Teaching both undergraduate and graduate phonology courses this quarter, and having just finished some revisions to a paper on opacity (ROA, lingBuzz; to appear in Phonology), I’ve had different types of rule interactions — in particular, ones that result in opacity — on the brain. In the paper, I describe several types of opaqueseeming rule interactions in detail and give them names like “selfdestructive feeding”. Now here’s a new one that’s not discussed in the paper because I don’t think there are any attested examples of it. Still, it’s an interesting type of case that I think is worth discussing. For reasons that’ll become clear as/if you read on below, I call it mutuallyassured destruction.
In Chapter 8 of my favorite textbook (“Rule Interaction”), Kenstowicz & Kisseberth review their first of two alternatives to “the orderedrule hypothesis” on pp. 291307: “the direct mapping hypothesis (DMH), which claims that the set of rules is applied directly to the UR to give as output the corresponding PR, as depicted in (1).”
The authors continue:
The essential claim of the DMH is that the input structure to each of the phonological rules is the underlying representation. The applicability of a rule is entirely a function of whether the underlying form meets the input requirements of that rule. The phonetic representation is then the result of applying the changes called for by the rule to the UR. The effects of a rule are necessarily irrelevant to the applicability of any other rule, since only the underlying form determines whether a rule applies or not. This view claims, then, that URs can be mapped into PRs without postulating intermediate levels of representation. Consequently, we refer to it as the directmapping hypothesis. It is often referred to as simultaneous application of the rules.
The remainder of the section cites examples of rule interactions that are consistent with the DMH, followed by examples that are not consistent with the DMH. One type of rule interaction that is consistent with the DMH is a typical case of counterbleeding. For example, consider the counterbleeding interaction between lowering and shortening in Yokuts (cribbed from p. 6 of my paper):
Here, shortening applies to the result of lowering; if shortening had applied first, it would have bled lowering (so: shortening counterbleeds lowering). But the same result can be guaranteed if both rules apply simultaneously to the UR, in accordance with the DMH, because the UR meets the structural descriptions of both rules. Obviously, though, the potential bleeding order between these same two rules (or any two rules, of course) that is possible to describe under the orderedrule hypothesis is not possible to describe under the DMH.
What K&K don’t discuss is a type of rule interaction that is possible to describe under the DMH but that is not possible to describe under the orderedrule hypothesis. Suppose you have two deletion rules like the following:
Rule 1. [son] → Ø / __ [+nas] (prenasal obstruent deletion)
Rule 2. [+son] → Ø / [cont] __ (poststop sonorant deletion)
Independent evidence for Rule 1 would come from derivations such as /pismo/ → [pimo] and evidence for Rule 2 would come from derivations such as /piglo/ → [pigo]. Evidence for their interaction would depend on the derivation of forms such as /pigmo/. Two possibilities arise under the orderedrule hypothesis: either Rule 1 precedes and bleeds Rule 2 and the result is /pigmo/ → [pimo], or Rule 2 precedes and bleeds Rule 1 and the result is /pigmo/ → [pigo]. Under the DMH, however, the result is mutuallyassured destruction: /pigmo/ → [pio]. Both consonants delete simultaneously, because the rules’ structural descriptions are both met in the underlying representation /pigmo/.
Mutuallyassured destruction is, of course, impossible to describe under the orderedrule hypothesis — probably a good thing, since this type of rule interaction is arguably unattested. (The “mutual bleeding” types of interactions between rules like Rule 1 and Rule 2 predicted by the orderedrule hypothesis are arguably attested.) This is thus potentially another argument not considered by K&K for the superiority of the orderedrule hypothesis over the DMH.
It is also impossible to describe mutuallyassured destruction with OT. Although an OT grammar derives forms in one “step” (putting aside possible stratal organization of a phonological grammar), it is of course not equivalent to the DMH; for example, counterbleeding as in Yokuts is known to be problematic for OT but not for the DMH (and counterbleeding is arguably an attested type of rule interaction). The reason why OT cannot describe mutuallyassured destruction is the same as the reason why it cannot describe counterbleeding (or selfdestructive feeding): deletion of both consonants is gratuitous, when deletion of only one will do to satisfy the likely types of constraints responsible for the consonant deletions that they occasion independently of each other.
A question worth investigating, I think, is whether versions of OT that correctly (attempt to) accomodate attested types of opacity also incorrectly accomodate unattested types such as mutuallyassured destruction. For example, take the version of OT in McCarthy’s first paper on the topic of opacity in OT (“Remarks on phonological opacity in Optimality Theory”, in J. Lecarme, J. Lowenstamm, & U. Shlonsky, eds., Studies in Afroasiatic Grammar. Papers from the Second Conference on Afroasiatic Linguistics, Sophia Antipolis, 1994, 215243. The Hague: Holland Academic Graphics, 1995; sometime since superceded on ROA). In this paper, McCarthy proposes that at least some types of markedness constraints can have the following form:
Under the “condition” column, each constraint specifies the values of α and β, the order between α and β (if any), and whether adjacency is strict, VtoV, etc. Under the “level” column, each constraint specifies whether the given condition must hold underlyingly, on the surface, or either (“indifferent”). The adjacency condition is not relevant to the discussion here, so I’ll ignore it from now on.
First, note how this constraint schema allows for the analysis of counterbleeding in Yokuts illustrated further above. In order to express the feature cooccurrence restriction like that responsible for long high vowel lowering, note that we must assume that there is a value of the linear order condition that requires α and β to hold of the same segment. The counterbleeding interaction between lowering and shortening is accomplished by making the lowering constraint indifferent to the level at which the β ([+long]) and linear order (α = β) conditions are applicable; all other relevant conditions apply to the surface:
Lowering.

Shortening.

Technically, a constraint like these is violated by any segment in the candidate surface form under evaluation that corresponds to a segment meeting the levelsensitive conditions stipulated by the constraint. (Correspondence being a reflexive relation, this includes the surface segment itself.) Because the lowering constraint is indifferent to the level at which length is represented on a surface high vowel, it can never be satisfied by shortening an underlying long high vowel. Thus, when both lowering and shortening conditions are met, both apply even though the application lowering appears on the surface to be gratuitous.
Now, the independent effects of Rule 1 and Rule 2 — the rules in a mutuallyassured destruction relation under the DMH — can be stated with two constraints:
Constraint 1.

Constraint 2.

Because both constraints are indifferent to the level at which their β and linear order conditions hold, and because both must be independently assumed to be satisfied by deletion (via domination of the antideletion constraint Max) they are in a mutuallyassured destruction relation: they can only be satisfied by deletion of both consonants in the case of /pigmo/ (Tableau 3 below), as opposed to by deletion of just the first consonant or deletion of just the second consonant in the cases of /pismo/ (Tableau 1) and /piglo/ (Tableau 2), respectively:
Tableau 1.
Input: /pis_{1}m_{2}o/  Constraint 1  Constraint 2  Max  Comment 
a. [pis_{1}m_{2}o]  * !  faithful candidate  
b. → [pim_{2}o]  *  C_{1} deletion  
c. [pis_{1}o]  * !  *  C_{2} deletion  
d. [pio]  ** !  C_{1} and C_{2} deletion 
Tableau 2.
Input: /pig_{1}l_{2}o/  Constraint 1  Constraint 2  Max  Comment 
a. [pig_{1}l_{2}o]  * !  faithful candidate  
b. [pil_{2}o]  * !  *  C_{1} deletion  
c. → [pig_{1}o]  *  C_{2} deletion  
d. [pio]  ** !  C_{1} and C_{2} deletion 
Tableau 3.
Input: /pig_{1}m_{2}o/  Constraint 1  Constraint 2  Max  Comment 
a. [pig_{1}m_{2}o]  * !  * !  faithful candidate  
b. [pim_{2}o]  * !  *  C_{1} deletion  
c. [pig_{1}o]  * !  *  C_{2} deletion  
d. → [pio]  **  C_{1} and C_{2} deletion 
Assuming that mutuallyassured destruction is in fact unattested and not something that we want phonological theory to allow, this is a major deficiency of the theory advanced by McCarthy (1995). Can mutuallyassured destruction also be described by other opacityfriendly versions of OT?