Spooky Memory Models Jerome Busemeyer Indiana Univer Univ sity er

Transcription

1 Spooky Memory Models Jerome Busemeyer Indiana University

2 Associative Memory star t = planet c=universe space Mars Notice: No return from space or mars to planet

3 Failure of Spreading Activation Model Why should associative links activate a target when there are no links back to target? Recall predicted best by simply summing all associative strengths for a target. Recall not well predicted d by summing product of links starting and returning to target

4 Intra list cue Collapse of Meaning train on paired associates (space, universe) this helps specify the meaning of space cue with universe retrieve space eliminates effects of associate links Extra list cues train on list without knowing purpose p target word space remains ambiguous sizeable effects of associative links on recall

5 Quantum Memory Retrieval Model

6 Short History 1920 s Heisenberg, Schroedinger, Born, Dirac ect. Quantum Mechanics 1930 s Von Neumann, Quantum Logic and Quantum Probability 1950 s Feynman Quantum computing 2008 Doug Nelson Nl wins Nobel lin Physics for Quantum Memory Theory

7 Probability Theory There are at least two mathematical approaches for building probabilistic systems Classic Kolmogorov probabilities Quantum probabilities Almost all previous theorizing i in Cognitive Science is based on classic probability systems This is an exploration of an alternative Quantum approach

8 Representation n Dimensional Hilbert Space A basis is formed by n orthonormal vectors { f1, f2,, fn } f i f j = 0 pairwise orthogonal f i f i = 1 unit length

9 2 z y x A feature Space

10 Superposition State Experience prepares a state for each memory memory j = Σψ i. f i memory k = Σ φ i. f i ect. Tensor product space, state combination memory j memory k..

11 2 z v y x V is a memory state

12 A Simple Retrieval Cue If presented with a cue, do you retrieve a target word? cue j = α i f i cue k = β i f j ect. P[ ] j d i 2 Pr[ yes ] = cue j word i 2 Squared inner product between vectors

13 2 z w v y x V is a memory, y, W is a retrieval cue

14 Retrieval as Projection Proj(cue j) = cue j cue j outer product projects memory onto cue j

15 z is memory x is a cue x1 = recall x2 = no recall z Pr(x1) = P x1 z 2 x1 x2 P x1 z x1

16 Proj(cue j) mem i = cue j cue j mem mem j = cue j cue j memory i = cue j memory i cue j Pr[yes] = Proj(cue j) mem i 2 = cue j memory i 2

17 Collapse of State State before making measurement by a cue memory j = Σψ i. f i State following a measurement of yes to a cue cue j = α i f i

18 Uncertainty Principle y1 x2 x1 = recall using x as cue x2 = fail using x as cue x1 = recall using x as cue, x2 = fail using x as cue y1 = recall using y as cue, y2= fail using y as cue y2 x1

19 Quantum Logic A quantum event (such as the event L x ) is defined geometrically as a subspace (e.g. a line or plane or hyperplane) of a Hilbert (vector) space H. Null event is the zero point, 0. Universal event is H itself. New events can be formed from other events by A complement operation, denoted dll x, which h is defined as the subspace that is orthogonal to L x. A meet operation x y which is defined by intersection of two subspaces L x L y. A join operation x y defined as the span of two subspaces L x, L y.

20 2 1.5 w z Failure of distributive axiom of Boolean logic v 0 u y x (Lu Lw) (Lx Ly Lz)) = Lu LwLw (Lu Lw) (Lx Ly) (Lu Lw) LzLz = Lu

21 Projective Probabilities Probabilities for events are determined from a unit length state vector z Hwith z z = 1. Each event L x corresponds to a projection operator P x that projects vectors in H onto L x The probability bili of an event L x is equal to Pr(x) = P x z 2, the squared length

22 Complement of a Cue Proj(~cue) = I cue cue = ~cue ~cue [I cue cue ] cue = I cue cue cue cue = 0 cue [I cue cue ] ~cue = I ~cue cue cue ~cue = 1 ~cue

23 Interference Effects: Condition 1 present pese tcue 1, retrieve e e associate ate (yes or no) ; followed by present cue 2, retrieve associate (yes or no) Pr[ yes1, yes2] = cue 2 cue 1 2 cue 1 ws 2 Pr[ no1, yes2] = cue 2 ~cue 1 2 ~cue 1 ws 2 Pr[ yes2] = Pr[ yes1, yes2] + Pr[ no1, yes2] Obeys Law of total probability

24 Interference Effects: Condition 2 Present cue 2 alone Pr[Yes 2] = cue2 wordstate 2 = cue2 I wordstate 2 = cue 2 ( cue1 cue1 + ~cue1 ~cue1 ) ws 2 = cue2 cue1 cue1 ws + cue2 ~cue1 ~cue1 ws 2

25 Interference Condition 1: Measure 2 cues Pr[yes2] = cue 2 cue 1 2 cue 1 ws 2 + cue 2 ~cue 1 2 ~cue 1 ws 2 Condition 2: Measure only second cue Pr[yes2] = cue2 cue1 cue1 ws + cue2 ~cue1 ~cue1 ws 2

26 Conclusion Quantum memory has great power quantum logic useful as a search engine Quantm memory is still under construction as a model of human memory Quantum memory models are not so spooky after all Maybe Doug can win a Nobel prize after all?