2.1 Unbounded circumambient processes

A precise definition of an unbounded circumambient process is given in (2). Crucially, this property is atheoretical, and thus agnostic to specific theories of representation and processes.

1. (2)

   

To illustrate the concept, a rewrite rule representation of this type of process is given in (3). In (3), X and Y are non-empty, they surround the target and there is no bound on the distance between them.

1. (3)

   

An example is the possible rule in (4).

1. (4)

   

In (4), whether a vowel becomes [+back] depends on the presence of [+back] vowels on both sides of the target. However, these [+back] vowels may be separated from the target by strings U and V. Because U and V can be of any length, the process must be able to ‘look back’ for a [+back] segment over any distance in the left context (i.e. over U) and ‘look ahead’ over any distance for a [+back] vowel in the right context (i.e. over V).

Importantly, X or Y might contain ‘blocking’ information which prevents the process from applying. A possible such rule is given in (5).

1. (5)

   

Here, the target will nasalise only when it is preceded by a [+nasal] segment and not followed by another [+nasal] segment. While (5) is given as a rule, such conditions are more intuitively expressed with Optimality Theory constraints (Prince & Smolensky Reference Prince and Smolensky1993).

Furthermore, the definition of unbounded circumambient applies also to autosegmental representations. From an autosegmental standpoint, the ‘target’ referred to in (2) is any unit affected by the changing of association lines, and the distance from the ‘trigger’ is measured on the timing tier. These choices will become clear in §4.4.

Because ‘unbounded’ is critical to the definition in (2), we must set the criteria for an unbounded process. Intuitively, an unbounded process is one which operates over multiple units, like segments or tone-bearing units (TBUs), for which the correct generalisation does not refer to a bound on how many units over which it may operate. As linguists may differ as to what constitutes evidence for a process being ‘unbounded’, I use the criteria in (6).

1. (6)

   

On its own, criterion (6a) does not constitute strong evidence for a process being unbounded, and thus no processes are included here which only meet (6a).

For some, (6b) is enough evidence that a process is unbounded, especially if there are examples of it operating over three or more units – as Kenstowicz (Reference Kenstowicz1994: 372) puts it, ‘phonological rules do not count past two’. In contrast, other researchers may consider only (6c) sufficient. However, the evidence presented here shows there is a typological asymmetry, regardless of which criterion one considers. By either (6b) or (6c), unbounded circumambient processes are far more common in tonal phonology than in segmental phonology.

2.2 Unbounded tonal plateauing

Having established the criteria for classifying a process as unbounded and circumambient, we can now discuss individual unbounded circumambient processes. This section surveys eight languages with some form of unbounded tonal plateauing, in which any number of L-toned or unspecified (Ø) TBUs (the TBU will be assumed to be the mora) surface as H if they are between two Hs, but as L otherwise.

Kisseberth & Odden (Reference Kisseberth, Odden, Nurse and Philippson2003: 67) motivate UTP as a repair for a constraint against ‘toneless moras between Hs’. Hyman & Katamba (Reference Hyman and Katamba2010) formalise UTP in Luganda (which they refer to as ‘H-tone plateauing’) as in (7).

1. (7)

   

UTP thus fits the definition in §2.1 of an unbounded circumambient process, because whether or not the process applies depends on two Hs that (a) are on both sides of the affected TBUs, and (b) can be arbitrarily far away from any one of the affected TBUs.Footnote ¹

Using data from a number of sources, the following subsections confirm Hyman's (Reference Hyman, Goldsmith, Riggle and Yu2011) claim that UTP is a well-attested tonal process, and that by criterion (6b) it is unbounded in all cases and by (6c) in most.

2.3 Unbounded circumambient processes in segmental phonology

Turning to segmental phonology, there are the only two potential attestations of unbounded circumambient processes of which I am aware. One is Sanskrit n-retroflexion. The other is ‘plateauing harmony’ in Yaka, which also has UTP, as explained in §2.2. Both satisfy the unboundedness criterion in (6b), but neither satisfy (6c).

2.4 Empirical summary

The preceding sections presented eight attestations of UTP, an unbounded circumambient process in tonal phonology, and two examples of separate unbounded circumambient processes in segmental phonology. All satisfied criterion (6b), with the greatest distance between target and trigger being five TBUs. Seven examples of UTP were shown to operate over domains extended by morphology or syntax, thus satisfying criterion (6c). Neither segmental process satisfied (6c), and in Yaka, the distribution of the process was quite limited. Both satisfied (6b), although in Yaka the greatest attested distance between trigger and target was three vowels, and in Sanskrit there was no evidence for blocking beyond the syllable following the target.

Most importantly, to the best of my knowledge, these are the only examples of such processes in segmental phonology. This is notable, given the wide attestation of long-distance segmental processes in general, as documented in the comprehensive surveys on feature-spreading harmony (Rose & Walker Reference Rose and Walker2004), vowel harmony (Baković Reference Baković2000, Nevins Reference Nevins2010), consonant harmony (Rose & Walker Reference Rose and Walker2004, Hansson Reference Hansson2001, Reference Hansson2010) and consonant disharmony (Suzuki Reference Suzuki1998, Bennett Reference Bennett2013). In fact, Wilson (Reference Wilson2003, Reference Wilson2006b), reviewing the typologies of nasal, emphasis and vowel harmonies, characterises segmental spreading processes as ‘myopic’. This means that even segmental spreading processes which affect multiple segments proceed in a local fashion, never ‘looking ahead’ beyond immediately adjacent segments. As unbounded circumambient processes by definition depend on information unboundedly far away from the target, any myopic process is not an unbounded circumambient process, and any unbounded circumambient spreading process is necessarily not myopic. Yaka vowel harmony and Sanskrit n-retroflexion are thus exceptions to the myopic spreading generalisation; this highlights how atypical these two processes are.Footnote ⁶

There is thus a typological asymmetry between tonal and segmental processes: unbounded circumambient patterns are extremely rare in segmental processes, but widely attested in tonal phonology, as evidenced by the variants of UTP discussed. A comparison between the proportions of circumambient unbounded processes found in typological surveys (such as those just mentioned) of segmental and tonal processes would be ideal, but comparable surveys of tonal processes, or even particular kinds of processes, do not exist (to the best of my knowledge). Regardless, the evidence reviewed in this paper clearly shows an asymmetry, despite the absence of such surveys for tone.

It should be noted that bidirectional spreading processes, in which a feature spreads outwards from a single trigger in two directions, are common in segmental harmony. An example is Arabic emphasis spreading, in which an emphatic gesture spreads in an unbounded fashion in both directions from an underlying emphatic segment. In the following example from Southern Palestinian Arabic (Al Khatib Reference Al Khatib2008: 2), emphasis spreads to the left (26a), to the right (b) and to both the left and right (c) (spreading is blocked by high front segments, such as /j/).Footnote ⁷

1. (26)

   

Other bidirectional processes are nasal spread in Capanahua and Southern Castillian (Safir Reference Safir1982) and cases of stem control in vowel harmony (Baković Reference Baković2000) and consonant harmony (Hansson Reference Hansson2001, Reference Hansson2010). While these processes apply in an unbounded fashion, and operate in two directions, they only have one trigger, and are thus not circumambient.

2.5 The sour-grapes pattern in tonal phonology

One final piece of evidence for the unbounded circumambient asymmetry involves the unattested ‘sour-grapes’ vowel-harmony pattern (Baković Reference Baković2000, Wilson Reference Wilson2003, McCarthy Reference McCarthy, Goldsmith, Hume and Wetzels2010, Heinz & Lai Reference Heinz, Lai, Kornai and Kuhlmann2013).Footnote ⁸ The sour-grapes pattern is a non-myopic harmony pattern predicted to exist by ranking permutations of classic OT with Agree constraints, but not attested in segmental harmony. The sour-grapes pattern is also unbounded circumambient, and, as will be discussed momentarily, a sour-grapes-like process appears in tonal phonology.

Sour-grapes harmony works as follows. Given a spreading [+F] feature (which targets underlying [–F] segments) and a blocking feature [!F], [–F] segments become [+F] after another [+F], provided that there is no blocking feature following in the word, as shown in (27).

1. (27)

   

In other words, [+F] spreads if and only if it can spread all the way.Footnote ⁹ Note that sour-grapes spreading is not myopic: the spread of the [+F] has to ‘look ahead’ to see if there is a blocker before it can apply. As such, it is also an unbounded circumambient pattern, because the presence or absence of both [+F] and [!F] segments, which can be any distance apart, bears on the realisation of [–F] segments in between (this blocking aspect makes it circumambient in a similar way to Sanskrit n-retroflexion).

However, a sour-grapes-like pattern does exist in tonal phonology. In Copperbelt Bemba (Bickmore & Kula Reference Bickmore and Kula2013, Kula & Bickmore Reference Kula and Bickmore2015), underlying H tones undergo one of two spreading processes, bounded ternary spreading or unbounded spreading. The latter is blocked by the presence of another H tone, as shown in (28). In phrase-final forms, unbounded spreading applies to the rightmost H.

1. (28)

   

Bounded spreading occurs when another H appears to the right, as in (29). Bounded spreading obeys the Obligatory Contour Principle; it will spread up to two additional TBUs, maintaining at least one L TBU before the second H. All other intervening TBUs surface with a L tone.

1. (29)

   

The formalisations in (30) summarise the facts. When no Hs are present, as in (a), all TBUs surface as L (cf. (28a): [ù-kù-tùl-à]). When one H is present, it spreads to all remaining TBUs in the domain (and the rest surface as L, as in (28d): [tù-kà-páápáátík-á]). When two Hs are present, the first only spreads to the next two TBUs (cf. (29c): [tù-kà-bélééng-él-àn-à kó]).

1. (30)

   

Note that, modulo the bounded spreading, the formalisations in (30) are almost identical to the sour-grapes generalisations in (27). In other words, in Copperbelt Bemba the second H can be seen as a blocker for unbounded spread. This makes it an unbounded circumambient process, because the realisation of unspecified TBUs depends on the presence or absence of Hs on both sides which can be arbitrarily far away. As a tonal process, it does not seem particularly aberrant, in contrast to Sanskrit n-retroflexion and Yaka vowel harmony. Thus it provides more evidence for the unbounded circumambient asymmetry.

3.1 Formal language complexity and cognitive complexity

A formal language is a set of strings, and FLT studies the relationships between formal languages and the expressive power of grammars that describe them. FLT characterisations of natural language patterns have been argued to reflect domain-specific cognitive biases for and against patterns of a certain level of complexity. For example, the regular class of formal languages is insufficient to describe English syntax, and English syntax is at least context-free (Chomsky Reference Chomsky1956). Phonology, on the other hand, appears to be at most regular (Johnson Reference Johnson1972, Kaplan & Kay Reference Kaplan and Kay1994).Footnote ¹⁰ This notion of complexity has been explicitly linked to cognitive complexity (Folia et al. Reference Folia, Uddén, Vries, Forkstam and Petersson2010, Rogers & Hauser Reference Rogers, Hauser and Hulst2010, Rogers & Pullum Reference Rogers and Pullum2011), and results from artificial language learning experiments provide evidence in support of the psychological reality of the context-free/regular division between syntax and phonology (Lai Reference Lai2015).

The primary goal of this paper is to use this notion of complexity to characterise the unbounded circumambient asymmetry discussed in §2. Instead of formal languages, however, the following sections study the relationships between regular maps, where a map is a relation between strings in which an input string is paired with at most one output string.Footnote ¹¹ Analogous to the regular/context-free distinction for formal languages, maps can be classified according to their complexity. The distinctions important for this paper centre around the property of subsequentiality (Mohri Reference Mohri1997), which can be defined in terms of finite-state transducers (FSTs). FSTs are idealised machines that match pairs of strings; they are described in more detail below. While FSTs in general can describe any regular map, subsequential FSTs describe a more restricted set of maps.

3.2 Overview: formal language complexity and phonology

The classes of interest in this paper are left- and right-subsequential maps, weakly deterministic maps and regular maps. Figure 1 depicts the relationships between these classes as a nested hierarchy, in which more complex classes properly include lesser ones. Left- and right-subsequential maps are provably less complex than weakly deterministic maps (Heinz & Lai Reference Heinz, Lai, Kornai and Kuhlmann2013) and regular maps (Mohri Reference Mohri1997), and weakly deterministic maps have been conjectured to be less complex than regular maps (Heinz & Lai Reference Heinz, Lai, Kornai and Kuhlmann2013). Assuming this conjecture to be true (see further below), this means that all weakly deterministic maps are also regular maps, but not all regular maps are weakly deterministic. Regular maps conjectured not to be weakly deterministic can be referred to as fully regular.

Figure 1 Phonological processes in the subregular hierarchy.

Figure 1 also depicts where typological research examining segmental processes as maps place these processes in the complexity hierarchy. This work has found all of them to be within the weakly deterministic class, with most in the less complex left- and right-subsequential classes. This has led to the hypothesis that that the weakly deterministic class forms a bound on the complexity of phonology. As Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013) discuss, this hypothesis is supported by the absence of sour-grapes patterns, which they prove to be neither left- nor right-subsequential, and conjecture to not be weakly deterministic.

In addition, Fig. 1 shows the other unbounded circumambient processes discussed in this paper belonging to the fully regular region. As will be explained in detail in §4, this is because UTP and other unbounded circumambient processes are, like sour-grapes harmony, neither left- nor right-subsequential, and there is no known weakly deterministic characterisation of them. By revising the above hypothesis, then, we have a characterisation for the typological asymmetry established earlier: segmental phonology, but not tonal phonology, is weakly deterministic. The existence of Sanskrit n-retroflexion and Yaka harmony are admittedly exceptions, but as already discussed in §2, they are not only rare, but also exceptions to Wilson's myopia generalisation. These exceptions will be discussed in §5.3.

3.3 Finite-state transducers, subsequentiality and determinism

This subsection gives an informal introduction to FSTs, and discusses subsequential transducers, a type of FST which is strictly less expressive than the fully regular non-deterministic transducers (Mohri Reference Mohri1997). As an example, consider the regressive assimilation generalisation ‘nasals become labial before a labial consonant’. A rule for this generalisation is given in (31).

1. (31)

   

To simplify the notation, let us abstract away from place features other than [labial] as follows. ‘C’ represents non-labial, non-nasal consonants and ‘P’ labial consonants. A FST corresponding to the generalisation in (31) is given in Fig. 2. It comprises a set of states (the circles labelled with a q) and transitions between the states (the labelled arrows). The FST can be interpreted as reading an input and writing an output as follows. Beginning from the start state (marked with the unlabelled arrow; q0 in Fig. 2), it traverses the transition for each symbol (indicated to the left of the colon in a transition label) in the input string, each time writing out the corresponding output (indicated to the right of each label).Footnote ¹² At the end of the input string, it appends to the output the output for the current state (indicated to the right of the colon on the state label). The empty string λ indicates that there is no output. In this way, the transitions and states define which input/output string pairs are accepted by a machine. As an example, the derivation in (32) shows how an input CVnC corresponds to an output CVnC (i.e. no change occurs). The derivation highlights each state, input symbol and output as the machine reads CVnC.

Figure 2 A deterministic finite-state transducer for nasal place assimilation.

1. (32)

   

Note that, while on the initial C and V the FST simply loops on state q0, outputting C and V respectively, on the following ‘n’ it makes a transition to state q1, outputting nothing. This is because the machine cannot yet ‘decide’ the output for this ‘n’, as it could be ‘n’ or ‘m’, depending on whether or not the following input symbol is a P. Let us thus call state q1 a ‘wait’ state. As the next symbol is a C, the machine takes the C:nC transition from q1 to q0, outputting nC. If the input is instead CVnP, the derivation is similar, but in the last step the machine takes the P:mP transition from state q1, as in (33).

1. (33)

   

In this way, state q1 allows the machine to ‘look ahead’ one symbol in the input. The reader can verify that this machine will take any permutation of C, V, n and P as an input, outputting ‘n’ as ‘m’ only before a P. Thus Fig. 2 describes exactly the map as (31).Footnote ¹³

For all states in Fig. 2, there is only one possible transition, given a particular input symbol. This means the FST in Fig. 2 is deterministic. Not all FSTs are deterministic; it is possible to write a FST such that a state may have two separate transitions on a particular output. Figure 3 is such a machine. Here, there are no outputs specified for the states, and q2 is a ‘non-accepting’ state (indicated by the single, rather than double, circle), meaning that the machine rejects an input/output pair for which the input ends on that state.

Figure 3 A non-deterministic finite-state transducer for nasal place assimilation.

Non-deterministic FSTs are more powerful than deterministic FSTs: any map describable by a deterministic FST can be described by a non-deterministic FST, but not vice versa (Mohri Reference Mohri1997). The ability to be described by a deterministic FST is central to the definition of the classes of left- and right-subsequential maps (Mohri Reference Mohri1997). The regular class of maps are those maps which can be described by any non-deterministic FST. Thus the left- and right-subsequential maps form a strict subset of the regular class.

UTP is a concrete example of a map which cannot be described with a deterministic FST, and will be described in detail in §4. For now, the restriction which determinism places on the subsequential maps can be understood in terms of ‘wait’ states, as in Fig. 2, which allow deterministic FSTs to look ahead in the input. Because the number of states is finite, there can only ever be a finite number of wait states, and so a deterministic FST has bounded look-ahead. Non-determinism, in contrast, allows a FST to ‘postpone’ a decision about a particular input symbol indefinitely.

3.4 Subsequentiality and segmental phonology

Computational analyses of typologies of segmental processes have shown that they are describable with deterministic FSTs, with some variation regarding directionality. Mohri (Reference Mohri1997) described two kinds of subsequential maps: right-subsequential and left-subsequential, which are describable by deterministic FSTs reading an input string left-to-right and right-to-left respectively. Collectively, they can be referred to as subsequential maps.

The discussion in the previous section suggests how local processes are subsequential. For a thorough survey of the subsequentiality of local processes, including epenthesis, deletion, metathesis, substitution and partial reduplication, see Chandlee (Reference Chandlee2014). Work on long-distance segmental processes such as vowel harmony (Gainor et al. Reference Gainor, Lai and Heinz2012, Heinz & Lai Reference Heinz, Lai, Kornai and Kuhlmann2013) and dissimilation (Payne Reference Payne2014) has also found them to be largely left- or right-subsequential. To see how, imagine a long-distance progressive consonantal harmony process in which a feature [–F] becomes [+F] after some other consonant specified [+F], no matter how early in the word this consonant appeared (an example is /l/ in Yaka becoming [n] in suffixes attaching to a root containing a nasal). Such a map, given in (34), is describable with the deterministic FST in Fig. 4 (the tailed arrow indicates an input/output pair that belongs to a map).

Figure 4 A deterministic finite-state transducer for progressive harmony (C = consonant, V = vowel segments not participating in harmony).

1. (34)

   

This is impossible for a regressive harmony process, in which [–F] becomes [+F] before some [+F] segment an unspecified distance later in the word, as in (35).

1. (35)

   

This requires unbounded look-ahead from left to right; in order to determine the output for a target [–F] in the input, a FST reading from left to right would have to wait indefinitely to see if a trigger [+F] appeared later in the string. However, this map is right-subsequential, because reading the input from right to left requires no look-ahead. Reading from right to left can be thought of as reversing the input, feeding it into the FST and then reversing the output (Heinz & Lai Reference Heinz, Lai, Kornai and Kuhlmann2013). If we reverse the strings in the map in (35), we get the same map as in (34). We can thus describe the reverse of (35) with a deterministic FST, so it is right-subsequential.

Thus local processes and unidirectional long-distance processes are either left- or right-subsequential. However, there are two classes of phonological processes which are neither left- nor right-subsequential: unbounded circumambient processes and bidirectional spreading processes. These processes are the focus of the following two sections.

4.1 Unbounded tonal plateauing as a map

To analyse UTP using the computational framework for studying string maps outlined above, this section uses a string-based representation which marks associations to H tones on each TBU. The use of this representation will be discussed in §4.4. For now, we can say that if UTP is viewed with this kind of string-based representation, then it is neither left- nor right-subsequential.

The unbounded tonal plateauing generalisation was formalised in (7) in §2. Its string-based counterpart is given in (36), where H represents a TBU associated to a H tone, and Ø represents an unspecified TBU. Superscript m, n and p represent any natural number.

1. (36)

   

The linear map in (36) makes explicit every possible situation in UTP. In (a) and (b), in which there are fewer than two H-toned TBUs in the underlying form, no plateauing occurs. Plateauing occurs instead when there are two or more Hs in the input (e.g. (10b) above from Luganda). This is seen in (36c): all TBUs between the first and last H-toned TBUs surface as H. The notation {Ø,H}n denotes a string of n TBUs, either H or Ø.

4.2 The non-subsequentiality of unbounded tonal plateauing

The UTP map in (36) is neither left- nor right-subsequential. As §5.3 will discuss in detail, this holds for any unbounded circumambient process. This section presents an informal illustration of the proof; for the full proof, see the online appendix.

The UTP map in (36) is regular, as it can be modelled with a non-deterministic FST. This FST is given in Fig. 5, where underlying Ø TBUs are H in the output in state q1, from which a final state can only be reached if there is another input H following; one can think of q1 as the ‘plateau’ state.

Figure 5 A non-deterministic finite-state transducer for unbounded tonal plateauing.

This non-determinism is necessary, as there is no way to capture this map with a deterministic FST. Recall that a deterministic FST must have at most one transition per input symbol at every state. We cannot determinise the FST in Fig. 5. To see why not, let us adopt the ‘wait’ strategy employed in the FST in Fig. 2 in §3.3. There are many ways to try this; this discussion follows one. The proof in the appendix ensures that all will fail, but the discussion here is intended to give an intuition as to why.

In the deterministic FST in Fig. 6, q2 is a wait state representing the knowledge that a sequence HØ – which may be a plateauing environment – has been seen in the input. Here, it ‘waits’ one symbol to see whether the next symbol in the input is a H or a Ø. It will thus correctly transform ØHØH, with only one intervening Ø TBU, to ØHHH. However it incorrectly maps inputs for which the second H is further away; for example, the input ØHØØH would be mapped to ØHØØH (itself). Thus, because there is only one wait state, the machine in Fig. 6 can only describe plateaus of at most three Hs.

Figure 6 First attempt at a deterministic finite-state transducer for unbounded tonal plateauing.

The machine in Fig. 7 adds an additional wait state to try to remedy this situation. Here the FST is much like the one in Fig. 6, except that it has two wait states, q2 and q3, which represent the look-ahead necessary to capture the behaviour of one and two Øs following an input H. Thus, Fig. 7 correctly maps ØHØH↦ØHHH and ØHØØH↦ØHHHH.However, it incorrectly maps inputs like ØHØØØH, where two Hs are separated by three or more Øs, to themselves. By now it is perhaps obvious that we are on a wild-goose chase; any ‘wait n symbol’ strategy will fail for any mapping in the UTP relation whose input string includes the sequence HØn+1H. However, given the restriction of determinism, ‘wait n symbols’ is the best we can do. Simply reversing the string, in an attempt to create a right-subsequential transducer, will not help; the position of the first triggering H is just as arbitrarily far to the right as the second is to the left. Thus a deterministic FST representation of the UTP map is impossible, and so it lies outside both the left- and right-subsequential classes.

Figure 7 Second attempt at a deterministic finite-state transducer for unbounded tonal plateauing.

4.3 Unbounded circumambient processes in general

That UTP is neither left- nor right-subsequential follows from its unbounded circumambient nature: as triggers may lie any distance away on either side of a given target, a FST describing the map requires unbounded look-ahead in both directions.

Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013) prove this is also true for sour-grapes harmony, for the same reasons. For these processes, the fate of an input segment that can potentially assimilate rests on whether or not a trigger appears to one side and whether or not a blocker appears to the other. They show that this means it cannot be described by a deterministic FST reading in either direction. We can then see why no unbounded circumambient process can be subsequential. For unbounded circumambient processes, it is definitional that crucial information may appear on either side of the target, unboundedly far away. This means any unbounded circumambient process will require unbounded look-ahead in both directions, and will not be subsequential. This result is then key to characterising the unbounded circumambient asymmetry in terms of computational complexity, as discussed further in §5.

As stated at the outset, this result depends on a particular kind of string-based representation. Before we compare unbounded circumambient processes with bidirectional spreading in §5, it is therefore necessary to address the potential objection that these results are invalid because they hinge on this string-based representation for UTP.

4.4 Autosegmental representations and linear representations

Tone has been widely analysed in terms of autosegmental representations (though for alternatives see Cassimjee & Kisseberth Reference Cassimjee, Kisseberth and Kaji2001 and Shih & Inkelas Reference Shih, Inkelas, Kingston, Moore-Cantwell, Pater and Staubs2014), while the notion of subsequentiality is defined in terms of strings. The preceding analysis used a string-based representation of UTP, even though it is a tonal process. This section explains and defends two interrelated facts about such string-based representations, formulated in (37).

1. (37)

   

(37a) is a common assumption about the relationship between string-based representations and autosegmental representations, and is the one implicit in the string-based representations used in the computational literature cited in §3.4. As for (37b), measuring distance on the timing tier as opposed to the melody tier is a representational choice. Indeed, Kornai (Reference Kornai1995) compares different ways of encoding string-based representations of autosegmental representations. The following sections explain and justify string-based representations based on (37).

5.1 Bidirectional spreading and the weakly deterministic class

The class of unbounded circumambient processes is not the only one which is not left- or right-subsequential. In patterns of stem-control harmony, or cases of bidirectional spreading, such as the Arabic emphasis spreading discussed in §2.5, a feature spreads outward both to the right and the left, as in (41).

1. (41)

   

Such a map requires unbounded look-ahead in either direction: a target may follow or precede the trigger, in any direction. Thus, as Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013) also show, such cases (which will henceforth be referred to under the umbrella term bidirectional spreading) are not subsequential. However, there is a crucial difference between bidirectional spreading and unbounded circumambient processes. Bidirectional spreading hinges on a single trigger, whose influence spreads outward. In contrast, unbounded circumambient processes hinge on two triggers/blockers, whose targets lie between.

Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013) observe that a bidirectional spreading process is essentially the same unidirectional map applied left-to-right and then right-to-left. They propose a superclass of the subsequential maps, called weakly deterministic maps, which includes this kind of process. A weakly deterministic map can be decomposed into a left- and right-subsequential map, such that the left-subsequential map is not allowed to change the alphabet or increase the length of the string.

The process in (41) can be decomposed into two left- and right-subsequential maps, describable by the consonant harmony FST in Fig. 4. First, the input string is read by the FST left-to-right (applying the left-subsequential map), then the resulting output is fed back into the FST right-to-left (applying the right-subsequential map to the output). This decomposition is schematised in (42).

1. (42)

   

This composition of the two maps is special, because it does not change the alphabet or increase the length of the string. Intuitively, this is because these bidirectional processes can be thought of as one unidirectional process operating in two directions. This is highlighted by the fact that both sub-maps use the same FST. Thus, bidirectional spreading is weakly deterministic, as first seen in Fig. 1.

The weakly deterministic class is defined as such by Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013) as a restriction on Elgot & Mezei's (Reference Elgot and Mezei1965) result that any regular map can be decomposed into a left-subsequential and right-subsequential map, as long as the left-subsequential map is allowed to enlarge the alphabet. This result holds because the left-subsequential map can ‘mark up’ the string with extra symbols in the first map, and then erase them in the second.Footnote ¹⁴ However, no attested segmental maps studied in the literature cited above require such a mark-up.

5.2 Unbounded circumambient processes and the weakly deterministic class

In contrast to attested segmental processes, Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013) argue that sour-grapes harmony requires such a mark-up, and thus is not weakly deterministic. This is because it is not simply the application of the same process in two directions. Because the sour-grapes pattern is unbounded circumambient, for any decomposition into two sub-maps the left-subsequential process must somehow encode whether or not a [+F] has been seen to the right of the [–F] targets. It is difficult to see how this can be done without intermediate mark-up, and thus Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013) conjecture that sour-grapes harmony is not weakly deterministic. The exact same arguments apply to UTP.

UTP can be decomposed into two subsequential maps with an augmented alphabet, as in (43). First, the left-subsequential process marks all Ø following a H as ?, and then the right-subsequential process changes all ? preceding a H to H (and all ? not preceding a H to Ø).

1. (43)

   

Crucially, this decomposition relies on the intermediate ? symbols to ‘carry forward’ the information that a H appears to the left in the string. This allows the right-subsequential map to correctly apply without any unbounded look-ahead. However, as in the case of sour-grapes processes, it is hard to see how there could be a similar decomposition which uses only H and Ø, maintains string length in the left-subsequential sub-map and obtains the exact same map. This is because using the same alphabet to create an encoding will always lead to distinctions between input strings being lost, and so such an encoding is bound to fail.

Again, this is based on the unbounded circumambient nature of the process: because there is crucial information on either side of the targets, it is necessary to mark targets to the right of the left trigger in order for the right-subsequential function to correctly process them without unbounded look-ahead. Thus, Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013)’s conjecture applies not just to UTP, but also to any unbounded circumambient process.

An important implication of Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013)’s conjecture is that there exist fully regular maps outside of the weakly deterministic class, and thus that the weakly deterministic class of maps is a proper subclass of the regular class of maps, as depicted in Fig. 1. This section has argued that unbounded circumambient processes fall into this fully regular class, as in Fig. 1, and that this distinguishes them in complexity from bidirectional spreading.

5.3 The weakly deterministic hypothesis

The conjecture that the weakly deterministic class is a proper subclass of the regular class provides a way to capture the unbounded circumambient asymmetry. Heinz & Lai (Reference Heinz, Lai, Kornai and Kuhlmann2013) propose that phonology is weakly deterministic. This hypothesis correctly predicts the absence of unbounded circumambient processes like the sour-grapes pattern in the typology of vowel harmony. However, this paper has shown that unbounded circumambient processes are widely attested in tonal phonology, and has proposed that the weakly deterministic bound only applies to segmental phonology. This accurately predicts that fully regular maps such as UTP and Copperbelt Bemba H-tone spreading exist in tonal phonology. Furthermore, the proposal that tone is more computationally complex than segmental phonology is in line with the assertion of Hyman (Reference Hyman, Goldsmith, Riggle and Yu2011) and others that tone can ‘do more’ than phonology.

An explanation for how the weakly deterministic bound is manifested in the phonological system will be left for future work, although it is possible to speculate on a few points. For example, formal language complexity correlates with an increase in the computational resources necessary for parsing and generation. It could be that tone has access to such resources because prosodic information more commonly interacts with syntax (see e.g. Hyman & Katamba Reference Hyman and Katamba2010), and thus requires more powerful computation. This issue of computational power can also be directly related to learning, as empirical work suggests that formal complexity constrains phonological learning (Moreton & Pater Reference Moreton and Pater2012, McMullin & Hansson Reference McMullin and Hansson2014, Lai Reference Lai2015).

5.4 Explaining the exceptions to the weakly deterministic hypothesis

The weakly deterministic hypothesis in its strongest form cannot account for the rare cases of unbounded circumambient segmental processes discussed in this paper, i.e. Sanskrit n-retroflexion and Yaka vowel harmony. There are a few ways to reconcile these exceptions with the weakly deterministic hypothesis.

One possibility is that the accounts of these processes in the literature are incorrect in classifying them as unbounded. As mentioned in §2.3, Ryan (forthcoming) observes that blocking of Sanskrit n-retroflexion may be bound to the adjacent syllable. In Yaka harmony, the greatest attested distance is only three vowels, and is restricted to a particular morphological context.

Another explanation comes from potentially interfering factors. For example, the presence of an unbounded circumambient process in the tonal phonology may license an unbounded circumambient process in segmental phonology. This may be the case for Yaka, which as noted in §2.2, also has UTP. It could be that Yaka speakers, having first internalised the plateauing process in tonal phonology, may then be able to generalise it to their segmental phonology.

Finally, it may simply be that the constraint against fully regular maps in segmental phonology is not categorical, but in some sense gradient, and thus can admit exceptions. One way in which this may be manifested is through a learning bias in which individuals are more receptive to some patterns than others (Wilson Reference Wilson2006a, Moreton Reference Moreton2008). Children may show a strong preference for weakly deterministic maps when learning segmental phonology, but change to a more general learner in the face of sufficient data. As Staubs (Reference Staubs2014) shows, gradient typological generalisations may also result from the transmission of patterns among multiple learning agents. It is likely that non-weakly deterministic patterns require more kinds of data to learn, and perhaps this data is only available in tonal phonology, which is known to operate over much larger domains than segmental phonology.

Hence there are a number of reasons why the cases of Sanskrit n-retroflexion and Yaka vowel harmony do not immediately invalidate the proposal offered here. Even if they cannot ultimately be explained away, they are exceptional with respect to other characterisations of segmental phonology, such as Wilson's (Reference Wilson2003, 2006b) myopia generalisation. Finally, Sanskrit n-retroflexion and Yaka vowel harmony appear to be the only such cases – Hyman (Reference Hyman, Goldsmith, Riggle and Yu2011: 218) states, for example, that Yaka harmony is the ‘only one example’ of such a process of which he is aware. As such, the development of the potential explanations above will be left to future research.

In sum, this paper has characterised the unbounded circumambient asymmetry in terms of computational complexity, and has identified the weakly deterministic boundary as the relevant difference between tonal and segmental phonology which captures this asymmetry.