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ACL-OCL / Base_JSON /prefixS /json /scil /2020.scil-1.56.json
Benjamin Aw
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 {
"paper_id": "2020",
"header": {
"generated_with": "S2ORC 1.0.0",
"date_generated": "2023-01-19T15:38:39.137970Z"
},
"title": "Metrical Grids and Generalized Tier Projection",
"authors": [
{
"first": "Yiding",
"middle": [],
"last": "Hao",
"suffix": "",
"affiliation": {
"laboratory": "",
"institution": "Yale University New Haven",
"location": {
"region": "CT",
"country": "USA"
}
},
"email": "[email protected]"
}
],
"year": "",
"venue": null,
"identifiers": {},
"abstract": "This paper formalizes metrical grid theory (MGT, Prince, 1983; Hayes, 1995) and studies its expressive power. I show that MGT analyses of a certain form can describe stress systems beyond the input tier-based input strictly local functions proposed by Hao and Andersson (2019), but conjecture that such analyses do not describe systems beyond the input tier-based strictly local languages of Baek (2018). These results reveal fundamental differences between the three formalisms.",
"pdf_parse": {
"paper_id": "2020",
"_pdf_hash": "",
"abstract": [
{
"text": "This paper formalizes metrical grid theory (MGT, Prince, 1983; Hayes, 1995) and studies its expressive power. I show that MGT analyses of a certain form can describe stress systems beyond the input tier-based input strictly local functions proposed by Hao and Andersson (2019), but conjecture that such analyses do not describe systems beyond the input tier-based strictly local languages of Baek (2018). These results reveal fundamental differences between the three formalisms.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Abstract",
"sec_num": null
}
],
"body_text": [
{
"text": "The problem of unbounded stress has proven challenging for subregular phonology. Baek (2018) has recently shown that unbounded stress provides a counterexample to the weak subregular hypothesis (Heinz, 2018) , which claims that phonological phenomena can be represented by tier-based strictly local languages (TSL, Heinz et al., 2011) when viewed as decision problems that accept grammatical utterances. To remedy this, Baek proposes that the decision to project a symbol to the tier may be conditioned by local contextual information such as the presence of word boundaries. This proposal has been extended to a generalized tier projection system in which tier projection is implemented by arbitrary deleting functions (Mayer and Major, 2018; Graf and Mayer, 2018) .",
"cite_spans": [
{
"start": 81,
"end": 92,
"text": "Baek (2018)",
"ref_id": "BIBREF2"
},
{
"start": 194,
"end": 207,
"text": "(Heinz, 2018)",
"ref_id": "BIBREF17"
},
{
"start": 309,
"end": 334,
"text": "(TSL, Heinz et al., 2011)",
"ref_id": null
},
{
"start": 720,
"end": 743,
"text": "(Mayer and Major, 2018;",
"ref_id": "BIBREF22"
},
{
"start": 744,
"end": 765,
"text": "Graf and Mayer, 2018)",
"ref_id": "BIBREF10"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Viewing unbounded stress as a transduction problem that maps underlying forms without prosodic representation to surface forms marked for primary stress, Hao and Andersson (2019) show that unbounded stress systems are similarly handeled by generalized tier projection, but that they fall outside the class of tier-based input strictly local functions (TISL, Chandlee, 2014) . Hao and Andersson demonstrate that default-to-oppositeside (DO) systems can be captured by adapting generalized tier projection to subregular functions. However, it turns out that the \"bidirectional\" nature of default-to-same-side (DS) systems cannot be implemented by subsequential functions, which allow only a single unidirectional pass over the input. They instead propose that DS systems should be understood as interaction-free weakly deterministic functions in the sense of McCollum et al. (2018) .",
"cite_spans": [
{
"start": 154,
"end": 178,
"text": "Hao and Andersson (2019)",
"ref_id": "BIBREF13"
},
{
"start": 351,
"end": 373,
"text": "(TISL, Chandlee, 2014)",
"ref_id": null
},
{
"start": 376,
"end": 383,
"text": "Hao and",
"ref_id": null
},
{
"start": 857,
"end": 879,
"text": "McCollum et al. (2018)",
"ref_id": "BIBREF27"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "These results raise conceptual questions regarding the treatment of stress in subregular phonology. In particular, the mechanism of generalized tier projection intuitively seems powerful and ad-hoc. The basic elements of subregular phonology, namely strict locality and the traditional tier projection system of Heinz et al. (2011) , can be viewed as formalizations of rule and tier systems that are well-established in phonological theory (Chandlee, 2014; Chomsky and Halle, 1968; Goldsmith, 1976) . While generalized tier projection enables Baek (2018) and Hao and Andersson (2019) to define classes of finite-state machines that capture unbounded stress, it is unclear whether generalized tier projection is similarly grounded in existing phonological constructs. Reflecting on this issue, Hao and Andersson observe that the generalized tier-projection mechanism they use to produce Dybo's Rule (Dybo, 1977) , a model of the stress system in Abkhaz, bears a striking resemblance to the syllable tier used in a standard analysis of that system within metrical grid theory (MGT, Prince, 1983; Hayes, 1995) . From that observation we might hypothesize that stress systems computed using generalized tier projection naturally correspond to those described by MGT.",
"cite_spans": [
{
"start": 312,
"end": 331,
"text": "Heinz et al. (2011)",
"ref_id": "BIBREF18"
},
{
"start": 440,
"end": 456,
"text": "(Chandlee, 2014;",
"ref_id": "BIBREF5"
},
{
"start": 457,
"end": 481,
"text": "Chomsky and Halle, 1968;",
"ref_id": "BIBREF7"
},
{
"start": 482,
"end": 498,
"text": "Goldsmith, 1976)",
"ref_id": "BIBREF9"
},
{
"start": 543,
"end": 554,
"text": "Baek (2018)",
"ref_id": "BIBREF2"
},
{
"start": 559,
"end": 583,
"text": "Hao and Andersson (2019)",
"ref_id": "BIBREF13"
},
{
"start": 793,
"end": 800,
"text": "Hao and",
"ref_id": null
},
{
"start": 898,
"end": 910,
"text": "(Dybo, 1977)",
"ref_id": "BIBREF8"
},
{
"start": 1074,
"end": 1093,
"text": "(MGT, Prince, 1983;",
"ref_id": null
},
{
"start": 1094,
"end": 1106,
"text": "Hayes, 1995)",
"ref_id": "BIBREF14"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "In this paper, I will argue that this intuition does not hold for the transduction problem, though it may hold for the decision problem. To that end, I define a formal model of MGT in Section 4, and show in Section 5 that the stress systems described by MGT do not correspond to those represented by functions defined using generalized tier projection. I also give evidence to suggest that decision problems described by MGT can be represented using generalized tier projection even when their corresponding transduction problems cannot. These results imply not only that functions based on generalized tier projection are not grounded in MGT, but also that the typological predictions they make about the range of possible stress systems differ from those made by MGT and by decision problems.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "Technical definitions used in this paper are given in Section 2, and Section 3 reviews the existing results on unbounded stress in subregular phonology. Section 6 concludes.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Introduction",
"sec_num": "1"
},
{
"text": "In this paper, uppercase Greek letters denote finite alphabets not including the boundary symbols o and n. The length of a string x is denoted by |x|, and denotes the empty string. Alphabet symbols are identified with strings of length 1, and individual strings are identified with singleton sets of strings. For k 0, \u21b5 k denotes \u21b5 concatenated with itself k-many times, \u21b5 <k denotes S k 1 i=0 \u21b5 i , \u21b5 \u21e4 denotes S 1 i=0 \u21b5 i , and \u21b5 + denotes \u21b5\u21b5 \u21e4 . The longest common prefix of a set of strings A is the longest string lcp(A) such that every string in A begins with lcp(A).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "For sets A and B, the notation f : A ! B means that f is a function with domain A and codomain B. The range of f is the set {y|9x.f",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "(x) = y} \u2713 B. A function f : A ! B is injective if for every x, y 2 A, f (x) = f (y) if and only if x = y. A function f : \u2303 \u21e4 ! \u21e4 is same-length if and only if for all x 2 \u2303 \u21e4 , |f (x)| = |x|.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "A subsequential finite-state transducer (SFST) is a 6-tuple T = hQ, \u2303, , q 0 , !, !i, where",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u2022 Q is the set of states, with q 0 2 Q being the start state;",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u2022 \u2303 and are the input and output alphabets, respectively;",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u2022 ! : Q \u21e5 \u2303 ! Q \u21e5 \u21e4 is the transition function; and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u2022 ! : Q ! \u21e4 is the final output function.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "For x 2 \u2303 \u21e4 ; y 2 \u21e4 ; and q, r 2 Q, the notation q",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "x:y ! r means that T emits y to the output stream and transitions to state r if it reads x in the input stream while it is in state q. Letting f :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u2303 \u21e4 ! \u21e4 , we say that T computes f if for every x 2 \u2303 \u21e4 , f (x) = y!(q), where q 0 x:y ! q.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "A function is subsequential if it is computed by an SFST.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "For a string x 6 = , I use the following indexing notation.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u2022 For 1 \uf8ff i \uf8ff j \uf8ff |x|, x[i : j] is the substring of x such that x = wx[i : j]y,",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "where |w| = i 1 and |y| = |x| j.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "\u2022",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "For |x| \uf8ff u, v \uf8ff |x| and 1 \uf8ff i \uf8ff j \uf8ff |x|, x[u : v] = x[i : j] if u \u2318 i mod (|x| + 1) and v \u2318 j mod (|x| + 1). \u2022 For each i, x[i] := x[i : i]; x[i :] := x[i : |x|]; and x[: i] := x[1 : i].",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "The remainder of this section reviews the algebraic characterization of subsequential functions as well as tier projection and strict locality.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Preliminaries",
"sec_num": "2"
},
{
"text": "Independently of SFSTs, the subsequential functions can be characterized using two operations on string functions. Definition 1. Let f : \u2303 \u21e4 ! \u21e4 . We define the function f :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Subsequential Functions",
"sec_num": "2.1"
},
{
"text": "\u2303 \u21e4 ! \u21e4 by f (x) := lcp ({f (xy)|y 2 \u2303 \u21e4 }) . For any x, y 2 \u2303 \u21e4 , f ! x (y) denotes the string such that f (xy) = f (x)f !",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Subsequential Functions",
"sec_num": "2.1"
},
{
"text": "x (y). We refer to f ! x as the translation of f by x and to f as f top.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Subsequential Functions",
"sec_num": "2.1"
},
{
"text": "The translations of a subsequential function may be used to construct the minimal SFST for that function, analogously to the Nerode-Myhill construction for the minimal finite-state automaton of a regular language.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Subsequential Functions",
"sec_num": "2.1"
},
{
"text": "Theorem 2 (Raney, 1958) . A function f : \u2303 \u21e4 ! \u21e4 is subsequential if and only if the set {f !",
"cite_spans": [
{
"start": 10,
"end": 23,
"text": "(Raney, 1958)",
"ref_id": "BIBREF31"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Subsequential Functions",
"sec_num": "2.1"
},
{
"text": "x |x 2 \u2303 \u21e4 } is finite. For a subsequential function f with minimal SFST T , the translations of f are in bijection with the states of T . After reading input x, T outputs f (x) and enters the state corresponding to f !",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Subsequential Functions",
"sec_num": "2.1"
},
{
"text": "x .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Subsequential Functions",
"sec_num": "2.1"
},
{
"text": "This paper will frequently make use of a class of functions known as homomorphisms.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Homomorphisms",
"sec_num": "2.2"
},
{
"text": "Definition 3. A function h :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Homomorphisms",
"sec_num": "2.2"
},
{
"text": "\u2303 \u21e4 ! \u21e4 is a homomorphism if for every x, y 2 \u2303 \u21e4 , h(xy) = h(x)h(y).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Homomorphisms",
"sec_num": "2.2"
},
{
"text": "Intuitively, homomorphisms are functions that replace each symbol of \u2303 with a string in \u21e4 . As such, homomorphisms are completely determined by their values on the input alphabet.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Homomorphisms",
"sec_num": "2.2"
},
{
"text": "Proposition 4. Let h, g : \u2303 \u21e4 ! \u21e4 be homomorphisms. If h(x) = g(x) for each x 2 \u2303, then h = g.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Homomorphisms",
"sec_num": "2.2"
},
{
"text": "Tier projections are functions that delete certain symbols in an input string x. A tier \u2327 can be used to enhance notions of locality defined by grammars, automata, and transducers by having local dependencies be enforced between adjacent symbols in \u2327 (x) instead of x, effectively ignoring symbols deleted by \u2327 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Locality and Tier Projection",
"sec_num": "2.3"
},
{
"text": "A tier projection is a function \u2327 :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "\u2303 \u21e4 ! \u2303 \u21e4 such that \u2327 ( ) = and for all x 2 \u2303 + , \u2327 (x) = y 1 y 2 . . . y |x| , where for each i, y i is either x[i]",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "or . If \u2327 is a homomorphism, then we identify \u2327 with the subset \u2713 \u2303 such that for all 2 , \u2327 ( ) = .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "Symbols not deleted by a tier projection are said to be projected to the tier. Tier-based strictly local functions are defined to be functions computed by minimal SFSTs whose states record the most recent k 1 symbols projected to some tier, for some k > 0. In this paper, we assume that the states only record symbols from the SFST input projected to the tier; variants of the definitions below where the tier projects symbols of the output have also been defined (Chandlee, 2014; Chandlee et al., 2015; Burness and McMullin, 2019) .",
"cite_spans": [
{
"start": 464,
"end": 480,
"text": "(Chandlee, 2014;",
"ref_id": "BIBREF5"
},
{
"start": 481,
"end": 503,
"text": "Chandlee et al., 2015;",
"ref_id": "BIBREF6"
},
{
"start": 504,
"end": 531,
"text": "Burness and McMullin, 2019)",
"ref_id": "BIBREF4"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "Definition 6. Let k > 0, and let \u2327 :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "\u2303 \u21e4 ! \u2303 \u21e4 be a tier projection. A function f : \u2303 \u21e4 ! \u21e4 is generalized input strictly k-local on tier \u2327 (k- GTISL on tier \u2327 ) if for all x, y 2 \u2303 \u21e4 , \u2327 (x)[: 1 k] = \u2327 (y)[: 1 k] implies f ! x = f ! y .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "We say that f is",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "\u2022 input strictly k-local (k-ISL) if \u2327 is the identity function; 1",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "\u2022 input strictly k-local on tier \u2327 (k-TISL on tier \u2327 ) if \u2327 is a homomorphism; and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "\u2022 j-input strictly k-local on tier \u2327 (j-I-k-TISL on tier \u2327 ) if \u2327 is j-TISL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "Remark 7. Homomorphisms are 1-ISL functions. Tier-based strictly local languages are defined to be sets of strings whose images under some tier projection only contain substrings deemed permissible.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "Definition 8. Let k > 0, and let \u2327 :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "\u2303 \u21e4 ! \u2303 \u21e4 be a tier projection. A language L \u2713 \u2303 \u21e4 is generalized strictly k-local on tier \u2327 (k-GTSL on tier \u2327 ) if there exists S \u2713 (\u2303 [ {o, n}) k such that for all x 2 \u2303 \u21e4 , x 2 L if and only if every length-k substring of o k 1 \u2327 (x)n k 1 is in S. We say that L is \u2022 strictly k-local (k-SL) if \u2327 is the identity function;",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "\u2022 strictly k-local on tier \u2327 (k-TSL on tier \u2327 ) if \u2327 is a homomorphism; and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "\u2022 j-input strictly k-local on tier \u2327 (j-I-k-TSL on tier \u2327 ) if \u2327 is j-ISL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Definition 5.",
"sec_num": null
},
{
"text": "Stress is a phonological process in which syllables are assigned varying levels of prominence (i.e., primary stress, secondary stress, or no stress) with respect to one another. Stress is culminative, meaning that each word contains exactly one maximally-prominent syllable. Stress is usually represented by marking syllables with their prominence levels, leaving all other information about those syllables intact. This section introduces the formalism I use to represent stress and reviews the current results on stress in subregular phonology. Throughout this paper, I treat syllables as atomic units, and I represent them using symbols drawn from an alphabet \u2303. Words, being strings of syllables, are elements of \u2303 \u21e4 . When a syllable 2 \u2303 is assigned primary stress, I mark this syllable with a diacritic\u00b4 . I do not mark syllables for secondary stress. Thus, the set\u2303 := (\u2303 [ {\u00b4 | 2 \u2303}) \u21e4 is the complete alphabet of symbols used to discuss stress.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "Using this representation, stress in a particular language can be construed in two ways. Firstly, we may think of a stress system as a function mapping words without stress marking to words with stress marking. This formalizes the transduction problem for stress. Definition 9. A stress system is a same-length function s : \u2303 \u21e4 !\u2303 \u21e4 such that for every x 2 \u2303 + , there exists i > 0 and 2 \u2303 such that",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "\u2022 x[i] = and s(x)[i] =\u00b4 ; and \u2022 for all j > 0, if j 6 = i, then s(x)[j] = x[j].",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "Remark 10. All stress systems are injective.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "Secondly, we can think of a stress system as the set of all words in which stress has been assigned correctly. This formalizes the decision problem for stress.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "Definition 11. A stress constraint is any subset C \u2713\u2303 such that C is the range of some stress system s :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "\u2303 \u21e4 !\u2303 \u21e4 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "These formalizations are \"equivalent\" in the sense that we can easily convert between them.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "Definition 12. Let s : \u2303 \u21e4 !\u2303 \u21e4 be a stress system. The stress constraint given by s is the range of s.",
"cite_spans": [],
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"eq_spans": [],
"section": "Stress in Subregular Phonology",
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"text": "Definition 13. Let C \u2713\u2303 \u21e4 be a stress constraint. The stress system given by C is the stress system s C : \u2303 \u21e4 !\u2303 \u21e4 whose range is C.",
"cite_spans": [],
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"section": "Stress in Subregular Phonology",
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"text": "A well-known example of a stress system is the leftmost heavy otherwise rightmost (LHOR) system (Hayes, 1995) . In this system, syllables are either light or heavy. The leftmost heavy syllable in a word receives primary stress. If there are no heavy syllables in a word, then the rightmost (light) syllable receives primary stress. In Kwak'wala, for example, heavy syllables are those that contain a long vowel or a vowel with a coda consisting of [m], [n], or [l] (Bach, 1975) . Some illustrative examples are given below.",
"cite_spans": [
{
"start": 96,
"end": 109,
"text": "(Hayes, 1995)",
"ref_id": "BIBREF14"
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{
"start": 465,
"end": 477,
"text": "(Bach, 1975)",
"ref_id": "BIBREF1"
}
],
"ref_spans": [],
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"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "(14) LHOR stress in Kwak'wala (Boas et al., 1947; Bach, 1975) a. [ts\u00ba@ma\"a:tud] 'to melt away something in the ear' b.",
"cite_spans": [
{
"start": 30,
"end": 49,
"text": "(Boas et al., 1947;",
"ref_id": "BIBREF3"
},
{
"start": 50,
"end": 61,
"text": "Bach, 1975)",
"ref_id": "BIBREF1"
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"ref_spans": [],
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"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "[\"ba:bagw@m] 'boys' c.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "[g@g@\"n@m] 'wives'",
"cite_spans": [],
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"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "LHOR stress can be formalized using the following stress system. ",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "v = H uwv, uvw 2 L + .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "It is easy to see that the transduction problem for LHOR is I-TISL but not TISL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "Proposition 16. LHOR is not TISL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "Proof. Fixing k > 0 and homomorphic tier \u2327 , let us show that LHOR is not k-TISL on \u2327 . Suppose L / 2 \u2327 . Then, observe that su\u21b5",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "k 1 (\u2327 ( )) = su\u21b5 k 1 (\u2327 (L)) = o, but LHOR ! ( ) = , while LHOR ! L ( ) =\u0139. Thus, LHOR is not k- TISL on \u2327 if L / 2 \u2327 . But if L 2 \u2327 , then we have LHOR ! HL k (H) = H and LHOR ! L k (H) = LH even though su\u21b5 k 1 (HL k ) = su\u21b5 k 1 (L k ) = L k 1 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "Therefore, LHOR is also not k-TISL on \u2327 if L 2 \u2327 , so we conclude that it is not TISL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "Proposition 17. LHOR is 2-I-2-TISL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
},
{
"text": "Proof. Consult Figure 1 .",
"cite_spans": [],
"ref_spans": [
{
"start": 15,
"end": 23,
"text": "Figure 1",
"ref_id": "FIGREF1"
}
],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "Similarly, as Baek (2018) shows, the decision problem for LHOR is I-TSL but not TSL.",
"cite_spans": [
{
"start": 14,
"end": 25,
"text": "Baek (2018)",
"ref_id": "BIBREF2"
}
],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "Proposition 18. C LHOR is not TSL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "Proof. Fix k > 0, and suppose C LHOR is k-TSL on homomorphic tier \u2327 . It is clear thatH,\u0139 2 \u2327 ; otherwise, we would haveHH,\u0139\u0139 2 C LHOR . Furthermore, we must have H 2 and L 2 , since otherwise we would have HH 2 C LHOR and LL 2 C LHOR , respectively. Thus, every symbol of \u2303 is projected to tier \u2327 , so C LHOR is k-SL. Now, let S \u2713 (\u2303, o, n) k be the set of substrings that are permitted to appear in strings of C LHOR , and let x :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
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"text": "= L kH L k . Since x 2 C LHOR , every length-k substring of o k 1 xn k 1 is in S. However, observe that every length-k substring of o k 1 xxn k 1 = o k 1 L kH L 2kH L k n k 1",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "is also a substring of o k 1 xn k 1 , and is therefore also in S. Thus, we have deduced that xx 2 C LHOR , contradicting the definition of C LHOR .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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{
"text": "Proposition 19. C LHOR is 2-I-2-TSL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "Proof. Let be the homomorphic tier projection given by {H,H,\u0139}, and let \u2327 be defined by",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Stress in Subregular Phonology",
"sec_num": "3"
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"text": "\u2327 (x) := ( (x)L, x 2 \u2303 \u21e4 L (x), otherwise.",
"cite_spans": [],
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"eq_spans": [],
"section": "Stress in Subregular Phonology",
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"text": "In other words, \u2327 is the same as , except the last symbol of the input is always projected. It is easy to see that \u2327 is 2-ISL. Now, observe that C LHOR is 2-I-2-TSL on tier \u2327 with permissible substrings oH, o\u0139, HH, HL, Hn,HH,HL,Hn,\u0139n, Ln, and on.",
"cite_spans": [],
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"sec_num": "3"
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"text": "o start H L :\u0139 L : L : L H : LH L : L, H : H H :H o start L H L : L L : H : H H : H H : H L :",
"cite_spans": [],
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"section": "Stress in Subregular Phonology",
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"text": "As these examples illustrate, the I-TISL functions and I-TSL languages form the current subregular complexity bounds for attested subsequential stress systems and their associated stress constraints (Baek, 2018; Hao and Andersson, 2019 ). 2 These results extend those of Heinz (2009) , Rogers et al. (2013) , and Heinz (2014), who observed that stress constraints belong to restrictive subclasses of the regular languages. Other ways of refining the subregular hierarchy for stress have been proposed; Rogers and Lambert (2019), for example, define the strictly piecewise local and the piecewise locally testable language classes. The remainder of this paper will seek to compare metrical grid theory against the benchmarks I-TISL and I-TSL benchmarks.",
"cite_spans": [
{
"start": 199,
"end": 211,
"text": "(Baek, 2018;",
"ref_id": "BIBREF2"
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{
"start": 212,
"end": 235,
"text": "Hao and Andersson, 2019",
"ref_id": "BIBREF13"
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{
"start": 271,
"end": 283,
"text": "Heinz (2009)",
"ref_id": "BIBREF15"
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{
"start": 286,
"end": 306,
"text": "Rogers et al. (2013)",
"ref_id": "BIBREF32"
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"text": "Treatments of stress in phonological theory are typically based on the intuition that phonemes are organized into hierarchical structures, each 2 Hao and Andersson (2019) and Koser and Jardine (To appear) show that some stress systems are not subsequential; I do not consider such systems in this paper. level of which imposes prominence relations on its elements. While current approaches in Optimality Theory (OT) use constraints on the shapes of prosodic units and the prominence relations they impose (Prince and Smolensky, 1993, 2004; , 1996 , metrical theory has provided several frameworks for understanding stress outside of OT. 3 This section reviews and formalizes metrical grid theory (MGT, Prince, 1983; Halle and Vergnaud, 1987; Idsardi, 1992; Halle and Idsardi, 1995; Hayes, 1995) , a classic example of such a framework.",
"cite_spans": [
{
"start": 144,
"end": 145,
"text": "2",
"ref_id": null
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{
"start": 175,
"end": 204,
"text": "Koser and Jardine (To appear)",
"ref_id": null
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{
"start": 505,
"end": 516,
"text": "(Prince and",
"ref_id": "BIBREF30"
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{
"start": 517,
"end": 539,
"text": "Smolensky, 1993, 2004;",
"ref_id": null
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{
"start": 540,
"end": 546,
"text": ", 1996",
"ref_id": "BIBREF26"
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{
"start": 702,
"end": 715,
"text": "Prince, 1983;",
"ref_id": "BIBREF30"
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{
"start": 716,
"end": 741,
"text": "Halle and Vergnaud, 1987;",
"ref_id": "BIBREF12"
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{
"start": 742,
"end": 756,
"text": "Idsardi, 1992;",
"ref_id": "BIBREF19"
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{
"start": 757,
"end": 781,
"text": "Halle and Idsardi, 1995;",
"ref_id": "BIBREF11"
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{
"start": 782,
"end": 794,
"text": "Hayes, 1995)",
"ref_id": "BIBREF14"
}
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"ref_spans": [],
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"section": "Metrical Grid Theory",
"sec_num": "4"
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{
"text": "\u21e4 Word \u21e4 \u21e4 \u21e4 \u21e4 Foot \u21e4 \u21e4 \u21e4\u21e4 \u21e4\u21e4 \u21e4 \u21e4\u21e4 \u21e4 \u21e4 Syllable L LH H L H L L \u21e4 Word \u21e4 Foot \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 Syllable L L L L L L L\u0139",
"cite_spans": [],
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"section": "Metrical Grid Theory",
"sec_num": "4"
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{
"text": "According to MGT, prosodic relations are represented using diagrams like the ones that appear in Figure 2 . Each syllable is associated with a continuous stack of asterisks. The height of each stack represents the prosodic prominence of its associated syllable, with the tallest stack marking primary stress and the second-tallest stack(s) marking secondary stress. Each layer of asterisks represents a level of the prosodic hierarchy: the bottom asterisks, the syllable layer, mark the location of each syllable; the middle asterisks, the foot layer, mark syllables that are prominent within their respective feet; and the top asterisk, the word layer, marks the syllable with the greatest prominence in the word.",
"cite_spans": [],
"ref_spans": [
{
"start": 97,
"end": 105,
"text": "Figure 2",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Metrical Grid Theory",
"sec_num": "4"
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"text": "The placement of asterisks within the diagram is determined as follows. In the syllable layer, all light syllables receive an asterisk, while all heavy syllables receive two asterisks (\u21e4\u21e4). Thus, the syllable layer serves to record which syllables are heavy and which are light. In the foot layer, asterisks are placed by applying one or more of the following rules.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Metrical Grid Theory",
"sec_num": "4"
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"text": "\u2022 Quantity Sensitivity (QS): Place an asterisk directly above each \u21e4\u21e4 in the syllable layer.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Metrical Grid Theory",
"sec_num": "4"
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"text": "\u2022 Perfect Grid: Place an asterisk in every second position, starting from the first (PG(odd)) or the second (PG(even)) position.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Metrical Grid Theory",
"sec_num": "4"
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"text": "\u2022 End Rule: Place an asterisk in the first (ER(foot, )) or last (ER(foot, \u21e4)) position.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Metrical Grid Theory",
"sec_num": "4"
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"text": "In Figure 2 , for example, foot-level asterisks are assigned according to QS and ER(foot, \u21e4).",
"cite_spans": [],
"ref_spans": [
{
"start": 3,
"end": 11,
"text": "Figure 2",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Metrical Grid Theory",
"sec_num": "4"
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"text": "Applying both rules means that an asterisk is added to a position if and only if either QS or ER(foot, \u21e4) adds an asterisk to that position. Finally, the single word-layer asterisk is assigned according to the following End Rule.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Metrical Grid Theory",
"sec_num": "4"
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"text": "\u2022 End Rule: Place an asterisk directly above the first (ER(word, )) or the last (ER(word, \u21e4)) asterisk in the foot layer.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Metrical Grid Theory",
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"text": "Using these rules, LHOR is implemented in MGT as follows: the foot-level asterisks are assigned using QS and ER(foot, \u21e4), and the word-level asterisk is assigned using ER(word, ). In words containing a heavy syllable, such as L 2H HLHL 2 , the leftmost asterisk on the foot level occurs directly above the leftmost H in the word. Thus, ER(word, ) assigns primary stress to the leftmost H. In words without a heavy syllable, such as L 8 , QS does not place any asterisks on the foot layer, so the leftmost asterisk of the foot layer is the single asterisk placed by ER(foot, \u21e4). This occurs at the right word boundary, so ER(word, \u21e4) assigns primary stress to the rightmost syllable.",
"cite_spans": [],
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"section": "Metrical Grid Theory",
"sec_num": "4"
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"text": "Let us now give a precise definition of the system we have informally described. To represent stacks of asterisks, I annotate alphabet symbols with a subscript indicating the number of asterisks above that symbol. Since \u21e4\u21e4 only occurs in the syllable layer directly above an H, I do not distinguish between a single position in the grid that contains \u21e4\u21e4 and a single position that contains only one asterisk. For example, the upper grid in Figure 2 is represented by the string",
"cite_spans": [],
"ref_spans": [
{
"start": 440,
"end": 448,
"text": "Figure 2",
"ref_id": "FIGREF2"
}
],
"eq_spans": [],
"section": "Formalizing MGT",
"sec_num": "4.1"
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{
"text": "L 2 1 H 3 H 2 L 1 H 2 L 1 L 2 .",
"cite_spans": [],
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"section": "Formalizing MGT",
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"text": "Definition 20. Let \u2303 be any alphabet, and for every 2 \u2303, let 1 , 2 , . . . be symbols not in\u2303. Let \u2303 0 := \u2303, and for i 0, define the alphabet \u2303 i := { i | 2 \u2303}, with 0 = for each 2 \u2303. Let \u2303 \uf8ffj := S j i=0 \u2303 i and \u2303 \u21e4 := S 1 i=0 \u2303 i . In this formalization, each rule must be associated with a particular level in the prosodic hierarchy. A rule associated with level i, where the syllable layer is level 1, takes as input a grid whose tallest stack of asterisks is at most i levels tall, and increments the height of stacks ending at the previous level by one.",
"cite_spans": [],
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"text": "start L 1 : L 1 , H 1 : H 2",
"cite_spans": [],
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"sec_num": "4.1"
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"text": "Definition 21. For i > 0, a level-i rule is a samelength subsequential function \u21e2 :",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formalizing MGT",
"sec_num": "4.1"
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"text": "\u2303 \u21e4 \uf8ffi ! \u2303 \u21e4 \uf8ffi such that for each x 2 \u2303 \u21e4",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formalizing MGT",
"sec_num": "4.1"
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"text": "\uf8ffi and for each position j,",
"cite_spans": [],
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"text": "\u2022 if x[j] = i 1 for some 2 \u2303, then either \u21e2(x)[j] = i 1 or \u21e2(x)[j] = i ; \u2022 otherwise, \u21e2(x)[j] = x[j].",
"cite_spans": [],
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"text": "Example 22. Figure 3 shows an SFST implementing QS as a level-2 rule. Since L 1 represents a column with a single asterisk and H 1 represents a column with \u21e4\u21e4, this SFST simply changes all H 1 s to H 2 . ER(i, ) is represented by the following level-i rule:",
"cite_spans": [],
"ref_spans": [
{
"start": 12,
"end": 20,
"text": "Figure 3",
"ref_id": "FIGREF3"
}
],
"eq_spans": [],
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"sec_num": "4.1"
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"text": "ER i (x) = ( y i z, x = y i 1 z and y 2 \u2303 \u21e4 \uf8ffi 2",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "Formalizing MGT",
"sec_num": "4.1"
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"text": "x, otherwise.",
"cite_spans": [],
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"sec_num": "4.1"
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"text": "ER(i, ) places an asterisk above the leftmost asterisk on level i 1. Symbols in \u2303 i [ \u2303 i 1 represent syllables with an asterisk on level i 1; symbols in \u2303 i represent syllables with an asterisk on both level i 1 and level i. If the first symbol of x in \u2303 i [ \u2303 i 1 is of the form i 1 2 \u2303 i 1 , then this symbol is incremented to i . If the first symbol of x in \u2303 i [ \u2303 i 1 is of the form i 2 \u2303 i , then this symbol is left unchanged: ER(i, ) is still understood to add an asterisk on the ith level, but an asterisk has already been added there by another rule. If x does not contain any symbols of \u2303 i [ \u2303 i 1 , then ER(i, ) does not add any asterisks. Observe that ER i is 2-TISL on tier",
"cite_spans": [],
"ref_spans": [],
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"sec_num": "4.1"
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"text": "\u2303 i [ \u2303 i 1 .",
"cite_spans": [],
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"section": "Formalizing MGT",
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"text": "The mapping of input words to their metricalgrid representations is simply the composition of a sequence of rules. Since rules can only place asterisks on top of existing asterisks from the previous layer, the rules in the sequence are required to be monotonically increasing in their associated level of the hierarchy. 4 Definition 23. For i > 0, an i-level metrical grid is a function \u21e2 :",
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{
"start": 320,
"end": 321,
"text": "4",
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"text": "\u2303 \u21e4 ! \u2303 \u21e4 \uf8ffi such that \u21e2 = \u21e2 n \u21e2 n 1 \u2022 \u2022 \u2022 \u21e2 0",
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"text": "for some n > 0, where",
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"text": "\u2022 \u21e2 0 : \u2303 \u21e4 ! \u2303 \u21e4",
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"text": "1 is the homomorphism given by \u21e2( ) = 1 for all 2 \u2303;",
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"text": "\u2022 \u21e2 n is a level-i rule; and",
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"text": "\u2022 for all j, if \u21e2 j is a level-k rule, then \u21e2 j+1 is either a level-k rule or a level-(k + 1) rule.",
"cite_spans": [],
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"sec_num": "4.1"
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"text": "From an i-level metrical grid, we recover the stress system described by the grid by assuming that asterisks on level i represent primary stress.",
"cite_spans": [],
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"section": "Formalizing MGT",
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"text": "Definition 24. Let \u21e2 be an i-level metrical grid. The stress system induced by \u21e2 is the stress system",
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"section": "Formalizing MGT",
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"text": "s \u21e2 := s i \u21e2, where s i : \u2303 \u21e4 \uf8ffi !\u2303 \u21e4 is the homomorphism given by s i ( j ) := (\u00b4 , j = i , j < i.",
"cite_spans": [],
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"section": "Formalizing MGT",
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"text": "Trivially, the version of MGT formalized in Subsection 4.1 can express any subsequential stress system s: since level-i rules are allowed to be arbitrary subsequential functions, it suffices to construct a grid consisting of a level-2 rule that places an asterisk above the syllable assigned primary stress by s. In this section, I show that MGT is strictly more expressive than the I-TISL functions. The example that separates MGT from the I-TISL functions is motivated by Hao and Andersson's (2019) formalization of Dybo's Rule (Dybo, 1977) , a description of unbounded stress in Abkhaz. I review Hao and Andersson's implementation of Dybo's Rule both as a stress system and as a 3-level grid in Subsection 5.1. In Subsection 5.2, I show that a slight modification of Hao and Andersson's stress system is in fact not I-TISL, even though the ability of MGT and I-TSL languages to describe the system is not affected by the change in representation, as will be shown in Subsection 5.3.",
"cite_spans": [
{
"start": 474,
"end": 500,
"text": "Hao and Andersson's (2019)",
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"start": 530,
"end": 542,
"text": "(Dybo, 1977)",
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"sec_num": "5"
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"text": "In Abkhaz, syllables are lexically marked as being dominant or recessive. Dybo's Rule is an LHOR stress system in which dominant syllables not followed by other dominant syllables are considered to be heavy, and all other syllables are considered to be light. This is illustrated by the following examples, where dominant syllables are underlined.",
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"sec_num": "5.1"
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"text": "(25) Dybo's Rule in Abkhaz (Spruit, 1986) 25b, and [ra] in (25c) are heavy, since they are not followed by another dominant syllable. In (25a) and (25c), the sole heavy syllable receives primary stress. In (25b), the first of the two heavy syllables receives primary stress. 25ddoes not have any heavy syllables, so the last syllable receives primary stress by default. Hao and Andersson (2019) In the two-letter Dybo's Rule, dominant syllables are represented by D and recessive syllables are represented by R. The first D not followed by an R receives primary stress. This stress system turns out to be an I-TISL function.",
"cite_spans": [
{
"start": 27,
"end": 41,
"text": "(Spruit, 1986)",
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"start": 370,
"end": 394,
"text": "Hao and Andersson (2019)",
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"section": "Dybo's Rule",
"sec_num": "5.1"
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"text": "Proposition 27 (Hao and Andersson, 2019) . The two-letter Dybo's Rule is 2-I-3-TISL.",
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{
"start": 15,
"end": 40,
"text": "(Hao and Andersson, 2019)",
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"sec_num": "5.1"
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"text": "Hao and Andersson implement this system in MGT using a 3-level grid of the form ER 3 \u21e2 1 \u21e2 0 , where \u21e2 0 is as defined in Definition 23 and \u21e2 1 is given by the 2-ISL SFST shown in the right panel of Figure 4 . Following the MGT analysis of LHOR stress, \u21e2 1 serves to mark all heavy syllables, as well as the last syllable, with an asterisk on level 2. Thus, \u21e2 1 places an asterisk above all Ds followed by an R, along with the last syllable. ",
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{
"start": 199,
"end": 207,
"text": "Figure 4",
"ref_id": "FIGREF5"
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"sec_num": "5.1"
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"text": "\u21e4 Level 3 \u21e4 \u21e4 \u21e4 Level 2 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 Level 1 R DD R D D R D \u21e4 Level 3 \u21e4 Level 2 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 \u21e4 Level 1 R R R R R R R\u0154 o start R 1 D 1 : R 2 : D 2 R 1 : R 1 : R 1 D 1 : D 1 : D 1 D 1 : R 1 R 1 : D 2",
"cite_spans": [],
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"sec_num": "5.1"
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"text": "Let us now introduce the following variant of the two-letter Dybo's Rule. Proof. Suppose is i-I-j-TISL on tier \u2327 . Observe that",
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"sec_num": "5.2"
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"text": "! E i D (R) =DR ! E i E (R) =\u00c9R ! E i R (R) = R ! E i (R) =\u00c9R, thus \u2327 (E i D), \u2327 (E i R)",
"cite_spans": [],
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"sec_num": "5.2"
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"text": ", and \u2327 (E i ) must all be distinct. Let t := \u2327 (E i ), and for 2 \u2303, let t be such that \u2327 (E i ) = tt . Clearly, t 6 = for every .",
"cite_spans": [],
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"sec_num": "5.2"
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"text": "Let q 0 be the start state of the minimal SFST T for \u2327 , let q be the state of T corresponding to \u2327 ! E i , let ! be the transition function of T , and let r, y, and z be such that",
"cite_spans": [],
"ref_spans": [],
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"sec_num": "5.2"
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"text": "q 0 E i :y ! q DE i :z ! r.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
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"sec_num": "5.2"
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"text": "Since \u2327 is i-ISL and",
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"sec_num": "5.2"
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"text": "E i [: 1 i] = E i DE i [: 1 i] = E i 1 , we must have \u2327 ! E i DE i = \u2327 ! E i , thus r = q. It follows that for every k 0, \u2327 (E i (DE i ) k ) = yz k .",
"cite_spans": [],
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"eq_spans": [],
"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "Since t E z, it must be the case that |yz k | k.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "Now, observe that",
"cite_spans": [],
"ref_spans": [],
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"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "\u2327 (DRE i (DE i ) j )[: 1 j] = \u2327 (DRE i )z j [: 1 j] = yz j [: 1 j] = \u2327 (E i (DE i ) j )[: 1 j]. Therefore, \u2327 ! DRE i (DE i ) j = \u2327 ! E i (DE i ) j .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "However, this contradicts the fact that",
"cite_spans": [],
"ref_spans": [],
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"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "\u2327 ! DRE i (DE i ) j (R) = R 6 =\u00c9R = \u2327 ! E i (DE i ) j (R),",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "so we conclude that is not i-I-j-TISL on tier \u2327 for any i, j, or \u2327 .",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
},
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"text": "The 2-I-3-TISL SFST given by Hao and Andersson (2019) for the two-letter Dybo's Rule projects DR sequences to the tier. When the SFST encounters a contiguous block of Ds, it must delay its output by one time step, as shown below, because it is unknown whether or not the current D should be assigned stress. Stress is not assigned until the SFST encounters an R or the end of the input string has been reached. Once the tier contains a full DR sequence, the SFST knows that stress has already been assigned, and therefore does not assign stress for the remainder of its computation.",
"cite_spans": [
{
"start": 29,
"end": 53,
"text": "Hao and Andersson (2019)",
"ref_id": "BIBREF13"
}
],
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"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "With the three-letter Dybo's Rule, the state needs to record the identity of the most recent input symbol in order to delay the output by one time step. The only way to do this with an I-TISL SFST is to project the most recent input symbol to the tier.",
"cite_spans": [],
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"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "oo D: ! oD E:D ! DE D:E ! ED ! . . .",
"cite_spans": [],
"ref_spans": [],
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"sec_num": "5.2"
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"text": "Since an ISL tier projection cannot distinguish between the first block of dominant syllables in its input and subsequent blocks of dominant syllables, the schema shown above requires every block of dominant syllables to be projected to the tier. These syllable blocks overflow the memory provided by the tier, thus preventing it from recording whether or not stress has already been assigned.",
"cite_spans": [],
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"section": "MGT vs. I-TISL Functions",
"sec_num": "5.2"
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"text": "Despite the fact that the three-letter Dybo's Rule is not I-TISL, the stress constraint it induces is I-TSL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
{
"text": "Proposition 30. C is 2-I-3-TSL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
{
"text": "Proof. Let \u2327 be the 2-ISL tier projection that projects",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
{
"text": "\u2022 all instances ofD,\u00c9, and\u0154;",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
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"text": "\u2022 all instances of DR, D\u0154, ER, E\u0154,DR,D\u0154, ER, and\u00c9\u0154; and",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
{
"text": "\u2022 the last symbol of the input. Now, observe that C is 2-I-3-TSL on tier \u2327 , with the following permissible substrings: oo\u00b4 , oo\u0154, oon, o\u00b4 R, o\u00b4 n, o\u0154n, onn, R , Rn, R R, Rnn,\u00b4 R ,\u00b4 Rn,\u00b4 nn, and\u0154nn, where , 2 {D, E}.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
{
"text": "The tier projection described here is similar to the tier projection used for Hao and Andersson's (2019) 2-I-3-TISL implementation of \u21b5. Like the 2-I-2-TSL grammar for C LHOR , the 2-I-3-TSL grammar for C projects all heavy syllables and stressed syllables to the tier, along with the last syllable of the input. Unlike the grammar for C LHOR , the grammar for C also projects recessive syllables following dominant syllables. This allows the grammar to ensure that all stressed dominant syllables are dominant: they must be immediately followewd by either R or n.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
{
"text": "Because neither an ITSL grammar nor a metrical grid needs to produce the surface form as output, the problem of using the tier to delay computation does not arise for the ITSL implementation of C or for the MGT analysis of . While there is still a discrepancy between layer 2 of the MGT analysis and the tier projection used for C , I conjecture based on this observation that MGT describes ITSL decision problems.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
{
"text": "Conjecture 31. Let \u21e2 = ER 3 \u21e2 1 \u21e2 0 be a 3level metrical grid. If \u21e2 1 is ISL, then C s\u21e2 is I-TSL.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "MGT vs. I-TSL Languages",
"sec_num": "5.3"
},
{
"text": "In comparing the I-TISL implementation of \u21b5 with the MGT analysis, Hao and Andersson (2019) express the intuition that generalized tier projections and MGT are similar in that both systems use intermediate representations in order to compute stress. The analysis of Section 5 has revealed that this similarity is superficial because the computations carried out by I-TISL functions, I-TSL languages, and metrical grids are fundamentally different from one another. The most prominent of the differences discussed here is that systems implementing the transduction problem need to transfer a substantial amount of information about the underlying form to the surface form, while systems implementing the decision problem only need to retain enough information to distinguish a grammatical string from an ungrammatical one. Thus, the transduction problem may be viewed as conceptually more difficult than the decision problem. 5 While metrical grids compute transductions, their memory capabilities are enhanced by the fact that rule composition allows state information to be encoded in intermediate layers. Conjecture 31 suggests that this enhanced memory may be sufficient for MGT to bridge the gap between the transduction problem and the decision problem.",
"cite_spans": [
{
"start": 67,
"end": 91,
"text": "Hao and Andersson (2019)",
"ref_id": "BIBREF13"
}
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"section": "Conclusion",
"sec_num": "6"
},
{
"text": "In conclusion, the comparison of generalized tier projection with metrical grids provides an instructive example of an analytical toolintermediate representations of prominence relations-that behaves differently depending on the formalism in which it is instantiated. This approach could potentially offer a way to compare different theoretical frameworks in terms of how they accommodate superfically similar proposals. I leave the exploration of such ideas to future work.",
"cite_spans": [],
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"eq_spans": [],
"section": "Conclusion",
"sec_num": "6"
},
{
"text": "In the automata theory literature, k-ISL functions are known as k-local functions(Vaysse, 1986). SeeSakarovitch (2009, pp. 661-664) for an overview.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "See Kager (1995) for a survey overview of various approaches in metrical theory.",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "This requirement is known in the phonological literature as the continuous column constraint(Hayes, 1995).",
"cite_spans": [],
"ref_spans": [],
"eq_spans": [],
"section": "",
"sec_num": null
},
{
"text": "This asymmetry mirrors the relationship between search problems and decision problems in computational complexity theory (seeArora and Barak, 2009, pp. 54-55 for an overview). The conjecture that NP ) P captures the intuition that the search problem is the more difficult one.",
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"section": "",
"sec_num": null
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"num": null,
"uris": null,
"text": "Heavy syllables are represented by the symbol H, while light syllables are represented by the symbol L. Definition 15. Let \u2303 := {H, L}. The LHOR system is defined as follows. For u 2 L \u21e4 , v 2 \u2303, and w 2 \u2303 \u21e4 , LHOR(uvw) = ( uvw,"
},
"FIGREF1": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "A 2-I-2-TISL SFST for LHOR (top) and a 2-ISL SFST for its tier projection (bottom)."
},
"FIGREF2": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "Sample metrical grids for the LHOR system."
},
"FIGREF3": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "A homomorphic SFST implementing QS."
},
"FIGREF4": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "represent Dybo's Rule using the following stress system. Definition 26. Let \u2303 := {D, R}. The two-letter Dybo's Rule is the stress system \u21b5 : \u2303 \u21e4 !\u2303 \u21e4 defined as follows. For u 2 R \u21e4 D \u21e4 , v 2 \u2303, and w 2 \u2303 \u21e4 , \u21b5(uvw) := ( uvw, v = D and w / 2 D\u2303 \u21e4 uwv, uvw 2 R \u21e4 ."
},
"FIGREF5": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "Left: Sample metrical grids for the two-letter Dybo's Rule. Right: 2-ISL SFST implementing the level-2 rule for the two-letter Dybo's Rule."
},
"FIGREF6": {
"type_str": "figure",
"num": null,
"uris": null,
"text": "Let \u2303 := {D, E, R}. The threeletter Dybo's Rule is the stress system :\u2303 \u21e4 !\u2303 \u21e4 defined as follows. For u 2 R \u21e4 {D, E} \u21e4 , v 2 \u2303, and w 2 \u2303 \u21e4 , (uvw) := ( uvw, v 6 = R and w / 2 {D, E}\u2303 \u21e4 uwv, uvw 2 R + .The three-letter Dybo's Rule is exactly like the two-letter Dybo's Rule, except that there are two alphabet symbols representing dominant syllables: D and E. The MGT analysis of the two-letter Dybo's Rule can be easily adapted to the threeletter Dybo's Rule just by identifying E with D. The I-TISL implementation of the two-letter Dybo's Rule, however, cannot be applied to the three-letter Dybo's Rule.Theorem 29. The three-letter Dybo's Rule is not i-I-j-TISL on tier \u2327 for any i, j, or \u2327 ."
}
}
}
}