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| "paper_id": "2020", | |
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| "date_generated": "2023-01-19T15:39:34.098444Z" | |
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| "title": "Questioning to Resolve Transduction Problems", | |
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| { | |
| "first": "Eric", | |
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| "last": "Meinhardt", | |
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| "institution": "Rutgers University", | |
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| "first": "Anna", | |
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| "institution": "Rutgers University", | |
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| { | |
| "first": "Eric", | |
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| "last": "Bakovi", | |
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| "institution": "Rutgers University", | |
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| { | |
| "first": "Adam", | |
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| "last": "Mccollum", | |
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| { | |
| "first": "U", | |
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| "text": "show that any non-deterministic regular function (NDRF) :\u2303 < \u00f4 < can be decomposed into the composition \u21e2\u02dd of two subsequential functions (SSQs) that proceed in opposite directions; crucially, the first function to apply must behave as unbounded lookahead for the second. We henceforth refer to such decompositions \u21e2\u02dd as 'EM decompositions'. Recent work in computational phonology has shown the utility of such decompositions for analyzing and comparing the minimum expressivity required for iterative, bidirectional, (non-)myopic, and other long-distance phonological processes that require greater expressivity than that supplied by SSQ functions. Existing work has identified the (interaction-free) weakly deterministic functions (IF-WDRFs; McCollum et al. 2018 , Hao & Andersson 2019 and the NDRFs as salient lower and upper bounds on the complexity of such processes (Heinz & Lai 2013 , Jardine 2016 . Because unbounded lookahead is a key feature of this region, we suggest that understanding it is crucial for picking out additional phonologically interesting subclasses within this region. In this work, we identify several concepts useful for describing lookahead in decomposed NDRFs and o er a set of necessary and su cient properties for a composition \u21e2\u02dd to be an EM decomposition of a non-SSQ NDRF . We then use these ideas to outline a set of functions in between the IF-WDRFs and proper NDRFs, organized in terms of a precise notion of the degree of lookahead that can provide for \u21e2.", | |
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| "text": "(IF-WDRFs;", | |
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| "text": "McCollum et al. 2018", | |
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| "text": ", Hao & Andersson 2019", | |
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| "text": "(Heinz & Lai 2013", | |
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| "text": ", Jardine 2016", | |
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| "text": "For present purposes, 1 a question may be identified with a partition Q over a set of possible worlds W (e.g. a formal language L) into equivalence classes ('cells'), and a resolving answer or observation is information that picks out (with respect to some background knowledge -e.g. prior knowledge of L and information gleaned from an observed prefix of a current input string) the cell q k of the partition that the actual world (total string, unseen su x, etc.) falls into. While two distinct answers a i , a j may resolve a question in the same way by picking out the same cell, entailment defines a (partial) ordering on the informativeness of answers or observations: if a i and a j pick out the same cell q k , but a i is strictly more specific than a j , then a i \u00d9 a j but both resolve Q in the same way. Similarly, refinement can be used to define an analogous ordering on questions: if every cell of Q 0 is a subset of some cell of Q 1 , then any resolving answer to Q 0 is also a resolving answer to Q 1 . An agent faced with choosing the next action sequence ('output string') u \u00c0 < given its current knowledge about the state of the world is faced with a decision problem that induces a partition on W : each cell is associated with the ('optimal') action sequence that the agent should take at the current timestep if it thinks the actual world currently is in that cell.", | |
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| "text": "A non-SSQ NDRF at some point while reading the prefix x of a string xy faces a(t least one) 'decision problem': 2 exactly what the incremental output of the prefix x should be depends on which of at least two cells q k , q l some a priori unboundedly distant portion of the as-yet unseen su x y falls into. Consider the hypothetical 'sour grapes' pattern entertained by McCollum et al., based on Turkish and dubbed 'Zurkish': [+round] spreads left to right from initial U , changing I to U , unless there is a low vowel A anywhere in the word, in which case there is no spreading at all. 3 Thus input strings of the form U I n are mapped to U U n , but input strings of the form U I < A + X < (X = {I, A}) remain unchanged. Whether a given prefix x = U I n maps to U U n or to U I n depends on whether the su x y contains an A. If \u21e2\u02dd is an EM decomposition of , then it must be the case that reads input strings xy from the far end relative to \u21e2, 4 identifies which cell the su x y belongs to, remembers this long enough to recognize where within x it should transform the input string (be it via markup symbols, length-increasing codes, or 'phonotactic' codes; McCollum et al., Smith & O'Hara 2019), and creates an intermediate string (xy) = x \u00ae y \u00ae such that reading the transformed prefix x \u00ae from the other end is su cient to resolve 's decision problem -i.e. identify which cell the su x of the original string belongs to and therefore what output string should be emitted. Thus in hypothetical Zurkish, reads input strings from right to left and \u21e2 reads the output of from left to right. If the su x y contains an A, then transforms the input string such that all instances of I between A and the beginning of the string are marked to not be changed by \u21e2; otherwise, all instances of I after initial U will in fact be changed by \u21e2. This thus resolves 's decision problem for Zurkish. A further constraint on 's rewriting is that \u21e2 must be able to recognize this transformed prefix and thereby infer the associated cell at a particular point in time, viz. by the time it reads the input symbol (or within an a priori bounded distance after) associated with 's decision problem. Finally, \u21e2's output for the symbol associated with the decision problem must then depend on the information about y that has injected into x \u00ae .", | |
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| "text": "Our work synthesizes the results of Elgot & Mezei with those of McCollum et al. and Heinz & Lai. First, we explicate the notions of 'information smuggling' and lookahead left informal in McCollum et al.'s discussion of 'interacting' compositions; thus equipped, we can formally articulate for any non-SSQ \u00c0 NDRF the properties that any potential EM decomposition \u21e2\u02dd must have in order for it to su ce as an EM decomposition of . Second, it follows clearly and explicitly from our analysis of EM decompositions that the IF-WDRFs \u25ca NDRFs. Third, we conjecture that the framework we present o ers a useful way of defining and comparing functions with more expressivity than the interaction-free WDRFs but less than the full set of NDRFs. We sketch our current model of such functions below.", | |
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| "text": "In this hierarchy of 'lookahead-constrained' ('LoCo') weakly deterministic regular functions, 5 interaction is possible, but the 'questions' the lookahead pass in an EM decomposition can 'answer' for \u21e2 are qualitatively constrained in some way -e.g. might be OSL or I-TISL (Hao & Andersson). For any two potential lookahead functions f , g, we can ask whether the question partition of one is a refinement of the other. We conjecture that this can be extended to classes of functions to compare how relatively fine or coarse the questions each can answer when employed as a lookahead function in an EM decomposition. Finally, we can also use the analysis of EM decompositions described above to identify substrings where \u21e2\u02dd interact, but where the change in behavior of \u21e2 on a given substring cannot be be associated with a strict increase in knowledge about the unseen su x. ", | |
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| "text": "These concepts are adapted from literature on the meaning of questions and the value of questions and information (see e.g. van Rooy 2003), but no familiarity with such literature is necessary.2 We have not yet considered multiple decision problems per NDRF , especially incomparable ones.3 In actual Turkish, [+round] spreading proceeds up to A, which blocks further spread.", | |
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| "text": "For clarity, we use 'prefix' here from the view of \u21e2: i w = xy and \u21e2 sees x first, x is a prefix.5 To be precise: we can define a bounded lattice (organized by refinement of questions) of non-SSQ NDRFs, with IF-WDRFs at the bottom, otherwise-unrestricted NDRFs at the top, and LoCo WDRFs in between.", | |
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| "text": "References. \u2022 Elgot, C. & J. Mezei. 1965. On relations defined by generalized finite automata. IBM Journal of Research and Development 9(1). 47-68. \u2022 Hao, Y. & S. Andersson. 2019. Unbounded Stress in Subregular Phonology. In SIGMORPHON 16, 135-143. ACL. \u2022 Heinz, J. & R. Lai. 2013. Vowel harmony and subsequentiality. In MoL 13, 52-63. \u2022 Jardine, A. 2016. Computationally, tone is di erent. Phonology 33(2). 247-283. \u2022 McCollum, A. G., E. Bakovi , A. Mai & E. Meinhardt. 2018. The expressivity of segmental phonology and the definition of weak determinism. lingbuzz/004197 . \u2022 van Rooy, R. 2003. Questioning to resolve decision problems. Linguistics and Philosophy 26(6). 727-763. \u2022 Smith, C. & C. O'Hara. 2019. Formal characterizations of true and false sour grapes. In Proceedings of SCiL 2019, vol. 2 1, 338-341.", | |
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