Consider the following example of feeding, based on a Finnish example (Kiparsky, 1973, 1993). The raising rule in (1a) changes word-final e to i, and the assibilation rule in (1b) changes ti sequences to si sequences². The feeding interaction is apparent in (1c) only raising is applicable to vete, resulting in veti; assibilation is now newly applicable, yielding vesi.

(1) Raising feeds assibilation

In previous work (Baković & Blumenfeld, 2017, 2018a), we show how a classical feeding interaction like this one is composed of two more basic relations between the two rules. This decomposition is illustrated in Figure 2, with attention limited³ to a set of minimal substrings standing in for the equivalence classes required to reveal the more basic relations ⁴.

Because these types of figures and the relations they illustrate are unfamiliar, we will take some time to explain Figure 2 in such a way that later figures may be read more easily.

The earlier-ordered rule involved in the interaction —in this case, the feeding raising rule ‘rse’— is displayed horizontally (input te⋉ on the left, output ti⋉ on the right, and a ‘rse’-labeled arrow between them), and the later-ordered rule —the fed assibilation rule ‘asb’— is displayed vertically (input ti⋉ above, output si⋉ below, and an ‘asb’-labeled arrow between them). To more fully represent the extensions of the rules involved in the interaction, we explicitly indicate that si⋉ is also an independently-predicted nonvacuous output of raising by adding the relevant input se⋉ to its left and a ‘rse’-labeled arrow between them. Finally, to highlight the starting point of the path through the figure that intuitively corresponds to the feeding interaction itself, the initial input of raising is underlined: te⋉.

The first, perhaps more obvious relation is that raising provides inputs to assibilation (‘rse i-provides asb’), because for at least one nonvacuous ⟨input, output⟩ pair of raising, ⟨ te⋉, ti⋉⟩, the output, ti⋉, is a possible input to assibilation, but the input, te⋉, is not. The second relation is that raising also provides outputs to assibilation (‘rse o-provides asb’), because for at least one nonvacuous ⟨input, output⟩ pair of raising, ⟨se⋉, si⋉⟩, the output, si⋉, is a possible output of assibilation, but the input, se⋉, is not.

This output-provision relation is less obvious, but crucial. It ensures that the application of assibilation to ti⋉ does not change the fact that the resulting form, si⋉, is also a possible direct result of raising, albeit from a different input. If this output-provision relation did not hold, the application of assibilation to the output of raising would result in a form that does not look like raising should have applied to it. In such a scenario, the application of raising would not be surface apparent, a well-known property of counterbleeding interactions (McCarthy, 1999) —and also of self-destructive feeding interactions (Baković, 2007), of which this hypothetical example would be one, as discussed in §5.2 below.

Another way to understand these two relations is in terms of the intersections of the string sets that constitute the two rules’ inputs and outputs (Baković, 2013b). Let’s call the set of input strings to raising rse_i, the set of output strings of raising rse_o, the set of input strings to assibilation asb_i, and the set of output strings of assibilation asb_o. Limiting our attention again to minimal relevant substrings, we have the string set intersections in (2).

(2) String set intersections of rse_x and asb_y

Raising thus nonvacuously produces both outputs that are inputs to assibilation and outputs that are outputs of assibilation; that is, raising both i-provides and o-provides assibilation.

Consider now the following example of bleeding, based on a Lamba example (Doke, 1938; Kenstowicz & Kisseberth, 1979) ⁵ . Both lowering and palatalization are applicable to kosika, but once lowering applies, yielding koseka, palatalization is no longer applicable.

(3) Lowering bleeds palatalization

A classical bleeding interaction like this one is also composed of two more basic relations between the two rules (Baković & Blumenfeld, 2017, 2018a). This is illustrated in Figure 3, again with attention focused on just the minimal substrings relevant to the interaction.

Again, the earlier-ordered rule involved in the interaction — the bleeding lowering rule ‘low’— is displayed horizontally (input osi on the right, output ose on the left, ‘low’-labeled arrow between them), and the later-ordered rule — the bled palatalization rule ‘pal’ — is displayed vertically (input osi above, output oʃi below, ‘pal’-labeled arrow between them). And again, to more fully represent the exten-sions of the rules involved in the interaction, we explicitly indicate that oʃi is a predicted nonvacuous input to lowering by adding the relevant output oʃe to its left and a ‘low’-labeled arrow between them ⁶ . And finally, the starting point of the path through the figure that intuitively corresponds to the bleeding interaction itself is highlighted by underlining the initial input of lowering: osi.

The first, more obvious relation is that lowering removes inputs from palatalization (‘low i-removes pal’), because for at least one nonvacuous ⟨input, output⟩ pair of lowering, ⟨osi, ose⟩, the input, osi, is a possible input to palatalization but the output, ose, is not. The second relation is that lowering also removes outputs from palatalization (‘low o-removes pal’), because for at least one nonvacuous ⟨input, output⟩ pair of lowering, ⟨oʃi, oʃe⟩, the input, oʃi, is a possible output of palatalization but the output, oʃe, is not.

Like the output-provision relation of feeding, the less-than-obvious output-removal relation of bleeding is crucial. Its presence ensures that the application of palatalization to osi does not change the fact that lowering is still applicable to the resulting form, oʃi. If this output-removal relation did not hold, then the output of palatalization would no longer be eligible for lowering, and in that case palatalization would bleed lowering just as lowering bleeds palatalization. This is mutual bleeding, as discussed in §5.1 below.

And again, from the point of view of string set intersection:

(4) String set intersections of rse_x and pal_y

Lowering thus nonvacuously changes both inputs that are inputs to palatalization and inputs that are outputs of palatalization to outputs that are neither inputs nor outputs of palatalization; that is, lowering both i-removes and o-removes palatalization.

The classical feeding and bleeding rule interactions can each be decomposed into two more basic relations between rules, provision and removal of inputs and outputs. Provision and removal are respectively defined more explicitly in (5) and (6).

(5) Provision of inputs and outputs (feeding)

(6) Removal of inputs and outputs (bleeding)

In the case of feeding, the earlier-ordered feeding rule both i-provides and o-provides the later-ordered fed rule; in the case of bleeding, the earlier-ordered bleeding rule both i-removes and o-removes the later-ordered bled rule.

Generalizing somewhat from the relatively specific (albeit simplified) examples discussed above, let us imagine three rules of the following schematic and possibly familiar form.

Given a starter input cade, P feeds Q and bleeds R, as shown in Figure 4 and Figure 5.

These figures are intended as schematic visual representations of crucial formal properties of each interaction. Some properties depend on rule order: the distinction between feeding and counterfeeding depends on the order of P and Q, and the distinction between bleeding and counterbleeding depends on the order of P and R. Order-dependent properties are illustrated by the dimensional orientation of the arrows in the figures: the earlier-ordered rule is displayed horizontally, while the later-ordered rule is displayed vertically. This dimensional distinction will be manipulated in §3 below to represent the result of the swapping operation.

Other properties are independent of order, and are encoded in the topology of the arrows. The structure → . → . ← corresponds to both feeding and counterfeeding (e.g. CADE → CBDE → CBDF ← CADF), and the structure ← . → . → corresponds to both bleeding and counterbleeding (e.g. CBDE ← CADE → CADF → CBDF). Mnemonically, when the two outermost arrows point inward, we have in-flow (feeding and counterfeeding); when those two arrows point outward, we have out-flow (bleeding and counterbleeding).

Consider first the case of mutual bleeding in (9), based on a Russian example (Kenstowicz & Kisseberth, 1979). Bleeding is mutual here because if l-drop were to apply first, it would be the rule that bleeds and deletion would be the rule that is bled: metl →l−d met ↛del *met.

(9) Mutual bleeding of deletion and l-drop

The illustrative figure for this case of mutual bleeding thus looks like Figure 12, with deletion and l-drop mutually removing inputs from each other, and with no output-removal.

It is crucial that there is no deletion arrow originating in l⋉, and no l-drop arrow originating in t⋉: the absence of an arrow in this figure is as significant as its presence. The topology of Figure 12 is thus ← . →, without a third arrow.

Generalizing somewhat to our schematic rules and interactions, imagine that P is further specified such that it more specifically changes cade to cbde, and call this new rule crop(P). ⁷ (R is repeated for ease of reference below.)

This constitutes a ‘cropping’ of P in that the extension of crop(P) is a narrower, proper subset of P‘s extension. Most critically, crop(P) doesn’t apply to the output cadf of R, which the original P does apply to. The effect of narrowing P ‘s extension in this way is to excise the rightmost arrow of the (counter)bleeding topology ← . → . →, converting it to mutual bleeding. ← . → The generalized figure for mutual bleeding is thus Figure 13: just like P bleeds R in Figure 5, except that the output-removal arrow of P has been excised.

The cropping operation is intended to encompass both excision and addition of an arrow, thus mimicking the inherent symmetry of the swapping and flipping operations.⁸ Thus classical bleeding can be converted via cropping to mutual bleeding by excision of the rightmost, o-removal arrow of the topology, and mutual bleeding can likewise be converted via cropping to classical bleeding by addition of this o-removal arrow.

To see how cropping works with more substantive examples, let’s first see what it takes to convert the example based on Lamba from bleeding to mutual bleeding. Excising the o-removal arrow of the lowering rule from Figure 3 results in Figure 14.

In order to achieve this result, the context of lowering must be changed such that the last of the zero or more consonants intervening between the trigger mid vowel and the target i is not ʃ. ⁹ (Recall now the relevance of the discussion in fn. 6 about whether the s~ʃ alternation in Lamba is allophonic or the neutralization of an underlying contrast.) This condition excludes rather than includes a natural class, and so must be stated negatively:

(11) Mutual bleeding of crop(low) & pal

Cropped lowering and palatalization mutually bleed each other: if cropped lowering applies first to osi, yielding ose, then palatalization is no longer applicable, and if palatalization applies first, yielding oʃi, then cropped lowering is no longer applicable.

Now let’s see how to convert the example based on Russian from mutual bleeding to bleeding. Adding an o-removal arrow to the deletion rule in Figure 12 results in Figure 15.

The minimal change necessary to achieve this result is for the context of deletion to become a disjunctive one, such that coronal stops delete before l or at the end of the word:

(12)Cropped deletion bleeds l-drop

Cropped deletion bleeds l-drop, but not vice-versa: if cropped deletion applies first to tl⋉, yielding l⋉, then l-drop is no longer applicable, but if l-drop applies first, yielding t⋉, then cropped deletion is still applicable, yielding the maximally deleted ⋉.

The contextual disjunction technically means that l-drop ‘transfuses’ cropped deletion, in the sense defined in an unpublished paper by Don Churma entitled ‘Rule interactions’ and cited by Zwicky (1987, p. 93) ¹⁰ “[I]f one rule transfuses another, the string to which the second rule applies is different from what it would be if the first rule didn’t apply — either because the first rule removes some material to which the second could apply but also supplies new places for the second rule to apply in, or because the first rule changes one string to which the second is applicable into a different string to which the second is applicable[.]” We discuss transfusion and similar complex interactions in Baković & Blumenfeld, in prep.

Consider now the case of self-destructive feeding (Baković, 2007) shown in (13), based on a Turkish example. The feeding is ‘self-destructive’ because applying deletion to the output of epenthesis results in a form that is itself not a possible output of epenthesis. Following Baković & Blumenfeld (2018b), we henceforth refer to this interaction type as seeding.

(13) Epenthesis seeds deletion

Epenthesis i-provides deletion: deletion is inapplicable to the input to epenthesis, bebekn, but is applicable to its output, bebekin. Unlike classical feeding, however, epenthesis does not also o-provide deletion: there is no application of epenthesis that leads directly to the output bebein. The illustrative figure for this case of seeding is thus Figure 16, with epenthesis i-providing but not o-providing deletion. Furthermore, since the application of deletion results in a form that is not in the set of possible outputs of epenthesis, deletion in turn o-removes epenthesis. (Seeding shares this feature in common with counterbleeding; see §6.)

The topology of seeding, → . →, is distinct from other topologies we’ve seen thus far. But much like mutual bleeding, there are only two arrows; the absence of a deletion arrow originating at ekn⋉ and of an epenthesis arrow ending at ein⋉ in Figure 16 is crucial.

Generalizing somewhat again to our schematic rules and interactions, recall crop(P) in (10a), a → b / c __ de, compared to the original P in (7a), a → b / c __ d. In this case, what’s relevant about the difference between P and crop(P) is that crop(P ) doesn’t apply to an input that would result in the output cbdf of Q. The effect of narrowing P ‘s extension in this way is to excise the rightmost arrow of the (counter)feeding topology, → . → . ← converting it into seeding → . → . The generalized figure for seeding is thus Figure 17: just like P feeds Q in Figure 4, except that the o-provision arrow of P has been excised.

Again, the cropping operation is intended to be symmetrical, encompassing both excision and addition of an arrow. Thus classical feeding can be converted by cropping to seeding by excision of the rightmost, o-provision arrow of the topology, and seeding can likewise be converted by cropping to classical feeding by addition of this o-provision arrow.

To see how cropping in the feeding context works with more substantive examples, let’s first see what it takes to convert the example based on Finnish from feeding to seeding. Excising the o-provision arrow of the raising rule from Figure 2 results in Figure 18.

In order to achieve this result, the context of raising must be changed such that there not be an s to the left of the raising target e. Because this condition excludes rather than includes a natural class, it must be stated negatively:

(14) Cropped raising seeds assibilation

Cropped raising seeds rather than feeds assibilation, because assibilation destroys the conditions that were necessary for the application of cropped raising in the first place.

Now let’s see how to convert the example based on Turkish from seeding to feeding. Adding an o-provision arrow to the epenthesis rule in Figure 16 results in Figure 19.

This result can be achieved by changing the context of epenthesis such that i is inserted before any word-final consonant, without regard for what may appear to its left:

(15) Cropped epenthesis feeds deletion

Cropped epenthesis feeds deletion, but deletion does not in turn destroy the context that enabled the application of cropped epenthesis: the form resulting from the feeding interaction is itself a possible direct output of cropped epenthesis.

Just as with classical feeding and bleeding, the rules involved in mutual bleeding and seeding interactions can be swapped — in other words, Figure 13 and Figure 17 can be rotated (clockwise) 90 degrees. The results of these rotations are shown in Figure 20 and Figure 21.

In the case of Figure 20, the result is simply that R is now the bleeding rule and P is now the bled rule. In the case of Figure 21, on the other hand, the result is a counterseeding interaction. Such an interaction is shown in (16), based on an example in Lomongo. If deletion were to apply first, deletion would seed gliding: obina →del oina →gld *wina.

(16) Deletion counterseeds gliding:

Deletion i-provides but does not also o-provide gliding: there is no input to which deletion can apply that leads directly to the output wina that would result from deletion seeding gliding. The illustrative figure for this case of counterseeding is thus Figure 22, with deletion i-providing but not o--providing gliding, and with gliding consequently o-removing deletion.

The interaction in Lomongo has been described simply as counterfeeding in e.g. Baković (2011, 2013a), but proper attention to the classically-ignored output relations reveals that it is formally distinct from counterfeeding. In order for this interaction to be converted to counterfeeding, an o-provision crop(del)-arrow would need to be added from input wbi to output wi in Figure 22 above, thus generalizing deletion such that it applies when the preceding segment is a vowel or a glide (=[-Cons] ) , not just a vowel (= V /[+syll]).

Finally, note once again that our illustrative figures highlight the difference between order-dependent and order-independent properties. Swapping (as figure rotation) converts between seeding and counterseeding as order-dependent categories; cropping (as arrow excision/addition) converts between (counter)feeding and (counter)seeding as order-independent categories.

In the same way that flipping P converts be-tween (counter)feeding and (counter)bleeding, flipping crop(P) converts between mutual bleeding and (counter)seeding. Figure 13, Figure 17, Figure 20 and Figure 21 thus become Figure 23, Figure 24, Figure 25 and Figure 26, pairwise respectively.

The orientations of these figures differ from those of their unflipped originals, but the results are the same three interaction types: seeding, mutual bleeding, and counterseeding.