Sergio’s thoughts on computationalism evoked a big response. Here is his considered reaction, complete with bibliography…


A few weeks ago, Peter was kind enough to publish my personal reflections on computational functionalism, I made it quite clear that the aim was to receive feedback and food for thought, particularly in the form of disagreements and challenges. I got plenty, more than I deserve, so a big “thank you” is in order, to all the contributors and to Peter for making it all possible. The discussions that are happening here somehow vindicate the bad rep that the bottom half of the internet gets. Over the last week or so, I’ve been busy trying to summarise what I’ve learned, what challenges I find unanswerable and what answers fail to convince. This is hard to do, as even if we have all tried hard to remain on topic, our subject is slippery and comes with so much baggage, so apologies if I’ll fail to address this or that stream, do correct me whenever you feel the need.

This post is going to be long, so I’ll provide a little guidance here. First, I will summarise where I am now, there may be nuanced differences with the substance of original post, but if there are, I can’t spot them properly (a well-known cognitive limitation: after changing your mind, it’s very difficult to reconstruct your previous beliefs in a faithful way). There certainly is a shift of focus, which I hope is going to help clarify the overall conceptual architecture that I have in mind. After the summary, I will try to discuss the challenges that have been raised, the ones that I would expect to receive, and where disagreements remain unresolved. Finally, I’ll write some conclusive reflections.

Computational Functionalism requires Embodiment.

The whole debate, for me revolves around the question: can computations generate intentionality? Critics of computational functionalism say “No” and take this as a final argument to conclude that computations can’t explain minds. I also answer “No” but reach a weaker conclusion: computations can never be the whole story, we need something else to get the story started (provide a source for intentionality), but computations are the relevant part of how the story unfolds.

The missing ingredient comes from what I call “structures”: some “structures” in particular (both naturally occurring and artificial) exist because they effectively measure some feature of the environment and translate it into a signal, more structure then use, manipulate and re-transmit this signal in organised ways, so that the overall system ends up showing behaviours that are causally connected to what was measured. Thus, I say that such systems (the overall structure) are to be understood as manipulating signals about certain features of the world.

This last sentence pretty much summarises all I have to say: it implies computations (collecting, transmitting, manipulating signals and then generating output). It also implies intentionality: the signals are about something. Finally, it shows that what counts are physical structures that produce, transmit and manipulate signals: without embodiment, nothing can happen at all. I conclude that such structures can use their intrinsic intentionality (as long as their input, transmission & integration structures work, they produce and manipulate signals about something), and build on it, so that some structures can eventually generate meanings and become conscious.

Once intentionality is available (and it is from the start), you get a fundamental ingredient that does something fundamental, which is over an beyond what can be achieved by computations alone.

Say that you find an electric cable, with variable current/voltage that flows in it. You suspect it carries a signal, but as Jochen would point out, without some pre-existing knowledge about the cable, you have no way of figuring out what it is being signalled.

But what precedes brains already embodies the guiding knowledge: the action potentials that travel on axons from sensory periphery to central brain are already about something (as are the intracellular mechanisms that generate them at the receptor level). Touch, temperature, smells, colours, whatever. Their mapping isn’t arbitrary or optional, their intentionality is a quality of the structures that generate them.

Systems that are able to generate meanings and become conscious are, as far as we know (and/or for the time being), biological, so I will now move to animals and human animals in particular. Specifically, each one of us produces internal sensory signals that already are about something, and some of them (proprioception, pain, pleasure, hunger, and many more) are about ourselves.

Critics should say that, for any given functioning body, you can in theory interpret these signals as encoding the list of first division players of whichever sport you’d like, all you need is to design a post-hoc encoding to specify the mapping from signal to signified player – this is equivalent to our electrical cable above: without additional knowledge, there is no way to correctly interpret the signals from a third-party perspective. However, within the structure that contains these signals, they are not about sports, they really are about temperature, smell, vision and so forth: they reliably respond to certain features of the world, in a highly (and imperfect) selective way.

Thus, you (all of us) can find meaningful correlations between signals coming from the world, between the ones that are about yourself, and all combinations thereof. This provides the first spark of meaning (this smells good, I like this panorama, etc). From there, going to consciousness isn’t really that hard, but it all revolves around two premises, which are what I’m trying to discuss here.

Premise one: intentionality comes with the sensory structure. I hope that much is clear. You change the structure, you change what the signal is about.

Premise two: once you have a signal about something, the interpretation isn’t arbitrary any more. If you are a third-party, it may be difficult/impossible to establish what a signal is signalling exactly, but in theory, the ways to empirically get closer to the correct answer can exist. However, these ways are entirely contingent: you need to account for the naturally occurring environment where the signal-bearing structure emerged.

If you accept these premises, it follows that:

a) Considering brains, they are in the business of integrating sensory signals to produce behavioural outputs. They can do so, because the system they belong to includes the mapping: signals about touch are just that. Thus, from within, the mapping comes from hard-constraints, in fact, you would expect that these constraints are enough to bootstrap cognition.
b) To do so, highly developed and plastic brains use sensory input to generate models of the world (and more). They compute in the sense that they manipulate, shuffle symbols around, and generate new ones.
c) Because the mapping is intrinsic, the system can generate knowledge about itself and the world around it. “I am hungry”, “I like pizza”, “I’ll have a pizza” become possible, and are subject-relative.
d) thus, when I say “The only facts are epistemological” I actually mean two things (plus one addendum below): (1) they relate to self-knowledge, and (2) they are facts just the same (actually, I’d say they are the only genuine facts that you can find, because of (1)).

Thus, given a system, from a third-party perspective, in theory, we can:

i. Use the premises and conclusion a) to make sensible hypotheses on what the intrinsic mapping/intentionality/aboutness might be (if any).
ii. Use a mix of theories, including information (transmission) theory (Shannon’s) and the concept of abstract computations to describe how the signals are processed: we would be tapping the same source of disambiguation – the structure that produces signals about this but not that.
iii. This will need to closely mirror what the structures in the brain actually do: at each level of interpretational abstraction, we need to keep verifying that our interpretation keeps reflecting how the mechanisms behave (e.g. our abstractions need to have verifiable predictive powers). This can be done (and is normally done) via the standard tools of empirical neuroscience.
iv. If and only if we’ll be able to build solid interpretations (i.e. ones that make reliable predictions) and we’ll be able to cover all the distance between original signals, consciousness and then behaviour, we will have a second map: from the mechanisms as described in third-person terms, to “computations” (i.e. abstractions that describe the mechanisms in parsimonious ways).

Buried within the last passage, there is some hope that, at the same time, we will learn something about the mental, because we have been trying to match, and remain in contact with, the initial aboutness of the input. We have hope that computations can describe what counts of the mechanism we are studying because the mechanisms themselves rely on (exist because of) their intrinsic intentionality. This provides a third meaning to “the only facts are epistemological” (3): we would have a way to learn about otherwise subjective mental states. However, in this “third-party” epistemological route, we are using empirical tools, so, unlike the first-party view, where some knowledge is unquestionable (I think therefore I am, remember?), the results we would get will always be somewhat uncertain.

Thus: yes, there is a fact to be known on whether you are conscious (because it is an epistemological fact, otherwise it would not qualify as fact), but the route I’m proposing to find what this fact is is an empirical route and therefore can only approximate to the absolute truth. This requires to grasp the concept that absolute truths depend on subjective ones. If I feel pain, I’m in pain, this is why a third-party can claim there is an objective matter on the subject of my feeling pain. This me is possible because I am made of real physical stuff, which, among other things, collects signals about the structure it entails and the world outside.

At this point it’s important to note that this whole “interpretation” relies on noting that the initial “aboutness” (generated by sensory structures) is possible because the sensory structures generate a distinction. I see it as a first and usually quite reliable approximation of some property of the external environment: in the example of the simple bacterium, a signal about glucose is generated, and it becomes possible to consider it to be about glucose because, on average, and in the normal conditions where the bacterium is to be found, it will almost always react to glucose alone. Thus, intentionality is founded on a pretence, a useful heuristic, a reliable approximation. This generates the possibility of making further distinctions, and distinctions are a pre-requisite for cognition.

This is also why intentionality can sustain the existence of facts: once the first approximations are done, and note that in this context I could just as well call them “the first abstractions”, conceptual and absolute truths start to appear. 2+2 = 4 becomes possible, as well as “cogito ergo sum” (again).

This more or less concludes my positive case. The rest is about checking if it has any chance of being accepted by more than one person, which leads us to the challenges.

Against Computational Functionalism: the challenges.

To me, the weakest point in all this is premise one. However, to my surprise, my original post got comments like “I don’t know if I’m ready to buy it” and similar, but no one went as far as saying “No, the signal in your fictional bacterium is not about glucose, and in fact it’s not even a signal“. If you think you can construct such a case, please do, because I think it’s the one argument that could convince me that I’m wrong.

Moving on the challenges I have received, Disagreeable Me, in comment #82 remarks:

If your argument depends on […] the only facts about consciousness or intentionality are epistemological facts, then rightly or wrongly that is where Searle and Bishop and Putnam and Chalmers and so on would say your position falls apart, and nearly everyone would agree with them. If youre looking for feedback on where your argument fails to persuade, Id say this is it.

I think he is right in identifying the key disagreement: it’s the reason I’ve re-stated my main argument above, unpacking what I mean with epistemological fact and why I do.

In short: I think the criticism is a category error. Yes there are facts, but they subsist only because they are epistemological. If people search for ontological facts, they can’t find them, because there aren’t any: knowledge requires arbitrary distinctions and at the first level, only allows for useful heuristics. However, once these arbitrary distinctions are done and taken for granted, you can find facts about knowledge. Thus: there are facts about consciousness, because consciousness requires to make the initial arbitrary distinctions. Answering “but in your argument, somewhere, you are assuming some arbitrary distinctions” doesn’t count as criticism: it goes without saying.

This is a problem in practice, however: for people to accept my stance, they need to turn their personal epistemology head over feet, so once more, saying “this is your position, but it won’t convince Searle [Putnam, Chalmers, etc…]” is not criticism to my position. You need to attack my argument for that, otherwise you are implying “Searle is never going to see why your position makes sense”, i.e. you are criticising his position, not mine.

Similarly, the criticism that stems from Dancing with Pixes (DwP: the universal arbitrariness of computations) doesn’t really apply. This is what I think Richard Wein has been trying to demonstrate. If you take computations to be something so abstract that you can “interpret any mechanism to perform any computation” you are saying “this idea of computation is meaningless: it applies to everything and nothing, it does not make any distinction”. Thus, I’ve got to ask: how can we use this definition of computation to make distinctions? In other words, I can’t see how the onus of refuting this argument is in the computationalist camp (I’ve gone back to my contra-scepticism and I once more can’t understand how/why some people take DwP seriously). To me, there is something amiss in the starting definition of computation, as the way it is formulated (forgetting about cardinality for simplicity’s sake) allows drawing no conclusions about anything at all.

If any mechanism can be interpreted to implement any computation, you have to explain me why I spend my time writing software/algorithms. Clearly, I could not be using your idea of computations because I won’t be able to discriminate between different programs (they are all equivalent). But I have a job, and I see the results of my work, so, in the DwP view, something else, not computations, explain what I do for a living. Fine: whatever that something is, it is what I call computations/algorithms. Very few people enjoy spending time in purely semantic arguments, so I’ll leave it to you to invent a way to describe what programming is all about, while accepting the DwP argument. If you’ll be able to produce a coherent description, we can use that in lieu of what I normally refer to computation. The problem should be solved and everyone may save his/her own worldview. It’s also worth noting how all of this echoes the arguments that Chalmers himself uses to respond to the DwP challenge: if we start with perfectly abstract computations, we are shifting all the work on the implementation side.

If you prefer a blunt rebuttal, this should suffice: in my day job I am not paid to design and write functionless abstractions (computations that can be seen everywhere and nowhere). I am ok with my very local description, where computations are what algorithms do: they transform input into outputs in precise and replicable ways. A so and so signal gets in this particular system and comes out transformed in this and that way. Whatever systems show the same behaviour are computationally equivalent. Nothing more to be said, please move on.

Furthermore, what Richard Wein has been trying to show is indeed very relevant: if we accept an idea of computations as arbitrary interpretations of mechanisms, we are saying “computations are exclusively epistemological tools” they are, after all interpretations. Thus, interpretations are explicitly something above and beyond their original subject. It follows that they are abstract and you can’t implement them. Therefore, a computer can’t implement computations: whatever it is that a computer does, can be interpreted as computations, but that’s purely a mental exercise, it has no effect on what the computer does. I’m merely re-stating my argument here, but I think it’s worth repeating. We end up with no way to describe what our computers do: Richard is trying to say, “hang on, this state of affairs in a CPU reliably produces that state” and a preferential way to describe this sort of transitions in computational terms does exist: you will start describing “AND”, “OR”, “XOR” operators and so on. But doing so is not arbitrary.

If I wanted to play the devil’s advocate, I would say: OK, doing so is not arbitrary because we have already arbitrarily assigned some meaning to the input. Voltage arriving through this input is to be interpreted as a 1, no voltage is a 0, same for this other input. On the output side we find:

1,1 -> 1, while 0 is returned for all other cases. Thus this little piece of CPU computes an AND operation. Oh, but now what happens if we invert the map on both the inputs and assume that “no voltage equals 1″?

This is where I find some interest: computers are useful, as John Davey remarked, because we can change the meaning of what they do, that’s why they are versatile.

The main trouble is that computers cant be held to represent anything. And that trouble is precisely the reason they were invented numbers (usually 1s and 0s) are unlimited in the scope of things that they can represent […].

This is true, and important to accept, but does not threaten my position: if I’m right, intentional systems process signals that have fixed interpretations.

Our skin has certain types of receptors that when activated send a signal which is interpreted as “danger! too hot!” (this happens when something starts breaking up the skin cells). If you hold dry ice in your hand, after one or two seconds you will receive that signal (because ice crystals form in your skin and start breaking cells: dry ice is Very Cold!) and it will seem to you that the ice you’re holding has abruptly become very hot (while still receiving the “too cold” signal as well). It’s an odd experience (a dangerous one – be very careful if you wish to try it), which comes in handy: my brain receives the signal and interprets it in the usual way, the signal is about (supposed) too hot conditions, it can misfire, but it still conveys the “Too hot” message. This is what I was trying to say in the original post: within the system, certain interpretations are fixed, we can’t change them at will, they are not arbitrary. We do the same with computers, and find that we can work with them, write one program to play chess, another one for checkers. It’s the mapping in the I/O side that does the trick…

Moving on, Sci pointed to a delightful article from Fodor, which challenges Evolutionary Psychology directly, and marginally disputes the idea that Natural Selection can select for intentionality. I’m afraid that Fodor is fundamentally right in everything he says, but suffers from the same kind of inversion that generates the DwP and the other kind of criticism. I’ll explain the details (but please do read the article: it’s a pleasure you don’t want to deny yourself): the central argument (for us here) is about intentionality and the possibility that it may be “selected for” via natural selection. Fodor says that natural selection selects, but it does not select “for” any specific trait. What traits emerge from selection depends entirely from contingent factors.

On this, I think he is almost entirely right.

However, at a higher level, an important pattern does reliably emerge from blind selection: because of contingency, and thus unpredictability of what will be selected (still without the for), what ultimately tends to accumulate is adaptability per-se. Thus, you can say that natural selection, weakly selects for adaptability. Biologically, this is defensible: no organism relies on a fixed series of well-ordered and tightly constrained events to happen at precise moments in order to survive and reproduce. All living things can survive a range of different conditions and still reproduce. The way they do this is by, surprise! sample the world, and react to it. Therefore: natural selection selects for the seeds of intentionality because intentionality is required to react to changing conditions.

Now, on the subject of intentionality, and to show that natural selection can’t select for intentions, Fodor uses the following:

Jack and Jill

Went up the hill

To fetch a pail of water

Jack fell down

And broke his crown

And thus decreased his fitness.

(I told you it’s delightful!)

His point is that selection can’t act on Jack’s intention of fetching water, but only on the contingent fact that Jack broke his crown. He is right, but miles from our mark. What is selected for is Jack’s ability to be thirsty, he was born with internal sensors that detect lack of water: without them he would have been long dead before reaching the hill. Mechanisms to maintain homeostasis in an ever changing world (within limits) are not optional, they exist because of contingency: their existence is necessary because the world out there changes all the time. Thus: natural selection very reliably selects for one trait: intentionality. Intentionality about what, and how intentionality is instantiated in particular creatures is certainly determined by contingent factors, but it remains that intentionality about something is a general requirement for living things (unless they live in a 100% stable environment, which is made impossible by their very presence).

However, Fodor’s argument works a treat when it’s used to reject some typical Evolutionary Psychology claims such as “Evolution selected for the raping-instinct in human males”, such claims might be pointing into something which isn’t entirely wrong, but are nevertheless indefensible because evolution directly selects for things that are far removed from complex behaviours. Once intentionality of some sort is there, natural selection keeps selecting, and Fodor is right in explaining why there are no general rules on what it selects for (at that level): when we are considering the details, contingency gets the upper hand.

What Fodor somehow manages to ignore is the big distance between raw (philosophical) intentionality (the kind I’m discussing here – AKA aboutness), and fully formed intentions (as desires and similar). We all know the two are connected, but they are not the same: it’s very telling that Fodor’s central argument revolves around the second (Jack’s plan to go and fetch some water), but only mentions the first in very abstract terms. Once again: selection does select for the ability to detect the need for a given resource (when this need isn’t constant) and for the ability to detect the presence/availability of needed resources (again, if their levels aren’t constant). This kind of selection for is (unsurprisingly) very abstract, but it does pinpoint a fundamental quality of selection which is what explains the existence of sensory structures, and thus of intrinsic intentionality. What Fodor says hits the mark on more detailed accounts, but doesn’t even come close to the kind of intentionality I’ve been trying to pin-down.

The challenges that I did not receive.

In all of the above I think one serious criticism is missing: we can accept that a given system collects intentional systems, but how does the systems “know” what these signals are about? So far, I’ve just assumed that some systems do. If we go back to our bacterium, we can safely assume that such a systems knows exactly nothing: it just reacts to the presence of glucose in a very organised way. It follows that some systems use their intrinsic intentionality in different ways: I can modulate my reactions to most stimuli, while the bacterium does not. What’s different? Well, to get a glimpse, we can step up to a more complex organism, and pick one with a proper nervous system, but still simple enough. Aplysia: we know a lot about these slugs, and we know they can be conditioned. They can learn associations between neutral and noxious stimuli, so that after proper training they would react protectively to the neutral stimulus alone.

Computationally there is nothing mysterious (although biologically we still don’t really understand the relevant details): input channel A gets activated and carries inconsequential information, after this, channel B starts signalling something bad and an avoidance output is generated. Given enough repetitions the association is learned, and input from channel A short-cuts to produce the output associated with B. You can easily build a simulation that works on the hypothesis “certain inputs correlate” and reproduces the same functionality. Right: but does our little slug (and our stylised simulation) know anything? In my view, yes and no: trained individuals learn a correlation, so they do know something, but I wouldn’t count it as fully formed knowledge because it is still too tightly bound, it still boils down to automatic and immediate reactions. However, this example already shows how you can move from bare intentionality to learning something that almost counts as knowledge. It would be interesting to keep climbing the complexity scale and see if we can learn how proper knowledge emerges, but I’ll stop here, because the final criticism that I haven’t so far addressed can now be tackled: it’s Searle’s the Chinese room.

To me, the picture I’m trying to build says something about what’s going on with Searle in the room, and I find this something marginally more convincing than all the rebuttals of the Chinese room argument that I know of. The original thought experiment relies on one premise: that it is possible to describe how to process the Chinese input in algorithmic terms, so that a complete set of instructions can be provided. Fine: if this is so, we can build a glorified bacterium, or a computer (a series of mechanisms) to do Searle’s job. The point I can add is that even the abilities of Aplysiae exceed the requirements: Searle in the room doesn’t even need to learn simple associations. Thus, all the Chinese room shows us is that you don’t need a mind to follow a set of static rules. Not news, isn’t it? Our little slugs can do more: they use rules to learn new stuff, associations that are not implicit in the algorithm that drives their behaviour, what is implicit is that correlations exist, and thus learning them can provide benefits.

Note that we can easily design an algorithm that does this, and note that what it does can count as a form of abstraction: instead of a rigid rule, we have a meta-rule. As a result, we have constructed a picture that shows how sensory structures plus computations can account for both intentionality and basic forms of learning; in this picture, the Chinese room task is already surpassed: it’s true that neither Searle nor the whole room truly know Chinese, because the task doesn’t require to know it (see below). What is missing is feedback and memories of the feedback: Searle chucks out answers, which count as output/behaviour, but they don’t produce consequences, and the rules of the game don’t require to keep track neither of the consequences nor of the series of questions.

Follow me if you might: what happens if we add a different rule, and say that the person who feeds in the questions is allowed to feed questions that build on what was answered before? To fulfil this task Searle would need an additional set of rules: he will need instructions to keep a log of what was asked and answered. He would need to recall associations, just like the slugs. Once again, he will be following an algorithm but with an added layer of abstraction. Would the room “know Chinese” then? No, not yet: feedback is still missing. Imagine that the person feeding the questions is there with the aim of conducting a literary examination (they are questions about a particular story) and that whenever the answers don’t conform to a particular critical framework, Searle will get a punishment, when the answers are good he’ll get a reward. Now: can Searle use the set of rules from the original scenario and learn how to “pass the exam”? Perhaps, but can he learn how to avoid the punishments without starting to understand the meanings of the scribbles he produces as output? (You guessed right: I’d answer “No” to the second question)

The thing to note is that the new extended scenario is starting to resemble real life a little more: when human babies are born, we can say that they will need to learn the rules to function in the world, but also that such rules are not fixed/known a-priori. Remember what I was saying about Fodor and the fact that adaptability is selected for? To get really good at the Chinese room extended game you need intentionality, meaning, memory and (self) consciousness: so the question becomes, not knowing anything about literary theory, would it be possible to provide Searle with an algorithm that will allow him to learn how to avoid punishments? We don’t know, so it’s difficult to say whether, after understanding how such an algorithm may work, we would find it more or less intuitive that the whole room (or Searle) will learn Chinese in the process. I guess Peter would maintain that such an algorithm is impossible, in his view, it has to be somewhat anomic.

My take is that to write such an algorithm we would “just” need to continue along the path we have seen so far: we need to move up one more level of abstraction and include rules that imply (assume) the existence of a discrete system (self), defined by the boundaries between input and output. We also need to include the log of the previous question-answer-feedback loop, plus of course, the concepts of desirable and undesirable feedback (anybody is thinking “Metzinger” right now?). With all this in place (assuming it’s possible), I personally would find it much easier to accept that, once it got good at avoiding punishments, the room started understanding Chinese (and some literary theory).

I am happy to admit that this whole answer is complicated and weak, but it does shed some light even if you are not prepared to follow me the whole way: at the start I argue that the original task is not equivalent to “understanding Chinese”, and I hope that what follows clarifies that understanding Chinese requires something more. This is why the intuition the original Chinese room argument produces is so compelling and misleading. Once you imagine something more life-like, the picture starts blurring in interesting ways.

That’s it. I don’t have more to say at this moment.

Short list of the points I’ve made:

Computations can be considered as 100% abstract, thus they can apply to everything and nothing. As a result, they can’t explain anything. True, but this means that our hypothesis (Computations are 100% abstract) needs revising.

What we can say is that real mechanisms are needed, because they ground intentionality.

Thus, once we have the key to guide our interpretation, describing mechanisms as computations can help to figure out what generates a mind.

To do so, we can start building a path that algorithmically/mechanistically produces more and more flexibility by relying on increasingly abstract assumptions (the possibility to discriminate first, the possibility to learn from correlations next, and then up to learning even more based on the discriminations between self/not-self and desirable/not desirable).

This helps addressing the Chinese room argument, as it shows why the original scenario isn’t depicting what it claims to depict (understanding Chinese). At the same time this route allows to propose some extensions that start making the idea of conscious mechanisms a bit less counter-intuitive.

In the process, we are also starting to figure out what knowledge is, which is always nice

I Hope youve enjoyed the journey as much as I did! Please do feel free to rattle my cage even more, I will think some more and try to answer as soon as I can.


Bishop, J. M. (2009). A cognitive computation fallacy? cognition, computations and panpsychism. Cognitive Computation, 1(3), 221-233.

Chalmers, D. J. (1996). Does a rock implement every finite-state automaton?. Synthese, 108(3), 309-333.

Chen, S., Cai, D., Pearce, K., Sun, P. Y., Roberts, A. C., & Glanzman, D. L. (2014). Reinstatement of long-term memory following erasure of its behavioral and synaptic expression in Aplysia. Elife, 3, e03896.

Fodor, J. (2008). Against Darwinism. Mind & Language, 23(1), 1-24.

Searle, J. R. (1980). Minds, brains, and programs. Behavioral and brain sciences, 3(03), 417-424.

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