Tweedledum and Tweedledee:
Symmetry in Behavior Analysis
M. Jackson Marr
Georgia Institute of Technology
mm27@prism.gatech.edu
Maxwell’s Equations
• Symmetry between electricity and magnetism: Explained
the nature of light as electromagnetic waves moving at
speed c.
Einstein’s Principles of Relativity
• (1) Laws of physics independent of states of motion.
• (2) Constancy of the speed of light, independent of
states of motion.
• One implication: Equivalence of mass and energy.
E = mc2
SYMMETRY AND CONSERVATION LAWS
• Emmy Noether’s Theorem: For every symmetry
exhibited by a physical law, there is a corresponding
observable quantity that is conserved.
• Momentum: translations in space (including rotations).
• Energy: translations in time.
• Many other examples of symmetry/conservation, e.g., in
particle physics (e.g., CPT—charge, parity, time—all
together.).
SYMMETRIES IN BEHAVIOR ANALYSIS
Positive and negative reinforcement
Positive and negative punishment
Reinforcement and punishment
Discrimination and generalization
Shaping dynamics
Three-term contingency
Equivalence and n-term contingency
Schedule performance
Generally, invariances in behavior-analytic principles
(contingencies, species, intra- and inter-individual, and
socio-cultural behaviors)
MEN ACT UPON THE WORLD,
AND CHANGE IT, AND ARE
CHANGED IN TURN BY THE
CONSEQUENCES OF THEIR
ACTION.
B. F. Skinner, Verbal Behavior, 1957, p. 1
Operant Conditioning
O-rules,
functional relations
B = f (r)
r:
B:
feedback
output
E-rules,
feedback functions
r = g (B)
Figure 1. The behavior-environment feedback system
Operant Conditioning
Three-term Contingency
SD
SD: discriminative
stimulus
SR
Ro
Ro: operant class
SR: reinforcer
Newton’s Second Law of
Motion
F m




dv 
dt 
F: force
m: mass
dv/dt: acceleration
The Equivalence Relation:
Ordered pairs of all
elements that participate in
the reinforcement
contingency
EQUIVALENCE RELATION
GIVEN SETS: A, B, & C
A = A; B = B; C = C (reflexivity or matching)
AND
A = B → B = A; B = C → C = B (symmetry)
AND
IF A = B and B = C, then A = C (transitivity)
Operant Conditioning
Operant Conditioning
Scale Invariance in FI Schedules with  = 1
If
f (x) = c xα
then,
log f (x) = log c + α log x.
This is a linear function on a loglog scale, with slope  (the
scaling factor).
Scaling Rate and Response Number in FI Schedules
Scaling in IRT>t Schedules
IRT>t Scheduled Value
MATCHING LAW
• R1 / R2 = r1 / r2
Herrnstein
• R1 / R2 = b (r1 / r2)a
Baum
HERRNSTEIN’S HYPERBOLA
Conservation of Behavior:
Exclusive choice is conserved; either you do one
thing or another, each action with differential
consequences (r and r0), and total behavior in
the situation is constant (k).
B  r krro
FI 10 (FR 20: Sp)
Symmetry, as wide or as narrow as
you define its meaning, is one idea
by which man through the ages has
tried to comprehend and create
order, beauty and perfection.
Hermann Weyl, Symmetry (1952, p.
5)