Quant-Markov by stariya

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									 Distinguishing quantum and Markov models of human decision making
   Jerome R. Busemeyer1, Efrain Santuy1, Ariane Lambert Mogiliansky2
                                                    1
                                                  Cognitive Science, Indiana University
                                             1101 E. 10th Street, Bloomington Indiana, 47405
                                              jbusemey@indiana.edu ellorent@indiana.edu

                                                2
                                                 PSE Paris-Jourdan Sciences Economiques
                                                         ariane.LM@gmail.com



                           Abstract                                     models are Markov models, and they are also closely
  A general property for empirically distinguishing Markov              related to the optimal Bayesian sequential sampling model
  and quantum models of dynamic decision making is                      (Bogatz, et. al., 2006).
  derived. A critical test is based on measuring the decision
  process at two distinct time points, and recording the                Recently, we (Busemeyer, Townsend, & Wang, 2006)
  disturbance effect of the first measurement on the second.            developed a quantum dynamic model for the signal
  The test is presented within the context of a signal detection        decision task. Thus an important question is what
  paradigm, in which a human or robotic operator must decide            fundamental properties distinguish these two very different
  whether or not a target is present on the basis of a sequence         classes of models?
  of noisy observations. Previously, this task has been
  modeled as a random walk (Markov) evidence accumulation
  process, but more recently we developed a quantum                     First we describe the formal characteristics of a signal
  dynamic model for this task. Parameter free predictions are           detection task. Second, we describe Markov and quantum
  derived from each model for this test. Experimental                   models of signal detection. Third we derive theoretical
  methods for conducting the proposed tests are described,              properties from each model that provide a parameter free
  and previous empirical research is reviewed.                          test of the two models. Finally, we summarize
                                                                        experimental results that provide initial evidence for
Interference is a signature of quantum processes. One way               interference effects in human decision making.
to produce interference effects is to perturb a target
measurement by an earlier probe. In this article we develop             The Signal Detection Task
Markov and quantum models of human decision making                      Signal detection can be a very complex task. But our goal
and prove that only the latter is capable of producing this             is to describe a simple setting for testing models of this
type of interference effect. The decision models are                    task. Imagine, for example, a security agent who is
developed within the context of a decision paradigm called              checking baggage for weapons. With this idea in mind, we
the signal detection paradigm.                                          assume that a decision maker is faced with a series of
                                                                        choice trials. On each trial, a noisy stimulus is presented
Signal detection is a fundamentally important decision                  for some fixed period of time, tf, and at the end of the fixed
problem in cognitive and engineering sciences. In this                  time period, the decision maker has to immediately decide
dynamic decision situation, a human or robotic operator                 whether or not a target was present. The evidence
monitors rapidly incoming information to decide whether                 generated by the stimulus is assumed to be stationary
or not a potential target is present (e.g., missiles in the
                                                                        during the duration of the stimulus. The amount of time
military, or cancer in medicine, or malfunctions in
                                                                        given to observe the stimulus can be experimentally
industry). The incoming information provides uncertain
and conflicting (noisy) evidence, but at some point in time,            manipulated.
a decision must be made, and the sooner the better.
Incorrectly deciding that the target is present (i.e. a false           The operator can respond by choosing one of 2n+1 ordered
alarm) could result in deadly consequences, but so could                levels of confidence (n is assumed to be an even number).
the failure to detect a real target (i.e. a miss).                      For example, n = 2 produces five levels that might be
                                                                        labeled: 2 = no target with high confidence, 1 = no
In statistics, Bayesian sequential sampling models provide              target with low confidence, 0 = uncertain, +1 = yes target
an optimal model for this task (DeGroot, 1970). During the              with low confidence, +2 = yes target with high
past 35 years, cognitive scientists (see, Ratcliff & Smith,             confidence). The choice probabilities of the 2n+1
2004, for recent review) have developed random walk                     categories are estimated from each person pooled across
model and diffusion models to describe human                            responses made on several thousand choice trials.
performance on this task. The random walk/diffusion
Models of Signal Detection                                          from state |j. The intensities must satisfy kij  0 for i  j
                                                                    and  i kij = 0 to guarantee that T(t) remains a transition
A random walk (Markov) model                                        matrix.
To construct a random walk (Markov) model for this task,
we postulate a set of (2m+1) states of confidence about the         The random walk model is a special case of a Markov
                                                                    process which assumes that intensities are positive only
presence or absence of the target:
   { |m, |m+1 , …, |1, |0, |+1, …, |+m1, |+m }.          between adjacent states: kij = 0 for |ij| > 1. For |j| < m, we
The state |j can be interpreted as a (2m+1) column vector          make the following assumptions: If the target is present
                                                                    then we assume that kj+1,j > kj1,j; if the target is absent then
with zeros everywhere, except that it has 1.0 located at the
row corresponding to index j. Positive indices, j > 0,              we assume that kj1,j > kj+1,j. At the boundaries, we set
represent a state of evidence favoring target present;              km1,+m > 0 and km+1,m > 0 (reflecting boundaries).
negative indices, j < 0, represent a state of evidence for
target absent, and zero represents a neutral state of               Note that the discrete state random walk model is closely
evidence. The number of states could be as small as the             related to a continuous state diffusion model. If we allow
                                                                    the number of states to become arbitrarily large, and at the
number of response categories (m = n), but the participant
may be able to employ a more refined scale of confidence,           same time let the increment between states become
and so the number could also be much larger (m = 10n).             arbitrarily small, then the distribution produced by the
                                                                    random walk model converges to the distribution produced
At the start of a single trial, the process starts in some          by the diffusion model (see Bhattacharya & Waymire,
particular state, for example the neutral state |0. Then as        1990, pp. 386-388).
time progresses within the trial, the process either steps up
one index, or steps down one index, or stays at the same            The response measured at time tf is determined by the
state, depending on the information or evidence that is             following choice rule. The entire set of states is partitioned
                                                                    into 2n+1subsets, with the first subset define by Rn = { |j
sampled at that moment. The process continues moving up
or down the evidence scale until the fixed time period tf           | m  j  cn), and subsequent subsets defined by Rk = { |j
ends and a decision is requested. At that point, a choice           | ck-1  j  ck) with cn = +m. If the state of confidence at
                                                                    time tf equals |jRk, then the response category k is
and a rating are made based on the state existing at the time
of decision.                                                        selected. The probability of this event is simply
                                                                       Pr[R(tf) = k] =  j Rk Pj(tf).
The initial state of the decision maker on each choice trial
is not known, and this initial state may even change from           For later comparisons with the quantum model, it will be
trial to trial. The probability of starting in state |j is         helpful to redefine the computation of the response
                                                                    probabilities in matrix notation. Define Mk as a projection
denoted Pj(0), and the (2m+1) column vector
   P(0) =  j Pj(0)|j                                             matrix with elements Mjj = 1 if |jRk, and zero otherwise.
represents the initial probability distribution over states so      Note that  k=1,n Mk = I, where I is the identity matrix. Then
                                                                    MkP(tf) is the projection of the probability vector P(tf) onto
that  j Pj(0) = 1.
                                                                    the subspace defined by the states that are assigned to
During an observation period of time t of a single trial, the       category k. Define 1 as a (2m+1) row vector with all entries
                                                                    equal to one. Then the desired probability equals the sum
probability distribution over states, P(t), evolves in a
direction guided by the incoming evidence from the                  of the elements in the projection:
stimulus. This evolution is determined by a transition                 Pr[R(tf) = k] = 1MkP(tf) .
matrix T(t) as follows:
    P(t) = T(t)  P(0).                                             A quantum dynamic decision model
The entry in row i and column j of T(t), Tij(t), determines         To construct a quantum model for this task, we again
the probability of transiting to state |i from state |j after a   postulate a set of (2m+1) states of confidence about the
period of time t. The transition matrix must satisfy 0             presence or absence of the target:
Tij(t)  1 and  i Tij(t) = 1 to guarantee that P(t) remains a        { |m, |m+1 , …, |1, |0, |+1, …, |+m1, |+m }.
probability distribution.                                           In this case, we assume that these states form an
                                                                    orthonormal basis for a (2m+1) dimensional Hilbert space.
The transition matrix satisfies the group property                  More specifically, i|j = 0 for every i  j and i|i = 1 for all
  T(tf) = T(tf ti)T(ti),                                          i, where i|j denotes the inner product between two
and from this it follows that it satisfies the Kolmogorov           vectors. As before, states with positive indices represent
forward equation:                                                   evidence favoring target present, states with negative
  dT(t)/d(t) = KT(t),                                              indices represent evidence for target absent, and zero
which has the solution                                              represents a neutral state of evidence.
  T(t) = exp(tK).
The matrix K is called the intensity matrix with element Kij        The initial state of the quantum system is represented by a
determining the rate of change in probability to state |i          superposition of the confidence states
   |(0) =  j |jj|(0) =  j j(0)|j.                          H = P2/2m + V(x).
The coefficient j(0) = j|(0) is the probability amplitude     The off diagonal elements of the Hamiltonian correspond
of being in state |j at start of the trial. The complex          to the P2 operator, and the diagonal elements correspond to
(2m+1) column vector (0) represents the initial                  the potential function V(x) (see Feynman et al. (1966, Ch.
probability amplitude distribution over the states. The           16).
coordinate in the j-th row of (0) is j(0) = j|(0). The
initial state vector has unit length so that (0)|(0) =        The quantum response probabilities measured at time tf are
(0)†(0) = 1.0. The initial probability amplitude                determined as follows. Once again the entire set of states is
distribution over states represents the decision maker’s          partitioned into 2n+1subsets, with the first subset define by
initial beliefs at the beginning of the trial.                    Rn = { |j | m  j  cn), and subsequent subsets defined
                                                                  by Rk = { |j | ck-1 < j  ck) with cn = +m. The probability
During an observation period of time t of a single trial, the     of choosing category k at time tf is simply
probability amplitude distribution over states, (t), evolves        Pr[R(tf) = k] =  j Rk |j(tf)|2.
in a direction guided by the incoming evidence from the
stimulus. This evolution is determined by a unitary matrix        As before, it will be helpful to describe these computations
U(t) as follows:                                                  in matrix formalism. Define Mk as a projection matrix with
    (t) = U(t)  (0).                                           elements Mjj = 1 if |jRk, and zero otherwise. Note again
The entry in row i and column j of U(t), Uij(t), determines       that  k=1,n Mk = I, where I is the identity matrix. Then
the probability amplitude of transiting to state |i from state   Mk(tf) is the projection of the probability amplitude
|j after a period of time t. The unitary matrix must satisfy     vector (tf) onto the subspace defined by the states that are
U(t)†U(t) = I (where I is the identity matrix) to guarantee       assigned to category k. Then the probability is given by
that (t) remains unit length.                                    the squared length of the projection:
                                                                     Pr[R(tf) = k] = |Mk(tf)|2
The unitary matrix satisfies the group property                      = (tf)†Mk†Mk(tf) = (tf)†Mk(tf).
  U(tf) = U(tf ti)U(ti),
and from this it follows that it satisfies the Schrödinger        The disturbance effect of measurement.
equation:
  dU(t)/d(t) = iHU(t),                                         We modify the signal detection task by asking the decision
which has the solution                                            maker to report a confidence rating at two time points, ti
  U(t) = exp(itH).                                             initially and later tf. The initial confidence measurement
The complex number i = 1 is needed to guarantee that            will cause a ‘state collapse’ with both the Markov and the
U(t) is unitary. The matrix H is called the Hamiltonian           quantum models. But how does this change the final
matrix with element Hij determining the rate of change in         distribution of probabilities? This simple manipulation has
probability amplitude to state |i from state |j. The            profoundly different effects on the two models. The
Hamiltonian matrix must be Hermitian (H = H†) to                  quantum model exhibits an interference effect that does not
guarantee that U(t) remains unitary.                              occur with the Markov model.

To construct a random walk analogue, we assume that               The general idea is to compare the results from two
Hamiltonian elements are non zero only between adjacent           experimental conditions at time tf. Under the single
states: Hi,j = 0 for |ij| > 1. For rows |j| < m, we make the     measurement condition (C1), we can estimate the
following assumptions: Hj1,j = Hj+1,j =  and Hjj = j. If       probability distribution over the category responses at time
the target is present, then we assume that j+1 > j ; if the     tf , that is Pr[ R(tf) = k | C1]. Under the double measurement
target is absent then we assume that j+1 < j. At the            condition (C2), we can obtain another estimate of the
boundaries we set Hm1,+m =  = Hm+1,m and Hm,m = m         probability distribution over the category responses at time
Hmm = m. This choice of Hamiltonian corresponds to a             tf using the law of total probability
crystal model discussed in Feynman et al. (1966, Ch. 16).             Pr[ R(tf) = k | C2]
                                                                      = k’ Pr[ R(tf) = k  R(ti) = k’ ],
Note that the discrete state quantum walk model is closely            = k’ Pr[ R(tf) = k | R(ti) = k’ ] Pr[ R(ti) = k’ ].
related to a continuous state quantum model. Once again, if
we allow the number of states to become arbitrarily large,        Effects of measurement on the Markov model.
and at the same time let the increment between states             According to the Markov model, the distribution over the
become arbitrarily small, then the distribution produced by       confidence states immediately before measurement at time
the discrete quantum model converges to the distribution          ti will be
produced by the continuous quantum model (see Feynman                P(ti) = T(ti)  P(0).
et al., 1966, Ch. 16). Also note that the Hamiltonian             The probability of choosing category k’ is
defined above is closely related to the Hamiltonian that is          Pr[ R(ti) = k’] = 1Mk’P(ti) .
commonly used to model the one dimensional movement
of a particle in physics:
If category k’ is selected at this point, then the probability        The projection of this final distribution on the basis states
distribution over states collapses to a new distribution after        of response category k at time tf equals
measurement                                                                 = MkU(tf  ti)(ti|k’)
    P(ti|k’) = Mk’P(ti) / 1Mk’P(ti).                                     = MkU(tf  ti)[Mk’(ti)/|Mk’(ti)|].
The joint probability of selecting category k’ and then               The joint probability of selecting category k’ and then
reaching one of the (2m+1) final states at time tf equals             selecting category k at time tf equals
   [T(tf  ti)P(ti|k’)]Pr[R(ti) = k’] = T(tf  ti)  Mk’ P(ti) .      Pr[ R(tf) = k  R(ti) = k’ ]
The joint probability of selecting category k’ and then                     = Pr[ R(ti) = k ‘]|MkU(tf  ti)[Mk’(ti)/|Mk’(ti)|]|2
selecting category k at time tf equals                                      = |Mk’(ti)|2(|MkU(tf  ti)Mk’(ti)|2)/|Mk’(ti)|2
   Pr[ R(tf) = k  R(ti) = k’ ] = 1MkT(tf  ti)Mk’P(ti).                = |MkU(tf  ti)Mk’(ti)|2
The marginal probability of selecting category k at time tf                 = |MkU(tf  ti)Mk’U(ti)(0)|2 .
equals                                                                The marginal probability of selecting category k at time tf
   Pr[ R(tf) = k | C2]                                                equals
      = k’ Pr[ R(tf) = k  R(ti) = k’ ]                                 Pr[ R(tf) = k | C2]
      = k’ 1MkT(tf  ti)Mk’ P(ti)                                      = k’ Pr[ R(tf) = k  R(ti) = k’ ] ,
      = 1MkT(tf  ti)k’ Mk’P(ti)                                       = k’ |MkU(tf  ti)Mk’U(ti)(0) |2 ,
      = 1MkT(tf  ti)( k’ Mk’)P(ti)                                     | k’ MkU(tf  ti)Mk’U(ti)(0) |2 ,
      = 1MkT(tf  ti)IP(ti)                                             = | MkU(tf  ti) k’ Mk’U(ti)(0) |2 ,
      = 1MkT(tf  ti)P(ti)                                               = | MkU(tf  ti) (k’ Mk’)U(ti)(0) |2 ,
      = 1MkT(tf  ti)T(ti)P(0)                                          = | MkU(tf  ti) I U(ti)(0) |2 ,
      = 1  Mk T(tf )P(0)                                                 = | MkU(tf  ti)U(ti)(0) |2 ,
      = Pr[ R(tf) = k | C1].                                                = | MkU(tf )(0) |2 ,
Thus the first measurement has no effect on the marginal                    = Pr[ R(tf) = k | C1] .
distribution of the second measurement. The law of total              The two results are not equal, and the first measurement
probability is satisfied. However, the first measurement              does effect the marginal distribution of the second
does influence the distribution of the second measurement,            measurement. The law of total probability is not satisfied.
conditioned on the observed value of the first                        As noted above, it is important for us to compare models
measurement. So it is important for us to compare models              using the marginal distribution from the two conditions.
using the marginal distribution.                                      The difference between the two marginal distributions,
                                                                         d(tf ti) = Pr[ R(tf) = k | C2]  Pr[ R(tf) = k | C1],
It is important to note that the above results hold quite             is called the interference effect, and it depends strongly on
generally. It holds for any number of confidence states (as           the time lag (tf ti).
long as this number is equal or greater than the number of
response categories). It holds for any transition matrix T(t)         It is important to note that the above results hold quite
and so it is not restricted to any particular form of intensity       generally. Once again, it holds for any number of
matrix. Finally, it holds for any initial probability                 confidence states (as long as this number is equal or greater
distribution across the confidence states. This property              than the number of response categories). It holds for any
tests a basic assumption of linearity that is implicit in the         unitary matrix U(t) and so it is not restricted to any
Markov model:                                                         particular form of Hamiltonian matrix. Finally, it holds for
       1  Mk T(t)[pP1(0) + qP2(0)] ,                             any initial probability amplitude distribution across the
   = p[1  Mk T(t) P1(0)] + q[1  Mk T(t) P2(0)].               confidence states. This property tests a basic assumption of
                                                                      nonlinearity that is implicit in the quantum model:
Effects of measurement on the quantum model.                                 |Mk U(t)[pP1(0) + qP2(0)]|2 ,
According to the quantum model, the probability amplitude                 p|Mk U(t) P1(0)|2 + q|Mk T(t) P2(0)|2.
distribution over the confidence states immediately before
measurement at time ti equals                                         Example of interference effects.
   (ti) = U(ti)  (0).
                                                                      To illustrate the predicted interference effect of the
The probability of choosing category k’is
   Pr[ R(ti) = k’] = |Mk’(ti)|2 .                                   quantum model, the predictions for the single and double
                                                                      conditions were computed using the following parameters:
If category k’ is selected at this point, then the amplitude
                                                                      (2m+1) = 51, 2K+1 = 3, c1 = 9, c+1 = +8, 1(0) = 0(0)
distribution over states collapses to a new distribution after
measurement                                                           = +1(0) = 1/3,  = .5, j = j/51, tf = ti+50. The differences
                                                                      between the double versus the single conditions are shown
    (ti|k’) = Mk’ (ti)/ |Mk’(ti)| .
                                                                      in the table below for various ti. As can be seen in this
The probability amplitude distribution over the (2m+1)
final states at time tf given k’ observed at time ti equals           table, the interference effects can be quite large.
      = U(tf  ti)(ti|k’),
      = U(tf  ti)[Mk’(ti)/|Mk’(ti)|] .
                  Interference Effect                            Shafir and Tversky (1992) tested a property called the sure
Initial time   k = 1          k=0            k = +1             thing principle, which could be re-interpreted as a test of
ti = 0         0               0              0                  interference effects (Busemeyer, Matthew, Wang, 2006).
ti = 100       .0385           .0251          -.0636             Decision makers were asked to play a prisoner dilemma
ti = 200       .0177           .0865          -.1043             game under three conditions: Knowing the opponent has
ti = 300       .0592           .1509          .2101             defected, knowing the opponent has cooperated, and not
Note: These predictions were computed from the quantum           knowing the opponent’s action. This paradigm can be used
equations using Matlab’s matrix exponential function.            to determine the interference effect produced by
                                                                 knowledge of the opponent on the choice of the decision
Proposed method for investigating the interference               maker. Decision makers defected 97% of the time when
effect. An experiment is underway to experimentally              the other was known to defect; 84% of the time when the
investigate the disturbance effect of measurement. A signal      other was known to cooperate; and 63% of the time when
detection task was designed that requires an average             the other’s action was unknown. Busemeyer et al. (2006)
decision time of at least 1 second or more. Within this          showed that these results also violate the predictions of a
amount of time, a probe measurement can be made, say             Markov model, and they explained these findings using a
within 500 msec, and a final measurement can obtained            quantum dynamic decision model.
after 1000 msec. The participants are asked to view a
complex visual scene and decide whether or not a target          Conte et al. (2006) conducted an experiment to test
object (e.g. a bomb) is present or not, similar to the task      interference effects in perception. The task required
faced by security agents at airports or government               individual’s to judge whether or not two lines were
buildings. The time interval between probe and final             identical. The lines were in fact identical, but they were
measurement will be manipulated, as well as the                  presented within a context that is known to produce a
discriminability of the stimulus and prior probability of the    perceptual illusion of a difference. Two conditions were
target. According to our quantum model, the time interval        examined: In one condition, line judgment task B was
would affect the delay parameter (tfti), the discriminability   presented alone; in another condition, line judgment task A
would affect the potential of the Hamiltonian, j, and the       preceded line judgment task B. The results showed
prior probability would affect the initial state, (0).          significant differences in the response proportions to task
                                                                 B. These results were interpreted within a general quantum
Related empirical tests of interference effects                  measurement framework developed by Khrennikov (2007).
in human decision making.
                                                                 Atmanspacher, Filk, & Romer (2004) examined a quantum
                                                                 zeno effect in a bi-stable perception task. The task involves
Although we are not aware of any experimental tests of
                                                                 the presentation of a Necker cube, which is the projection
interference effects that were conducted using the single
                                                                 of a cube onto a plane. The projection can be perceptually
and double measurement conditions described above,
                                                                 interpreted as being viewed from a top or bottom
several related lines of evidence have been reported by past
                                                                 viewpoint, and the viewer experiences a spontaneous
researchers.
                                                                 switch from one view to another. The time period between
                                                                 switches in experienced viewpoints is the primary
Townsend, Silva, & Spencer – Smith (2000) conducted a
                                                                 measurement of interest. According to quantum theory,
closely related test of interference. Decision makers were
                                                                 this time period can be extended by repeated measurements
presented faces belonging to one of two categories (e.g.,
                                                                 of the perception. Atmanspacher, et. al. (2004) report some
good guys, bad guys) and they were asked to decide to
                                                                 experimental evidence related to this effect, and they
choose between two actions (e.g. attack or withdraw). Two
                                                                 explain these results using a quantum dynamic model.
conditions were examined: In the category  decision task
they were asked to first categorize the face, and then decide
how to act; in the decision only task, they were only asked
                                                                    Comparing Markov and quantum models
to decide how to act and no categorization was requested.
                                                                 There is a surprising amount of similarity between the
This paradigm was used to investigate the interference
                                                                 Markov and quantum models. The basic equations appear
effect of the category task on the decision task. Townsend
                                                                 almost the same if one simply replaces probabilities with
et al. (2000) reported that 72 out of 276 participants
                                                                 complex amplitudes. The initial states of both models can
produced statistically significant deviations from the
                                                                 be represented as a linear combination of basis states
predictions of a Markov model. Busemeyer & Wang
                                                                 (mixed versus superposition). The transition operators that
(2006) explained these deviations using a quantum
                                                                 evolve the states both obey simple group properties which
dynamic decision model.
                                                                 lead to deterministic linear differential equations
                                                                 (Kolmogorov versus Schrödinger). The solutions to the
                                                                 differential equations are matrix exponentials for both
models. Measurement produces a collapse of the states for          All of the above applications follow a quantum decision
both models.                                                       program of research that uses only the mathematical
                                                                   principles of quantum theory to explain human decision
Based on these similarities, one might suspect that the two        making behavior (Aerts, et al., 2003; Atmanspacher, et al.,
models are indistinguishable. This is not the case because         2002; Khrennikov, 2007). No attempt or assumptions are
there are several key differences between the models. First,       made at this point about the possible neural basis for these
the Markov model operates on real valued probabilities             computations. This program differs radically from a more
(bounded between zero and one), whereas the quantum                reductionist program that attempts to explain neural
model operates on complex probability amplitudes.                  computations using quantum physical models (Penrose,
Second, the intensity matrix for the Markov model obeys            1989; Pribram, 1993; Woolf & Hammeroff, 2001)
different constraints than the Hamiltonian for the quantum
model. Third, the Schrödinger equation introduces a                                  Acknowledgments
complex multiplier to maintain a unitary operation. Finally,       This research was supported by NIMH R01 MH068346 to
the Markov model uses a linear projection to determine the         the first author.
final probabilities, but the quantum model uses a nonlinear
operation (the squared projection) to determine the final
probabilities. The latter is crucial for the interference effect                         References
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