Vowel Harmony and Subsequentiality Jeffrey Heinz and Regine Lai { - - PowerPoint PPT Presentation

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Characterizing Phonology Subsequentiality Harmony Results Discussion Vowel Harmony and Subsequentiality Jeffrey Heinz and Regine Lai { heinz,rlai } @udel.edu University of Delaware MOL @ Sofia, Bulgaria August 9, 2013 1 / 39

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Vowel Harmony and Subsequentiality

Jeffrey Heinz and Regine Lai {heinz,rlai}@udel.edu

University of Delaware

MOL @ Sofia, Bulgaria August 9, 2013

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Characterizing Phonology Subsequentiality Harmony Results Discussion

This talk

  • In this talk we propose the tightest computational

characterization currently known for vowel harmony patterns, and by extension, for phonological patterns more generally.

  • Specifically, we show how ‘pathological’ phonological

patterns can be distinguished from attested ones with subregular computational boundaries.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Outline

Characterizing Phonology Subsequentiality Harmony Results Discussion

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Characterizing Phonology Subsequentiality Harmony Results Discussion

The computational nature of phonological generalizations Phonological processes can be modeled with mappings from underlying lexical representations to surface representations.

Question

  • What kind of maps are these?

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Characterizing Phonology Subsequentiality Harmony Results Discussion

First answer: They are regular (Johnson 1972, Koskiennimi 1983, Kaplan and Kay 1994)

Important!

While this result was shown with SPE-style and two-level grammars, the fact remains: The mappings themselves are regular regardless

  • f the grammatical formalism used (SPE, 2-level,

OT, GP)

(at least until a bonafide phonological pattern is found that is not describable with SPE or 2-level grammars)

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Classifying Sets of Strings

Computably Enumerable

Context- Sensitive Mildly Context- Sensitive Context-Free Regular Finite

Figure: The Chomsky hierarchy

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Classifying Sets of Strings

Computably Enumerable

Context- Sensitive Mildly Context- Sensitive Context-Free Regular Finite Yoruba copying Kobele 2006 Swiss German Shieber 1985 English nested embedding Chomsky 1957 English consonant clusters Clements and Keyser 1983 Kwakiutl stress Bach 1975 Chumash sibilant harmony Applegate 1972

Figure: Natural language patterns in the hierarchy.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Second Answer: They are subregular.

Context- Sensitive Mildly Context- Sensitive Context-Free Regular Finite

Subregular

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Why do we want stronger characterizations?

Better characterizations of phonological patterns

  • Leads to stronger universals
  • Leads to new hypotheses regarding what a humanly

possible phonological pattern is, which is in principle testable with artificial language learning experiments (Lai 2012, J¨ ager and Rogers 2012)

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Why do we want stronger characterizations?

Payoffs for better understanding learning

  • These computational properties can help solve the learning

problem (Heinz 2009, 2010).

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Why do we want stronger characterizations?

Payoffs for natural language processing

  • Insights can be incorporated into NLP algorithms
  • Factoring and composition may occur with lower

complexity

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Overview of Results

Non-regular Regular Weakly deterministic Left Subsequential Right Subsequential × PH × RH × DR × SC ?? × SG × MR

Figure: Hierarchies of transductions with the results of this paper

  • shown. PH=progressive harmony, RH=regressive harmony,

DR=dominant/recessive harmony, SC=stem control harmony, SG=sour grapes harmony, and MR=majority rules harmony.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Related Work

It has been shown that the following are left or right subsequential.

  • Nevins’ 2010 actual vowel harmony analyses in his VH

typology (Gainor et al. 2012)

  • synchronically attested metathesis patterns in Beth Hume’s

database, including long-distance ones, (Chandlee et al. 2012, Chandlee and Heinz 2012)

  • the typology of partial reduplication in Riggle (2006)

(Chandlee and Heinz 2012)

  • All local phonological patterns whose trigger and target fall

within a span of length k (Chandlee, in progress)

  • long distance consonantal harmony and disharmony (Luo

2013 MS, Payne 2013 MS) The only robust exception seems to be unbounded tone plateauing (Jardine 2013, MS)—but this only establishes Yip’s (2002) and Hyman’s (2011) point that tone is different from segmental phonology.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Outline

Characterizing Phonology Subsequentiality Harmony Results Discussion

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Ostensive definition of ‘subsequential’

Informally, subsequential transducers are weighted acceptors that are deterministic on the input, and where the weights are strings and multiplication is concatenation. 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer which recognizes iterative, progressive harmony.

(Sch¨ utzenberger 1977, Mohri 1997, Roche and Schabes 1999) 12 / 39

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Ostensive definition of ‘subsequential’

input +

  • state

→ 2 → 2 → 2

  • utput

0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer which recognizes iterative, progressive harmony.

(Sch¨ utzenberger 1977, Mohri 1997, Roche and Schabes 1999) 12 / 39

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Ostensive definition of ‘subsequential’

input +

  • state

→ 2 → 2 → 2

  • utput

+ 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer which recognizes iterative, progressive harmony.

(Sch¨ utzenberger 1977, Mohri 1997, Roche and Schabes 1999) 12 / 39

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Ostensive definition of ‘subsequential’

input +

  • state

→ 2 → 2 → 2

  • utput

+ + 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer which recognizes iterative, progressive harmony.

(Sch¨ utzenberger 1977, Mohri 1997, Roche and Schabes 1999) 12 / 39

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Ostensive definition of ‘subsequential’

input +

  • state

→ 2 → 2 → 2

  • utput

+ + + 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer which recognizes iterative, progressive harmony.

(Sch¨ utzenberger 1977, Mohri 1997, Roche and Schabes 1999) 12 / 39

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Ostensive definition of ‘subsequential’

input +

  • state

→ 2 → 2 → 2

  • utput

+ + + + 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer which recognizes iterative, progressive harmony.

(Sch¨ utzenberger 1977, Mohri 1997, Roche and Schabes 1999) 12 / 39

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Ostensive definition of ‘subsequential’

input +

  • state

→ 2 → 2 → 2

  • utput

+ + + + λ 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer which recognizes iterative, progressive harmony.

(Sch¨ utzenberger 1977, Mohri 1997, Roche and Schabes 1999) 12 / 39

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Left and Right subsequential

Definition (Left subsequential)

The class of functions recognized by subsequential transducers are called left subsequential. Denote this class LSF.

Definition (Right subsequential)

The reverse of f is f r = {xr, yr) | (x, y) ∈ f}. A function f is right subsequential iff f r is left subsequential. Denote this class RSF.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Notation

For any right subsequential function f, there exists a subsequential transducer T which recognizes f reading and writing the input and output string from right to left.

Lemma

Let f r be right subsequential. Then there exists T recognizing f such that (∀x ∈ X∗)[f r(x) = T(xr)r]. (1)

  • If T reads and writes left-to-right then we write −

→ T .

  • If T reads and writes right-to-left then we write ←

− T .

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Some facts

Theorem (Mohri 1997)

The following hold:

  • 1. LSF, RSF RR

(RR denotes the class of regular relations).

  • 2. RSF r = LSF.
  • 3. LSF and RSF are incomparable.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Canonical forms and learnability

Left subsequential functions have canonical transducers determined by sets of “good tails.” TLf(x) =

  • (y, v) | f(xy) = uv ∧

u = lcp(f(xX∗))

  • .

(2)

Theorem (Oncina et al. 1993)

f ∈ LSF ⇔ {TLf(x) | x ∈ X∗} has finite cardinality.

Theorem (Oncina et al. 1993)

Left subsequential functions are identifiable in the limit from positive data.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Weak Determinism

Theorem (Elgot and Mezei 1965)

Let T : X∗ → Y ∗ be a function. Then T ∈ RR iff there exists L : X∗ → Z∗ ∈ LSF, and R : Z∗ → Y ∗ ∈ RSF with X ⊆ Z such that T = R ◦ L.

Definition

A regular function T is weakly deterministic iff there exists L : X∗ → X∗ ∈ LSF, and R : X∗ → X∗ ∈ RSF such that L is not length-increasing and T = R ◦ L. The class of weakly deterministic functions is denoted WD.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Why alphabet-preservation alone isn’t sufficient

X∗ X∗ Z∗ X∗ T L′ = h ◦ L L R R′ = R ◦ h−1 h h−1

Figure: Decompositions of regular function T with X ⊆ Z.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Corollary to Elgot and Mezei 1965

Corollary

LSF, RSF ⊆ WD ⊆ RR. Vowel harmony patterns analyzed will be witnesses separating LSF, RSF from WD and will suggest a separation for WD and RR.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Outline

Characterizing Phonology Subsequentiality Harmony Results Discussion

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Ostensive example of Vowel Harmony

noun genitive gloss a. ip ip-in rope b. el el-in and c. son son-un end d. pul pul-un stamp

Table: Examples illustrating a fragment of the Vowel harmony from Turkish (Nevins 2010:32).

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Example phonological analysis

(a) w f(w) /ip-un/ [ip-in] /el-un/ [el-in] /son-un/ [son-un] /pul-un/ [pul-un] . . . (b) w f(w) /−C+C/ [−C−C] /C+C+C/ [C+C+C] . . . Table: Examples showing fragments of the phonological function describing Turkish back harmony assuming the underlying genitive morpheme is /-un/.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Illustrating the mappings examined here

w PH(w) RH(w) DR(w) SG(w) MR(w) a. /+ − −/ [+ + +] [− − −] [+ + +] [+ + +] [− − −] b. /− + +/ [− − −] [+ + +] [+ + +] [− − −] [+ + +] c. /− − −/ [− − −] [− − −] [− − −] [− − −] [− − −] d. /− + −/ [− − −] [− − −] [+ + +] [− − −] [− − −] e. /+ − ⊟/ [+ + ⊟] [− − ⊟] [+ + ⊟] [+ − ⊟] [− − ⊟] f. /+ ⊖ −/ [+ ⊖ +] [− ⊖ −] [+ ⊖ +] [+ ⊖ +] [− ⊖ −]

Table: Example mappings of underlying forms (w) given by progressive harmony (PH), regressive harmony (RH), dominant/recessive harmony (DR), and ‘pathological’ sour grapes harmony (SG), and majority rules harmony (MR). Symbols [+] indicates a [+F] vowel and [−] indicates a [−F] vowel where “F” is the feature harmonizing. Symbols [⊟] and [⊖] are [−F] vowels that are opaque and transparent, respectively.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Coverage of these examples.

  • Nevins (2010) provides a typological survey of dozens of

VH patterns and concludes they all can be analyzed as progressive or regressive harmony with underspecification. Gainor et al. (2012) confirm his analyses are left or right subsequential mappings.

  • Other linguists prefer analyzing VH patterns as progressive
  • r regressive harmony without underspecification.
  • Others still prefer to analyze VH in terms of

dominant/recessive or stem control.

  • Only one non-subsequential VH example has come to our

attention (which is NOT included in Nevins 2010.) This is Yaka (Hyman 1998).

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Coverage of these examples.

  • Nevins (2010) provides a typological survey of dozens of

VH patterns and concludes they all can be analyzed as progressive or regressive harmony with underspecification. Gainor et al. (2012) confirm his analyses are left or right subsequential mappings.

  • Other linguists prefer analyzing VH patterns as progressive
  • r regressive harmony without underspecification.
  • Others still prefer to analyze VH in terms of

dominant/recessive or stem control.

  • Only one non-subsequential VH example has come to our

attention (which is NOT included in Nevins 2010.) This is Yaka (Hyman 1998).

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Note

Two additional properties of these harmony patterns:

  • 1. They are same-length
  • 2. They are sequential: They can be described by a

subsequential transducer whose output function for every state is λ While both of these (independent) properties are strong; they are not true of other phonological processes such as epenthesis, and deletion (and even substitution) and therefore we make no use of these properties here.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Outline

Characterizing Phonology Subsequentiality Harmony Results Discussion

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Progressive and Regressive Harmony

0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer T . − → T recognizes iterative, progressive harmony and ← − T recognizes iterative, regressive harmony.

Theorem

PH is left subsequential and RH is right subsequential.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Illustrative Example

PH = − → T input +

  • state

→ 2 → 2 → 2

  • utput

+ + + + λ 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer T .

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Illustrative Example

RH = ← − T ⇔ ∀x ∈ X∗[← − T = (− → T (xr))r] input x − ⊟ − − + 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer T .

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Illustrative Example

RH = ← − T ⇔ ∀x ∈ X∗[← − T = (− → T (xr))r] input x − ⊟ − − + xr + − − ⊟ − 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer T .

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Illustrative Example

RH = ← − T ⇔ ∀x ∈ X∗[← − T = (− → T (xr))r] input x − ⊟ − − + xr + − − ⊟ − − → T (xr) + + + ⊟ − 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer T .

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Illustrative Example

RH = ← − T ⇔ ∀x ∈ X∗[← − T = (− → T (xr))r] input x − ⊟ − − + xr + − − ⊟ − − → T (xr) + + + ⊟ − (− → T (xr))r − ⊟ + + + 0,λ 1,λ 2,λ −, ⊟ +, ⊞ ⊞ ⊟ C C, −, ⊟, + :− C, +, ⊞, − :+

Figure: A subsequential transducer T .

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Characterizing Phonology Subsequentiality Harmony Results Discussion

The regular boundary

MR(w) = +|w| if |w|+F > |w|−F −|w| if |w|−F > |w|+F (3)

Theorem

Majority Rules is not regular.

Proof.

We show that the intersection of a regular set with the image of the inverse of MR is not regular. This seems to be widely known. For example, see Riggle (2004).

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Characterizing Phonology Subsequentiality Harmony Results Discussion

The subsequential boundary

SG(+−n) = + +n ∧SG(+ −n ⊟) = + −n ⊟ (4)

Theorem

Sour Grapes is regular but neither left nor right subsequential.

Proof (sketch).

The proof shows that for all distinct n, m ∈ N the tails of +n is not the same as the tails of +−m, which implies that the canonical left subsequential transducer would have infinitely many states.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

A nondeterministic transducer for SG

1 2 3 4 + −:+ − ⊟ −:+ −

Figure: A non-deterministic transducer which recognizes this important fragment of SG harmony.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

The weakly deterministic boundary

∀w ∈ {+, −}∗, DR(w) = +|w| if (∃ 0 ≤ i ≤ |w|)[wi = +] −|w|

  • therwise

(5)

Theorem

DR harmony is neither left nor right subsequential.

Proof (sketch).

As with SG, one can show for all distinct n, m ∈ N the tails of n is not the same as the tails of −m, which implies infinitely many states.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

DR is weakly deterministic

Theorem

DR harmony is weakly deterministic.

Proof.

We show that for all w ∈ {+, −}∗ it is the case that DR(w) = ← − − − TP HP ◦ − − − → TP HP (w). 0,λ 1,λ 2,λ − + + C − +, − :+

Figure: The subsequential transducer TP HP which recognizes iterative, progressive harmony where only the + value spreads.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Contrasting DR with SG

Derivation of DR(− − − + − − −)

input − − − + − − −

Derivations of SG(+ − − − −⊟) and SG(+ − − − −)

input + − − − −⊟ + − − − −

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Contrasting DR with SG

Derivation of DR(− − − + − − −)

input − − − + − − − − − − → TP HP − − − + + + +

Derivations of SG(+ − − − −⊟) and SG(+ − − − −)

input + − − − −⊟ + − − − −

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Contrasting DR with SG

Derivation of DR(− − − + − − −)

input − − − + − − − − − − → TP HP − − − + + + + ← − − − TP HP + + + + + + +

Derivations of SG(+ − − − −⊟) and SG(+ − − − −)

input + − − − −⊟ + − − − −

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Contrasting DR with SG

Derivation of DR(− − − + − − −)

input − − − + − − − − − − → TP HP − − − + + + + ← − − − TP HP + + + + + + +

Derivations of SG(+ − − − −⊟) and SG(+ − − − −)

input + − − − −⊟ + − − − − − → T + ? − ? − ? − ? − ⊟ + ? − ? − ? − ? −

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Contrasting DR with SG

Derivation of DR(− − − + − − −)

input − − − + − − − − − − → TP HP − − − + + + + ← − − − TP HP + + + + + + +

Derivations of SG(+ − − − −⊟) and SG(+ − − − −)

input + − − − −⊟ + − − − − − → T + ? − ? − ? − ? − ⊟ + ? − ? − ? − ? − ← − T ′ + − − − −⊟ + + + + +

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Outline

Characterizing Phonology Subsequentiality Harmony Results Discussion

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Review of Results

Non-regular Regular Weakly deterministic Left Subsequential Right Subsequential × PH × RH × DR × SC ?? × SG × MR

Figure: Hierarchies of transductions with the results of this paper

  • shown. PH=progressive harmony, RH=regressive harmony,

DR=dominant/recessive harmony, SC=stem control harmony, SG=sour grapes harmony, and MR=majority rules harmony.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Open Questions

  • Is Sour Grapes weakly determinstic or not?
  • Is there a hierarchy of ‘weak determinism’ depending on

how much mark up is allowed?

  • Is Yaka truly non-subsequential? Hyman’s analysis of Yaka

is to to somewhat controversial and appears to be unique (Hyman, p.c.). It merits further study.

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Conclusion

  • 1. It is possible to characterize nearly all known

analyses of vowel harmony as weakly deterministic; the analyses of some linguists would characterize them as either left or right subsequential.

  • 2. If correct, this is significant computational

constraint on what a possible vowel harmony pattern is, and we would expect payoffs in many related areas including

  • learning
  • NLP
  • psycholinguistic research

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Characterizing Phonology Subsequentiality Harmony Results Discussion

Almost time for lunch - Thank you!

Non-regular Regular Weakly deterministic Left Subsequential Right Subsequential × PH × RH × DR × SC ?? × SG × MR

Figure: Hierarchies of transductions with the results of this paper

  • shown. PH=progressive harmony, RH=regressive harmony,

DR=dominant/recessive harmony, SC=stem control harmony, SG=sour grapes harmony, and MR=majority rules harmony.

39 / 39