Iterating Semantic Automata - Stanford University · Generalized Quantiﬁers Semantic Automata...

Generalized Quantifiers Semantic Automata Iterating Semantic Automata Experimental Results

Iterating Semantic Automata

Shane Steinert-Threlkeld(Joint work with Thomas Icard)

May 21, 2013


Outline

1 Generalized QuantifiersIntroduction to GQTIterated Quantifiers

2 Semantic AutomataBasic DefinitionsGeneralized Quantifiers and Regular LanguagesGeneralized Quantifiers and Context-Free Languages

3 Iterating Semantic AutomataIterating Two DFAsIterating With One or More PDAs

4 Experimental ResultsExperiments for Single GQsExperiments for Nested GQsAutomata and LinguisticsReferences


Introduction to GQT

Definition of GQ

Definition ([Mos57], [Lin66])

A generalized quantifier Q of type 〈n1, . . . , nk〉 is a class of modelsM = 〈M,R1, . . . ,Rk〉 closed under isomorphism, where eachRi ⊆ Mni . A generalized quantifier is monadic if ni = 1 for all i ,and polyadic otherwise.

We write QM R1 . . . Rk as shorthand for 〈M,R1, . . . ,Rk〉 ∈ Q.Usually the subscripted M is omitted for readability. Thus, e.g., fortype 〈1, 1〉 we write Q A B , where A and B are sets. 1

1For more complete introductions to the theory of generalized quantifiers,see [BC81, B86, Wes89, Kee96, KW11, PW06].


Introduction to GQT

Examples of GQs

This connects with the more familiar definition given in linguisticsemantics as can be seen by the following examples:

all = {〈M,A,B〉 | A ⊆ B}

some = {〈M,A,B〉 | A ∩ B 6= ∅}

more than four = {〈M,A,B〉 | |A ∩ B | ≥ 4}

The isomorphism closure condition (partially) captures theintuition that quantifiers are sensitive only to the size of therelevant subsets of M and not the identity of any particularelements or the order in which they are presented.


Introduction to GQT

Definability of GQs

Definition

A generalized quantifier Q of type 〈1, 1〉 is first-order definable ifand only if there is a first-order language L and an L-sentence ϕwhose non-logical vocabulary contains only two unary predicatesymbols A and B such that for any model M = 〈M,A,B〉,

QMAB ⇔ 〈M,A,B〉 |= ϕ.

The generalization to higher-order definability (and other typesthan 〈1, 1〉) is obvious.


Introduction to GQT

Definability of GQs

Definition

A generalized quantifier Q of type 〈1, 1〉 is first-order definable ifand only if there is a first-order language L and an L-sentence ϕwhose non-logical vocabulary contains only two unary predicatesymbols A and B such that for any model M = 〈M,A,B〉,

QMAB ⇔ 〈M,A,B〉 |= ϕ.

The generalization to higher-order definability (and other typesthan 〈1, 1〉) is obvious.

First-order: some, all, more than n, between m and n, . . .

Higher-order: an even/odd number of, most, less than m/n


Introduction to GQT

Constraints on GQs

CONS QMAB iff QMA(A ∩ B).

EXT QMAB iff QM′AB for every M ⊆ M ′.

Lemma

A quantifier Q satisfies CONS + EXT iff, for all M = 〈M,A,B〉:

QMAB ⇔ QAA(A ∩ B).


Introduction to GQT

Constraints on GQs

CONS QMAB iff QMA(A ∩ B).

EXT QMAB iff QM′AB for every M ⊆ M ′.

Lemma

A quantifier Q satisfies CONS + EXT iff, for all M = 〈M,A,B〉:

QMAB ⇔ QAA(A ∩ B).

Theorem

A quantifier Q satisfies CONS and EXT if and only if for everyM,M ′ and A,B ⊆ M, A′,B ′ ⊆ M ′, if |A− B | = |A′ − B ′| and|A ∩ B | = |A′ ∩ B ′|, then QMAB ⇔ QM′A′B ′.


Introduction to GQT

GQs as Binary Relations on N

In other words, quantifiers that satisfy CONS and EXT can besummarized succinctly as binary relations on natural numbers.Given Q we define:

QcMxy ⇔ QMAB and |A− B | = x , |A ∩ B | = y .

Standard generalized quantifiers can thus be seen as particularsimple cases.


Introduction to GQT

GQs as Binary Relations on N

In other words, quantifiers that satisfy CONS and EXT can besummarized succinctly as binary relations on natural numbers.Given Q we define:

QcMxy ⇔ QMAB and |A− B | = x , |A ∩ B | = y .

Standard generalized quantifiers can thus be seen as particularsimple cases.

every cM xy ⇔ x = 0

somecM xy ⇔ y > 0

at least threecM xy ⇔ y ≥ 3

mostcM xy ⇔ y > x

an even number of cM xy ⇔ y = 2n for some n ∈ N


Iterated Quantifiers

Embedded Quantification in Natural Language

To handle sentences such as

(1) (a) One of our neighbors stole all but four of the sunflowers.(b) Three explorers discovered most of the islands.

in which quantified phrases appear both in object position andsubject position, we need to look at so-called polyadic lifts ofmonadic quantifiers. Intuitively, these sentences express complexproperties of the respective transitive verbs. Since these verbs taketwo arguments, it will be impossible to give truth-conditions usingmonadic predicates.


Iterated Quantifiers

Iterating Type 〈1, 1〉 GQs

If Q1 and Q2 are type 〈1, 1〉, then It(Q1,Q2) will be of type〈1, 1, 2〉, defined:

It(Q1,Q2) A B R ⇔ Q1 A {x | Q2 B Rx}

where Rx = {y | Rxy}.We will sometimes use the alternative notation Q1 · Q2 forIt(Q1,Q2).

2

2This is a special case of a general definition for iterating quantifiers. Fordetails, see Chapter 10 of [PW06].


Outline






Basic Definitions

Main Idea

We will associate each GQ with a formal language in {0, 1}∗. First,every finite model will get mapped to a string s by

Starting with a model M = 〈M,A,B〉

Enumerating A as ~a

Writing a 0 for each element of A \ B and a 1 for eachelement of A ∩ B

From the earlier results, we have that

M ∈ Q ⇔ 〈#0(s),#1(s)〉 ∈ Qc ,


Basic Definitions

More Formally Defined

Definition

Let M = 〈M,A,B〉 be a model, ~a an enumeration of A, andn = |A|. We define τ (~a,B) ∈ {0, 1}n by

(τ (~a,B))i =

{

0 ai ∈ A \ B

1 ai ∈ A ∩ B

Thus, τ defines the string corresponding to a particular finitemodel.


Basic Definitions

More Formally Defined

Definition

Let M = 〈M,A,B〉 be a model, ~a an enumeration of A, andn = |A|. We define τ (~a,B) ∈ {0, 1}n by

(τ (~a,B))i =

{

0 ai ∈ A \ B

1 ai ∈ A ∩ B

Thus, τ defines the string corresponding to a particular finitemodel.

Definition

For a type 〈1, 1〉 quantifier Q, define the language of Q as

LQ = {s ∈ {0, 1}∗ | 〈#0(s),#1(s)〉 ∈ Qc}


Generalized Quantifiers and Regular Languages

Finite State Automata

The simplest class of automata are the (deterministic) finite-stateautomata:3

Definition

A deterministic finite-state automaton (DFA) is a tuple〈Q,Σ, δ, q0,F 〉:

Q a finite set of states

δ : Q × Σ → Σ a transition function

F ⊆ Q the set of accepting states

Σ a finite set of input symbols

q0 the start state

We denote the components of a DFA M by Q (M), Σ (M), etc.

3For a canonical reference on automata theory, see [HMU01].



Examples of GQ Automata

0

1

0

1

Figure : A finite state automaton for every.



Examples of GQ Automata

0

1

0

1

Figure : A finite state automaton for every.

10

0

1

0

1

Figure : A finite state automaton for at least two.



First-Order Definability

Not all quantifiers whose languages are accepted by DFAs are FOdefinable:

0 01

1

Figure : A cyclic finite state automaton for an even number of.



Characterizing First-Order Definable GQs

Theorem ([B86], p.156-157)

A quantifier Q is first-order definable iff LQ can be recognized by apermutation-invariant acyclic finite state automaton.



Characterizing First-Order Definable GQs

Theorem ([B86], p.156-157)

A quantifier Q is first-order definable iff LQ can be recognized by apermutation-invariant acyclic finite state automaton.

Lemma (Ehrenfeucht-Fraısse)

Q is FO-definable iff ∃k s.t.

M ≡k M′ ⇒ M ∈ Q iff M′ ∈ Q

where ≡k means Dup has k-round winning strategy in E-F game.

In our case:

A ∼k B ⇔ |A| = |B | = n < k or |A|, |B | ≥ k

and M ≡k M′ iff A ∩ B ,A \ B ,B \ A,M \ (B ∪ A) all bear ∼k totheir primed counterparts.



Proof of Definability Theorem

Proof.⇒:

Interpret E-F “threshold” in “Tree of Numbers”




Proof.⇒:


Transform Tree of Numbers into FSA




Proof.⇒:



By previous remarks, accepts the right language

By inspection, acyclic and permutation-invariant




Proof.⇒:



By previous remarks, accepts the right language

By inspection, acyclic and permutation-invariant

⇐: a p.i. acyclic FSA has a finite number of accepting states,each of which has a finite number of paths leading to it, each ofone of the forms: =i 1/ =j 0,=i 1/ ≥j 0,≥i 1/ =j 0,≥i 1/ ≥j 0.Each of these conditions is FO-definable. Take disjunction for allsuch paths for each accepting state. Then take disjunction overthe accepting states.



Characterizing GQs with Regular Languages

The type 〈1〉 divisibility quantifier Dn is defined:

〈M,A〉 ∈ Dn iff |A| is divisible by n.

Theorem ([Mos91])

Finite state automata accept exactly the class of quantifiers of type〈1, . . . , 1〉 definable in first-order logic augmented with Dn for all n.


Generalized Quantifiers and Context-Free Languages

Pushdown Automata

Definition

A (non-deterministic) pushdown automaton (PDA) is a tuple〈Q,Σ, Γ, δ, q0,Z0,F 〉:

Q is a finite set of states

Γ is a finite stack alphabet

q0 is the start state

Z0 is the start symbol

Σ is a finite set of input symbols

δ : Q × (Σ ∪ {ǫ})× Γ → P (Q × Γ∗) is a transition function

F is the set of accepting states



Example: Less Than Half

ǫ, 0/ǫ

0, ǫ/01, ǫ/11, 0/ǫ0, 1/ǫ0, 0/001, 1/11

ǫ,Z0/Z0

ǫ, 0/ǫ

Figure : A pushdown automaton for less than half.



Characterizing the PDA Quantifiers

Definition

A quantifier Q is first-order additively definable if there is aformula ϕ in the first-order language with equality and an additionsymbol + such that

QcMab ⇔ 〈N,+, a, b〉 |= ϕ(a, b)

Theorem ([B86], p.163-165)

LQ is computable by a pushdown automaton if and only if Q isfirst-order additively definable.


Outline






Iterating Two DFAs

Translation of Models

Definition

Let M = 〈M,A,B ,R〉 be a model, ~a and ~b enumerations of A andB , with n = |A|, m = |B |. We overload notation (see Definition 5)by allowing τ to take a relation as an extra argument:

τ(

~a, ~b,R)

=(

τ(

~b,Rai

)

�

)

i≤n

where τ(

~b,Rai

)

is the translation given in the single-quantifier

case. The operation (·)i≤n concatenates instances of (·) for0, . . . , n. The � functions as a separator symbol in a way that willshortly be made precise.


Iterating Two DFAs

Example

Example

To see this translation in a concrete example, consider a model〈M,A,B ,R〉 where M = {x , y , z}, A = {x , y}, B = M, and

R = {〈x , y〉, 〈y , x〉, 〈y , y〉, 〈y , z〉}

Let the enumerations ~a and ~b be given alphabetically. Then

τ(

~a, ~b,R)

will be

010 � 111� (1)


Iterating Two DFAs

Example

Example

To see this translation in a concrete example, consider a model〈M,A,B ,R〉 where M = {x , y , z}, A = {x , y}, B = M, and

R = {〈x , y〉, 〈y , x〉, 〈y , y〉, 〈y , z〉}

Let the enumerations ~a and ~b be given alphabetically. Then

τ(

~a, ~b,R)

will be

010 � 111� (1)

Proposition

Let M = 〈M,A,B ,R〉 be a model and Q1,Q2 quantifiers of type〈1, 1〉. Then for any enumerations ~a and ~b of A and B,

τ(

~a, ~b,R)

∈ LQ1·Q2⇔ 〈M,A,B ,R〉 ∈ It(Q1,Q2)


Iterating Two DFAs

Iterated Languages

Definition

Let Q1 and Q2 be quantifiers of type 〈1, 1〉. We define thelanguage of Q1 ·Q2 by

LQ1·Q2= {w ∈ (wi�)∗ | i ≤ n, wi ∈ {0, 1}∗ and

⟨

card ({wi | wi /∈ LQ2}) , card ({wi | wi ∈ LQ2

})⟩

∈ Qc1}

For w ∈ (wi�)∗, we write numsep (w) for the number of �

symbols in w .


Iterating Two DFAs

Examples of Iterated Languages

Example

Here are a few examples using some, every, and most. We omitsome of the conditions (e.g., that i ≤ n) to enhance readability.

w ∈ Levery ·some ⇔⟨

card ({wi | wi ∈ Lsome}) ,

card ({wi | wi /∈ Lsome})⟩

∈ every c

⇔ card ({wi | wi /∈ Lsome}) = 0

⇔ card ({wi | #(1) = 0}) = 0


Iterating Two DFAs


Example





∈ every c


⇔ card ({wi | #(1) = 0}) = 0

w ∈ Lsome·every ⇔ card ({wi | #(0) = 0}) > 0

One can see that the translation given in (1) will be in Lsome·every .


Iterating Two DFAs


Example





∈ every c


⇔ card ({wi | #(1) = 0}) = 0

w ∈ Lsome·every ⇔ card ({wi | #(0) = 0}) > 0

w ∈ Lmost·some ⇔ card ({wi | #(1) > 0}) > card ({wi | #(1) = 0})

One can see that the translation given in (1) will be in Lsome·every .


Iterating Two DFAs

Preliminary Definitions

Definition

Let M be a DFA. The pushdown reader of M, Mp is defined by

Q (Mp) = Q (M), q0 (Mp) = q0, F (Mp) = F (M) ;

Σ (Mp) = ∅ ;

Γ (Mp) = Σ (M) ;

δ (Mp) = {〈q1, ǫ, r , ǫ, q2〉 | 〈q1, r , q2〉 ∈ δ (M)}.


Iterating Two DFAs

Preliminary Definitions

Definition

Let M be a DFA. The pushdown reader of M, Mp is defined by

Q (Mp) = Q (M), q0 (Mp) = q0, F (Mp) = F (M) ;

Σ (Mp) = ∅ ;

Γ (Mp) = Σ (M) ;

δ (Mp) = {〈q1, ǫ, r , ǫ, q2〉 | 〈q1, r , q2〉 ∈ δ (M)}.

For an automaton M, let the sign of q ∈ Q (M) be given by

sgn (q) =

{

1 q ∈ F

0 q /∈ F

We define the sign of M as

sgn (M) = sgn (q0 (M))


Iterating Two DFAs

Central Definition

Definition (Iteration Automaton)

Let Q1 and Q2 be two DFAs accepting LQ1and LQ2

, respectively.The PDA It (Q1,Q2) is given by:

Q = {qI} ∪ Q(

Qp1

)

∪ Q (Q2)

Σ = {0, 1,�}

Γ = {0, 1}

q0 = qI

F = F(

Qp1

)


Iterating Two DFAs

Central Definition

Definition (Iteration Automaton, cont.)

Transition function:

δ = δ(

Qp1

)

∪ {〈qI , ε, x , sgn (Q2) x , q0 (Q2)〉 | i ≤ n}

∪ {〈q1, 1, x , x , q2〉 | 〈q1, 1, q2〉 ∈ δ (Q2) and sgn (q1) = sgn (q2)}

∪ {〈q1, 0, x , x , q2〉 | 〈q1, 0, q2〉 ∈ δ (Q2) and sgn (q1) = sgn (q2)}

∪ {〈q1, 1, x , xc , q2〉 | 〈q1, 1, q2〉 ∈ δ (Q2) and sgn (q1) 6= sgn (q2)}

∪ {〈q1, 0, x , xc , q2〉 | 〈q1, 0, q2〉 ∈ δ (Q2) and sgn (q1) 6= sgn (q2)}

∪ {〈q,�, x , x , qI 〉 | q ∈ Q (Q2)}

∪{

〈qI , ε, x , x , q0(

Qp1

)

〉}


Iterating Two DFAs

Example of Iterated Automaton

Example

Here is a PDA for computing some · every .

qI

0, 1/01, x/x

1, 0, x/x

ǫ, x/1x

�, x/x

�, x/x

ǫ, x/x ǫ, 1/ǫ

ǫ, 0/ǫ

Q2

Qp1

Figure : A pushdown automaton for some A R every B.


Iterating Two DFAs

Key Lemma

Lemma

Let Q1 and Q2 be quantifiers corresponding to regular languagesand wi ∈ {0, 1}∗. Abbreviate It (Q1,Q2) by M.

(1) If wi ∈ LQ2, then (qI ,wi � w ,X ) ⊢∗

M (qI ,w , 1X ) for anyX ∈ Γ∗, w ∈ Σ∗.

(2) If wi /∈ LQ2, then (qI ,wi � w ,X ) ⊢∗

M (qI ,w , 0X ) for anyX ∈ Γ∗, w ∈ Σ∗.

In other words, for any string wi , there is a wi� path through theiterated PDA such that a 1 or 0 is pushed onto the stack accordingto whether or not wi ∈ LQ2

.

Proof.

By induction on the length of wi .


Iterating Two DFAs

Iterated Machines Accept Iterated Languages

Theorem

Let Q1 and Q2 be quantifiers corresponding to regular languages.The language accepted by It (Q1,Q2) is LQ1·Q2

.

Proof.

⊆: by induction on numsep(w) for w ∈ LQ1·Q2, show that

(qI ,w ,Z0) ⊢∗M (qI , ǫ,X ) where

card ({wi | wi ∈ LQ2}) = #1 (X )

card ({wi | wi /∈ LQ2}) = #0 (X )

⊇: numsep(w) applications of previous lemma and the definitionof iterated PDA


Iterating Two DFAs

Are PDAs necessary?

Proposition

If LQ1and LQ2

are regular, then so too is LQ1·Q2.

Proof.

Basically closure of regular languages under substitution.


Iterating Two DFAs

Are PDAs necessary?

Proposition

If LQ1and LQ2

are regular, then so too is LQ1·Q2.

Proof.

Basically closure of regular languages under substitution.

Proposition

If LQ1and LQ2

are star-free, then so too is LQ1·Q2.

Proof.

The star-free langauges are exactly those definable in FO[<].a Usethe � separator to change quantifiers over characters to quantifiersover words.

aSee [DG07].


Iterating With One or More PDAs

Some Redefining

Definition

Let M be a PDA. The pushdown reader of M, Mp, is a two-stackpushdown automaton defined by

Q (Mp) = Q (M), q0 (Mp) = q0, F (Mp) = F (M)

Σ (Mp) = ∅

Γ (Mp) = Σ (M)

δ (Mp) = {〈q1, ǫ,X , r , q2, γ, ǫ〉 | 〈q1, r ,X , q2, γ〉 ∈ δ (M)}

The central definition goes exactly as before.



Example of Iterating With a PDA

Example

Here is a two-stack PDA accepting Lnmost·some .

qI

1, ǫ/ǫ, 0/10, ǫ/ǫ

1, 0, x/x , x/x

ǫ, ǫ/ǫ, x/0x

�, ǫ/ǫ, x/x

�

ǫ ǫ, 1/ǫ, x/x

ǫ, ǫ/0, 0/ǫǫ, ǫ/1, 1/ǫǫ, 0/ǫ, 1/ǫǫ, 1/ǫ, 0/ǫǫ, 0/00, 0/ǫǫ, 1/11, 1/ǫ

ǫ,Z0/Z0, x/x

ǫ, 1/ǫ, x/x



Do We Need Two Stacks

Proposition

If LQ1and LQ2

are context-free, then so too is LQ1·Q2.

Proof.

Basically closure of context-free languages under substitution.


Outline






Experiments for Single GQs

Primary Experimental Results

That some quantifiers are recognizable by FSAs while othersrequire PDAs suggests a difference in cognitive demand: the lattermay require more executive resources, such as working memory.




That some quantifiers are recognizable by FSAs while othersrequire PDAs suggests a difference in cognitive demand: the lattermay require more executive resources, such as working memory.[MCM+05] test this:

12 volunteers at UPenn

120 grammatical sentences, of two types, half of each true

at least three, all, someless than half, an odd/even number

Measured accuracy and did fMRI




Main findings:

Behavioral: Higher-order significantly less accurate thanfirst-order

Imaging: significant differential activation in dorsolateralprefontal and inferior frontal cortices bilaterally.

These areas previously found to be involved in workingmemory.




Main findings:

Behavioral: Higher-order significantly less accurate thanfirst-order

Imaging: significant differential activation in dorsolateralprefontal and inferior frontal cortices bilaterally.

These areas previously found to be involved in workingmemory.

[SZ10a, SZ10b, SZ11] corroborate these results (using behavioralmethods) and also tease apart higher-order quantiers from thoseneeding PDA.


Experiments for Nested GQs

Preliminary Findings for Iterated Quantifiers

Can these techniques be used to tease apart FSA from PDArepresentations of iterated quantifiers? In [SSTZI13], we presentthe first experimental results.



Preliminary Findings for Iterated Quantifiers

Can these techniques be used to tease apart FSA from PDArepresentations of iterated quantifiers? In [SSTZI13], we presentthe first experimental results.

21

Figure : Examples of stimuli used in the study. Sentence ’Every circle isconnected with some square’ is true in situation 1. Sentence ’Some circleis connected with every square’ is true in situation 2.



Preliminary Findings

Table : Means (SD) of all variables and correlations between iterationtask and other cognitive measures

Control Memory Prop15 acc Prop17 acc Mean (SD)

Every-some -.12 -.04 -.14 .02 5019 (2520)readSome-every -.05 -.01 -.15 -.07 4738 (2140)readEvery-some .11 .11 -.01 .01 2506 (1129)verSome-every .10 -.13 -.02 -.10 2079 (867)verEvery-some -.06 .02 .10 -.05 6.48 (1.76)accSome-every -.38** .29* .32** .45** 7.61 (.92)accControl -.26* -.33** -.29* 95.68 (38.22)

Memory .25* .30** 20.89 (2.15)

Prop15 acc .49** 6.86 (1.22)

Prop17 acc 6.88 (1.24)

* p < 0.05** p < 0.01

Note Read - reading time; ver - verification time; acc - accuracy; prop15 - proportional

quantifiers presented with 15 objects, prop17 - proportional quantifiers presented with

17 objects.



Preliminary Findings, cont.

Table : Means (SD) of iterated sentences in true and false conditions,and their correlations with other cognitive measures

Control Memory Prop15 acc Prop17 acc Mean (SD)

Every-some .12 .05 -.02 .05 2231 (870)ver falseEvery-some .09 .20 -.02 -.06 2781 (1569)ver trueSome-every .03 -.12 .10 .05 2468 (1315)ver falseSome-every .16 -.20 -.21 -.19 1690 (752)ver trueEvery-some -.06 .18 .03 .03 3.5 (0.80)acc falseEvery-some -.05 -.09 .05 -.11 2.96 (1.35)acc trueSome-every -.30** .24* .23* .41** 3.72 (0.62)acc falseSome-every -.38** .28* .31** .36** 3.90 (0.42)acc true


Automata and Linguistics

How to Understand the Role of Semantic Automata

In [STI13], we propose understanding semantic automata in termsof Marr’s [Mar82] levels of explanation for theories incomputational psychology:

(1) Computational

(2) Algorithmic

(3) Implementation

Automata can be seen as existing at a level 1.5, specifying abstractcontrol structure of algorithms without particular details aboutprocessing order, saliency features, and the like. This also invites asemantic competence/performance distinction.



General Conclusion

We hope to have shown that there can still be very fruitfulinteraction between

logic

language

computation

cognition



References I

[BC81] Jon Barwise and Robin Cooper. Generalized Quantifiersand Natural Language. Linguistics and Philosophy,4(2):159–219, 1981.

[B86] Johan van Benthem. Essays in Logical Semantics. D. ReidelPublishing Company, Dordrecht, 1986.

[DG07] Volker Diekert and Paul Gastin. First-order definablelanguages. In Jorg Flum, Erich Gradel, and Thomas Wilke,editors, Logic and Automata: History and Perspectives,Texts in Logic and Games, pages 261–306. AmsterdamUniversity Press, Amsterdam, 2007.

[HMU01] John E. Hopcroft, Rajeev Motwani, and Jeffrey D.Ullman. Introduction to Automata Theory, Languages, andComputation. Addison Wesley, Boston, 2nd edition, 2001.



References II

[Kee96] Edward L. Keenan. The Semantics of Determiners. InShalom Lappin, editor, The Handbook of ContermporarySemantic Theory, number 1989, chapter 2, pages 41–63.Blackwell, Oxford, 1996.

[KW11] Edward L. Keenan and Dag Westerstahl. GeneralizedQuantifiers in Linguistics and Logic. In Johan van Benthemand Alice ter Meulen, editors, Handbook of Logic andLanguage, pages 859–910. Elsevier, second edition, 2011.

[Lin66] Per Lindstrom. First order predicate logic with generalizedquantifiers. Theoria, 32:186–195, December 1966.

[Mar82] David Marr. Vision. Freeman, San Francisco, 1982.



References III

[MCM+05] Corey T. McMillan, Robin Clark, Peachie Moore,Christian Devita, and Murray Grossman. Neural basis forgeneralized quantifier comprehension. Neuropsychologia,43(12):1729–1737, January 2005.

[Mos57] Andrzej Mostowski. On a generalization of quantifiers.Fundamenta Mathematicae, 44:12–36, 1957.

[Mos91] Marcin Mostowski. Divisibility Quantifiers. Bulletin of theSection of Logic, 20(2):67–70, 1991.

[PW06] Stanley Peters and Dag Westerstahl. Quantifiers inLanguage and Logic. Clarendon Press, Oxford, 2006.

[SSTZI13] Jakub Szymanik, Shane Steinert-Threlkeld, MarcinZajenkowski, and Thomas F. Icard III. Automata andComplexity in Multiple-Quantifier Sentence Verification. InProceedings of the 12th International Conference onCognitive Modeling, 2013.



References IV

[STI13] Shane Steinert-Threlkeld and Thomas F. Icard III.Iterating Semantic Automata. Linguistics and Philosophy,2013.

[SZ10a] Jakub Szymanik and Marcin Zajenkowski. Comprehensionof simple quantifiers: empirical evaluation of acomputational model. Cognitive Science, 34(3):521–532,April 2010.

[SZ10b] Jakub Szymanik and Marcin Zajenkowski. Quantifiers andWorking Memory. Lecture Notes in Artificial Intelligence,6042:456–464, 2010.

[SZ11] Jakub Szymanik and Marcin Zajenkowski. Contribution ofworking memory in parity and proportional judgments.Belgian Journal of Linguistics, 25(1):176–194, January 2011.



References V

[Wes89] Dag Westerstahl. Quantifiers in Formal and NaturalLanguages. In D Gabbay and F Guenthner, editors,Handbook of Philosophical Logic (vol. 4), pages 1–132. D.Reidel Publishing Company, Dordrecht, 1989.



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