Part 4
Locality and the subregular
hierarchy
The Chomsky hierarchy
Any computation
Context-sensitive
Context-free
Regular
The subregular hierarchy
Regular
Star free
Tier-based
strictly local
Strictly local
The subregular hierarchy
Regular
Star free
Tier-based
strictly local Strictly
piecewise?
Strictly local
The subregular hierarchy
Regular
Star free
Neighborhood Tier-based
distinct?
strictly local
Strictly local
The subrational hierarchy
Rational
Subsequential
Input strictly local
Output strictly local
Star free
A set of strings is star free or aperiodic if there is some k
> 0 such that, if uvkw > 0 is in the set then so is if uvk+iw for
all i, for all strings u, v, w. The set of strings containing an
even number of a’s is regular but not star free.
aabbaba
aaaaabbab
aabb
aba
Claim
Graf 2010
All sets of licit surface forms in languages are star free.
The Creek stress pattern appears to violate this
generalization.
Creek
Haas 1977, Martin and Johnson 2002
Creek has a stress system realized as pitch countours.
According to the description, it has a property that is
entirely possible in metrical theory, but that no other
stress patter n appears to have (except the
questionable “Cairene Arabic” pattern)
Creek
Haas 1977, Martin and Johnson 2002
Nonfinite forms have a single main stress, realized as a
level tone
ifá
ifóci
amifocí
aktopá
alpatóci
cáːlo
pocóswa
hoktí:
LL
LLL
LLLL
HLL
HLLL
HL
LHL
LH
Creek
Haas 1977, Martin and Johnson 2002
The usual analysis of this is as alternating stress, with
quantity sensitive iambic feet assigned left to right and
main stress assigned to the rightmost foot
Creek
Haas 1977, Martin and Johnson 2002
The usual analysis of this is as alternating stress, with
quantity sensitive iambic feet assigned left to right and
main stress assigned to the rightmost foot
The problem is that the secondary stresses do not
appear on the surface
Creek
Haas 1977, Martin and Johnson 2002
The usual analysis of this is as alternating stress, with
quantity sensitive iambic feet assigned left to right and
main stress assigned to the rightmost foot
The problem is that the secondary stresses do not
appear on the surface
This means that, without the intermediate
representation, the pattern has to count, and is thus not
star free
Subsequentiality
A function on strings is subsequential if all possible
inputs x belong to one of a finite set of equivalence
classes with respect to what output we add once we add
z in f (xz).
ababb → ababb
abbbb → abbbba
bbabbb → bbabbba
aaaaaa → aaaaaa
aaaaa → aaaaaa
String (x)
Tails
ε, aa, aba, baa, babba, ...
ε→ε, b→b, ba→baa, ...
a, abaa, abaa, bababa, ...
ε→a, b→ba, ba→ba, ...
Subsequentiality
A function on strings is subsequential if all possible
inputs x belong to one of a finite set of equivalence
classes with respect to what output we add once we add
z in f (xz).
aabbb → aaaaa
aabbba → aabbba
babbb → baaaa
babbbab → baaaaaa
String (x)
Tails
aabbb, ...
b→bbbb, ba→aaaaa, bba→aaaaaa,...
aabbbb, ...
b→bbbbb, ba→aaaaaa, bba→aaaaaaa,...
Pathological harmony
Padgett 1995
Vowel harmony often has certain segments that block
harmony (e.g., Turkish non-high vowels block roundness
harmony: kul + lɑr + ɯ / *u); but the pattern of blocking
just seen (“sour grapes”) is unattested.
aabbb → aaaaa
aabbba → aabbba
Claim
Heinz and Lai 2013
All phonological processes are subsequential or rightsubsequential (subsequential with the segment order
reversed).
Strictly local sets
A set of strings is strictly local if it has a maximum length,
k, for dependencies: the presence in the set of any pair of
strings xyz and uyv with |y| = k-1 guarantees the presence
of xyv.
Strictly local sets
A set of strings is strictly local if it has a maximum length,
k, for dependencies: the presence in the set of any pair of
strings xyz and uyv with |y| = k-1 guarantees the presence
of xyv.
a*b*
ab
aabb
aabbb
aaaaabbb
Strictly local sets
A set of strings is strictly local if it has a maximum length,
k, for dependencies: the presence in the set of any pair of
strings xyz and uyv with |y| = k-1 guarantees the presence
of xyv.
a*b*
ab
aabb
aabbb
aaaaabbb
Strictly 1-local?
ab, aabb
aaabb, abaabb
Strictly local sets
A set of strings is strictly local if it has a maximum length,
k, for dependencies: the presence in the set of any pair of
strings xyz and uyv with |y| = k-1 guarantees the presence
of xyv.
a*b*
ab
aabb
aabbb
aaaaabbb
Strictly 1-local?
ab, aabb
aaabb, abaabb
Strictly 2-local?
ab, aabb
abb, a
Strictly local sets
A set of strings is strictly local if all its strings’
subsequences are drawn from a finite set of strings
(augmented with beginning and end symbols) of length k,
for some k.
a*b*
ab
aabb
aabbb
aaaaabbb
2-factors:
{«», «a, «b, aa, ab, bb, a», b»}
Strictly local sets
A set of strings is strictly local if all its strings’
subsequences are drawn from a finite set of strings
(augmented with beginning and end symbols) of length k,
for some k.
Set of all strings over {a,b} with at least one b
ab
aabb
baaaba
aaaaabaa
Not strictly local
{... aaaaaaaaa, aaaaaaaab, aaaaaaaba ...}
e.g., k = 9
Strictly local sets
A set of strings is strictly local if none of its strings’
subsequences are allowed to be drawn from a certain
finite set of strings (augmented with beginning and end
symbols) of length k, for some k.
a*b*
ab
aabb
aabbb
aaaaabbb
Forbidden 2-factors:
{ba}
Tier-based strictly local sets
A set of strings is strictly local on a tier T if all
subsequences of subsequences onto T are from a finite
set of strings (augmented with beginning and end
symbols) of length k, for some k.
Navajo sibilant harmony
ʃi + lĩːʔ
ʃi + taːʔ
si + ts’aːʔ
si + zid
dʒi + ʒ + ɣiʃ
dzi + z + tĩ
Forbidden 2-factors on the [anterior] tier:
{sʃ, stʃ, sʒ, stʃ’, sdʒ, ... ,
ʃs, ʃts, ʃz, ʃts’, ʃdz, ... }
Claim
Heinz 2007, Heinz, Rawal, and Tanner 2011
All segmental phonological restrictions that can be stated
as surface (long-distance or local) phonotactics are either
strictly local or tier-based strictly local with a small k.
Experimental results
Lai 2015
Subjects presented orally with:
Tested on preference for:
ʃ ... ʃ ... ʃ
s ... s ... s
Sibilant harmony
s ... ʃ ... s
s ... s ... s
First-last assimilation
s ... ʃ ... s
s ... s ... ʃ
ʃ ... s ... ʃ
s ... ʃ ... s
Experimental results
Lai 2015
First-last discrimination
Harmony discrimination
Input strictly local functions
A function f over strings is input strictly local if it has a
maximum length, k, for dependencies on the input: for any
two input strings xy and uy with |y| = k-1, any string z will
be mapped to the same output in f (xyz) as in f (uyz).
[-sonorant] → [+voice] / [+nasal] —
a
n
t
a
o
r
u
n
t
a
a
n
d
a
o
r
u
n
d
a
Output strictly local functions
A function f over strings is output strictly local if it has a
maximum length, k, for dependencies on the output: for
any two output strings xy = f (A) and uy = f (B) with |y| =
k-1, any string z will be mapped to the same output in f
(Az) as in f (Bz).
[-sonorant] → [+nasal] / [+nasal] —
a
n
t
a
o
r
u
n
t
a
a
n
n
a
o
r
u
n
n
a
Conjecture
Chandlee 2014
All segmental processes are input or output strictly local,
or a tier-wise equivalent (not yet developed) with a small k.
Possible exceptions
The nearest things to being exceptions are diachronic,
except for three possible cases of long-distance
displacement of a segment where the data we have is
insufficient to tell if the movement is unbounded.
La Huerta Diegueño
Shuswap
/ʔnʲ + m + ka + náp/ → nʲmkaʔnáp
/x + kʔwul + wʔs + m/ → xkʔwulʔwsm
Initial glottal stop moves to
before the root
Glottalization on affixes moves to
the end of the root
Colville
/c + pʕas + áyaʔ/ → cpsʕáyaʔ
Pharyngealization in the root
moves to a stressed lexical
suffix
Phonology
Any computation
Context-sensitive
Phonology
Context-free
Regular