On the Cognitive Complexity of Phonotactic Constraints James Rogers - PDF document

Stony Brook—CogComp 2018 1 On the Cognitive Complexity of Phonotactic Constraints James Rogers Dept. of Computer Science Earlham College Slide 1 jrogers@cs.earlham.edu http://cs.earlham.edu/~jrogers/slides/stonybrook.ho.pdf Joint work with Jeff Heinz (UDel), Sean Wibel, Maggie Fero and Dakotah Lambert (EC) Some simple patterns (1) Primary stress falls on the final syllable Slide 2 (2) Primary stress falls on the antepenultimate syllable (3) In words of five or more syllables primary stress falls on the antepenultimate syllable

Stony Brook—CogComp 2018 2 Some simple patterns (4) Primary stress falls on the initial syllable if it is heavy, else the peninitial syllable. Slide 3 (5) Primary stress falls on the leftmost heavy syllable (6) Secondary stress falls on every third syllable counting left from the antepenultimate syllable. Some simple patterns (7) Final syllable is heavy Slide 4 (8) All heavy syllables get some stress (9) There are always an odd number of heavy syllables

Stony Brook—CogComp 2018 3 Some simple patterns (10) Primary stress falls on some syllable. (At least one) Slide 5 (11) Primary stress falls on at most one syllable. (12) Primary stress falls on exactly one syllable. Complexity of Simple Patterns (7) Sequences of ‘ L ’s and ‘ H ’s which end in ‘ H ’: S 0 − → LS 0 , S 0 − → HS 0 , S 0 − → H H L H ( L + H ) ∗ H L Slide 6 (9) Sequences of ‘ L ’s and ‘ H ’s which contain an odd number of ‘ H ’s: S 0 − → LS 0 , S 0 − → HS 1 , S 1 − → LS 1 , S 1 − → HS 0 , S 1 − → ε L L H ( L ∗ HL ∗ HL ∗ ) ∗ L ∗ HL ∗ H

Stony Brook—CogComp 2018 4 Some More Simple Patterns (10) Sequences of ‘ σ ’s and ‘´ σ ’s which contain at least one ‘´ σ ’: S 0 − → σS 0 , S 0 − → ´ σS 1 , S 1 − → σS 1 , S 1 − → ´ σS 1 , S 1 − → ε σ, ´ σ ´ σ σ σ ∗ ´ σ ) ∗ σ ( σ + ´ Slide 7 (12) Sequences of ‘ σ ’s and ‘´ σ ’s which contain exactly one ‘´ σ ’: S 0 − → σS 0 , S 0 − → ´ σS 1 , S 1 − → σS 1 , S 1 − → ε σ, ´ σ σ σ ´ σ σ ´ σ ∗ ´ σσ ∗ Cognitive Complexity from First Principles What kinds of distinctions does a cognitive mechanism need to be sensitive to in order to classify an event with respect to a pattern? Slide 8 Reasoning about patterns • What objects/entities/things are we reasoning about? • What relationships between them are we reasoning with?

Stony Brook—CogComp 2018 5 Some Assumptions about Linguistic Behaviors • Perceive/process/generate linear sequence of (sub)events • Can model as strings—linear sequence of abstract symbols Slide 9 – Discrete linear order (initial segment of N ). – Labeled with alphabet of events Partitioned into subsets, each the set of positions at which some event occurs. Word models �D , ⊳, <, P σ � σ ∈ Σ (+1) �D , ⊳, P σ � σ ∈ Σ ( < ) �D , <, P σ � σ ∈ Σ D — Finite — Linear order on D < Slide 10 — Successor wrt < ⊳ P σ — Subset of D at which σ occurs ( P σ partition D ) CCV C = � { 0 , 1 , 2 , 3 } , {� i, i + 1 � | 0 ≤ i < 3 } , { 0 , 1 , 3 } C , { 2 } V � � D ⊳ P C P V �

Stony Brook—CogComp 2018 6 An Alphabet for Stress Patterns Syllable Weight Stress • L = Light • σ = Unstressed • H = Heavy • σ ´ = Primary Stress Slide 11 • S = Super Heavy • σ ` = Secondary Stress + • σ = Arbitrary • σ = Some Stress • σ = Not Primary Stress ∗ • σ = Arbitrary Stress HL` ´ eg: H Local Constraints • Blocks of adjacent symbols – k -factors • End markers: ‘ ⋊ ’, ‘ ⋉ ’ Slide 12 F 2 ( ⋊ σσ ´ σ ⋉ ) = { ⋊ σ, σσ, σ ´ σ, ´ σ ⋉ } F 3 ( ⋊ σσ ´ σ ⋉ ) = { ⋊ σσ, σσ ´ σ, σ ´ σ ⋉ } F 6 ( ⋊ σσ ´ σ ⋉ ) = { ⋊ σσ ´ σ ⋉ }

Stony Brook—CogComp 2018 7 Strictly k -Local Constraints • Co-occurrence of negative atomic local constraints – Conjunctions of negated k -factors (1) Primary stress falls on the final syllable Slide 13 ¬ σ ⋉ (SL 2 ) (2) Primary stress falls on the antepenultimate syllable σ ∗ σ ∗ σ ∗ σ ∗ ¬ ´ σ ∧ ¬ ´ σ ⋉ ∧ ¬ ´ σ ⋉ (SL 4 ) Cambodian 1) In words of all sizes, primary stress ¬ σ ⋉ ∧ ¬ ` σ ⋉ (SL 2 ) falls on the final syllable. σ ∗ 1b) Primary stress does not fall before ¬ ´ σ (SL 2 ) the final syllable. Slide 14 2) In words of all sizes, secondary ¬ H (SL 1 ) stress falls on all heavy syllables. ∗ ∗ ∗ 3) Light syllables occur only immedi- ¬ ⋊ L ∧ ¬ L L (SL 2 ) ately following heavy syllables. ¬ ⋊ ´ [ 4) Light monosyllables do not occur. L ⋉ (SL 3 ) ] Cambodian stress is SL 2 .

Stony Brook—CogComp 2018 8 Scanners a b a b a b a b a a b a b a b a b a · · · · · · b k k T Set Start G : · · · ∈ Q Slide 15 Clear · · · a F a · · · b b · · · a b · · · k Recognizing an SL k stringset requires only remembering the k most recently encountered symbols. Character of Strictly k -Local Sets Theorem (Suffix Substitution Closure): A stringset L is strictly k -local iff whenever there is a string x of length k − 1 and strings w , y , v , and z , such that k − 1 ���� w · x · y ∈ L v · x · z ∈ L Slide 16 then it will also be the case that w · x · z ∈ L ⋆ CCC is SL 3 But ⋆ CCC is not SL 2 : V · CC · V C ∈ ⋆ CCC C · C · V C ∈ ⋆ CCC · · ∈ ⋆ CCC · · ∈ ⋆ CCC CV CC V V C CV V · CC · V ∈ ⋆ CCC C · C · CV �∈ ⋆ CCC

Stony Brook—CogComp 2018 9 Alawa • In words of all sizes, primary stress falls on the penultimate syllable. • [ —Except in monosyllables ] G Alawa = { ⋊ σσ, ⋊ σ ´ σ, ⋊ ´ σσ, σσσ, σσ ´ σ, σ ´ σσ, Slide 17 σσ ⋉ , ⋊ ´ ´ σ ⋉ } ´ ´ ⋊ σ σ σ ⋉ ⋊ σ σ σσ ⋉ ´ ⋊ ´ ⋊ σ ⋉ σ σ ⋉ ⋆ ⋊ σ σ ´ ⋉ ⋆ ⋊ σ σ ⋉ Alawa stress is in SL 3 − SL 2 . SL Hierarchy Theorem 1 (SL-Hierarchy) SL 1 � SL 2 � SL 3 � · · · � SL i � SL i +1 � · · · � SL Slide 18 Every Finite stringset is SL k for some k : Fin ⊆ SL. There is no k for which SL k includes all Finite stringsets. SL k is learnable in the limit from positive data. SL is not.

Stony Brook—CogComp 2018 10 Cognitive interpretation of SL • Any cognitive mechanism that can distinguish member strings from non-members of a (properly) SL k stringset must be sensitive, at least, to the length k blocks of consecutive events that occur in the presentation of the string. • Any cognitive mechanism that is sensitive only to the Slide 19 co-occurrence of length k blocks of consecutive events in the presentation of a string will be able to recognize only SL k stringsets. Sequential: This corresponds to being sensitive, at each point in the string, to the immediately prior sequence of k − 1 events. Parallel: This corresponds to being sensitive to the presence of simple contiguous blocks in the string. Strictly Local Stress Patterns StressTyp2 Database (2015)—699 languages, 106 formally distinct patterns 9 are SL 2 Abun West, Afrikans, . . . Cambodian,. . . Maranungku 44 are SL 3 Alawa, Arabic (Bani-Hassan),. . . Slide 20 23 are SL 4 Dutch,. . . 3 are SL 5 Asheninca, Bhojpuri, Hindi (Fairbanks) 1 is SL 6 Icua Tupi 26 are not SL Amele, Bhojpuri (Shukla Tiwari), Ara- bic (Classical), Hindi (Kelkar), Yidin,. . . 75% are SL, all k ≤ 6. 50% are SL 3 .

Stony Brook—CogComp 2018 11 Obligatoriness: Some-´ σ k − 1 � �� � ⋊ σ σ · · · σ σ ⋉ ` k − 1 � �� � ⋊ ´ σ · · · σ σ σ ⋉ k − 1 Slide 21 � �� � σ · · · σ ⋆ σ σ ⋉ Some-´ σ �∈ SL How can any stress pattern be SL? Locally definable stringsets def f ∈ F k ( ⋊ · Σ ∗ · ⋉ ) w | = f ⇐ ⇒ f ∈ F k ( ⋊ · w · ⋉ ) def ϕ ∧ ψ w | = ϕ ∧ ψ ⇐ ⇒ w | = ϕ and w | = ψ def ¬ ϕ w | = ¬ ϕ ⇐ ⇒ w �| = ϕ Slide 22 ϕ ∨ ψ ≡ ¬ ( ϕ ∧ ψ ) L = L ( ϕ ) def = { w ∈ Σ ∗ | w | = ϕ } � SL k ≡ [ ¬ f i ] � LT k f i �∈G

Stony Brook—CogComp 2018 12 Some-´ σ again Some-´ σ = L (´ σ ) Slide 23 Some-´ σ ∈ LT 1 NKL • Primary stress falls on the final syllable if it is Heavy • Else on the initial syllable if it is Light • Else on the penultimate syllable Slide 24 ϕ NKL = ´ H ⋉ final syllable if it is Heavy ∨ ( ¬ ´ H ⋉ ∧ ⋊ ´ L) Else on the initial if it is Light σ ∗ ∨ ( ¬ ´ H ⋉ ∧ ¬ ⋊ ´ L ∧ ´ σ ⋉ ) Else on the penultimate syllable

Stony Brook—CogComp 2018 13 LT Automata a b a b a b a b a a b a b a b a b a a b b a � b Accept a a Yes Boolean a b � Slide 25 Network No b a � Reject b b a b � Membership in an LT k stringset depends only on the set of k -Factors which occur in the string. Recognizing an LT k stringset requires only remembering which k -factors occur in the string. Character of Locally Testable sets Theorem 2 ( k -Test Invariance) A stringset L is Locally Testable iff there is some k such that, for all strings x and y , if ⋊ · x · ⋉ and ⋊ · y · ⋉ have exactly the same set of k -factors Slide 26 then either both x and y are members of L or neither is. Definition 1 ( k -Local Equivalence) k v def w ≡ L ⇐ ⇒ F k ( ⋊ w ⋉ ) = F k ( ⋊ v ⋉ ) . LT 1 � LT 2 � LT 3 � · · · � LT i � LT i +1 � · · · � LT