Computation of metrical structure

1. Method 1: Application of an ordered sequence of language-specific rules (e.g. Chomsky and Halle 1968)

Main stress rule:

(1) V -> [1 stress] / X - C0]NAV

Alternating stress rule:

(2) V -> [1 stress] / - C0 V C0 V1 C0]

Lexical entry: hurricAn => hurricA1n [Main stress rule] => hu1rricA1n [Alternating stress rule] => hu1rricA2n [Stress subordination]

Advantages: explicitness; it works.
Disadvantages: a) the rules are language-specific; b) the rule ordering is language-specific; c) there are many possible rule orders for the learner to consider; learning is hard. d) Also, parsing - working out the prosody and recovering the underlying lexical form from a surface form - is hard (Bear 1990), but not that hard (Maxwell 1994).

2. Method 2: Parameters and principles (e.g. Halle and Vergnaud 1987, Dresher and Kaye 1990)

a) Principles (e.g. "feet are made of syllables") are true for all (or almost all) languages; innate; no learning is necessary.
b) Parameters are like paired rules, of which the learner must select one option. Again, parameters are held to be innately specified, and so the learner has less work to do than learning a rule system.
c) However, the way in which parameters and principles are applied in generating words still employs an explicit ordering. For example, assignment of stress to syllables assumes that syllabification is performed first; computation of quantity-sensitive stress requires that the affiliation of consonants to one syllable or another has been determined.

2.1. Syllabification (Roca and Johnson ch. 10)

Universal bottom-up syllabification algorithm (after Kahn 1976):

1) Project syllabics (vowels, including long vowels and diphthongs, and syllabic consonants) to nuclei, as far as the language permits;
2) Parse segments preceding the nucleus into the onset, as far as the language permits;
3) Parse unsyllabified segments following the nucleus into the coda, as far as the language permits;
 
N N O N O N O N C O N C
| | | | /\ | | | | /\ | |
e.g. /pamflIt/ =>1 p a m f l I t =>2 p a m f l I t =>3 p a m f l I t

4) Join nucleus+coda sequences into rimes;
5) Project any remaining nuclei into rimes;
6) Join onset+rime into syllables.
 
/\ /\
O R O R O R O R O R O R
| /\ /\ /\ | /\ /\ /\ | /\ /\ /\
=>4 p N C f l N C =>5 p N C f l N C =>6 p N C f l N C
| | | | | | | | | | | |
a m I t a m I t a m I t
2.2. Stress assignment (Roca and Johnson ch. 12)

p. 360 (24) English stress algorithm (no. 5).

0) Mark each syllable with an asterisk - assumes that syllabification rules have already applied.
1) ENDSTRESS[Right/Left]: Project the rightmost/leftmost asterisk.
2) Extrametricality[Right/Left]: Make the rightmost/leftmost asterisk extrametrical.
Flavours of (2):
English extrametricality: "... in nouns and suffixed adjectives."
(10b) Compound Extrametricality [Right/Left]
(13a) French Extrametricality: "schwa" on the right edge (if there is one)
p. 335: Footing is mentioned, but not defined.
p. 343: Line Conflation: delete line 1. (Uh oh ... no more feet!)
 
* * * * * * * <*>
[sIg nI fI k+nt]A  =>Projection [sIg nI fI k+nt]A  =>Extrametricality [sIg nI fI k+nt]A
* * * * * *
[tIm pan zii]N =>Projection [tIm pan zii]N =>Extrametricality [tIm pan zii]N
*
* (*) (* .) (*) (* .)
* * * <*> * * * <*> * * * <*>
=>Accent [sIg nI fI k+nt]A  =>Footing [sIg nI fI k+nt]A  =>EndStress[Right] [sIg nI fI k+nt]A
*
* * * (*) (*) (*) (*) (*) (*)
* * * * * * * * *
=>Accent [tIm pan zii]N  =>Footing [tIm pan zii]N  =>EndStress[Right] [tIm pan zii]N
*
* * * <*>
=>Conflation [sIg nI fI k+nt]A
*
* * *
=>Conflation [tIm pan zii]N

The principle weakness of the bottom-up method is that it employs multiple passes through the word. Also, it requires that whole word to be present (it cannot work as it goes along from start to end of the word), making it psychologically implausible. Also, although bottom-up processing is intuitively appealing, it does not deal well with evidence of top-down processing, such as "tip-of-the-tongue" states.

3. Method 3: constraint filtering in Optimality Theory

GEN: a (random? exhaustive?) generator of possible structures for lexical strings.
CON: a ranked/ordered set of constraints
EVAL: a mechanism for determining which of many possible structures is the correct, preferred, or least objectionable analysis.

3.1. Syllabification (Roca and Johnson ch. 19)

Constraints:
 
ONS syllables must have onsets
NOCODA syllables must not have codas
*COMPLEX clusters/diphthongs are prohibited
PARSE lexical segments must be parsed into syllable positions
FILLNuc nucleus positions must be filled (with a V)
FILLOns onset positions must be filled (with a C) 

The most optimal satisfaction of these constraints is to parse all syllables as just CV. Obviously, not all languages do this. If they do not have onsets in every syllable, it may be because faithfulness or alignment constraints override ONS. If they allow codas or clusters, it may be for similar reasons, or because PARSE is ranked higher than NOCODA or *COMPLEX.

Example: syllabification of /pamflIt/
 

/pamflIt/ ONS FILLNuc NOCODA *COMPLEX
pa.mflIt     * **
pam.flIt     ** *
pamf.lIt     ** *
pamfl.It *   ** ***
pa.mf.lI.t   **    

In this example, it remains unclear why /pam.flIt/ is preferred to /pamf.lIt/, since both syllables have a coda and one cluster.

3.2. A result: Infixation

Tagalog um+abot => umabot; um+tawag => t­um­awag; um+gradwet =>gr­um­adwet.

Optimality theoretic analysis: the constraint requiring the prefix um­ to be aligned with the left edge of the word is weaker than the NOCODA constraint. Note that with a vowel­initial stem such as abot, the /m/ of um­ can be parsed as an onset.

3.3. Stress assignment

In many languages, feet are binary (FT­BIN) and all syllables are parsed into feet, if possible (PARSE­). In words with an even number of syllables these two constraints are easily simultaneously satisfied. In words with an odd number of syllables, however, they cannot both be satisfied: either a) at least one syllable will not be parsed into a foot, i.e. will be extrametrical, or b) one foot must have an extra syllable, or c) there may be a one­syllable foot.

a) Pintupi (mála)(wàna), (púli)(kàla)tju; Warao (yàpu)(rùki)(tàne)(háse), e(nàho)(ròa)(hàku)(tái).

c) Maranungku (lángka)(ràte)(tì)

In Optimality Theory it is said that for a) FT­BIN has priority over PARSE­ (FT­BIN » PARSE­), whereas for c), PARSE­ » FT­BIN.

3.4. Alignment

In Pintupi, the extra syllable is word­final; in Warao, word­initial. Garawa is similar to Pintupi, except that the extra syllable follows the primary foot: e.g. (káma)la(rài), (ánki)ri(kìrim)(pàyi). An Optimality-Theoretic analysis of Garawa holds that:

a) The left edge of every word is aligned with the left edge of a foot: ALIGN(PRWD­L, FT­L).

b) The right edge of every foot is aligned with the right edge of the word: ALIGN(FT­R, PRWD­R).

(b) will be violated by every foot except the last, but non­final feet will be even further away from the end of the word if the extra syllable is anything other than leftmost. Thus, the extra syllable will tend to fall at or near to the beginning of the word. Specifically, if (b) is prioritized over (a), the extra syllable will fall in absolute word­initial position, as in Warao. If (a) is prioritized over (b), the extra syllable will fall as close as possible to the word­initial position, but there will be a foot in absolute word­initial position. This is the Garawa pattern. (See McCarthy and Prince 1993 and Kenstowicz 1995 for other interesting examples.)

3.5. Example (Roca and Johnson 605-6)

"FT-TYPETROCHEE" is shorthand for ALIGN(s­L, FT­L).
"ALIGN-RIGHT" is shorthand for ALIGN(FT­R, PRWD­R)
 

/dIvelp/
 
V/µ
 
FT­BIN
 
FT-TYPETROCHEE
 
ALIGN-RIGHT
 
 
 
 *
 
 
 
  
 
 
 
 
 
 
 
 
µ
 
(µ
 
µ)
 
 
 
 
 
 
 
 
 
 
|
 
|
 
|
 
 
 
 
 
 
 
 
 
d
I
v
e
l
p
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
*
                          
 
 µ
 
µ
 
µ
 
 
 
 
 
 
 
 
 
 
|
  
|
  
|
        
*!
        
d
I
v
e
l
p
                
                             
         
*
                  
 
µ
  
(µ
  
µ)
                  
 
|
  
|
  
|
            
*!
    
d
I
v
e
l
p
                
                             
 
*
                          
 
(µ
  
µ)
  
µ
                  
 
|
  
|
  
|
                
*!
d
I
v
e
l
p
                

Strengths: strong universalism.

Weaknesses:
i) Inexplicitness: what's in GEN? If generation of structures is random (as Roca and Johnson p. 594  states), how do we ensure that the right analysis is in the candidate set? If generation is exhaustive, the mechanism becomes psychologically implausible. (But no more than most other phonological theories.)
ii) Is it plausible that absolutely all constraints are universal? Optimality Theory has no interesting way to deal with some kinds of language­specific constraints. For examples, according to Dell and Elmedlaoui's analysis of Tashlhit Berber, every vowel and consonant is a possible nucleus, but /a/ is always a nucleus (FILLNuca). To guarantee that this constraint is exceptionless, it must be ranked higher than all the other syllable parsing constraints: FILLNuca » PARSE » ONS » FILL » FILLNuc » NOCODA. This is a language­specific fact about Tashlhit Berber. There is a more general fact, however: FILLNuca is more specific than FILLNuc, so naturally it takes priority; otherwise it would not do any work in the grammar! This is a familiar instance of the Elsewhere Condition, a.k.a. subsumption.

4. Method 4: phrase structure parsing in Declarative Phonology (e.g. Church 1983, Coleman 1990, 2000)

To define a set of legal structures, a grammar is needed. To define hierarchical structures, context-free phrase structure grammar is a good starting point.
 
Syllable structure rules Metrical structure rules
1) H -> O RH 12) PrWdLat -> (L) (Fw) Fs (C[+em])
2) L -> O RL 13) PrWdLat -> (L) (Fw) Fs ([+em])
3) RH -> N Co 14) PrWdGer -> Fs *
4) RH -> NH 15) F ->  L
5) RL -> NL 16) F -> H
6) NL -> V
7) NH -> V V
8) O -> /s/ (C) (G)
9) O -> C
10) Co -> C
11) Co -> C C

Parsing: there are several proven algorithms for parsing any context-free grammar (see e.g. Jurafsky and Martin 2000, ch. 10). One of the most common approaches is a left-to-right, top-down, nondeterministic method that makes an interesting contrast to the usual bottom-up method found in phonology textbooks.

e.g. parsing /dIvelp/Lat by generation:

PrWdLat =>12 (L) (Fw) Fs (C[+em])
=>2 (O RL) (Fw) Fs (C[+em])
=> d I (Fw) Fs (C[+em])
=>15 d I L (C[+em]) ...ignoring Fs
=>2 d IO RL L (C[+em])
=>5 d Iv NL L (C[+em])
=>6 d Iv V L (C[+em])
=> d Iv e L (C[+em])
=>2 d Iv e O RL (C[+em])
=> d Iv e l RL (C[+em])
=>5 d Iv e l NL (C[+em])
=>6 d Iv e l V (C[+em])
=> d Iv e l  (C[+em])
=> d Iv e l  p

In Declarative Phonology, the metrical grammars of different languages are just different, though there are certain similarities. For examples (other analyses are possible):
 
Pintupi: PrWd -> Fs Fw* (w) F -> s w
Warao: PrWd -> (w) Fw* Fs F -> s w
Maranungku: PrWd -> Fs Fw* (w) F -> s w
Garawa: PrWd -> Fs Fw* Fs -> s w (w) Fw -> s w

Since context-free grammars are nondeterministic, sometimes there is more than one possible analysis of a string: ambiguity. If only one analysis is required, many strategies may be used to pick one:

1) Employ the rules in a particular order: e.g. big onsets first (cf. 8, 9 above); empty/small coda rules first (10, 11; rules 3 and 4 would need to be reversed).
2) Add a trans-syllabic constraint preventing filled codas with empty onsets.
3) Allow/accept variation!
4) Calculate the probabilities of rules and derivations, and pick the most likely analysis (Coleman 2000).

References

Bear, J. (1990) Backwards phonology. In Karlgren (1990), Vol. 3, 13-20.

Church, K. W. (1983) Phrase-Structure Parsing: a method for taking advantage of allophonic constraints. MIT PhD thesis, distributed by Indiana University Linguistics Club.

Coleman, J. S. (1990) Unification Phonology: Another look at "synthesis-by-rule". In Karlgren (1990) Vol. 2, 79-84.

Coleman, J. S. (2000) Candidate selection. The Linguistic Review 17, 167-179.

Dresher, B. E. and J. Kaye (1990) A computational learning model for metrical phonology. Cognition 34, 137-195.

Halle, M. and J.-R. Vergnaud (1987) An Essay on Stress. MIT Press.

Jurafsky, D. and J. H. Martin (2000) Speech and Language Processing: an introduction to natural language processing, computational linguistics, and speech recognition. Upper Saddle River, NJ: Prentice Hall.

Kahn, D. (1976) Syllable-based generalizations in English phonology. MIT PhD dissertation, distributed by Indiana University Linguistics Club.

Karlgren, H. (ed.) (1990) COLING-90. Papers Presented to the Thirteenth International Conference on Computational Linguistics. Helsinki: University of Helsinki.

Kenstowicz, M. (1995) Cyclic vs. non­cyclic constraint evaluation. Phonology 12. 397-436.

Maxwell, M. (1994) Parsing using linearly ordered phonological rules. Computational Phonology: First Meeting of the ACL Special Interest Group in Computational Phonology. Proceedings of the Workshop. Bernardsville, NJ: Association for Computational Linguistics. 59-70.

McCarthy, J. and A. Prince (1993) Generalized alignment. Yearbook of Morphology 1993. 79-153.