Attendee questionnaire

1
Attendee questionnaire

• Name
• Affiliation/status
• Area of study/research
• For each of these subjects
• Linguistics (Optimality Theory)
• Computation (connectionism/neural networks)
• Philosophy (symbolic/connectionist debate)
• Psychology (infant phonology)
• please indicate your relative level of
• interest (for these lectures) 1 least, 5
  most
• background 1 none, 5 expert
• Thank you

2
Optimality in Cognition and Grammar

• Paul Smolensky
• Cognitive Science Department
• Johns Hopkins University

3
Optimality in Cognition and Grammar

• Paul Smolensky
• Cognitive Science Department, Johns Hopkins
  University
• Plan of lectures
• Cognitive architecture
• Symbols and neurons
• Symbols in neural networks
• Optimization in neural networks
• Optimization in grammar I HG ? OT
• Optimization in grammar II OT
• OT in neural networks

4
Cognitive architecture

• Central dogma of cognitive science
• Cognition is computation
• But what type of computation?
• What exactly is computation, and what work must
  it do in cognitive science?

5
Computation

• Functions, cognitive
• Pixels ? objects ? locations low- to high-level
  vision
• Sound stream ? word string phonetics
• Word string ? parse tree syntax
• Underlying form ? surface form phonology
• petit copain /p?tit kop?/ ? p?.ti.ko.p?
• petit ami /p?tit ami/ ? p?.ti.ta.mi
• Reduction of complex procedures for evaluating
  functions to combinations of primitive operations
• Computational architecture
• Operations primitives combinators
• Data

6
Symbolic Computation

• Computational architecture
• Operations primitives combinators
• Data
• The Pure Symbolic Architecture (PSA)
• Data strings, (binary) trees, graphs,
• Operations
• Primitives
• Concatenate (string, tree) cons
• First-member(string) left-subtree(tree) ex0
• Combinators
• Composition f(x) def g(h(x)))
• IF(x A) THEN ELSE

7
Passive

• Few leaders are admired by George
• ? admire(George, few leaders) 

(s) cons(ex1(ex0(ex1(s))),
cons(ex1(ex1(ex1(s))), ex0(s)))

• But for cognition, need a reduction to a very
  different computational architecture

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The cognitive architecture The connectionist
hypothesis
At the lowest computational level of the
mind/brain
PDP Computation

• Representations Distributed activation patterns
• Primitive operations (e.g.)
• Multiplication of activations by synaptic weights
• Summation of weighted activation values
• Non-linear transfer functions

• Combination Massive parallelism

9
Criticism of PDP (e.g., neuroscientists)

• Much too simple
• Misguided. Relevant complaint
• Much too complex
• Target of computational reduction must be within
  the scope of neural computation.
• Confusion between two questions

10
The cognitive questionfor neuroscience

• What is the function of each component of the
  nervous system?
• Our question is quite different.

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The neural question for cognitive science

• How are complex cognitive functions computed by a
  mass of numerical processors like neuronseach
  very simple, slow, and imprecise relative to the
  components that have traditionally been used to
  construct powerful, general-purpose computational
  systems? How does the structure arise that
  enables such a medium to achieve cognitive
  computation?

12
The ICS Hypothesis

• The Integrated Connectionist/Symbolic Cognitive
  Architecture (ICS)
• In higher cognitive domains, representations and
  fuctions are well approximated by symbolic
  computation
• The Connectionist Hypothesis is correct
• Thus, cognitive theory must supply a
  computational reduction of symbolic functions to
  PDP computation

13
PassiveNet
14
The ICS Isomorphism
Tensor product representations
Tensorial networks
?
15
Within-level compositionality
(s) cons(ex1(ex0(ex1(s))),
cons(ex1(ex1(ex1(s))), ex0(s)))

• W Wcons0Wex1Wex0Wex1
  Wcons1Wcons0(Wex1Wex1Wex1)Wcons1(Wex0)

Between-level reduction
16
Levels
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The ICS Architecture
dogs
d?gz
A
18
Processing I Activation

• Computational neuroscience
• Key sources
• Hopfield 1982, 1984
• Cohen and Grossberg 1983
• Hinton and Sejnowski 1983, 1986
• Smolensky 1983, 1986
• Geman and Geman 1984
• Golden 1986, 1988

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Processing I Activation
Processing spreading activation is
optimization Harmony maximization
20
The ICS Architecture
cat
kæt
A
21
Processing II Optimization

• Cognitive psychology
• Key sources
• Hinton Anderson 1981
• Rumelhart, McClelland, the PDP Group 1986

Processing spreading activation is
optimization Harmony maximization
22
Processing II Optimization
Processing spreading activation is
optimization Harmony maximization
23
Processing II Optimization

• The search for an optimal state can employ
  randomness
• Equations for units activation values have
  random terms
• pr(a) ? eH(a)/T
• T (temperature) randomness ? 0 during search
• Boltzmann Machine (Hinton and Sejnowski 1983,
  1986) Harmony Theory (Smolensky 1983, 1986)

24
The ICS Architecture
cat
kæt
A

25
Two Fundamental Questions
? Harmony maximization is satisfaction of
parallel, violable constraints

• 2. What are the constraints?
• Knowledge representation
• Prior question
• 1. What are the activation patterns data
  structures mental representations evaluated
  by these constraints?

26
Representation

• Symbolic theory
• Complex symbol structures
• Generative linguistics (Chomsky Halle 68 )
• Particular linguistic representations
• Markedness Theory (Jakobson, Trubetzkoy, 30s )
• Good (well-formed) linguistic representations
• Connectionism (PDP)
• Distributed activation patterns
• ICS
• realization of (higher-level) complex symbolic
  structures in distributed patterns of activation
  over (lower-level) units (tensor product
  representations etc.)
• will employ local representations as well

27
Representation
28
Tensor Product Representations

• Representations

29
Tensor Product Representations
?
30
Tensor Product Representations
31
Local tree realizations

• Representations

32
stopped
33
The ICS Isomorphism
Tensor product representations
Tensorial networks
?
34
Tensor Product Representations
35
Tensor Product Representations

• Mental representations are defined by the
  activation values of connectionist units. When
  analyzed at a higher level, these representations
  are distributed patterns of activity activation
  vectors. For core aspects of higher cognitive
  domains, these vectors realize symbolic
  structures.
• Such a symbolic structure s is defined by a
  collection of structural roles ri each of which
  may be occupied by a filler fi s is a set of
  constituents, each a filler/role binding fi/ri.
• The connectionist realization of s is an activity
  vector
• s Si fi Ä ri
• In higher cognitive domains such as language and
  reasoning, mental representations are recursive
  the fillers or roles of s have themselves the
  same type of internal structure as s. And these
  structured fillers f or roles r in turn have the
  same type of tensor product realization as s.

36
Representation Combination
37
The ICS Architecture
cat
kæt
A
38
Two Fundamental Questions
? Harmony maximization is satisfaction of
parallel, violable constraints

• 2. What are the constraints?
• Knowledge representation
• Prior question
• 1. What are the activation patterns data
  structures mental representations evaluated
  by these constraints?

39
Representation
40
Two Fundamental Questions
? Harmony maximization is satisfaction of
parallel, violable constraints

• 2. What are the constraints?
• Knowledge representation
• Prior question
• 1. What are the activation patterns data
  structures mental representations evaluated
  by these constraints?

41
Constraints
NOCODA A syllable has no coda Maori
H(as k æ t) sNOCODA lt 0
42
The ICS Architecture
cat
kæt
A
43
The ICS Architecture
cat
kæt
A
NEXT LECTURE HG, OT