Why Computation is Medium Independent

Many thanks again to Andre Curtis-Trudel, Marcin Milkowski, Danielle Williams, and Mazviita Chirimuuta for their commentaries on our recent book The Physical Signature of Computation: A Robust Mapping Account. The medium independence of computation seems to have caused trouble for the latter three commentators, so we take this opportunity to clarify it. We will then respond to some of Milkowski and Chirimuuta’s more specific remarks. (We responded to Curtis-Trudel and Williams’s specific remarks, and some of Milkowski’s, below their commentaries. Curtis-Trudel’s commentary was focused on other issues.)

Chirimuuta implies that medium independence is a notion “inherited from digital computation”. But medium independence applies to all computation, digital or non-digital, and many other things besides computation.

Something is medium-independent (aka substrate-neutral) just in case it can be implemented (or realized) in different media. For example, messages in Morse code can be transmitted using any medium that allows for short signals, long signals, and pauses between signals. Medium independence should not be confused with multiple realizability, which is the notion of something that always involves the same medium but can be realized (or implemented) in different ways—more precisely, a causal power is multiply realizable just in case there are different kinds of mechanisms for realizing it (Piccinini and Maley 2014). For example, there are different ways of transporting people from one location to another, involving different types of mechanisms. Thus, the notion of vehicle (for people) is multiply realizable, and yet all such vehicles manipulate the same medium (people). Medium independence entails multiple realizability, but not vice versa (Piccinini 2020).

Computation is medium-independent because it can be implemented in different media. As we point out in the book, the notion of computation is the notion of methods for solving instances of a general mathematical problem that work for all instances of the problem and do not require insight, creativity, or guesses. Solving mathematical problems does not require using any particular medium. It can be done using paper and pencil, an artificial computer, or even, in some cases, in one’s head. If a medium can encode the problem and solution and implement the operations that lead from each problem instance to its solution, it can be used to solve the problem. In this important sense, computation is medium-independent.

Of course, any given implementation of a computation is medium-dependent, because it involves a specific medium that constraints how the computation is implemented. Any given medium constrains how the computational variables can be physically encoded, how primitive computational operations can be carried out, how different operations can be organized together, how many computing and supporting noncomputing components can be packed into a given space, how much energy is needed for each operation, how much heat is dissipated, and so forth.

For example, suppose we want to solve a differential equation by using integration. How that works depends in part on the type of computation we choose. If we choose analog computation, a key primitive computational operation is integration—primitive because it does not decompose into simpler computational operations. Still, analog integration can be implemented in different media, and each medium may involve different mechanisms for implementing it. (Maley 2024 pointed this out but, from this, inferred that analog computation itself is medium dependent (cf. also Maley 2021) — for a more detailed argument that this is a non sequitur, see Sect. 4 of Piccinini 2022). If, instead, we choose digital computation, integration is a computationally complex operation, which is carried out by performing a numerical algorithm. Still, any numerical algorithm decomposes into primitive digital operations (such as AND, OR, or NOT), each primitive digital operation can be implemented in different media, and each medium may involve different mechanisms for implementing it. Thus, any specific computational implementation is necessarily medium-dependent, while the computation that gets implemented is medium-independent.

The main idea we defend in our book is the robust mapping account of implementation, according to which a physical system PS implements a computing system CS just in case PS bears CS’s physical signature. This means that there is a physical-to-computational mapping from (a constitutional and thus medium-dependent description of) PS to a computational definition of CS and the mapping satisfies several precise and strict criteria that include physical-computational-equivalence. Any number of physical systems – made of different materials, types of components, and employing different types of forces – may well map onto the same computing system in the requisite way. The same point applies to physical processes and the computations they implement: any number of physical processes may well map onto the same computation in the requisite way. Since computing systems (and the computations they support) can be implemented by physically different systems (and the processes they undergo), computing systems (and computations) are medium-independent according to the robust mapping account. The robust mapping account applies to any kind of computation, whether digital, analog, neural, quantum, etc. Therefore, the robust mapping account deems computation in general to be medium-independent, as it should.

Milkowski on Physical Descriptions and Medium Independence

Milkowski expresses doubts about our distinction between structural and constitutive physical descriptions. He also expresses skepticism about medium independence, claiming that it is unnecessary for our account of computation. He discusses a chemical reaction involving hydrogen and oxygen molecules to illustrate the reasons for his doubts. 

To address both, we first note that it is specifically computation — not chemical reactions or any other physical phenomenon — that we claim is inherently medium-independent. For us, computations are procedures or methods for solving mathematical (including logical) problems via sequences of operations on variables (which may be, but need not be, numerical). No specific physical media, whether described structurally or constitutively, are involved in the definition and specification of a computation. Physical media and their descriptions enter the picture only when computations are regarded as being implemented in these media. Mappings between a medium’s physical states and state transformations on the one hand, and the states and state transitions that define a particular computation on the other, spell out the relationships between the physical and the computational that would underly any claim that the specified computation is implemented in the specified physical medium.

So, while Milkowski is correct that the physical descriptions of media that map onto the abstract definitions of computations in our physical-to-computational mappings may be structural or constitutive, our account presumes that the computations onto which physical descriptions map are themselves medium independent. Were this not the case, it would be impossible for any given computation to be implemented in more than one medium — even in principle — and we certainly have examples of digital and analog computation where this is demonstrably the case. Of course, there may be types of computations — perhaps in the brain — that are extremely complex, incompletely understood, and presently known to be fully implemented only in one type of medium, and are implemented in that medium via processes that are idiosyncratic to that medium. Such examples are not in conflict with the notion that the implemented computations are themselves medium-independent.

There is not much to say about Milkowski’s example involving the chemical reaction, as we make no claims of medium independence for such purely physical processes. Having said that, we can discuss this example in terms of structural and constitutive physical descriptions as we define them in the book to hopefully provide some clarity on that point. 

In our (fairly standard) view of physical description, we have physical theories that provide general descriptions of broad classes of systems and phenomena. A physical description of a physical system (PDPS) results from tailoring that theory to the specifics of a given system. If our system of interest was a handful of molecules, and we wish to describe and compare the energetics of the system of charged particles involved as arranged in two different configurations (pre- and post-reaction), we might select, say, quantum mechanics as our fundamental physical theory. The formalism is very general and, when cast in forms appropriate for systems with finite or infinite numbers of degrees of freedom, can be expressed in a purely structural manner. We would then tailor that general formalism to the cases at hand, through which the number of degrees of freedom would be recognized, and the masses and charges of the interacting particles and the specific nature of their interactions would enter the picture. The result of doing this for the each of the two configurations would be a PDPS for each. The resulting equations can be expressed constitutively, with numerical values and units substituted for the mass and charge parameters, and solved to reveal the energy differences between the pre- and post-reaction configurations. Or they might be expressed structurally, with those parameters as dimensionless placeholders perhaps lumped together as constants, and the forms of the solutions studied for different ranges of parameter values. The two types of description serve different purposes: comparison with experimental results clearly requires numerical results from constitutive PDPSs, but investigations of structural PDPSs can and do provide their own insights. Note that this brief discussion is merely intended to clarify the distinction between structural and constitutive physical descriptions within the context of Milkowski’s example; we have neglected the formidable difficulties of actually solving the relevant equations.

In computing contexts, to which Milkowski’s example does not belong, PDPSs of differently constituted systems that have similar mathematical forms can map onto the same CDCS, accounting for the multiple, medium-specific implementations of the same medium-independent computation specified by the CDCS. But, again, we make no claims of medium independence for anything relevant to Milkowski’s example, with the possible exception that some Laws of Nature — articulated at a sufficiently general level — are expressed in terms that do not refer to specifics of the physical systems that will be governed by these laws.

Chirimuuta on Neural Signaling and Medium Independence

Chirimuuta argues that neural signaling and information processing are medium-dependent, because the efficiency of neural information processing was optimized to the properties of neurobiological systems via evolution. Her main points are the following:

“(1) A computer simulation of a brain will always be an abstraction away from at least some of the brain’s cognitively relevant properties.

(2) The brain is not like a digital computer in which a sparse abstraction can capture all of the computationally relevant properties.” (Chirimuuta, personal communication)

In a footnote, she leaves open whether neural information processing involves computation.

In our book, we defend the mainstream view among computational neuroscientists and artificial neural network researchers that at least some important aspects of what neurocognitive systems do are computational. We offer two arguments for this conclusion (Sect. 9.5.2). The first is that some functionally relevant aspects of neural processes – spike frequency and timing – are multiply realizable, and processes that manipulate multiply realizable variables are medium independent; since neurocognitive systems operate in a rule-governed way and rule-governed medium-independent processes are computations, neurocognitive processes involve computations. The second argument is that neural signals encode physically disparate stimuli and responses by means of physically similar internal variables, which suggests that what matters for such encoding is multiply realizable aspects of the internal variables; from here the second argument mirrors the first one. We also illustrate the importance of neural computation for cognition by sketching the story of how discoveries about information processing in the cat visual cortex inspired the development of convolutional (artificial) neural networks, which are both the basis for recent successes in AI and can provide successful models of object recognition in neurocognitive systems (Sect. 9.5.3). This is just one example among many of how our understanding of biological neural information processing and artificial neural network computation mutually influence and shed light on one another. This is an additional reason to conclude that neural information processing involves computation (cf. Colombo and Piccinini 2023, Ritchie and Piccinini 2024 for more details).

Chirimuuta seems to think that there is something wrong with these arguments because neural signaling and information processing are medium-dependent (cf. Chirimuuta 2022). On the contrary, what we take Chirimuuta to be showing is that the implementation of neural computation in neurobiological systems is medium dependent – as any physical implementation of any computation must be. This is consistent with our point that insofar as neural information processing is computation, there is something medium independent about it. That explains why it can be reproduced at least to a degree in artificial systems that are not made of biological neurons.

We will now address some of Chirimuuta’s more specific points in more detail.

One of us has long argued that neural computation is not digital (Piccinini and Bahar 2013). Thus, we agree with Chirimuuta that “[t]he brain is not like a digital computer in which a sparse abstraction can capture all of the computationally relevant properties”. Simulating neural computations in digital computers requires specialized, “non-sparse” abstractions. Reproducing (as opposed to simulating) neural computations requires the construction of neuromorphic chips.

The main purpose of our discussion of neural computation in the book is to argue that consciousness involves non-computational aspects (Sects. 9.6-9.7). If consciousness is involved in cognition, it follows that there are aspects of cognition that are not entirely computational. If this is the case, Chirimuuta is right that “[a] computer simulation of a brain will always be an abstraction away from at least some of the brain’s cognitively relevant properties.” At the very least, if our argument that consciousness involves non-computational aspects is sound, a computer simulation of a brain will not include its conscious aspects.

Chirimuuta makes the point that in (artificial) digital computers, computationally relevant degrees of freedom are demarcated from computationally irrelevant degrees of freedom by design, which is true. But the robust mapping account is agnostic as to whether a system is an artifact or is natural, how many computationally irrelevant degrees of freedom might be at play, or how neatly demarcated they might be from computationally relevant degrees of freedom.

Chirimuuta writes that in digital computers “most of the physical properties of the system are ‘wasted’ (i.e. not utilised for information processing).” It may be true that in digital computers most physical properties do not encode computationally relevant information, but to characterize them as “wasted” implies that they could have been used to encode computational variables. Many and perhaps most properties of digital computers do not encode computational variables, although many of these may still play important supporting roles.  Indeed, our account accommodates roles for “computationally supporting” subsystems that may be essential while not encoding computational variables directly, and where present in artificial systems such supporting subsystems are hardly “wasted” in the broader sense that they contribute to the functioning of the system. It seems that many physical properties in the nervous system similarly support and sustain the cognitive functioning of the brain without themselves being used for information processing. If the brain does perform computation, then its many non-computational degrees of freedom support those computations, and we would not consider those supporting degrees of freedom “wasted.”

Chirimuuta writes that medium independence is “never a thrifty solution to information processing needs,” in that it “does not maximize the information processing potential of the system.” This implies that medium independence is a kind of solution to a problem. It is not. It is simply a characteristic of a process or phenomenon such that it can be implemented in different media.

Chirimuuta makes the widely accepted point that neural systems evolved for certain functions and that they perform these functions efficiently. This is recognized by scientists and engineers who design artificial “neuromorphic” systems whose functionality and architecture more closely mimic natural neural systems than do digital computers — and do so to great effect. Indeed, prototype systems that perform tasks such as image recognition—and that perform these tasks much more efficiently than do “parallelized digital processors” — have been realized. 

However, the claim that neural computations are medium-independent in no way requires that we dispute this fact, whether these computations are implemented in the inorganic media in which artificial neuromorphic systems are realized or the neurobiological systems that inspired them. Indeed, one could fairly characterize the designers of artificial neuromorphic systems as aiming to implement computations like those performed in biological systems — as they understand them — in non-biological media, thus presuming that the computations implemented in natural neural systems are medium-independent. While nature has indeed evolved systems that efficiently perform crucial tasks using the media that it can evolve, it does not follow that these tasks must be themselves medium-dependent. Chirimuuta seems to be claiming that it does follow — that because nature evolved particular implementations of neural systems from particular biological media, any computation that would be implemented in these systems must be medium-dependent. But she does not say why this must be the case, or why the neurobiological systems that nature was able to evolve cannot be implementing medium-independent computations.

The question of whether some structure composed in some physical medium implements a computation is a question of that medium’s structure, properties, and capacities—not its origin story. It is obviously significant if nature co-evolved a unique species of computation together with the biological systems that implement them — could it have done otherwise? — but that is consistent with those same computations being implementable in different media and thus being medium independent.

Chirimuuta writes that “[r]esource efficiency is only ever achieved by leveraging the nuanced, scale-dependent properties of a physical system. If a code maps to physical properties in such a way that it can be implemented at any scale, it will not be an energy-efficient solution at any one of those scales.” Again, medium independence is an essential feature of computation — not something that could be concluded from the energy efficiency of a particular type of implementation in a particular medium. Of course, the efficiency and efficacy of any particular implementation are scale-dependent, both in natural and artificial systems. But to recognize that fact does not require that the computation being performed could not be implemented by different media at a different scale with different efficiency, i.e. it does not require that that computation itself be medium-dependent.

— Neal Anderson and Gualtiero Piccinini

References

Chirimuuta, M. (2022). “Comment on Neurocognitive Mechanisms by Gualtiero Piccinini.”  Journal of Consciousness Studies 29 (7-8): 185-94.

Colombo, M., and Piccinini, G. (2023). The Computational Theory of Mind. Cambridge: Cambridge University Press.

Maley, C. J. (2021). “The physicality of representation.” Synthese 199, 14725–14750.

Maley, C. J. (2024). “Computation is not essentially medium independent.” Feb 2024. American Philosophical Association Central Meeting. New Orleans, LA.

Piccinini, G. (2020). Neurocognitive Mechanisms: Explaining Biological Cognition. OUP.

Piccinini, G. (2022). “Neurocognitive Mechanisms: Some Clarifications.” Journal of Consciousness Studies 29 (7-8): 226-50.

Piccinini, G. and C. J. Maley (2014). “The Metaphysics of Mind and the Multiple Sources of Multiple Realizability.” In M. Sprevak and J. Kallestrup (eds.) New Waves in the Philosophy of Mind (125–52). London: Palgrave Macmillan.

Ritchie, J. B., and G. Piccinini (2024). “Cognitive Computational Neuroscience.” In Advances in Neurophilosophy. Edited by N. Heinzelmann, 151-178. London: Bloomsbury.