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Introduction to ACT-R
Tutorial
21st Annual Conference
Cognitive Science Society
John R. Anderson
Christian Lebiere
Psychology Department
Carnegie Mellon University
Pittsburgh, PA 15213
ja+@cmu.edu
cl+@cmu.edu
ACT-R Home Page:
Dieter Wallach
Institut fur Psychologie
Universitaet Basel
Bernoullistr. 16
CH-4056 Basel
wallachd@ubaclu.unibas.ch
http://act.psy.cmu.edu
Tutorial Overview
1. Introduction
2. Symbolic ACT-R
Declarative
Procedural
Learning
3. Subsymbolic Performance in ACT-R
Activation (Declarative)
Utility (Procedural)
4. Subsymbolic Learning in ACT-R
Activation (Declarative)
Utility (Procedural)
5. ACT-R/PM
Note: For detailed (40-100 hrs) tutorial, visit ACT-R Education link.
For software visit ACT-R Software link.
For models visit Published ACT-R Models link.
Unified Theories of Cognition
ACT-R exemplifies what Newell meant when he spoke of a unified
theory of cognition – i.e., a single system within which we can
understand the wide range of cognition.
Arguments against Unified Theories
1. Modularity – behavioral and neural evidence.
2. Need for specialization - Jack of all trades, master of none.
Argument for Unified Theories
1. System Organization - We need to understand how the overall
mental system works in order to have any real
understanding of the mind or any of its more specific
functions.
2. Mental plasticity – ability to acquire new competences.
Newell’s Constraints on a
Human Cognitive Architecture
(Newell, Physical Symbol Systems, 1980)
+ 1. Behave as an (almost) arbitrary function of the environment
(universality)
+ 2. Operate in real time
+ 3. Exhibit rational, i.e., effective adaptive behavior
+ 4. Use vast amounts of knowledge about the environment
+ 5. Behave robustly in the face of error, the unexpected, and the
unknown
+ 6. Use symbols (and abstractions)
+ 7. Use (natural) language
- 8. Exhibit self-awareness and a sense of self
+ 9. Learn from its environment
+ 10. Acquire capabilities through development
- 11. Arise through evolution
+ 12. Be realizable within the brain
The Missing Constraint: Making Accurate
Predictions about Behavioral Phenomena.
ACT-R is explicitly driven to provide models for
behavioral phenomena. The tasks to which ACT-R has been
applied include:
1. Visual search including menu search
2. Subitizing
3. Dual tasking including PRP
4. Similarity judgements
5. Category learning
6. List learning experiments
7. Paired-associate learning
8. The fan effect
9. Individual differences in working memory
10. Cognitive arithmetic
11. Implicit learning (e.g. sequence learning)
12. Probability matching experiments
13. Hierarchical problem solving tasks including
Tower of Hanoi
14. Strategy selection including Building Sticks Task
15. Analogical problem solving
16. Dynamic problem solving tasks including military
command and control
17. Learning of mathematical skills including interacting
with ITSs
18. Development of expertise
19. Scientific experimentation
20. Game playing
21. Metaphor comprehension
22. Learning of syntactic cues
23. Syntactic complexity effects and ambiguity effects
24. Dyad Communication
A priori ACT-R models can be built for new domains taking knowledge
representations and parameterizations from existing domains. These
deliver parameter-free predictions for phenomena like time to solve
an equation.
History of the ACT-framework
Predecessor
HAM
(Anderson & Bower 1973)
Theory versions
ACT-E
ACT*
ACT-R
ACT-R 4.0
(Anderson, 1976)
(Anderson, 1978)
(Anderson, 1993)
(Anderson & Lebiere, 1998)
GRAPES
(Sauers & Farrell, 1982)
PUPS
(Anderson & Thompson, 1989)
ACT-R 2.0
ACT-R 3.0
ACT-R 4.0
(Lebiere & Kushmerick, 1993)
ACT-R/PM
(Byrne, 1998)
Implementations
(Lebiere, 1998)
ACT-RInformation
: Information Flow
ACT-R:
Flow
ACT-R
Goal
Stack
(Frontal Cortex)
Pop
Push
Conflict
Resolution
Current
Goal
Retrieval
(Cortical
Result
Activation)
Transform
Popped
Goal
Goal
Production
Procedural
Memory
(Basal Ganglia
& Frontal Cortex)
Action
Compilation
Declarative
Memory
Retrieval
Request
(Hipp ocampus
& Cortex)
Perception
OUTSIDE WORLD
ACT-R: Knowledge Representation
Declarative-Procedural Distinction
Declarative Knowledge: Chunks
Configurations of small numbers of elements
addend1
Three
sum
Addition-Fact
Seven
addend2
Four
Procedural Knowledge: Production Rules
for retrie ving chunks to solve problems.
336
+848
4
IF the goal is to add n1 and n2 in a column
and n1 + n2 = n3
THEN set as a subgoal to write n3 in that column.
Productions serve to coordinate the retrieval of
information from declarative memory and the enviroment
to produce transformations in the goal state.
ACT-R: Assumption Space
Performance
Declarative
Symbolic
Subsymbolic
Retrieval of
Chunks
Application o f
Production Rules
Noisy Activ ations
Control Speed and
Accuracy
Noisy Utili ties
Control Choice
Learning
Declarative
Symbolic
Subsymbolic
Procedural
Encoding
Environment and
Caching Goals
Bayesian
Learning
Procedural
Compilation from
Example and Instruction
Bayesian
Learning
Chunks: Example
(
(N
CHUNK-TYPE
NAME SLOT1
(F
EWCHUNK
SLOT2
SLOTN
ACT3+4
isa
NAME
isa
ADDITION-FACT
SLOT1
Filler1
ADDEND1
THREE
SLOT2
Filler2
ADDEND2
FOUR
SLOTN
FillerN
SUM
SEVEN
)
)
)
Chunks: Example
(CLEAR-ALL)
(CHUNK-TYPE addition-fact addend1 addend2 sum)
(CHUNK-TYPE integer value)
(ADD-DM (fact3+4
isa addition-fact
addend1 three
addend2 four
sum seven)
(three
isa integer
value 3)
(four
isa integer
value 4)
(seven
isa integer
value 7)
Chunks: Example
ADDITION-FACT
3
VALUE
7
isa
ADDEND1
THREE
VALUE
SUM
FACT3+4
SEVEN
ADDEND2
isa
FOUR
isa
INTEGER
4
VALUE
isa
Chunks: Exercise I
Fact:
The cat sits on the mat.
Encoding:
proposition
(Chunk-Type proposition agent action object)
(Add-DM
(fact007
isa proposition
agent cat007
action sits_on
object mat)
)
isa
cat007
agent
fact007
action
sits_on
object
mat
Chunks: Exercise II
The black cat with 5 legs sits on the mat.
Fact
Chunks
(Chunk-Type proposition agent action object)
(Chunk-Type cat legs color)
cat
(Add-DM
(fact007 isa proposition
agent cat007
action sits_on
object mat)
(cat007 isa cat
legs 5
color black)
proposition
isa
isa
legs
5
cat007
fact007
agent
color
)
black
mat
object
action
sits_on
Chunks: Exercise III
(Chunk-Type proposition agent action object)
(Chunk-Type prof money-status age)
(Chunk-Type house kind price status)
Fact
The rich young professor buys a
beautiful and expensive city
house.
proposition
prof
agent
money- prof08
status
age
young
house
isa
isa
rich
Chunk
fact008
action
buys
expensive
price
isa
object
status
beautiful
obj1001
kind
city-house
(Add-DM
(fact008 isa proposition
agent prof08
action buys
object house1001
)
(prof08 isa prof
money-status rich
age young
)
(obj1001 isa house
kind city-house
price expensive
status beautiful
)
)
Productions
set of productions, organized
through reference to goals
procedural
memory
•
•
•
•
productions
Structure of productions
( p
modularity
abstraction
goal factoring
conditional asymmetry
name
<Goal pattern>
<Chunk retrieval >
condition part
delimiter
==>
action part
)
<Goal Transformation>
<External action>
Psychological reality of productions
Taken from: Anderson, J.R. (1993). Rules of the mind. Hillsdale, NJ: LEA.
Error rates: Data & Model
Taken from: Anderson, J.R. & Lebiere, C. (1998). The atomic components of thought. Hillsdale, NJ:
Add-numbers
(p add-numbers
head/slot separator
= fact >
variable prefix
production name
=goal>
isa add-column
num1 =add1
num2 =add2
result nil
goal pattern
=fact>
isa addition-fact
addend1 =add1
addend2 =add2
sum =sum
chunk retrieval
= =>
=goal>
result =sum
)
action description
Add-numbers
3
+4
?
(p add-numbers
IF
the goal
is to add numbers in a column
and =add1 is the first number
and =add2 is the second number
and you remember
an addition fact that
=add1 plus
=add2 equals
=sum
Then
=goal>
isa add-column
num1 =add1
num2 =add2
result nil
=fact>
isa addition-fact
addend1 =add1
addend2 =add2
sum =sum
==>
note in the goal that the
result is =sum
=goal>
result =sum
)
(first-goal
isa add-colomn
num1 three
num2 four
result nil)
(fact3+4
isa addition-fact
addend1 three
addend2 four
sum seven)
(first-goal
isa add-colomn
num1 three
num2 four
result seven)
Pattern matching
left-hand side
negation
—
addend1
=goal>
isa find-sum
addend2 =num2
sum =sum
=fact>
isa add-fact
addend1 zero
addend2 =num2
sum =sum
goal
(goal1 isa find-sum
addend1 nil
addend2 two
sum
four
)
declarative memory
(fact2+3 isa add-fact
addend1 two
addend2 three
sum five)
(fact3+1 isa add-fact
addend1 three
addend2 one
sum four)
(fact0+4 isa add-fact
addend1 zero
addend2 four
sum four)
(fact2+2 isa add-fact
addend1 two
addend2 two
sum four)
Counting Example
First-Goal 0.000
isa COUNT-FROM
start 2
end 5
(P increment
=goal>
ISA
count-from
start
=num1
=count>
ISA
count-order
first
=num1
second
=num2
==>
!output!
( =num1)
=goal>
start
=num2)
(P stop
=goal>
ISA
start
end
==>
!output!
!pop!)
count-from
=num
=num
( =num)
(add-dm
(a ISA count-order first 1 second 2)
(b ISA count-order first 2 second 3)
(c ISA count-order first 3 second 4)
(d ISA count-order first 4 second 5)
(e ISA count-order first 5 second 6)
(first-goal ISA count-from start 2 end 5))
Web Address:
ACT-R Home Page
Published ACT-R Models
Counting Example
Goal Stack
G3
G1
Initial state
G2
G1
!push! =G2
!push! =G3
G2
G4
G1
G1
!pop!
!focus-on! =G4
stack-manipulating actions
Tower of Hanoi Demo
Start-Tower
IF the goal is to move a pyramid of size n to peg x
and size n is greater than 1
THEN set a subgoal to move disk n to peg x
and change the goal to move a pyramid of size n-1 to peg x
Final-Move
IF the goal is to move a pyramid of size 1 to peg x
THEN move disk 1 to peg x
and pop the goal
Subgoal-Blocker
IF the goal is to move disk of size n to peg x
and y is the other peg
and m is the largest blocking disk
THEN post the goal of moving disk n to x in the interface
and set a subgoal to move disk m to y
Move
IF the goal is move disk of size n to peg x
and there are no blocking disks
THEN move disk n to peg x
and pop the goal
Web Address:
ACT-R Home Page
Published ACT-R Models
Atomic Components of Thoughts
Chapter 2
Model for Ruiz
Tower of Hanoi: Data & Models
Taken from: Anderson, J.R. & Lebiere, C. (1998). The atomic components of thought. Hillsdale, NJ: LEA
Subsymbolic level
Summary
Computations on the subsymbolic level are responsible for
•
•
•
•
which production ACT-R attempts to fire
how to instantiate the production
how long the latency of firing a production is
which errors are observed
As with the symbolic level, the subsymbolic level is not a
static level, but is changing in the light of experience to
allow the system to adapt to the statistical structure of the
environment.
Chunks & Activation
A DDITION -F A CT
isa
(p add-numbers
=goal>
isa add-column
num1 =add1
num2 =add2
result nil
=fact>
isa addition-fact
addend1 =add1
addend2 =add2
sum =sum
THREE
Wj
addend1
Sji
F A CT 3+4
Bi
Sji
sum
Sji
addend2
F OUR
(goal1
isa add-column
num1 Three
num2 Four
result nil
)
Wj
S
Ai=Bi+ WjSji
S EVEN
Chunk Activation
Context activation
activation
=
base
activation
+
(
Ai = Bi +
source
activation
SW
j
j
*
associative
strength
)
* Sji
Activation makes chunks available to the degree that past experiences
indicate that they will be useful at the particular moment:
•
•
Base-level: general past usefulness
Context: relevance in the current context
Base-level Activation
activation =
Ai
=
+
base
activation
Bi
(
+
source
activation
*
associative
strength
SW
*
Sji
j
)
The base level activation Bi of chunk Ci reflects a contextindependent estimation of how likely Ci is to match a production, i.e.
Bi is an estimate of the log odds that Ci will be used.
Two factors determine Bi:
• frequency of using Ci
• recency with which Ci was used
Bi = ln
(
P(Ci)
P(Ci) )
Base-Level Activation & Noise
Basel-level activation fluctuates and decays with time
initial expected base-level
activation
decay with time, parameter
d denotes the decay rate
B(t) = - d * ln(t) + 1 + 2
random noise in initial baselevel activation
1 at creation time
transient noise 2, reflecting
moment-to-moment fluctuations
Source Activation
activation
Ai
=
=
base
activation
Bi
+
+
(
source
activation
SW
j
j
*
associative
strength
*
Sji
The source activations Wj reflect the amount of attention
given to elements, i.e. fillers, of the current goal. ACT-R
assumes a fixed capacity for goal elements, and that each
element has an equal amount (W= S Wi = 1).
(1) constant capacity for source activations
(2) equally divided among the n goal elements: constant/n
(3) W reflects an individual difference parameter
)
Associative strength
activation
Ai
=
=
base
activation
Bi
+
(
source
activation
*
SW
+
associative
strength
j
*
)
Sji
The association strength Sji between chunks Cj and Ci is a measure of
how often Ci was needed (retrieved) when Cj was element of the goal,
i.e. Sji estimates the log likelihood ratio of Cj being a source of
activation if Ci was retrieved.
Sji = ln
(
P(Ni Cj)
P(Ni)
)
= S - ln(P(Ni|Cj))
Retrieval time
Chunks i to instantiate production p are retrieved sequentially
Retrieval-timep =
S Time
i
ip
Time to retrieve a chunk as function of match score Mip and
strength of matching production Sp
-f(Mip + Sp)
Timeip = Fe
Retrieval time is an exponential function of the sum of match
score of the chunk and the production strength
Retrieval time
Fan effect
Lawyer
Park
In
Church
Fireman
Doctor
Bank
Fan Effect Demo
Retrieve-by-Person
If the goal is to retrieve a sentence involving a person and a location
and there is a proposition about that person in some location
Then store that person and location as the retrieved pair.
Retrieve-by-Location
If the goal is to retrieve a sentence involving a person and a location
and there is a proposition about some person in that location
Then store that person and location as the retrieved pair.
Mismatch-Person
If the retrieved person mismatches the probe
Then say no.
Mismatch-Location
If the retrieved location mismatches the probe
Then say no.
Match-Both
Web Address:
If the retrieved person and location both match the probe
ACT-R Home Page
Then say yes.
Published ACT-R Models
Atomic Components of Thought
Chapter 3
Fan Effect Model
Fan Effect
Threshold
Chunks with an activation lower than threshold
can not be retrieved
Retrieval probability =
1
-(A-)/s
1+e
(A- )/s
Equivalently: Odds of recall = e
recall is an exponential function of the distance between
Activation Ai of Chunk Ci and threshold , scaled by activation
noise s.
odds of recall decreases as a power function of time
Partial matching
Errors of Omission
These occur when the correct chunk falls below the activation threshold
for retrieval and the intended production rule therefore cannot fire.
==>
Errors of Commission
These occur when some wrong chunk is retrieved instead of the correct
one and so the wrong instantiation fires.
==>
Partial matching
partial matching is restricted to chunks with the same type as
specified in a production’s retrieval pattern
an amount reflecting the degree of mismatch Dip to a retrieval
pattern of production p is subtracted from the activation level Ai
of a partially matching chunk i. The match score for the match
of chunk i to production p is:
Mip = Ai - Dip
Dip is the sum for each slot of the degree of mismatch between
the value of the slot in chunk i and the respective retrieval pattern
Probability of retrieving chunk i as a match for production p:
eMip/t
Mjp/t
Se
j
t=
6
=
2 s
SUGAR FACTORY
SUGAR FACTORY
Sugar productiont = 2 * workerst - sugar productiont-1
[+/- 1000]
Negative correlation between knowledge and performance
workers
100 200 300 400 500 600 700 800 900 1000 1100 1200
s
u
g
a
r
p
r
o
d
u
c
t
i
o
n
1000 1000 3000 5000 7000 9000 11000 12000 12000 12000 12000 12000 12000
2000 1000 2000 4000 6000 8000 10000 12000 12000 12000 12000 12000 12000
3000 1000 1000 3000 5000 7000 9000 11000 12000 12000 12000 12000 12000
4000 1000 1000 2000 4000 6000 8000 10000 12000 12000 12000 12000 12000
5000 1000 1000 1000 3000 5000 7000 9000 11000 12000 12000 12000 12000
6000 1000 1000 1000 2000 4000 6000 8000 10000 12000 12000 12000 12000
7000 1000 1000 1000 1000 3000 5000 7000 9000 11000 12000 12000 12000
8000 1000 1000 1000 1000 2000 4000 6000 8000 10000 12000 12000 12000
9000 1000 1000 1000 1000 1000 3000 5000 7000 9000 11000 12000 12000
10000 1000 1000 1000 1000 1000 2000 4000 6000 8000 10000 12000 12000
11000 1000 1000 1000 1000 1000 1000 3000 5000 7000 9000 11000 12000
12000 1000 1000 1000 1000 1000 1000 2000 4000 6000 8000 10000 12000
Similarities: example
12
2
11
Ratio Similarities:
sim(a, b)
min a, b
max a, b
10
.09
0.1
.08
3
0.5
0.33
1
0.11
9
0.25
0.125
0.2
0.16
8
4
5
0.14
7
6
D = Mismatch Penalty * (1-sim(a, b))
Retrieval of encoded chunks
(p retrieve-episode
==>
)
=goal>
isa transistion
state =state
production =production
(GOALCHUNK
isa transition
state 2000
production 9000
worker nil)
=episode>
isa transition
state =state
production =production
worker =worker
(Episode007
isa transition
state 1000
production 8000
worker 5)
goal>
worker =worker)
Match
Partial Match
Lebiere, C., Wallach, D. & Taatgen, N. (1998). Implicit and explicit learning in ACT-R. In F. E. Ritter
And R. Young (Eds.) Proceedings of the Second European Conference on Cognitive Modeling, pp. 183-189.
Nottingham: Nottingham University Press.
Control performance
25
Target
States
20
Trial 41-80
15
10
Trial 1-40
5
0
ACT-R
Experiment
D&F
Concordance
Transition from computation to retrieval
Conflict resolution
In general, conflict resolution gives answers to two questions:
Which production out of a set of matching productions
is selected?
Goal factoring
Success probability
Costs
expected gain
Which instantiation of the selected production is fired?
Sequential instantiation
No backtracking
activation
Conflict resolution
Expected Gain =
P
* G –
Probability of
goal achievement
Goal value
goal-specific
production-specific
C
Cost of
goal achievement
Selection of Productions
Expected Gain =
P
* G
Probability of
goal achievement
q
•
r
Goal value
C
–
Cost of
goal achievement
a
+
b
Probability of Goal Achievement
q
P
*
probability of the
production working
successfully
Production's matching/actions/subgoals
have the intended effect
r
probability of achieving
the goal if the production
works successfully
Goal accomplished and popped
successfully.
Achieving a goal depends on the joint probability of the
respective production being successful and subsequent rules
eventually reaching the goal.
Costs of a production
a
amount of effort
(in time) that a production will take
Production's costs of
matching/actions/subgoals
C
+
b
estimate of the amount
of effort from when a production completes until
the goal is achieved
Costs of future productions
Production costs are calculated as the sum of the effort
associated with production pi and (an estimate of) the
effort that subsequent productions pj..n take on the way
to goal achievement.
Conflict resolution
P
q
*
{
{
{
a
…
{
Intended
next
state
current
state
r
+
C
b
goal
state
Goal value
G=20
p3
!push!
p3 parameters:
q: 1
r: .9
a: .05
b: 1
G'=17
G' = rG-b = .9 * 20 - 1 = 17
ACT-R values a goal less the more deeply it is embedded
in uncertain subgoals
ACT-R pops the goal with failure if no production above the
utility threshold (default: 0) can match (goal abandonment)
Noise in Conflict Resolution
Remember:
Evaluation Ei of production i = P(i)*G-C(i)
Boltzmann Equation
Probability of choosing i among n
applicable productions with Evaluation Ej
Ei/t
e
Ej/t
Se
j
t = 2
2-person Matrix Game
Players
Player1, Player2
Actions
Actions A, B ...
Payoff matrix
A1
B1
A2
3, 7
4, 6
B2
8, 2
1, 9
Data sets
Erev & Roth (1998)
“ There is a danger that investigators will treat the
models like their toothbrushes, and each will use
its own model only on his own data.”
Diverse data sets re-analyzed
2x2
4x4
5x5
Suppes & Atkinson (1960) [SA2, SA8, SA3k, SA3u]
Erev & Roth (1998)
[SA3n]
Malcom & Liebermann (1965)
O'Neill (1987)
Rapoport & Boebel (1992) [R&B10, R&B15]
Model
(p player1-A
=goal>
isa decide
player1 nil
==>
=goal>
player1 A
)
Productions
Chunk
game12
isa decide
player1 A
player2 B
(p player1-B
=goal>
isa decide
player1 nil
==>
=goal>
player1 B
)
24
33
60
15
1/3 2/3 1
0
1/2 1/2 1/6 5/6
Best Fits – Random Games
1-Parameter
Reference
Erev & Roth (1998)
point=xmin
Parameter S(1)=15
Data set
Random games
100*MSD
game
game
game
game
game
game
game
game
game
game
#1
#2
#3
#4
#5
#6
#7
#8
#9
#10
1.3
0.304
0.89585
0.8994
0.28565
2.0305
0.6485
2.201
0.4287
3.589
1.7195
Reference point
=0
1.087
0.4725
0.4403
1.235
0.4074
1.181
1.298
0.6303
1.18
2.523
1.504
Model
ACT-R;
Average
Par.
Data
set
100*MSD
game
game
game
game
game
game
game
game
game
game
priors=53
Random games
1
2
3
4
5
6
7
8
9
10
0.471
0.288
0.163
0.289
0.325
0.447
0.903
0.626
0.204
1.006
0.546
Conflict resolution
(1)
selection
(2)
(3)
Goal
test
goal pattern
Match
Ep = 13.95
evaluate
conflict set
Ep = 18.95
Ep = 17.30
Ep = 18.95
Match
(4)
retrieve
chunk(s)
fire
production
Learning as Subsymbolic Tuning to
the Statistics of the Environment
1. Lael Schooler: Statistical structure of the
demands on declarative memory posed by the
environment.
2. Christian Lebiere: Consequences for 20 years of
practicing arithmetic facts.
3. Marsha Lovett: Selection among production
rules is also sensitive to both the features of the
current problem and the rule’s past history of
success.
Lael Schooler
Declarative Memory:
Statistical Tuning
1. The goal of declarative memory is to m ake
most available those memory chunks that are
most likely to be needed at a particular point
in time.
2. The probability of a memory chunk being
relevant depends on its past history of usage
and the current context.
n
d
tj
j1
3. Log Odds = Log
+ Context
Odds = .14 T
-.73
Log Odds= - 1.95 - 0.73 Log Days
R^2 = 0.993
0.2
-1
(a) New York Times Retention
(d) New York Tim es Retention
Log Need Odds
Probabilitity on Day 101
-2
0.1
-3
-4
-5
0.0
0
20
40
60
80
Days since Last Occurrence
100
-6
0
1
2
3
Log Days
4
5
Odds = .18 T
-.77
Log Odds = - 1.70 - 0.77 Log Utterances
R^2 = 0.984
0.12
-2
(b) Parental Speech Retention
(e) Parental Speech Retention
-3
0.08
Log Need Odds
Probability in Utterance 101
0.10
0.06
0.04
-5
0.02
0.00
-4
0
20
40
60
80
Utterances since Last occurrence
100
-6
0
1
2
3
Log Utterances
4
5
Odds = .34 T
-.83
Log Odds = - 1.09 - 0.83 Log Days
R^2 = 0.986
0.3
0
(c) Mail Sources Retention
(f) Mail Sources Retention
0.2
Log Need Odds
Probability on Day 101
-1
0.1
-2
-3
-4
0.0
0
20
40
60
80
Days since Last Occurrence
100
-5
0
1
2
3
Log Days
4
5
Parameter learning:
n
log(S tj-d)
j=1
Lael Schooler’s Research
p(AIDS) = .018
New Y ork T imes
Associates
virus
spre ad
patients
health
p(AIDS|associate)
.75
.54
.40
.27
p(AIDS|associate)
p(AIDS)
41.0
29.4
21.8
14.6
Parental Speech
p(play) = .0086
p(play|game) p(play|game)
p(play)
.41
47.3
Environmental Analyses of Context and Recency
(b) New York Times
standard
(a) CHILDES
standard
0.45
0.45
0.40
0.40
0.35
0.30
0.30
need odds
need odds
0.35
strong context
weak context
0.25
0.20
0.15
0.10
0.25
0.20
0.15
0.10
0.05
0.05
0.00
0 10 20 30 40 50 60 70 80
retention in utterances
0.00
0
10 20 30
40 50 60 70 80
retention in days
0
-1
-1
-2
-2
log need odds
log need odds
0
(d) New York Times
power
(c) CHILDES
power
-3
-4
-5
-3
-4
-5
-6
-6
-7
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
retention in log utterances
-7
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
retention in log days
Lael Schooler
Retrieval Odds Mirrors Odds of Occurring
Conclusions from Environmental Studies:
Log Odds = Log
n
d
t
j
j1
+ Context
Proposal for ACT-R’s Declarative Memory:
- Activation reflects Log Odds of Occurring
Eight
W
j
addend1
S
ji
addit ion-fact
B
i
S
ji
sum
Twelve
S
ji
addend2
Four
W
j
- Learning works as a Bayesian inference scheme to try to
identify the right values of the factors determining odds of
recall.
Declarative Equations
Activation Structure
Ai = Bi + Wj Sji
j
n
d
t j
j1
Activation Equation
Bi = ln
Base-Level Learning
Sji= S - ln((P(i|j))
Strength Learning
Performance Structure
Mi = Ai - Dp
Match Equation
e
Probability =
M /t
i
e
M /t
j
Chunk Choice
j
Timei = Fe-fMi
Retrieval Time
What happens when the probabilistic world of information
retrieval hits the hard and unforgiving world of mathematics?
Christian Lebiere’s
Simulation of Cognitive Arithmetic
Over 100,000 problems of e ach type (1+1 to 9+9; 1x1 to 9x9) over 20
years.
Retrieve
Compute
Addition
IF the goal is to find a + b
and a + b = c
THEN the answer is c
Multiplication
IF the goal is to find a * b
and a * b = c
THEN the answer is c
IF the goal is to find a + b
IF the goal is to find a * b
THEN set a subgoal to count THEN set a subgoal to add
b units past a
a for b times
Critical Phenomena:
Transition from computation to retrieval
Errors due to partial matching and noise
Errors due to retrieving wrong answers
Effects of frequency distribution 1+1 is about three times
more frequent than 9+9
Problem Size Effect over Time
Model
Problem Size Effect (Data)
Problem Size Effect over Time
10
10
8
Response Time (sec)
8
1st
4th
7th
10th
College
RT (sec)
6
4
2
0
1st
6
4th
7th
10th
College
4
2
Small
Large
0
Small
Large
Problem Size
Problem Size
Effect of Argument Size on Accuracy
For Addition (4 year olds)
Data
Model
Percentage Correct for Addition Retrieval
in the First Cycle (1000 Problems)
Addition Retrieval
80
80
Augend
Addend
60
60
50
40
30
20
Augend
Addend
70
Percentage Correct
Percentage Correct
70
50
40
30
0
1
2
3
4
5
Operand
Percentage of Correct Retrieval per Operand
6
20
0
1
2
3
4
5
Operand
Percentage Correct in Simulation
6
Effect of Argument Size on Accuracy
For Multiplication (3rd Grade)
Data
Model
Error Percentage for Multiplication
Computation in Cycle 3 (~4th Grade)
Multiplication Computation
50
50
Multiplicand
Multiplier
Multiplicand
40
Error Percentage
Error Percentage
40
30
20
10
0
Multiplier
30
20
10
0
2
4
6
Argum ent
8
10
Percentage of Correct Computations per Operand
2
4
6
Argument
8
10
Percentage Errors in Multiplication Simulation
Conclusions about
Cognitive Arithmetic
Subsymbolic learning mechanisms that yield adaptive
retrieval in the world at large are behind the 20 year
struggle that results in the mastery of cognitive
arithmetic. Part of the reason why it is a struggle is
that there is n oise in the system. However, more
deeply, two things about the arithmetic domain fail
to match up with the assumptions our memory
system makes about the world:
1. Precise matching is required.
2. High interference between competing memories.
Procedural Learning
Making Choices: Conflict Resolution
P is expected probability of success
G is value of goal
C is expected cost
Expected Gain = E = PG-C
Probability of choosing i =
e
Ei /t
e
E /t
j
t reflects noise in evaluation
and is like temperature in
the Bolztman equation
j
Successes
P = Successes + Failures
Successes = + m
Failures = + n
is prior successes
m is experienced successes
is prior failures
n is experienced failures
Building Sticks Task (Lovett)
INITIAL STATE
desired:
current:
building:
a
c
b
possible first moves
desired:
current:
building:
a
desired:
current:
c
b
UNDERSHOOT
Looks
Undershoot
building:
a
desired:
current:
b
c
building:
a
OVERSHOOT
b
c
UNDERSHOOT
Undershoot
Overshoot
More Successful
More Successful
10 Undershoot
10 (5) Undershoot
0 Overshoot
10 (15) Overshoot
Looks
10 (15) Undershoot
Overshoot
10 (5) Overshoot
0 Undershoot
10 Overshoot
Proportion Choice More Successful Operator
Lovett & Anderson, 1996
Observed Data
1
3
1
0
0.8
3
1
0.7
0.6
3
1
3
0.3
1
0.2
0
3
3
0
1
0.9
0.8
0
0.5
0
3
3
1
0.4
1
1
0
3
0.7
0.6
0.5
0.1
3
1
1
0
0.3
0.2
0
0.1
0
Proportion Choice More Successful Operator
(5/6)
1
0.9
0.4
Extreme-Biased Condition
(2/3)
Biased Condition
0
0
0
High
Low
Neutral
Low
High
Against Against
Toward Toward
Test Problem Bias
High
Low
Neutral
Low
High
Against Against
Toward Toward
Test Problem Bias
Predictions of Decay-Based ACT-R
1
1
0.9
3
1
0
0.8
0.7
3
1
0.6
0
0.5
0.4
0.3
0.2
3
1
0
3
1
3
1
0
0
0.1
3
1
0
0.9
0.8
3
1
0.7
0.6
0.5
0.4
3
1
0
3
1
3
1
0
0
0
0.3
0.2
0.1
0
0
High
Low
Neutral
Low
High
Against Against
Toward Toward
Test Problem Bias
High
Low
Neutral
Low
High
Against Against
Toward Toward
Test Problem Bias
Build Sticks Demo
Decide-Under
If the goal is to solve the BST task
and the undershoot difference is less
th an th e overshoot difference
Then choose undershoot.
Decide-Over
If the goal is to solve the BST task
and the overshoot difference is less
than th e undershoot difference
Then choose overshoot.
Force-Under
If the goal is to solve the BST task
Then choose undershoot.
Force-Over
If the goal is to solve the BST task
Then choose overshoot.
Web Address:
ACT-R Home Page
Published ACT-R Models
Atomic Components of Thought
Chapter 4
Building Sticks Model
ACT-R model probabilities before and after
problem-solving experience in Experiment 3
(Lovett & Anderson, 1996)
Production
Force-Under
More Successful
Prior
Final Value
Probability
of Success 67% Condition 83% Condition
.50
.60
.71
.50
.38
.27
.96
.98
.98
.96
.63
.54
Context Free
Force-Over
Less Successful
Context Free
Decide-Under
More Successful
Context Sensitive
Decide-Over
Less Successful
Context Sensitive
Decay of Experience
Note: Such temporal weighting is critical in the real world.
Credit-Assignment in ACT-R
• But, what happens when there is more than one
critical choice per problem?
-How is credit/blame assigned by human problem
solvers?
-How well does ACT-R's learning mechanism handle
this more complex case?
-In ACT-R all choices leading to goal resolution are
equally weighted.
-But, is there evidence for a goal gradient?
Building Sticks Task 2 Levels
INITIAL STATE
desired:
current:
building:
a
add b
b
c
add c
75%
OVERSHOOT
UNDERSHOOT
desired:
current:
desired:
current:
building:
a
b
building:
a
c
desired:
current:
desired:
current:
delete a
MAINTAIN
building:
a
building:
a
c
b
c
building:
a
b
c
add a
desired:
current:
b
c
building:
a
building:
a
b
c
c
building:
a
building:
a
b
c
delete a
desired:
current:
desired:
current:
b
c
desired:
current:
add a
desired:
current:
b
add c
add a
REVERSE
MAINTAIN
75%
desired:
current:
desired:
current:
delete a
building:
a
b
delete c
75%
REVERSE
desired:
current:
c
add c
delete c
building:
a
b
b
c
building:
a
b
c
Choice Learning
Adapting to a Variable
and Changing World
It would be trivial to create a system that would do well at this
task simply by eliminating the noise and getting rid of the
discounting of past experience. However, this again makes the error of
assuming that the human mind evolved for optimal performance at
our particular laboratory task.
In the real world noise is important both to explore other options and to
avoid getting caught in traps.
The discounting of experience also allows us to rapidly update in the
presence of the changing world.
Christian Lebiere and Robert West have shown that these features are
critical to getting good performance in games as simple as rocks-papersscissors.
ACT-R/PM
Martin-Emerson-Wickens Task
Perform compensatory tracking,
keeping the crosshair on target
(Dual-)
Task
Respond to choice stimuli as
rapidly as possible
Choice stimulus appears at
various distances from target
(vertical separation)
Zur Anzeige wird der QuickTime™
Dekompressor “Photo - JPEG”
benötigt.
Model
Tracking requires eye to be on
the crosshair
Eye must be moved to see stimulus
Martin-Emerson
& Wickens (1992):
The vertical visual
field and implications
for the head-up
display
Choice response & tracking movements are bottlenecked through
single motor module
MEW Productions
Find-Target-Oval
IF the target hasn't been located
and the oval is at location
THEN mark the target at location
Attend-Cursor
IF the target has been found and the state has not been set
and the pointer is at location and has not been attended to
and the vision module is free
THEN send a command to move the attention to location
and set the state as "looking"
Attend-Cursor-Again
IF the target has been found and the state is "looking"
and the pointer is at location and has not been attended to
and the vision module is free
THEN send a command to move the attention to location
Start-Tracking
IF the state is "looking"
and the object focused on is a pointer
and the vision module is free
THEN send a command to track the pointer
and update the state to "tracking"
Move-Cursor
IF the state is "tracking" and the target is at location
and the motor module is free
THEN send a command to move the cursor to location
Stop-Tracking
IF the state is "tracking"
and there is an arrow on screen
that hasn't been attended to
THEN move the attention to that location
and update the state to "arrow"
Right-Arrow
IF the state is "arrow"
and the arrow is pointing to the right
and the motor module is free
THEN send a command to punch the left index finger
and clear the state
Left-Arrow
IF the state is "arrow"
and the arrow is pointing to the left
and the motor module is free
THEN send a command to punch the left middle finger
and clear the state
Schedule chart for Schumacher, et al. (1997) perfect time-sharing model. VM = visualManual ask, AV = auditory-verbal task, RS = response selection.
ACT-R/PM simulation of Schumacher, et al. (1997) perfect time-sharing results.
