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Quanti�er-free least �xed pointfunctions for phonologyJane ChandleeHaverford
and Adam JardineRutgers
NECPhon 2018Nov 4, 2018
MIT
Introduction• What kind of functions are phonological UR-SR maps?
• Automata-theoretic characterizations have focused onsubsequentiality (Heinz and Lai, 2013; Payne, 2017; Chandlee and Heinz, 2018)
• Logical characterizations of sets providerepresentation-independent complexity hypotheses
• No previous logical characterizations of functions approachsubsequentiality
2
• Chandlee and Lindell (forthcoming) capture inputstrictly-local (ISL) functions with quanti�er-free (QF) logic
• We generalize this with least �xed-point extension of QFfunctions (QFLFP)
• QFLFP o�ers recursive, output-based de�nitions offunctions
• This is a (proper?) subclass of the subsequential functionsthat tightly �ts the typology of phonological functions
3
Logical de�nitions of functions
⋊1 a2 b3 b4 a5 b6 ⋉7
• Model of a string over Σ:– D = {1, 2, ..., n} D = {1, 2, 3, 4, 5, 6, 7}
– Pσ ⊆ D for each σ ∈ Σ,⋊,⋉ Pb = {3, 4, 6}
– A predecessor function p p(2) = 1, etc.
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⋊1 a2 b3 b4 a5 b6 ⋉7
• QF logic of strings:– Terms are
• variables x, y, ..., z range over D
• p(t) for term t
– Pσ(t) for σ ∈ Σ,⋊,⋉ and term t
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⋊1 a2 b3 b4 a5 b6 ⋉7
• QF logic of strings:– Terms are
• variables x, y, ..., z range over D
• p(t) for term t
– Pσ(t) for σ ∈ Σ,⋊,⋉ and term t
– E.g., Pb(x)
5
⋊1 a2 b3 b4 a5 b6 ⋉7
• QF logic of strings:– Terms are
• variables x, y, ..., z range over D
• p(t) for term t
– Pσ(t) for σ ∈ Σ,⋊,⋉ and term t
– E.g., Pb(x), Pb(p(x))
5
⋊1 a2 b3 b4 a5 b6 ⋉7
• QF logic of strings:– Syntax:
Pσ(t) | ¬ϕ | ϕ ∨ ψ | ϕ ∧ ψ | ϕ→ ψ
– E.g., Pb(x) ∧ Pb(p(x))
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⋊1 a2 b3 b4 a5 b6 ⋉7
a2′ b3′ c4′ a5′ b6′
• A logical transduction de�nes an output structure in thelogic of the input structure (Courcelle, 1994; Courcelle et al., 2012)
P ′a(x)
def= Pa(x)
P ′b(x)
def= Pb(x) ∧ ¬(Pb(p(x))
P ′c(x)
def= Pb(x) ∧ (Pb(p(x))
• b→ c / b7
⋊ p a m p a ⋉
p a m b a
• Chandlee and Lindell (forthcoming): QF transductionscapture ISL functions (Chandlee, 2014; Chandlee and Heinz, 2018)
P ′a(x)
def= Pa(x)
P ′m(x)
def= Pm(x)
P ′p(x)
def= Pp(x) ∧ ¬(Pp(p(x))
P ′b(x)
def= Pp(x) ∧ (Pm(p(x))
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• Long-distance patterns are not QF
• Iterative spreading, e.g. nasal spread in Malay (Onn, 1980)
/mawa/→ [mãwã]⋊ m a w a ⋉
m a w a
P ′a(x)
def= Pa(x) ∧ nasal(p(x))
P ′a(x)
def= Pa(x) ∧ ¬nasal(p(x))
• nasal(x) def= Pm(x) ∨ P′a(x) ∨ P
′w(x)
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• Long-distance patterns are not QF
• L-D harmony, e.g. nasal harmony in Kikongo (Ao, 1991)
/mala/→ [mana] /makala/→ [makana]
⋊ m a l a ⋉
m a n a
⋊ m a k a l a ⋉
m a k a l a
P ′n(x)
def= Pn(x) ∨
(
Pl ∧ ¬nasal(p(p(x))))
P ′l (x)
def= Pl(x) ∧ ¬nasal(p(p(x)))
• nasal(x) def= Pm(x) ∨ Pn(x)
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• Least-�xed point logic allows:
– reference to output structures;
– de�nition of precedence from predecessor (p)
• Restriction to QF keeps logic weak
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Least �xed point logic• An operator on D is a function f : P(D) → P(D)
D
X
f (X)f
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• The least �xed point of f is lfp(f ) =⋃
iXi, where
X0 = ∅, X i+1 = f (X i)
D
X1X2
X3
. . .
f
f f
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• ϕ(A, x) with a special predicate A(x) induces an operator
fϕ(X) ={
d ∈ D∣
∣ ϕ(A, x) is satis�ed with A 7→ X, d 7→ x}
Example
⋊1 a2 b3 a4 a5 a6 c7 a8 ⋉9
ϕ(A, x) = Pa(x) ∧(
Pb(p(x)) ∨ A(p(x)))
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• ϕ(A, x) with a special predicate A(x) induces an operator
fϕ(X) ={
d ∈ D∣
∣ ϕ(A, x) is satis�ed with A 7→ X, d 7→ x}
Example
⋊1 a2 b3 a4 a5 a6 c7 a8 ⋉9
ϕ(A, x) = Pa(x) ∧(
Pb(p(x)) ∨ A(p(x)))
fϕ(∅) = {4}
14
• ϕ(A, x) with a special predicate A(x) induces an operator
fϕ(X) ={
d ∈ D∣
∣ ϕ(A, x) is satis�ed with A 7→ X, d 7→ x}
Example
⋊1 a2 b3 a4 a5 a6 c7 a8 ⋉9
ϕ(A, x) = Pa(x) ∧(
Pb(p(x)) ∨ A(p(x)))
fϕ(∅) = {4}fϕ({4}) = {4, 5}
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• ϕ(A, x) with a special predicate A(x) induces an operator
fϕ(X) ={
d ∈ D∣
∣ ϕ(A, x) is satis�ed with A 7→ X, d 7→ x}
Example
⋊1 a2 b3 a4 a5 a6 c7 a8 ⋉9
ϕ(A, x) = Pa(x) ∧(
Pb(p(x)) ∨ A(p(x)))
fϕ(∅) = {4}fϕ({4}) = {4, 5}
fϕ({4, 5}) = {4, 5, 6}
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• ϕ(A, x) with a special predicate A(x) induces an operator
fϕ(X) ={
d ∈ D∣
∣ ϕ(A, x) is satis�ed with A 7→ X, d 7→ x}
Example
⋊1 a2 b3 a4 a5 a6 c7 a8 ⋉9
ϕ(A, x) = Pa(x) ∧(
Pb(p(x)) ∨ A(p(x)))
fϕ(∅) = {4}fϕ({4}) = {4, 5}
fϕ({4, 5}) = {4, 5, 6}fϕ({4, 5, 6}) = {4, 5, 6}
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• ϕ(A, x) with a special predicate A(x) induces an operatorfϕ(X) =
{
d ∈ D∣
∣ ϕ(A, x) is satis�ed with A 7→ X, d 7→ x}
Example
⋊1 a2 b3 a4 a5 a6 c7 a8 ⋉9
ϕ(A, x) = Pa(x) ∧(
Pb(p(x)) ∨ A(p(x)))
fϕ(∅) = {4} X1
fϕ({4}) = {4, 5} X2
fϕ({4, 5}) = {4, 5, 6} X3
fϕ({4, 5, 6}) = {4, 5, 6} X4 = X5 = ...
lfp(fϕ) = {4, 5, 6}
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• ϕ(A, x) with a special predicate A(x) induces an operatorfϕ(X) =
{
d ∈ D∣
∣ ϕ(A, x) is satis�ed with A 7→ X, d 7→ x}
• QFLFP is QF extended with predicates of the form[
lfpϕ(A, x)
]
(x)
for some ϕ(A, x) in QF extended with A(x)
[
lfpPa(x) ∧
(
Pb(p(x)) ∨ A(p(x)))]
(x)
⋊1 a2 b3 a4 a5 a6 c7 a8 ⋉9
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Example: iterative spreading with blocking
baaa 7→ bbbbbaaca 7→ bbbcabaacaba 7→ bbbcabb
P ′b(x)
def= [lfp(Pb(x) ∨ (A(p(x)) ∧ ¬Pc(x)))](x)
⋊1 b2 a3 a4 c5 a6 b7 a8 ⋉9
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Example: iterative spreading with blocking
baaa 7→ bbbbbaaca 7→ bbbcabaacaba 7→ bbbcabb
P ′b(x)
def= [lfp(Pb(x) ∨ (A(p(x)) ∧ ¬Pc(x)))](x)
⋊1 b2 a3 a4 c5 a6 b7 a8 ⋉9
15
Example: iterative spreading with blocking
baaa 7→ bbbbbaaca 7→ bbbcabaacaba 7→ bbbcabb
P ′b(x)
def= [lfp(Pb(x) ∨ (A(p(x)) ∧ ¬Pc(x)))](x)
⋊1 b2 a3 a4 c5 a6 b7 a8 ⋉9
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Example: iterative spreading with blocking
baaa 7→ bbbbbaaca 7→ bbbcabaacaba 7→ bbbcabb
P ′b(x)
def= [lfp(Pb(x) ∨ (A(p(x)) ∧ ¬Pc(x)))](x)
⋊1 b2 a3 a4 c5 a6 b7 a8 ⋉9
b2 b3 b4 c5 a6 b7 b8
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Long-distance agreement
cbccca 7→ cbcccb
P ′b(x)
def= [lfp(Pb(x) ∨ A(p(x)))](x) ∧ ¬Pc(x)
⋊1 c2 b3 c4 c5 c6 a7 ⋉8
c2 b3 c4 c5 c6 b7
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Spreading with blocking:
P ′b(x)
def= [lfp(Pb(x) ∨ (A(p(x)) ∧¬Pc(x)))](x)
LD agreement:
P ′b(x)
def= [lfp(Pb(x) ∨ A(p(x)))](x) ∧¬Pc(x)
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QFLFP is (probably) subsequential• Subsequential functions have some deterministic�nite-state transducer (Schützenberger, 1977; Mohri, 1997)
• Reading left-to-right, we immediately know the output ateach position in the input
0 1
m:m
a:a,w:w
p:p
a:ã,w:w
m:mp:p
⋊ m a w a ⋉
m a w
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• For any ϕ(x) ∈ QFLFP, whether a position satis�es ϕ(x)depends entirely on the preceding information in the input
• Reading left-to-right, we immediately know the output ateach position in the input
Pw(x)def=
[
lfp+son(x) ∧
(
Pm(p(x)) ∨ A(p(x)))]
(x)
⋊ m a w a ⋉
m a w
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Subsequential is (probably) not QFLFP• Keeping track of even and odd-numbered elements of aparticular type over arbitrary distances is subsequential
0 1
b : b
c : c
b : b
c : c
a : da : da : d
a : aa : aa : a
bbbbbbaaa 7→ bbbbbbddd
abbbbbaaa 7→ dbbbbbaaa
• We cannot think of a QFLFP de�nition for this function
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• This is a good phonological prediction of QFLFP;functions like “odd-numbered sibilants harmonize” are notattested.
• But, QFLFP can capture ‘local’ even/odd counting (for, e.g.,iterative stress)
[
lfp⋊ (p(x)) ∨ A(p(p(x))
]
(x)
⋊1 a2 a3 a4 a5 a6 a7 a8 ⋉9
21
The general picture (probably)
SUBSEQ
LSUBSEQ RSUBSEQ
QFLFPpQFLFPpQFLFPp QFLFPsQFLFPsQFLFPs
LOSL ROSLISL
OSL = output strictly local functions (Chandlee, 2014; Chandlee et al., 2015)
22
Discussion• QFLFP is a restrictive theory for phonology based onrecursive de�nitions of local structures
• If QFLFP ⊆ SUBSEQ, then it is learnable (Oncina et al., 1993)
• Abstract de�nition of QFLFP?
• More e�cient/plausible learner for QFLFP?
23
• Logic can be applied to non-string structures:– Features
– Autosegmental representations
– Metrical structure
– Others?
• What do we get with two-place predicates and QFLFP (Koseret al., AMP)?
24
Conclusion• QFLFP combines the restrictiveness of QF with the ability torecursively reference the output structure.
• Allows us to model non-ISL phenomena such as LDagreement and iterative spreading.
• This class of functions appears to cross-cut severalsubregular classes that have been applied to the modelingof phonological processes.
• If/as a subset of subsequential, it is also learnable.
25
AcknowledgementsThanks to helpful thoughts and discussion from Je� Heinz, Bill
Idsardi, Steve Lindell, Jim Rogers, Jon Rawski, and the audience atthe �rst Rutgers computational phonology workshop
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