Potential Distinguishing Characteristics of Human Aural Pattern - PDF document

1 RecHul’07 Potential Distinguishing Characteristics of Human Aural Pattern Recognition James Rogers † and Marc D. Hauser ‡ † Dept. of Computer Science Earlham College Slide 1 this work completed, in part, while at the Radcliffe Institute for Advanced Study ‡ Depts. of Psychology, Oganismic & Evolutionary Biology and Biological Anthropolgy Harvard University We hypothesize that FLN only includes recursion and is the only uniquely human component of the Slide 2 faculty of language. Hauser, Chomsky and Fitch, Nature , v. 298, 2002.

2 RecHul’07 The Comparative Approach to Language Evolution • Shared vs. unique Slide 3 – Homologous vs. analogous • Gradual vs. saltational • Continuity vs. exaption Three Hypotheses 1. FLB is strictly homologous to animal communication Slide 4 2. FLB is a derived, uniquely human adaptation for language 3. Only FLN is uniquely human

3 RecHul’07 Empirical support for the comparative method • Across species Slide 5 • Domains other than (just) communication • Spontaneous and trained behaviors Contrasting ( AB ) n with A n B n • Finite State vs. Context-Free • { ( ding dong ) n } vs. { people n left n } Slide 6 • vs. { those people who were left ( by people who were left ) n left } • vs. { those people who were left ( by people who were left ) 2 n left }

4 RecHul’07 Dual Characterizations of Classes of Patterns Descriptive characterizations – Nature of the information about strings – Independent of mechanism – Support conclusions about abstract properties of Slide 7 mechanisms Grammar- and automata-theoretic characterizations – Concrete algorithm s – Support reasoning about the structure of stringsets – Guide experimental design Strictly Local Stringsets 2-factors: G ( AB ) n = { ⋊ A, AB, BA, B ⋉ } ⋊ A B ABAB ABBAB ⋉ ⋊ ⋉ Strictly k -Local Definitions Slide 8 G ⊆ F k ( { ⋊ } · Σ ∗ · { ⋉ } ) = G def w | ⇐ ⇒ F k ( ⋊ · w · ⋉ ) ⊆ G L ( G ) def = { w | w | = G} Membership in an SL k stringset depends only on the individual k -factors which do and do not occur in the string.

5 RecHul’07 Scanners a b a b a b a b a a b a b a b a b a · · · · · · b k k Q D G : · · · ∈ Slide 9 φ · · · a a · · · b b · · · a · · · b k Recognizing an SL k stringset requires only remembering the k most recently encountered symbols. Character of Strictly 2-Local Sets Theorem (Suffix Substitution Closure): A stringset L is strictly 2-local iff whenever there is a word x and strings w , y , v , and z , such that · · ∈ L w x y · · ∈ L v x z Slide 10 then it will also be the case that w · x · z ∈ L Example: The dog · likes · the biscuit ∈ L · · ∈ L Alice likes Bob · · ∈ L The dog likes Bob

6 RecHul’07 Probing the SL Boundary Some-B def = { w ∈ { A, B } ∗ | | w | B ≥ 1 } A . . . A · A . . . A · BA . . . A ∈ Some-B � �� � k − 1 · · ∈ Some-B A . . . AB A . . . A A . . . A � �� � Slide 11 k − 1 A . . . A · A . . . A · A . . . A �∈ Some-B � �� � k − 1 In Out ( AB ) n ( AB ) i + j +1 ( AB ) i AA ( AB ) j SL A m B n A i + k B j + l A i B j A k B l A i BA j A i + j +1 non-SL Some-B Locally k -Testable Stringsets Some-B: ¬ ( ¬ ⋊ B ∧ ¬ AB ) (= ⋊ B ∨ AB ) k -Expressions def f ∈ F k ( ⋊ · Σ ∗ ⋉ ) w | ⇐ ⇒ f ∈ F k ( ⋊ · w · ⋉ ) = f def ϕ ∧ ψ w | = ϕ ∧ ψ ⇐ ⇒ w | = ϕ and w | = ψ Slide 12 def ¬ ϕ w | = ¬ ϕ ⇐ ⇒ w �| = ϕ Locally k -Testable Languages (LT k ): L ( ϕ ) def = { w | w | = ϕ } Membership in an LT k stringset depends only on the set of k -Factors which occur in the string.

7 RecHul’07 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 a a Slide 13 Boolean a b Network b a b b a b φ Recognizing an LT k stringset requires only remembering which k -factors occur in the string. Character of Locally Testable Sets Theorem ( k -Test Invariance): A stringset L is Locally Testable iff Slide 14 there is some k such that, for all strings x and y , if ⋊ · x · ⋉ and ⋊ · y · ⋉ have exactly the same set of k -factors then either both x and y are members of L or neither is.

8 RecHul’07 Probing the LT Boundary Some-B = { w ∈ { A, B } ∗ | w | = ⋊ B ∨ AB } ( ∈ LT 2 ) One-B def = { w ∈ { A, B } ∗ | | w | B = 1 } �∈ LT A k BA k ∈ One-B A k BA k BA k �∈ One-B Slide 15 F k ( ⋊ A k BA k ⋉ ) = F k ( ⋊ A k BA k BA k ⋉ ) In Out A i BA j A i + j +1 LT Some-B A i BA j + k +1 A i BA j BA k non-LT One-B FO( +1 ) (Strings) | = ( ∀ x )[ A ( x ) ∨ B ( x )] ∧ ( ∃ x )[ B ( x )] AABA �D , ⊳, P σ � σ ∈ Σ D E AABA = { 0 , 1 , 2 , 3 } , {� i, i + 1 � | 0 ≤ i < 3 } , { 0 , 1 , 3 } A , { 2 } B First-order Quantification (over positions in the strings) def x ⊳ y w, [ x �→ i, y �→ j ] | = x ⊳ y ⇐ ⇒ j = i + 1 Slide 16 def P σ ( x ) w, [ x �→ i ] | = P σ ( x ) ⇐ ⇒ i ∈ P σ . . ϕ ∧ ψ . . . ¬ ϕ . def ( ∃ x )[ ϕ ( x )] w, s | = ( ∃ x )[ ϕ ( x )] ⇐ ⇒ w, s [ x �→ i ] | = ϕ ( x )] for some i ∈ D FO(+1)-Definable Stringsets: L ( ϕ ) def = { w | w | = ϕ } .

9 RecHul’07 Character of the FO( +1 ) Definable Stringsets Definition 1 (Locally Threshold Testable) A set L is Locally Threshold Testable (LTT) iff there is some k and t such that, for all w, v ∈ Σ ∗ : if for all f ∈ F k ( ⋊ · w · ⋉ ) ∪ F k ( ⋊ · v · ⋉ ) either | w | f = | v | f or both | w | f ≥ t and | v | f ≥ t , Slide 17 then w ∈ L ⇐ ⇒ v ∈ L . Theorem 1 (Thomas) A set of strings is First-order definable over �D , ⊳, P σ � σ ∈ Σ iff it is Locally Threshold Testable . Membership in an FO(+1) definable stringset depends only on the multiplicity of the k -factors, up to some fixed finite threshold, which occur in the string. Probing the LTT Boundary One-B = { w ∈ { A, B } ∗ | w | = ( ∃ x )[ B ( x ) ∧ ( ∀ y )[ B ( y ) → x ≈ y ] ] } ( ∈ LTT) B-before-C def = { w ∈ { A, B, C } ∗ | at least one B precedes any C } �∈ LTT A k BA k CA k and A k CA k BA k have exactly the same number of Slide 18 occurrences of every k -factor. In Out A i BA j + k +1 A i BA j BA k LTT One-B A i BA j CA k A i CA j BA k non-LTT B-before-C

10 RecHul’07 FO( < ) (Strings) = ( ∃ x )[ C ( x ) → ( ∃ y )[ B ( y ) ∧ y ⊳ + x ] ] ABACA | �D , ⊳, ⊳ + , P σ � σ ∈ Σ First-order Quantification over positions in the strings def x ⊳ y w, [ x �→ i, y �→ j ] | = x ⊳ y ⇐ ⇒ j = i + 1 def x ⊳ + y = x ⊳ + y Slide 19 w, [ x �→ i, y �→ j ] | ⇐ ⇒ i < j def P σ ( x ) w, [ x �→ i ] | = P σ ( x ) ⇐ ⇒ i ∈ P σ . . ϕ ∧ ψ . . . ¬ ϕ . def ( ∃ x )[ ϕ ( x )] w, s | = ( ∃ x )[ ϕ ( x )] ⇐ ⇒ w, s [ x �→ i ] | = ϕ ( x )] for some i ∈ D Locally Testable with Order (LTO k ) LT k plus = ϕ • ψ def ϕ • ψ w | ⇐ ⇒ w = w 1 · w 2 , w 1 | = ϕ and w 2 | = ψ. B-before-C: ( ⋊ B ∨ AB • ⋊ C ∨ AC ) ∨ ¬ ( ⋊ C ∨ AC ∨ BC ) Definition 2 (Star-Free Set) The class of Star-Free Sets (SF) is the smallest class of languages satisfying: Slide 20 • ∅ ∈ SF, { ε } ∈ SF, and { σ } ∈ SF for each σ ∈ Σ . • If L 1 , L 2 ∈ SF then: L 1 · L 2 ∈ SF , L 1 ∪ L 2 ∈ SF , L 1 ∈ SF . Theorem 2 (McNauthton and Papert) A set of strings is First-order definable over �D , ⊳, ⊳ + , P σ � σ ∈ Σ iff it is Star-Free .

11 RecHul’07 Character of FO( < ) Definable Sets Theorem (McNaughton and Papert): A stringset L is definable by a set of First-Order formulae over strings iff it is recognized by a finite-state automaton that is non-counting (that has an aperiodic syntactic monoid), that is, iff: there exists some n > 0 such that Slide 21 for all strings u, v, w over Σ if uv n w occurs in L then uv n + i w , for all i ≥ 1, occurs in L as well. E.g. those people who were left ( by people who were left ) n left ∈ L those people who were left ( by people who were left ) n +1 left ∈ L A Characterization via ANNs Binary valued Artificial Neural Nets Buzzer-free: no inhibitory feedback. Almost loop-free: no loops including more than one neuron or delay greater than one. Slide 22 1 Theorem (McNaughton and Papert): A stringset L is definable by a set of First-Order formulae over strings iff it is representable by a buzzer-free, almost loop-free ANN.

12 RecHul’07 Probing the LT0 Boundary B-before-C = { w ∈ { A, B } ∗ | w | = ( ∃ x )[ C ( x ) → ( ∃ y )[ B ( y ) ∧ y < x ] ] } ( ∈ LTO) Even-B def = { w ∈ { A, B } ∗ | | w | B = 2 i, 0 ≤ i } �∈ LTT Slide 23 A i B n B n ∈ Even-B A i B n +1 B n �∈ Even-B but In Out A i BA j CA k A i CA j BA k LTO B-before-C B 2 i B 2 i +1 non-LTO Even-B MSO (Strings) �D , ⊳, ⊳ + , P σ � σ ∈ Σ Slide 24 First-order Quantification (positions) Monadic Second-order Quantification (sets of positions) ⊳ + is MSO-definable from ⊳ .