Toy Models of Top Down Causation
Adrian Kent




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      Mathematical Sciences, University of Cambridge, Wilberforce Road,
      Cambridge, CB3 0WA, U.K.Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON N2L 2Y5, Canada.A.P.A.Kent@damtp.cam.ac.uk
    
    Models in which causation arises from higher level structures
    as well as from microdynamics may be relevant to unifying  
    quantum theory with classical physics or general relativity. 
    They also give a way of defining a form of panprotopsychist property dualism, in
    which consciousness and material physics causally affect one another. 
    I describe  probabilistic toy models based on cellular automata that
    illustrate possibilities and difficulties with
    these ideas.
    Toy Models of Top Down Causation
        Adrian Kent
        September 2019; updated September 2020
    ==========================================
    
    
    
    § INTRODUCTION
    
    
    The reductionist paradigm for theoretical physics suggests that 
    the properties of complex structures, including their dynamics, can be understood as a 
    consequence of those of their elementary components.
    It is not easy to characterise precisely what this means in all cases.
    Is a space-time or the vacuum state of a quantum field a complex
    structure, for example?  And if so, what are their elementary
    components?    Is a bare quark an elementary component or a 
    mathematical fiction?   Is quantum entanglement a counter-example to 
    reductionism or just an illustration that the concept needs to 
    be framed more carefully?   
    
    Nonetheless, it is widely agreed that some appropriately nuanced 
    and qualified version of reductionism has been extremely successful, so much so
    that many theorists seek unified theories in which
    all of physics is characterised by some theory of the initial conditions
    together with relatively simple (though seemingly probabilistic) dynamical laws.    
    
    We should distinguish this strong but quite arguable stance
    from the stronger and fairly indefensible one that understanding the 
    fundamental laws is the only really important task in science.   
    As Anderson compellingly argued in his classic essay <cit.>, solid-state physics, 
    chemistry, biology, psychology and other higher-level theories 
    produce new behaviours and new laws that require great inspiration
    and creativity to find and understand.   But Anderson was nonetheless 
    a card-carrying reductionist: 
     The reductionist hypothesis may still be a topic for controversy
    among philosophers, but among the great majority of active
    scientists I think it is accepted without question.   The workings
    of our minds and bodies, and of all the animate or inanimate matter
    of which we have any detailed knowledge, are assumed to be 
    controlled by the same set of fundamental laws, which except
    under certain extreme conditions we feel we know pretty well.<cit.> 
    
    Chalmers' <cit.> distinction between types of emergence is very helpful 
    here.    High-level phenomena are  weakly emergent when they are unexpected, but deducible in
    principle (even if not currently in practice) from fundamental
    theories.    
    They are  strongly emergent if they are not deducible even in
    principle. 
    Reductionists aim for a relatively simple universal theory in which there are
    no examples of strong emergence, but should be comfortable with weak
    emergence. 
    A representative survey would be very interesting,
    but my guess is that Anderson's characterisation still holds true
    today: most scientists believe 
    we already know enough of the fundamental theory to understand non-extreme regimes,
    and in particular would deny that the emergence of 
    (quasi-)classicality from quantum theory, or (pace Chalmers
    <cit.>)
    of consciousness
    from classical or quantum physics, are clear examples of
    strong emergence.   On the other hand, these questions are hotly
    debated among scientists working on consciousness and on 
    quantum foundations.   
    
    Although the boundaries of reductionism may be slightly fuzzy, we can
    certainly produce models or theories that are clearly beyond them and unambiguously
    anti-reductionist.  One example would be a theory that predicts 
    qualitatively different dynamical equations for different 
    types of molecule.[For the theory to be unambiguously
    anti-reductionist, it should not be possible to derive these
    equations from some simpler unifying principle.   
    For example, dynamical collapse models <cit.>
    are not anti-reductionist, 
    although in a sense they predict different behaviours 
    for molecules of low and high mass: these predictions all follow
    from the same stochastic differential equation.] 
    Another would be a vitalist theory that
    predicts that living creatures disobey some conservation
    law of classical mechanics. 
    The general consensus is that we should assign low priors
    to such theories, not only because reductionism has been successful, 
    but also because reductionist theories tend to be more elegant, and
    aligning credence with elegance has also been a very successful
    scientific methodology.       
    
    There is, though, more serious interest in better motivated
    models that do not fit the current mainstream reductionist paradigm.  
    Consciousness, the topic of this special issue,  seems 
    to give one than one causal narrative – mind and matter  seem to affect
    both themselves and each other.
    Yet the causal effects of matter on matter also seem enough for a
    complete description of the material world: there is no physical evidence that 
    the known fundamental laws of physics don't suffice to describe 
    the behaviour of the brain. 
    There are (controversial) mainstream reductionist stances on this
    (e.g. <cit.>), but 
    also well-known (also
    controversial) arguments (e.g. <cit.>)
    against the reducibility of consciousness to known physics. 
    There has been an upsurge of interest lately in 
    exploring alternative ideas involving new (or non-standard) 
    physical hypotheses
    (e.g. <cit.>). 
    Several of these have drawn inspiration and motivation
    from work on “integrated information theory” (IIT)
    <cit.>
    which, although open to many criticisms (e.g. <cit.>, gives a mathematical
    framework to explore and generalise as well as a connection to empirical
    data.  
    
    
    Ideas of top-down causation have been mooted in the
    context of quantum theory and more broadly (e.g. <cit.>). 
    The scope for top-down causal models of
    consciousness has not been extensively explored, and 
    even the meaning and scope of top-down causation is not fully elaborated. 
    This paper aims to give one framework for discussion,
    by describing some simple toy models, inspired by cellular automata,
    which illustrate possible ways in which higher level structures 
    act causally on the microscopic components, as well as vice versa.  
    It should be stressed that these are not
    meant to capture most realistic features of the world.
    The aim is to illustrate scope for more realistic models that use
    a similar mechanism to combine types of causation. 
    
    
    
    § CELLULAR AUTOMATON 110
    
    
    Our toy models are based on cellular automata, but are not meant in 
    the spirit of the well-known research
    programmes aiming to describe nature at a fundamental level in terms of cellular
    automata <cit.>.    We use cellular automata
    simply as convenient illustrations.   
    
    Wolfram <cit.> classified the
    256 elementary 
    one-dimensional cellular automata.   These are defined by binary states 
    with a time step rule in which the states of cell n at time t
    are determined by those of cells n-1,n,n+1 at time (t-1). 
    He noted the particularly interesting and complex behaviour of the automaton
    defined by rule 110 in his classification.  
    Again, we pick this out not out of any fundamental preference –
    higher-dimensional cellular automata such the Game of Life
    <cit.> could equally well be used,
    for example – but for simplicity of illustration.   
    
    Rule 110 is defined by 
    
    
    Wolfram had previously suggested <cit.> that rule 110 is Turing complete,
    a result subsequently proved by Cook <cit.>; another discussion
    of the result is given in Ref. <cit.>. 
    We review here some of its known properties, using results and images 
    generated by Wolfram Mathematica and resources <cit.>
    from the Wolfram Data Repository, which helpfully includes routines that
    reproduce many interesting diagrams originally given in Ref. <cit.>. 
    
    The rule generates a regular repeating array with period 14, 
    known as the “ether”:
    
    We will take this to be the analogue of a “vacuum” or “background
    state” in the toy models defined below.
    
    Some finite dislocations in the lattice-like 1D structure of a row of the
    ether can propagate regularly.   These so-called “gliders” generate quasi-particle-like tracks 
    in the ether.   Cook  <cit.> classified a variety of rule 110 gliders, including some
    infinite classes:
      
    Colliding gliders undergo “interactions” that are superficially
    [Of course, these are deterministic classical interactions,
    not complex quantum interaction amplitudes.]
    reminiscent of Feynman
    diagrams, typically producing a finite number of stable new gliders
    after a finite interaction time:
    
    (For more discussion of the general phenomena of glider “particles”
    and background “domains” in cellular automata, see e.g.
    Refs. <cit.>.) 
    
    We can define a very simple model of errors or noise in these
    structures by considering the possibility of a single bit flip
    on the first row.   One might motivate this by supposing that 
    there is something particular about the system at t=0 that
    makes errors or noise possible on the first row, and only that row,[Or so much more likely that
    we can neglect the possibility of errors on other rows.] with
    error probability low enough that we can neglect the possibility
    of two or more errors arising 
    
    If we consider a glider propagating in the ether, with a single 
    bit of the initial state flipped at a site in the ether that is
    far from the glider, the effect tends to be simply to cause some
    ripples in the ether that propagate for some time and then peter 
    out without interacting with the glider.   The glider's propagation
    is thus unaffected, as Fig. <ref> illustrates. 
    
    
    However, if we flip a bit close to a glider, it can interact in a way 
    that permanently alters 
    the number and type of gliders.   Fig. <ref> shows the same glider states
    with one initial bit flipped.      Only the second is asymptotically unaffected.     
    
    
    
    Perturbations affect interacting gliders similarly.  A perturbation
    distant from interacting gliders generally peters out without
    affecting them.   However, perturbations in the vicinity of one or
    more interacting gliders may alter the types and/or number of gliders 
    in the final or asymptotic state.    
    
    Fig. <ref> shows a pair of interacting gliders with a 
    single flipped bit, whose site runs sequentially through 21   
    sites initially located between the gliders.   
    Of these perturbations, the 1st, 4th, 12th, 13th, 15th
    and 18th leave the final glider states, highlighted, are asymptotically unchanged.
    
    The 5th, 9th and 17th, highlighted in Fig. <ref>,
    all produce the same new asymptotic
    final state, consisting of a single glider.   
    
    
    § PROBABILISTIC MODELS BASED ON CELLULAR AUTOMATA AND TOP-DOWN CAUSATION
    
    
     §.§ A simple probabilistic model
    
    
    We can formalise the model above as a 1D cellular automaton
    whose states are defined on sites labelled by the integers,
    at times also labelled by the integers.   
    When the error probability p is zero, it is a deterministic 
    type 110 automaton.   The ether, or the ether with a
    single glider, then propagate indefinitely without
    perturbation.   A pair of gliders may approach one another
    from infinity, interact, and produce some number of 
    outgoing gliders, which then propagate indefinitely. 
    
    We may also take the model to have finitely many spatial
    sites, with periodic boundary conditions.   In this case,
    with appropriate numbers of sites, the ether and a single
    glider state may still propagate indefinitely.    
    If a pair of gliders with different velocities are evident during some time
    interval, they will, so to speak, interact in both the past and
    future.   If the interaction products contain two or more gliders
    with different velocities, they in turn will interact, and the
    asymptotic behaviour may be quite complex. 
    We can avoid this by defining the model only for a finite time interval, short
    compared to N/v, where N is the number of sites and v the 
    maximum glider speed.      
    
    We suppose that there is some probability p>0 of an error occurring
    on the row of sites at t=0.   An error flips the bit value of 
    a single site, so that it takes the opposite value to that predicted
    by the deterministic dynamics from the state at t=-1.  
    To simplify, we suppose that there is no probability of more than
    one error, and that errors are restricted to sites x, where
    -M ≤ x ≤ M, where (2M+1) ≤ N if there are finitely many
    (N) sites.   The errors in this region have uniform probability,
    so that each site in the region has error probability
    p/2M+1. 
    
    The discussion of the previous section then applies: errors
    sufficiently far from any gliders at t=0 will typically
    peter out before interacting and have no effect on the final
    or asymptotic late time glider states;
    errors close to gliders can alter the number and type 
    of final or asymptotic late time gliders.     
    
    
    
     §.§ Incorporating top down causation
    
    
    We now consider modifying the dynamics by assigning probability
    weight factors to final glider states conditional on initial glider
    states.   
    
    One simple rule is to assign probability weight factor 
    1 to final states that are the same as the initial state for single glider propagation and
    two glider interactions, and weight factor 0 to distinct states. 
    Formally, 
    
        p_ mod ( G_f | G_i )    =    C w (G_f | G_i ) p (G_f | G_i )   , 
        
        p_ mod ( G^1_f , G^2_f , … , G^n_f | G_i )    =    0
             for  n ≠ 1   , 
        
        p_ mod ( G^1_f , G^2_f | G^1_i , G^2_i )    =    C' w (G^1_f , G^2_f | G^1_i , G^2_i )
        p (G^1_f , G^2_f | G^1_i , G^2_i )   , 
        
        p_ mod ( G^1_f , G^2_f , … , G^n_f | G^1_i , G^2_i )    =    0
             for  n ≠ 2   .
    
    Here 
    
        w (G_f | G_i ) = δ_G_f , G_i   .
    
    Multiple glider states are listed from left to right and so
    
        w (G^1_f , G^2_f | G^1_i , G^2_i ) = δ_G^1_f , G^2_iδ_G^2_f  ,
           G^1_i
    
    for colliding gliders, while
    
        w (G^1_f , G^2_f | G^1_i , G^2_i ) = δ_G^1_f , G^1_iδ_G^2_f  ,
           G^2_i
    
    for gliders that never collide. 
    The expression p(G_f | G_i ) is the probability of the final state containing
    (only) the single glider G_f when the initial state contains glider G_i in the model of the
    last subsection; p (G^1_f , G^2_f | G^1_i , G^2_i ) is the probability
    of the final state containing (precisely) the pair G^1_f ,G^2_f when the
    initial state contains the pair G^1_i , G^2_i; C and C' are rescaling factors chosen so that the
    probabilities of all possible final states sum to 1 for a given
    initial state.   
    
    This rule is understood as applying to the system as a whole.
    It does not alter the deterministic dynamics of rule 110, 
    and so its effect is to alter the probabilities of errors in the 
    state at t=0, which are the only probabilistic feature of the
    toy model.   
    For example, for an initial state containing a single glider G, 
    it slightly increases the probability of errors at sites
    (such as those far from the glider) where they do not affect
    the glider propagation, increases the probability of no error,
    and eliminates the possibility of errors occurring at sites
    where they would alter the asymptotic glider propagation.   
    It has similar effects for initial states containing two
    gliders G and G'. 
    Effectively, the rule acts to suppress errors in glider
    propagation, ensuring the stability of one and two glider
    states, which would otherwise be menaced by possible errors in the 
    microdynamics.
    
    A variation is to assign probability weight 1 to specified 
    final state outcomes of two glider interactions, and 0 otherwise,
    while retaining the weights above for single glider states
    Thus
    
        w (G_f | G_i ) = δ_G_f , G_i
    
    as above but w ( G^1_f , G^2_f , … , G^n_f  | G^1_i , G^2_i ) may have a more general
    form.
    For example, we might take w (G_f | G^1_i , G^2_i ) = 1 for some
    specified final state G_f, and zero for all other final states.
    This ensures that
    initial glider states G^1_i ,G^2_i always produce final state G_f.   
    however small the unmodified probability of this outcome is,
    so long as it is nonzero.  
    
    
    
     §.§ Compatibility with standard temporal and Minkowski causation
     
    
    Framed as above, these modified toy models may appear to
    involve something like instantaneous action at a distance,
    since the probability of error at a given site at t=0 effectively
    depends on the type and number of gliders at distant sites 
    at the same time.  
    If we think of the models as capturing the behaviour of
    particles (modelled by gliders) propagating in a background
    (the ether) with stochastic fluctuations (the errors), 
    in some non-relativistic limit of a theory in relativistic space-time,
    this may seem to involve retrocausation: the probability of an error at a site
    depends on the final glider states in regions in its causal future. 
    
    While the models certainly could represent features of 
    theories with non-standard causation, they are compatible with
    standard causation, even for relativistic theories.    
    We can take the relevant glider speeds to be below light speed
    in such theories.   The gliders contained in the state at t=0 depend
    deterministically on those contained in the states at t<0.  
    We can thus equally well understand the probability of error
    of any site at t=0 as determined by the glider states
    at suitably large negative t, when the gliders are within the site's
    past light cone.    
    Interpreted in this way, errors at t=0 are causally determined by
    glider states at large negative t, according to laws that ensure
    specific glider states at large positive t.   
    For example, one might imagine the models as capturing essential features
    of some deeper theory in which this causal determination is 
    made more explicit, by degrees of freedom that
    carry information away non-superluminally from negative time glider
    states to sites throughout the ether and influence the
    error probabilities at t=0 appropriately.   
     
    
    
    § DISCUSSION
    
    
     §.§ Panprotopsychist models of consciousness
    
    
    There are reasons to consider physical models of consciousness
    that feature top down causation (although, as with every approach
    to consciousness, there are also problems and counterarguments).
    One line of argument runs as follows.   There 
    are evidently physical correlates of consciousness, namely human
    brains.   If there is a fundamental physical law associating conscious
    states to physical systems, it seems unlikely that it associates
    consciousness to brains and to nothing simpler:   
    brains seem too complex as physical systems to be the elementary
    referents of such a law. 
    Full-blown panpsychism, in which every elementary particle has 
    an associated elementary consciousness, is a possible alternative, 
    but comes with many problems <cit.> 
    and does not seem to fit naturally with
    neuroscientific data and our conscious self-observations or those
    reported by others.    
    The intermediate option of panprotopsychism <cit.>, according to which
    elementary consciousness is associated with some
    physical systems (whose nature remains to be specified) larger than elementary particles and smaller than
    brains, shares some of the problems of panpsychism, but allows more
    possibilities that might fit with empirical observation.
    Taking panprotopsychism seriously means accepting  some sort of new
    physical law(s) associating the relevant systems with consciousness.
    
    Our probabilistic models based on cellular automata can be taken as toy
    models of interactionist panprotopsychism. 
    In these toy models, the elementary bits at each site
    are meant to correspond to elementary components, and
    the deterministic dynamics of rule 110 and unmodified error probability
    rules correspond to the elementary microdynamics.    The gliders represent
    physical systems associated with elements of consciousness, which
    we might take to be quales or (if we stretch the present models even more
    unrealistically in order to illustrate how the idea might be extended) thoughts that 
    we can represent by a sentence such as “I see blue”.   
    The first modified versions of the model, in which the error probabilities
    are redefined to ensure that single gliders and pairs of gliders
    propagate unaffected by errors, then correspond to toy models in
    which panprotopsychist consciousness ensures error suppression 
    at the level of consciousness, in the sense that quales (or thoughts)
    propagate unaffected in the substrate, despite errors in
    the microdynamics.   
    The second modified versions redefine the error probabilities to ensure that pairs of gliders produce specified outcomes
    that would not arise in the absence of errors.   These correspond to toy
    models in which panprotopsychist consciousness is equipped with its
    own dynamics, which overrides the dynamics of the substrate when 
    the two conflict.     
    
    An argument in favour of something like this picture is that, if
    particular physical structures are indeed singled out as having an
    elementary proto-consciousness by fundamental physical laws, it is
    arguably natural that these physical structures should also feature in the fundamental
    dynamical laws.  One might even speculate that nature has
    probabilistic laws because of the need to combine dual causalities, of
    matter and mind. 
    
    It is helpful to compare the pros and cons of this line of thought
    with those of a similarly panprotopsychist form of epiphenomenalism.
    This would associate consciousness in a lawlike 
    way to specified physical structures, without modifying the
    microdynamics.   
    The problem with this and other types of epiphenomenalism, as William
    James first pointed out <cit.>, 
    is that they leave all the apparently evolutionarily adaptive properties of consciousness
    unexplained.   If physical laws of consciousness do not affect the
    microdynamics, then we and other creatures would function equally well
    if we were unconscious zombies, or if pleasure and pain were
    uncorrelated with evolutionary advantage, or if our consciousnesses
    were focussed on information that had no relevance to our survival
    or well-being, or if we had “locked-in” consciousnesses disconnected
    from any of our communications.  
    On this view, we have to accept that not only the existence of
    consciousness, but the apparent fine-tuning of its specific features,
    are just astonishingly convenient coincidences.     
    
    In contrast, there is scope for more convincing explanations
    of the evolution of consciousness and of some of its features if dynamics give it a genuinely
    causal role in behaviour.    It seems plausible, for example, 
    that effectively coupling two types of dynamical rule 
    allows evolution to more easily produce stimulus-response circuits that are more
    stable or follow higher-level reasoning.  
    It also seems plausible that evolution would use this coupling to
    allow creatures to communicate their conscious states to one another.
    This would allow them to coordinate their behaviour better than 
    communications that are influenced only by the microdynamics of their
    physical substrates, since in these models their behaviour may be
    directly affected by their conscious states.    
    Models in which consciousness acts causally, via laws involving its
    complex physical correlates, also seem to offer some scope
    for explaining the correlation of pleasure (pain) with evolutionary (dis)advantage.
    A pain is something a conscious mind attempts to avoid, arguably by
    definition, and if the dynamics of conscious states reflect this, 
    then evolution could naturally exploit this dynamics if
    disadvantageous physical situations created physical (brain)
    states that involved subsystems associated (via the hypothesized
    laws) with avoidant conscious states.   
    
    Even if these arguments can be made convincing, it would still seem a surprising and fortunate coincidence that, somewhere
    in the evolutionary chain, and perhaps very early, life took a
    material form that had proto-consciousness, and that matter and
    consciousness were associated in such a way that evolution was able
    to make use of the dynamical rules that give consciousness causal
    effect (via its material correlates) on matter.  
    A priori, one might imagine that, if there are simple laws of
    psychophysical parallelism and simple associated dynamical laws,
    they need have nothing to do with self-replicating chemicals 
    or organic information processing systems.   
    So it is fair to ask how much fine-tuning interactionist
    panprotopsychist theories could explain,
    and how much they would still leave unexplained.  
    Still, a partial explanation is better than none, and we also
    need to be clear whether we could possibly hope for a fuller 
    explanation given our present conceptual frameworks. 
    After all, we are conscious. 
    Anyone who finds conceivability arguments <cit.> persuasive has to 
    accept this, and all the features of our consciousnesses, as marvellous yet 
    contingent features of our universe.  
    On this view, we might hope that relatively simple laws characterise our
    consciousness and explain its evolution, but we can't
    hope for an argument that the laws must take the form they do. 
    
    Closer analysis of all these arguments would undoubtedly be valuable. 
    It would also be interesting to develop more sophisticated toy models, in which we can see
    rudimentary creatures evolving in a simple environment 
    via modified dynamics.
      
    
    
    
     §.§ Quantum theory, gravity and classical physics
    
    
    As these toy models illustrate, probabilistic theories of microdynamics can be simply modified so
    that structures at two or more levels play roles in 
    the fundamental laws.   This makes it easy to build and 
    explore models with top down causation.   
    Such models could also potentially be relevant to unifying quantum theory and gravity.
    For example, one could imagine space-time emerging from a
    fundamentally quantum theory, within a theory in which it is equipped with its own
    independent dynamical laws; in such a theory, both space-time and  
    its quantum constituents would causally affect one another, with neither reducible to 
    the other.      
    The same type of relationship is possible between classical and
    quantum degrees of freedom within a fixed background space-time.
    Classical physics is normally thought of as emerging from and 
    reducible to quantum theory (by Everettians; see e.g. <cit.>) or
    some extension of quantum theory that does not radically alter the
    dynamics (by non-Everettians who believe some
    extension is needed to resolve the measurement problem). 
    The latter looks plausible (e.g. <cit.>) and the simplest 
    possibility, but it is interesting to ask how strongly
    empirical evidence constrains more general theories
    <cit.> 
    that support this type of dual causation. 
    
    
    
    § ACKNOWLEDGEMENTS
    
    I am very grateful to the organisers and participants of 
    the Oxford 2019 “Models of Consciousness” conference,
    at which this work was presented; many of their 
    comments and criticisms were very helpful. 
    I would also like to thank anonymous referees for constructive
    criticisms and suggestions.    
    This work was supported by an FQXi grant, by UK Quantum
    Communications Hub grant no. EP/T001011/1 and by Perimeter Institute
    for Theoretical Physics. 
    Research at Perimeter Institute is supported
    by the Government of Canada through Industry Canada and by the
    Province of Ontario through the Ministry of Research and Innovation.
    
    
    
    
    § REFERENCES
    unsrtnat
