Counterfactuals, Computation, and Consciousness

 

Mark Muhlestein
Tucson, AZ
email: mark@muhlestein.com
blog: humbleinquirer.blogspot.com

Introduction
Background
Whole brain emulation
Woody as a cellular automaton
Experiments with counterfactual Sensitivity
Objections
Concluding observations
References

 

Published in the journal Cognitive Computation © 2012 — DOI: 10.1007/s12559-012-9155-2. The final publication is available at www.springerlink.com.

 

Abstract

Using a specially designed cellular automaton capable of emulating completely the function of a human brain, we explore ways in which counterfactual sensitivity, i.e. the ability of a computational system to respond to any valid inputs, relates to discrete state machine consciousness. In this computational framework, the distinction between a computation and the recording of a computation can be blurred arbitrarily, yet the physical implementation of the computation itself is unchanged. From this, we conclude that a purely computational account of consciousness is unsatisfactory.

 

Keywords  consciousness, computationalism, counterfactuals, whole brain emulation, cellular automata

 

“Well, Dex, if droids could think, we wouldn’t be here, would we.” — Obi-Wan Kenobi in Star Wars Episode II (George Lucas)

“Dave, my mind is going. I can feel it.” — HAL 9000 in 2001: A Space Odyssey (Arthur C. Clarke)

 

Introduction

The question of whether consciousness could be realized in a purely computational system has been widely discussed, but any kind of consensus on the question is still remote. We assume that consciousness is produced by the brain, but is it by virtue of purely computational processes? If so, an emulation of those computational processes should result in consciousness, since an emulation of a computation is just another, equivalent, computation. But if consciousness is not the result of purely computational processes, we must seek explanations elsewhere, starting with the physical components of a brain considered as physical entities, rather than abstract representations of computational states.

In this paper we meet “Woody,” a discrete computational system which implements an emulation of a human brain, and consider a number of experiments that could be done with such a system, experiments that call into question one of the requirements that has been postulated for any conscious computation. The reader can test his or her intuitions regarding the effects of these experiments on the putative subjective state of Woody, our volunteer experimental subject. Recognizing that there is already a large literature on the subject of computational functionalism, the modest hope is that these thought experiments will provide a more intuitive framework for discussing some of these ideas.

 

Background

Before going further, it will be useful to review related work by Tim Maudlin, Hilary Putnam, and Mark Bishop, as their ideas, and attempts to refute those ideas, are germane to this discussion, and we will be referring to them throughout. We will start with Maudlin [1]. This brief overview can hardly do justice to Maudlin’s paper, but we will review some key points needed for our purposes. Maudlin sets up a scenario with a mechanical computer he calls “Klara” which, when fed some appropriate input tape τ, will compute a set of states and outputs which a computationalist argues would instantiate a conscious computation. Once the computation has run, a new computer, which he names “Olympia”, is constructed, which processes the same input τ, but in all places where a conditional state transition would occur, the unused branches are effectively replaced with “hard coded” state transitions that ignore any inputs that differ from τ. The computationalist will insist that in spite of transitioning through the same set of states as Klara, what Olympia is doing is too trivial to even label a computation, much less a conscious computation. The missing element is what we will call counterfactual sensitivity, which we can define as the ability of a computational system to correctly respond to arbitrary inputs. A computational system has full counterfactual sensitivity only when all components of that system are capable of operating as designed when presented with any valid input.

To remedy Olympia’s missing counterfactual sensitivity, Maudlin asks us to imagine placing an instance of the fully capable Klara next to each conditional state of Olympia, arranged so that if any inputs not identical to τ were to arrive, a Klara would take over the computation. This restores counterfactual sensitivity, and therefore presumably consciousness, even though the Klaras never actually do anything, since we continue to use only τ as input. Maudlin goes further and has us imagine that some of the Klaras have rusty components so that, were the Klara to actually run, it might not work. He also considers a scenario where the Klaras are physically blocked so that they couldn’t run. But since the Klaras are never actually invoked when using input τ, Maudlin argues that whatever their condition, they can make no difference to the physical implementation of the computation. From this he concludes that computationalism implies that consciousness arises from (supervenes on) something more than the physical activity taking place in a computation, since only the original “trivial” Olympia is actually doing anything. Maudlin proposes that rejecting computationalism is more tenable than rejecting physicalism.

Mark Bishop [2][3] takes a different tack, but arrives at a similar conclusion. Again, our review is brief, and the original works should be studied for more detailed explanations and justifications. Extending ideas first put forward by Putnam [4], Bishop notes that any discrete state machine M with fixed, known inputs can have its conditional state transitions replaced with direct transitions to the state that is selected based on that particular fixed input. This process prunes the combinatorial explosion of possible state transitions down to a single set of simple state transitions (which we can call U for unrolled) that corresponds to the trace of the execution of M with a given set of inputs. Now we can take any system that generates discrete, non-cyclic state transitions (e.g. a simple counter, C) and map the “unrolled” states of M to the states of C by a simple correspondence. The interpretation of any state is arbitrary, so we choose to associate via a mapping function each state Ct with state Ut for all states t in U. At any time, then, we can both find the current state of U, and predict the next states of U, by mapping the current and next states of C to U. In short, any time C is running, we may consider U to be running.

Now suppose that we have a computation that a computationalist claims is conscious, say the experience of a robot seeing a red square. Bishop’s construction allows us to translate the computation plus its inputs into a sequence of states U that correspond to the actual state transitions that occurred during the conscious computation. Again, this translated sequence can then be considered to run whenever the state machine C is running, meaning that the operation of a simple counter can be interpreted as the execution of a program resulting in consciousness. Worse still, Putnam puts forward an argument claiming to show that any open physical system, say a rock, can be considered as passing through a series of discrete, non-cyclic states that we could also associate with the states of U. If that is true, computationalism must imply a sort of panpsychism, since the interpretation of these states is arbitrary, and must include states that correspond to every sort of conscious experience imaginable. Putnam and Bishop understandably reject this possibility. However, once again, counterfactual sensitivity figures large in this argument, because constructing U is only possible when the inputs are fixed. Critics of this line of reasoning can once again point to the requirement of counterfactual sensitivity for anything we might call a real computation, and since the discrete state machine U does not satisfy this requirement, Bishop’s and Putnam’s arguments lose their force. [5] Bishop has responded to this, concluding as Maudlin does that the lack of counterfactuals can only affect the putative consciousness of the computation if we are prepared to abandon physicalism.

Clearly, the notion of counterfactual sensitivity is crucial to the question of whether a computation is, or could be, conscious. But as Maudlin and Bishop maintain, exactly how the presence or lack of counterfactual sensitivity might affect consciousness has not been satisfactorily explained. The rest of this paper considers thought experiments which explore some of these ideas further, in a framework that may be more intuitive for some readers.

One important point to make at the outset: for the following thought experiments it is not necessary to draw fine distinctions between the different types of consciousness that a human brain can experience. Because the whole brain is emulated, everything from creature consciousness, to dreaming, to higher order beliefs can be considered. And if one takes the position that consciousness is an illusion, one may freely substitute “the illusion of consciousness”, or “the illusion of subjective experience”, wherever desired in what follows. Regardless of whether it is an illusion, the phenomenon of subjective experience, i.e., whatever it is that vanishes during anesthesia and slow-wave sleep, is the phenomenon in question here.

 

Whole brain emulation

For the purposes of this paper, let us assume that the problems of scanning the necessary structures of a human brain and modeling the relevant behaviors of those structures have both been solved: we have a computational system which emulates the behavior of an individual human brain sufficiently well that the system tells us that it has normal human phenomenal experience [6]. The level of emulation required (whether neural circuit, neuron, macromolecule, atom, or even string) does not matter for the purposes of this thought experiment; we only require that the emulation must accurately reproduce the behavior over time of a physical brain (in its normal embodied state). Obviously, the level of detail required for the emulation greatly affects the quantity of computational resources needed, but we will assume that we have enough to run the emulation.

For concreteness, let’s say that we have scanned the brain of our friend, Woody, who has volunteered to undergo the procedure. Once the scanning is complete, we run the emulation for a while, feeding it inputs that correspond to environmental events such as sights and sounds. We interpret the outputs as signals to the muscles of the tongue, arms, etc., which in turn can be interpreted as speech, gestures, and the like. The complete input stream should of course include everything needed to keep a human brain functioning normally, such as proprioception, and additionally, the outputs must be acted on by the world (virtual in our case), and fed back into the emulation. And there might well be a need to pause the emulation from time to time while staging and preparing the input stream. But these and a myriad other details need not occupy us here.

Once we have satisfied ourselves (and presumably both the human Woody and the emulated Woody) that everything is working as expected, the human Woody exits the scene, so from this point on, the name “Woody” refers to the emulated Woody. To begin our experimentation, we take a checkpoint of the entire state of the computational system, then we directly ask Woody, “Are you conscious? And do you smell anything?” At the same time, we have arranged for inputs corresponding to the smell of roses to arrive at Woody’s emulated olfactory center. The system runs for a while, then Woody answers, “Yes, I’m conscious, and I smell a lovely rose.”

As previously stated, it is the premise of these thought experiments that this whole brain emulation does result in normal conscious behavior, so there can be no question of whether the existence of inputs corresponding to smells would result in behavior appropriate to the perception of sound, say, instead of odor. The issue of embodiment does not arise, given that the scenario postulates a complete emulation of an entire virtual body and environment. The emulated environment provides all necessary inputs and feedback for normal operation of a human brain in a human body.

At this point it is necessary to mention one detail about the implementation: whenever the computation requires a random number, it retrieves it from a table of random numbers which were previously generated by an appropriate source, say a quantum device. This table of random numbers is part of the initial configuration. That means we can reset the state of the computer to the checkpoint, feed it the same inputs, and observe that the system behaves identically whenever the emulation runs, since the computation is perfectly deterministic. We can run this particular computation as many times as we like, and in each instance, the result will be a pattern corresponding to “Yes, I’m conscious, and I smell a lovely rose.” Nothing has changed from the first run, so we must conclude that if the computation had a subjective experience the first time, it would have the identical experience in subsequent runs. This is an important distinction between physical brains and emulated brains, since it is not possible to set up a physical brain that goes through a given set of states more than once (if indeed the notion of a sequence of states even makes sense for physical brains).

 

Woody as a cellular automaton

For the next set of experiments we choose a new computational system, a two-dimensional array of unit cells which together implement Conway’s Game of Life [7]. Such an array is an example of a cellular automaton (CA), and it has been shown that a sufficiently large array of these Game of Life cells can implement any computation that can be performed by a general-purpose computer [8].

All the unit cells of this system are identical, made up of three parts (Figure 1):

• input photodetectors, which respond to the presence of photons arriving from the eight immediately neighboring unit cells,

• a logic unit, which receives the information from the input detectors and sets the next output state based on the cell’s current output state plus the neighboring cells’ on/off states, following Conway’s simple rules[1], and

• a light source which turns on and off, based on the output of the logic unit.

Figure 1: Unit cell components

Figure 1.  Unit cell components.

 

One further component is needed for the array: a system-wide clock signal that controls when the inputs are to be sampled. We arrange things so that the inputs are sampled in the middle of the cycle, leaving adequate time for the next state to be computed and for the light to reliably indicate each cell’s state before the next clock cycle.

The array needed would doubtless be enormous, but in principle, the original Woody emulation could be set up on such a cellular automaton. With identical inputs, staged to arrive exactly as they had in the original setup, and with appropriate interpretation of outputs, the CA implementation would be computationally equivalent. If we have done the conversion properly, after the computation has run, we should see outputs corresponding to “Yes, I’m conscious, and I smell a lovely rose.” We expect this because the computation is deterministic, and computational equivalence guarantees that if a computation results in a given set of outputs on one computational system, it will result in the same set of outputs on any other equivalent system. Computationalism goes further and asserts that since the first computation creates a feeling of subjective awareness, the identical feeling should arise any time that computation is performed, regardless of the computational substrate.

 

Experiments with counterfactual sensitivity

In what follows, we will call this most recently described system the “free-running” version of Woody, for reasons that will become obvious. We can now use the Woody CA to test our intuitions regarding counterfactual sensitivity. As explained above, counterfactual sensitivity is a reasonable precondition for anything which might be called a computation; it allows us to distinguish between, on the one hand, the mere replaying of a recording of states, versus a computational process which actually generates states, on the other. Many scholars (e.g. Chalmers [9], Hardcastle [10], Chrisley [11]) have advanced counterfactual sensitivity as a requirement for any conscious computation. David Chalmers, for example, states in [9], “Cognition requires at least the possibility of functioning in more than one way”.

To explore this idea, let us now imagine that as usual we run the “Are you conscious, and do you smell anything?” computation on the Woody CA, but now we

1. film or otherwise record the pattern of lights of the computation,

2. reset the array back to the original checkpoint,

3. run the computation again,

4. project the results of the prior, identical computation back on top of the Woody CA as it runs.

In this case of projecting the results of a prior computation onto the array, given that the projection will register exactly with the computation, nothing changes in terms of the computation and local information flows, and naturally, at the end, the expected output shows up. But would anything happen to the feeling of subjective awareness, assuming there had been one in the unprojected case?

If one insists on full counterfactual sensitivity for full consciousness, one might suspect that the moment the projection of the previous computation turns on, consciousness would vanish, because suddenly the counterfactual sensitivity of the system would vanish. Why? Because some of the photons from the projector that hit a unit cell’s output light will scatter into neighboring cells’ photodetectors. When projecting light onto unit cell A, unit cell A’s neighbors will detect photons from both sources of light: photons internally generated by unit cell A, and scattered photons from the projection onto unit cell A. So in terms of the definition of counterfactual sensitivity given above, no unit cell that was projected on would be able to signal “off” if the projection were signaling “on”. That component would not be capable of operating as designed, thus violating our rule. Further, if any different inputs were introduced while the projector was on, the computation would almost certainly fail, due to the brittle nature of cellular automata: any lights that were on in the projection, but not on in the new computation, would disrupt the pattern, and doubtless in short order the entire array would cease to represent the computation of anything meaningful.

Counterfactual sensitivity is definitely removed by projecting the results of the previous computation, but the conclusion that projecting onto the array obliterates consciousness when the input stream is not changing seems rather mysterious, since physically the computation proceeds as usual. This scenario is distinguished from both Maudlin’s trivial Olympia and Bishop’s unrolled discrete state machine U, in that at the level of the basic computational elements (the unit cells), everything is physically present and operating normally: the photons (or lack thereof) that show up on unit cell A’s photodetectors continue to set the next output state according to the logic unit’s rules. It happens that with these inputs the output will always agree with the projection, but nothing physically different is going on in terms of the local information flows and logical processing.

Concluding that consciousness is lost when the projection is turned on can be made even more dubious by supposing that we turn on the projection and leave it on, but now we add a simple input checker which will monitor the input stream, comparing it to the original inputs. If any different inputs arrive, the input checker will immediately turn off the projector so that it will not disrupt the new computation. Now the system as a whole does have full counterfactual sensitivity, so presumably consciousness returns, but we are once again left with the question of how this could affect the consciousness of the system, since the actual computation that results in conscious behavior relies only on the information flows and other causal interactions of the computational elements, which are unchanged by the addition of the input checker. This is analogous to Maudlin’s installing of Klaras to restore counterfactual sensitivity to Olympia, except that here the presence of the input checker does not have anything to do with the computation going on in the array. It merely switches off the projection if different inputs are detected.

At this point, someone inclined toward a computational account of consciousness might be tempted to give up on the requirement for counterfactual sensitivity, and argue that it is only the causal structure that counts for consciousness. But this cannot be done lightly, because as discussed above, without the requirement for counterfactual sensitivity, one is led to results that are uncomfortable for computationalism, such as those pointed out by Maudlin and Bishop.

One way to preserve the requirement for counterfactual sensitivity would be to show some meaningful physical difference between the projected system and the free-running system, with the hope that this could explain why only the free-running computational system actually experiences anything (e.g. Chrisley [11]). In our case, someone could point out that the system with projection is physically not the same as the system without projection. That is true, but so would be two different unprojected arrays implementing the same computation, but with the clock running at half speed on one array, or with the lights twice as bright on one of them. A hallmark of computationalism is that the substrate does not matter, so just being different is not enough. The differences must actually affect the physical processes that underlie the computation such that the second system is not logically equivalent to the first. In our scenario, at the level of the unit cells, all components are operating normally, in both the projected and non-projected cases.[2]

Furthermore, to claim (as a consistent computationalist must) that the addition of an input checker rescues consciousness is in some ways even more problematic, since it is causally distant from the low-level processes. For example, suppose that instead of installing an automatic input checker, we, the experimenters, leave the projection on, but because of a side-effect of the design of the machine we are forced to turn the projector off before we can reach the switch used to change the inputs. Now the system has full counterfactual sensitivity, but only by accident. Or suppose we just promise to turn off the projector before any new inputs are given. Now we are part of the system, and Woody is presumably conscious to the degree that we keep our promises.

With no explanation why the projected Woody should fail to be conscious in spite of having no counterfactual sensitivity, it seems a committed computationalist has no choice but to accept that projected Woody is conscious, and try to live with the consequences, or if that is too unpalatable, perhaps to seek other explanations for conscious experience.

 

Objections

Regardless of whether one is convinced by arguments like Maudlin’s, Putnam’s and Bishop’s, the idea of a conscious system without counterfactual sensitivity may well strike one as extremely counterintuitive, if not absurd. The notion feels similar in plausibility to ascribing consciousness to images of actors projected on a movie screen. Let’s consider some objections that could be raised.

Objection 1: The projected Woody system is not conscious because it lacks counterfactual sensitivity.

It is the burden of anyone offering this objection to point out a mechanism for the projected system to detect that counterfactual sensitivity is missing. It cannot be at the level of the logic units, since at that level, inputs, outputs, and computations are fully equivalent to the free-running Woody. The laws of physics assure us that all photons of a given wavelength are identical, so what could be the other source of information which causes subjective awareness to arise or not? If one wants to claim that it is an emergent property of the lower levels, how could any emergent property be different, given that all lower-level behaviors are identical? Again, the question here is not whether there are physical differences between the free-running and projected systems: clearly they are different. But if consciousness arises from a computation, and if the computational elements are operating in an identical manner with and without projection, the defender of the requirement for counterfactual sensitivity must explain clearly how these physical differences affect subjective awareness.

Objection 2: The argument is too powerful: if we substitute the notion “computing 6×7 = 42” for “digital emulation of a brain yields consciousness” in the foregoing thought experiments, we should presumably be led to doubt whether a computational account can be given of multiplication.

This objection fails immediately because it begs the question: the substitution only works if we already accept that the feeling of consciousness arises from (or just is) the right sort of computation. It is granted at the outset in these thought experiments that the digital emulation of a brain will result in “the right answer” (conscious-like behavior), but the question is whether subjective awareness is also produced. A small parable might help make this clear: suppose there are people who claim, for what are to them good reasons, that whenever 6 and 7 are multiplied, the answer 42 results, of course, but that in addition, a wonderful phenomenon arises called Life-the-Universe-and-Everything-ness, or LUE-ness. Our thought experiments might trouble these believers, because their claim is that LUE-ness happens only when that computation is actually carried out, so by blurring the distinction between a computation and the recording of a computation, we call into question the LUE-ness that results, even though we always get the right answer. Other people accept the existence of LUE-ness, but are skeptical that multiplying 6 and 7 produces it, preferring instead an account involving specific arrangements of physical entities rather than abstract representations. Still others deny that LUE-ness exists, in spite of finding themselves in the middle of it.

Objection 3: None of these systems is conscious because they all use a previously generated table of random numbers.

As long as the numbers are generated from a genuine random source, it seems unreasonable to insist that one set of random numbers is better for consciousness than another. Again, the burden is on the person bringing this objection to demonstrate how a computation could feel different because of the “freshness” of the random numbers used in the computation.

Objection 4: Since the end state is always the same, it doesn’t matter whether Woody was conscious during the intervening states, regardless of any projection of the previous computation.

It doesn’t matter much in the scenario where we are merely asking Woody if he is conscious, and if he smells anything. It might matter immensely if the experiment were to see what would happen if all of Woody’s pain sensors were made to fire at the maximum rate for an extended subjective time. If this is just lights turning off and on with a certain pattern, fine. If not, a real moral question arises.

 

Concluding observations

A great many variations are possible in this Woody “Game of Life” cellular automaton framework, only a tiny fraction of which we have explored here. For example, we can consider scenarios where

These are just a few possibilities, some of which allow us to explore variations in both the counterfactual sensitivity and the causal structure of these computations. All of them raise difficult questions about the prospects for computationalism by blurring in various ways the distinction between a computation and the recording of a computation.

So what are we to make of the conscious awareness of the Woody array when it is not in a free-running state? The most parsimonious explanation is that since projecting onto the array can have no effect whatever on the physics of the computation, it would have no effect on any subjective awareness produced by the computation. But since the projection definitely removes the counterfactual sensitivity of the system, if we insist that counterfactual sensitivity is a requirement for subjective awareness, we must conclude that there is some non-physical explanation for the lack of awareness when the Woody CA is not free-running. We can (reluctantly) abandon counterfactual sensitivity as a requirement for computationalism, but then we find ourselves in the awkward position of granting consciousness to too many entities, including Maudlin’s trivial Olympia, and Bishop’s and Putnam’s trivial state machines. So we reject computationalism, thereby avoiding these problems, since if the original, unprojected Woody does not have conscious awareness, we would not expect the projected version to have awareness either.

But rejecting computationalism is not without its own issues. Assuming we are someday able to construct something like Woody, what do we make of it? We will have built a system that can answer any question about the consciousness which the brain it is emulating would be experiencing, were it a brain, and not an emulation of a brain. And among the outputs of the system are states that correspond to behaviors that are indistinguishable from those of human beings. Evolution has not prepared us for interactions with an entity which exhibits conscious behavior but which is not in fact conscious; we would find it tempting, perhaps overwhelmingly so, to grant full rights and responsibilities to an entity which can pass every imaginable test of sentience. But if Woody says, “Come on in! The water’s fine!” and suggests that you submit to a destructive brain scan in order to upload to a computational substrate, you would be wise to do so only if the nature of consciousness is clearly understood, and if you have included as part of your upload package any hardware (or wetware) necessary for conscious experience.

 

Acknowledgment

Though I cannot say I am persuaded by his conclusions, (e.g. that all reality results from the operation of a universal computer whose mere Platonic existence implies a sort of modal realism), Marchal [12] discusses a thought experiment with projection onto a two dimensional computer which was an important inspiration for some of the ideas in this paper.

 

References

1. Maudlin T. Computation and consciousness. J Philos. 1989 Aug;86(8):407-32.

2. Bishop JM. A cognitive computation fallacy? Cognition, computations and panpsychism. Cogn Comput. 2009 Sep;1(3):221–33.

3. Bishop JM. Counterfactuals cannot count: a rejoinder to David Chalmers. Conscious Cogn. 2002 Dec;11(4):642-52.

4. Putnam HW. Representation and reality. Cambridge (MA): Bradford Books; 1991. 154 p.

5. Chalmers DJ. Response to articles on my work: Mark Bishop, Counterfactuals cannot count [Internet]. Canberra: David Chalmers; [cited 2011 Jul 14]. Available from: http://consc.net/responses.html#bishop

6. Sandberg A, Bostrom N. Whole brain emulation: a roadmap [Internet]. Oxford (UK): Future of Humanity Institute, Oxford University; 2008. Technical Report No.:2008-3. Available from: http://www.fhi.ox.ac.uk/__data/assets/pdf_file/0019/3853/brain-emulation-roadmap-report.pdf

7. Gardner M. Mathematical games: the fantastic combinations of John Conway's new solitaire game "life". Sci Am. 1970 Oct;223(4):120-3

8. Berlekamp ER, Conway JH, Guy RK. Winning ways for your mathematical plays. Vol. 2. New York: Academic Press; 1982. 472 p.

9. Chalmers DJ. Does a rock implement every finite-state automaton? Synthese. 1996;108:309-33.

10. Hardcastle VG. Conscious computations. Electron J Anal Philos [Internet]. 1993 Aug [cited 2008 Jul 23];1(4). Available from: http://ejap.louisiana.edu/EJAP/1993.august/hardcastle.html

11. Chrisley R. Counterfactual computational vehicles of consciousness. Paper presented at: Tucson VII. 7th Biannual Toward a Science of Consciousness; 2006 Apr 4-8; Tucson, AZ, USA.

12. Marchal B. Mechanism and personal identity. In: De Glas M, Gabbay DM, editors. Proceedings of WOCFAI 91; 1991 Jul 1-5; Paris. [place unknown]: Angkor; c1991. p. 335-45.



 

[1] Next output state is “on” if exactly three neighboring cells are on, unchanged if exactly two neighboring cells are on, otherwise “off”.

[2] To remove any question of whether there are “extra” photons in the projected case, we can arrange things so that when projecting onto a unit cell, the total light is the same as the unprojected case, with either source individually brighter than the detection threshold. So when not projecting, each unit cell could output, say, 4× the photons needed for detection by the neighbors, but when the projector is on, each unit cell could output only 2× the threshold, with the projector also outputting 2× the threshold, so that the total brightness is the same for both the projected and unprojected cases. It is also important to remember that the construction of the CA involves the use of a clock signal that indicates when each unit cell is to sample its inputs, and this signal changes only after both photon sources have fully mingled. This means that there can be no notion of which photons “really” indicated the state, since the laws of physics guarantee that all photons of the same wavelength are identical.