Quantifiers in TIME and SPACE.
Computational Complexity of Generalized Quantifiers in Natural Language
Jakub Szymanik
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.