1. Introduction
This discussion emerges from two fundamental questions: What is physically computable? and What is the relationship between Turing computability and physical computability? As Turing computability is the central force of computability theory, the former question is often posed in terms of the latter (e.g., Arrighi and Dowek Reference Arrighi and Dowek2012; Cotogno Reference Cotogno2003; Hogarth Reference Hogarth1994; Shagrir and Pitowsky Reference Shagrir and Pitowsky2003; Ziegler Reference Ziegler2009, among countless others). Piccinini’s (Reference Piccinini2011, Reference Piccinini2018) discussion of the physical Church–Turing thesis follows this format. He argues that if notions of computability are to be linked to what is epistemically useful to finite observers, a modest version of the physical Church–Turing thesis likely holds. This modest physical Church–Turing thesis states that what is Turing computable acts as an upper limit for what is physically computable, given some constraints on what is considered a physical computation. These constraints are meant to restrict discussion to physical computations that could be epistemically useful to finite observers.
Though the modest physical Church–Turing thesis may seem plausible, we will see that the account of what counts as an epistemically useful physical computation that Piccinini uses to argue for this thesis requires more explicit conceptual grounding. In particular, I argue that it begs the question regarding what physical processes one considers possible computational operations and implicitly fills this gap with the assumption that physical computational operations correspond to processes that can be executed by physical systems that can be built using classical (Turing) computational processes on physical systems.
Trying to determine which physical processes can correspond to epistemically useful computational operations leads to a bootstrapping problem: we want to limit epistemically useful computations to those with a physically buildable manifestation for finite observers like ourselves; however, our criteria for determining which physical processes finite observers can implement as part of a computational process require us already to know which physical processes finite observers can implement to build physical systems to implement that computation. This undermines the usefulness of buildability as a constraint for determining which physical computations are epistemically useful.
Piccinini implicitly assumes a classical (Turing) computation-based position in which finite observers can implement Turing-computable processes to build computing systems. Although this classical starting point is not without merit—an assumed starting point is required, and I will argue in section 4.2 that a classical computation-based one is a good default position—it is also not the only possible position, and it might even be ill suited to describing what is possible for finite observers in some physical theories. Additionally, this position may lead us to overestimate the likelihood that the modest physical Church–Turing thesis holds.
Section 2 discusses what is meant by physical computability in the context of this article. Section 3 discusses what a model of physical computation is in light of Piccinini’s conditions for ensuring that physical computational models are epistemically useful to finite observers. Section 4 argues that, given a physical theory, there is an inherent freedom for selecting physical operations to ground what counts as a model of computation in that theory. This results in a bootstrapping problem, unless we impose a selection of what physically possible processes to consider possible building blocks for models of computation in a theory. The selection, implicitly made by Piccinini, derived from a classical computational understanding of operations, though a particularly compelling response to this freedom, is not without its weaknesses. Section 5 concludes.
2. Physical computability questions
Using Turing machines in classical universes whose spatial dimensions are described by hyperreal, rather than real, lines, Aitken and Barrett (Reference Aitken and Barrett2009, Reference Aitken and Barrett2010) show that what is computable using a Turing machine depends on the physical theory in which it resides. They conclude that, generally, what is physically computable depends on the physical theory. Additionally, they propose that questions of physical computing power can be answered only relative to a physical theory and a computational model constructed relative to this theory. Thus questions about physical computability should be answered relative to a particular computational model constructed relative to a particular physical theory.
On its own, this proposal tells us nothing about how we should select models of computation relative to physical theories. But determining the limits of physical computability and whether different versions of the physical Church–Turing thesis hold relies on some assessment of what counts as a model of physical computation in some physical theory. If we do not properly restrict how computational states and operations can be mapped to physical states and operations, then we can end up with some version of pancomputationalism in which every physical system performs either some computation or every computation. Footnote 1
One might reasonably hope that appropriate physical computational models can be selected in a principled way, ideally by catering our selection of computational model to leverage a physical theory’s properties while reflecting what finite observers governed by this theory could possibly achieve. We will explore such an approach in section 4.1 and find that it succumbs to a bootstrapping problem without supplementary assumptions that do not arise from the physical theory itself. Thus Piccinini’s usability constraint requires additional specification of what operations finite observers can utilize that do not arise from the physical theory or the usability constraint itself. The necessity of additional constraints that do not arise from a particular physical theory on allowed computational operations and arguments for and against Piccinini’s position are discussed in section 4.2.
3. Models of physical computation
For the purposes of this article, a physical theory
${\cal T}$
will specify sets of objects J, possible physical states
${S_j}$
for these objects, and dynamical rules
${\cal R}$
that describe how the states of these objects evolve and interact. For instance, the objects of J may be particles or perturbations of a field and will constitute the physical systems used to construct a model of computation. The states
${S_j}$
will describe the properties of these objects, such as location or energy level, and the dynamical rules
${\cal R}$
determine how these objects and states can evolve. This definition is meant to be broad enough to encompass a wide variety of physical theories.
Given an appropriate physical theory
${\cal T}$
, a model of computing in that theory can be constructed in an immense number of ways. Any process can be considered to be a computation of almost anything if one is liberal enough with what one considers a computation, as the pancomputationalism literature demonstrates.
Footnote 2
We will want to consider models of physical computation that follow Aitken and Barrett’s (Reference Aitken and Barrett2009, Reference Aitken and Barrett2010) prescription in light of Piccinini’s epistemological concerns. To give us a tractable starting point, I will define a model of computation as follows.
A model of computation relative to
${\cal T}$
,
${{\cal M}_{\cal T}}$
, will consist of finite numbers of units of information (“bits” of some sort)
${\cal B}$
and operations
${\cal O}$
that can be performed on these units of information; these will arise from information-carrying objects
$B \subseteq J$
and operator systems
$O \subseteq J$
that act on the elements of B according to dynamical rules
${\cal R}$
to change their corresponding physical states. There may not be a one-to-one correlation between units of information
${\cal B}$
and elements or even subsets of B. For instance, a unit of information may be carried by one particle or multiple particles, perhaps even within a single computer. Importantly, the physical states of B serve to transmit information, and the evolution of these states according to the dynamical rules
${\cal R}$
of the theory and through the influence of operator systems allows computation.
Piccinini proposes a usability constraint to restrict discussion of physical computation to those physical computational models that would be epistemically useful for finite observers:
Usability constraint. If a physical process is a computation, it can be used by a finite observer to obtain the desired values of a function Footnote 3 (Piccinini Reference Piccinini2011).
This constraint is broken down into four subconstraints on physical processes (Piccinini Reference Piccinini2011, 741):
-
1. “An executable physical process is one that a finite observer can set in motion to generate the values of a desired function until it generates a readable result.”
-
2. “An automatic physical process is one that runs without requiring intuitions, ingenuity, invention, or guesses.”
-
3. “A uniform physical process is one that doesn’t need to be redesigned or modified for different inputs.”
-
4. “[A] reliable physical process is one that generates results at least some of the time, and, when it does so, its results are correct.”
Our concern will be a particular aspect of the executability criterion. To be executable, a physical process must have inputs and outputs that are readable for a finite observer, the ability to solve problems that can be “defined independently of the processes that compute them,” the ability to be repeated, the ability to set the computing system into a particular initial state, and a physically constructible manifestation (Piccinini Reference Piccinini2011). The constructibility criterion implies a distinction between physically possible objects and processes and the physical processes and objects that could be constructed by some physically possible finite agent. This limits discussion of physical computation to things that finite observers could execute and find epistemically useful.
The last component of the executability condition—that the physical process corresponds to some system that can be physically constructed—will be the main target of our concern, because an investigation of this criterion reveals an unrecognized assumption that guides the rest of Piccinini’s work on the modest physical Church–Turing thesis. What we now want to consider is how this restriction would play out when developing models of computation relative to a physical theory. For the sake of avoiding confusion with other concepts, we will use “build” where Piccinini uses “construct.”
4. Determining what is buildable
We will see that Piccinini’s executability subconstraint, owing to ambiguities in how to determine what is buildable, can leave open a wider range of possibilities for computational operations than seems desirable and that seems to have been acknowledged in the literature. By investigating ways one can describe what counts as “physically constructible” (what we call “physically buildable” to avoid confusion with other ideas tied to the word constructible), we will see that Piccinini implicitly supplements this concept with a notion of what operations finite observers can use which is based in classical computation, rather than on the physical theory itself, in contrast with the prescription Aitken and Barrett (Reference Aitken and Barrett2009, Reference Aitken and Barrett2010) describe. Nonetheless, I will argue that adapting the implicit classical notion underpinning Piccinini’s discussion of his usability constraint to a particular physical theory is a strong starting point for investigating physical computation in that theory, though we should be explicitly aware that we are making an assumption.
4.1. Buildable physical systems and a bootstrapping problem
If our concern is what a physical theory allows finite observers to do, not all physically possible system states may be useful for computation. For a model of physical computation to pass Piccinini’s executability condition, finite observers must be able to reliably build devices that manifest it. One then needs to understand what processes finite observers can reliably use to build physical systems to act as computers. If a finite observer cannot systematically put a system in (or extremely near to) a state, then that state cannot be part of a model of computation that is useful for finite observers, even if the state can be realized in a world governed by theory
${\cal T}$
. This applies to both the information-carrying systems and the operator systems: we need systematic processes within the theory to build the elements of B and O and to set them up for a computation; otherwise, these states cannot be epistemically usefully harnessed. These restrictions are motivated by epistemic considerations—they are meant to limit usable models of physical computation to those that are not just physically possible simpliciter but are physically possible ways for an observer within the physical reality to use other systems within the same physical reality to solve some problem (disregarding surmountable technological limitations). Ideally, we would be able to define a notion of “buildability” that emerges from an account of finite observers in a particular physical theory alone. However, we will see that the situation seems not to be so simple and that deciding what counts as buildable requires us to make stipulations that do not arise from the physical theory itself.
What can be built within different physical theories obviously varies; for instance, quantum mechanics allows the construction of Hadamard gates, whereas classical mechanics may not. Also, what building procedures are available may vary from one theory to another. One theory may allow infinite-step processes to be completed, whereas another does not, or, more typically, one theory may allow access to different physical systems and different ways physical systems can be manipulated than another. If theories allow different processes to manipulate physical systems, they may also allow different operator and bit systems to be built and perhaps different
${\cal B}$
and
${\cal O}$
, which can be used in computational models relative to the theories. But finite physical systems can implement many processes that should not be included in an account of epistemically useful physical computation. The task for understanding useful physical computing is determining which of the physical processes that finite systems can implement can be used to build operator and bit systems for use in physical computers.
Suppose we want to determine what models of computation finite observers can build relative to some physical theory. To determine what can be built for use in a model of computation relative to our physical theory, we must first determine what physical processes in the theory can be used for building computer components. But here we run into a problem—building some component for a computer requires a set of operations we can perform on systems to change their states in order to build that component. But this set of operations we can perform on systems is precisely what we are attempting to discover, as these are the operations that can be used as part of a model of physical computation. So we have a bootstrapping problem—we want to know which physical processes a physical theory allows finite observers to reliably implement (that would be used in computers), but to discover this, we must already know what physical processes the physical theory allows finite observers to reliably implement (that would be used in building computers).
Let’s make this more concrete. Imagine we have some physical theory
${\cal T}$
with a set of objects J that have possible physical states
${S_j}$
and dynamical rules
${\cal R}$
, which describe how these states evolve. We want to know what bits
${\cal B}$
, corresponding with the physical states
${S_j}$
, and operations
${\cal O}$
, corresponding with dynamical operations on these states from
${\cal R}$
, could be used in epistemically useful models of computation
${{\cal M}_{\cal T}}$
in this physical theory. To find this out, we must know what physical systems finite observers can build to act as bit systems B and operator systems O in computing systems. That is, to determine what B and O can be in an epistemically useful physical computer, we must know how finite observers can systematically manipulate the states
${S_{}}$
of physical systems J to build B and O. But knowing how finite observers can systematically manipulate physical states for use in computing requires us to already know what operations these finite observers can perform on the states of physical systems. We need a way to select which physical processes that finite observers can implement to include in our account of buildability in a particular physical theory.
Buildability needs supplementation to work as a useful constraint on what we consider epistemically useful physical computation. Though epistemically useful physical computation must in fact fit the buildability criterion, this criterion itself has significant epistemic problems when we try to use it; we would need a full account of what physical operations and states finite observers can and cannot reliably use in a particular physical theory, but this list does not directly fall out of the laws of a physical theory, as finite observers (acting as finite systems) can implement all sorts of operations that we would not want to consider part of a computational model in that theory. Footnote 4 For instance, a system acting as a unitary operator in a real quantum computer that we build will, with probability 1, correspond to some operator that is classically noncomputable. Yet these operations are not considered part of quantum computing; they can only be approximated, for example, with the Solovay–Kitaev algorithm (Dawson and Nielsen Reference Dawson and Nielsen2006; Kitaev Reference Kitaev1997). We turn to Geroch’s (Reference Geroch2009) more thorough discussion of this topic in the following section. We need additional rules for determining which operations pass the buildability criterion.
Can the rest of Piccinini’s usability constraint free us from this problem? Let’s first look at the rest of the executability subconstraint. Again, this requires that there be readable inputs and outputs, the ability to solve problems defined independently of the processes that compute them, repeatability, and the ability to set computing systems into particular initial states, in addition to having a physically buildable manifestation. Whether a process is repeatable and whether a computing system can be set to a particular initial state are consequences of what is physically buildable—if we can build components to act out some physical process, then we can do so repeatedly, and we can do so to initialize the state of a computing system. The ability to solve problems defined independently of the processes that compute them is a matter of what computations can be said to compute but not of what basic operations can be implemented on physical systems. Requiring readable inputs and outputs seems to show the most promise, because presumably, finite observers cannot access the complete dynamical consequences of all their actions. For instance, I may be able to throw a ball some classically noncomputable distance (in meters), but I certainly cannot read off the real number corresponding to this distance. But readability itself is at least partially dependent on buildability, in that finite observers may be able to use physical processes to change an output they cannot read into one they can.
The other subconstraints will be of no help. Whether a purported computing system can be considered “automatic” will depend on what operations are considered to be automatic processes, which in turn depends on what can be built to act as a computing component and thus cannot be considered to be acting with intuition or ingenuity. Whether a process is Uniform over different inputs and whether a process is capable of getting correct results are properties of a purported computation as a whole, not of its individual components.
Thus Piccinini’s executability criterion leaves us with a bootstrapping problem: we want to figure out which physical processes we can implement on physical systems (as part of a model of epistemically useful computation), but to determine this, we need to already know what these operations are (for use in building computer components to implement a model of computation). In the following section, we discuss how to avoid this bootstrapping problem and one type of solution adapted from Geroch’s discussion of quantum computation—namely, to take operations corresponding to classical computational operations as a starting point—and why this type of solution is likely the best starting point for investigating physical computation, despite some weaknesses.
4.2. Avoiding the bootstrapping problem by stipulating constraints
Developing a notion of what is buildable in a physical theory requires us to already have a concept of what operations and systems are allowed in a notion of buildability. Because of the bootstrapping problem, we cannot simply read a notion of buildability off of a physical theory’s treatment of finite systems. Instead, we must stipulate on the theory some external criteria to determine which physical processes finite systems can implement to be considered relevant to physical computation. These criteria do not arise directly from the physical theory or its properties but instead are external constraints on which physical processes are allowed in building procedures and hence which operations and states are considered buildable. This helps us determine what to count as an epistemically useful model of physical computation in a particular physical theory.
To avoid the bootstrapping problem, we could appeal to classical computational processes as a starting point. That is, we would assume that analogues to classical computational operations can be executed in a physical theory, namely, we can execute finite-step processes where steps involve discrete changes in a physical system’s state, and these discrete changes are classically (Turing) computable. For instance, a rotation of a state through an angle described by a classically computable number is allowed, whereas a rotation of a state through a classically noncomputable angle is not—unless we are able to reliably build an operator system that rotates a state through a classically noncomputable angle by using a finite number of classically derived operations. This solution allows us to bypass the bootstrapping problem by selecting the physical processes used to build systems to act as computational states and operations.
Geroch describes a version of this strategy in his discussion of plausible limits to quantum computability that the quantum computational literature implicitly assumes. He argues that unitary operators that rotate a quantum state through a classically noncomputable angle, though they are allowed by the laws of quantum mechanics, should not be permitted in a model of quantum computation, because a classical, stepwise method of building systems that perform these operations would require hypercomputation (see Geroch Reference Geroch2009, chap. 8). Though finite quantum systems can execute classically noncomputable unitary operations, Geroch argues for their exclusion from quantum computation based on an appeal to classical computational principles. Similarly, Piccinini assumes this classical computational strategy in his use of physical constructibility, as seen in his discussion of why “unconstrained appeals to real-valued quantities” should not be considered part of epistemically useful physical computation: “there is no reason to believe that a finite observer can use the Turing-uncomputable operations … to compute in the epistemological sense that motivates CT [the modest physical Church–Turing thesis] in the first place” (Piccinini Reference Piccinini2011, XX).
One reason for adopting the classical starting point for buildability—at least regarding the exclusion of real-valued quantities—noted by Piccnini (Reference Piccinini2011, XX) is that “there is no reason to believe that … unbounded precision is available to a finite observer.” Relatedly, the original arguments for what Piccinini calls the “Mathematical Church–Turing thesis” Footnote 5 —that “any function that is computable by following an effective procedure is Turing computable”—may bolster this position. Particularly, Turing argues in favor of the notion of computability given by Turing machines on the grounds that they limit the set of computational states and operations to those corresponding to what a human (with pen and paper) would be able to use (Turing Reference Turing1936–37, repr. in Turing Reference Turing and Davis1965). This suggests that Turing machines and Turing (classical) computable processes probably capture the types of processes that (human) finite observers can reliably execute and thus are likely the best starting point for developing physical computation in some physical theory, even though the classical computational processes were not developed to most optimally leverage the properties of that particular physical theory.
Also, appealing to classical computational operations is a particularly salient strategy when investigating physical theories, such as quantum mechanics or relativity, that are supposed to subsume physical theories that give rise to classical computation, such as classical physics. We can often assume that a physical theory at least allows classical finite state automata, which then provide us with information on some, if not all, B and O systems we may be able to build. This strategy also allows us to utilize an enormous amount of information about the processes we can use to build B and O systems in a theory, as we have a robust understanding of what classical computational processes look like and what they can do.
Though the strategy of appealing to classical computational processes is strong, its limitations should be noted. Most significantly, it is not a strategy that emerges from considerations of a particular physical theory at hand. Restricting ourselves to computational systems that can be built using classical processes and not theory-specific processes may unnecessarily limit the models of computation that we can build relative to a theory. Operations that the theory allows but that cannot be implemented by systems built using classical procedures cannot be used, and there may exist some systematic theory-specific methods for finite observers to build systems that carry out these operations. There may be an operating system O that can be reliably built by finite observers but could not be built by finite observers limited to construction processes using only classically computable operations. So although the classically grounded approach may be a powerful starting point in theories that are amenable to classical computational operations, it may also unduly constrain the models of computation we can build relative to a theory because it may not allow us to fully leverage the unique properties of that theory. Ruling out a building process that cannot emerge from classically computable steps for finite observers in any physical theory is a significant assumption that we should make consciously.
These problems are particularly important because we live in a universe that is not classical. To better utilize the particular properties of a physical theory, such as quantum mechanics, we may find we want to select some plausible theory-specific operations as our starting point, instead of or in addition to the classical operations. The bit systems B and operator systems O that we deem credible will thus be whatever can be built using some preselected set of operations allowed by the theory’s dynamics, similar to the classical solution but not necessarily reliant on concepts or operations from classical computing models.
The nonclassicality of our world already throws the modest physical Church–Turing thesis in doubt; for instance, quantum mechanics allows a genuine random number generator by using multiple measurements of superposition states. Perhaps we will find that we are able to harness quantum operations in a way that surpasses the limits of Geroch’s classical building approach, or perhaps some other physical theory will allow us to surpass this classical framework for building processes. In any case, we should be mindful of the assumptions in our discussions of physical computation, which we may want to drop when exploring epistemically useful computation in other physical theories.
5. Remarks
Tying our accounts of physical computation to what would in fact be epistemically useful for finite observers requires restrictions on what models of physical computation are considered. Piccinini gives a usability constraint that is meant to achieve this goal. But part of this constraint lacks adequate conceptual grounding, in lieu of which Piccinini has implicitly appealed to classical computing notions to determine which physical processes can be included in epistemically useful models of computation. Classical computation so deeply underlies discussions of computation that this assumption may be intuitive to the point of escaping detection, but we should be aware when we are making it in discussions of physical computation, especially as the world we inhabit is not, in fact, classical.
As Aitken and Barrett (Reference Aitken and Barrett2009, Reference Aitken and Barrett2010) note, what is physically computable depends on the selection of a model of computation relative to a particular physical theory, so we may hope that our understanding of what is buildable, and hence potentially usable for finite observers, in a physical theory will emerge from the physical theory and its descriptions of finite systems. However, attempting to derive a notion of buildability directly from an account of finite systems in a physical theory succumbs to a bootstrapping problem, in which our notion of what is “buildable” is meant to allow us to determine which physical processes finite observers can reliably implement (as part of a computation), but determining what counts as buildable requires that we already know which physical processes finite observers can reliably implement (as part of building the components of computing systems). We are then left with a choice of how to select what physically possible states and operations we will include in building processes. In this situation, the classical-based approach used by Piccinini and explicitly stated by Geroch is an especially strong starting point for investigating what models of computation could be epistemically useful in a physical theory, though we should be aware that it is one of many possible selections we can make when discussing physical computation. Although a classically founded notion of buildability is a strong default starting point for investigating what models of computation could be epistemically useful in a physical theory, it limits what physical processes are included without attending to the particular properties of a given physical theory, and adopting it without being aware that it is an assumption we are making may lead us to pick sides in debates regarding physical computation without adequate conceptual grounding.
Open access
