Causation and determinism

In this post I will try to disentangle the notions of determinism and causality, and suggest a different way to think of them. I came to think of these issues via the following informal assertion

The decay of a radioactive atom has no cause

I will not be discussing hidden variable theories nor Bell’s inequalities; I will assume outright that the phenomenon of radioactive decay is intrinsically random (as opposed to “apparent randomness” induced by ignorance), its quantum mechanical description is the most complete model possible; said model assigns probabilities to outcomes. With that out of the way, the usual argument that arrives at the above statement is

1) Radioactive decay is intrinsically random, indeterminate and cannot be predicted

2) There is no physical factor which determines whether a radioactive atom decays or not

3) Therefore, that a specific atom decays has no cause

Although the argument makes sense I am hesitant to accept 3) as is, and what it implies about how we think of causality.

Causality has been confusing minds for hundreds of years, it is a very difficult subject as evidenced by the volumes that have been written on it. So there’s not much point in trying to figure out what causality means exhaustively, via conceptual analysis in the tradition of analytic philosophy. Instead we will just quickly define causality using the mathematics of causal models, and see where that takes us for a specific scenario. In the context of these models, we will define causality according to two complementary questions:

A) what is meant by “the effect of a cause”

B) what is meant by “the cause of an effect”

Two types of causal models have been developed over the last thirty years, causal bayesian networks and structural causal models. These two formalisms are largely equivalent[4]; both make use of graph representations. Vertices in these graphs correspond to the variables under study whereas edges represent causal influences between the variables.  Guided by these graphs, one follows precise procedures to obtain mathematical expressions for causal queries over variables. These expressions are cast in the language of probability.

From [Pearl2000] page 15

In this post I will refer to structural causal models as described in Pearl’s Causality: Models, Reasoning, and Inference [Pearl2000]. To begin, we have the definition[2]

causal_model

In the above, D is a directed graph whose edges represent causal influences; these influences are quantitavely specified by functions (structural equations) on variables. Finally, probabilities are assigned to variables not constrained by functions, these are exogenous variables.

The effect of a cause

Given a structural causal model, question A) can be answered with the following result

causal_effect

alternatively

The difference E(Y | do(x’)) – E(Y | do(x”)) is sometimes taken as the definition of “causal effect” (Rosenbaum and Rubin 1983)

The causal effect of changing the variable x’ => x” on y is defined as the difference in expectation of the value that y will take. Note how the formalism includes explicit notation for interventions, do(x).

The cause of an effect

Question b) looks at it from a different point of view. Instead of asking what the effects of some cause are, we ask what the cause of some effect is; it’s a question of attribution. These questions naturally assume the form of counterfactuals (wikipedia):

Counterfactuals

A counterfactual conditional, is a conditional (or “if-then”) statement indicating what would be the case if its antecedent were true.

For example

If it were raining, then he would be inside.

Before you run off screaming “metaphysics!”, “non-falsifiability!” or other variants of hocus pocus, rest assured: counterfactuals have a clear empirical content. In fact, what grants counterfactuals their empirical content is the same assumption that allows confirmation of theories via experiments: that physical laws are invariant. Counterfactuals make predictions just the same way as experiments validate hypothesis. If I say

“if you had dropped the glass it would have accelerated downwards at g”

I am also saying that

“If you now drop the glass, it will accelerate downwards at g”

Given that I make the assumption that all relevant factors remain equal (ie, gravity has not suddenly disappeared).

The following result allows us to answer queries about counterfactuals[3]:

counterfactuals

Once we have expressions for counterfactuals, we can answer questions of type B), with the following results. Note that these results are expressed in terms of counterfactuals, which is why one needs theorem 7.1 as a prerequisite.

necessity

This completes the brief listing of key results for the purposes of the discussion.

Consequences

So what was the point of pasting in all these definitions without going into the details? The point is that given the formalisms of these models and their associated assumptions[5], we can think quantitatively about questions A) and B), without going into the nightmare of trying to figure out what causality “really means” from scratch. Our original criteria now have assumed a quantitative form:

A) The difference in expectation on some value Y when changing some variable X

B1)The probability that some variable X is a necessary requirement for the value of some observed variable Y

B2) The probability that some variable X is a sufficient requirement for the value of some observed variable Y

Thankfully our example of a radioactive atom is very simple compared to the applications causal models were designed for; for our purposes we do not need to work hard to identify the structure nor the probabilities  involved, these are given to us by the physics of nuclear decay.

Feynman diagram for Beta- decay (wikipedia)

Having said this, we construct a minimal model for eg. negative beta decay with the following two variables

r: The neutron-proton ratio, with values High, Normal, Low (using some arbitrary numerical threshold)

d: Whether β- decay occurs at some time t, with values True, False

Our questions, then, are

Q1) What is the causal effect of r=High on d?

Q2) What is the probability of necessity P(N) of r = High, relative to the observed effect d = True?

Q3) What is the probability of necessity P(S) of r = High, relative to the observed effect d = True?

In order to interpret the answers to the above questions we must first go into some more details about causality and the models we have used.

General causes, singular causes, and probabilities

Research into causality has distinguished two categories of causal claims:

General (or type-level) causal claims:

Drunk driving causes accidents.

Singular (or token-level) causal claims:

The light turning on was caused by me flipping the switch.

General claims describe overall patterns in events, singular claims describe specific events. This distinction brings us to another consideration. The language in which causal models yield expressions is that of probability. We have seen probabilities assigned to the value of some effect, as well as probabilities assigned to the statement that a cause is sufficient, or is necessary. But how do these probabilities arise?

Functional causal models are deterministic; the structural equations that describe causal mechanisms (graph arrows) yield unique values for variables as a function of their influences. On the other hand, the exogenous variables, those that are not specified within the model, but rather are inputs to it, have an associated uncertainty. Thus probabilities arise from our lack of knowledge about the exact values that these external conditions have. The epistemic uncertainty spreads from exogeneous variables throughout the rest of the model.

[Pearl2000] handles the general/singular dichotomy elegantly: there is no crisp border, rather there is a continuous spectrum as models range from general to specific, corresponding to how much uncertainty exists in the associated variables. A general causal claim is one where exogenous variables have wide probability distributions; as information is added these probabilities are tightened and the claim becomes singular. In the limit, there is no uncertainty, the model is deterministic.

We can go back to question 1) whose answer can be interpreted without much difficulty.

Q1) What is the causal effect of r=High (high nuclear ratio) on d (decay)?

If physics is correct, having a certain values for r will increase the expectaton of d being equal to True, relative to some other value for r. This becomes a general causal claim,

A1) High nuclear ratio causes Beta- decay

So, relative to our model, high nuclear ratio is a cause of Beta- decay. Note that we can say this despite the fact that decay is intrinsically indeterministic. Even though the probabilities are of a fundamentally different nature, the empirical content is indistinguishable from any other general claim with epistemic uncertainty. Hence, in this particular case determinism is not required to speak of causation.

The more controversial matter is attribution of cause for a singular indeterministic phenomenon, which is where we began.

3) A specific atom decay has no cause

This is addressed by questions 2) and 3).

Q2) What is the probability of necessity P(N) of r = High, relative to the observed effect d = True?

Q3) What is the probability of necessity P(S) of r = High, relative to the observed effect d = True?

Recall, functional causal models assign probabilities that arise from uncertainty in exogenous variables; this is what we see in definitions 9.2.1 and 9.2.2. The phrase “probability of sufficiency/necessity” conveys that sufficiency/necessity is a determinate property of the phenomenon, it’s just that we don’t have enough information to identify it. Therefore, in the singular limit these properties can be expressed as logical predicates

Sufficiency(C, E): Cause => Effect

Necessity(C, E): Effect => Cause

In the case of the decay of a specific atom at some time the causal claims become completely singular, definitions 9.2.1 and 9.2.2 reduce to evaluations of whether the above predicates hold. If we assume that atoms with low nuclear ratio do not undergo Beta- decay, our answers are:

A2) High nuclear ratio is a necessary cause of Beta- decay

A3) High nuclear ratio is not a sufficient cause of Beta- decay

Thus the truth of the statement that the decay of a radioactive atom has no cause depends on whether you are interested in sufficiency or necessity. In particular, that the atom would not have decayed were it not for its high nuclear ratio suggests this ratio was a cause of its decay.

But let’s make things more complicated, let’s say there is a small probability that atoms with low nuclear ratios show Beta- decay. We’d have to say that (remember, relative to our model) the decay of a specific atom at some time has no cause, because neither criterion of sufficiency or necessity is met.

The essence of causality, determinism?

We can continue to stretch the concept. Imagine that a specific nuclear ratio for a specific atom implied a 99.99% probability of decay at some time t, and also that said probability of decay for any other nuclear ratio were 0.001%. Would we still be comfortable saying that the decay of such an atom had no cause?

Singular indeterministic events are peculiar things. They behave according to probabilities, like those of general causation, but are fully specified, like instances of singular deterministic causation. Can we not just apply the methods and vocabulary of general causation to singular indeterministic events?

In fact, we can. We can modify functional causal models such that the underlying structural equations are stochastic, as mentioned in [Pearl2000] section 7.2.2. Another method found in [Steel2005] is to add un-physical exogenous variables that account for the outcomes of indeterministic events. Both of these can be swapped into regular functional models. This should yield equivalent definitions of 9.2.1 and 9.2.2, where probability of sufficiency and necessity are replaced with degrees, giving corresponding versions of A2) and A3).

In this approach, singular causation is not an all or nothing property, it is progressive. Just as general causal claims are expressed with epistemic probabilities, singular causal claims are expressed in terms of ontological probabilities. In this picture, saying that a particular radioactive decay had no cause would be wrong. Instead, perhaps we could say that a specific decay was “partially” or “mostly” caused by some property of that atom, rather than that there was no cause.

I believe this conception of causality is more informative. Throwing out causation just because probabilities are not 100% is excessive and misleading, it ignores regularities and discards information that has predictive content. The essence of causation, I believe, is not determinism, but counterfactual prediction, which banks on regularity, not certainty. It seems reasonable to extend the language we use for general causes onto singular ones, as their implications have the same empirical form. Both make probabilistic predictions, both can be tested.

No cause

What would it mean to say that some event has no cause, according to this interpretation? It would mean that an event is entirely unaffected by, and independent of, any of the universe’s state; no changes made anywhere would alter the probabilities we assign to its occurence. Such an event would be effectively “disconnected” or “transparent”.

Pollock – Lavender Mist Number 1

We could even imagine a completely causeless universe, where all events would be of this kind. It is not easy to see how such a strange place would look like. The most obvious possibility would be a chaotic universe, with no regularities. If we described such a universe as an n-dimensional (eg 3 + 1) collection of random variables, a causeless universe would exhibit zero interaction information, and zero intelligibility, as if every variable resulted of an independent coinflip. But it is not clear to me whether this scenario necessarily follows from a causeless  universe assumption.


References

[Pearl2000] http://www.amazon.com/Causality-Reasoning-Inference-Judea-Pearl/dp/0521773628

[Pearl2009] http://ftp.cs.ucla.edu/pub/stat_ser/r350.pdf

[Steel2005] http://philoscience.unibe.ch/documents/causality/Steel2005.pdf

[2] http://bayes.cs.ucla.edu/BOOK-2K/ch2-2.pdf

[3] http://www.cs.ucla.edu/~kaoru/ch7-final

[4] http://www.mii.ucla.edu/causality/?p=571

[5] See eg causal markov condition, minimality, stability

[6] ftp://ftp.cs.ucla.edu/pub/stat_ser/r393.pdf

The essential ingredient of causation, as argued in Pearl (2009:361) is responsiveness, namely, the capacity of some variables to respond to variations in other variables, regardless of how those variations came about.

[7] Laurea and her tolerance

Occam’s razor in a cellular physics universe

A cellular automaton (http://www.noyzelab.com/)

cellular automaton (CA) is an algorithm acting on cells in  a grid at discrete time steps. The cells can be typically in two states on or off. At each step, the CA computes what the new state of the cells are,  as a function of the state of its neighbors. Here is a simple example of how the new cells are calculated from the old ones:

in this example, the new cell is shown below, where the input cells (neighbors) are the three above. The image at the top of this post shows the evolution of a CA, by displaying new cells at each row. In other words, time flows vertically downwards.

CA’s were discovered in the 1940’s by Stanislaw Ulam and John von Neumann, who were working together at Los Alamos National Laboratory. Perhaps the most famous automaton is the Game of Life, invented by John Conway in 1970.

In this post we will consider a model of a universe based on cellular automata and see what it says about Occam’s razor and the problem of induction. The idea that the universe is describable by a cellular automaton is not new

many scholars have raised the question of whether the universe is a cellular automaton.[68] Consider the evolution of rule 110: if it were some kind of “alien physics”, what would be a reasonable description of the observed patterns?[69]

If you didn’t know how the images were generated, you might end up conjecturing about the movement of some particle-like objects (indeed, physicist James Crutchfield made a rigorous mathematical theory out of this idea proving the statistical emergence of “particles” from CA). Then, as the argument goes, one might wonder if our world, which is currently well described by physics with particle-like objects, could be a CA at its most fundamental level.

This idea is a specific variant of a more general perspective known as digital physics

In physics and cosmology, digital physics is a collection of theoretical perspectives based on the premise that the universe is, at heart, describable by information, and is therefore computable.

Note that we are not claiming digital physics here, but rather constructing a model based on some initial postulates and seeing where it leads us.

Given this background we can consider the problem of induction in a CA universe. The properties of this model are:

1) The universe consists of an n-dimensional infinite grid of cells

2) The time evolution of cells is governed by a cellular automaton

Let’s add our scientist. An agent in this universe makes observations and must formulate hypothesis as to what natural laws describe reality. If we accept a bayesian model, the problem of induction is how to construct a prior on possible theories such that inference is possible. But what form do theories have in this model?

From CA’s to Boolean functions

Although not immediately obvious, typical (2-state) CA’s are equivalent to boolean functions. This is something I noticed when I came across the equation that describes the number of CA’s as a function of states and neighbors:

The general equation for such a system (CA) of rules is kks , where k is the number of possible states for a cell, and s is the number of neighboring cells

This has the same shape as the expression 22k, which is the number of boolean functions for arity k . The connection is simple: a CA with 2-state cells that takes n neighbors as inputs to produce a new output cell (again 2-state) is equivalent to a function

ƒ : Bk → B, where B = {0, 1}

which is precisely the definition of a k-arity boolean function. In this CA -> Boolean Function correspondence the arity is given by the CA’s dimensionality and neighborhood. Below is a one-dimensional CA, each cell’s new value is a function of its two adjacent neighbors plus its own value (arity of 3).

Rule 179

This CA is known as rule 179 because that number encodes the binary specification of the boolean function. You can see this by looking at its truth table (I’m using bexpred):

rule179

Truth table specification for Rule 179

The table shows the output of the 3-ary function, inputs A,B,C. If you read the output bits bottom up you get 10110011 which in decimal is 179.

Boolean functions, expressions and trees

Besides the equivalence with CA’s, boolean functions are in general described by boolean algebra and are specified with boolean expressions or formulas. In this algebra variables take on the values true (T), false (F),  and the operators are disjunction (v), conjunction (^) and negation (~). For example, Rule 179 above can be formulated as

(A^C) v ~B

where A is the left neighbor cell, B is the center, and C is the right neighbor; you can check that this in fact corresponds to the CA by applying the formula on cells: doing this repeteadly would result in the pattern in the image above.

The nature of boolean is expressions is such that you can represent them as trees. For example (from D. Gardy[1]), the expression

x ^ (y v z v ~y v y) ^ (~y v t v (x ^ ~v) v u)

can be represented as

booleantree

Image taken from [1]

this representation of is very similar to that of boolean circuits, in which boolean expressions are represented as directed acyclic graphs. This representation allows classifying boolean circuits in terms of their computational complexity:

In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of Boolean circuits that compute them.

There are two measures of complexity, depth and circuit-size complexity. In this post we will use a boolean expression analog of circuit-size complexity, which measures the computational complexity of a boolean function by the number of nodes of the minimal circuit that computes it.

L(f) = length of shortest formula (boolean expression) computing f

With this last piece we can revisit our model and add some further detail:

1) The universe consists of an n-dimensional infinite grid of cells

2) The time evolution of cells is governed by some 2-state cellular automaton describable by a boolean tree of complexity L(f)

We can also answer the question posed earlier:

What form do theories have in this model?

The theories our scientist constructs take the from of boolean expressions or equivalently boolean trees. As stated before, the problem of induction in a bayesian setting is about constructing priors over theories. In our model this now translates into constructing a prior over boolean expressions.

Finally, we will postulate two desirable properties such a prior must have, following the spirit of work on algorithmic probability[3][4]. One is Epicurus’ Principle of Multiple Explanations:

Epicurus: if several theories are consistent with the observed data, retain them all

The other is the Principle of Insufficient Reason

when we have no other information than that exactly mutually exclusive events can occur, we are justified in assigning each the probability 1/N.

These last three epistemological characteristics complete our model:

3) Theories take the form of boolean expressions with tree complexity L(f)

4) A-priori all theories are consistent with evidence (Epicurus)

5) A-priori all theories are equally likely

A uniform prior on boolean expressions

Per the characteristics of our model we wish to construct a prior probability distribution over boolean expressions such that

a) The distribution’s support comprises all boolean expressions for some n-dimensional 2-state CA

b) All boolean expression are assigned equal probability

In order to achieve this we turn to results by Lefmann and Savicky[1] et al. on a specific tree representation of boolean formulas, And/Or trees:

We consider such formulas to be rooted binary trees.. each of the inner nodes .. is labeled by AND or OR. Each leaf is labelled by a literal, i.e. a variable or its negation

Note that these properties of And/Or trees do not reduce their expressiveness: any boolean expression can be formulated as an And/Or tree.

We wish to construct a uniform (a) probability distribution over all (b) And/Or trees, which are infinite. Lefmann and Savicky (see also Woods[6]) proved that such a probability distribution exists as an asymptotic limit of a uniform finite distribution:

theorem2.3

Finally we will use two results (later improved in Chauvin[7]) which relate the probability P(f) and the boolean expression complexity L(f) in the form of probability bounds:

theorem3.1

and

theorem3.5

establishing upper and lower bounds. Note the L(f) term in both cases.

Implications

Let’s recap. We defined a toy universe governed by a variant of CA physics, then showed the equivalence between these CA’s, boolean functions, expressions and finally trees. After adding two epistemological principles we recast the problem of induction in this model in terms of constructing a uniform prior over boolean expressions (theories). Further restrictions (And/Or tree representation of theories) allowed us to use existing results to establish the existence of, and then provide upper and lower bounds on, our uniform prior.

The key characteristic in these bounds is the term for the boolean function’s complexity. In theorem 3.1, the L(f) term appears as a positive exponential on a number < 1. In theorem 3.5, L(f) appears as a negative exponential on a number > 1. This means that the complete bounds are monotonically decreasing with increasing expression complexity. This is essentially equivalent to Occam’s razor.

Thus we have shown[8] that Occam’s razor emerges automatically from the the properties of our model; we get the razor “for free”, without having to add it as a separate assumption. Our scientist would therefore be justified in assigning higher probabilities to simpler hypothesis.

As an example, we can see concrete values, not just bounds, for the prior distribution in Chauvin [7], for the specific case of n = 3 (This would correspond with a 2-state 1-dimensional CA).

table

Sample P(f) for n = 3 (taken from [7])


The column of interest is labelled P(f). We can see how probabilities decrease with increasing boolean expression complexity. Refer to section 2.4 of that paper to see the corresponding increasing values of L(f).

Generalizations

Although we have reviewed the basic steps that outline how Occam’s razor follows from our simple model’s properties, we have not discussed the details as to how and why this happens. In a future post we’ll discuss these details, and the possibility that the mechanism at work may (or may not) generalize to other formalizations of universe-theory-prior.


Notes/References

[1] D. Gardy. Random Boolean expressions. In Colloquium on Computational Logic and Applications, volume AF, pages 1–36. DMTCS Proceedings, 2006.

[2] H. Lefmann and P. Savicky. Some typical properties of large And/Or Boolean formulas. Random Structures and Algorithms, 10:337351, 1997.

[3] Principles of Solomonoff Induction and AIXI 

[4] A Philosophical Treatise of Universal Induction

[5] http://www.scholarpedia.org/article/Algorithmic_probability#Bayes.2C_Occam_and_Epicurus

[6]  A. Woods. Coloring rules for finite trees, and probabilities of monadic second order sentences. Random Structures and Algorithms, 10:453485, 1997.

[7] B. Chauvin, P. Flajolet, D. Gardy, and B. Gittenberger. And/Or trees revisited. Combinatorics Probability and Computing, 13(4 5):475497,July-September 2004

[8] We are leaving out some technical details here. One is that monotonically decreasing bounds do not imply a monotonically decreasing probability. There may be local violations of Occam’s razor, but the razor must holds besides minor fluctuations. In the sample results for n=3 in Chauvin[7], probabilities are in fact monotonically decreasing.

Two, the asymptotics for P(f) for fixed m and P(f) for trees <= m are the same, see [7] 2.1 and [1] 3.3.3

Another detail is the assumption that ceteris paribus, a minimal expression computing f1 corresponding to expression e1 will be shorter than the minimal expression computing f2 corresponding expression e2, if e1 < e2. I e1 < e2 implies on average L(f1) < L(f2).

Finally, it is worth nothing that it is the syntactic prior over boolean expressions that induces an occamian prior over boolean functions. What makes this work is that formula reductions[9] produce multiplicities in the syntactic space for any given element in semantic space. A uniform prior over boolean functions would not yield Occam, this would have to be added separately (ie, the problem of induction)

[9] Boolean expressions may be reduced (simplified) using the laws of boolean algebra. Here is an example boolean reduction

The image above shows a reduction of the 3-ary boolean expression

(!A*!B*!C)+(A*!B*!C)+(!A*!B*C)+(A*!B*C)+(A*B*C)

which yields

A*C + !B

Which is in fact the boolean function corresponding to Rule 179

Causal entropy maximization and intelligence

CEM

Taken from Causal Entropic Forces

Recently I was referred to a paper titled Causal Entropic Forces published in Physical Review Letters that attempts to link intelligence and entropy maximization. You can find reviews of this paper here and here. The paper starts with

Recent advances in fields ranging from cosmology to computer science have hinted at a possible deep connection between intelligence and entropy maximization….In this Letter, we explicitly propose a first step toward such a relationship in the form of a causal generalization of entropic forces that we show can spontaneously induce remarkably sophisticated behaviors associated with the human ‘‘cognitive niche,’’ including tool use and social cooperation, in simple physical systems.

The authors then go on to define a causal path version of entropy. Briefly, this is a generalization from standard entropy, a measure of how many states a system can be in at a specific point in time, to causal path entropy, a measure of how many paths that system can follow during a given time horizon. In technical language, microstates are mapped to paths in configuration space, and macrostates are mapped to configuration space volumes:

In particular, we can promote microstates from instantaneous configurations to fixed-duration paths through configuration space while still partitioning such microstates into macrostates according to the initial coordinate of each path

In other words, an initial coordinate establishes a volume in configuration space which represents possible future histories starting at that point. This is the macrostate (depicted as a cone in the image above)

Having defined this version of entropy, the authors then add the condition of entropy maximization to their model; this is what they call causal entropic forcing. For this to have a net effect, some macrostates have volumes which are partially blocked off for physical reasons. Consequently these macrostates have less available future paths, and less causal path entropy. The result is that different macrostates with different entropies can be differentially favored by condition of causal entropy maximization:

there is an environmentally imposed excluded path-space volume that breaks translational symmetry, resulting in a causal entropic force F directed away from the excluded volume.

Note that, contrary to actual thermodynamical systems that naturally exhibit entropy maximization for statistical reasons, causal entropic forcing is not physical, it is a thermodynamics inspired premise the authors add to their model as a “what if” condition, to see what behaviour results. So, what happens when systems are subject to causal entropic forcing?

 we simulated its effect on the evolution of the causal macrostates of a variety of simple mechanical systems: (i) a particle in a box, (ii) a cart and pole system, (iii) a tool use puzzle, and (iv) a social cooperation puzzle…The latter two systems were selected because they isolate major behavioral capabilities associated with the human ‘‘cognitive niche’’

Before you get excited, the “tool use puzzle” and “social cooperation puzzle” are not what one would imagine. They are simple “toy simulations” that can be interpreted as tool use and social cooperation. In any case, the result was surprising. When running these simulation the authors observed adaptive behaviour that was remarkably sophisticated given the simplicity of the physics model it emerged from. What’s more, not only was the behaviour adaptive, but it exhibited a degree of generality; the same basic model was applied to both examples without specific tuning.

The remarkable spontaneous emergence of these sophisticated behaviors from such a simple physical process suggests that causal entropic forces might be used as the basis for a general—and potentially universal—thermodynamic model for adaptive behavior.

How does this fit in with intelligence?

I see two different ways one can think about this new approach. One, as an independent definition of intelligence from very simple physical principles. Two, in terms of existing definitions of intelligence, seeing where it fits in and if it can be shown to be equivalent or recovered partially.

Defining intelligence as causal entropy maximization (CEM) is a very appealing as it only requires a few very basic physical principles to work. In this sense it is a very powerful concept. But as all definitions it is neither right nor wrong, its merit rests on how useful it is. The question is thus how well does this version of intelligence capture our intutions about the concept, and how well it fits with existing phenomena that we currently classify as intelligent[1]. Ill consider a simple example to suggest that intelligence defined this way cannot be the entire picture.

That example is unsurprisingly life, the cradle of intelligence. The concept that directly collides with intelligence defined as CEM is negentropy. Organisms behave adaptively to keep their biological systems within the narrow space of parameters that is compatible with life. We would call this adaptive behaviour intelligent, and yet its effect is precisely that of reducing entropy. Indeed, maximizing causal entropy for a living being means one thing, death.

One could argue that the system is not just the living organism, but the living organism plus its environment, and that in that case the entropy perhaps would be maximized[2]. This could resolve the apparent incompatibility, but CEM still seems unsatisfying. How can a good definition of intelligence leave out an essential aspect of intelligent life: the entropy minimization that all living beings must carry out. Is this local entropy minimization implicit in the overall causal entropy maximization?

CEM, intelligence and goals

Although there is no single correct existing definition of intelligence, it can be said that current working definitions share certain common features. Citing [3]

If we scan through the definitions pulling out commonly occurring features we find that intelligence is:

• A property that an individual agent has as it interacts with its environment or environments.

• Is related to the agent’s ability to succeed or profit with respect to some goal or objective.

• Depends on how able to agent is to adapt to different objectives and environments.

In particular, intelligence is related to the ability to achieve goals. One of the appealing characteristics of CEM as defining intelligence is that it does not need to define goals explicitly. In the simulations carried out by the authors the resulting behaviour seemed to be directed at achieving some goal that was not specified by the experimenters. It could be said that the goals emerged spontaneously from CEM.  But it remains to be seen whether this goal directed behaviour results automatically in real complex environments. For the example of life I mentioned above, it looks to be just the opposite.

So in general, how does CEM fit in with existing frameworks where intelligence is the ability to achieve goals in a wide range of environments[3]? Again, I see two possibilities:

  • CEM is a very general heuristic[4] that aligns with standard intelligence when there is uncertainty in the utility of different courses of action
  • CEM can be shown to be equivalent if there exists an encoding that represents specific goals via blocked off regions in configuration space (macrostates)

The idea behind the first possibility is very simple. If an agent is faced with many possibilities where it is unclear which one will lead to achieving its goals, then maximizing expected utility would seek to follow courses of action that allow it to react adaptively and flexibly when more information becomes available. This heuristic is just a version of keep your options open.

The second idea is just a matter of realizing that CEM’s resulting behaviour depends on how you count possible paths to determine the causal entropy of a macrostate. If one were to rule out paths that result in low utility given certain goals, then CEM could turn out to be equivalent to existing approaches to intelligence. Is it possible to recover intelligent goal directed behaviour as an instance of CEM given the right configuration space restrictions?


References

[1] Our intuitions about intelligence exist prior to any technical definition. For example, we would agree that a monkey is more intelligent than a rock, and that a person is more intelligent than a fly. A definition that does not fit these basic notions would be unsatisfactory.

[2] http://prd.aps.org/abstract/PRD/v76/i4/e043513

[3] http://www.vetta.org/documents/A-Collection-of-Definitions-of-Intelligence.pdf

[4] This seems related to the idea of basic AI drives identified by Omohundro in his paper http://selfawaresystems.files.wordpress.com/2008/01/ai_drives_final.pdf. In particular 6. AIs will want to acquire resources and use them efficiently. Availability of resources translates to the ability to follow more paths in configuration space, paths that would be unavailable otherwise.