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Transcript of Introduction to ACT-R Tutorial 21st Annual Conference Cognitive Science Society ACT-R Home Page: ...
Introduction to ACT-RTutorial
21st Annual Conference Cognitive Science Society
ACT-R Home Page: http://act.psy.cmu.edu
John R. AndersonChristian Lebiere
Psychology DepartmentCarnegie Mellon
UniversityPittsburgh, PA 15213
[email protected][email protected]
Dieter WallachInstitut fur Psychologie
Universitaet Basel Bernoullistr. 16 CH-4056 Basel
Tutorial Overview1. Introduction
2. Symbolic ACT-RDeclarativeProceduralLearning
3. Subsymbolic Performance in ACT-RActivation (Declarative)Utility (Procedural)
4. Subsymbolic Learning in ACT-RActivation (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.
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.
Unified Theories of Cognition
+ 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
Newell’s Constraints on a Human Cognitive Architecture
(Newell, Physical Symbol Systems, 1980)
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 search2. Subitizing3. Dual tasking including PRP 4. Similarity judgements5. Category learning6. List learning experiments7. Paired-associate learning8. The fan effect9. Individual differences in working memory 10. Cognitive arithmetic11. Implicit learning (e.g. sequence learning)12. Probability matching experiments
The Missing Constraint: Making Accurate Predictions about Behavioral Phenomena.
13. Hierarchical problem solving tasks including Tower of Hanoi 14. Strategy selection including Building Sticks Task15. Analogical problem solving16. Dynamic problem solving tasks including military command and control17. Learning of mathematical skills including interacting with ITSs18. Development of expertise19. Scientific experimentation20. Game playing21. Metaphor comprehension22. Learning of syntactic cues23. Syntactic complexity effects and ambiguity effects24. 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 (Anderson, 1976)ACT* (Anderson, 1978)ACT-R (Anderson, 1993)ACT-R 4.0 (Anderson & Lebiere, 1998)
Implementations GRAPES (Sauers & Farrell, 1982)
PUPS (Anderson & Thompson, 1989)
ACT-R 2.0 (Lebiere & Kushmerick, 1993)ACT-R 3.0ACT-R 4.0 (Lebiere, 1998)ACT-R/PM (Byrne, 1998)
ACT-R : Information Flow
ConflictResolution
Retrieval Request
Transform
Goal
CurrentGoal
(Cortical
Activation)
ProceduralMemory
(Basal Ganglia& Frontal Cortex)
DeclarativeMemory
(Hippocampus
& Cortex)
GoalStack
(Frontal Cortex)
Retrieval
Result
PopPush
Production
Compilation
ACT-R
OUTSIDE WORLD
Action Perception
Popped
Goal
ACT-R: Information Flow
Addition-FactThree Seven
Four
addend1 sum
addend2
Declarative-Procedural Distinction
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.
Declarative Knowledge: Chunks
Configurations of small numbers of elements
ACT-R: Knowledge Representation
PerformanceDeclarative Procedural
SymbolicRetrieval of
ChunksApplication of
Production Rules
SubsymbolicNoisy ActivationsControl Speed and
Accuracy
Noisy UtilitiesControl Choice
LearningDeclarative Procedural
SymbolicEncoding
Environment andCaching Goals
Compilation fromExample and Instruction
SubsymbolicBayesianLearning
BayesianLearning
ACT-R: Assumption Space
NAME
SSLOT1LOT1 Filler1Filler1
SSLOT2 LOT2 Filler2Filler2
SSLOTLOTNN
NEWCHUNK(
FillerNFillerN )
isa ADDITION-FACT
AADDENDDDEND11 TTHREEHREE
AADDENDDDEND22 FFOUROUR
SSUM UM
FACT3+4(
SSEVENEVEN )
isa
Chunks: Example
CHUNK-TYPE NAME SSLOT1LOT1 SSLOT2LOT2 SSLOTNLOTN( )
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)
ADDITION-FACT
FACT3+4ADDEND1 SUM
ADDEND2
THREE
FOUR
SEVEN
isa
isa
INTEGER
isa
VALUE VALUE
3 7
isa
Chunks: Example
VALUE
4
Chunks: Exercise I
Fact:
Encoding:
(Chunk-Type proposition agent action object)
The cat sits on the mat.
proposition
action
cat007
sits_on
mat
isa
fact007agent object
(Add-DM (fact007
isa proposition
agent cat007
action sits_on
object mat) )
Chunks: Exercise IIFact The black cat with 5 legs sits on the mat.
Chunks(Chunk-Type proposition agent action object)(Chunk-Type cat legs color)
(Add-DM (fact007 isa proposition
agent cat007action sits_onobject mat)
(cat007 isa catlegs 5
color black) )
proposition
action
cat007
sits_on
mat
isa
fact007agent object
cat
isa
color
5
black
legs
Chunks: Exercise III
Fact
ChunkThe rich young professor buys a beautiful and expensive city house.
(Chunk-Type proposition agent action object)(Chunk-Type prof money-status age)(Chunk-Type house kind price status)
(Add-DM (fact008 isa proposition
agent prof08action buysobject house1001
) (prof08 isa prof
money-status richage young
) (obj1001 isa house
kind city-houseprice expensivestatus beautiful
))
proposition
action
buys
isa
fact008agent object
prof
isa
prof08
age
young
rich
house
kind
city-house
obj1001
price
expensive
isa
status
beautiful
money-status
proceduralmemory
set of productions, organizedset of productions, organizedthrough reference to goalsthrough reference to goals
productions
• • modularitymodularity• • abstractionabstraction• • goal factoringgoal factoring• • conditional asymmetryconditional asymmetry
Productions
( p
==>
)
<Goal Transformation>
<External action>
condition part
delimiter
action part
name
<Goal pattern><Chunk retrieval >
Structure of productions
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: LEA.
chunk retrievalvariable prefix
(p add-numbers
=goal> isa add-column num1 =add1 num2 =add2 result nil
=fact> isa addition-fact addend1 =add1 addend2 =add2 sum =sum
= =>
=goal> result =sum)
production name
action description
goal pattern
>
head/slot separator
fact=
Add-numbers
note in the goal that the result is =sum
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
??Add-numbers
IF
Then
(p add-numbers
=goal> isa add-column num1 =add1 num2 =add2 result nil
=fact> isa addition-fact addend1 =add1 addend2 =add2 sum =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)
3+4
declarative memory
goal
left-hand side
Pattern matching
(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)
=goal> isa find-sum addend2 =num2 sum =sum
(fact2+2 isa add-fact addend1 two addend2 two sum four)
(goal1 isa find-sum addend1 nil addend2 two sum four )
=fact> isa add-fact addend1 zero addend2 =num2 sum =sum
negation
— addend1
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 count-from start =num end =num==> !output! ( =num) !pop!)
(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))
Counting Example
Web Address: ACT-R Home Page
Published ACT-R Models Counting Example
Initial state
stack-manipulating actions
Goal Stack
!focus-on! =G4
G4
G1
!pop!
G1
G2G1
!push! =G2
G2
!push! =G3
G1
G3
Tower of Hanoi Demo
Web Address: ACT-R Home Page Published ACT-R Models Atomic Components of Thoughts Chapter 2 Model for Ruiz
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
Tower of Hanoi: Data & Models
Taken from: Anderson, J.R. & Lebiere, C. (1998). The atomic components of thought. Hillsdale, NJ: LEA.
Subsymbolic levelSummary
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.
(goal1 isa add-column num1 Three num2 Four result nil )
FACT3+4Bi
FOUR
ADDITION-FACT
SEVEN
add
Chunks & Activation
end2
isa
Sji
Wj
THREE
Wj
=fact> isa addition-fact addend1 =add1 addend2 =add2 sum =sum
(p add-numbers =goal> isa add-column num1 =add1 num2 =add2 result nil
addend1 sum
Sji Sji
Ai=Bi+WjSji
Chunk Activation
baseactivatio
n
associativestrength
sourceactivation
activation ( )= +
Ai = Bi + Wj * Sji
Context activation
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
j
*
Base-level Activation
The base level activation Bi of chunk Ci reflects a context-independent 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
BBii = ln = ln (( ))P(CP(Cii))P(CP(Cii))
baseactivation
associativestrength
sourceactivation
activation ( )= +*
Ai = Bi + Wj * Sji
Base-Level Activation & Noise
B(t) = - d * ln(t) + 1 + 2
Basel-level activation fluctuates and decays with time
initial expected base-level activation
decay with time, parameterd denotes the decay rate
transient noise 2, reflectingmoment-to-moment fluctuations
random noise in initial base-
level activation 1 at creation time
Source Activation
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= Wi = 1).
(1) constant capacity for source activations(1) constant capacity for source activations(2) equally divided among the n goal elements: (2) equally divided among the n goal elements: constant/nconstant/n(3) W reflects an individual difference parameter(3) W reflects an individual difference parameter
baseactivatio
n
associativestrength
sourceactivation
activation
( )= + *
Ai = Bi + Wj * Sjij
Associative strength
The association strength SThe association strength Sjiji between chunks C between chunks Cjj and C and Ci i is a is a measure of how often Cmeasure of how often Cii was needed (retrieved) when C was needed (retrieved) when Cjj was was element of the goal, i.e. Selement of the goal, i.e. Sjiji estimates the log likelihood ratio of estimates the log likelihood ratio of CCjj being a source of activation if C being a source of activation if Ci i was retrieved.was retrieved.
baseactivatio
n
associativestrength
sourceactivation
activation
( )= +
( )P(Ni Cj)P(Ni)
Sji = ln
*
= S - ln(P(Ni|Cj))
Ai = Bi + Wj * Sji
Retrieval time
Chunks i to instantiate production p are retrieved sequentially
Time to retrieve a chunk as function of match score Mip andstrength of matching production Sp
Retrieval time is an exponential function of the sum of matchscore of the chunk and the production strength
TimeipRetrieval-timep =
i
Timeip = Fe-f(Mip + Sp)
Retrieval time
Fan effect
Lawyer
Church
Park
Fireman
In
Doctor Bank
Fan Effect DemoRetrieve-by-PersonIf the goal is to retrieve a sentence involving a person and a location and there is a proposition about that person in some locationThen store that person and location as the retrieved pair.
Retrieve-by-LocationIf the goal is to retrieve a sentence involving a person and a location and there is a proposition about some person in that locationThen store that person and location as the retrieved pair.
Mismatch-PersonIf the retrieved person mismatches the probeThen say no.
Mismatch-LocationIf the retrieved location mismatches the probeThen say no.
Match-BothIf the retrieved person and location both match the probeThen say yes.
Web Address: ACT-R Home Page 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
1 + e-(A-)/s
Equivalently: Odds of recall = e(A- )/s
recall is an exponential function of the distance betweenActivation Ai of Chunk Ci and threshold scaled by activationnoise s.
odds of recall decreases as a power function of time
These occur when the correct chunk falls below the activation thresholdfor retrieval and the intended production rule therefore cannot fire.
Errors of Omission
These occur when some wrong chunk is retrieved instead of the correctone and so the wrong instantiation fires.
Errors of Commission
Partial matching
==>
==>
partial matching is restricted to chunks with the same type asspecified in a production’s retrieval pattern
Partial matching
an amount reflecting the degree of mismatch Dip to a retrievalpattern of production p is subtracted from the activation level Ai
of a partially matching chunk i. The match score for the matchof chunk i to production p is:
Mip = Ai - DipDip 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/tej
t = 6 = 2 s
SUGAR FACTORY
SUGAR FACTORY
Sugar productiont = 2 * workerst - sugar productiont-1 [+/- 1000]
Negative correlation between knowledge and performance
100 200 300 400 500 600 700 800 900 1000 1100 1200
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
workers
production
sugar
Similarities: example
sim(a,b)=mina,b( )maxa,b( )
D = Mismatch Penalty * (1-sim(a, b))
aa
1221110 345678
9 1.08.090.10.110.1250.140.160.20.250.330.5Ratio Similarities:
Retrieval of encoded chunks
(GOALCHUNK isa transition state 2000 production 9000 worker nil) (Episode007 isa transition state 1000 production 8000 worker 5)MatchPartial Match
(p retrieve-episode =goal> isa transistion state =state production =production =episode> isa transition state =state production =production worker =worker==> goal> worker =worker))Lebiere, C., Wallach, D. & Taatgen, N. (1998). Implicit and explicit learning in ACT-R. In F. E. RitterAnd R. Young (Eds.) Proceedings of the Second European Conference on Cognitive Modeling, pp. 183-189.Nottingham: Nottingham University Press.
Control performance
0510152025
ACT-R D & FExperiment
Trial 41-80Trial 1-40
TargetStates
Concordance
1
baselinecorrectwrong0
.5
.25
.75
Problem solving vs. questionaire
ACT-R Experiment D & F
Transition from computation to retrieval
8060402000.00.20.40.60.81.0
Trials
Conflict resolution
In general, conflict resolution gives answers to two questions:
Which production out of a set of matching productionsis selected?
Which instantiation of the selected production is fired?
activationSequential instantiationNo backtracking
expected gainGoal factoringSuccess probabilityCosts
goal-specific
Expected Gain = –G
Probability ofgoal achievement Goal value
Cost ofgoal achievement
P C*
production-specific
Conflict resolution
Expected Gain =
q r• a b+
–GProbability of
goal achievement Goal valueCost of
goal achievement
P C*
Selection of Productions
q rP*
Probability of Goal Achievement
probability of the production working successfully
probability of achieving the goal if the production works successfully
Achieving a goal depends on the Achieving a goal depends on the joint probability joint probability of of the respective production being successful the respective production being successful andand subsequent rules eventually reaching the goal.subsequent rules eventually reaching the goal.
Production's matching/actions/subgoals Goal accomplished and popped have the intended effect successfully.
Costs of a production
amount of effort (in time) that a pro- duction will take
estimate of the amount of effort from when a pro- duction completes until the goal is achieved
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.
a bC+
Production's costs ofmatching/actions/subgoals Costs of future productions
currentstate
Intendednextstate
goalstate…
{ {{ {
Conflict resolution
q rP*
a bC+
Goal valueG=20
p3
G'=17
!push!
G' = rG-b = .9 * 20 - 1 = 17
p3 parameters:q: 1r: .9a: .05b: 1
ACT-R values a goal less the more deeply it is ACT-R values a goal less the more deeply it is embeddedembeddedin uncertain subgoalsin uncertain subgoalsACT-R pops the goal with failure if no production above ACT-R pops the goal with failure if no production above the utility threshold (default: 0) can match (goal the utility threshold (default: 0) can match (goal abandonment) abandonment)
Noise in Conflict Resolution
Evaluation Ei of production i = P(i)*G-C(i)
Probability of choosing i among n applicable productions with Evaluation Ej
eEi/t
Ej/tej
2t =
Remember:
Boltzmann Equation
2-person Matrix Game
Players
Actions
Payoff matrix A2 B2A1 3, 7 8, 2B1 4, 6 1, 9
Player1, Player2
Actions A, B ...
Data sets
Erev & Roth (1998)
“ There is a danger that investigators will treat themodels like their toothbrushes, and each will use
its own model only on his own data.”
Diverse data sets re-analyzed
2 x 2 4 x 45 x 5
Suppes & Atkinson (1960) [SA2, SA8, SA3k, SA3u]Erev & Roth (1998) [SA3n]Malcom & Liebermann (1965)O'Neill (1987)Rapoport & Boebel (1992) [R&B10, R&B15]
(p player1-A =goal> isa decide player1 nil==> =goal> player1 A)
(p player1-B =goal> isa decide player1 nil==> =goal> player1 B)
game12 isa decide player1 A player2 B
Productions
Chunk
Model
2 4 6 03 3 1 5
1/3 2/3 1 01/2 1/2 1/6 5/6
Best Fits – Random Games
1-ParameterErev & Roth (1998)
Referencepoint=xmin
Reference point= 0
Model ACT-R; Average
Parameter S(1)=15 Par. priors=53Data set Random games Data
setRandom games
100*MSD 1.3 1.087 100*MSD 0.471game #1 0.304 0.4725 game 1 0.288game #2 0.89585 0.4403 game 2 0.163game #3 0.8994 1.235 game 3 0.289game #4 0.28565 0.4074 game 4 0.325game #5 2.0305 1.181 game 5 0.447game #6 0.6485 1.298 game 6 0.903game #7 2.201 0.6303 game 7 0.626game #8 0.4287 1.18 game 8 0.204game #9 3.589 2.523 game 9 1.006game #10 1.7195 1.504 game 10 0.546
Conflict resolution
Goal Match
testgoal pattern(1)
fire production(4)
Ep = 18.95Ep = 18.95
Match
retrieve chunk(s)
(3)
sele
ctio
n
Ep = 13.95
Ep = 17.30
evaluateconflict set(2)
Ep = 18.95Ep = 18.95
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.
Learning as Subsymbolic Tuning to the Statistics of the Environment
Lael SchoolerDeclarative Memory:
Statistical Tuning
1. The goal of declarative memory is to makemost available those memory chunks that aremost likely to be needed at a particular pointin time.
2. The probability of a memory chunk beingrelevant depends on its past history of usageand the current context.
3. Log Odds = Log t
j
− d
j = 1
n
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟ + Context
1008060402000.0
0.1
0.2
(a) New York Times Retention
Days since Last Occurrence
Probabilitity on Day 101
543210-6
-5
-4
-3
-2
-1
(d) New York Times Retention
Log Days
Log Need Odds
Log Odds= - 1.95 - 0.73 Log Days R^2 = 0.993
Odds = .14 T-.73
1008060402000.00
0.02
0.04
0.06
0.08
0.10
0.12
(b) Parental Speech Retention
Utterances since Last occurrence
Probability in Utterance 101
543210-6
-5
-4
-3
-2(e) Parental Speech Retention
Log Utterances
Log Need Odds
Log Odds = - 1.70 - 0.77 Log Utterances R^2 = 0.984
Odds = .18 T-.77
1008060402000.0
0.1
0.2
0.3
(c) Mail Sources Retention
Days since Last Occurrence
Probability on Day 101
543210-5
-4
-3
-2
-1
0
(f) Mail Sources Retention
Log Days
Log Need Odds
Log Odds = - 1.09 - 0.83 Log Days R^2 = 0.986
-.83Odds = .34 T
Parameter learning:log( tj
-d) n
j=1
Lael Schooler’s Research
p(AIDS) = .018 New York Times p(AIDS| associate) p(AIDS| associate)
p(AIDS)virus .75 41.0
Associates spread .54 29.4patients .40 21.8health .27 14.6
Parental Speech p(play) = .0086p(play|game) p(play|game)
p(play).41 47.3
807060504030201000.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
strong contextweak context
(a) CHILDES standard
retention in utterances
need odds
807060504030201000.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
(b) New York Times standard
retention in days
need odds
Environmental Analyses of Context and Recency
5.04.54.03.53.02.52.01.51.00.50.0-7
-6
-5
-4
-3
-2
-1
0
retention in log utterances
log need odds
(c) CHILDES power
4.54.03.53.02.52.01.51.00.50.0-7
-6
-5
-4
-3
-2
-1
0
(d) New York Times power
retention in log days
log need odds
Lael Schooler Retrieval Odds Mirrors Odds of Occurring
Conclusions from Environmental Studies: Log Odds = Log
Proposal for ACT-R’s Declarative Memory: - Activation reflects Log Odds of Occurring
- Learning works as a Bayesian inference scheme to try to identify the right values of the factors determining odds of recall.
t j−d
j=1
n∑
⎛
⎝ ⎜
⎞
⎠ ⎟ + Context
TwelveEight
Four
addition-factsumaddend1
addend2
W SB
Sj ji ji
S
W
ji
j
i
Activation Structure
Ai = Bi + ∑j
Wj Sji Activatio n Equation
Bi=ln t
j
− d
j = 1
n
∑
⎛
⎝
⎜
⎜
⎞
⎠
⎟ Base-Level Learning
Sj i= S - ln((P(i| j )) Stre ngth Learning
Performa nce Str ucture
Mi = Ai - Dp Match Equat ion
Probability = e
M
i
/ t
e
M
j
/ t
j
∑
Chunk Choice
Time i = Fe-fMi Retrieval Time
Declarative Equations
What happens when the probabilistic world of informationretrieval hits the hard and unforgiving world of mathematics?
Christian Lebiere’sSimulation of Cognitive Arithmetic
Over 100,000 problems of each type (1+1 to 9+9; 1x1 to 9x9) over 20years.
Addition MultiplicationIF the goal is to find a + b and a + b = cTHEN the answer is c
IF the goal is to find a * b and a * b = cTHEN the answer is c
IF the goal is to find a + bTHEN set a subgoal to count
b units past a
IF the goal is to find a * bTHEN set a subgoal to add
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
Retrieve
Compute
Small Large0
2
4
6
8
10
1st
4th
7th
10th
College
Problem Size Effect (Data)
Problem Size
RT (sec)
Problem Size Effect over Time
Small Large0
2
4
6
8
10
1st
4th
7th
10th
College
Problem Size Effect over Time
Problem Size
Response Time (sec)
Model
Effect of Argument Size on AccuracyFor Addition (4 year olds)
654321020
30
40
50
60
70
80
AugendAddend
Percentage Correct for Addition Retrieval in the First Cycle (1000 Problems)
Operand
Percentage Correct
654321020
30
40
50
60
70
80
Augend
Addend
Addition Retrieval
Operand
Percentage Correct
Percentage of Correct Retrieval per Operand Percentage Correct in Simulation
Data Model
Effect of Argument Size on AccuracyFor Multiplication (3rd Grade)
1086420
10
20
30
40
50
Multiplicand
Multiplier
Multiplication Computation
Argument
Error Percentage
Percentage of Correct Computations per Operand Percentage Errors in Multiplication Simulation
1086420
10
20
30
40
50
Multiplicand
Multiplier
Error Percentage for MultiplicationComputation in Cycle 3 (~4th Grade)
Argument
Error Percentage
Data Model
Conclusions aboutCognitive Arithmetic
Subsymbolic learning mechanisms that yield adaptiveretrieval in the world at large are behind the 20 yearstruggle that results in the mastery of cognitivearithmetic. Part of the reason why it is a struggle isthat there is noise in the system. However, moredeeply, two things about the arithmetic domain failto match up with the assumptions our memorysystem makes about the world:
1. Precise matching is required.2. High interference between competing memories.
Making Choices: Conflict Resolution
Expected Gain = E = PG-C
Probability of choosing i = e
E
i
/ t
e
E
j
/ t
j
∑
P=
Successe =s α+mFailure =s +n
SuccessesSuccesses +Failures
P is expected probability of successG is value of goalC is expected cost
t reflects noise in evaluation and is like temperature in the Bolztman equation
α is prior successesm is experienced successes is prior failuresn is experienced failures
Procedural Learning
Undershoot
More Successful
Overshoot
More Successful
Looks
Undershoot
10 Undershoot
0 Overshoot
10 (5) Undershoot
10 (15) Overshoot
Looks
Overshoot
10 (15) Undershoot
10 (5) Overshoot
0 Undershoot
10 Overshoot
INITIAL STATE
desired:
current:
building:
UNDERSHOOT UNDERSHOOTOVERSHOOT
desired:
current:
building:
desired:
current:
building:
desired:
current:
building:
possible first moves
a b c
a b c a b c a b c
Building Sticks Task (Lovett)
0
0
0
0
0
1
1
1
1
1
3
3
3
3
3
High
Against
Low
Against
Neutral Low
Toward
High
Toward
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Proportion Choice More Successful Operator
Test Problem Bias
Observed Data
Biased Condition
Extreme-Biased Condition
0
0
0
0
0
1
1
1
1
1
3
3
3
3
3
High
Against
Low
Against
Neutral Low
Toward
High
Toward
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Test Problem Bias
0 0
0
00
1 1
1
11
3 3
3
3 3
High
Against
Low
Against
Neutral Low
Toward
High
Toward
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Proportion Choice More Successful Operator
Test Problem Bias
0 0
0
0 0
1 1
1
1 1
3 3
3
3 3
High
Against
Low
Against
Neutral Low
Toward
High
Toward
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Test Problem Bias
Predictions of Decay-Based ACT-R
(2/3) (5/6)
Lovett & Anderson, 1996
Build Sticks DemoDecide-Under If the goal is to solve the BST task and the undershoot difference is less than the overshoot differenceThen choose undershoot.
Decide-Over If the goal is to solve the BST task and the overshoot difference is less than the undershoot differenceThen choose overshoot.
Force-Under If the goal is to solve the BST taskThen choose undershoot.
Force-Over If the goal is to solve the BST taskThen 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 afterproblem-solving experience in Experiment 3
(Lovett & Anderson, 1996)
ProductionPrior
Probabilityof Success
Final Value
67% Condition 83% Condition
Force-Under
More Successful
Context Free
.50 .60 .71
Force-Over
Less Successful
Context Free
.50 .38 .27
Decide-Under
More Successful
Context Sensitive
.96 .98 .98
Decide-Over
Less Successful
Context Sensitive
.96 .63 .54
Successes ( t ) = t
j
− d
j = 1
m
∑ Success Discounting
Failures ( t ) = t
j
− d
j = 1
n
∑ Failure Discounting
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
U, U O, U U, O O, O
Outcome on Trial N-2, N-1
Data
Decay Model
No Decay Model
Decay of Experience
Note: Such temporal weighting is critical in the real world.
• But, what happens when there is more than onecritical choice per problem?
-How is credit/ blame assigned by human problemsolvers?
-How well does ACT-R's learning mechanism handlethis more complex case?
-In ACT-R all choices leading to goal resolution areequally weighted.
-But, is there evidence for a goal gradient?
Credit-Assignment in ACT-R
INITIAL STATE
UNDERSHOOTOVERSHOOT
desired:
current:
building:
desired:
current:
building:
desired:
current:
building:
ab
c
ab
c a b c
add cadd b
delete c add c
desired:
current:
building:
ab
c
desired:
current:
building:
a b c
delete a
desired:
current:
building:
a b c
delete a
desired:
current:
building:
a b c
delete c
desired:
current:
building:
a b c
add a
desired:
current:
building:
a b c
add cadd a
desired:
current:
building:
a b c
add a
desired:
current:
building:
a b c
desired:
current:
building:
a b c
desired:
current:
building:
ab
c
delete a
MAINTAIN REVERSEMAINTAIN REVERSE
75%
75% 75%
Building Sticks Task 2 Levels
Choice Learning
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
1 2 3 4
Block
1st - 100% 1st-75%2nd-75% 2nd 50%1st-25% 2nd-25%
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 arecritical to getting good performance in games as simple as rocks-papers-scissors.
Adapting to a Variable and Changing World
AC
T-R
/PM
CognitionLayer
Perceptual/Motor Layer
DeclarativeMemory
ProductionMemory
Environment
VisionModule
icon
MotorModule
AuditionModule
audicon
SpeechModule
attentiontarget of attention(chunks)
pixels
raw audio
clicks,keypresses,
etc.
attentiontarget ofattention(chunks)
audio
Martin-Emerson-Wickens Task
Zur Anzeige wird der QuickTime™ Dekompressor “Photo - JPEG”
benötigt.
Martin-Emerson& Wickens (1992):The vertical visual
field and implicationsfor the head-up
display
Perform compensatory tracking,keeping the crosshair on target
Respond to choice stimuli asrapidly as possible
Choice stimulus appears at various distances from target(vertical separation)
Tracking requires eye to be onthe crosshair
Eye must be moved to see stimulus
Choice response & tracking move-ments are bottlenecked throughsingle motor module
(Dual-)Task
Model
Find-Target-Oval IF the target hasn't been located and the oval is at locationTHEN 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 freeTHEN 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 freeTHEN 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 freeTHEN send a command to track the pointer and update the state to "tracking"
MEW Productions
Move-Cursor IF the state is "tracking" and the target is at location and the motor module is freeTHEN 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 toTHEN 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 freeTHEN 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 freeTHEN send a command to punch the left middle finger and clear the state
VM RS
+ AV P
VM Init
AV RS Cognition
Motor
Vision
Audition
VM
AV
VM Feature Prep
AV Feature Prep AV Init AV Detect Speech
VM Detect
Schedule chart for Schumacher, et al. (1997) perfect time-sharing model. VM = visual-Manual ask, AV = auditory-verbal task, RS = response selection.
Location Discrim. Tone Discrim.
0
50
100
150
200
250
300
350
400
450
500
Task
Single-task
Dual-task
ACT-R/PM simulation of Schumacher, et al. (1997) perfect time-sharing results.