formal typology: explanation in optimality theory paul smolensky cognitive science department johns...
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Formal Typology: Explanation in Optimality
TheoryPaul Smolensky
Cognitive Science Department Johns Hopkins University
Géraldine LegendreDonald Mathis
Melanie Soderstrom
Alan PrinceSuzanne Stevenson
Peter Jusczyk†
with:
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Blackwell 2002 (??) Develop the Integrated
Connectionist/Symbolic (ICS) Cognitive Architecture
Apply to the theory of grammar
The Harmonic Mind:
From neural computation to optimality-theoretic grammar
Paul Smolensky & Géraldine Legendre
Chomsky 1988
“1. What is the system of knowledge? 2. How does this system of
knowledge arise in the mind/brain? 3. How is this knowledge put to use? 4. What are the physical mechanisms
that serve as the material basis for this system of knowledge and for the use of this knowledge?” (p. 3)
Responsibilities of Grammatical Theory
Chomsky’s “Big 4” questions concerning
knowledge of grammar
Structure
Acquisition
Processing
Neuro-genetics
Nativist hypothesis
OT①
①②
③
④
Not new to Chomsky or generative grammar …
Jakobson’s Program
Linguistic theory is not just for theoretical linguists
The same principles that explain formal cross-linguistic and language-internal distributional patterns can also explain• Acquisition• Processing• Neurological breakdown
Jakobson’s Program
Markedness enables a Grand Unified Theory for the cognitive science of language: Avoid α①Structure
Inventories lack αAlternations eliminate α
②Acquisitionα is acquired late
③Processingα is processed poorly
④NeuralBrain damage most easily disrupts α
Talk Plan
① Structure
② Acquisition
③ Processing
④ Neuro- genetics
OT Explanation
Formal re
sult(
s)
Jakobso
n’s program
Question
Achieves g
oal
Empirical in
sights
Responsibilities of Grammatical Theory
Chomsky’s “Big 4” questions concerning
knowledge of grammarStructure
Structure of UG: Captured in a general formalism for grammars and their variation
OT①
⇒Possible strong version – Explanatory Goal ①:
Analysis of phenomenon Φ in
language L
Universal typology of phenomenon Φ
①
Inherent typology
Acquisition
Processing
Neuro-genetics
From Markedness to OT
Formalizing markedness ⋯ OT• Markedness constraints• Faithfulness constraints• Competition• Strict domination• Strong universality & Richness of the
Base
Structure: Formal Result
Formalizing Markedness: Two Problems
Goal: Change epiphenomenal explanatory status of markedness • Markedness “explains grammars (e.g.,
rules)”; informal commentary about grammar vs.
• Markedness IS grammar: markedness-grammars formally determine languages
Structure: Formal Result
Formalizing Markedness: Two Problems
Problem 1: Multidimensional integration
Each dimension of linguistic structure independently has its own marked pole, but how do these dimensions combine?
Turns out to be related to another fundamental problem:
Structure: Formal Result
Formalizing Markedness: Two Problems
“α is marked” ⇝ “Avoid α” But when & how does “avoidance” happen?
Problem 2: Pervasive variability in “avoidance”• Inventories: If [θ] is absent in French “because it is
marked” how can it be present in English “despite being marked”?
¿The grammar of every language turns on or off: “No α ” = *α — a markedness constraint. OT: More subtle version that also solves:
• Alternations: If in environment E, α β “because α is more marked than β”, how do we explain that in E α ̷ β “even though” α is more marked than β?
Structure: Formal Result
Formalizing Markedness Most crudely: Why aren’t unmarked
elements always avoided? Something must oppose markedness
forces. Markedness cannot be the sole basis
of a formal grammatical theory: it is only one half of the complete story.
Structure: Formal Result
The Great DialecticPhonological representations serve two masters
Phonological Representation Lexico
nPhoneti
cs
Phonetic interface
[surface form]
Often: ‘minimize effort (motoric & cognitive)’;
‘maximize discriminability’
Locked in eternal conflict
Lexical interface
/underlying form/
‘be this invariant form’
FAITHFULNESSMARKEDNESS
Structure: Formal Result
The Core Constraints of Con MARKEDNESS: *α (“minimize effort; maximize distinctiveness”)
• “constraint *α Con” α meets empirical criteria for ‘marked’
• Freedom? Empirically constrained by universal patterns FAITHFULNESS (“be this invariant form”):
• /input/ [output] is the identity map, i.e., • elements /x/ and [x] are in one-to-one correspondence
and identical (McCarthy & Prince ’95)
• Constraints: MAX(x), DEP(x), IDENT(x), …• Essentially determined by elements {x} of
representation• Freedom? Representations — as always: empirically
constrained to allow statement of markedness constraints
¿ “In OT you can invent any constraint you want” ?
Structure: Formal Result
Conflict Dialectic: MARK vs. FAITH conflict
• Why aren’t marked elements always avoided? Because sometimes MARK is over-ruled by FAITH
• Why aren’t words always pronounced in their invariant, lexical form?Because sometimes FAITH is over-ruled by MARK
1 over-rules (dominates) 2: 1 ≫ 2
Whether M gets violated (whether marked elements fail to ‘be avoided’) varies by • Language (in some, M ≫ F; in others, F ≫ M)• Context (in some, M ≫ F2; in others F1 ≫ M)
Structure: Formal Result
Conflict Dialectic: MARK vs. FAITH conflict Whether M gets violated (whether marked
elements fail to ‘be avoided’) varies by • Language (in some, M ≫ F; in others, F ≫ M)• Context (in some, M ≫ F2; in others F1 ≫ M)
Why is there cross-linguistic variation?• Phonetic Lexical ~ MARK FAITH Dialectic
gets resolved differently • Typology by re-ranking: Factorial Typology
{possible human languages} {rankings of Con}
(n constraints give n! rankings — many are equivalent)
Structure: Formal Result
Formalizing Markedness
Problem 1: ‘Avoidance of the marked’ is pervasively variable; exactly where does marked material appear?• Solution: Constraint ranking
— MARK w.r.t. FAITH
Will now see this also solves: Problem 2: Multidimensional markedness
• Solution: single constraint ranking for all constraints in a given language
Structure: Formal Result
Formalizing Markedness Markedness is multidimensional
• Each dimension has its universally marked pole• How do dimensions combine? (M1, *M2) vs. (*M1, M2)
CV ́C.CV (STRESSHEAVY, *MAINSTRESSRIGHT) vs. CVC.CV ́• Integrate via a common markedness currency:
Harmony Numerical: *M1 = 3.2; *M2 = 2.8 Symbolic: *M1 absolutely worse than *M2
see below
OT:•For a given language, there is a single constraint ranking for all constraints•Strict domination hierarchy: markedness on higher-ranked constraints can never be compensated for by unmarkedness on lower-ranked ones
Structure: Formal Result
Competition for Optimality Given an input, an OT grammar does not
provide a procedure for how to construct the output — bur rather a description of the output: the structure that best-satisfies the constraint ranking
Best-satisfies is a comparative criterion; outputs compete and the grammar identifies the winner: the optimal — grammatical — highest Harmony — output for that input
Structure: Formal Result
Harmonic Competition Numerical Harmony
Stress is on the initial heavy syllable iff the number of light syllables n obeys
Pathological grammars
“Grammars can’t count”
TRESS EAVY
AIN TRESS IGHT
S H
M S Rany number
wn
w
Candidates STRESSHEAVY MAINSTRESSRIGHT Harmony
a. σHσ…σσ n
**…* n
n(wMAINSTRESSRIGHT)
b. σHσ…σσ n
* wSTRESSHEAVY
´
´
Candidates STRESSHEAVY MAINSTRESSRIGHT
a. σHσ…σσ
n
**…* n
b. σHσ…σσ n
*!
´
´
Structure: Formal Result
Harmonic Competition Symbolic Harmony: Strict domination
• STRESSHEAVY ≫ MAINSTRESSRIGHT
Stress the initial heavy syllable
Stress the final syllable
• MAINSTRESSRIGHT ≫ STRESSHEAVY
´
Candidates MAINSTRESSRIGHT STRESSHEAVY
a. σHσ…σσ
n
**…*! n
b. σHσ…σσ n
*
´
´ Strict domination “Grammars can’t count”
Structure: Formal Result
OT: ‘Formal’ definition Gen: Specifies candidate outputs for any
given input Con: The constraint set A grammar: A hierarchical ranking of Con H-Eval: Given two candidates and a
ranking, a formal definition employing strict domination of which has higher Harmony — which better-satisfies the ranking
I O mapping: I The maximal-Harmony candidate[s] in Gen(I)
Structure: Formal Result
Richness of the Base Universality: All systematic cross-linguistic
variation arises from differences in constraint ranking
Therefore: • Con is universal; H-Eval is universal• Gen is universal, including the space of possible inputs
as well as possible outputs i.e.: No systematic cross-linguistic variation is due to
differences in inputs e.g.: Languages with no surface codas cannot get this property
from limitations on the lexicon (e.g., a morpheme structure constraint *Cwd]) — but rather from the ranking
i.e.: The grammar must have the property that even if there were C-final inputs, there would still be no surface codas
Aside
Richness of the Base is a principle for inducing a grammar (generalizing) from a set of grammatical items
It can be justified by the central principle of John Goldsmith’s presentation: Maximize the probability of the data
Structure: Conceptual “Question”
Explanatory Power“OT is as unexplanatory as extrinsically-
ordered rule-theory”Stipulating ranking ~ stipulating ordering
Actually, OT achieves Explanatory Goal ①, Inherent Typology: In the analysis of phenomenon Φ in one language is inherent a typology of Φ in all languages
Structure: Explanatory Goal
Inherent Typology
Structure: Conceptual “Question”
Analytic Restrictiveness“You can make up any constraint you want in OT ”
Actually, in OT, positing in the analysis of a language L necessarily has a huge number of empirically falsifiable implications (one consequence of Inherent Typology)
E.g., Two pervasive patterns generated by ‘ Con’
Structure: Explanatory Goal
Robust Falsifiability
Structure: Explanatory Goal
Consequences of ‘ Con’ – I: The Subordination
Pattern E.g., = NOCODA
Recall: • If ‘No codas’ is in UG, why do codas ever appear?• Conflict
With faithfulness constraintsWith other markedness constraints – other dimensions
of markedness
Cross-linguistic variation: codas are less and less restricted as NOCODA is subordinated to more and more conflicting constraints (i.e., dimensions of markedness)
Structure: Empirical Application
Subordination Pattern: Codas
No codas at all
Codas only in stressed syllables
… + Geminate codas
Codas unrestricted …except prohibited inter-vocalically [~V.CV~]
STRESS-TO-WEIGHT
MAXμ
MAX
NOCODA
Structure: Conceptual “Question”
Multiplicity of ConstraintsFor second pervasive pattern generated by ‘ Con’:
“Any framework which leads to the morass of constraints found in OT analyses in phonology cannot possibly be explanatorily adequate.”
Actually, OT interaction-via-domination replaces many rules by fewer constraints
Structure: Explanatory Goal
Factorial Interaction
Structure: Explanatory Goal
Consequences of ‘ Con’ – II: Factorial Interaction
‘Factorial interaction’: with varying interaction (re-ranking), n simple modular constraints correspond to• Multiplicity of rules (many more than n) • Complex, non-modular rules• Rules + representational/notational
tricks• Rules + constraints
E.g., = NOCODA
Structure: Empirical Application
Factorial Interaction: Codas
Consider Con {MAX} ↪ {MAX, DEP} Number of constraints increases by 1 Number of corresponding rules
doubles as set of ‘repairs’ now includes epenthesis as well as deletion:NOCODA ≫ MAX ~ CØ/—σ]
↪ NOCODA ≫ DEP ~ Ø V/Cσ]—
ONSET ≫ MAX ~ VØ/[σ—
↪ ONSET ≫ DEP ~ Ø C/[σ—V
Structure: Empirical Application
Factorial Interaction: Codas
MARKEDNESS ≫ FAITHFULNESS
MARKEDNESS
NOCODA ONSET
FAITHFUL-NESS
MAX CØ/—σ] VØ/[σ—
DEP Ø V/Cσ]— Ø C/[σ—V
In general, the number of comparable rules increases much faster than the number of constraints
Structure: Explanatory Goal
Consequences of ‘ Con’ – II: Factorial Interaction
‘Factorial interaction’: with varying interaction (re-ranking), n simple modular constraints correspond to• Multiplicity of rules (many more than n) • Complex, non-modular rules• Rules + representational/notational
tricks• Rules + constraints
E.g., = NOCODA
Structure: Empirical Application
Factorial Interaction: Codas
STRESS-TO-WEIGHT ≫ NOCODA • Codas only in stressed syllables• CØ/—σ̆] segmental rule sensitive to foot structure
[‘non-modular rules’] ANCHOR-R ≫ NOCODA
• Codas only word-finally• CØ/—σ] plus final-C extrametricality
[‘representational trick’] MAXμ ≫ NOCODA
• Only geminate codas — /Cμ/• CØ/—σ] plus Hayes’ exclusivity of association
[‘notational trick’]
Structure: Empirical Application
Factorial Interaction STRESS-TO-WEIGHT ≫ NOCODA Codas only in stressed
syllables• STRESS-TO-WEIGHT ≫ *Cμ • Geminates only after stressed V• μØ/—σ̆]
ANCHOR-R ≫ NOCODA Codas only word-finally• ANCHOR-R ≫ *[+voi,son] • Obstruent devoicing except word-finally• [+voi][voi]/[—, son] plus ?? to block word-finally
MAXμ ≫ NOCODA Only geminate codas; /C μ/
• MAXμ ≫ WEIGHT-TO-STRESS
• Geminates are the only codas in unstressed syllables• CØ/—σ̆] plus exclusivity of association
Structure: Jakobson’s ProgramMarkedness + Faithfulness =
HarmonyIn summary: Jakobson’s key insight concerning linguistic
structure: the central organizing principle of grammar is: Minimize Markedness
OT formalizes this as Maximize Harmony OT formalizes Markedness via violable constraints OT adds the crucial notion of Faithfulness – the
other (lexical) half of the phonological dialectic OT Harmony combines Markedness with
Faithfulness; their conflict is adjudicated via ranking
Ranking unifies multiple dimensions of markedness
Structure: Summary
OT achieves the explanatory goals of• Changing the epiphenomenal status
of markedness in grammatical theory: markedness is now in grammar, not about grammar
• A strongly universalist formalism exhibiting Inherent Typology
• Robust falsifiability
Responsibilities of Grammatical Theory
Chomsky’s “Big 4” questions concerning
knowledge of grammarStructure
Acquisition
Processing
Neuro-genetics
Nativist hypothesis
OT①
①②
Possible strong version – Explanatory Goal ②:
⇒②
General Learning Theory
Substantive structure (①) of a UG module
governing phenomenon Φ
Acquisition theory — initial state, learning
algorithm — for phenomenon Φ
Acquisition: Formal Result I
Learning Theory Learning algorithm
• Provably correct and efficient (when part of a general decomposition of the grammar learning problem)
• Sources:Tesar 1995 et seq. Tesar & Smolensky 1993, …, 2000** See for how to exploit the analogy to
‘weighted OT’ (Goldsmith, today)
• If you hear A when you expected to hear E, increase the Harmony of A above that of E by minimally demoting each constraint violated by A below a constraint violated by E
in +possible
Candidates
FaithMark (NPA)
☹ ☞ Einpossibl
e *
A impossibl
e *
Faith
*☺ ☞
If you hear A when you expected to hear E, increase the Harmony of A above that of E by minimally demoting each constraint violated by A below a constraint violated by E
Correctly handles difficult case: multiple violations in E
Acquisition: Formal Result I
Constraint Demotion Algorithm
Acquisition: Conceptual “Question”
Large Grammar Space “Huge number of grammars” —
“OT is too unrestrictive”
Acquisition: Explanatory Goal
General Learning Theory Actually, OT achieves Explanatory Goal
②: General Learning Theory: A theory-general, UG-informed learning algorithm, provably correct and efficient (under strong assumptions)
Acquisition: Formal Result II
Learnability & the Initial State
M ≫ F is learnable with /in+possible/→impossible• ‘not’ = in- except when followed by …• “exception that proves the rule”: M = NPA
M ≫ F is not learnable from data if there are no ‘exceptions’ (alternations) of this sort, e.g., if no affixes and all underlying morphemes have mp: M and F, no M vs. F conflict, no evidence for their ranking
Thus must have M ≫ F in the initial state, ℌ0
Acquisition: Empirical Application
Initial State: Experimental Test Collaborators
Peter Jusczyk Theresa Allocco (Elliott Moreton, Karen Arnold)
Here, only a thumbnail sketch (more in the OT Workshop Thursday)
Acquisition: Empirical Application
Initial State: Experimental Test Linking hypothesis:
More harmonic phonological stimuli ⇒ Longer listening time
More harmonic: M ≻ *M, when equal on F F ≻ *F, when equal on M• When must chose one or the other,
more harmonic to satisfy M: M ≫ F M = Nasal Place Assimilation (NPA)
15.36
12.31
0
2
4
6
8
10
12
14
16
18
20
Faithfulness Markedness M ≫ F
Tim
e (s
ec)
Higher HLower H
4.5 Months (NPA)Higher
HarmonyLower Harmony
um…ber…umber
um…ber… iŋgu
p = .006 (11/16)
Acquisition: Empirical
Application
15.2315.36
12.7312.31
0
2
4
6
8
10
12
14
16
18
20
Faithfulness Markedness M ≫ F
Tim
e (s
ec)
Higher HLower H
Higher Harmony
Lower Harmony
um…ber…umber
un…ber…unber
p = .044 (11/16)
4.5 Months (NPA) Acquisition:
Empirical Application
15.2315.36
12.7312.31
0
2
4
6
8
10
12
14
16
18
20
Faithfulness Markedness M ≫ F
Tim
e (s
ec)
Higher HLower H
4.5 Months (NPA) Markedness * Faithfulness
* Markedness Faithfulness
un…ber…umber
un…ber…unber
???
Acquisition: Empirical
Application
16.75
15.2315.3614.01
12.7312.31
0
2
4
6
8
10
12
14
16
18
20
Faithfulness Markedness M ≫ F
Tim
e (s
ec)
Higher HLower H
4.5 Months (NPA)Higher
HarmonyLower Harmony
un…ber…umber
un…ber…unber
p = .001 (12/16)
Acquisition: Empirical
Application
Acquisition: Jakobson’s ProgramMarkedness = Distance from Initial
State X is universally more marked than Y ~ In addition to the constraints M1, M2, …, Mk
violated by Y, X also violates markedness constraints M1, M2, …, Mn
Y will be acquired – become admitted into the child’s inventory – after M1, M2, … Mn are all demoted below relevant faithfulness constraints
These demotions are all necessary for X to be acquired, and additional demotions of M1, M2, …, Mn are also required ~
X will require more time to be acquired
Responsibilities of Grammatical Theory
Chomsky’s “Big 4” questions concerning
knowledge of grammarStructure
Acquisition
Processing
Neuro-genetics
Nativist hypothesis
OT①
①②
③
Possible strong version – Explanatory Goal ③ :
⇒③
General Processing Theory
Substantive structure (①) of a UG module
governing phenomenon Φ
Processing theory — e.g., parsing algorithm — for
phenomenon Φ
Processing: Formal Results
Context-Free Parsing Algorithm
Theorem (Tesar 1994, 1995b, a, 1996). Suppose• Gen parses a string of input symbols into structures
specified via a context-free grammar• Con constraints meet a tree-locality condition and
penalize empty structure
Then a given dynamic programming algorithm is• Left-to-right • General (any such Gen, Con)• Guaranteed to find the optimal outputs• As efficient as parsers for conventional context-free
grammars.
Processing: Formal Results
Finite-State Parsing Algorithm
Theorem (Ellison 1994). Suppose• Gen(I) is representable as a (non-deterministic)
finite-state transducer (particular to I) mapping the input string to a set of output candidates
• Con constraints are reducible to multiply-violable binary constraints each representable as a finite-state transducer mapping an output candidate to a sequence of violation marks
Then composing the Gen(I) and rank-sequenced constraint-transducers yields a transducer that • Directly maps I to its optimal outputs • Can be efficiently pruned by dynamic
programming
Processing: Formal Results
Complexity of Violable Constraints
Theorem (Frank and Satta 1998). Suppose• Gen is representable as a (non-deterministic) finite-
state transducer mapping an input string to a set of output candidates
• Con: the set of structures incurring n violations of each constraint is generable by a finite-state machine, and n can be finitely bounded for each constraint
Then the mapping from inputs to optimal outputs has the complexity of a finite-state transducer.
Theorem (Hiller 1996, Smolensky 1997). If n is unbounded there are (extremely simple) OT
grammars with greater computational complexity.
Processing: Conceptual “Question”
Processing (Symbolic): Theory “Infinite candidate set uncomputable”
Actually, achieves Explanatory Goal ③ (computational)
Processing: Conceptual “Question”
Processing (Symbolic): Theory
⇒③
General Processing Theory
Substantive structure (①) of a UG module
governing phenomenon Φ
Processing theory — e.g., parsing algorithm — for
phenomenon Φ
Processing: Empirical Application
Sentence Processing Because an OT grammar assigns a
parse to any input, no additional principles (e.g., ‘parsing heuristics’) are needed for parsing the initial, incomplete segment of a sentence
Linking hypothesis:Processing difficulty arises when previously established structure needs to be abandoned in the face of further input
Processing: Empirical Application
PP AttachmentThe servant of the actress who… (Cuetos & Mitchell 88)
[Assuming who is ambiguous for Case.]
Violates: *NOM, LOCALITY2
Violates: *NOM, AGRCASE
Violates: *GEN
who [+nom]
NP PP
P NP
NP
of theactress [+gen]
theservant
who [+nom]
who [+gen]
• LOCALITY: If XP c-commands YP, then XP precedes YP.• AGRCASE: A relative pronoun must agree in Case with the modified NP.• *CASE: *GEN ≫ *DAT ≫ *ACC ≫ *NOM (universal)
Processing: Empirical Application
PP AttachmentThe servant of the actress who… (Cuetos & Mitchell 88)
• If *GEN, AGRCASE ≫ LOCALITY2, then : attach high• If LOCALITY2 ≫ *GEN or AGRCASE, then or : attach low
NP PP
P NP
NPwho [+nom]
who [+nom]
who [+gen]
Violates: *NOM, LOCALITY2
Violates: *NOM, AGRCASE
Violates: *GENof theactress
[+gen]
theservant
Processing: Empirical Application
PP Attachment Preliminary result: A cross-linguistic
typology of PP attachment patterns (across differences in case and embedding depth)
Empirically promising, but not perfect Unclear yet how rankings determining
parsing preferences relate to rankings in the pure ‘competence grammar’
Processing: Jakobson’s Program
Processing and Markedness
Phonological analogy: Incrementally parse C…V…C…• /C/ [C]�• /CV/ [CV]• /CVC/ [CV][C]�
Now ‘expect’ a V … if get it, no ‘reanalysis’• But if get a C, need reanalysis difficulty:• /CVCC/ [CVC][C]�
Processing marked material (coda C) creates difficulty because it is initially analyzed as unmarked (as an onset)
Processing: Conceptual “Question”
Processing (Symbolic): Theory “OT not psychologically plausible”
Actually, achieves Explanatory Goal ③ (empirical perspective): a competence theory automatically entails an empirically fruitful performance (processing) theory
Processing: Conceptual “Question”
Processing (Symbolic): Theory
Responsibilities of Grammatical Theory
Chomsky’s “Big 4” questions concerning
knowledge of grammarStructure
Acquisition
Processing
Neuro-genetics
Nativist hypothesis
OT①
①②
③
④
Possible strong version –Explanatory Goal ④:
⇒④
General Biological Realization
Substantive structure (①) of a UG module M
Neural network instantiating M (nativism:
with genetic encoding)
Neuro-genetics: Formal Results
Neural Representations (Gen)
Activation patterns: cat and its constituents
-1 4 9 14
Unit (Area = activation level)
k/r0
æ/r01
t/r11
σ/rε
[σ k [æ t]]
σ
ktæ
{ / }i i if r i ii f r
OT & Connectionism
OT derives from the numerical formalism, derived from connectionist Harmony maximization, of• Harmonic Grammar (Legendre,
Miyata, & Smolensky, 1990)
Neuro-genetics: Formal Results
Neural Constraints (Con)NOCODA: A syllable has no codaσ
ktæ
* violation
W
* H(a[σ k [æ t]) =
–sNOCODA < 0
a[σ k [æ t ]] *
* violation
Neuro-genetics: Formal Results
UGenome for CV Theory The game: take a first shot at a concrete
example of a genetic encoding of UG in a Language Acquisition Device
¿ Proteins ⇝ Universal grammatical principles ?
Case study: Basic CV Syllable Theory Introduce an ‘abstract genome’ notion
parallel to (and encoding) ‘abstract neural network’
Collaborators• Melanie Soderstrom• Donald Mathis
Neuro-genetics: Formal Results
PARSE
C
V
3 3
3
3
33
1
11
1
1
1
3 3
3
3
33
3 3
3
3
33
All connection coefficients are +2
Neuro-genetics: Formal Results
Constraint: PARSE
CV
3 33
3
33
111
11
1
3 33
3
33
3 33
3
33
Input units grow south and connect Output units grow east and connect Correspondence units grow north & west
and connect with input & output units.
Neuro-genetics: Formal Results
Connectivity Genome Contributions from ONSET and PARSE:
Source:
CI VI CO VO CC VC xo
Projec-tions:
S LCC S L VC E L CC E L VC
N&S S VO
N S x0
N L CI
W L CO
N L VI
W L VO
S S VO
Key: Direction Extent Target
N(orth) S(outh)E(ast) W(est)F(ront) B(ack)
L(ong) S(hort)
Input: CI VI
Output: CO VO x(0)
Corr: VC CC
Φ
Ψ
Neuro-genetics: Formal Results
Processing
11 0R c
[P1] ∝ s1
1 1 11 1w [ ]P R s c
W = wii
22 0R c
Φ
Ψ
Neuro-genetics: Formal Results
Learning
2 22 2 2[ ]P K L G c
1 11 1 1
When and are simultaneously active,
[ ] is P K L G c
1 11L G c
11 1K L c
1 1[ ]P K
(during phase P+; reverse during P )
Neuro-genetics: Formal Results
Learning Behavior A simplified system can be solved
analytically Learning algorithm turns out to ≈
si() = [# violations of constrainti
P ]