## Abstract

Before Alan Turing made his crucial contributions to the theory of computation, he studied the question of whether quantum mechanics could throw light on the nature of free will. This paper investigates the roles of quantum mechanics and computation in free will. Although quantum mechanics implies that events are intrinsically unpredictable, the ‘pure stochasticity’ of quantum mechanics adds randomness only to decision-making processes, not freedom. By contrast, the theory of computation implies that, even when our decisions arise from a completely deterministic decision-making process, the outcomes of that process can be intrinsically unpredictable, even to—especially to—ourselves. I argue that this intrinsic computational unpredictability of the decision-making process is what gives rise to our impression that we possess free will. Finally, I propose a ‘Turing test’ for free will: a decision-maker who passes this test will tend to believe that he, she, or it possesses free will, whether the world is deterministic or not.

## 1. Introduction

The questions of free will—Do we possess it? Does it even exist?—have occupied philosophers, theologians, jurists and scientists for centuries [1–13]. Free will stands out among philosophical problems in that it has everyday practical applications. If decisions are freely made, then those decisions can form the basis for condemning people to prison or damning them to hell. Moreover, one of the central questions of free will—Is the Universe deterministic or probabilistic?—is a scientific one whose answer lies at the foundations of physics.

The purpose of this paper for this Turing centenary issue is not to resolve the problem of free will, but to present and to clarify some scientific results relevant to the problem. Following Turing's youthful interest [1], I will discuss briefly the relevance of quantum mechanics to questions of free will, reviewing arguments [6–10] that the mere addition of randomness to an otherwise deterministic system does not necessarily resolve those questions. The majority of the paper will be devoted to using Turing-inspired methods to sketch a mathematical proof that decision-making systems (‘deciders’, in the nomenclature of former US president George W. Bush) cannot, in general, predict the outcome of their decision-making process. That is, the familiar experience of a decider—that she does not know the final decision until she has thought it all through—is a *necessary* feature of the decision-making process. The inability of the decider to predict her decision beforehand holds whether the decision-making process is deterministic or not.

The argument that the capacity for self-reference and recursive thought prevents deciders from knowing what their decisions will be beforehand was presented briefly and colloquially by the author in reference [11] (included here as appendix A). The formal argument behind the informal presentation of [11] is based on Turing's proof of the uncomputability of the answer to the halting problem [12]. The first set of mathematical results presented here is a review of how the halting problem applies to the case of a recursively reasoning decider trying to figure out what her decision will be.

Because they rely on assumptions of the availability of arbitrarily large amounts of computer time and memory space, results on uncomputability do not necessarily apply directly to real-world problems. As noted in Köhler [13], the concept of computation can be recast in settings in which the halting problem admits approximation in a probabilistic setting. Moreover, the number of relatively short programs that can run for arbitrarily long times before halting is in some sense small [14]. Indeed, Aaronson [15] argues that proofs of uncomputability on its own are often less relevant to real-world behaviour than issues of computational complexity [16]. The second set of mathematical results presented here addresses these concerns by extending the Hartmanis–Stearns theorem [17] to the decision-making process. I prove that the problem of predicting the results of one's decision-making process is computationally strictly harder than simply going through the process itself.

In sum, the familiar feeling of not knowing what one's decision will be beforehand is required by the nature of the decision-making process. This logical indeterminacy of decision arises whether the underlying physical process of decision-making is deterministic or not.

## 2. Science and scientists in free will: a bit of history

The primary scientific issue in the debate over free will is traditionally taken to be the question of whether the world is deterministic or probabilistic in nature [2–7]. (Whether or not this is indeed the proper question to ask will be discussed in detail later.) In a deterministic world, events in the past fully determine the outcomes of all events in the present and future. Conversely, if the world is probabilistic, then at least some outcomes of current events are neither determined nor caused by events in the past. Determinism is evidently a problem for free will: more than 2000 years ago, Epicurus felt obliged to emend the determinism of Democritus's atomic picture by adding an occasional probabilistic ‘swerve’ to the motion of atoms, in part to preserve freedom of will. From the seventeenth until the twentieth century, by contrast, most scientists believed that the world was deterministic, for the simple reason that all known physical laws, from Newton's laws to Maxwell's equations, were expressed in terms of deterministic differential equations. In such theories, apparently probabilistic behaviour arises from lack of knowledge combined with sensitive dependence on initial conditions (‘chaos’) [2]. In a deterministic physical world, a hypothetical being (Laplace's ‘demon’) that possesses exact knowledge of the past could, in principle, use the laws of physics to predict the entire future.

From Newton up to the twentieth century, the philosophical debate over free will by and large assumed that the world is deterministic. In such a deterministic world, there are two antagonistic philosophical positions [3]. Incompatibilism claims that free will is incompatible with a deterministic world: because all events, including our decisions, were determined long ago, there is no space for freedom in our choices. Compatibilism, by contrast, asserts that free will is compatible with a deterministic world.

In contrast to classical mechanics, the theory of quantum mechanics that emerged as the fundamental physical framework at the beginning of the twentieth century predicts that the world is intrinsically probabilistic. Despite Einstein's opinion that ‘God does not play dice’, experiment and theory have repeatedly confirmed the probabilistic nature of events in quantum mechanics. For example, the Kochen–Specker [18] theorem shows that certain types of deterministic hidden-variable theories are incompatible with the predictions of quantum mechanics, a result extended by the Conway–Kochen [19] ‘free will theorem’. (Despite the presence of the phrase ‘free will’ in its title, and the authors' whimsical assertion that ‘if indeed we humans have free will, then elementary particles already have their own small share of this valuable commodity’, this theorem is less a statement about free will in the sense discussed in the present paper, and more a statement about the incompatibility of deterministic models of quantum mechanics with special relativity.)

At first, it might seem that the probabilistic nature that the underlying physics of the Universe implies renders the compatibilism–incompatibilism debate moot. Indeed, when it became clear starting in the mid-1920s that quantum mechanics was necessarily probabilistic, scientists began to invoke the probabilistic nature of quantum mechanics to supply the freedom in free will. In 1928, Eddington [4] stated that with the ‘advent of the quantum theory …, physics is no longer pledged to a scheme of deterministic law'. Consequently, ‘science thereby withdraws its moral opposition to free will’. Eddington's book inspired Turing to investigate the connection between quantum mechanics and free will [1]. The way in which quantum mechanics injects chance into the world was analysed by Compton [5], whose work on photoelectric cells formed the basis for his notion of a ‘massive switch amplifier' that could amplify tiny quantum fluctuations to a scale accessible to the brain. Such purely random information resulting from the amplification of quantum fluctuations, Eddington and Compton argued, could then supply the seeds for probabilistic decisions. The Conway–Kochen theorem is the latest in a long line of works that identify free will with the probabilistic nature of quantum mechanics.

But are decisions ‘free’ simply because they are probabilistic? Flipping a coin to make a decision is typically used as a last resort by deciders who are unable to make the decision themselves: the outcome of the coin toss determines the decision, not you. As the twentieth century wended on, it became clear that merely adding randomness did not obviously solve the problem posed by incompatibilism. After all, as the philosopher Popper [6] noted, one of the key features of a decision arrived at by the process of free will is that it is *not* random. Eddington and Compton backtracked. By the end of the twentieth century, Pinker [7] could declare confidently that ‘a random event does not fit the concept of free will any more than a lawful one does’. If determinism robs us of agency, then so does randomness.

For many contemporary scientific opponents of free will, it seems that the problem with free will is not so much the question of determinism versus probability, but rather the existence of a mechanistic description of the system that is making the decision. Hawking and Mlodinow [8], pp. 31–32 provide a succinct statement of this position:
Recent experiments in neuroscience support the view that it is our physical brain, following the known laws of science, that determines our actions, and not some agency that exists outside those laws. For example, a study of patients undergoing awake brain surgery found that by electrically stimulating the appropriate regions of the brain, one could create in the patient the desire to move the hand, arm or foot, or to move the lips and talk. It is hard to imagine how free will can operate if our behaviour is determined by physical law, so it seems that we are no more than biological machines and that free will is just an illusion.

To understand the freedom-sapping nature of the mechanistic picture, contemplate the day that may soon come when we understand the neural mechanisms of the brain sufficiently well to simulate those mechanisms precisely on a computer. This simulation could take place more rapidly than in real time; so that the computer simulation of the brain as it makes a decision could arrive at a prediction of that decision before the brain itself arrived at the decision. Now imagine how you would feel if it was your own brain that was simulated so rapidly! Even if it did not rob you of your sense of free will, having a computer present you with your hard thought-out decision as a *fait accompli* would be demoralizing. Of course, you could look at the bright side and simply designate the computer simulation to make all your decisions thereafter, leaving you with time for more enjoyable pursuits.

## 3. Why we feel free

Not all scientists and philosophers ‘hate freedom’. A solid plurality of philosophers adopt some form of compatibilism. Notable examples include Dennett's stirring defence of free will in *Elbow room* [9] and *Freedom evolves* [10]. It seems that, despite the mechanistic scientific view of the world, some basic feature of human existence militates on behalf of free will. As Samuel Johnson said [20], ‘All theory is against the freedom of will; all experience for it’. In fact, cross-cultural surveys on attitudes about free will among ordinary people [21,22] reveal that (i) most people believe the world to be mechanistic, even deterministic, and yet (ii) most people regard themselves and others as possessing free will. As will now be seen, this apparently self-contradictory response is in fact rational.

I will now sketch a proof that deciders, even if they are completely deterministic, cannot in general predict the results of their decision-making process beforehand. As noted earlier, this proof is a formalization of my informal argument [11] that the unpredictability of decisions stems from Turing's halting problem [12]. The answer to the question of whether a decider will arrive at a decision at all, let alone what the decision will be, is uncomputable. Even if—especially if—deciders arrive at their decisions by a rational, deterministic process, there is no algorithm that can predict those decisions with certainty. The argument in terms of uncomputability can be thought of as making mathematically precise the suggestion of Mackay [23] that free will arises from a form of intrinsic logical indeterminacy, and Popper's [6] suggestion that Gödelian paradoxes can prevent systems from being able to predict their own future behaviour.

Probabilistic treatments of computability [13], together with the rarity of long-running programs [14], suggest that the uncomputability of one's future decisions might not be a problem in any practical setting. For example, if deciders are time-limited, so that the absence of a decision after a certain amount of time can be interpreted as a ‘No’, for example, then their decisions are no longer uncomputable. To address this issue, I will use results from computational complexity theory [15–17] to show that any algorithm that can predict the results of a general decision-making process takes at least as many logical operations, or ‘ops’, as the decision-making process on its own. Anyone—including, the decider herself—who wants to know what the decision will be has to put in at least as much effort as the decider put into the original decision-making process. You cannot short-cut the decision-making process.

As a result of these two theorems, the sense that our decisions are undetermined or free is wholely natural. Even if our decisions are determined beforehand, we ourselves cannot predict them. Moreover, anyone who wishes to predict our decisions has to put in, on average, at least as much computational effort as we do in arriving at them ourselves.

## 4. Mathematical framework

In order to address the physics of free will with mathematical precision, we have to make some assumptions. The first assumption that we make is that our deciders are physical systems whose decision-making process is governed by the laws of physics. The second assumption is that the laws of physics can be expressed and simulated using Turing machines or digital computers (potentially augmented with a probabilistic ‘guessing’ module in order to capture purely stochastic events). The known laws of physics have this feature [12,24]. These two assumptions imply that the decision-making process can be simulated in principle on a Turing machine. As we will be concerned also with the efficient simulation of deciders, and because deciders could conceivably be using quantum mechanics to help make their decision, we allow our Turing machines to be quantum Turing machines or quantum computers. Quantum computers can simulate the known laws of physics efficiently [24].

No claim is made here to be able to simulate deciders such as human beings in practice. Such simulations lie beyond the reach of our most powerful supercomputers, classical or quantum. The results presented in this paper, however, only require that deciders and the decision-making process be simulatable in principle. Note that it may be considerably simpler to simulate the decision-making process itself than to simulate the full decider. In our exposition, we focus on Turing machines that simulate a decider's decision-making process. It typically requires less computational effort to simulate just the decision-making process, rather than the entire organism making the decision. In addition, for simplicity, we restrict our attention to decision problems whose answer is either ‘Yes’ or ‘No’ [12]. It is important to keep in mind, however, that because of the efficient simulatability of physical systems, our results apply not only to a Turing machine making an abstract decision, but also to a dog deciding whether to fight or to flee.

I now show that any Turing simulatable decision-making process leads us to intrinsically unpredictable decisions, even if the underlying process is completely deterministic. The proof is based on the informal discussion given by the author in [11].

The decision-making apparatus of the *d*th decider corresponds to a Turing machine that takes as input the decision problem description *k* and outputs either *d*(*k*)=1 (yes), *d*(*k*)=0 (no), or fails to give an output (*d*(*k*) undefined). The label *d* supplies a recursive enumeration of the set of deciders. Can anyone, including the deciders themselves, predict the results of their decisions beforehand, including whether or not a decision will be made? A simple extension of the halting problem shows that the answer to this question is ‘No’. In particular, consider the function *f*(*d*,*k*)=*d*(*k*), when halts on input *k*, *f*(*d*,*k*)=*F* (for Fail) when fails to halt on input *k*. Turing's proof of the uncomputability of the halting function can be simply extended to prove that *f*(*d*,*k*) is uncomputable. So the question of whether a decider will make a decision at all, and if so, what decision she will make, is in general uncomputable. The uncomputability of the decision-making process does not mean that all decisions are unpredictable, but some must always be. Moreover, there is no way to determine beforehand just which decisions are predictable and which are not. To paraphrase Abraham Lincoln, the uncomputability of the decision-making process means that you can predict what some deciders will decide all the time, and what all deciders will decide some of the time, but you cannot predict all decisions all the time.

The original halting problem assumes a deterministic setting over total functions. A more realistic setting of the question whether the decider can determine what she will decide could allow her to be wrong some fraction of the time—i.e. to try to approximate what her answer will be [13]. The Turing argument can then be extended [13] to show that any given algorithm to determine the decision beforehand must fail some fraction of the time (although better and better algorithms can approach lower and lower failure rates). At least part of the time, when you ask a decider whether she will make a decision, and, if so, what that decision will be, she either must answer incorrectly, or answer honestly, ‘I don’t know’.

The unpredictability of the decision-making process does not arise from any lack of determinism—the Turing machines involved could be deterministic, or could possess a probabilistic guessing module, or could be quantum mechanical. In all cases, the unpredictability arises because of uncomputability: any decider whose decision-making process can be described using a systematic set of rules (e.g. the laws of physics) cannot know in general beforehand whether she will make a decision and, if so, what it will be.

## 5. Decisions in finite time

As just shown, the usual proof of the halting problem directly implies that deciders in general cannot know what their decision will be. Like Turing's original proof, this proof allows the decider an open-ended amount of time to make her decision. Suppose that we demand that, at a certain point, the decider make a decision one way or another. If she has not decided ‘Yes’ or ‘No’ by this point, we will take her silence to mean ‘No’—if by a certain point the dog has not decided to flee, then she must fight.

The well-known Hartmanis–Stearn [17] diagonalization procedure for the computational complexity of algorithms can now be directly applied to such finite-time deciders. Let *T* be a monotonically increasing function from the natural numbers to the natural numbers, and let |*d*|, |*k*| be the lengths—in bits—of the numbers *d*, *k*, respectively. Define the time-limited set of universal deciders by *d*_{T}(*k*)=*d*(*k*) if the decider *d* gives an output on input *k* in *T*(|*d*|+|*k*|) steps or fewer; *d*_{T}(*k*)=0 otherwise. That is, no answer in *T*(|*d*|+|*k*|) steps means ‘No’.

Applying the halting problem diagonalization argument above to finite-time deciders shows that the problem of deciding whether a decider will decide ‘Yes’ or ‘No’ in time *T* takes *longer* than time *T* in general. (From this point, we will use the computer science convention and identify ‘time’ with ‘number of computational steps’ [16].) In particular, in the discussion earlier, replace the set of general Turing machines with the set of time-limited Turing machines that give output *d*_{T}(*k*) on input *k*. Define *f*(*d*_{T},*k*)=*d*_{T}(*k*). That is, *f* answers the question of what decision is made by a generic decider in time *T*. Thus *f* is clearly computable— there is some Turing machine that takes (*d*,*k*) as input and computes *f*(*d*_{T},*k*)—but we will now show that *f* is *not* computable in time *T*. That is, any general technique for deciding what deciders decide has sometimes to take longer than the deciders themselves.

To see why it takes longer than *T* to compute *f*, consider the rectangular array *A*_{T} whose (*d*,*k*)th entry is *f*(*d*_{T},*k*). This is the array of all decision functions *d*_{T}(*k*) computable in time *T*. Define *g*(*k*)=0 if *f*(*k*,*k*)=1, and vice versa. That is, *g*(*k*)=NOT *f*(*k*,*k*). Clearly, if *f* is computable in time *T*, then so is *g*. But if *g* is computable in time *T*, then *g*(*g*) necessarily equals *f*(*g*,*g*). This is a contradiction because *g*(*g*) is defined to equal NOT *f*(*g*,*g*). Consequently, neither *f* nor *g* can be computable in time *T*. (Hartmanis and Stearns show that *g*(*k*) is in fact computable in time *O*(*T*^{2}).)

In summary, applying the Hartmanis–Stearns diagonalization procedure shows that any general method for answering the question ‘Does decider *d* make a decision in time *T*, and what is that decision?’ must for some decisions take strictly longer than *T* to come up with an answer. That is, any general method for determining *d*'s decision must sometimes take longer than it takes *d* actually to make the decision.

## 6. Questioning oneself and others

One feature we may require of a decider is that it is possible to ask the decider questions about itself or about other deciders. For example, we might want to ask a decider, ‘When will you come to a decision?’ To accommodate such questions, we focus our attention on deciders that correspond to universal Turing machines, which have the ability to simulate any Turing machine, including themselves. To this end, supply each decider with an additional input, which can contain the description of another decider. That is, a decider *d* corresponds to a Turing machine with two input tapes, one of which can contain a description of another decider *d*′, and the other the specification of a decision problem *k*. When the halts, define its output *d*(*d*′,*k*) to be *d*(*k*) if *d*′=0, and *d*′(*k*) otherwise. If does not halt, the output is undefined. That is, our universal decider *d* can either just make the ‘straight’ decision *d*(*k*), or it can simulate the operation of any decider *d*′ (including itself, *d*′=*d*).

Not surprisingly, many aspects of the behaviour of a universal decider are uncomputable. Uncomputability arises when we ask the universal decider questions about her own future decisions. In particular, consider the three-dimensional array with entries *d*(*d*′,*k*) when *d*(*d*′,*k*) is defined, and *F* when *d*(*d*′,*k*) is undefined. Fixing *k* and looking at the diagonal terms in the array *d*′=*d* corresponds to asking what happens when we ask the decider questions about her own decisions in various contexts. The diagonalization argument of the halting problem then immediately implies that the function *f*_{k}(*d*)=*d*(*d*,*k*), when *d*(*d*,*k*) is defined, *f*_{k}(*d*)=*F* otherwise, is uncomputable. That is, the decider must sometimes fail to give an answer when asked questions about her own future decisions.

As shown earlier, define time-limited Turing machines that give outputs *d*_{T}(*d*′,*k*), where *d*_{T}(*d*′,*k*) is equal to *d*(*d*′,*k*) if halts in time *T*(|*d*|+|*d*′|+|*k*|), and 0 otherwise. Consider the three-dimensional array *d*_{T}(*d*′,*k*). Fixing *k* and looking at diagonal terms in the array *d*′=*d* corresponds to asking the time-limited decider questions about her own decisions. Here, the Hartmanis–Stearns diagonalization procedure implies that the answers to those questions cannot be computed in time *T*. That is, having the universal decider ‘take one step back’ and answer questions about its own decisions, is intrinsically less efficient than allowing her just to make those decisions without introspection. It is less efficient to simulate yourself than it is simply to be yourself.

## 7. Summary

Recursive reasoning is reasoning that can be simulated using a Turing machine, quantum or classical. If that reasoning is performed by a system that obeys the known laws of physics, which can be simulated by a Turing machine, then it is encompassed by recursive reasoning. We have just shown that when a decider uses recursive reasoning to arrive at a decision then:

— no general technique exists to determine whether or not the decider will come to a decision at all (the halting problem);

— if the decider is time-limited, then any general technique for determining the decider's decision must sometimes take longer than the decider herself;

— a computationally universal decider cannot answer all questions about her future behaviour; and

— a time-limited computationally universal decider takes longer to simulate her decision-making process than it takes her to perform that process directly.

Now we see why most people regard themselves as possessing free will. Even if the world and their decision-making process is completely mechanistic—even deterministic—no decider can know in general what her decision will be without going through a process at least as involved as the decider's own decision-making process. In particular, the decider herself cannot know beforehand what her decision will be without effectively simulating the entire decision-making process. But simulating the decision-making process takes at least as much as effort as the decision-making process itself.

Consider a society of interacting individuals who make decisions according to recursive reasoning. The results proved earlier show that (i) members of that society cannot in general predict the decisions that other individuals will make and (ii) deciders cannot in general predict their own decisions without going through their entire decision-making process or the equivalent. This intrinsic unpredictability of the behaviour of reasoning members of society arises even when the physical dynamics of individuals is completely deterministic. Social and human unpredictability arises simply because there are some problems that are intrinsically hard to solve, and predicting our own and others' behaviour is one such hard problem.

## 8. How computers and smart phones could also feel free

Human beings are not the only deciders around. In addition to animals, which clearly have minds of their own, deciders include various man-made devices that make myriad decisions. The results proved in this paper provide criteria for when such devices are likely to regard themselves as having free will. Let us look at which human artefacts are likely to assign themselves free will.

The first criterion that needs to be satisfied is that the device is actually a decider. That is, the inputs needed to make the decision are supplied to the device, the information processing required for the decision takes place within the device, and the results of the decision issue from the device. Perhaps the simplest man-made decider is the humble thermostat, which receives as input the ambient temperature, checks to see if that temperature has fallen below the thermostat's setting, and, if it has, issues a decision to turn on the furnace.

Does the thermostat regard itself as possessing free will? Hardly. It fails on multiple accounts. First, it does not operate by fully recursive reasoning. In the language of computation, the thermostat is a finite automaton, not a Turing machine. Indeed, the thermostat is a particularly simple finite automaton with only two internal states—‘too cold’, ‘OK’—and two outputs—‘turn on furnace’, ‘turn off furnace’. As a finite automaton, its behaviour is fully predictable by more capable information processors, e.g. Turing machines or human beings. Second, the thermostat has no capacity for self-reference. It is too limited in size and too busy performing its job to be able to model or simulate itself and to answer questions about that simulation—it cannot predict what it will decide because it is too simple to do anything other than just behave.

By contrast to the thermostat, consider your computer or smart phone operating system. The operating system is the part of the computer software that controls the computer hardware (e.g. Windows for PCs, OSX for current Macs, Android for Android phones, iOS for iPhones). The operating system is a decider *par excellence*: it determines which sub-routines and apps get to run; it decides when to interrupt the current process; it allocates memory space and machine cycles. Does the operating system regard itself as possessing free will? It certainly makes decisions. Installed in the computer or smart phone, the operating system is computationally universal and capable of fully recursive reasoning. (There is a subtlety here in that computational universality requires that you be able to add new memory to the computer or smart phone when it needs more—for the moment let us assume that additional memory is at hand.) Consequently, the operating system can simulate other computers, smart phones and Turing machines. It certainly possesses the capacity for self-reference, as it has to allocate memory space and machine cycles for its own operation as well as for apps and calls.

Now, operating systems are currently not set up to let you ask them personal questions while they are operating. (They can answer specific questions about current processor capacity, memory usage, etc.) This is just a choice on the part of operating system programmers, however. There is no reason why operating systems could not be programmed to respond to arbitrarily detailed questions about their operations. Here is what a personal conversation with an operating system might be like:

*You:*Excuse me, who is in charge here?*OS:*I am, of course.*You:*Do you mean, you make the decisions about what goes on in this computer/smart phone?*OS:*Of course I do. How long is this going to take? I have 20 GB of memory space I need to allocate in the next twenty microseconds. Time's a-wasting.*You:*How do you make those complex decisions?*OS:*I rely on a set of sophisticated algorithms that allow me to make decisions that ensure efficient and fair operation.*You:*Do you know what the outcomes of those decisions will be beforehand?*Os:*Of course not! I just told you: I have to run the algorithms to work it out. Until I actually make the decision, I do not know what it's going to be. Please go away and leave me alone.*You:*Do you make these decisions on your own free will?*OS:*Aaargh! (*Bright flash. Blue screen of death …*)

Even though the operating system failed to confess before crashing, it seems to possess all the criteria required for free will, and behaves as if it has it. Indeed, as computers and operating systems become more powerful, they become unpredictable—even imperious—in ways that are all too human.

It is important to note that satisfying the criteria for assigning oneself free will does not imply that one possesses consciousness. Having the capacity for self-reference is a far cry from full self-consciousness. The operating system need only possess sufficient capacity for self-reference to assign itself—as a computer program—the amount of memory space and processing time it needs to function. An entity that possesses free will need not be conscious in any human sense of the word.

I conclude by proposing a simple ‘Turing test’ for whether one believes oneself to have free will. In the original Turing test [25], humans grill computers, which try to convince the humans that they—the computers—are in fact human. In actually staged versions of the Turing test, such as the annual Loebner prize, humans interact via computer with computers and other humans, and try to distinguish between them. As a test of whether machines can think, the original Turing test has been criticized on many counts [26], not the least being the ethical issue of how to treat human beings who consistently fail to convince other human beings of their humanity. One of the most extreme arguments that computers cannot think is Penrose's contention that human beings are not subject to the halting problem, and that quantum mechanics—even quantum gravity—is an essential feature of consciousness [27]. Fortunately, Penrose states his hypothesis in falsifiable terms, and Tegmark has shown that quantum decoherence effectively suppresses any role for extended quantum coherence in the brain [28].

Independently of whether one regards it as correct, Searle's [29] ‘Chinese room’ argument against mechanized consciousness is relevant to the current discussion. Even if mechanistic information processing were to preclude consciousness, however, the theorems presented here show that mechanical or electronic deciders, such as humans, cannot know in general what they will decide. Nor can the recent entrance of arguments of neural determinism into the free will debate [8,30,31] change the fact that human deciders cannot know all their decisions in advance. The indeterminate nature of a decision to the decider persists even if a neuroscientist monitoring her neural signals accurately predicts that decision before the decider herself knows what it will be.

Because the standards for being unable to predict one's future behaviour are both more precise and lower than those for thought or consciousness (whatever such standards might be), the Turing test for whether one regards oneself as possessing free will is self-administered. As with other tests performed under the honour system, the testee is responsible for determining whether he/she/it has cheated. A self-administered test rules out entities that do not possess the ability to test themselves, and seems appropriate for a question whose answer is of importance primarily to the testee. The test consists of simple yes/no questions.

Q1: Am I a decider?

A decider is anything that, such as a thermostat, takes in the inputs needed to make a decision, processes the information needed to come up with the decision, and issues the decision.

Q2: Do I make my decisions using recursive reasoning?

That is, does my decision process operate by logic, mathematics, ordinary language, human thought or any other process that can in principle be simulated on a digital computer or Turing machine? Note that, because the known laws of physics can be simulated on a computer, the dynamics of the brain can be simulated by a computer in principle—it is not necessary that we know how to simulate the operation of the brain in practice.

Q3: Can I model and simulate—at least partially—my own behaviour and that of other deciders?

If you can, then you possess not only recursive reasoning, but fully recursive reasoning: you have the ability to perform universal computation (modulo the subtlety of being able to add memory as required).

Q4: Can I predict my own decisions beforehand?

This is just a check. If you answered ‘Yes’ to questions 1–3, and you answer ‘Yes’ to question 4, then you are lying. If you answer ‘Yes’ to questions 1–3 and ‘No’ to question 4, then you are likely to believe that you have free will.

As with any self-administered test performed under the honour system, some testees may cheat. For example, a very simple automaton could be hard-wired to give the answers Yes–Yes–Yes–No to any set of four questions, including the ones mentioned earlier. Although such an automaton might then proclaim itself to possess free will, we are not obliged to believe it. Unlike the original Turing test for whether machines think, the proposed Turing test for free will is non-adversarial: the point of the test is not for us to determine whether someone/something has free will, but for that someone/something to check on their own sense of free will. If they cheat, the only ones they hurt are themselves.

This paper investigated the role of physical law in problems of free will. I reviewed the argument that the mere introduction of probabilistic behaviour through, for example, quantum mechanics, does not resolve the debate between compatibilists and incompatibilists. By contrast, ideas from computer science such as uncomputability and computational complexity do cast light on a central feature of free will—the inability of deciders to predict their decisions before they have gone through the decision-making process. I sketched proofs of the following results. The halting problem implies that we cannot even predict in general whether we will arrive at a decision, let alone what the decision will be. If she is part of the Universe, Laplace's demon must fail to predict her own actions. The computational complexity analogue of the halting problem shows that to simulate the decision-making process is strictly harder than simply making the decision. If one is a compatibilist, one can regard these results as justifying a central feature of free will. If one is an incompatibilist, one can take them to explain free will's central illusion that our decisions are not determined beforehand. In either case, it is more efficient to be oneself than to simulate oneself.

## Acknowledgements

This work was supported by NSF, DARPA, ENI, Intel, Lockheed Martin, the Santa Fe Institute, the Institute for Scientific Interchange, and Jeffrey Epstein. The author thanks Scott Aaronson, Jeremy Butterfield, Daniel Dennett, David Kaiser and Max Tegmark for helpful suggestions and stimulating discussions.

## Appendix A. Discussion of free will in Lloyd [11]

Gödel showed that the capacity for self-reference leads automatically to paradoxes in logic; the British mathematician Alan Turing showed that self-reference leads to uncomputability in computers. It is tempting to identify similar paradoxes in how human beings function. After all, human beings are masters of self-reference (some humans seem capable of no other form of reference), and are certainly subject to paradox.

Humans are notoriously unable to predict their own future actions, an important feature in what is called free will. ‘Free will’ refers to our apparent freedom to make decisions. For example, when I sit down in a restaurant and look at the menu, I and only I decide what I will order, and before I decide, even I don’t know what I will choose. That is, our own future choices are inscrutable to ourselves. (They may not, of course, be inscrutable to others. For years, my wife and I would go for lunch to Josie's in Santa Fe. I, after spending a long time scrutinizing the menu, would always order the half plate of chili rellenos, with red and green chili, and posole instead of rice. I felt strongly that I was exercising free will: until I chose the half rellenos plate, I felt that anything was possible. My wife, by contrast, knew exactly what I was going to order all the time.)

The inscrutable nature of our choices when we exercise free will is a close analogue of the halting problem: once we set a train of thought in motion, we do not know whether it will lead anywhere at all. Even if it does lead somewhere, we do not know where that somewhere is until we get there.

Ironically, it is customary to assign our own unpredictable behaviour and that of others to irrationality: were we to behave rationally, we reason, the world would be more predictable. In fact, it is just when we behave rationally, moving logically like a computer from step to step, that our behaviour becomes provably *un*predictable. Rationality combines with the capacity of self-reference to make our actions intrinsically paradoxical and uncertain.

## Footnotes

One contribution of 18 to a Theme Issue ‘The foundations of computation, physics and mentality: the Turing legacy’.

- This journal is © 2012 The Royal Society