Cognitive Complexity of Phonological Patterns James Rogers Dept. - PDF document

UDel Cognitive Science 1 Cognitive Complexity of Phonological Patterns James Rogers Dept. of Computer Science Earlham College jrogers@cs.earlham.edu Slide 1 http://cs.earlham.edu/~jrogers/slides/UDel.ho.pdf Joint work with Jeff Heinz, U. Delaware, Geoff Pullum and Barbara Scholz, U.Edinburgh, and a raft of Earlham College undergrads. Portions of this work completed while in residence at the Radcliffe Institute for Advanced Study Yawelmani Yokuts (Kissberth’73) ⋆ CCC Σ ∗ CCC Σ ∗ V Σ ∗ C C C V C CC CCC V V Slide 2 ⋆ C 2 i +1 Contrast: Definition 1 A finite-state stringset is one in which there is an a priori bound, independent of the length of the string, on the amount of information that must be inferred in distinguishing strings in the set from those not in the set. Regular = Recognizable = Finite-State

UDel Cognitive Science 2 Cognitive Complexity of Simple Patterns Sequences of ‘ A ’s and ‘ B ’s which end in ‘ B ’: S 0 − → AS 0 , S 0 − → BS 0 , S 0 − → B B A B ( A + B ) ∗ B A Slide 3 Sequences of ‘ A ’s and ‘ B ’s which contain an odd number of ‘ B ’s: S 0 − → AS 0 , S 0 − → BS 1 , S 1 − → AS 1 , S 1 − → BS 0 , S 1 − → ε A A B ( A ∗ BA ∗ BA ∗ ) ∗ A ∗ BA ∗ B Some More Simple Patterns Sequences of ‘ A ’s and ‘ B ’s which contain at least one ‘ B ’: S 0 − → AS 0 , S 0 − → BS 1 , S 1 − → AS 1 , S 1 − → BS 1 , S 1 − → ε A, B A B A ∗ B ( A + B ) ∗ Slide 4 Sequences of ‘ A ’s and ‘ B ’s which contain exactly one ‘ B ’: S 0 − → AS 0 , S 0 − → BS 1 , S 1 − → AS 1 , S 1 − → ε A, B A A B B A ∗ BA ∗

UDel Cognitive Science 3 Dual characterizations of complexity classes Computational classes • Characterized by abstract computational mechanisms • Equivalence between mechanisms • Tools to determine structural properties of stringsets Slide 5 Descriptive classes • Characterized by the nature of information about the properties of strings that determine membership • Independent of mechanisms for recognition • Subsume wide range of types of patterns 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 6 Reasoning about patterns • What objects/entities/things are we reasoning about? • What relationships between them are we reasoning with?

UDel Cognitive Science 4 Some Assumptions about Linguistic Behaviors • Perceive/process/generate linear sequence of (sub)events • Can model as strings—linear sequence of abstract symbols Slide 7 – 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 �D , ⊳ + , P σ � σ ∈ Σ (+1) �D , ⊳, P σ � σ ∈ Σ ( < ) D — Finite ⊳ + — Linear order on D Slide 8 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 �

UDel Cognitive Science 5 Adjacency—Substrings C VCVCV Definition 2 ( k -Factor) v is a factor of w if w = uvx for some u, v ∈ Σ ∗ . v is a k -factor of w if it is a factor of w and | v | = k . Slide 9  { v ∈ Σ k | ( ∃ u, x ∈ Σ ∗ )[ w = uvx ] }  if | w | ≥ k, F k ( w ) def = { w } otherwise .  F 2 ( CV CV CV ) = { CV, V C } F 7 ( CV CV CV ) = { CV CV CV } Strictly Local Stringsets—SL Strictly k -Local Definitions —Grammar is set of permissible k -factors G ⊆ F k ( { ⋊ } · Σ ∗ · { ⋉ } ) = G def w | ⇐ ⇒ F k ( ⋊ · w · ⋉ ) ⊆ G Slide 10 L ( G ) def = { w | w | = G} e.g.: L ( G ) = CV ( CV ) ∗ C G = { ⋊ C, CV, V C, C ⋉ } , Definition 3 (Strictly Local Sets) A stringset L over Σ is Strictly Local iff there is some strictly k -local definition G over Σ (for some k ) such that L is the set of all strings that satisfy G

UDel Cognitive Science 6 SL Hierarchy Definition 4 ( SL ) A stringset is Strictly k -Local if it is definable with an SL k definition. A stringset is Strictly Local (in SL) if it is SL k for some k . Slide 11 Theorem 1 (SL-Hierarchy) SL 2 � SL 3 � · · · � SL i � SL i +1 � · · · � SL Every Finite stringset is SL k for some k : Fin ⊆ SL. There is no k for which SL k includes all Finite languages. ⋆ CCC is SL 3 G ¬ CCC = F 3 ( { ⋊ } · Σ ∗ · { ⋉ } ) −{ CCC } ⋆ Slide 12 ⋊ CV V C C V ⋉ ⋊ VCCCV ⋉ Membership in an SL k stringset depends only on the individual k -factors which occur in the string.

UDel Cognitive Science 7 Scanners a b a b a b a b a a b a b a b a b a · · · · · · b k k D Q G : · · · ∈ Slide 13 φ · · · a a · · · b b · · · a b · · · k Recognizing an SL k stringset requires only remembering the k most recently encountered symbols. Scanners as FSA CC C C V ⋊ C C V CV C C ⋊ V V C V C V ⋊ V Slide 14 C V V V V M def = � Q, Σ , q 0 , δ, F � def F k − 1 (Σ ∗ ) ∪ { ⋊ } · � 0 ≤ i<k − 1 [ { F i<k − 1 (Σ ∗ ) } ] = Q def q o = ⋊ def δ ( σ · v, γ ) = u ⇔ u = v · γ ∈ Q ∨ u = σ · v · γ = ⋊ · v · γ ∈ Q def F = Q

UDel Cognitive Science 8 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 · · ∈ L v x z Slide 15 then it will also be the case that · · ∈ L w x z E.g.: But ⋆ CCC is not SL 2 : · · ∈ ⋆ CCC · · ∈ ⋆ CCC V V C CV C C V C C · V C · V C ∈ ⋆ CCC V · C · CV ∈ ⋆ CCC V · V C · V C ∈ ⋆ CCC C · C · CV �∈ ⋆ CCC Cognitive interpretation of SL • Any cognitive mechanism that can distinguish member strings from non-members of a (properly) SL k language must be sensitive, at least, to the length k blocks of consecutive events that occur in the presentation of the string. Slide 16 • If the strings are presented as sequences of events in time, then this corresponds to being sensitive, at each point in the string, to the immediately prior sequence of k − 1 events. • Any cognitive mechanism that is sensitive only to the length k blocks of consecutive events in the presentation of a string will be able to recognise only SL k languages.

UDel Cognitive Science 9 Cambodian 0 • In words of all sizes, primary stress L falls on the final syllable. ` H • In words of all sizes, secondary stress Slide 17 2 ´ ` H H falls on all heavy syllables. • Light syllables occur only immedi- ´ H ´ L ately following heavy syllables. • Light monosyllables do not occur. 1 Cambodian 0 ´ L L ` H L Slide 18 2 ´ ` H H ⋊ ` ´ H H ´ L ´ H 1

UDel Cognitive Science 10 Cambodian—Primary stress falls on the final syllable 0 ´ L L ` H L Slide 19 2 ´ ` H H ⋊ ` ´ H H ´ L ´ H 1 Cambodian—Light syllables occur only immediately following heavy syllables 0 ´ L L ` H Slide 20 L 2 ´ ` H H ⋊ ` H ´ H ´ L ´ H 1

UDel Cognitive Science 11 Cambodian—Minimized 0 ´ L 1 L ` H L 0 Slide 21 ´ 2 ` H H ⋊ ` ´ H H ´ 2 L ´ H 1 Alawa 0 ⋊ σ σ ´ σ ⋉ σ ⋊ σ ´ ⋉ σ ´ ⋆ ⋊ σ σ ´ ⋉ 3 σ Slide 22 ⋊ σ σ σσ ⋉ ´ 1 σ ´ ⋊ ´ σ σ ⋉ σ ⋆ ⋊ σ σ ⋉ 4 G Alawa = { ⋊ σσ, ⋊ σ ´ σ, ⋊ ´ σσ, σ σσσ, σσ ´ σ, σ ´ σσ, 2 ´ σσ ⋉ , ⋊ ´ σ ⋉ }

UDel Cognitive Science 12 Alawa 0 σσ σ 3 σ σ σ ´ σ ´ ⋊ σ 3 σ 4 σ ´ σ ´ σ 0 σ Slide 23 σ ⋊ 1 ´ σ ´ σ 2 σ σσ ´ σ ⋊ ´ σ 4 1 σ σ ´ ´ σ 2 Arabic (Bani-Hassan) 0 ` σ 0 σ 0 6 σ 2 ` σ 1 ` σ 0 ` σ 0 G ArabicBH = σ 2 ´ 7 ´ σ 1 {· · · } −{ σ ´ σ 0 ⋉ | σ ∈ σ 0 , σ 1 , σ 2 } σ 2 ` σ 0 ´ σ 0 Slide 24 σ 1 ` σ 2 ´ L ArabicBH = ` σ 1 σ 1 ´ 3 σ 2 ` σ 1 ´ L {··· } ∩ L σ ´ σ 0 ⋉ σ 0 ´ σ 2 ´ 4 1 σ 0 σ 0 σ 1 σ 1 8 5 σ 0 σ 0 σ 1 σ 1 2

UDel Cognitive Science 13 Arabic (Classical) 0 σ 1 σ 2 σ 0 σ 2 k − 1 σ 1 3 σ 0 � �� � σ 0 · · · σ 0 ´ ⋊ σ 1 σ 2 ⋉ ´ σ 1 σ 0 ´ σ 1 ´ k − 1 σ 2 ´ σ 2 ´ � �� � Slide 25 ⋊ ´ σ 2 σ 0 · · · σ 0 σ 1 ⋉ 4 k − 1 � �� � ⋆ ⋊ σ 1 σ 0 · · · σ 0 σ 1 ⋉ σ 0 σ 0 1 σ 1 σ 1 2 Strictly Local Stress Patterns Heinz’s Stress Pattern Database (ca. 2007)—109 patterns 9 are SL 2 Abun West, Afrikans, . . . Cambodian,. . . Maranungku 44 are SL 3 Alawa, Arabic (Bani-Hassan),. . . Slide 26 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), Ara- bic Classical, Hindi (Keldar), Yidin,. . . 72% are SL, all k ≤ 6. 49% are SL 3 .