I use set intersection, collecting the common members from a group of sets, to determine the underlying representation of the alternating vowels. Since the +high of the high realization (i, u) and the –high of the mid realization (e, o) are not identical, these two valued features are not in the end result of set intersection. I will then assume that these alternating vowels are not specified for the feature High but are specified for the other four features (Syllabic, Low, Back, and Round) and that they are shorthanded as the capitals E and O. (All the other non-alternating vowels, of course, are fully specified in terms of these five features.)
A rule consists of Search and Change (
(2)
basics of Search
INR: [±F1, ±F2…],
DIR: left(ward), right(ward)
TRM: [±G1, ±G2…],
Change, by default, happens to those INRs which successfully found their TRM, unless it is stated otherwise. The two intrasegmental changes mentioned in Bale et al. (
(3)
intrasegmental changes
a.
subtraction
A – B = {αX: αX ∈ A ∧ αX ∉ B}
b.
unification
Subtraction is (set-theoretic) difference: the elements in A which are shared with B are not the members of the difference of A and B. For example, {–sonorant, +voiced} – {+voiced} results in an obstruent without voicing specification ({–sonorant}). Unification of A and B returns with A alone if there is a feature from the universal features set ({F1,…,Fn}) and the plus-valued version and the minus-valued one of this feature are both the members of the union of A and B; otherwise, unification is the same as set union, collecting all the elements of each member of a group of sets. Let’s see the following example.
(4)
three scenarios of unification
a.
non-vacuous unification
A:
{+syllabic, –low}, B: {+high}
A
∐ B = A ∪ B: {+syllabic, –low, +high}
b.
vacuous unification
A′:
{+syllabic, –low, +high}, B: {+high}
A′
∐ B = A′ ∪ B: {+syllabic, –low, +high}
c.
unification failure
A″:
{+syllabic, –low, –high}, B: {+high}
A″
∐ B = A’’: {+syllabic, –low, –high}
No feature’s plus and minus versions are the members of the union of A and B (4a) and those of the union of A′ and B (4b); so, the unification is the same as the union. In (4c), the set {–high, +high} is a subset of the union of A″ and B {+syllabic, –low, –high, +high}; therefore, the unification returns with A″ only.
The first of the three rules to be proposed is now given.
This rule deals with the situation where the non-low vowel is followed by the high vowel i or u:
(5)
+high spec.
Search
Change
INR: [+syllabic, –low]; DIR: right; TRM: [+syllabic, +high, –low]
INR ∐ {+high}
‘Unify +high with every non-low vowel followed by a high vowel.’
In (5), the INR is derived by keeping what is shared between E and O: Back (–back vs. +back) and Round (–round vs. +round) are different, so only +syllabic and –low are retained. The same mechanism (set intersection) operates on i and u for the TRM. Observe that the resulted “[+syllabic, –low]” actually contains more than just E and O, so i, u, and (derived or underlying) e and o are included as well. Even though the e/o followed by a high vowel undergoes the change of (5), unification returns the e/o unaltered—{+syllabic, –high, –low,…} ∐ {+high} = {+syllabic, –high, –low,…}, an instance of unification failure (4c).
Those E/O’s which are not followed by a high vowel are realized as e/o, which leads us to the second of the three rules needed.
The rule below changes the non-low vowels unconditionally into the mid ones:
(6)
–high-g.
Search
Change
INR: [+syllabic, –low]
INR ∐ {–high}
‘Unify –high with every non-low vowel.’
Here we see how a context-free rule is phrased in Search: the parameters for DIR and TRM are left blank and not manifested. The change effect on the non-low vowels is as follows: i) for e and o, the effect is vacuous since they are already –high; ii) for E and O, they are turned into e and o non-vacuously; iii) for i and u, which are +high, they remain the same after the unification with –high.
The current rule has to be applied after +high spec. (+high spec. ≺ –high-g.) for both rules to contribute to the overall result. With only these two rules, the E/O of the string …E/O…æ…i/u… would have the following derivation: i) after +high spec., the E/O becomes i/u as E/O are indeed followed by i/u, irrespective of the intervening low front vowel; ii) after –high-g., the derived i/u is still i/u; iii) the output then is …i/u…æ…i/u…—this, however, is
Recall that when both the high vowel and the low front vowel appear after the non-low vowel, the former two’s positions to each other matter. If the high vowel precedes the low front vowel (…E/O…i/u…æ…), the non-low vowel becomes high due to +high spec. (…i/u…i/u…æ…). Should the order be reversed (…E/O…æ…i/u…), it is the mid vowel that shows up (…e/o…æ…i/u…). Since we have to pay attention to the two natural classes (the high vowels i/u and the low front vowel æ), we know beforehand that some connected searches will be involved, that is, searches whose INR and/or TRM parameters contain
(7)
two hypothesis-forming strings, where arrows stand for searches
a.
the nearer i/u wrongly ignored
b.
the farther i/u wrongly included
The analysis depicted in (7a) is straightforward: i) let E/O look rightward for æ; ii) let the found æ look rightward for i/u. But there are chances, as shown in (7a), that the nearer i/u are incorrectly missed. The remedy analysis described in (7b) solves the (7a) problem with another problem for itself: i) let E/O look rightward for æ; ii) let the found æ look leftward for i/u. But it is possible, as shown in (7b), that the i/u found by the æ is to the left of E/O and is thus irrelevant to the alternation in question. The rule to be proposed uses the following tripartite searches to overcome the aforementioned problems: i) let E/O look rightward for i/u; ii) let the same E/O look rightward for æ; iii) let the æ found from (ii) look rightward for the i/u found from (i). Since the searches (i-ii) start from the same E/O and both look rightward, we are certain that the i/u and the æ thus found, if present, are to the same side of E/O and that the problem encountered in (7b) is avoided. The search (iii) indicates how Search indirectly performs distance comparing: if the found æ can to its right farther find the found i/u, we then can be sure that æ precedes i/u. This comparison is valid because the previous two searches set the E/O as the point of origin (of a line) and establish that E/O precedes i/u and that E/O precedes æ.
(7a′)
with tripartite searches
Although there is indeed an i/u to the right of the æ in (7a′), it is
The above preparation when phrased formally and followed by Unify is as follows:
(8)
–high-sp.
Search
INR-a: [+syllabic, –low]; DIR-a: right; TRM-a: [+syllabic, +high, –low]
‘Let every non-low vowel look to its right for a high vowel.’
INR-b: INR-a; DIR-b: right; TRM-b: [+syllabic, –high, +low, –back, –round]
‘Let every INR-a look to its right for a low front unrounded vowel.’
INC-c: TRM-b; DIR-c: right; TRM-c: TRM-a
‘Let every TRM-b look to its right for a TRM-a.’
Change
INR-a ∐ {–high}
‘Unify –high with the INR-a’s whose Searches a to c are all successful.’
Unlike in (5) and (6), where the prose translation is for the whole rule, each of the subparts in (8) is accompanied by a prose translation. If all the three searches are successful, the configuration …E/O…æ…i/u… is present (ignoring the other non-low vowels for they will not have the non-vacuous effect), the unification then turns the E/O of this configuration into e/o.
All the other E/O’s in any other configurations will leave –high-sp. unchanged (still unspecified for the feature High). When +high spec. later applies, the E/O’s will be turned into i/u as long as they are followed by a high vowel. Finally, the still remaining E/O’s will become e/o after the application of –high-g. In sum, the ordering among the three rules is: –high-sp. ≺ +high spec. ≺ –high-g. From looking for the presence of the low front vowel and the high vowel, to doing so only for the high vowel, and to not finding, the rules must be applied in the above order for me (as an analyst) to postulate them and for their effects to be partially observable.
To make perspicuous what the aforementioned three rules yield, the schematic derivations depending on the existence of the high vowel or the low front one are provided. Although more segments are involved for the three rules because of the natural class [+syllabic, –low], the sole attention will be on E and O (merely for the sake of clarity with no theoretical implications intended):
(9)
E/O-focused schematic derivations, where the effect of the rule is non-vacuous is not shaded