Using a rigorous, computational notion of locality, this paper evaluates one of the central motivations for autosegmental representations (ARs)—that they reduce longdistance processes to local ones. We analyze a variety of tone processes using two computational notions of locality: input strict locality as defined by Chandlee (
Locality has long played a role in phonological theory as a restriction on possible grammars, with various proposals put forth over the years for what it means to be ‘local’. Preserving locality was in fact one of the major motivating factors behind the use of featuregeometric, autosegmental representations (ARs;
At the same time, recent work in theoretical computational linguistics has made it possible to rigorously study representations and locality. There are many ways in which locality can be conceptualized, but theoretical computer science provides one grounded in principles of computation that can be readily applied to natural language in general and phonology in particular (
(1)
For example, a *CLASH constraint against adjacent Htoned syllables (
(2)
a.
*
b.
c.
However, it is common for phonological elements to interact at a distance, and this is particularly prevalent in tone (
Indeed, the computational study of phonotactics has adapted autosegmental concepts to do exactly this. Recent work has shown the utility of tier projections (
(3)
a.
*HH
b.
c.
A largely unanswered question, however, is how these results can be extended to a local characterization of phonological
This paper expands on the results of Chandlee & Jardine (
Next we apply these four categories to a series of tone processes, which includes bounded and unbounded tone spread and tone shift, as well as several variants of Meeussen’s rule, establishing for each which categories it does and does not belong to. The results show that all four categories have utility in accommodating the range of attested tone processes (i.e., no one category subsumes the other). At the same time, it is not evident from this survey alone that all four categories play
These results have larger implications for our understanding of tone in particular and the phonological grammar more broadly in the following two ways. One, we are constructing a computational theory of tone, one grounded in locality and its interaction with representation. The analysis of tone processes in isolation suggests that in such a theory the phonological grammar has more than one option for asserting locality.
Two, regardless of one’s theoretical orientation, the computational framework provided here for investigating tone offers a means of comparing competing analyses of a process that differ in representational assumptions or classification as local or nonlocal. In other words, it offers a way of identifying the computational cost of differing assumptions as well as the (non)exceptionality of a given process in relation to others. We will see examples of this in our survey.
To briefly demonstrate the computational framework used in this paper, consider the process of unbounded tone shift in Zigula (
(4)  
a.  /kusongoloza/  [kusongoloza]  ‘to avoid’  
b.  /ásongoloza/  [asongolóza]  ‘he/she is avoiding’  
c.  /kulómbeza/  [kulombéza]  ‘to ask’  
d.  /kulómbezezana/  [kulombezezána]  ‘to ask for each other’ 
In (5) we show this shift for example (4b).
(5)
We here present a brief analysis of Zigula tone shift as an AISL function, with a more detailed presentation of the framework to follow in the next section. Following Chandlee & Jardine (
(6)
The conditions for association specified by this formula are broken down below. (Note that
an output position 

H( 

the tone 

the TBU 
This configuration is depicted visually as a graph in (7), with the positions of the structure depicted as nodes (circles), and the successor function depicted as edges (arrows) between the nodes. The target nodes
(7)
The righthand side of a formula such as (6) then identifies those positions in the input structure that satisfy certain conditions. The lefthand side asserts that the
(8)
The significance of the formula being quantifierfree (QF) is that global information that would require scanning the entire input (e.g., ‘there exists a position such that…’) is not available. Rather, the formula is limited to examining a fixed window of material around a particular position. In this way the transduction is required to operate in a local manner. As we show, the fact that AISL captures processes that would otherwise not be amenable to a local analysis confirms, from a formal perspective, that the power of ARs comes from the asynchronicity of distinct tiers and the manipulation of the association relation between them.
However, as already noted, the AISL class cannot capture processes that require
The remainder of the paper is as follows. §2 details the use of logical transductions to model phonological processes and defines the two types of locality, input locality in §2.1 and output locality in §2.2. In §3 we use this framework in analyses of several types of tone processes, including bounded and unbounded tone shift and tone spread in §3.1 and variants of Meeussens’s rule in §3.2. In §4 we compare these computational notions of locality to previous ones in theoretical phonology, and also discuss the implications of our results for a more comprehensive computational theory of tone processes. §5 then concludes.
The first function class we will make use of in our analyses is the input strictly local (ISL) functions, which are functions that determine an output string for a given input string based only on input substrings of bounded length (
Computing an ISL function. Here,
The property of being ISL then provides a precise and computational notion of what it means for a phonological process to be local. One goal for this paper is to assess how well this formal conception of locality aligns with the sense in which autosegmental representations enable a local treatment of otherwise nonlocal phenomena. In pursuit of this goal we will use logical characterizations of functions, which enable a more direct comparison among different representations (in our case strings versus ARs) compared to previous automatatheoretic characterizations.
Consider a process in which a tone shifts some fixed number of TBUs away from its underlying position. For example, in Rimi, a tone shifts one TBU to the right, as shown in (9). For convenience these examples are rewritten as strings of TBUs in (10). (More will be said about this representational choice in §3.)
(9)  
a.  /uhanga/  [uhanga]  ‘to meet’  
b.  /upú̧ma/  [upu̧má]  ‘to go away’  
c.  /muntu/  [muntu]  ‘person’  
d.  /rámuntu/  [ramúntu]  ‘of a person’  
e.  /uhuvii̧/  [uhuvii̧]  ‘belief’  
f.  /mutémi̧/  [mutemí]  ‘chief’ 
(10)
a.
b.
c.
d.
e.
f.
The logical characterization of ISL defines formulas in first order (FO) logic that pick out input elements that meet certain conditions. For example, the formulas
A map or transduction from an input to output string involves defining
(11)
Bounded shift (ISL) (part 1)
a.
b.
(11a) says that an output element
(12)
a.
b.
c.
Likewise, (13a) identifies those output elements that are unspecified for tone: namely, those elements whose predecessors in the input do
(13)
Bounded shift (ISL) (part 2)
a.
b.
A complete definition of a logical transduction includes output predicates for all possible output labels (and in the case of ARs also the association relation). In the interest of space our analyses will only provide output predicates for those aspects of the representation that actually change from input to output. All others can be assumed to be defined to retain their value from the input (e.g
As noted in the introduction, the transduction is forced to operate in a local way because all of the formulas are
Chandlee & Jardine (
It follows then that the ISL functions provide a means of formally distinguishing local from longdistance processes. Consider the example of longdistance consonant agreement found in Kikongo, in which the suffix
(14)  
/tunikidi/  [tunikini]  ‘we ground’ 
In terms of the FO characterization used in this paper, a formula that determines the nasality of the suffix consonant would require a quantifier to examine the entire stem. The use of embedded predecessor functions is not possible, as there is no upper bound on how many preceding segments must be examined to confirm the presence or absence of a stem nasal.
This formal means of distinguishing local from longdistance enables a similarly formal means of investigating how longdistance patterns are made local by ARs. We first must extend ISL to operate over ARs instead of strings. ARs consist of two strings (one of tones and one of TBUs), each with their own predecessor and successor functions.
In the case of Rimi, the transduction involves a change in the association relation, specified by (15a). This formula says that a TBU
(15)
Bounded shift (AISL)
a.
b.
The example in (16) demonstrates how this formula implements the tone shift. In this example,
(16)
A function such as this one we call AISL, to reflect the fact that it operates over ARs instead of strings. Both ISL and AISL require the formulas to be QF; they differ only in representations.
As is shown in the analyses below, however, the tier structure of ARs allows for some processes that operate over an arbitrary number of TBUs (and thus are not ISL) to be captured with QF formulas (and thus are AISL). To give a hypothetical example, imagine a Meeussen’s Ruletype process in which an H becomes L following another H, regardless of the number of intervening TBUs.
(17)
a.
b.
c.
This process is not ISL for the same reason that Kikongo nasal agreement isn’t: determining whether or not an underlying /
(18)
a.
b.
c.
This process is described exactly by the QF definition for an output L given in (19a).
(19)
Longdistance Meeussen’s rule (AISL)
a.
b.
However, note that this analysis depends on the assumption of a privative H/∅ system. If, for example, there could be any number of intervening M tones, as in (20), the triggering H would no longer be within a fixed distance of the target H.
(20)
NonAISL version of longdistance Meeussen’s rule
Such a map is
This example also highlights the limits to the expressive power of AISL. First, the map
(21)
(
A consequence of this theorem is that any modifications on the tonal tier are
In this section we propose two additional function classes that can reference output structure, one for strings and one for ARs. The function class for strings is based on the
For example, consider a process that spreads a high tone to the end of the string, as in the rule in (22), applied iteratively.
Computing an OSL function. Here,
(22)
Two example mappings for this rule are shown in (23).
(23)
a.
b.
Examples (23a) and (23b) contrast two input
The OSL functions have been defined in terms of formal language theory and finitestate automata, but no logical characterization currently exists. We instead propose a logical characterization of two function classes we are calling
Following Koser et al. (
(24)
Unbounded spreading
a.
b.
Notice that this formula is recursive, in that it references itself in the second disjunct. The first disjunct then starts the recursion with any underlying high tone, which will remain high in the output. The second disjunct then checks the
Over ARs instead of strings, unbounded spreading can be achieved using recursion in the output association relation, as in (25a). An example map is given in (26).
(25)
Unbounded spreading
a.
d.
(26)
In (26), the H and the initial syllable are associated in the input; thus, they satisfy the first disjunct (25b) and remain associated in the output. Because the initial syllable is the predecessor of the second syllable, the H and the
This use of recursion is the crucial difference between RSL and ISL, since the latter cannot use recursive formulas. Recursion is, however, a powerful mechanism, and so its use must be limited in order to maintain the restrictiveness that locality provides.
(27)
a.
The transduction includes either formulas that use only
b.
Whenever a formula uses
These conditions reflect the formal properties of the OSL functions in order to guarantee a similar degree of computational restrictiveness. We conjecture that the RSL functions will turn out to be the logical characterization of the OSL functions, though we leave a proof for future work.
The reason for condition (27a) is that output locality (as defined by OSL functions) is necessarily
The reason for condition (27b) is to limit the formula to only reference the input structure for the current position (i.e.,
Using these notions of input and output locality for string and AR maps, in the remainder of the paper we will survey a set of tone processes. We will classify these processes as ISL, AISL, RSL, and/or ARSL, according to (28).
(28)
a.
A process is
b.
A process is
c.
A process is
d.
A process is
The results of the survey will demonstrate the extent to which these notions of locality overlap and the extent to which they distinguish different tone processes in terms of the computations they require. Furthermore they will reveal the various mechanisms by which ARs render nonlocal processes local, as well as the factors that can prevent them from doing so.
In this section we present analyses of a range of tone processes using the logical notions of locality defined in the previous section. For each process, we first analyze it assuming a string representation, and then again assuming an AR. Our string representations will be strings of TBUs, by default strings of syllables (e.g.,
In addition, to simplify the diagrams, in the representations to follow (both strings and ARs) we will only include the word boundary symbols (#) when they are directly relevant to the pattern in question. This means that we largely abstract away from the question of which prosodic boundaries delineate the domain of the process. As a reviewer points out, this is an important consideration in tone, in particular as the often phrasal nature of tone poses a challenge for many theories of phonology (
Lastly, we would like to emphasize that in those cases where a process is amenable to more than one analysis (e.g., input and/or output local over strings and/or ARs), we will not be making an argument for which of those options is the best characterization. Identifying the ‘best’ analysis of any of the patterns presented here is not the goal. Rather it is to establish which computational properties the process does and does not have, in order to address the larger question of how representation interacts with these computational notions of locality.
The set of processes we analyze and their classifications are previewed in
Summary of analyses.
PROCESS  ISL  AISL  RSL  ARSL 

Bounded spread (Bemba)  ✓  ✓  ✗  ✗ 
Bounded shift (KukiThaadow)  ✓  ✓  ✗  ✗ 
Unbounded shift to penult (Zigula)  ✗  ✓  ✗  ✓ 
Unbounded spread to penult (Shambaa)  ✗  ✗  ✗  ✗ 
Unbounded Meeussen’s, deletion (Arusa)  ✗  ✓  ✗  ✗ 
Bounded Meeussen’s, lowering (Luganda)  ✓  ✗  ✗  ✗ 
Alternating Meeussen’s, lowering (Shona)  ✗  ✗  ✓  ✓ 
In bounded spread, an underlying tone spreads some fixed number of TBUs and no further. In Northern Bemba an H tone spreads exactly one TBU to the right; this is also referred to as ‘binary tone spread’ or ‘tone doubling’ (
(29)  
a.  /tulakaka/  [tulakaka]  ‘we tie up’  
b.  /bálakaka/  [bálákaka]  ‘they tie up’  
c.  /bákafika/  [bákáfika]  ‘they will arrive’  
d.  /bákabila/  [bákábila]  ‘they will sew’ 
This process is intuitively local, and indeed it is ISL. A QF formula is given in (30a), with an example mapping given in (31).
(30)
Bounded spread (ISL)
a.
b.
(31)
The first disjunct of (30a) (
With an AR, the spreading rule corresponds to the definition in (32a), which reads as follows:
Bounded spread formula in (32a) applied to an AR with three syllables.
(32)
b.
d.
In
Like (30a), the formula in (32a) is both QF and nonrecursive. Bounded spreading is thus both ISL and AISL. Note that this analysis assumes underlying TBUs are either hightoned or unspecified. But the classification as ISL and AISL does not depend on this assumption. If for example the language instead had a H/L contrast, we would simply include a formula that specifies output TBUs as Ltoned if they are Ltoned in the input and their predecessors are
In the remainder of the section we will continue to follow the representational assumptions of the original cited analyses in those cases where the classification does not depend on those assumptions. In those cases where the classification is representationdependent, however, we will make note of that fact.
As for output locality, bounded spread is in fact necessarily ISL, not RSL. This is because it requires keeping track of how far the spreading has gone, which in turn requires examining the input. Once the formula references the output structure the spreading will be necessarily unbounded, as the underlying trigger can no longer be distinguished from its targets. For similar reasons the process is also not ARSL.
We now turn to bounded shift. An example is found in KukiThaadow, in which a string of tones each associate to the following syllable, as in (33). The first and last tones also remain associated to their underlying TBUs.
(33)  
a.  /kà zóoŋ  lien  thúm/  [kà  zòoŋ  lien  thǔm]  ‘my three big monkeys’  
b. 
As a map over strings, we need four formulas, one for each possible output tone. Output positions that bear a low tone satisfy the formula in (34a), which identifies two scenarios. One, the TBU is the first TBU and is underlyingly low (i.e.,
(34)
Bounded shift in KukiThaadow (ISL) (part 1)
a.
b.
The parallel formula in (35a) designates those TBUs that bear a high tone in the output.
(35)
Bounded shift in KukiThaadow (ISL) (part 2)
a.
b.
The contour tones are handled by (36a) and (36c). (36a) says that a TBU bears a LH contour if it is the last TBU (#(
(36)
Bounded shift in KukiThaadow (ISL) (part 3)
a.
b.
c.
d.
Collectively these formulas establish the map as ISL. It is also AISL. An AR map for the example in (33) is shown in (37), and the formula defining it is given in (38a).
(37)
(38)
Bounded shift in KukiThaadow (AISL)
a.
b.
Informally, (38a) says that a tone
Bounded shift then satisfies both of our notions of input locality. It does not, however, satisfy the corresponding notions of output locality. The condition on RSL/ARSL formulas in (27b) forces them to look back to the output predecessor, but they need to see the input predecessor to determine which tone it bears. In addition, the formulas in (36) and (38a) violate condition (27a), as they reference both
We now turn to unbounded tone shift and spread. We already saw a case of unbounded shift in §1.1: in Zigula (
(39)  
a.  /kusongoloza/  [kusongoloza]  ‘to avoid’  
b.  /ásongoloza/  [asongolóza]  ‘he/she is avoiding’  
c.  /kulómbeza/  [kulombéza]  ‘to ask’  
d.  /kulómbezeza/  [kulombezéza]  ‘to ask for’  
e.  /kulómbezezana/  [kulombezezána]  ‘to ask for each other’ 
Not surprisingly, this map is not ISL. To see why, consider the contrast between the penultimate syllable in an underlyingly toneless word (40a) and toned words (40b).
(40)
a.
b.
To correctly place the high tone in the output we need a formula that is false for the penult in (40a) but true for the penults in (40b). But without a quantifier, this formula must rely on a finite sequence
With ARs, unbounded shift has been derived multiple ways, often by decomposing it into two steps such as spreading and delinking (see, for example,
(41)
Unbounded shift (AISL)
a.
b.
(42)
Note that (41a) does not depend on there actually being an H in the input. It only dictates that an
However, this classification as AISL does assume an underlying H/∅ contrast. If an unbounded number of other tones (i.e., L or M) can intervene between the final H and the word boundary, it would be impossible (without a quantifier) to determine that the H is in fact the last H.
Unbounded shift to penult is not RSL for a similar reason as why it’s not ISL: with a fixed window the function can’t distinguish a word with a high tone from a toneless word. In fact the situation is even more severe for RSL: the output will never contain a
Now consider unbounded spread, in which an underlying H spreads until it is blocked or reaches a certain position. In Shambaa (
(43)  
a.  /kuhanda/  [kuhanda]  ‘to plant’  
b.  /kufúmbati∫a/  [kufúmbátí∫a]  ‘to tie securely’  
c.  /kuhandijana/  [kuhandijana]  ‘to plant for each other’  
d.  /kufúmbati∫ijana/  [kufúmbátí∫íjána]  ‘to tie securely for each other’  
e.  /ku∫unt^{h}a/  [ku∫unt^{h}a]  ‘to wash’  
f.  /kut∫í∫unt^{h}a/  [kut∫í∫únt^{h}a]  ‘to wash’  
g.  /kuɣo∫oaɣo∫oa/  [kuɣo∫oaɣo∫oa]  ‘to do repeatedly’  
h.  /kut∫íɣo∫oaɣo∫oa/  [kut∫íɣó∫óáɣó∫óa]  ‘to do repeatedly’ 
Unbounded spread is not ISL, for the same reason that unbounded shift isn’t. Consider the penultimate syllables in (44). Whether each surfaces as
(44)
a.
b.
In contrast to unbounded shift, however, unbounded spread to penult is
It might seem that the answer will be found in output locality, since in §2.2 it was shown that unbounded spread to the end of the word is both RSL and ARSL. However, the version of unbounded spread found in Shambaa differs crucially in that the spread must not reach the final TBU. This difference is crucial because it means the function needs to look in both directions: backwards to see the output of the previous TBU and forwards to check if the current TBU is final or not. This violates condition (27a), regardless of whether or not we use ARs. Thus we have here a pattern that meets none of our conceptions of locality.
This result for Shambaa does, however, depend on our treatment of unbounded spreading to penult as a single map. As noted above and pointed out by an anonymous reviewer, spreading processes like that in Shambaa are typically analyzed as two generalizations: unbounded spread to the end of the word and something like tone retraction. We could likewise treat the Shambaa pattern as the
We reiterate here that our goal is not to argue for a particular analysis, nor for a particular factoring of generalizations. Rather it is to assess which of our notions of locality each process satisfies. The case of Shambaa does, however, exemplify the way in which the computational framework advocated for here allows us to identify the computational distinctions among competing analyses and sets of assumptions.
Next we consider variants of Meeussen’s rule. First, in Arusa, the last H in a phrase is deleted following another H, no matter the distance (
(45)  
a.  /sídáy/  [sídáy]  ‘good’  
b.  /enkér sídáy/  [enkér siday]  ‘good chair’  
c.  /olórika sídáy/  [olórika siday]  ‘good ewe’ 
In (45a), the underlying H’s in /sídáy/ ‘good’ surface faithfully, whereas in (45b) and (45c) they are deleted because of the presence of a H in the preceding word.
Not surprisingly, this process is not ISL, for reasons that at this point should be clear. Consider the schematic examples in (46). The predicate
(46)
a.
b.
c.
However, it is AISL. An example mapping is shown in (47). With logical definitions, ‘deletion’ results when an input element does not satisfy
(47)
(48)
Unbounded Meeussen’s Rule deletion in Arusa (AISL)
a.
b.
This formula is interpreted as follows. The output label of
Just as with the hypothetical longdistance Meeussen’s lowering rule presented in §2.1, however, this classification is dependent on an underlying H/∅ contrast. If the intervening TBUs between the two H’s are specified, it will no longer be possible to detect the presence of a triggering H using a finite number of calls to predecessor.
As for output locality, this process is not RSL for the same reason it is not ISL: the necessary information (preceding H tone) is not a bounded distance away regardless of whether you look at the input or output. And the formula in (48a) violates condition (27a) in that it needs to look both backwards (to identify a preceding H tone) and forwards (to determine if x is phrasefinal). So the process is also not ARSL.
Next are two bounded variants of Meeussen’s Rule. In Luganda (
(49)  
a.  /alába/  alába  ‘s/he sees’  
b.  /bálába/  bálàba  ‘they see’  
c.  /bálílába/  bálìlàba  ‘they will see’  
d.  /abátálílábilila/  abátàlìlàbilila  ‘they who will not look after’  
e.  /bákilába/  bákilába  ‘they see it’ 
This process is ISL. Schematic examples are given in (50). Note in particular example (50c), in which two
(50)
a.
b.
c.
d.
The logical transduction includes the predicates
(51)
Bounded Meeussen’s Rule in Luganda (ISL)
a.
b.
c.
d.
(51a) specifies that a TBU is output as Htoned iff it is Htoned in the input (
(52)
In contrast, TBU 2 instead satisfies (51c)
Over ARs, the desired map is as exemplified in (53).
(53)
a.
b.
Both the second H in (53a) and the second H in (53b) follow an H in the input, but the lowering should only apply to (53a). Here we recall the theorem in (21), which asserts that for a process to be AISL, any changes on a particular tier must
(54)
a.
b.
This is then our first case of a map that is ISL but not AISL, or in other words, there are local maps that ARs actually render
Again we note here that this analysis is representationdependent. For example, if the intervening unspecified TBUs receive some sort of representation on the tonal tier, the map would become AISL.
(55)
a.
b.
In (55b), an explicit unspecified symbol ∅ marks the intervening unspecified TBUs. Given this, the two examples can easily be distinguished again by simply examining the predecessor of the second H. Explicit unspecified marks on the tonal tier are not standard representation, and so we do not propose this as a solution to the problem. This example does, however, offer an interesting case study into the interaction of locality and representation. Given locality as defined presently, the Luganda process is local over strings regardless of underlying tone specifications, but it is only local over ARs given a particular representation of underlying tones. We will return to this point in the summary at the end of this section.
It was noted above that this process is necessarily inputoriented, in that the conditions for lowering need to be checked in the input form. Not surprisingly then, the process isn’t RSL or ARSL. Consider again example (50c). In a RSL analysis, the formula for
Lastly, we consider another variant of Meeussen’s that produces an alternating pattern of H and L tones. In Shona, an H is lowered following another H. In contrast to Luganda, however, an H tone to which Meeussen’s Rule has applied does not serve as a trigger for a following H tone. Examples are given in (56).
(56)  
a.  /hóvé/  [hóvé]  ‘fish’  
b.  /néhóvé/  [néhòvè]  ‘withfish’  
c.  /nééhóvé/  [néèhóvé]  ‘withoffish’  
d.  /néééhóvé/  [néèéhòvè]  ‘likewithoffish’ 
In (56), the underlying H tones in /hóvé/ ‘fish’ surface as L when following the Htoned prefix /né/ ‘with’ in (56b). This lowering is blocked when the Htoned prefix /é/ ‘of’ intervenes in (56c). Instead, the H tone of this prefix lowers.
This map is not ISL. To see why, compare the outputs for the second and third Htoned syllables in the four syllable word in (57).
(57)
a.
b.
The third syllable has an H tone in the input, and its predecessor and successor are also Htoned. But the exact same conditions hold for the second syllable, which instead is output as Ltoned. Clearly, the input string does not contain the needed information to distinguish these two output positions. For the same reason, this pattern is not AISL. As shown in (58), the input string of the tonal tier does not sufficiently distinguish the H’s that lower from those that do not.
(58)
a.
b.
Clearly, the crucial information is available provided we can access the output structure. As a map over strings then, alternating Meeussen’s is RSL. The formulas in (59) assert which syllables bear high and low tones.
(59)
Alternating Meeussen’s Rule in Shona (RSL)
a.
b.
c.
d.
In both cases these formulas determine the output tone by looking at the
Likewise, the map is also ARSL, as the changes that take place on the tone tier are essentially the same as in the stringbased map. The formulas just need to refer to H and L instead of
(60)
Alternating Meeussen’s Rule in Shona (ARSL)
a.
b.
c.
d.
Summary of analyses.
PROCESS  ISL  AISL  RSL  ARSL 

Bounded spread (Bemba)  ✓  ✓  ✗  ✗ 
Bounded shift (KukiThaadow)  ✓  ✓  ✗  ✗ 
Unbounded shift to penult (Zigula)  ✗  ✓  ✗  ✓ 
Unbounded spread to penult (Shambaa)  ✗  ✗  ✗  ✗ 
Unbounded Meeussen’s, deletion (Arusa)  ✗  ✓  ✗  ✗ 
Bounded Meeussen’s, lowering (Luganda)  ✓  ✗  ✗  ✗ 
Alternating Meeussen’s, lowering (Shona)  ✗  ✗  ✓  ✓ 
In addition, the survey identified three cases—Zigula, Arusa, and Luganda—in which the classification over ARs depended on another representational assumption, namely what and how tones are marked underlyingly. Such cases demonstrate how this computational framework allows us to clearly enumerate the options for preserving locality. For example, in Luganda, do we introduce an explicit unspecified symbol on the tone tier in order to preserve locality over ARs? Or do we dispense with ARs and instead use a string analysis? Option three of course is to dispense with locality altogether and recognize the process as a ‘truly’ nonlocal phenomenon. We will not provide an answer for Luganda itself here. Rather, its example demonstrates how the precision of this computational framework facilitates the demarcation of local versus nonlocal and clarifies how that line is affected by representational choices.
At the outset of the paper we noted a goal of narrowing down the necessary categories for a computational theory of tone. Doing so requires further investigation into the gaps in the table in terms of what classes overlap others. Specifically, what’s missing are examples of processes in the following categories. One, a process that is only RSL, as the one example of an RSL map is also ARSL. This would be a process that requires reference to the output structure, but for which the AR disrupts that outputlocality (as in Luganda, in which inputlocality over ARs was not possible without crosstier information). And two, a process that is only ARSL, as every ARSL map is also either AISL or RSL. This would be a process that again necessarily references the output structure, but for which the trigger is an unbounded number of positions away given a string representation.
These gaps may be filled in by further investigation, or they may be accidental. Further study will thus aim to expand on this catalog of tone processes. While the nonexistence of a particular process cannot be proven, as the catalog becomes more exhaustive the continued absence of processes in these categories will build confidence in a proposal that they can be omitted from a computational theory of tone.
Our survey of tone processes revealed that the traditional distinction between local and longdistance phenomena is more nuanced when assessed in terms of the computational notions of input and output locality defined at the outset of the paper. The ways in which locality interacts with representation is not as straightforward as one might think. This section considers the broader implications of these results for phonological theory. In §4.1, we review the import that locality has had in the phonological literature and compare our formal definitions of locality with previous conceptions. Then in §4.2, we discuss how the results presented here can be developed into a complete computational theory of tone processes.
Considerations of locality have loomed large in the literature on phonological representations. Odden (
A widely held desideratum in phonological theory—indeed much of the motivation for nonlinear phonology and one of the outstanding problems of linear phonology—is that rules should be ‘local.’ Though there are many unresolved problems in the locality issue, it is generally agreed that a local rule formulation would only allow specification of one element to the right and/or left of the focus.
As an example, Odden raises the process of unbounded spread to antepenultimate position in the Nguni languages, such as Ndebele (
(61)  
a.  /úkuhleka/  [úkúhleka]  ‘to laugh’  
b.  /úkuhlekisa/  [úkúhlékisa]  ‘to amuse (make laugh)’  
c.  /úkuhlekisana/  [úkúhlékísana]  ‘to amuse each other’ 
Odden notes that it is possible to posit the iterative rule in (62) for this process. The target for the rule is a syllable that is followed by two other syllables. As shown in (63), this will iterate up to but not including the penult, which is followed by only one syllable.
(62)
(63)
However, this rule is nonlocal according to Odden’s definition, because it refers to more than one syllable to the target’s right. Instead, Odden argues that an accentual analysis makes such rules local. If the antepenultimate syllable receives an accent, then we can formulate a toneaccent attraction rule, as in (64a), which associates the H and the antepenult. While Odden does not explicitly state how spreading would be accomplished, one could imagine that the WellFormedness Condition (WFC;
(64)
a.
b.
The rule in (64a) is local, according to Odden (
This concern for locality has been no less important for Optimality Theory (OT;
However, not all tone patterns can be motivated entirely by local output conditions. A case in point is that of bounded shift in Rimi (
(65)  
a.  /uhanga/  [uhanga]  ‘to meet’  
b.  /upú̧ma/  [upu̧má]  ‘to go away’  
c.  /muntu/  [muntu]  ‘person’  
d.  /rámuntu/  [ramúntu]  ‘of a person’  
e.  /uhuvii̧/  [uhuvii̧]  ‘belief’  
f.  /mutémi̧/  [mutemí]  ‘chief’ 
Crucial to capturing this shift is the position of the underlying H tone. To handle this, Myers (
(66)
LOCAL (
Importantly, this constraint is not a markedness condition, as it crucially refers to the input. In fact, it is more of a ‘twolevel’ constraint (
Furthermore, output constraints often invoked to explain tone patterns are not ‘local’ by the definitions already put forth. Much of the foundational work on tone in OT (
The results of our survey increase our understanding of how ARs relate to locality in two key ways: 1) they highlight
First AISL and ARSL do not cover all attested tone patterns. One example in our survey was local Meeussen’s rule in Luganda, which was shown in §3.2 not to be AISL because it required reference to both tiers at once. One interpretation of this result is that both ISL and AISL are necessary to capture tone. Indeed, what this makes clear is that local information on the TBU tier—specifically, the tone values of neighboring TBUs—is also necessary to capture at least some processes. Another way of interpreting this result is that autosegmental locality would be better defined using some representation other than our naïve implementation of ARs. As one reviewer points out, we could define some representation that includes both the autosegmental tonal tier and the local tonal information on the TBU tier. Indeed, Jardine (
Another example that escaped our definition of locality was the unbounded spreading pattern in Shambaa. We showed in §2.2 that ‘generic’ unbounded spreading is straightforwardly RSL and ARSL. But unbounded spread in Shambaa is not, because it requires ‘looking ahead’ to determine whether it has reached the penultimate syllable (and again, (A)RSL forbids any lookahead).
There are two possible solutions to this. One is to appeal to a function class that combines these two abilities, which is precisely what the
A second solution is to break the process down into the composition of an RSL function and an ISL function, as already discussed at the end of §3.1. Such a decomposition strategy may allow other processes to be brought into the AISL and ARSL fold. As discussed in 2.2, the unidirectional nature of RSL and ARSL exclude bidirectional processes such as those highlighted in Jardine (
This is related to another big question, which is how a computational theory can combine individual processes to generate a full phonological grammar. In Appendix A we show how multiple processes can be combined directly into a single logical transduction, but this is only a start. An obvious option for combining functions from different classes is functional composition. However, composition is potentially very powerful, as computational properties may not be preserved under composition. Neither the ISL class nor the OSL class as originally defined is closed under composition, meaning that the composition of two ISL processes is not always itself ISL. Future work can and should study under what conditions AISL and ARSL are preserved under composition and when they are not. Additionally, other operations for combining functions, such as priority union (
Lastly, as with segmental phonology, we could say that patterns like Shambaa’s unbounded spreading and unbounded tone plateauing are simply nonlocal and require a different mechanism for computing longdistance dependencies. This raises new questions in terms of whether and how to restrict those dependencies and what their existence predicts typologically. For example, a reviewer points out that ‘longdistance’ Meeussen’s (as in (19)) is only AISL if there is an underlying H/∅ contrast, as any intervening tones marked on the tone tier would disrupt the locality of the two H’s. If all tone patterns are local in the sense defined in this paper, this in turn predicts that such ‘longdistance’ effects are only possible in systems with privative tone. Whether such predictions turn out to be true is an interesting question we leave for future work.
This paper applied computational techniques to study, in a rigorous way, what exactly ARs contribute to phonological locality. The framework provided here allows us to formally distinguish which tone processes are and are not local, and which are only made local with ARs (e.g., Meeussen’s in Arusa is neither ISL nor RSL, but it is AISL) and those that are
The computational properties used in our analyses make precise what it means to be local over different representations and serve as the foundation for a computational theory of tone that requires more than one notion of locality. Such a theory will ultimately narrow in on the minimal number of computational categories needed to accommodate the full range of tone processes. What those categories are is a question we don’t yet have a complete answer to, because this investigation has raised some nontrivial questions that require further study. In addition, we have established here a framework through which the computational properties of competing analyses grounded in noncomputational theories can be assessed. If a particular process is local by our definitions under one set of assumptions but not another, that fact can be used—alongside several other considerations such as how that process fits into the larger grammar—to argue for or against a particular analysis.
Looking forward then, further study will aim to fit the complete typology of tone processes into the provided framework, investigating the full scope of what happens in tone. This includes those processes that have proven to be difficult for most if not all approaches, such as interactions between tone and segmental features (e.g.,
The additional file for this article can be found as follows:
A unified analysis of OCP effects in Shona. DOI:
We analyze processes in isolation in order to focus on the interaction of locality and representation, but this computational framework is not limited to singleprocess analyses. In Appendix A we demonstrate its broader utility with a unified analysis of multiple related processes in Shona.
We have opted to not call this class AOSL because we do not yet have a mathematical proof that it subsumes the existing class of OSL functions.
An anonymous reviewer points out that this is also true in some cases in syntax, where some constraints can be simplified when stated over strings instead of trees. For example, the
While Chandlee (
This might give the impression that we are ruling out metathesis, which is in fact ISL (when local) (see
We only consider ARs with two strings, leaving the analysis of ARs with more than two strings to future work.
Throughout the paper, we use dashed association lines to indicate associations that satisfy the output association formula.
We thank an anonymous reviewer for pointing this out.
Obviously if we look far enough back in the input string we will identify a
For a more comprehensive explanation of the desirability of computational restrictiveness in phonology readers are referred to Chandlee & Heinz (
There is a third condition, which is not relevant to any of the examples analyzed in this paper and is technically involved, and so we omit it in the interest of clarity and for space limitations. Essentially though it serves to prohibit longdistance agreement, which can be enacted using leastfixed point logic via a type of covert ‘markup’. See Chandlee & Jardine (
This is not the case with ISL functions; when referencing only input the same output string will result regardless of direction. Hence the logical characterization of ISL is free to combine
Readers are referred to StrotherGarcia (
Ternary spread (as in Copperbelt Bemba) is also ISL and AISL, as witnessed by the following formulas:
(i)
(ii)
These are simply the formulas in (30a) and (32a) with additional disjuncts for the syllable two syllables away. This analysis of ternary spread demonstrates how our notion of locality is not the same as the prior notion of ‘don’t count past two’. What makes a pattern local is not that it involves a window of size 2, it’s that an upper bound exists on the length of that window. For Copperbelt Bemba that length is 3. Formally, we’d say that Copperbelt Bemba’s pattern is a 3ISL or 3AISL function, while Northern Bemba’s is 2ISL or 2AISL.
Representing unbounded shift as a onestep process is reminiscent of previous OT analyses (e.g.,
The fact that both H’s are deleted from /sídáy/ is explained in an AR with a single H associated to two TBUs. The generalization would be complicated then in a stringbased representation, in which each TBU must bear its own H. We put aside this complication, however, since even without it the rule is not ISL.
Note the process only applies at the end of a phrase (
We thank an anonymous reviewer for pointing this out.
In contrast, an input like /
Zoll (
See Chandlee (
The authors have no competing interests to declare.