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The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Three subregular classes of formal languages for phonology Jeffrey Heinz heinz@udel.edu University of Delaware University of Pennsylvania

  1. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Logically Possible Computable Patterns Swiss German English nested embedding Chumash sibilant harmony Shieber 1985 Chomsky 1957 Applegate 1972 Yoruba copying Kobele 2006 Mildly Context- Regular Finite Context-Free Context- Sensitive Sensitive English consonant clusters Kwakiutl stress Clements and Keyser 1983 Bach 1975 Where is phonology? 22 / 69

  2. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Logically Possible Computable Patterns Mildly Context- Regular Finite Context-Free Context- Sensitive Sensitive Where is phonology? Johnson 1972, Kaplan and Kay 1994 22 / 69

  3. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Phonology is regular (Kaplan and Kay 1994) F 2 × . . . × F 1 × F n = P 1. Optional, left-to-right, right-to-left, and simultaneous application of rules A − → B / C D (where A,B,C,D are regular expressions) describe regular relations , provided the rule cannot reapply to the locus of its structural change. 2. Rule ordering is functional composition (finite-state transducer composition). 3. Regular relations are closed under composition. 4. SPE grammars (finitely many ordered rewrite rules of the above type) can describe virtually all phonological patterns. 5. Therefore, phonology is regular (both F i and P ). 23 / 69

  4. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Phonology is regular (Kaplan and Kay 1994) F 2 × . . . × F 1 × F n = P 1. Optional, left-to-right, right-to-left, and simultaneous application of rules A − → B / C D (where A,B,C,D are regular expressions) describe regular relations , provided the rule cannot reapply to the locus of its structural change. 2. Rule ordering is functional composition (finite-state transducer composition). 3. Regular relations are closed under composition. 4. SPE grammars (finitely many ordered rewrite rules of the above type) can describe virtually all phonological patterns. 5. Therefore, phonology is regular (both F i and P ). 23 / 69

  5. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Phonology is regular (Kaplan and Kay 1994) F 2 × . . . × F 1 × F n = P 1. Optional, left-to-right, right-to-left, and simultaneous application of rules A − → B / C D (where A,B,C,D are regular expressions) describe regular relations , provided the rule cannot reapply to the locus of its structural change. 2. Rule ordering is functional composition (finite-state transducer composition). 3. Regular relations are closed under composition. 4. SPE grammars (finitely many ordered rewrite rules of the above type) can describe virtually all phonological patterns. 5. Therefore, phonology is regular (both F i and P ). 23 / 69

  6. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Phonology is regular (Kaplan and Kay 1994) F 2 × . . . × F 1 × F n = P 1. Optional, left-to-right, right-to-left, and simultaneous application of rules A − → B / C D (where A,B,C,D are regular expressions) describe regular relations , provided the rule cannot reapply to the locus of its structural change. 2. Rule ordering is functional composition (finite-state transducer composition). 3. Regular relations are closed under composition. 4. SPE grammars (finitely many ordered rewrite rules of the above type) can describe virtually all phonological patterns. 5. Therefore, phonology is regular (both F i and P ). 23 / 69

  7. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Phonology is regular (Kaplan and Kay 1994) F 2 × . . . × F 1 × F n = P 1. Optional, left-to-right, right-to-left, and simultaneous application of rules A − → B / C D (where A,B,C,D are regular expressions) describe regular relations , provided the rule cannot reapply to the locus of its structural change. 2. Rule ordering is functional composition (finite-state transducer composition). 3. Regular relations are closed under composition. 4. SPE grammars (finitely many ordered rewrite rules of the above type) can describe virtually all phonological patterns. 5. Therefore, phonology is regular (both F i and P ). 23 / 69

  8. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Outline The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations 24 / 69

  9. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Hypothesis: Phonology is Subregular. 1. The individual factors and the whole phonologies cannot be any regular pattern. Instead they belong to well-defined subregular regions. 25 / 69

  10. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Logically possible and regular phonotactic patterns Attested Unattested and Weird Words do not have NT strings. Words do not have 3 NT strings (but 2 is OK). Words must have a vowel (or a Words must have an even syllable). number of vowels (or conso- nants, or sibilants, . . . ). If a word has sounds with [F] If the first and last sounds in a then they must agree with re- word have [F] then they must spect to [F] agree with respect to [F]. Words have exactly one pri- These six arbitrary words { w 1 , w 2 , w 3 , w 4 , w 5 , w 6 } mary stress. are well-formed. (Pater 1996, Dixon and Aikhenvald 2002, Bakovi´ c 2000, Rose and Walker 2004, Liberman and Prince 1977) 26 / 69

  11. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples of regular, but weird processes → > 1. t − ts / C D where C is all strings containing an odd number of vowels and D is all strings containing an even number of sibilants. / atiSos / → [ a> tsiSos ] but / apatiSos / → [ apatiSos ] 2. l − → r / #l[+seg]* # / lalitol / → [ lalitor ] but / palitol / → [ palitol ] 3. ‘Sour Grapes’ vowel harmony (Padgett 1995, Wilson 2003). The example below is based on Finnish front/back harmony except that /i,e/ now block vowel harmony as opposed to being transparent to it. / kurœpælœ / → [ kuropalo ] but / kurœpæli / → [ kurœpæli ] 27 / 69

  12. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples of regular, but weird processes → > 1. t − ts / C D where C is all strings containing an odd number of vowels and D is all strings containing an even number of sibilants. / atiSos / → [ a> tsiSos ] but / apatiSos / → [ apatiSos ] 2. l − → r / #l[+seg]* # / lalitol / → [ lalitor ] but / palitol / → [ palitol ] 3. ‘Sour Grapes’ vowel harmony (Padgett 1995, Wilson 2003). The example below is based on Finnish front/back harmony except that /i,e/ now block vowel harmony as opposed to being transparent to it. / kurœpælœ / → [ kuropalo ] but / kurœpæli / → [ kurœpæli ] 27 / 69

  13. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples of regular, but weird processes → > 1. t − ts / C D where C is all strings containing an odd number of vowels and D is all strings containing an even number of sibilants. / atiSos / → [ a> tsiSos ] but / apatiSos / → [ apatiSos ] 2. l − → r / #l[+seg]* # / lalitol / → [ lalitor ] but / palitol / → [ palitol ] 3. ‘Sour Grapes’ vowel harmony (Padgett 1995, Wilson 2003). The example below is based on Finnish front/back harmony except that /i,e/ now block vowel harmony as opposed to being transparent to it. / kurœpælœ / → [ kuropalo ] but / kurœpæli / → [ kurœpæli ] 27 / 69

  14. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Hypothesis: Phonology is Subregular. F 1 × F 2 × . . . × F n = P 1. The individual factors and the whole phonologies cannot be any regular pattern. Instead they belong to well-defined subregular regions. 2. We ought to characterize necessary and sufficient properties of these regions. 3. We ought to aim to prove that these regions are feasibly learnable (under various definitions). 4. We ought to investigate the empirical consequences from a psycholinguistic perspective. 28 / 69

  15. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is at stake if phonology is subregular? F 1 × F 2 × . . . × F n = P 1. We obtain more precise characterizations of possible phonological patterns. • We can decide whether some logically possible pattern is a possible phonological one. • We can cross-classify to help understand why this is so. For example, we can formulate more precise theories which ground phonology in (articulatory or perceptual) phonetics. 29 / 69

  16. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is at stake if phonology is subregular? F 1 × F 2 × . . . × F n = P 2. The computational complexity issues may resolve. • The complexity problems noticed by Barton et al., Eisner and Idsardi stem from the the known fact that the intersection/composition of arbitrarily-many arbitrary regular sets/relations is NP-Hard. • But if actual phonological patterns belong to more “well-behaved” subregular regions, these issues may disappear. (Barton et. al 1997, Eisner 1997, Idsardi 2006, Heinz et al. 2009) 30 / 69

  17. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is at stake if phonology is subregular? 3. The learning problems Mildly may become easier to Context- Finite Regular Context-Free Context- Sensitive Sensitive solve. • No superfinite class of languages is identifiable in the limit from positive data (or with probability p > 2 / 3) • The finite languages are not PAC-learnable. • While the class of r.e. languages and stochastic languages is identifiable from positive data from computable classes of texts, • these learners are not feasible, and • the learning criteria is much weaker than these others • But many non-superfinite classes of languages are feasibly learnable and include patterns found in natural language (proofs are often constructive) (Gold 1967, Horning 1969, Angluin 1980, 1982, 1988, Osherson et al. 1984, Wiehagen et. al 1984, Pitt 1985, Valiant 1984, Blum et. al 1989, Garcia et al. 1990, Muggleton 1990, Jain et. al 1999, Kearns and Vazirani 1994, Yokomori 2003, Clark and Thollard 2004, Oates et al. 2006, Niyogi 2006, Chater and Vitany´ ı 2007, Clark and Eryaud 2007, Heinz 2008, 2010, Yoshinaka 31 / 69 2008, Case et al. 2009, de la Higuera 2010)

  18. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is at stake if phonology is subregular? 3. The learning problems may become easier to Mildly Context- Regular Finite Context-Free Context- Sensitive Sensitive solve. Recursively Enumerable • No superfinite class of languages is identifiable in the limit from positive data (or with probability p > 2 / 3) • The finite languages are not PAC-learnable. • While the class of r.e. languages and stochastic languages is identifiable from positive data from computable classes of texts, • these learners are not feasible, and • the learning criteria is much weaker than these others • But many non-superfinite classes of languages are feasibly learnable and include patterns found in natural language (proofs are often constructive) (Gold 1967, Horning 1969, Angluin 1980, 1982, 1988, Osherson et al. 1984, Wiehagen et. al 1984, Pitt 1985, Valiant 1984, Blum et. al 1989, Garcia et al. 1990, Muggleton 1990, Jain et. al 1999, Kearns and Vazirani 1994, Yokomori 2003, Clark and Thollard 2004, Oates et al. 2006, Niyogi 2006, Chater and Vitany´ ı 2007, Clark and Eryaud 2007, Heinz 2008, 2010, Yoshinaka 31 / 69 2008, Case et al. 2009, de la Higuera 2010)

  19. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is at stake if phonology is subregular? 3. The learning problems may become easier to Mildly Context- Regular Finite Context-Free Context- Sensitive Sensitive solve. Recursively Enumerable • No superfinite class of languages is identifiable in the limit from positive data (or with probability p > 2 / 3) • The finite languages are not PAC-learnable. • While the class of r.e. languages and stochastic languages is identifiable from positive data from computable classes of texts, • these learners are not feasible, and • the learning criteria is much weaker than these others • But many non-superfinite classes of languages are feasibly learnable and include patterns found in natural language (proofs are often constructive) (Gold 1967, Horning 1969, Angluin 1980, 1982, 1988, Osherson et al. 1984, Wiehagen et. al 1984, Pitt 1985, Valiant 1984, Blum et. al 1989, Garcia et al. 1990, Muggleton 1990, Jain et. al 1999, Kearns and Vazirani 1994, Yokomori 2003, Clark and Thollard 2004, Oates et al. 2006, Niyogi 2006, Chater and Vitany´ ı 2007, Clark and Eryaud 2007, Heinz 2008, 2010, Yoshinaka 31 / 69 2008, Case et al. 2009, de la Higuera 2010)

  20. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is at stake if phonology is subregular? 3. The learning problems may become easier to Mildly Context- Regular Finite Context-Free Context- Sensitive Sensitive solve. Recursively Enumerable • No superfinite class of languages is identifiable in the limit from positive data (or with probability p > 2 / 3) • The finite languages are not PAC-learnable. • While the class of r.e. languages and stochastic languages is identifiable from positive data from computable classes of texts, • these learners are not feasible, and • the learning criteria is much weaker than these others • But many non-superfinite classes of languages are feasibly learnable and include patterns found in natural language (proofs are often constructive) (Gold 1967, Horning 1969, Angluin 1980, 1982, 1988, Osherson et al. 1984, Wiehagen et. al 1984, Pitt 1985, Valiant 1984, Blum et. al 1989, Garcia et al. 1990, Muggleton 1990, Jain et. al 1999, Kearns and Vazirani 1994, Yokomori 2003, Clark and Thollard 2004, Oates et al. 2006, Niyogi 2006, Chater and Vitany´ ı 2007, Clark and Eryaud 2007, Heinz 2008, 2010, Yoshinaka 31 / 69 2008, Case et al. 2009, de la Higuera 2010)

  21. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is at stake if phonology is subregular? 3. The learning problems may become easier to Mildly Context- Regular Finite Context-Free Context- Sensitive Sensitive solve. Recursively Enumerable • No superfinite class of languages is identifiable in the limit from positive data (or with probability p > 2 / 3) • The finite languages are not PAC-learnable. • While the class of r.e. languages and stochastic languages is identifiable from positive data from computable classes of texts, • these learners are not feasible, and • the learning criteria is much weaker than these others • But many non-superfinite classes of languages are feasibly learnable and include patterns found in natural language (proofs are often constructive) (Gold 1967, Horning 1969, Angluin 1980, 1982, 1988, Osherson et al. 1984, Wiehagen et. al 1984, Pitt 1985, Valiant 1984, Blum et. al 1989, Garcia et al. 1990, Muggleton 1990, Jain et. al 1999, Kearns and Vazirani 1994, Yokomori 2003, Clark and Thollard 2004, Oates et al. 2006, Niyogi 2006, Chater and Vitany´ ı 2007, Clark and Eryaud 2007, Heinz 2008, 2010, Yoshinaka 31 / 69 2008, Case et al. 2009, de la Higuera 2010)

  22. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is at stake if phonology is subregular? Mildly Context- Regular Finite Context-Free Context- Sensitive Sensitive Recursively Enumerable 4. The learning solutions can help explain the limits of phonological variation (Heinz 2009, 2010). 32 / 69

  23. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Three classes for phonology SL k Regular Finite TSL k SP k SL k means Strictly k -Local patterns. SP k means Strictly k -Piecewise patterns. T SL k means Tier-based SL k patterns. These classes are incomparable 33 / 69

  24. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Three classes for phonology Sour grapes First/Last Vowel Harmony Harmony disharmony A counting mod n pattern SL k Finite TSL k Regular SP k SL k means Strictly k -Local patterns. SP k means Strictly k -Piecewise patterns. T SL k means Tier-based SL k patterns. These classes are incomparable and demonstrably exclude many unattested, weird regular patterns. 33 / 69

  25. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Specific claims 1. Local processes belong to SL k 1.1 substitution (assimilation, dissimilation) 1.2 epenthesis 1.3 deletion 1.4 metathesis 1.5 bounded stress assignment 2. Long-distance processes belong to SL k or T SL k 2.1 consonsantal harmony ( SP k , T SL k ) 2.2 vowel harmony with no neutral vowels ( SP k , T SL k ) 2.3 vowel harmony with transparent vowels ( SP k ) 2.4 vowel harmony with opaque vowels ( T SL k ) 2.5 long-distance dissimilation ( T SL k ) 2.6 unbounded stress assignment ( SP k , T SL k ?) 3. Sets of surface forms derived from such processes. 34 / 69

  26. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Explaining the details 1. Subregular sets (Subregular Hierarchies) for phonotactic patterns • Strictly Local • Strictly Piecewise • Tier-based Strictly Local 2. Subregular relations for phonological alternations • Subsequential relations • Subclasses of the subsequential relations based on the above three classes 3. Of course there are senses in which even these three subregular classes are “too big”, but they provide a substantially tighter bound that what previously existed. 35 / 69

  27. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Outline The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations 36 / 69

  28. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Dual hierarchies of subregular sets (simplified) Regular NonCounting = Star-Free contiguous subsequences subsequences Locally Testable Piecewise Testable Locally Testable in the Strict Sense Piecewise Testable in the Strict Sense = Strictly Local = Strictly Piecewise • Each class has independent, equivalent characterizations from formal language theory, group theory, logic, and automata theory. (McNaughton and Papert 1971, Simon 1975, Rogers and Pullum 2007, Rogers et. al 2010) 37 / 69

  29. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Dual hierarchies of subregular sets (simplified) Regular NonCounting = Star-Free contiguous subsequences subsequences Locally Testable Piecewise Testable Locally Testable in the Strict Sense Piecewise Testable in the Strict Sense = Strictly Local = Strictly Piecewise • Decision procedures and closure properties under intersection, concatenation, etc. are known. (McNaughton and Papert 1971, Simon 1975, Rogers and Pullum 2007, Rogers et. al 2010) 37 / 69

  30. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Dual hierarchies of subregular sets (simplified) Regular NonCounting = Star-Free contiguous subsequences subsequences Locally Testable Piecewise Testable Locally Testable in the Strict Sense Piecewise Testable in the Strict Sense = Strictly Local = Strictly Piecewise • We introduce the Tier-based Strictly Local, which is properly Noncounting. (McNaughton and Papert 1971, Simon 1975, Rogers and Pullum 2007, Rogers et. al 2010) 37 / 69

  31. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Measures of complexity Sequences of As and Bs which Sequences of As and Bs with end in B an odd number of Bs ( A + B ) ∗ B ( A ∗ BA ∗ BA ∗ ) ∗ A ∗ BA ∗ Minimal deterministic Minimal deterministic finite-state automata finite-state automata A B A A B B 0 1 0 1 A B Conclusion: The size of the DFA as given by the Nerode equivalence relation doesn’t seem appropriate. 38 / 69

  32. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Strictly k-Local: Adjacency—Substrings C V C V ⋊ ⋉ Definition u is a factor of w iff w = xuy for some x, y ∈ Σ ∗ . u is a k-factor of w iff u is a factor and | u | = k . The container of u is C ( u ) = { w ∈ { ⋊ } Σ ∗ { ⋉ } : u is a factor of w } Note C ( u ) is the set of all words not containing the factor u . L ∈ SL k iff there exists a finite S ⊆ { ⋊ } Σ <k ∪ Σ ≤ k ∪ Σ <k { ⋉ } such that � L = C ( u ) u ∈ S 39 / 69

  33. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA illustration of SL 2 Let Σ = { a, b, c } . Forbidding factors is equivalent to considering subgraphs of the following. Examples. a a λ a c b a c b a b c c b b c 40 / 69

  34. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA illustration of SL 2 Let Σ = { a, b, c } . Forbidding factors is equivalent to considering subgraphs of the following. Examples. Suppose ac is forbidden. a a λ a c b a c b a b c c b b c 40 / 69

  35. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA illustration of SL 2 Let Σ = { a, b, c } . Forbidding factors is equivalent to considering subgraphs of the following. Examples. Suppose ac and c ⋉ is forbidden. a a λ a c b a c b a b c c b b c 40 / 69

  36. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA characterization of SL k languages L ∈ SL k iff it is a subgraph of the following automata • The states Q = Σ <k • The initial states are I = { λ } • The final states are F = Q • The transition function δ ( a 1 · · · a n , b ) = a 2 · · · a n b iff | a 1 · · · a n b | ≥ k and | a 1 · · · a n b | otherwise 41 / 69

  37. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples: Strictly K-Local Markedness Constraints F 1 × F 2 × . . . × F n = P 1. *a is SL 1 . 2. *NT is SL 2 . 3. *´ σ # is SL 2 . 4. *CCC is SL 3 . 42 / 69

  38. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations More Examples: Stress Patterns F 1 × F 2 × . . . × F n = P Edlefsen et. al (2008) classify the 109 patterns in the Stress Pattern Database (Heinz 2007,2009). 9 are SL 2 Abun West, Afrikans, Maranungku, Cambodian, . . . 44 are SL 3 Alawa, Arabic (Bani-Hassan), . . . 24 are SL 4 Arabic (Cairene), . . . 3 are SL 5 Asheninca, Bhojpuri, Hindi (Fairbanks) 1 is SL 6 Icua Tupi 28 are not SL Amele, Bhojpuri (Shukla Tiwari), Arabic Classical, Hindi (Keldar), Yidin, . . . 72% are SL k for k ≤ 6. 49% are SL 3 . 43 / 69

  39. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is not SL k For any k : 1. Unbounded stress patterns (because the primary stress may occur arbitrarily far from a word edge) 2. Long-distance harmony and disharmony patterns (because arbitrarily long material may occur between segments) 44 / 69

  40. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Strictly Piecewise (Rogers et al. 2010) S o o S t k Definition u is a subsequence of w iff u = a 0 a 1 · · · a n and w ∈ Σ ∗ a 0 Σ ∗ a 1 Σ ∗ · · · Σ ∗ a n Σ ∗ . u is a k-subsequence of w iff u is a subsequence of w and | u | = k . The shuffle ideal of u is SI ( u ) = { w : u is a subsequence of w } . Note SI ( u ) is the set of all words not containing the subsequence u . L ∈ SP k iff there exists a finite set S ⊂ Σ ≤ k such that � L = SI ( w ) w ∈ S 45 / 69

  41. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA representation of SP 2 . Let Σ = { a, b, c } . Forbidding subsequences is equivalent to removing transitions from the rightmost states in the acceptors below and then taking the intersection of these acceptors. Examples. c c b b a c a a0 a1 c b a a c b b0 b1 b b a a c c0 c1 46 / 69

  42. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA representation of SP 2 . Let Σ = { a, b, c } . Forbidding subsequences is equivalent to removing transitions from the rightmost states in the acceptors below and then taking the intersection of these acceptors. Examples. Suppose ac is forbidden. c c b b a c a a0 a1 c b a a c b b0 b1 b b a a c c0 c1 46 / 69

  43. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA representation of SP 2 . Let Σ = { a, b, c } . Forbidding subsequences is equivalent to removing transitions from the rightmost states in the acceptors below and then taking the intersection of these acceptors. Examples. Suppose ac and bb is forbidden. c c b b a c a a0 a1 c b a a c b b0 b1 b b a a c c0 c1 46 / 69

  44. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples: What is SP k (Heinz 2010) SP 2 includes phonotactic patterns derived from: 1. Asymmetric consonantal harmony • Sibilant Harmony in Sarcee (Cook 1978a,b, 1984) • E.g. forbidding the subsequence * sS 2. Symmetric consonantal harmony • Sibilant Harmony in Navajo (Sapir and Hojier 1967, Fountain 1998) • E.g forbidding the subsequences Ss and sS 3. Vowel harmony patterns with transparent vowels • Finnish, Korean sound-symbolic harmony, . . . 4. Unbounded stress patterns, once culminativity is factored out (Heinz, in prep). (wrinkle: Culminativity is properly Piecewise Testable.) 47 / 69

  45. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples: What is SP k (Heinz 2010) SP 2 includes phonotactic patterns derived from: 1. Asymmetric consonantal harmony • Sibilant Harmony in Sarcee (Cook 1978a,b, 1984) • E.g. forbidding the subsequence * sS 2. Symmetric consonantal harmony • Sibilant Harmony in Navajo (Sapir and Hojier 1967, Fountain 1998) • E.g forbidding the subsequences Ss and sS 3. Vowel harmony patterns with transparent vowels • Finnish, Korean sound-symbolic harmony, . . . 4. Unbounded stress patterns, once culminativity is factored out (Heinz, in prep). (wrinkle: Culminativity is properly Piecewise Testable.) 47 / 69

  46. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples: What is SP k (Heinz 2010) SP 2 includes phonotactic patterns derived from: 1. Asymmetric consonantal harmony • Sibilant Harmony in Sarcee (Cook 1978a,b, 1984) • E.g. forbidding the subsequence * sS 2. Symmetric consonantal harmony • Sibilant Harmony in Navajo (Sapir and Hojier 1967, Fountain 1998) • E.g forbidding the subsequences Ss and sS 3. Vowel harmony patterns with transparent vowels • Finnish, Korean sound-symbolic harmony, . . . 4. Unbounded stress patterns, once culminativity is factored out (Heinz, in prep). (wrinkle: Culminativity is properly Piecewise Testable.) 47 / 69

  47. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples: What is SP k (Heinz 2010) SP 2 includes phonotactic patterns derived from: 1. Asymmetric consonantal harmony • Sibilant Harmony in Sarcee (Cook 1978a,b, 1984) • E.g. forbidding the subsequence * sS 2. Symmetric consonantal harmony • Sibilant Harmony in Navajo (Sapir and Hojier 1967, Fountain 1998) • E.g forbidding the subsequences Ss and sS 3. Vowel harmony patterns with transparent vowels • Finnish, Korean sound-symbolic harmony, . . . 4. Unbounded stress patterns, once culminativity is factored out (Heinz, in prep). (wrinkle: Culminativity is properly Piecewise Testable.) 47 / 69

  48. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is not SP k 1. Vowel harmony with blocking, i.e. with opaque vowels 2. Long-distance dissimilation Example. Latin Liquid Dissimilation ([l,r] on own tier): a. /nav-alis/ nav-alis ‘naval’ b. /episcop-alis/ episcop-alis ‘episcopal’ c. /infiti-alis/ infiti-alis ‘negative’ d. /sol-alis/ sol-aris ‘solar’ e. /lun-alis/ lun-aris ‘lunar’ f. /milit-alis/ milit-aris ‘military’ However, the rule does not apply if an [r] intervenes. g. /flor-alis/ flor-alis ‘floral’ *flor-aris h. /sepulkr-alis/ sepulkr-alis ‘funereal’ *sepulkr-aris i. /litor-alis/ litor-alis ‘of the shore’ *litor-aris Note there are some exceptions; e.g. ‘fili-alis’ and ‘glute-alis’. 48 / 69

  49. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Tier-based SL k Patterns Let T be a subset of Σ. T is the tier. Definition First, define the Erasing function for all w = a 1 · · · a n : E T ( w ) = v 1 · · · v n where v i = a i iff a i ∈ T and v i = λ otherwise The tier-based container of u ∈ { ⋊ } T ∗ { ⋉ } is TC ( u ) = { w ∈ { ⋊ } Σ ∗ { ⋉ } : u is a factor of E T ( w ) } Note TC ( u ) is the set of all words not containing the factor u on tier T . L ∈ TSL k iff there exists a finite S ⊆ { ⋊ } Σ <k ∪ Σ ≤ k ∪ Σ <k { ⋉ } such that � L = TC ( u ) u ∈ S 49 / 69

  50. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA representation of TSL 2 Let Σ = { a, b, c } and T = { b, c } . Forbidding factors on the tier T is equivalent to considering subgraphs of the following. Examples. c b c c b b a a c b λ a 50 / 69

  51. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA representation of TSL 2 Let Σ = { a, b, c } and T = { b, c } . Forbidding factors on the tier T is equivalent to considering subgraphs of the following. Examples. Suppose bb is forbidden. c b c c b b a a c b λ a 50 / 69

  52. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations FSA representation of TSL 2 Let Σ = { a, b, c } and T = { b, c } . Forbidding factors on the tier T is equivalent to considering subgraphs of the following. Examples. Suppose bb and cc is forbidden. c b c c b b a a c b λ a 50 / 69

  53. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is TSL k Phonotactic patterns derivable from 1. Vowel harmony with opaque vowels. 2. Long-distance dissimilation patterns 51 / 69

  54. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is not TSL k Phonotactic patterns derivable from 1. Vowel harmony with transparent vowels (unless the transparent vowels are off the tier). 52 / 69

  55. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Learning Results for these Subregular Sets Theorem Given k , SL k and SP k are identifiable in the limit from positive data by an incremental learner which is efficient in the size of the sample and which has many other desirable properties. They are also PAC-learnable. (Garcia et al. 1990, Heinz 2010b, Kasprzik and K¨ otzing 2010) 53 / 69

  56. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Learning Results for these Subregular Sets Theorem Given k , and a sample S drawn according to a stochastic SL k ( SP k ) language, there is an efficient procedure which finds the parameter values which maximizes the likelihood of S w.r.t the SL k ( SP k ) family of distributions. (Jurafsky and Martin 2008, Heinz and Rogers 2010) 53 / 69

  57. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Learning Results for these Subregular Sets Theorem Given k and the tier T , TSL k is identifiable in the limit from positive data by an incremental learner which is efficient in the size of the sample and which has many other desirable properties. They are also PAC-learnable. (Heinz 2010b, Kasprzik and K¨ otzing 2010) 53 / 69

  58. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Interim Summary F 1 × F 2 × . . . × F n = P 1. Three subregular classes circumscribe virtually all attested phonotactic patterns ( F i ), and exclude many weird, regular ones. 2. These classes are learnable under established definitions, and provide better measures of pattern complexity than DFA size. 3. Noncounting is closed under intersection so if the interaction ( × ) is intersection ( ∩ ) then at worst P is Noncounting 54 / 69

  59. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Interim Summary F 1 × F 2 × . . . × F n = P 1. Three subregular classes circumscribe virtually all attested phonotactic patterns ( F i ), and exclude many weird, regular ones. 2. These classes are learnable under established definitions, and provide better measures of pattern complexity than DFA size. 3. Noncounting is closed under intersection so if the interaction ( × ) is intersection ( ∩ ) then at worst P is Noncounting 54 / 69

  60. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Interim Summary F 1 × F 2 × . . . × F n = P 1. Three subregular classes circumscribe virtually all attested phonotactic patterns ( F i ), and exclude many weird, regular ones. 2. These classes are learnable under established definitions, and provide better measures of pattern complexity than DFA size. 3. Noncounting is closed under intersection so if the interaction ( × ) is intersection ( ∩ ) then at worst P is Noncounting 54 / 69

  61. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Interim Summary F 1 × F 2 × . . . × F n = P 4. Some loose ends: • Graf 2009 (PLC) observes a counting stress pattern • Culminativity “exactly one” is properly Piecewise Testable. 5. Linguistically motivated open question: Optimization over regular constraints yield nonregular patterns (Frank and Satta 1998, Kartunnen 1998). What about optimization over constraints drawn from these classes? 6. What about regular relations describing phonological processes? 54 / 69

  62. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Interim Summary F 1 × F 2 × . . . × F n = P 4. Some loose ends: • Graf 2009 (PLC) observes a counting stress pattern • Culminativity “exactly one” is properly Piecewise Testable. 5. Linguistically motivated open question: Optimization over regular constraints yield nonregular patterns (Frank and Satta 1998, Kartunnen 1998). What about optimization over constraints drawn from these classes? 6. What about regular relations describing phonological processes? 54 / 69

  63. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Interim Summary F 1 × F 2 × . . . × F n = P 4. Some loose ends: • Graf 2009 (PLC) observes a counting stress pattern • Culminativity “exactly one” is properly Piecewise Testable. 5. Linguistically motivated open question: Optimization over regular constraints yield nonregular patterns (Frank and Satta 1998, Kartunnen 1998). What about optimization over constraints drawn from these classes? 6. What about regular relations describing phonological processes? 54 / 69

  64. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Outline The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations 55 / 69

  65. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Subregular relations Subsequential Regular Finite Subsequential functions are those describable by finite-state transducers that are deterministic on the input. 56 / 69

  66. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Subregular relations Finite Subsequential Regular Subsequential functions are identifiable in the limit from positive data (where they are defined) (Oncina et al. 1993), though the sample necessary for learning phonological patterns does not appear to be present in natural language corpora (Gildea and Jurafsky 1996). 56 / 69

  67. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Subregular relations SL k Finite TSL k Noncounting Subsequential Regular SP k We are defining subsequential counterparts to SL k , SP k , TSL k . 56 / 69

  68. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Subsequential Transducers One definition permits each state to be associated with a string. The action of “ending” in a state appends this string to the output. Example: Σ = { a, b, c, d, e } and the rule a − → b / c d a:a, b:b, d:d, e:e c:c b:b, d:d, e:e a: λ c:c q 2 , a q 0 , λ q 1 , λ c:ac d:bd a:aa, b:ab, e:ae 57 / 69

  69. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Subsequential Transducers One definition permits each state to be associated with a string. The action of “ending” in a state appends this string to the output. Example: Σ = { a, b, c, d, e } and the rule a − → b / c d /cad/ → [cbd] a:a, b:b, d:d, e:e c:c b:b, d:d, e:e a: λ c:c q 2 , a q 0 , λ q 1 , λ c:ac d:bd a:aa, b:ab, e:ae 57 / 69

  70. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Subsequential Transducers One definition permits each state to be associated with a string. The action of “ending” in a state appends this string to the output. Example: Σ = { a, b, c, d, e } and the rule a − → b / c d /ca/ → [ca] a:a, b:b, d:d, e:e c:c b:b, d:d, e:e a: λ c:c q 2 ,a q 0 , λ q 1 , λ c:ac d:bd a:aa, b:ab, e:ae 57 / 69

  71. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations What is and what is not subsequential Subsequential processes Non-subsequential processes 1. epenthesis, deletion, substitution, local 1. “sour grapes” vowel metathesis harmony process 2. consonantal harmony 2. unbounded long-distance metathesis (not even 3. long-distance regular) dissimilation 4. Vowel harmony patterns (verified for all in Nevins 2010 (46 languages) ) 58 / 69

  72. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Strictly k -Local Subsequential Relations A relation R ⊆ Σ ∗ 1 × Σ ∗ 2 is input-based Strictly k -Local iff (above the line is same as SL k sets) • The states Q = Σ <k • The initial states are I = { λ } • The final states are F = Q • The transition function δ ( a 1 · · · a n , b ) = a 2 · · · a n b iff | a 1 · · · a n b | ≥ k and | a 1 · · · a n b | otherwise • The output of each transition belongs to Σ ∗ 2 . • Final appending strings belong to Σ ∗ 2 . 59 / 69

  73. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples of Strictly k -Local Subsequential Relations Simultaneous application : 1. Substitution: rules of the form a − → b / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uav | . 2. Deletion: Rules of the form a − → ∅ / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uav | . 3. Epenthesis: Rules of the form ∅ − → b / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uv | . 4. Metathesis: Rules of the form ab − → ba / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uabv | . 60 / 69

  74. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples of Strictly k -Local Subsequential Relations Simultaneous application : 1. Substitution: rules of the form a − → b / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uav | . 2. Deletion: Rules of the form a − → ∅ / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uav | . 3. Epenthesis: Rules of the form ∅ − → b / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uv | . 4. Metathesis: Rules of the form ab − → ba / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uabv | . 60 / 69

  75. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples of Strictly k -Local Subsequential Relations Simultaneous application : 1. Substitution: rules of the form a − → b / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uav | . 2. Deletion: Rules of the form a − → ∅ / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uav | . 3. Epenthesis: Rules of the form ∅ − → b / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uv | . 4. Metathesis: Rules of the form ab − → ba / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uabv | . 60 / 69

  76. The Computational Perspective Subregular Phonology Subregular Phonotactics Subregular Alternations Examples of Strictly k -Local Subsequential Relations Simultaneous application : 1. Substitution: rules of the form a − → b / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uav | . 2. Deletion: Rules of the form a − → ∅ / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uav | . 3. Epenthesis: Rules of the form ∅ − → b / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uv | . 4. Metathesis: Rules of the form ab − → ba / u v where u and v are strings. • This alternation is Strictly k -Local with k = | uabv | . 60 / 69

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