The Generative Programs Framework

Usually a generative program will require some inputs which do not depend on anything else within the program, which we will refer to as ‘original vertices.’ For example, in traditional deterministic dynamical theories the initial conditions are ‘original vertices,’ as are the time-evolution laws, and the generative program determines the remaining course of history entirely from those original vertices. Similarly, in operational theories the agent is usually regarded as external to the physical theory and thus in an operational theory the decisions made by agents about actions like state preparations and measurements are ‘original vertices.’ On the other hand, physical theories with a more realist flavour may aspire to model agents within the theory, and therefore they may not treat the decisions made by agents as original vertices: instead an agent may be regarded as a physical system which takes inputs (such as sensory information and memories of the past) and produces outputs according to some internal process which can in principle be predicted within the physical theory.

By treating a vertex as an original vertex we are adopting an official attitude of agnosticism towards its origins, i.e. we are stipulating that its value is simply to be taken as given and there is nothing more to be said about where that value comes from. So for example, one should not imagine that the value of an original vertex $v$ is selected in a probabilistic way - that would involve postulating an objective probability distribution and measure outside of the generating program, but by stipulation if such a thing is known it should be included in the generating program, so a variable of this kind would not be original. However, of course we may still assign epistemic probability distributions over the values of the original vertices, provided we are clear that these distributions represent our own uncertainty about their values rather than any objective probabilistic generating process. And since there are by definition no known facts which could play a role in determining the value of $v$, we may invoke the principle of indifference to say that any agent making use of the generating program to arrive at predictions should assign a uniform epistemic probability distribution over the possible values of the original vertex. Here we stipulate that original vertices should always be defined in such a way that a uniform epistemic probability distribution over the values of the variables at the original vertex will induce epistemic probabilities downstream which match the predictions of the theory, ensuring empirical adequacy - in this context ‘uniform’ means simply ‘uniform according to the natural Lebesgue measure over the reals,’ but this is merely a convenient choice, since one could use any other range and measure provided that subsequent processing steps in the generating program are suitably adjusted to ensure the program predicts the expected probabilities.

Of course, there may also be cases where a generative program does postulate an objective probability distribution over the value of some variables, so we can also have probabilistic variables, which are different from original vertices. In this paper we will take it that probabilistic variables can be modelled by an original vertex $v_{1}$ combined with another vertex $v_{2}$ representing the probability distribution specified by the generative program - for example, if the program prescribes that a certain variable has value $a$ with probability 0.75 and value $b$ with probability 0.25, we could model this using an original vertex $v_{1}$ which can take any value in the range $[0,1]$,and an additional vertex $v_{2}$ whose value will here be set to $0.75$, and then we combine the two vertices to give a third vertex according to the prescription: if $v_{1}\leq v_{2}$ then $v_{3}=a$, otherwise if $v_{1}>v_{2}$ then $v_{3}=b$. Because $v_{1}$ is an original vertex, no objective probability distribution can be postulated over it, but as we have just specified, users of the program will associate it with a subjective probability distribution, which by convention we are taking to be a uniform distribution over the range $[0,1]$, and thus this program will reproduce the desired probability distribution over $a$ and $b$. Note that usually vertex $v_{2}$ will have other arrows going into it representing the prior factors which determine the probability distribution (here fully specified by just setting the value $v_{2}=0.75$) - for example, if the probabilistic event is a quantum measurement, $v_{2}$ will have arrows going into it from the quantum state and measurement direction which together determine the probability distribution for the measurement outcomes - see figure 4.

Throughout this article we will largely deal with generative programs which do not represent a whole universe but instead produce a complete physical description of some particular experimental scenario. In such cases we may have background knowledge that the original vertices are not ‘really’ original, since their values are fixed by processes in the world outside of the experimental scenario. But it is still appropriate to regard them as ‘original’ relative to the physical description of the experimental scenario because they depend only on parts of the universe not included in the scenario, so in the context of this specific scenario there is indeed nothing more to be said about where they come from. In the standard terminology of the causal modelling framework, the original vertices are ‘exogenous’ variables which are assumed not to depend on events taking place within the experimental scenario being studied. An experimental scenario or subregion of spacetime which can be given a self-contained description conditional on a set of variables treated as exogenous can essentially be regarded as something like an independent universe, with the original vertices playing the role of a ‘boundary’ for the universe and the dependent vertices making up the bulk.

Although all physical theories must provide some generative program in order to come up with some empirical data, there are different possible views that one might hold on the significance of these programs. For example, operationalists and constructive empiricists [25] do not in general commit to any ontology beyond the observable, so they will presumably not attribute any ontological significance to the generative programs they employ to arrive at empirical data - they will simply regard a generative program as a convenient codification of certain observable regularities. Realists, on the other hand, are committed to the existence of entities or structures which are not observable, and thus they may be inclined to attribute ontological significance to their generative programs. In particular, we suggest that in the most general case realists may think of generative programs as encoding information about ontological priority. ( [26, 27]).

Here we say that ‘A is ontologically prior to B,’ just in case B depends on A for its existence - for example, if one believes that the universe unfolds by a continuous evolution process involving ‘dynamical production,’ then one will naturally think that later states of the universe depend on earlier states of the universe for their existence, so earlier states are ontologically prior to later ones. In principle all sorts of things may depend on other things for their existence - for example, one might think that ‘the fact that A & B’ depends for its existence on ‘the fact that A’ and ‘the fact that B.’ However, in this article we are concerned specifically with ontological priority construed as a physical relation between distinct real entities, so we will not consider applications of it outside of the physical domain.

Ontological priority is often described metaphorically as representing the order in which god would have to create things in order to arrive at the universe - or in less theological terms, one could think of it as a computer program specifying the steps one would have to take in order to arrive at a complete universe [27, 28]. This metaphor makes it clear why it seems natural to interpret ‘generative programs’ in terms of ontological priority. Ontological priority is a reasonably common concept in philosophy, but has not yet made its way into physics in any explicit way, perhaps because physicists tend to be sceptical of notions which sound excessively ‘metaphysical.’ But really ontological priority (at least insofar as that term is understood as pertaining specifically to nomic or modal relations between distinct real entities) is not any more metaphysical than ‘causation,’ so in our view, anyone who is willing to accept ‘causation’ as falling in the domain of physics should also be willing to accept ‘ontological priority.’ After all, causation can be understood as simply a special case of ontological priority: if one thinks of the statement ‘A is the cause of B’ as an assertion that B depends for its existence on A, then this statement means that A is ontologically prior to B, and thus standard causal-dynamical physical theories would typically be associated with a generative program in which the initial condition of the universe is ontologically prior to the rest of history, which is generated from it by time evolution.

On the other hand, as shown by the diagrams in section 2, relations of ontological priority don’t have to coincide with our presuppositions about causal structure - for example, we typically imagine that causation goes forward in time, but we can certainly write down programs in which future events are ontologically prior to past ones. In this article we will use the term ‘strong causation’ to refer to relations which are self-evidently causal according to our common-sense understanding of causation (i.e. mostly relations between macroscopic events where the cause precedes the effect in time); there is probably a case to be made that other relations of ontological priority might also qualify as ‘causal’ according to some generalized notion of causality, but we will not use that terminology in this article. Now, those who espouse a direct-access approach to the flow of time typically believe that our common-sense notion of strong causation reflects our experience of a real structure of temporal becoming, and thus they may argue that DAGs in which relations of ontological priority disagree with intuitions about strong causation are not candidates to represent the actual world, since they don’t get our experiences of temporal becoming right. However, proponents of the perspectival approach to the flow of time in which experience supervenes only on the the nodes and not the arrows of a DAG will presumably take the view that a DAG in which relations of ontological priority disagree with common-sense intuitions about strong causation could be a candidate to represent the actual world.

Furthermore, relations of ontological priority can also apply to more general relata than causal relations, including relata which may not be located in spacetime, because relations of ontological priority may hold between any entities which play an active role in the way in which a generative program produces data, regardless of whether or not the associated theory is usually interpreted as saying that entities of this kind live in spacetime. For example, fundamental constants, mathematical objects like Lagrangians, or even laws of nature could stand in relations of ontological priority, even though these kinds of objects are arguably best understood as non-spatiotemporal.

There is a well-recognised connection between causation and ontological priority. Schaffer, for example, has noted that there is a close connection between causation and a philosphical concept known as ‘grounding,’ which is related to ontological priority: ‘the analogies run deep: both (causation and grounding) feel like relations of generation, both look something like partial orders, and both can back explanation’ [29]. Schaffer therefore argues that grounding can be represented by a system of structural equations, akin to the structural equations used to represent causation in causal modelling, and thus the same mathematical formalism can be used to understand them both.¹¹1Our approach has much in common with Schaffer’s, but ultimately has different subject matter. In particular, our definition of ontological priority is both narrower and broader than the usual definition of grounding: on the one hand, grounding is usually understood as subsuming all kinds of cases where one entity depends on another for its existence, so for example one might say ‘the fact that A & B’ is grounded by ‘the fact that A’ and ‘the fact that B,’ whereas we have specified that ontological priority applies only to physical relations between distinct real entities. But on the other hand, our definition entails that ontological priority includes causation as a special case, whereas Schaffer at least does not see causation as a special case of grounding. In addition Schaffer’s primary motivation is to better understand the concept of grounding (via a non-reductive analysis), whereas our focus here is on showing how ontological priority can be applied to real scientific problems and arguing that it can be a scientifically useful tool. This means we do not have to deal with concerns like that raised by Jansson [30], who contends that using structural equations to make inferences about grounding relations is problematic because we don’t have access to the kinds of a posteriori knowledge needed to decide which variables the DAG should include or what the appropriate contrast class for each of the variables is - this issue does not arise in our case, because we are not attempting to use structural equations to make inferences about relations of ontological priority, rather we simply take it that the generative program specified within a theory stipulates what the relevant ontological variables are and what possible alternative values they have, so the purpose of the DAG is to clarify the nature of these commitments rather than to figure out what they are.

Here we follow Schaffer in arguing that because ontological priority is naturally represented by a directed acyclic graph, we may apply some mathematical techniques developed in the study of DAGS representing causal models (e.g. see refs [3] and [4]) to our own DAGs encoding ontological priority. For example, as in Schaffer’s grounding case, we can use generative programs to investigate how relations of ontological priority ground counterfactual relations - the generative program prescribes relations of functional dependence between nodes, so we can make quantitative statements about the outcome of making changes to the value of a variable at one node of the graph while leaving other inputs to the program unchanged, and moreover we can do this regardless of whether or not the node in question is one on which realistic agents could actually intervene. For example, suppose we have a generative program where the value of some constant of nature (e.g. the cosmological constant) is an original vertex, and this vertex is used to determine the state $S$ of the world at some time $t$. Of course no real agent can intervene to vary the cosmological constant, but nonetheless we can use the generative program to imagine ‘intervening’ by changing the value of the cosmological constant whilst keeping the program and the values of the other original vertices constant: if the program is properly formulated we should be able to calculate a new state $S^{\prime}$ for the world at time $t$, so we can check whether it is the case that, according to this particular generative program, the state $S$ counterfactually depends on the value of the cosmological constant. The practice of studying the consequences of varying fundamental constants is actually very common in physics (see refs [31, 32]) and our notion of ontological priorty can be understood as formalizing the reasoning behind this practice.

To make mathematical statements about the dependence relations encoded in a generative program, we will make use of a probability distribution over original vertices (although as already noted, this distribution should be understood epistemically rather than ontologically). For example, we may wish to stipulate that part of what it is for a generative program DAG to be a correct representation of a possible generative program is that that program should satisfy the causal Markov condition relative to that DAG - i.e. when we consider the outputs of the generative program given independent random choices of values at the original vertices, we should find that for each vertex $v$, conditional on the set of its ‘parents’ in the graph (i.e. all vertices $x$ such that there is an arrow from $x$ to $v$ ) $v$ is statistically independent of all variables which are not its parents or its descendants [3]. Of course in practice the original vertices only take one value so the causal Markov condition can’t be verified using any actual empirical data, but nonetheless we can use this condition to check whether or not a proposed DAG is consistent with the generative program it is supposed to represent - if the program generates variables which fail to exhibit the right independence relations, this suggests that the generative program involves some further relations of ontological priority that we have failed to capture in the DAG.

Similarly, in causal modelling it is common to require that causal graphs should exhibit ‘faithfulness,’ i.e., when we consider independent random choices at the original vertices, two variables in the graph should be statistically independent only if they are ‘d-separated’ in the underlying graph, which is to say that any paths between them are ‘blocked’ by other variables playing a role somewhat like a common cause [3]. So in particular, a graph fails to exhibit faithfulness if there is a causal arrow from A to B but the distribution over values obtained from independent random inputs at the original vertices shows A as being statistically independent of B. And similarly, it seems natural to require that graphs of ontological priority should similarly obey faithfulness, for if there are unexpected independences in the empirical data, this may indicate that there is an opportunity to add something further to the generative program that has been proposed - we discuss this point further in section 4.1.

We can also apply the generative program framework to bring out some important consequences of taking an operational approach. We have argued that the founding tenet of operationalism is that all ‘empirically equivalent’ physical programs are equally good; evidently this means that the operationalist must be able to specify clearly which parts of a graph of ontological priority are to be regarded as empirical and which are not. As we have already noted, this is in general a non-trivial thing to do. Typically, operationalists seek to make this distinction by postulating an in-principle split between the agent and the rest of the world, so we are to imagine an external agent acting on reality and getting back certain responses, which are the data to be predicted by the theory²²2See refs [43, 44, 45] for some examples of operational formulations of theories. However, we emphasize that the authors of these papers may not necessarily identify as ‘operationalists’ - it is possible to adopt an operational approach to study the structure of theories without necessarily rejecting the possibility of an underlying realist story.. This allows the operationalist to identify what counts as ‘empirical’ in a somewhat more clean way - in particular, they don’t have to worry about which parts of the internal structure of the agent’s decision-making process should count as empirical. Evidently operationalists of this kind are restricted to choosing generative programs which represent their own choices as original vertices, although they will not be committed to any definite structure for the rest of the graph.

But what happens if we have two or more different agents? In that case, there are two options: either we adopt a first-person operationalism in which each agent aims to construct a generative program to reproduce only their own observations, or we adopt a collective operationalism in which the aim is to construct a generative program to reproduce the observations of all of the agents simultaneously. However, the latter approach seems quite hard to justify within an operationalist picture, because of course ‘the experiences of other agents’ are not directly observable or accessible any more than the other unobservable entities that the operationalist doesn’t want to commit to, so an operationalist who is committed to the reality of the observations made by other agents but not to any other unobservable entities seems to be making a somewhat arbitrary distinction. Thus, in fact operationalism is usually understood in a first-person way (this has been advocated explicitly in the context of pragmatic and operationally-inclined interpretations of quantum mechanics, as in refs [46, 47]), which is in accordance with the operationalist maxim that a scientific theory is just a tool that an agent can employ to predict the measurement results that they will witness.

What this means is that if Alice is an operationalist, her generative program will usually represents her own decisions as original vertices, but it need not represent Bob’s decisions as original vertices: he is just another physical system which she can perform measurements on and which may or may not include some original vertices. Furthermore, in the first-person approach Alice is committed to the view that all programs which predict the same empirical data for her are equally good, and therefore she cannot attach any special significance to programs which represent Bob’s decisions as original vertices: from the operationalist point of the view there is no fact of the matter about whether they are original vertices or not. So Alice necessarily has a description of the world which is different from Bob’s, because both of them give an account of reality which involves quotienting over a set of programs, and only a small subset of Alice’s programs will be in Bob’s set and vice versa. Thus the operationalist view is not compatible with the existence of a single generative program which generates the entire universe: rather operationalists must employ a different set of generative programs $p_{A}$ for each agent $A$, such that all of the programs in $p_{A}$ represent the actions of $A$ as a set of original vertices but the programs in $p_{A}$ do not as a general rule represent the actions of other agents as a set of original vertices.

The differences between the sets of programs associated with distinct agents are even more pronounced in operational approaches to quantum mechanics. For operationalists typically hold that an agent should apply unitary quantum mechanics universally to the external physical world, and when this prescription is applied in situations like a Wigner’s friend experiment [48], it has the consequence that different agents can disagree not only on the structure of their generative program DAGs but also on the empirical data generated by the generative programs: Wigner’s friend will use a generative program which specifies a unique outcome for his measurement, whereas Wigner will use a generative program which says that the measuring device remains in a coherent superposition so there is no measurement outcome at all. The Frauchiger-Renner experiment makes this even more explicit by showing that in a certain case the programs used by different agents will predict incompatible outcomes to one and the same measurement [49]. This case, and the related Local Friendliness arguments [50, 51] also highlight that while an agent chooses inputs and receives outputs which become part of a permanent record, they model other agents as having no permanent record of their experiences; so in the quantum context, we end up with a set of disconnected generative programs used by different agents, and though these sets of generative programs will display roughly similar statistical features they nonetheless represent different physical worlds with both different structure and also different empirical content. Thus in general operational theories cannot provide us with any unified account of an intersubjective reality shared by different agents; in addition to being counterintuitive, this also raises serious questions about the scientific method and empirical confirmation in the context of operational theories (see ref [52] for a discussion of the importance of intersubjectivity in empirical confirmation).

These problems with intersubjectivity also extend to intersubjective agreement between different versions of the same observer at different times - for in an operational context it is difficult to model agents persisting over time without undermining the basic premises of operationalism. After all, part of what it is to be an agent persisting over time is to retain memories and act in ways that are partly influenced by one’s past experiences, and this surely requires that the agent is involved in some kind of two-way interaction with the systems described by the theory: the external world must have something to do with the memories and experiences that define the stable character of the persisting agent. One might expect this to give rise to a generative program in which measurement outcomes in the past are used to generate an agent’s future choices, or measurement outcomes and future choices are generated jointly by the program as in an all-at-once model. But using a program of this kind amounts to accepting that the agent is not, after all, separate from the physics: at least some parts of the agent’s decision-making processes are in fact determined within the generative program generating the empirical data. So if we allow some kind of two-way interaction between agents and their surroundings, it becomes difficult to maintain the operationalist’s strict separation between agent and world, thus undermining the operationalist’s identification of what counts as ‘empirical’ and making it unclear exactly which generative programs we are supposed to be quotienting over.

The generative program framework is useful here because it compels the operationalist to make a choice about what their position is really saying about agency: either they must use a completely distinct generative program to predict the result of each individual experimental situation, leaving us with an impoverished and fragmented approach, or they must accept that the agent is, in at least some rudimentary way, modelled as part of the generative program, meaning that we are no longer adhering to a strict operationalist position. An example of such an intermediate possibility is given by ref [53], which is a version of relational quantum mechanics with an additional postulate aiming to secure the possibility of intersubjective agreement between different agents. In this approach, facts are regarded as being relative to ‘observers’ (where every physical system can be an observer) and thus each observer, including different versions of the same observer at different times, has their own perspective on reality such that these perspectives may sometimes contain incompatible empirical facts. But the added postulate of ‘cross-perspective links’ ensures that agents can come to share facts during certain kinds of interactions, meaning that their perspectives come to agree on at least certain kinds of facts. Thus the generative program DAG for this theory would have some disconnected parts but would also contain connections between the DAGs associated with different observers, with those connections generated in the course of appropriate physical interactions. It would be interesting to see this idea made more formal - indeed, trying to come up with a well-defined generative program DAG for this version of RQM could be an important test of its coherence, because this process would require us to resolve some open questions about how exactly the connections in this picture can be constructed.

The generative program framework can also be used to gain clarity on contested issues surrounding ‘fine-tuning.’ Specifically, we will say that a generative program is ‘fine-tuned’ if two or more of the original vertices have values which appear to match in a way that is surprising, given that original vertices are supposed to be independent of one another.

A variety of well-known examples of fine-tuning in physics fit this definition. For example, the flatness problem [58]: the current density of matter and energy in the universe appears to be very close to the critical value required for a flat universe. Furthermore, natural mechanisms ensure that deviations from the critical value tend to increase rapidly over time, and therefore the initial density must be chosen very carefully to match the critical density since otherwise it would be very far from the critical value by now. Thus if we propose a generative program in which the initial density and the critical value are separate original vertices, then this generative program is fine-tuned. However, it is now generally believed that the matching can be explained by the mechanism of inflation [59], which suggests an alternative generative program in which the density is not an original vertex since it is ‘generated’ during a period of inflation which naturally forces it towards the critical density, and therefore we can get rid of the fine-tuning by moving to a different generative program.

Similarly, concerns about ‘naturalness’ in particle physics pertain to the fact that in certain calculations for physical parameters there is supposedly a precise matching between the bare quantities and the vacuum fluctuation contributions such that they cancel up to many orders of magnitude [60]. For example, this occurs in the usual calculation for the cosmological constant - the cosmological constant is very close to zero, and in order to achieve this the contribution due to vacuum fluctuations, which is proportional to the fourth power of the cut-off scale, must very exactly cancel the ‘bare’ value of the cosmological constant [61]. Moreover it seems quite natural to think that both the bare cosmological constant and the cutoff length should be original vertices, as depicted in figure 5, and yet if we propose a generative program in which they are original vertices it will be fine-tuned.

Now, we should clarify what we mean by the claim that original vertices can be expected to be ‘independent,’ for as we noted in section 2, original vertices are not in general to be thought of as probabilistic. Discussions of fine-tuning in physics often presuppose (either implicitly or explicity) some choice of measure relative to which matches between the values of constants or other original vertices can be understood as ‘unlikely’ or ‘not typical,’ but this approach is controversial [62, 63] since there is no possibility of empirically verifying such a measure and it is not straightforward to understand what could possibly ground or justify such a thing. So the term ‘independent’ in our definition shouldn’t be understood in terms of probabilistic independence - it has a less precise meaning, referring simply to the fact that by writing some vertices as original vertices, we are stipulating that there is nothing more to be said about their origins - so in particular, there is no connection between them nor any mechanism which adjusts them to match each other. And thus, since there is ex hypothesi no connection between original vertices, it is reasonable for us to be surprised when we find that that their values are nonetheless very closely matched. The point of all this is that when we find this kind of fine-tuning, we may be inclined to suspect that there is a deeper explanation for the matching of these vertices - there may in fact be some mechanism which does adjust them to match each other, in which case we may wish to expand our generative program to include that mechanism. So fine-tuning of this kind can be regarded as an indication that some part of our model may still benefit from further development: trying to find explanations for the fact that parameters in our models match in surprising ways has historically been a useful way of making scientific progress, so for example, part of the motivation for Einstein’s formulation of GR was the desire to explain the apparent coincidence of the matching values of the gravitational and inertial mass [64]. Thus looking for ‘fine-tuning’ is potentially a valuable way to decide where to focus our research efforts.

However, we emphasize that in the absence of any objectively correct measure over the values of original vertices, fine-tuning must ultimately be understood as a subjective heuristic rather than an objective flaw of the model. For without a measure there is no way to quantify how close two original vertices are or how surprised we should be by their matching - we cannot make claims to the effect that their matching is ‘highly improbable,’ or ‘atypical’ because those terms assume a well-defined probability distribution over the values of the original vertices. Moreover, even if the vertices are suspiciously close, there is always a possibility that their apparent matching is really just a coincidence. So the presence of what appears to be ‘fine-tuning’ doesn’t guarantee that there will be a deeper explanation to be found - the only way to know for sure is to actually look for an explanation and see if that effort leads to scientific progress.

It is also helpful to consider the relation between fine-tuning in this sense and fine-tuning as it is used in the causal modelling paradigm. Causal graphs which fail to obey ‘faithfulness’ as described in section 2.3 are sometimes said to be ‘fine-tuned,’ and we can understand the connection between these two notions of fine-tuning by observing that in order to have a causal model where there is a causal arrow from one variable to another but those variables appear to be statistically independent, it is necessary to ‘match up’ a variety of supposedly independent parameters of the structural equations defining the causal model, in order to ensure that the causal influences exactly cancel out in every possible configuration. So if we write down a generative program in which the parameters of the causal model are considered as ‘original vertices,’ then fine-tuning of the causal modelling type entails that the generative program will be fine-tuned. Of course, in most real situations (particularly when we are using causal models to describe causal phenomena in the macroscopic world) the parameters of the causal model will not really be original vertices and so there may be opportunities for them to arise through some dynamical or non-dynamical process which explains the connection between them - for example, this occurs in ‘equilibration’ explanations of fine-tuning, as discussed in ref [65]. However, when we are dealing with ‘fine-tuning’ not in ordinary causal models but in graphs encoding ontological priority of other kinds, it may be more plausible that the parameters of the causal model should genuinely be considered as original vertices, so failures of faithfulness may indeed indicate that the generative program is fine-tuned.

We can also use the relation between these two types of fine-tuning to observe that when we have a fine-tuned generative program, we cannot always overcome that problem by simply switching to a model in which one of the original vertices is no longer original, because that is liable to give rise to a DAG which does not obey faithfulness and thus we will still have a fine-tuning problem. For example, suppose we try to solve the fine-tuning problem for the cosmological constant [61] by postulating a generative program as represented by the DAG shown in figure 6, where the bare cosmological constant $C_{o}$ is an original vertex but the cut-off scale $\Lambda$ is no longer an original vertex - instead it is a function of $C_{o}$, calculated in a way which ensures that $C$ will always remain small. Subsequently $C_{o}$ and $\Lambda$ are used to calculate $C$ in the usual way. But this DAG fails to obey faithfulness: although there are arrows of ontological priority from both $C_{o}$ and $\Lambda$ to $C$, $C_{o}$ and $C$ are statistically independent, and likewise $\Lambda$ and $C$ are statistically independent, since regardless of the value of $C_{o}$ and $\Lambda$, $C$ will always be very close to zero. So although we no longer have the problem that the values of $C_{o}$ and $\Lambda$ precisely match up despite the absence of any connection between them, we instead have the problem that the specific parameters used to calculate the relationships between $C_{o}$ and $\Lambda$ and $C$ have to match up in a highly specific way to make the dependence relations cancel out, so we have not in fact managed to eliminate fine-tuning by this simple stratagem.

If ‘determination’ is interpreted in causal terms, then one natural way of defining ‘free will’ is to say that an action is free if it is not caused by anything other than the will of the agent, which in turn is ‘free’ in the sense that it is not caused by anything at all. However, within the generative program framework we may instead wish to interpret ‘determination’ in terms of ontological priority, which leads to the idea that an action is ‘free’ if it is, in some appropriate way, associated with an original vertex in a DAG representing the actual ontological priority structure of our world, as depicted in figure 7. In this section, we will discuss various different ways in which this ‘undetermined action’ conception of free will could be implemented in the generative program framework, focusing on the question of whether there is a consistent way to come up with a DAG modelling a realistic physical theory in which human actions are associated with original vertices.

First, if we are using a graph of ontological priority which coincides with a standard causal-dynamical picture of reality where the initial state is ontologically prior to all later events, as those who espouse a direct-access approach to the flow of time will generally be obliged to do, then there are quite strong limitations on the extent to which actions can be ‘undetermined.’ Indeed, in a deterministic causal-dynamical picture it follows immediately that no actions can be free in this sense, since all of the original vertices are found at the very beginning of time when presumably there are no agents around to take actions, as depicted in figure 8.

On the other hand, if we now adopt a perspectival approach to the flow of time, in which the experience of conscious agents is understood to supervene only on the nodes and not on the arrows of the DAG, then our graph of ontological priority is not required to coincide with a standard causal-dynamical picture of reality, and therefore we have many more options. For example, consider again a deterministic and time-reversal invariant theory as depicted in figure 8. Because the theory is time-reversal invariant, it is equally the case that the final state together with the laws of nature can determine the course of history, so if we take it that agents don’t have direct access into the process of ‘coming-into-being’ then there is another possible choice of DAG representing ontological priority which gives rise to exactly the same course of history but which has its original vertices at the final condition and all arrows running backwards in time, as depicted schematically in figure 9. So although it is true that according to our usual notion of strong causation, ‘our acts are the consequences of the laws of nature and events in the remote past’ it need not be the case that this convention reflects the real structure of ontological priority.

Now, one might object that although there are two valid DAGs representing ontological priority with arrows going in opposite directions, there is no such DAG in which your action is an original vertex, so either way your action is not an original vertex. But note that in our deterministic and time-reversal invariant theory, the initial and final conditions are not in any way special: in fact the state on any time-slice together with the laws of nature fully defines the course of history, as we can simply evolve both forwards and backwards in time from the given time-slice. So if we are dealing with a theory of this kind we can always choose a time-slice $T$ containing your action and then formulate a DAG representing ontological priority such that all the original vertices are on the time-slice $T$, with arrows running both forwards and backwards in time from the vertices on $T$, as depicted in figure 10. So, again assuming that agents do not have direct accesss into the process of ‘coming-into-being,’ there is a possible description of the theory in which your current action is indeed an original vertex, so it is free in the ‘undetermined’ sense.

Now, this rewriting only gives you ‘free will’ at a single moment in time. However, it is possible to imagine an even more radical rewriting of the DAG where the original vertices are scattered throughout time: this can be done if we can find a set of events across all of history which together contain enough information that the full set of them, along with the time-evolution laws, is enough to determine (but not overdetermine!) the entire course of history. And possibly it could be the case that some such set of events might contain the set of all human actions, or at least a reasonably large subset of human actions. If so, then it may be possible to come up with a DAG in which all of our actions are associated with original vertices, as depicted in figure 11, and thus despite the deterministic and time-reversible evolution, our choices can truly be regarded as free in the ‘undetermined’ sense.

One potential danger of scattering ‘original vertices’ across spacetime is that this could potentially give rise to either logical contradictions or fine-tuning. Of course, if one starts from a pre-existing block universe and simply identifies a set of original vertices that can be used to summarise it, the result will never contain any contradictions. But if one imagines a generative program as starting from a set of independent vertices and then producing the block from them, then in principle contradictions could arise - for example, if two distinct original vertices both fully determine a third vertex and they assign it different values. And if the values of the original vertices are specifically chosen in such a way as to avoid contradictions, this is liable to look like a kind of fine-tuning, since the vertices will not really be independent any more. So if we are going to scatter original vertices across spacetime, ideally we would like to do this in such a way that the structure of the program guarantees there will not be contradictions.

One possible strategy is suggested by Loewer’s account of free will, which allows that an action taken in the present can ‘influence’ both the past and future, where event E ‘influences’ O if and only if O counterfactually depends on E. Loewer places strict limitations on the kinds of influences an action can exert on the past - he argues for an approach known as the ‘Mentaculus,’ where we evaluate counterfactuals by holding constant the macrostate at the beginning of time and the probability distribution over microstates compatible with that macrostate, but not the specific initial microstate [67]. This prescription entails that we can in general ‘influence’ microstates in the past, but the macroscopic past history is held constant. Loewer’s account is a compatibilist one, but inspired by his work we can imagine an ‘undetermined’ version of his view in which human actions are associated with original vertices giving rise to arrows pointing both into the future and to the past. Because of the way in which Loewer limits influences on the past, it follows that the influence of a human action on an event in the past of that action can never be observed by anyone, since human actions can influence only microstates, not macrostates, and if a macroscopic observer were to use a measurement to learn about these influences then the influences would also be changing a past macrostate, i.e. the macrostate of the observer’s brain. So in this kind of model it is impossible to learn about a past microstaste $M$ in advance of the action $A$ that influences it, which means we can’t perform an observation of $M$ and then use the observation to select $A$. Therefore despite the arrows going in both directions of time, the DAG for the ‘undetermined’ version of Loewer’s approach would remain acyclic. Loewer’s approach is presumably not the only possible way of achieving this, but it is a useful proof of principle that it is possible to come up with rules which define DAGs with unusual choices of original vertices without running into contradictions.

Of course, the mere existence of such possible DAGs does not lead straightforwardly to the conclusion that human actions are free, because if we adopt the perspectivalist position which suggest that conscious agents have no direct access into the process of coming-into-being, it follows that we cannot tell empirically which of the various possible ways of assigning ontological priority to events in spacetime is the correct one, and therefore we have no particular reason to think that the correct DAG is one in which our actions are original vertices. But we also have no particular reason to think that the correct DAG is the one where all the original vertices occur at the beginning of time, so for all we know, our actions could be the original vertices, in which case they could indeed be free choices. Thus this approach at least offers the possibility that our actions could be free in the ‘undetermined’ sense, even if it does not guarantee that this is the case. Indeed, since the perspectivalist will argue that changing the structure of the DAG while leaving the empirical data the same cannot change our experience in any way, they might be inclined to suggest that we should regard ourselves as always having free will, because our experience would be exactly the same if our actions were the original vertices.

This discussion demonstrates that there is still a possibility of ‘undetermined’ free will even in the context of deterministic laws. But of course our current understanding of quantum mechanics suggests that the laws of our actual world may not be deterministic, in which case we have even more possibilities for scattering original vertices across spacetime - for example, every time there is a random quantum event, we can have an original vertex which determines the precise outcome of a specific quantum event in order to get a DAG like the one in figure 7. So if it can be shown that random quantum events play an important role in the human brain, it would seem reasonable to assume that human actions always involve at least one original vertex and are thus ‘free’ in the undetermined sense. And in fact there are several ongoing research programs seeking to identify significant quantum effects in the brain - for example Hameroff and Penrose [68] have proposed models for long-term coherence in microtubule structures in the brain, while Kane [69] has argued that butterfly effects may magnify microscopic phenomena to higher scales so they can contribute to decision-making.

However there are some difficulties with simply associating free actions with original vertices - it is essentially an expression of event-causal libertarianism and thus is susceptible to the disappearing agent argument [70, 71, 72]. In the terminology of the generative program framework, this argument says that even if there is some original vertex which wholly or partly determines my action, it’s unclear that this original vertex has anything much to do with me, so I seem to ‘disappear’ at the moment of actually making the decision. After all, my character is in large part the result of the past events in the physical world that have shaped my life, together with genetic and biological facts which are likewise a part of the physical world: but any original vertices contributing to my actions are by definition completely independent of all of these things, so those original vertices are surely not in any meaningful sense a manifestation of me or my will. From this point of view, a ‘choice’ associated with an original vertex is no better than a completely random choice, since it has nothing to do with the actual person involved - they may determine the set of choices along with their relative probabilities, but not the choice itself. James made this point in 1907, arguing that ‘If a ‘free’ act be a sheer novelty, that comes not from me, the previous me, but ex nihilo, and simply tacks itself on to me, how can I, the previous I, be responsible? How can I have any permanent character that will stand still long enough for praise or blame to be awarded?’ [73] Similarly, various philosophers have argued that an action’s being caused by appropriate reasons is in fact the essence of free will - for example, Sartorio argues that when we act freely, our conduct is caused by reasons to act as we do and also the absences of reasons to do otherwise [74] - and from this point of view, an original vertex is not actually a meaningful expression of free will since it is by definition not caused by any reasons.

For those who are swayed by the disappearing agent argument, it would perhaps seem more natural to adopt an agent-causal libertarian approach. Here we will attempt to model Clarke’s agent-causal approach, which is based on the idea that physical events underlying beliefs and desires are not sufficient in themselves to bring about a free choice but require the agent’s exercise of agent-causal power to make those events causally efficient. This is represented in figure 12, where we add to the structure of ontological priority a ‘black box’ process, in which some information from previous physical events is synthesized together with some free volition in order to arrive at the decision made by the agent. This would still allow the decision to be at least somewhat ‘free’ in the undetermined sense.

Now, it is very natural to ask what exactly is supposed to be going on inside this black box. We can immediately conclude that the black box is not deterministic, because if it were the free volition would not actually be playing any role in deciding the output, and thus this would not represent a meaningful notion of freedom after all. So if we try to expand the contents of the box within the ontological priority framework, it seems we must represent it as something structurally similar to a probabilistic vertex: the inputs to the box are effectively determining a probability distribution, and then the ‘free volition’ is represented by an original vertex which plays the role of actually selecting the output in accordance with the specified probability distribution. But if that is the case then the DAG of figure 12 has the same information flow as the DAG of figure 11, which would entail that agent-causal libertarianism is not really meaningfully different from event-causal libertarianism once we clarify the nature of the information-flow in these models. Thus, if one wishes to maintain that agent-causal libertarianism is meaningfully different from event-causal libertarianism, it seems one must resist the move to expand the black box in this way. So the agent-causal libertarian must hold that whatever is going on inside the black box, it is neither deterministic nor random, so it apparently cannot be represented at all within the ontological priority framework described here.

From some points of view this may be regarded as a problem for the agent-causal libertarian: their account depends on postulating a mysterious ‘black-box’ form of information processing, but they cannot tell us anything about how this information processing actually works, and indeed since it is neither deterministic nor random it apparently doesn’t correspond to any known type of information-processing. People who are disinclined to believe in mysterious physical processes which can’t be clearly specified may take this as an argument against agent-causal libertarianism, at least until a better account of the operation of the black box can be provided. On the other hand, agent-causal libertarians might perhaps respond that consciousness is complex and mysterious and perhaps the exact nature of the information processing going on in the black box is just beyond the comprehension of agents like us with our limited cognitive capabilities. We will not attempt to adjudicate between these approaches here, but we think the two possible representations of agent-causal libertarianism given in figures 12 and 11 are helpful to clarify exactly what is at stake in this discussion and why one might wish for further clarification from the agent-causal libertarian.

A completely different approach to free will can be defined within ‘all-at-once’ style models, as advocated in refs [75, 76]. Models of this kind do not postulate temporal evolution or any other directed process by which the course of history is generated: instead the laws specify a set of nomically possible histories and some history from this set is selected and instantiated in an atemporal way. The selection from the set can be thought of as akin to the selection of an initial state in an ordinary time-evolution model: we simply select and actualise the whole history rather than just the initial state.

In this kind of picture, no event in spacetime is ontologically prior to any other, as depicted in figure 13, and no set of events is really ‘the cause’ of the rest of history: events at the beginning of time have no better claim to this title than any other set of events. Rather, events happening across all of history stand in mutual, reciprocal relations to one another: there is an underlying structure of ontological priority, which includes some original vertices, but none of the original vertices are located in spacetime.

So what does this mean for free will? Well, if one is committed to the ‘undetermined’ account of freedom then one will have to conclude that in this picture nobody really has freedom, since no actions are associated with original vertices. But on the other hand, if one’s concerns about free will stem largely from the idea that all our actions are determined by factors outside of our control in the far distant past, then the fact that in these models no direction of time corresponds to the objectively correct direction of ontological priority should assuage those concerns, for this entails that it is no more true to say that our actions are determined by the distant past than it is to say that the past is determined by our actions. This leads to a conception of freedom which emphasizes the participatory nature of agency: in this picture the course of history is selected in such a way that all moments of time play an equal role, so your current actions are no more or less important than the events at the beginning and end of time. Your choices play a real part in determining the course of history, because all other events must be adjusted to fit with your choices, just as your choices are mutually adjusted to fit other events. This way of thinking allows for your actions to be ‘free’ in a genuine, ontological sense while also allowing that meaningful choices are not random but rather are appropriately linked with the events in your past that have formed your identity.

Obviously, this approach in itself is not a definition of free will: the argument applies equally well to all events in history and every action that a person takes, regardless of whether the action corresponds to what we would ordinarily call a ‘free choice’ or whether it is the result of someone holding a gun to the agent’s head and making demands. So there is still a need to formulate criteria which can be used to make individual judgements about which actions count as free for moral, legal and religious purposes. However, this approach does serve to defuse the blanket challenge to free will posed by the idea that everything is ultimately determined by the distant past; it allows for the conceptual possibility that at least some human actions may count as ‘free,’ so we can go on to decide which particular actions we wish to regard as free⁴⁴4Our own view is that one should not seek a univocal criterion which divides all actions neatly into ‘free’ and ‘not free’ - rather one should acknowledge that human actions are subject to a range of different internal and external constraints, and it is better to focus on understanding the complex interplay of constraints in a specific case than to assign a simple binary label classifying the action as ‘free’ or ‘not free’..

For example, it is often said that we have free will if it is possible for us to act otherwise, and determinism is supposed to undermine this kind of free will because, if we condition on the microstate of the world up until the time of the action, then only one action is possible. But if no events in spacetime are really ontologically prior to any others, then there is no reason why the whole of the past must be held constant when one assesses what is possible at a given time. Rather we are free to make contextual choices about what factors to hold constant, so for example if we are assessing whether an action is free we will most likely hold constant factors like the presence or absence of external compulsion, intoxicants, phobias etc, but we need not hold constant the exact past microstate. Now, this is a fairly standard compatibilist argument - for example, List suggests that the relevant notion of possibility in ‘it was possible to act otherwise’ involves conditioning not on the past microstate but on what he calls ‘the agential state’ which describes various psychological factors, and in many cases if we condition only on the agential state then we can conclude it was possible for us to act otherwise, so we can distinguish between cases in which we have free will and cases in which we do not [77]. The account we have given here adds legitimacy to approaches of this kind, because it entails that these approaches are not simply sweeping under the carpet the fact that the action is really determined by the whole past microstate - in fact all events determine all other events and therefore any meaningful statement of physical possibility must involve a choice about what we are going to hold constant⁵⁵5This does not apply to statements about physical possibility which refer to the whole universe at once, e.g. saying which entire courses of history are physically possible or impossible. However, evidently such statements are far too general to be applied to questions about free will in any particular case.. From this point of view, holding only the ‘agential state,’ or some such thing constant is no more or less correct at a physical level than holding the past microstate fixed, and thus it is perfectly reasonable to say that in some circumstances it is possible for us to act otherwise.

Note that this notion of freedom is similar to the one featuring in the definition of superdeterminism which we presented in section 4. This definition uses the past empirical experiences of an agent to define the space of possible choices which must be considered as counterfactuals, which is similar to using List’s ‘agential state’ to determine which actions were ‘possible,’ or in the terms of [57] ‘empirically plausible’: in models which are superdeterministic according to this definition, the agent’s conception of which actions they are free to perform, holding constant only the ‘agential state,’ does not match the reality of what the generative program allows, and hence the program is ‘conspiratorial’ in the sense that the agent’s experiences deceive them into believing that some impossible choices are possible. That said, even with a superdeterministic generative program, there is still some intermediate sense in which the agent can be free; provided the response function for a given system allows at least two empirically plausible inputs, and specifies empirically plausible outputs for all of them, then the agent is still free to choose among those in this same sense, even if there are other empirically plausible choices that are in fact impossible in the given program.