Computational Complexity of Multiquantifier Sentences
, 2009
"... We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multiquantifier sentences. First, we show that the standard semantic constructions that turn simple quantifiers into complex ones ..."
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We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multiquantifier sentences. First, we show that the standard semantic constructions that turn simple quantifiers into complex ones, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into the operations yielding intractable natural language multiquantifier expressions: branching and Ramseyfication. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic distinctions between quantified reciprocal sentences. We show a computational dichotomy between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility to revise the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multiquantifier sentences. The paper not only contributes to the field of the formal semantics but also illustrates how the tools of computational complexity theory might be succesfully used in linguistics and philosophy with an eye towards cognitive science.