1 Introduction
The subregular program in phonology seeks to define subclasses of the regular languages and finitestate functions that describe attested phonotactic constraints and phonological processes. These subclasses provide a natural framework for typological classification of linguistic phenomena while allowing for the development of precise theories of language learning and processing. The traditional view in subregular phonology is that most phonotactic dependencies are described by tierbased strictly local languages (TSL, Heinz et al., 2011; McMullin and Hansson, 2016; McMullin, 2016), while most phonological process are described by strictly local functions (Chandlee, 2014; Chandlee et al., 2015, In prep). These classes of languages and functions are defined by a principle known as locality—that dependencies between symbols must occur over a bounded distance within the string. To account for longerdistance dependencies, Heinz et al. (2011) proposes a tier projection mechanism that allows irrelevant intervening symbols to be exempt from the locality condition.
Recent work in subregular phonology has identified a number of exceptions to the traditional view. On the language side, unbounded culminative stress systems (Baek, 2018), Uyghur backness harmony (Mayer and Major, 2018), and Sanskrit nretroflexion (Graf and Mayer, 2018) have been shown to lie outside the class of TSL languages. These observations have led to an enhancement of Heinz et al.’s (2011) tier projection system. On the function side, a number of processes, including bidirectional harmony systems (Heinz and Lai, 2013) and certain tonal processes (Jardine, 2016), have been shown to be not subsequential, and therefore not strictly local. At least two proposals, both known as the weakly deterministic functions, have been made in order to capture these processes (Heinz and Lai, 2013; McCollum et al., 2018).
This paper identifies rhythmic syncope as an additional example of a phonological process that is not strictly local. In rhythmic syncope, every second vowel of an underlying form is deleted in the surface form, starting with either the first or the second vowel. While rhythmic syncope cannot be expressed as a local dependency between symbols, it can be viewed as a local dependency between actions in the computation history of the minimal subsequential finitestate transducer (SFST). We formalize such dependencies by proposing the tierbased synchronized strictly local functions (TSSL). See Bowers and Hao (To appear) for a discussion of TSSL functions oriented towards the phonological literature.
This paper is structured as follows. Section 2 enumerates standard definitions and notation used throughout the paper, while Section 3 reviews existing work on strictly local functions. Section 4 introduces rhythmic syncope and shows that it is not strictly local. Section 5 presents two equivalent definitions of the TSSL functions—an algebraic definition and a definition in terms of a canonical SFST. Section 6 develops some formal properties of the TSSL functions, showing that they are incomparable to the full class strictly local functions. Section 7 compares our proposal to existing OT treatments of rhythmic syncope, and Section 8 concludes.
2 Preliminaries
As usual, denotes the set of nonnegative integers. and denote finite alphabets not including the left and right word boundary symbols and , respectively. The length of a string is denoted by , and denotes the empty string. Alphabet symbols are identified with strings of length , and individual strings are identified with singleton sets of strings. For , denotes concatenated with itself many times, denotes , denotes , and denotes . The longest common prefix of a set of strings is the longest string such that every string in begins with . The suffix of a string , denoted , is the string consisting of the last many symbols of .
A subsequential finitestate transducer (SFST) is a 6tuple , where

is the set of states, with being the start state;

and are the input and output alphabets, respectively;

is the transition function; and

is the final output function.
For ; ; and , the notation means that emits to the output stream and transitions to state if it reads in the input stream while it is in state . Letting , we say that computes if for every , , where . A function is subsequential if it is computed by an SFST.
An SFST is onward if for every state other than ,
Putting in onward form allows us to impose structure on the timing with which SFSTs produce output symbols.
Definition .
Let . We define the function by
For any , denotes the string such that . We refer to as the translation of by and to as top.^{1}^{1}1This terminology follows Sakarovitch (2009, pp. 692–693). In the transducer inference literature, Oncina et al. (1993) refer to as the tails of in , and Chandlee et al. (2015) refer to as the prefix function associated to .
Suppose computes . The following facts are apparent.

Fix and write and . If , then .

is onward if and only if for all , if , then .
These observations allow us to construct the minimal SFST for by identifying each state with a possible translation (Raney, 1958).
Let and be alphabets that are possibly infinite. A function is a homomorphism if for every , .
3 Background
The strictly local functions are classes of subsequential functions proposed by Chandlee (2014), Chandlee et al. (2015), and Chandlee et al. (In prep) as transductive analogues of the strictly local languages (McNaughton and Papert, 1971). Whereas phonotactic dependencies can usually be described using tierbased strictly local languages (Heinz et al., 2011; McMullin and Hansson, 2016; McMullin, 2016), Chandlee (2014) has argued that local phonological processes can be modelled as strictly local functions when they are viewed as mappings between underlying representations and surface representations. A survey overview of the related literature can be found in Heinz (2018).
Intuitively, strictly local functions are functions computed by SFSTs in which each state represents the most recent symbols in the input stream and the most recent symbols in the output stream along with the current input symbol, for some parameter values fixed. Such functions are “local” in the sense that the action performed on each input symbol depends only on information about symbols in the input and output streams within a bounded distance. In this paper, we augment strictly local functions with tier projection, a mechanism introduced by Heinz et al. (2011) and elaborated by Baek (2018), Mayer and Major (2018), and Graf and Mayer (2018) that allows the locality constraint to bypass irrelevant alphabet symbols, extending the distance over which dependencies may be enforced.
Definition .
For any alphabet , a tier on is a homomorphism such that for each , either or . In the former case, we say that is on ; in the latter case, we say that is off .
Chandlee (2014), Chandlee et al. (2015), and Chandlee et al. (In prep) give two definitions of the strictly local functions. Firstly, they state the locality condition in terms of the algebraic representation of minimal SFSTs.
Definition .
Fix and let be a tier on . A function is input–output strictly local on tier (TIOSL) if for all , if

and

,
then . A function is input strictly local on tier (TISL) if it is TIOSL on tier , and it is output strictly local on tier (TOSL) if it is TIOSL on tier .
Secondly, they define strictly local functions in terms of canonical SFSTs that directly encode suffixes of the input stream and suffixes of the output stream in their state names.
Definition .
Fix and let be a tier on . An SFST is input–output strictly local on tier (TIOSL) if the following conditions hold.

and .

If , then and .
An SFST is input strictly local on tier (TISL) if it is TIOSL on tier , and it is output strictly local on tier (TOSL) if it is TIOSL on tier .
These definitions turn out to be equivalent when the canonical SFSTs are required to be onward.
Theorem (Chandlee, 2014; Chandlee et al., 2015, In prep).
A function is TIOSL on tier if and only if it is computed by an onward SFST that is TIOSL on tier .
Example .
Rhythmic reduction is a phonological process in which alternating vowels in a word undergo reduction. The examples in (3) show rhythmic reduction in the Odawa variety of Ojibwe circa 1912, as documented by Edward Sapir. In our representation of reduction, vowels are reduced to @, starting from the first vowel. There is no reason to believe that @ appears in underlying forms.
Rhythmic reduction in Ojibwe circa 1912 (Rhodes et al., 2012)
/m2kIzIn2n/ [m@kIz@n2n] ‘shoes’
/gUtIgUmIn2gIbIna:d/
[g@tIg@mIn@gIb@na:d] ‘if he rolls him’
Figure 1 shows an SFST that implements the rhythmic reduction pattern illustrated in (3). We represent the pattern using an alphabet of three symbols: , representing consonants; , representing vowels that have not been reduced; and @, representing vowels that have been reduced. Observe that this SFST is onward and TOSL, with off the tier: each state represents the most recent vowel in the ouput stream.^{2}^{2}2For clarity, we omit the portions of the state names.
4 Rhythmic Syncope
Rhythmic syncope is a phonological process in which every second vowel in a word is deleted. The examples of (3) show rhythmic syncope in Macushi, in which deletion begins with the first vowel.^{3}^{3}3The synchronic status of rhythmic syncope is a matter of current discussion, as its development appears to push a phonological system into dramatic restructuring (Bowers, To appear). Rhythmic syncope in Macushi (Hawkins, 1950) /piripi/ [pripi] ‘spindle’ /wanamari/ [wnamri] ‘mirror’
In this section, we show that rhythmic syncope is not TIOSL. To see this, we formalize rhythmic syncope as a function over two alphabet symbols: , representing consonants, and , representing vowels. This idealization does not affect the argument that rhythmic syncope is not TIOSL, presented in Proposition 4.
Definition .
The rhythmic syncope function is defined as follows. For ,
where for each , if is even and if
is odd.
^{4}^{4}4While is defined on strings of phonemes with no prosodic symbols, phonological analyses often assume that the input is parsed into feet with iambic or trochaic stress. Such analyses are discussed in Section 7.The intuition underlying the argument below is that suffixes of the input and suffixes of the output do not contain information about whether vowels occupy even or odd positions within the input and output strings. Therefore, while an TIOSL SFST can record the most recent vowels read from the input stream and emitted to the output stream, this information is not sufficient for determining whether or not the SFST should delete a vowel.
Proposition .
The rhythmic syncope function is not TIOSL on tier for any and any .
Proof.
Let be even. Consider the strings and . Observe that ; thus . Now, if is on , then , and if is off , then . Thus, if is TIOSL on tier , then . However, letting , observe that
This means that but , so is not TIOSL on tier . ∎
5 Synchronized Strictly Local Functions
Proposition 4 raises the question of how to characterize the kind of computation that effects rhythmic syncope. To investigate this question, Figure 2 shows a natural SFST implementation of rhythmic syncope. The states in this SFST record the most recent action performed by the SFST. If the most recent action was to delete a vowel (), then the next vowel the SFST encounters is not deleted (); otherwise, the next vowel is deleted. This SFST is strikingly similar to the rhythmic reduction SFST in Figure 1. There, the special symbol @, which is not part of the input alphabet, indicates the location of a reduced vowel, effectively recording the previous action in the output. Since there is no way to mark the location of a deleted symbol, the SFST in Figure 2 explicitly records its previous action in its state names. Thus, the rhythmic syncope SFST may be seen as a generalization of the rhythmic reduction SFST. The goal of this section is to define a class of functions, known as the tierbased synchronized strictly local (TSSL) functions, based on this intuition. Following Section 2, we begin by defining the TSSL functions algebraically in terms of the minimal SFST, and then we define a canonical SFST format for the TSSL functions.
Recall that at each time step, an SFST must read exactly one input symbol while producing an output string of any length. Since the minimal SFST for a function must produce after reading the input string , we can determine the possible actions of by comparing with for arbitrary and .
Definition .
Let . The actions of are the alphabet defined as follows.
We denote elements of by .
Strings over represent computation histories of the minimal SFST for .
Definition .
Let and let . The run of on input is the string defined as follows.

If , then .

If , where and , then , where is the unique string such that .
The notation allows us to define the TSSL functions in a straightforward manner, highlighting the analogy to the TIOSL functions.
Definition .
Fix and let be a tier on . A function is synchronized strictly local on tier (TSSL) if for all , if , then .
Now, let us define the canonical SFSTs for TSSL functions. We define the actions of an SFST to be its possible transition labels.
Definition .
Let be an SFST. The actions of are the alphabet
We denote elements of by .
Again, the definition of the TSSL SFSTs is directly analogous to that of the TIOSL SFSTs.
Definition .
Fix and let be a tier on . An SFST is synchronized strictly local on tier (TSSL) if the following conditions hold.

and .

For every , if , then
As is the case with TIOSL SFSTs, TSSL SFSTs compute exactly the class of TSSL functions when they are required to be onward.
Theorem .
Fix , and let be a tier on . A function is TSSL on tier if and only if it is computed by an onward SFST that is TSSL on tier .
We leave the proof of this fact to Appendix A.
6 Properties of TSSL Functions
Having now defined the TSSL functions, this section investigates some of their formal properties. Subsection 6.1 compares the TSSL functions to the TISL, TOSL, and TIOSL functions. Subsection 6.2 observes that TSSL SFSTs compute a large class of functions when they are not required to be onward.
6.1 Relation to TIOSL Functions
A natural first question regarding the TSSL functions is that of how they relate to previouslyproposed classes of subregular functions. We know from the discussion of rhythmic syncope that the TSSL functions are not a subset of the TIOSL functions: we have already seen that the rhythmic syncope function is TSSL but not TIOSL for any . We will see in this subsection that the TIOSL functions are not a subset of the TSSL functions, though both function classes fully contain the TISL and TOSL functions. Therefore, the two function classes are incomparable, and offer two different ways to generalize the TISL and TOSL functions.
The fact that the TSSL functions contain the TISL and TOSL functions follows from the observation that actions contain information about input and output symbols. Remembering the most recent actions automatically entails remembering the most recent input symbols, and the most recent output symbols can be extracted from the most recent actions if deletions are ignored.
Proposition .
Fix . Every TISL function and every TOSL function is TSSL.
Proof.
Let , and let be a tier on . First, suppose that is TISL on tier . Let be a tier on defined as follows: an action is on if and only if is on . Now, suppose are such that . Write
Then, we have and . For all , , and therefore . But this means that , and since is TISL on tier , . We conclude that is TSSL on tier .
Next, suppose that is TOSL on tier . Let be a tier on defined as follows: an action is on if and only if . Now, suppose are such that . Write
Now, and . Again, for all we have , so . Observe that
where . Since is TOSL on tier , , so is TSSL on tier . ∎
This intuition does not carry over to the TIOSL functions. In Proposition 6.1, the proposed action tiers ignore symbols off the input and output tiers, thus ensuring that the relevant input and output symbols can always be recovered from the computation history. This approach encounters problems when an onward TIOSL SFST deletes symbols on the tier. Such SFSTs perform actions of the form , where is on the tier. These actions do not record any output symbols, but they must be kept on the tier in a TSSL implementation so that the input symbol can be recovered. If too many s are performed consecutively, they can overwhelm the memory of a TSSL SFST, causing it to forget the most recent output symbols. The following construction features exactly this kind of behavior.
Proposition .
There exists a function that is TIOSL for some but not TSSL for any .
Proof.
Let be the SFST shown in Figure 3, and let be the function computed by .^{5}^{5}5The angle brackets are omitted from the state names. Observe that is onward and TIOSL on tier , where and are on but and are not, so is TIOSL on tier . always copies the first symbol of its input to the output. Thereafter, behaves as follows: all s are deleted; a is changed to a if the most recent input symbol is the same as the first input symbol; a is changed to a otherwise. For example, .
Let , and let be a tier on . Suppose that either or is not on , and consider the strings and . Observe that and . Either if and is on , or if or is not on . However, but , so cannot be TSSL on tier .
Next, suppose that and is on . Consider the input strings and . Observe that and , thus
However, but , so is not TSSL on tier . ∎
6.2 NonOnward TSSL SFSTs
The equivalence between the two definitions of the TSSL functions presented in Section 5 crucially depends on the criterion that TSSL SFSTs be onward. In this subsection we show that without this criterion, TSSL SFSTs compute a rich class of subsequential functions. To illustrate how this is possible, let us consider an example that witnesses the separation between TSSL functions and TSSL SFSTs.
Proposition .
There exists a TSSL SFST that computes a function that is not TSSL for any .
Proof.
Consider the SFST in Figure 4. This SFST is clearly TSSL on a tier containing all actions, and the function it computes is given by , where and . Observe that for any , . Therefore, writing with for each ,
We need to show that is not TSSL for any and for any tier over .
Fix and . Suppose is on , and consider the input strings and . Observe that and , so
However, but , so is not TSSL on tier .
Next, suppose is not on , and consider the input strings and . We have and , so
However, but , so is not TSSL on tier . ∎
Let be the function described in Proposition 6.2. As discussed in the proof, an onward SFST computing must copy the current input symbol to the output stream during each time step. At the end of the computation, the final output function is responsible for adding the first input symbol to the end of the output string. Any onward TSSL SFST that attempts to compute will eventually forget the identity of the first input symbol, so the final output function cannot determine what to add to the output. The SFST in Figure 4 avoids this problem by exploiting its nononwardness. If the first symbol of its input is an , then behaves in an onward manner, copying the current input symbol at each time step. This can be seen in the left column of the state diagram. If the first symbol of ’s input is a , then alternates between producing no output and producing two symbols of output. Every time performs a nondeleting action , contains both the symbol that the onward SFST would produce at the current time step and the symbol that the onward SFST would have produced at the previous time step. This way, encodes the identity of the first symbol of its input using the manner in which it produces output—if produces output at every time step, then the first symbol is an , and if it produces output every two time steps, then the first symbol is a . In general, this kind of encoding trick can be applied to a wide range of SFSTs, including all SFSTs that do not perform deletions. Informally, we enumerate the states of by , and we construct a TSSL SFST that simulates by producing output at various frequencies. For each , produces output every time steps if is in state . If remembers at least many actions, then it can always deduce ’s state at any point in the computation, allowing it to simulate .
7 Rhythmic Syncope in Phonology
The view of rhythmic syncope we have presented here differs substantially in approach from existing treatments of rhythmic syncope in phonological theory. McCarthy (2008) identifies two major approaches to rhythmic syncope in Optimality Theory. In the pseudodeletion approach (e.g., Kager, 1997), the locations of symbols deleted by syncope are marked with blank symbols. This essentially makes rhythmic syncope identical to rhythmic reduction, which we have seen is TOSL. McCarthy himself proposes a Harmonic Serialism approach in which rhythmic syncope is implemented in multiple steps. Firstly, stress is assigned to every second vowel in the underlying form. Then, the unstressed vowels are deleted, resulting in syncope. This kind of derivation is illustrated in (6).
Rhythmic syncope in Harmonic Serialism (McCarthy, 2008)^{6}^{6}6The full derivation proposed by McCarthy (2008) includes syllabification and footing steps, which are omitted here for simplicity.
/wanamari/
Underlying Form
wanámarí
Stress
[wnámrí ]
Syncope
In both approaches, rhythmic syncope is decomposed into a TOSL function and a homomorphism. In the pseudodeletion approach, the TOSL function is rhythmic reduction, and the homomorphism removes the @s. In (6), the rhythmic stress step is TOSL, while the syncope step is a homomorphism. In general, this kind of approach is extremely powerful.
Proposition .
Every subsequential function can be written in the form , where is TOSL and is a homomorphism.
Proof.
Let be the minimal SFST for . Define as follows. Let . For , write
Then, . Next, define so that for any , . It is clear that for every . We now show that is TOSL on a tier containing the full output alphabet.
Fix . Observe that for all , . Therefore, suppose that . This means that and for some , so by definition. ∎
In both pseudodeletion and Harmonic Serialism, nonsegmental phonological symbols are used to encode state information in the output, making rhythmic syncope TOSL. Proposition 7 shows that this technique can be applied to arbitrary SFSTs, and therefore results in massive overgeneration. By contrast, we have already seen that the TSSL functions are a proper subset of the subsequential functions, making actionsensitivity a more restrictive alternative to current approaches to rhythmic syncope.
8 Conclusion
The classic examples of TIOSL phenomena in phonology are local processes and unidirectional spreading processes (Chandlee, 2014). Rhythmic syncope is qualitatively different from these phenomena in that it leaves no evidence that the process has occurred. As we have seen in Section 4, the fact that rhythmic syncope is not TIOSL is a consequence of this property. In defining the TSSL functions, we have proposed that rhythmic syncope should be viewed as a dependency between incremental steps in a derivation, here formalized as the actions of the minimal SFST.
A potential risk of such an analysis is that the notion of “action” is specific to the computational system used to implement rhythmic syncope, and therefore potentially subject to a broad range of interpretations. In this paper, we have used onwardness and the existence of the minimal SFST to formulate a notion of “actionsensitivity” that is both formalismindependent and implementationindependent. In Subsection 6.2, we have seen that actionsensitivity can be made very powerful if we relax our assumptions about the nature of the computation. This means that if actionsensitivity is to be incorporated into phonological analyses of rhythmic syncope, then care should be taken to avoid loopholes like the one featured in Proposition 6.2. Based on Proposition 7, a similar warning can be made regarding the composition of phonological processes. When decomposing phonemena into several processes, as McCarthy (2008) does in the Harmonic Serialism analysis, care should be taken to ensure that theoretical proposals do not allow for overgeneration.
Outstanding formal questions regarding the TSSL functions include their closure properties and the complexity of learning TSSL functions. We leave such questions to future work.
References
 Baek (2018) Hyunah Baek. 2018. Computational representation of unbounded stress: Tiers with structural features. In Proceedings of CLS 53 (2017), volume 53, pages 13–24, Chicago, IL, USA. Chicago Linguistic Society.
 Bowers (To appear) Dustin Bowers. To appear. The Nishnaabemwin Restructuring Controversy: New Empirical Evidence. Phonology.
 Bowers and Hao (To appear) Dustin Bowers and Yiding Hao. To appear. Rhythmic Syncope in Subregular Phonology. In Proceedings of the 42nd Annual Penn Linguistics Conference, volume 26.1 of Penn Working Papers in Linguistics, Philadelphia, PA, USA. Penn Graduate Linguistics Society.
 Chandlee (2014) Jane Chandlee. 2014. Strictly Local Phonological Processes. PhD Dissertation, University of Delaware, Newark, DE, USA.
 Chandlee et al. (2015) Jane Chandlee, Rémi Eyraud, and Jeffrey Heinz. 2015. Output Strictly Local Functions. In Proceedings of the 14th Meeting on the Mathematics of Language, pages 112–125, Chicago, IL, USA. Association for Computational Linguistics.
 Chandlee et al. (In prep) Jane Chandlee, Rémi Eyraud, and Jeffrey Heinz. In prep. Input–output strictly local functions and their efficient learnability.
 Graf and Mayer (2018) Thomas Graf and Connor Mayer. 2018. Sanskrit nRetroflexion is Input–Output TierBased Strictly Local. In Proceedings of the Fifteenth Workshop on Computational Research in Phonetics, Phonology, and Morphology, pages 151–160, Brussels, Belgium. Association for Computational Linguistics.
 Hawkins (1950) W. Neil Hawkins. 1950. Patterns of Vowel Loss in Macushi (Carib). International Journal of American Linguistics, 16(2):87–90.
 Heinz (2018) Jeffrey Heinz. 2018. The computational nature of phonological generalizations. In Larry M. Hyman and Frans Plank, editors, Phonological Typology, number 23 in Phonology and Phonetics, pages 126–195. De Gruyter Mouton, Berlin, Germany.
 Heinz and Lai (2013) Jeffrey Heinz and Regine Lai. 2013. Vowel Harmony and Subsequentiality. In Proceedings of the 13th Meeting on the Mathematics of Language (MoL 13), pages 52–63, Sofia, Bulgaria. Association for Computational Linguistics.
 Heinz et al. (2011) Jeffrey Heinz, Chetan Rawal, and Herbert G. Tanner. 2011. Tierbased Strictly Local Constraints for Phonology. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies, pages 58–64, Portland, OR, USA. Association for Computational Linguistics.
 Jardine (2016) Adam Jardine. 2016. Computationally, tone is different. Phonology, 33(2):247–283.
 Kager (1997) René Kager. 1997. Rhythmic vowel deletion in Optimality Theory. In Iggy Roca, editor, Derivations and Constraints in Phonology, pages 463–499. Clarendon Press, Oxford, United Kingdom.
 Mayer and Major (2018) Connor Mayer and Travis Major. 2018. A Challenge for TierBased Strict Locality from Uyghur Backness Harmony. In Formal Grammar 2018, 23rd International Conference, FG 2018, Sofia, Bulgaria, August 1112, 2018, Proceedings, volume 10950 of Lecture Notes in Computer Science, pages 62–83, Berlin, Germany. Springer Berlin Heidelberg.
 McCarthy (2008) John J. McCarthy. 2008. The serial interaction of stress and syncope. Natural Language & Linguistic Theory, 26(3):499–546.
 McCollum et al. (2018) Adam McCollum, Eric Baković, Anna Mai, and Eric Meinhardt. 2018. The expressivity of segmental phonology and the definition of weak determinism. LingBuzz, lingbuzz/004197.
 McMullin and Hansson (2016) Kevin McMullin and Gunnar Ólafur Hansson. 2016. LongDistance Phonotactics as TierBased Strictly Local Languages. In Proceedings of the 2014 Annual Meeting on Phonology, Proceedings of the Annual Meetings on Phonology, pages 13–24, Cambridge, MA, USA. Linguistic Society of America.
 McMullin (2016) Kevin James McMullin. 2016. TierBased Locality in LongDistance Phonotactics: Learnability and Typology. PhD Dissertation, University of British Columbia, Vancouver, Canada.
 McNaughton and Papert (1971) Robert McNaughton and Seymour A. Papert. 1971. CounterFree Automata. Number 65 in Research Monograph. MIT Press, Cambridge, MA, USA.

Oncina et al. (1993)
José Oncina, Pedro Garcia, and Enrique Vidal. 1993.
Learning Subsequential Transducers for Pattern Recognition Interpretation Tasks.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(5):448–458.  Raney (1958) George N. Raney. 1958. Sequential Functions. Journal of the Association for Computing Machinery, 5(2):177–180.
 Rhodes et al. (2012) Richard A. Rhodes, Karl S. Hele, and J. Randolph Valentine. 2012. Algonquian Trade Languages Revisited. In Papers of the Fortieth Algonquian Conference/Actes Du Congrès Des Algonquinistes, Papers of the Algonquian Conference, pages 358–369, Albany, NY, USA. State University of New York Press.
 Sakarovitch (2009) Jacques Sakarovitch. 2009. Elements of Automata Theory. Cambridge University Press, Cambridge, United Kingdom.
Appendix A Proof of Theorem 16
This appendix proves the equivalence between TSSL functions and onward TSSL SFSTs. We begin by showing how to construct an onward TSSL SFST computing any given TSSL function.
Definition .
Let be TSSL on tier . Define the SFST transducer as follows.

and .

For each , , where .

For each , let be such that , and let be such that . We define , where .

Fix If , then . Otherwise, we define , where .
Remark .
is TSSL on tier .
Note that in the third and fourth bullet points of Definition A, the action and the string only depend on and not on , since is TSSL on tier . We now need to show that computes and that it is onward.
Lemma .
Let be TSSL on tier , and write . For every , if , then .
Proof.
Let us induct on . For the base case, suppose . Then, by definition.
Now, fix , and suppose that if and , then . Fix and , and suppose that . By the induction hypothesis, . The definition of states that is the unique string such that . Thus, , and the proof is complete. ∎
Lemma .
Let be TSSL on tier , and write . For all , if , then .
Proof.
Let us induct on . For the base case, suppose . Since , by definition .
Now, fix , and suppose that if and , then . We need to show that for all and , if , then . The induction hypothesis gives us . Since , by the definition of ,
(1) 
as desired. ∎
Proposition .
If is TSSL on tier , then computes f.
Proof.
We need to show that for every , outputs on input . Write
Comments
There are no comments yet.