DS-2009-01:
Szymanik, Jakub
(2009)
*Quantifiers in TIME and SPACE. Computational Complexity of Generalized Quantifiers in Natural Language.*
Doctoral thesis, University of Amsterdam.

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## Abstract

In the dissertation we study the complexity of generalized quantifiers

in natural language. Our perspective is interdisciplinary: we combine

philosophical insights with theoretical computer science, experimental

cognitive science and linguistic theories.

In Chapter 1 we argue for identifying a part of meaning, the so-called

referential meaning (model-checking), with algorithms. Moreover, we

discuss the influence of computational complexity theory on cognitive

tasks. We give some arguments to treat as cognitively tractable only

those problems which can be computed in polynomial time. Additionally,

we suggest that plausible semantic theories of the everyday fragment

of natural language can be formulated in the existential fragment of

second-order logic.

In Chapter 2 we give an overview of the basic notions of generalized

quantifier theory, computability theory, and descriptive complexity

theory.

In Chapter 3 we prove that PTIME quantifiers are closed under

iteration, cumulation and resumption. Next, we discuss the

NP-completeness of branching quantifiers. Finally, we show that some

Ramsey quantifiers define NP-complete classes of finite models while

others stay in PTIME. We also give a sufficient condition for a Ramsey

quantifier to be computable in polynomial time. We end this chapter

with a question about the complexity dichotomy between Ramsey

quantifiers.

In Chapter 4 we investigate the computational complexity of polyadic

lifts expressing various readings of reciprocal sentences with

quantified antecedents. We show a dichotomy between these readings:

the strong reciprocal reading can create NP-complete constructions,

while the weak and the intermediate reciprocal readings do

not. Additionally, we argue that this difference should be

acknowledged in the Strong Meaning Hypothesis.

In Chapter 5 we study the definability and complexity of the

type-shifting approach to collective quantification in natural

language. We show that under reasonable complexity assumptions it is

not general enough to cover the semantics of all collective

quantifiers in natural language. The type-shifting approach cannot

lead outside second-order logic and arguably some collective

quantifiers are not expressible in second-order logic.

As a result, we argue that algebraic (many-sorted) formalisms dealing

with collectivity are more plausible than the type-shifting approach

. Moreover, we suggest that some collective quantifiers might not be

realized in everyday language due to their high computational

complexity. Additionally, we introduce the so-called second-order

generalized quantifiers to the study of collective semantics.

In Chapter 6 we study the statement known as Hintikka's thesis: that

the semantics of sentences like ``Most boys and most girls hate each

other'' is not expressible by linear formulae and one needs to use

branching quantification. We discuss possible readings of such

sentences and come to the conclusion that they are expressible by

linear formulae, as opposed to what Hintikka states. Next, we propose

empirical evidence confirming our theoretical predictions that these

sentences are sometimes interpreted by people as having the

conjunctional reading.

In Chapter 7 we discuss a computational semantics for monadic

quantifiers in natural language. We recall that it can be expressed in

terms of finite-state and push-down automata. Then we present and

criticize the neurological research building on this model. The

discussion leads to a new experimental set-up which provides empirical

evidence confirming the complexity predictions of the computational

model. We show that the differences in reaction time needed for

comprehension of sentences with monadic quantifiers are consistent

with the complexity differences predicted by the model.

In Chapter 8 we discuss some general open questions and possible

directions for future research, e.g., using different measures of

complexity, involving game-theory and so on.

In general, our research explores, from different perspectives, the

advantages of identifying meaning with algorithms and applying

computational complexity analysis to semantic issues. It shows the

fruitfulness of such an abstract computational approach for

linguistics and cognitive science.

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-2009-01 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 2009 |

Subjects: | Computation Language Logic |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/2071 |

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