# [time 1120] Re: [time 1118] Re: [time 1113] interactions, windows and Monads (Re: [time 1105])

Thu, 16 Dec 1999 00:25:34 +0900

Dear Stephen and friends,

Stephen Paul King <stephenk1@home.com> wrote:

Subject: [time 1118] Re: [time 1113] interactions, windows and Monads (Re: [time
1105])

> Dear Hitoshi and Friends,
>
> Oh, I have been creative. I hope this makes some sense! :-)
>
> >
> > Dear Stephen,
> >
> > Thanks for your opinions and information. I am not clear yet how the western
> > philosophy comes to the problem of dualism between mind and matter or its
> > negation by Leibniz. Let me make some elementary questions below.
> >
> > Stephen Paul King <stephenk1@home.com>
> >
> > Subject: [time 1111] Re: [time 1109] Monads (Re: [time 1105])
> snip
> [KM]
> > > > > It seems to me that Leibniz would lose nothing even if his monad is
> > > > > allowed to have a tiny window through which to see the outside nearby.
> [SPK]
> > > I am afraid that the allowance of windows, however small, would bring
> > > into the model of Local Systems a problem that would ruin it
> > > consistency. We need to look carefully what it means to make
> > > observations!
> [HK]
> > Could you explain what inconsistencies arise in more detail?
>
> If we consider that the allowance of "windows" through which causal
> connections could occur, e.g. the exchange of substances, it would seem
> to be tantamount to allowing iterations (e.g. observations) to occur
> between quantum mechanical systems that are by definition in a "pure"
> state, we would be causing ourselves problems.

I understand this. No observation is possible if the observer's local system is
separated from the observed system.

> It seems that one is
> trapped in the tar pit of mechanistic explanations. :-( For a possible
> alternative see: http://www.hpl.hp.com/techreports/97/HPL-97-122.html
> Is it necessary to have actual physical contacts between Local Systems?

In a sense, I think so: Your message through e-mails would come to my eyes as
emission of photons from CRT. No communication would be possible without
physical contacts or with complete separation between the observer and the
observed.

> What if it were possible to show that all of the properties of
> iterations could be given by a method that is used to model how
> computational systems interact in a distributed asynchronous concurrent
> network? Perhaps I am being a bit idealistic, but if we consider how it
> is that our minds "seem" to be able to have causal influences on each
> other, even though our minds can not "touch" each other, this may not be
> so confusing and difficult. Also, this might give us an explanation for
> the strange phenomena that seems to be hinted at by resent studies of
> "Consciousness-Related Anomalies In Random Physical Systems".
>
> http://WWW.Princeton.edu/~pear/preamble.pdf
> http://WWW.Princeton.edu/~pear/finalcap.pdf
> http://WWW.Princeton.edu/~pear/publist.html
> etc.
>
> The idea I have is to consider that the interaction of minds is more of
> a sort of "morphic-resonance" type of interaction.

A local system as a sum of the observer's system and the observed system is the
interaction itself between them. In this sense, I agree that interaction is a
kind of "morphic-resonance."

> I am considering LSs
> as a good representation of minds in that they are indivisible wholes
> and are not mechanical "windmills". :-) David Bohm has written about
> this...
> Minds that are similar in their external observational behavior would
> be able to simulate each other's internal behavior if there exist some
> consistent means of relating internal scattering dynamics with external
> classical motions. We see this occurring tacitly in the interactions of
> humans. The key is the possibility of an equivalence relation between
> "processes".
>
> snip
> [SPK]
> > > I would like to direct our attention to the following web site:
> > >
> > > http://plato.stanford.edu/archives/win1997/entries/leibniz-mind/
> [HK]
> >
> > The last paragraph:
> >
> > "He seems to think that causal interaction between two beings requires the
> > transmission or transposition of the parts of those beings. But substances
are
> > simple unextended entities which contain no parts. Thus, there is no way to
> > explain how one substance could influence another. Unfortunately, however,
this
> > line of reasoning would seem to also rule out one case of inter-substantial
> > causation which Leibniz allows, viz., God's causal action on finite simple
> > substances. "
> >
> > seems to be an explanation that Leibniz' monads do not have windows. And
this
> > seems a natural consequence of Leibniz' definition of monads. I do not see
> > problems here insofar as we neglect the following two points raised in
> > http://plato.stanford.edu/archives/win1997/entries/leibniz-mind/:
> >
> > "Here Leibniz gives a reason tied to his complete concept theory of
substance,
> > according to which "the nature of an individual substance or of a complete
being
> > is to have a notion so complete that it is sufficient to contain and to
allow us
> > to deduce from it all the redicates of the subject to which this notion is
> > attributed" (Discourse on Metaphysics, ec. 8). But there are, it seems, at
least
> > two problems with this explanation. First, Leibniz moves rather quickly from
a
> > conceptual explanation of substance in terms of the complete concept theory,
to
> > the conclusion that this consideration is sufficient to explain the activity
of
> > concrete substances. Second, even if conceptual considerations about
substances
> > were sufficient to explain their apparent causal activity, it does not seem
to
> > follow that substances do not interact--unless one is assuming that causal
> > overdetermination is not a genuine possibility. Leibniz seems to be assuming
> > just that, but without argument. "
>
> Yes, this is one reason why I was very happy that I found this article
> on-line. :-) Leibniz may have been very happy with the implications of
> Quantum Mechanics as it allows for "acausal" behavior in the classical
> local sense...
> But, is this "acausality" truly randomness or is it perhaps merely the
> local trace of a global causal situation such as illustrated by the
> "secondary observers" that discussed by Peter Wegner?

I think the latter is the case. I.e. the "acausality" would be the "local trace
of a global causal situation" in your words.

>
> [SPK]
> > > The discussion is directed at the issue at hand! :-) The one comment
> > > that I have of it is that the negation of dualism that Leibniz espouses
> > > is a bit misguided. The use of an ab initio "pre-established harmony"
> > > ("created minds and bodies are programmed at creation such that all
> > > their natural states and actions are carried out in mutual
> > > coordination.") to explain the facts of psycho-physical parallelism is
> > > subject to the same criticisms as the notion of a priori Cauchy
> > > hypersurfaces (cf. http://xxx.lanl.gov/abs/gr-qc/9310031)used in GR to
> > > fix the initial conditions of the Universe.
> > >
> > > "Leibniz's account of mind-body causation was in terms of his famous
> > > doctrine of the preestablished harmony. According to the latter, (1) no
> > > state of a created substance has as a real cause some state of another
> > > created substance (i.e. a denial of inter-substantial causality); (2)
> > > every non-initial, non-miraculous, state of a created substance has as a
> > > real cause some previous state of that very substance (i.e. an
> > > affirmation of intra-substantial causality); and (3) each created
> > > substance is programmed at creation such that all its natural states and
> > > actions are carried out in conformity with all the natural states and
> > > actions of every other created substance."
> > > http://plato.stanford.edu/archives/win1997/entries/leibniz-mind/#Noin
> > >
> > > If we are to mind the consequences of the Uncertainty Principle (UP),
> > > we must dismiss this assumption on absolute initiality.
> [HK]
> > I did not find time to see the pages you quoted below (I might have seen
them
> > before but am not sure). But if the initiality were an "absolute"
initiality, it
> > might be free from UP.
>
> Umm, I disagree.

I agree with your disagreement :-) I just thought the usage of the word
"absolute" seemed to be inappropriate here in your context :-)

>It would make the situation far worse since the notion
> of absolute initiality would necessitate a unique initial point of time
> for all existence, e.g. a metaphysical "ex nihilo" creation. Koichiro's
> discussion of the problems of absolute synchronization is another
> example of the problems that such implies. This is why I say that the
> Universe, as the totality of existence, merely exists. It is without
> beginning or end or extension or duration, it is merely itself. The
> duality of observer and object only enters when we allow for a division
> of the whole and this is implicit in the definition of an observer.
> The initiality is a matter of "fixing" a point in time as a lower bound
> to the observers possible observations, as you delineate in the
> explanation of the Hubble expansion in
>
> "We remark that the expansion' in this classical sense is different
> from the stationary universe $\phi$ in our context of quantum
> mechanical sense. The former expansion' is the result of an observation
> activity with fixing one observer's coordinate system, e.g., in the
> above explanation we have assumed a synchronous coordinate system, which
> explains why the universe looks expanding for all observers. The latter
> quantum mechanical stationary universe $\phi$ is the inner structure
> of its own and is independent of the observer's coordinate system.
> Theorem 2 guarantees that these two views are consistent with each
> other, and Axiom 6 predicts that this framework would explain and
> resolve the problems related with the actual observations. In the
> present problem of Hubble's law and expansion' of the universe, these
> phenomena are the consequences of the {\bf observation} with one
> coordinate system fixed. In other words, they are 'appearance,' so to
> speak, which the universe makes under the interference' of the observer
> to try to reveal its figure or shape. More philosophically, the past or
> the future does not exist unless one fixes the time coordinate. The Big
> Bang' is an imagination under this {\bf assumption} of the {\it a
> priori} existence of time coordinate. Unless it is observed with
> assuming the existence of a time coordinate, the universe is no more
> than a stationary state, which does not change and is correlated within
> itself as a whole."
>
> The finitude of the "actuality" that any given LS could experience

The "actuality" may be finite, but at the same time any LS is infinite in the
sense that the LS is connected to the whole universe.

> is
> tied, I believe, to the finite scattering dynamics of the LS and,
> specifically, the amount of information that it can encode.

Any LS has an infinite amount of information as it can have it as the
information that the complement of the LS can have. Any LS is equivalent to its
complement by the stationary nature of the total universe.

>What I need
> to find out is how is it possible to quantitate the amount of
> information that an LS is capable of encoding with its scattering state!
> :-) The fact that the precise amount of information is not decidable a
> priori is not a problem! All we need is some form of upper bound, or
> "maximality" and some iteration condition, which I believe, would be
> given by the scattering equation of the LS and some way of quantifying
> the information content thereof. Perhaps the BREMERMANN'S LIMIT
> (http://pespmc1.vub.ac.be/ASC/Bremer_limit.html) or the
> Bekenstein-Hawking formula
> (http://www.math.ucr.edu/home/baez/week111.html) would help.
>
> On a parallel note, here is a quote from Peter's paper that applies:
>
> "Mathematical Models of Interactive Computing 26/44
> Mappings m_t : S $B%e(B> G(A, S), where A are transactions that span > multiple time intervals, correspond to an extension of dynamical systems > from Markov to non$B%e(BMarkov processes that view time as an active rather
> than passive variable in specifying system evolution. The dependence of
> the mapping on uncontrollable inputs of other streams introduces both
> nonlinearity of nondeterminism (section 4.2).
> Multi$B%e(Bagent behavior cannot be unfolded by iteration of a stationary > mapping m such that m^infinity = M. Temporal decomposition of behavior > is not in general possible either for Multi$B%e(Bagent computers or for
> histories of a distributed interconnected world. Though time progresses
> linearly, multi-agent (distributed) behavior cannot be linearly
> described). Nondecomposition of behaviors into mappings for discrete
> time steps corresponds loosely to nonlinearity and could in principle be
> specified by set$B%e(Btheoretic axioms that specify solutions of nonlinear > equations." > > snip > [HK] > > > > My interpretation is that a monad in the context of Leibniz is a > > > > local system without disturbance in my context. Simpleness > > > > which Leibniz requires monads does not contradict the plurality > > > > of the elements in a local system: A local system becomes a > > > > different local system if it is divided, so it is indivisible as > > > > local systems and is an elementary unit of existence. "Monads > > > > have no windows, by which anything could come in or go out." is > > > > true for local systems in the sense that: a local system becomes > > > > a different local system if "anything could come in or go out" > > > > with respect to the local system, and therefore, as far as a > > > > local system remains the same, it has "no windows." > [SPK] > > > Yes, it is very important to note that any observation whatsoever of a > > > Quantum Mechanical Local System (LS) implies that it is perturbed by > > > the act and thus we could consider such as implying a change of the LS. > > > We might consider that the act of observation of a LS is an act of > > > selection from an equivalence class (defined using ZFC- set theory. cf. > > > http://bugs.cs.wcupa.edu/~lizhang/Thesis/thesis/abstract.html) of > > > permitted LSs. > [HK] > > >From the page you quoted: > > "In 1917, Mirimanoff first stated the fundamental difference between > > well-founded and non-well-founded sets. He called sets with no infinite > > descending membership sequence ordinary, and others extraordinary. In 1988, > > Peter Aczel introduced a uniform terminology. He replaced the Foundation Axiom > > (FA) with the Anti-Foundation Axiom (AFA). Aczel's AFA states that every graph, > > well founded or not, pictures a unique set. This results in Hyperset Theory or > > ZFC-. In ZFC-, a bisimulation determines whether two hypersets are equivalent > > and consequently makes the classification of hypersets possible. " > > > > What does the equivalence mean here, i.e. how is "bisimulation" defined and how > > does it determine the required equivalence relation? And how is that > > equivalence relation related with the following?: > > Ah, this is were we must get into the dirty mathematical details! :-) > Since I am not a mathematician, I will try to explain this the best I > can using metaphors and quotes over the course of our on-going > discussions. > The best on-line definition that I have found is: > > "An equivalence relation, defined in the context of process algebras, > which is a finer equivalence relation than trace equivalence and > distinguishes states based on branching properties." > from http://hissa.ncsl.nist.gov/~black/CRCDict/HTML/bisimulequiv.html > > I am trying to find a better definition! Section 3.4 of > http://www.cs.brown.edu/~pw/papers/math1.ps has the best definition. > > "3.4 Bisimulation > > Equivalence is a subtle concept that may be progressively specialized > from equality (of all properties) to similarity (equivalence of some > properties) and simulation (dynamic equivalence of behavior). Symmetry > of equivalence gives rise to bisimilarity and bisimulation that capture > two-way step$B%e(Bby$B%e(Bstep simulation of processes. Bisimulation captures > mutual two$B%e(Bway dynamic behavior simulation between two systems,
> and is the natural extension of static equivalence to dynamic sequential
> interaction. Bisimulation is a coinductive equivalence relation between
> non$B%e(Bwell$B%e(Bfounded sets that models the behavioral equivalence of
> streams. The mathematical question when do two equations have the same
> solution?'' models the computational question when do two systems have
> the same behavior?''. For coalgebras, this question becomes
> `when do two coalgebras have the same final coalgebra?''
> Equivalence for sets is specified by the $principle of extensionality$.
> Two sets S, T are equal (S = T) if:
>
> a) for every s \element S there is a t \element T such that s = t
> b) for every t \element T there is an s \element t S such that s = t
>
> Equality of sets is recursively defined in terms of equality of subsets
> down to an arbitrary recursive level. This recursion always terminates
> for well$B%e(Bfounded sets, giving us an inductive approach to proving set > equality. For non$B%e(Bwell$B%e(Bfounded sets, extensionality yields a circular, > coinductive form of extensionality called strong extensionality [BM] > that transcends inductive extensionality of finite structures. Strong > extensionality of non$B%e(Bwell$B%e(Bfounded sets determines equivalence of > infinite structures by interactive dynamic simulation processes. > Two sets S, T are equivalent if there exists a$bisimilarity relation$> R \subset S x T involving all members of S and T; R is recursively > defined as follows: > for all s \element S and t \element T, R(s,t) iff s and t are atomic and > s=t, or > a) for every s' \element s there is a t' \element t such that R(s',t') > b) for every t' \element t there is an s' \element s such that R(s',t') > > The primary difference between bisimilarity and the earlier definition > of equivalence is the replacement of extensional (inductively defined) > equality "='' by a coinductively defined relation R. More than one > bisimilarity R \subset S x T may exist for a given pair of sets S,T. > However, the union of all such R is unique, and is the greatest > bisimilarity. When bisimilarity is interpreted as equivalence of system > behavior for all states s \element S and t \element T, the greatest > bisimilarity includes all pairs of states that preserve behavior for > every possible action a \element A. This greatest bisimilarity expresses > coinductive maximality and specifies coinductive equivalence for > non$B%e(Bwell$B%e(Bfounded sets and the systems that they model. > When S and T are state sets of systems and R(s, t) means that s and t > have equivalent behavior, then bisimilarity expresses simulation of each > system by the other, and bisimilarity of sets becomes bisimulation of > systems. Bisimulation relations R model mutual on-line simulation of > sequences of actions in one system by sequences of actions in the other. > Bisimulation of systems is a specialized form of bisimilarity for > behavior equivalence between evolving systems about which we have > incomplete knowledge. > Bisimulation for coalgebras is defined by mutual simulation of their > system evolution functions. > Bisimulation of coalgebras: Two coalgebras CS = (S, m:S$B%e(B>\Lamda(S)) and
> CT = (T, m':T\$B%e(B>\Lamda(T)) are related by a bisimulation relation R
> \subset S x T if for each s \element S and each evolution step of CS
> there is a t \element T and evolution step of CT that preserves R, and
> conversely."

I see the equivalence or equality itself requires a careful argument in
non-well-founded set theory, although it is elementary.

>
> some other online sources:
>
>
http://boole.stanford.edu/~rvg/pub/abstracts/axiomst.:_Axiomatising_ST-bisimulat
ion_equivalence.html
> http://www.brics.dk/RS/98/22/
> http://www.cis.upenn.edu/~bcpierce/types/archives/1997-98/msg00010.html
> http://www.cl.cam.ac.uk/Research/Reports/TR334-lcp-final.coalgebra.pdf
> http://theory.lcs.mit.edu/~dmjones/LICS/References/mislovemo1989:263.html
> http://theory.stanford.edu/~rvg/abstracts.html#11
> http://theory.stanford.edu/~rvg/abstracts.html#13
>
> Umm, it appears that what I need to find out is if it there exists an
> isomorphism between the scattering dynamics of an LS and an automata:
> http://theory.stanford.edu/~rvg/hda
>
> I apologize, but there is so much here that is very technical! I will
> have to proceed very slowly!
> This on-line essay by Onar Aam may be a good intuitive starting point!
>
> http://www-diotima.math.upatras.gr/mirror/prncyb-l/0316.html
>
> Let us meditate on how each LS "reflects" the Universe onto and into
> each other! How is it that the LSs can make concrete representations of
> each other? Consider the Turing Test that Peter Wegner discusses in his
> papers. For example:
>
> "The key intuition is that the class of things that a finite agent can
> observe is greater than the class of things that an agent can construct.
> We can formalize this intuition by showing that the class of things that
> an agent can construct is enumerable, while the set of situation that an
> agent can observe is nonenumerable. More over, we can show that
> constructible sets can be specified by induction, while observable sets
> require a stronger inference rule called conduction.
> Bertrand Russell in the 1900s and Hilbert and Goedel in the 1920s made
> a fundamental mathematical mistake in assuming that induction was the
> strongest form of definition and reasoning. They were misled by the
> paradoxes of set theory and mistakenly thought that circular reasoning
> needed to describe observation processes was inconsistent. In fact,
> circular reasoning is consistent and allows stronger forms of definition
> and reasoning than is possible through induction. Though induction is
> sufficient to describe construction processes stronger forms of
> reasoning are needed to express observation processes. Turing machines
> turn out to be the strongest form of computation possible by inductive
> reasoning but are not strong enough to express interactive computations
> of finite agents that observe an incompletely known environment, which
> are modeled by circular reasoning.
> Inductive definitions of set theory and logic define minimal fixed
> points which exclude everything that is not explicitly definable, while
> coinductive definitions of non-well-founded set theory include
> everything that is not explicitly excluded.
> Observers who consider all possible worlds not ruled out by
> observations are using the coinductive maximal fixed-point principle.
> The maximal class of things not explicitly excluded by a set of
> observations is fundamentally larger than the minimal class of things
> constructible from a set of primitives and allows us to build richer
> kinds of models.
> Constructive models that employ induction can cope with only enumerable
> situations, while observation-based models that employ circular
> reasoning can cope with nonenumerable situations. Turing machines have
> only enumerable input strings and can perform only an enumerable number
> of computations, while interaction machines just like people can make
> nonenumerable distinctions about their environment.
> Interactive systems can handle nonenumerable environments while
> noninteractive systems can handle only enumerable environments. The
> existence of a mathematical foundation for interactive computing
> provides a mathematical basis for interactive models of objects and
> distributed systems."
> from: http://www.cs.brown.edu/people/pw/papers/ecoop99_speech.pdf

This problem may not be a problem: As mentioned any finite LS can cope with
nonenumerable observables by the existence itself of those nonenumerable
observables. Any LS can use the total universe as its "memory" by the stationary
nature of the total universe.

>
> snip
> [HK]
> > > > I agree with Koichiro:
> [KM]
> > > > > It seems to me that Leibniz would lose nothing even if his monad is
> > > > > allowed to have a tiny window through which to see the outside nearby.
> [HK]
> > > > in the point that no local system is observable if it does not change by
the
> > > > perturbation associated with the observation. In so far as we consider
> > > > observation of local systems, they have windows. However, being a true
atom
> > > > remains valid in the internal world of each local system, where no
outside
> > > > is considered and no disturbance is from the outside.
> [SPK]
> > > Yes, but here we are approaching the difficult issue! :-)
> [HK]
> > I should appreciate it if you would explain the difficulties.
>
> Is it possible for LSs to observe each other without the perturbation?
> They can not! So what is they merely are "guessing" or "simulating" each
> the state other's inner world?

I understand the difficulties and I agree with you on the fundamental nature of
observation. But I do not think it necessary that an LS needs calculate or
simulate the outside only by the "tools" inside itself; It can use the outside
as its tools as well.

>
> This involves my idea that we should model interactions using the
> notion of bisimulations between independent Local Systems. To do this we
> will need some way of thinking of an LS as a computational system, e.g.
> a system that is capable of "simulating" the behavior of other systems.
> Perhaps the idea that one could define a surjective (?) isomorphism
> (into mapping) between some subset of the configuration space of one LS
> and another. (This is called an "infomorphism")
> The idea here is to show that some aspect of the evolution of one LS_i
> is identical to some aspect of the evolution of another LS_j which is in
> general disjoint to LS_i.
> Using the metaphor of persons making observations, we would say that
> there exist a pair of observers that have some subset of their class of
> observables that can be smoothly transformed into each other using a
> Lorentzian transform, even though there exists another observer that
> would not have such. More simply, there exist a pair of observational
> agents (LSs) that can "kick a stone" that has properties that they can
> agree upon, but there also exist another pair that can not agree with
> either of the first pair.
> When we move from the observations of one LS to another in this
> situation, we might notice that their is a change in the "meaning giving
> context" that corresponds beautifully to Koichiro's notion of migrating
> inconsistencies. :-)
>
> snip
> [SPK]
> > > It appears to me that the habituation of Western thought following a
> > > materialistic and reductionistic paradigm is one root of this problem.
> [HK]
> > I am interested in how this "materialistic and reductionistic paradigm" came
> > into the western thoughts.
>
> Perhaps the emphasis on the replication and implementation of
> industrial mechanisms... We see the concepts of "time is money" and
> "consumerism" as exemplifying this idea...

I think industrial mechanism came after the introduction of the spirit of Modern
age in the 17th century or around. "Time is money" and "consumerism" may be
just the result of that spirit and express almost the final state of the Modern
age, a final stage in the sense that it is the beginning of the next age.

>
> snip
> [SPK]
> > > So, what do we do in order to proceed? :-) Perhaps Leibniz offers a
> > > clue:
> > > (http://plato.stanford.edu/archives/win1997/entries/leibniz-mind/)
> > >
> > > "...it is his view that the world consists solely of one type of
> > > substance, though there are infinitely many substances of that type.
> > > These substances are partless, unextended entities, some of which are
> > > endowed with thought and consciousness, and others of which found the
> > > phenomenality of the corporeal world. "
> > > The question them becomes: What quality is it that distinguishes those
> > > monads that are "endowed with thought and consciousness" and the "others
> > > of which found the phenomenality of the corporeal world"?
> [HK]
> > Given the problem this may be a solution, but I need to understand how the
> > western philosophy comes to the problem of mind and matter, without
> > understanding which I think we could not go further.
>
> This will take some effort that may be beneficial to all of us! Is the
> key question: "How do remote objects, situations and events carry
> information about one another without any substance moving between
> them?"

This question is explained, in my opinion, due to the stationary nature
of the total universe.

> This is the subject of Information Flow: The logic of distributed
> systems by Jon Barwise and Jerry Seligman, Cambridge Univ Pr; ISBN:
> 0521583861 (July 1997)
>
http://www.amazon.com/exec/obidos/ASIN/0521583861/qid=945194521/sr=1-6/002-28659
56-9281866
> And guess what, it is the formalism of non-well funded sets that comes
> to the rescue! :-)
>
> Consider the "Other Minds Problem" for example:
> http://members.home.net/stephenk1/Outlaw/othermind.html; the use of
> coinductive abduction could help us break free of the prison of the
> argument from inductive analogy. See:
> http://www.cs.brown.edu/~pw/papers/math1.ps section 1.3.
>
> Kindest regards,
>
> Stephen
>

Best wishes,
Hitoshi

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