# Causal Leverage Density: A General Approach to Semantic Information

Stuart J Bartlett
Division of Geological and Planetary Sciences
California Institute of Technology
Pasadena, CA 91125
United States  1200 E California Blvd
Pasadena
CA 91125<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

I introduce a new approach to semantic information based upon the influence of
erasure operations (interventions) upon distributions of a system’s future
trajectories through its phase space. Semantic (meaningful) information is
distinguished from syntactic information by the property of having some
intrinsic causal power on the future of a given system. As Shannon famously
stated, syntactic information is a simple property of probability
distributions (the elementary Shannon expression), or correlations between two
subsystems and thus does not tell us anything about the meaning of a given
message. Kolchinsky and Wolpert (2018) introduced a powerful framework for
computing semantic information, which employs interventions upon the state of
a system (either initial or dynamic) to erase syntactic information that might
influence the viability of a subsystem (such as an organism in an
environment). In this work I adapt this framework such that rather than using
the viability of a subsystem, we simply observe the changes in future
trajectories through a system’s phase space as a result of informational
interventions (erasures or scrambling). This allows for a more general
formalisation of semantic information that does not assume a primary role for
the viability of a subsystem (to use examples from Kolchinsky and Wolpert
(2018), a rock, a hurricane, or a cell). Many systems of interest have a
semantic component, such as a neural network, but may not have such an
intrinsic connection to viability as living organisms or dissipative
structures. Hence this simple approach to semantic information could be
applied to any living, non-living or technological system in order to quantify
whether a given quantity of syntactic information within it also has semantic
or causal power.

Quantifying meaning, or, the meaning of meaning has been as thorny an issue in
philosophy as an overgrown rose bush at a balloon party. It’s high time this
nut was finally cracked. While Shannon and others introduced powerful
frameworks for measuring information content, communication channel
capacities, mutual information, transfer entropy, etc. (Shannon, 1948), it has
been notoriously difficult to find precise, mathematical formalisms that can
quantify semantic or meaningful information (Froese and Taguchi, 2019;
Kiverstein et al., 2022; Kolchinsky and Wolpert, 2018; Putnam, 1975). However,
the need for such a formalism is great due to widespread application domains
including linguistics, communication, biological semiotics, machine learning,
information thermodynamics, control theory, and others. In such disciplines we
often need to establish and compute the amount of information in a given
system that has causal power over the future of that system. In fact, this is
a defining feature of life as compared to non-life: it’s incredible power to
exert large-scale influence using a relatively small amount of information
(e.g., a single strand of DNA) (Bartlett and Wong, 2020; Davies and Walker,
2016; Marzen and Crutchfield, 2018; Nurse, 2008; Seoane and Solé, 2018; Zenil
et al., 2012).

A primary example of this was the covid-19 pandemic, in which the entire world
was plunged into despair as widespread death, suffering and disruption ensued.
All of this chaos owed its causal origins to the peculiar informational
properties of a handful of molecules that are so small they make a miniscule
water droplet seem like an ocean. In this case the virus itself was selected
for, or had the emergent ‘goal’ of self-replication and proliferation, which
for several years it achieved with ruinous effectiveness. Thus this example is
in line with the definition of semantic information introduced in the seminal
work of Kolchinsky and Wolpert (2018), whom define it as: “the syntactic
information that a physical system has about its environment which is causally
necessary for the system to maintain its own existence”. The information
contained in the RNA sequence of the first covid-19 virions to infect humans
clearly had significant causal power. On the one hand these molecules
contained syntactic information that ensured the viability of many future
virus generations. This information also had massive causal influence on the
anthroposphere, disrupting the lives of almost every human on earth. In
general, it is safe to assume that all similar examples also hold, i.e., the
genetic sequence information of biological cells and viruses has significant
causal power to support the future viability of the cell, virus or lineage.

However, we may want to generalise this idea to situations where the viability
of a local subsystem is not the primary variable of interest, where the
relevance or quantification of viability is unclear, or situations where we
are simply interested in how a quantity of information has causal power over
the future unfolding of a system. For example, let us consider the writings
and teachings of the Buddha (or the Buddha’s subset of the ‘Dataome’ (Scharf,
2021)). This information has had vast causal influence on the world, bringing
peace and contentment to the lives of many, perhaps sparing nations from wars,
and provoking philosophical debate and inspiration to this day. There is
significant ‘meaning’ in this information, but it is not clear whether all of
this meaning and causal power is directly linked to the Buddha’s viability, or
the viability of his lineage (which, according to historical records, probably
ceased with his son).

We could imagine a hypothetical thought experiment where the Buddha’s early
inspirations are tragically ‘scrambled’, and he instead grows up as a regular
prince instead of the great spiritual master that he in fact was. The world
2.5 thousand years later would likely look very different, given the global
influence of the Buddha’s teachings. In the language of statistical physics,
the trajectory of the earth system through its accessible phase space would
likely have diverged considerably from the trajectory that we are familiar
with today. If we assume that in both cases the Buddha’s lifespan and the
extent of his lineage were the same, then here we have a case of significant
meaningful information (the Buddha’s teachings), which, when scrambled, has a
vast impact on the phase space trajectory of the earth system, while being
essentially disconnected from the viability of the subsystem containing that
information. A similar logic could apply to any great scholar, scientist,
thought leader, etc. They leave a significant legacy of influence upon the
world, which in the long term does not have a significant connection to their
viability (or in other words, their viability would likely have been similar
even in the absence of their influencing the world at large).

It seems that even though the viability of the human species may have been
enhanced (or at least affected) by such influential figures, the primary
variable of interest with respect to semantic information is the future causal
influence due to the syntactic information that such people introduced to the
world.

Another example could be the amino acid sequence of a protein that has been
extracted from the last member of a now extinct plant species. Whether this
sequence contains meaningful information clearly depends upon whether the
functional characteristics of the protein are relevant to the goals and
interests of molecular biology and medical engineering. A primary goal of
modern medicine is to eliminate cancer cells, for example. How might we
objectively quantify the meaningful information of the protein’s sequence,
while embracing the contextual and contingent features of the esoteric
environment that the protein is being studied within (which is of course very
different from the environment that it evolved in)? The medical engineers
might quantify the efficacy of the protein sequence by the fraction of cancer
cells it neutralises. But this is clearly too specific, if we are seeking a
definition of semantic information.

The unifying feature of this and the previous examples is whether the
information at hand (the amino acid sequence), has causal influence on the
future unfolding of the wider system the information is embedded within. If
the protein turns out to kill all cancer cells that it binds with, then it
would revolutionise medicine and the future trajectory of the Anthroposphere,
saving millions from death and suffering. Hence the sequence of amino acids in
this protein contains a vast quantity of semantic information (or rather it
has a high causal leverage density (CLD)), since it would change the course of
human evolution, putting us on a very different trajectory to the one where
cancer continues killing millions every year. In addition, in physical and
informational terms, it is very ‘small’ (a small number of bits and a small
physical instantiation).

We can even go all the way back to the origins of life on earth. For those who
ascribe to an ‘unlikely event’ approach to this process, we can imagine the
scenario in Star Trek The Next Generation, where the character Q takes Captain
Picard back to the moment when life began on earth. If Q, the prankster that
he is, had scrambled the information content of that first protocell or
progenote, life would have been literally stopped dead in its tracks, and the
earth would have remained sterile for its entire lifespan. Thus in this
scenario, unimaginable semantic information (CLD) was present in that
progenator pre-lyfeform.

Given these examples, and taking significant inspiration from the approach of
Kolchinsky and Wolpert (2018), we can formalise the notion of semantic
information by comparing phase space trajectories of a given system under
interventions that scramble syntactic information. Kolchinsky and Wolpert
(2018) compare the original system with its intervened counterparts using a
metric of viability (whether the subsystem of interest survives or persists).
However, in this work I would like to generalise beyond viability by simply
comparing the phase space trajectories of the original system with its
intervened counterparts. If we assume a known initial state, $R_{0}$ (point in
the system’s phase space at time $t=0$), a known complete set of accessible
states, and stochastic dynamics (intrinsic uncertainties in the future
unfolding of the system’s state as described by stochastic differential
equations, for example), we can in principle map out a distribution of
trajectories that the system could follow as a function of time, weighted by a
set of probabilities. We assume that we are interested in some pre-defined
time scale $\tau$, for which we allow the system to evolve under its
stochastic dynamics.

Let us now perform a similar intervention to those suggested in Kolchinsky and
Wolpert (2018), i.e., we scramble a portion of the syntactic information in
the system. Due to this intervention, the system now has a different initial
state, $\hat{R}_{0}$ (point in the system’s accessible phase space). However,
the stochastic dynamics remain the same, and we can now allow another set of
trajectories to unfold through the system’s accessible phase space. It is a
set rather than a single trajectory due to the stochastic nature of the
dynamics, and after we reach time $t=\tau$, we have a new distribution of
final states (points) in phase space, with each point weighted by a
probability.

At this stage we have two distributions over final states due to the system’s
stochastic evolution from time $t=0$ to $t=\tau$. The final, un-intervened
distribution $p$, will likely be different from its intervened counterpart,
$\hat{p}$, but could be very similar or even identical. If $p$ and $\hat{p}$
are similar, that implies the information that was scrambled during the
intervention did not have a high semantic content or causal power. However, if
the intervened distribution of states is very different from the un-
intervened, that implies a large semantic or causal power of the information
that was scrambled.

In order to quantify this notion, we can use standard tools from information
theory to quantify the difference between these two distributions of states.
For example, we can employ the symmetric metric for comparing probability
distributions, the Jensen-Shannon (J-S) divergence (Lin, 1991):

$D_{JS}(p\|\hat{p})=\frac{1}{2}D_{KL}(p\|m)+\frac{1}{2}D_{KL}(\hat{p}\|m)$

where $m=\frac{1}{2}(p+\hat{p})$, and $D_{KL}$ denotes the Kullback-Leibler
divergence, defined as:

$D_{KL}(p\|q)=\sum_{x}p(x)\log_{2}\left(\frac{p(x)}{q(x)}\right)$

Thus, the full expression for the J-S divergence can be written as:

$D_{JS}(p\|\hat{p})=\frac{1}{2}\sum_{x}p(x)\log_{2}\left(\frac{p(x)}{m(x)}\right)+\frac{1}{2}\sum_{x}\hat{p}(x)\log_{2}\left(\frac{\hat{p}(x)}{m(x)}\right),$

where $m(x)=\frac{1}{2}(p(x)+\hat{p}(x))$.

At this stage we have a measure of the divergence of our un-intervened and
intervened systems, capturing the degree to which the information that was
scrambled in the intervention has causal power over the evolution of the
system. We can now normalise this metric into a dimensionless density, the
CLD, by dividing by the number of bits that were erased during the
intervention:

$\chi_{LD}=\frac{D_{JS}(p\|\hat{p})}{\Omega_{scr}},$

where $\Omega_{scr}$ represents the number of bits that were scrambled during
the intervention. Hence $\chi_{LD}$ is a dimensionless representation of the
causal power that the information erased during the intervention process has
on the stochastic evolution of our system over the characteristic time $\tau$.
If $\chi_{LD}\approx 0$, we can assume that the information erased during the
intervention has no causal influence or semantic content on the unfolding of
the system’s state over the time scale $\tau$. However, large values of
$\chi_{LD}$ imply a significant ‘CLD’. Note that the metric has no upper
bound, but is bounded from below at 0 for the case described above.

So we now have a general metric for semantic information which encapsulates
the idea of causal power on the future evolution of a system due to some
quantity of syntactic information within a subsystem. While the metric is
dimensionless, being a ratio of two informational quantities with units of
bits, it does require the selection of a relevant time scale $\tau$ and the
choice of what information-bearing degrees of freedom in the chosen subsystem
to randomise. These choices were also addressed by Kolchinsky and Wolpert
(2018), and in general it is expected that they will be pre-determined by
whomever wishes to compute whether there is semantic information present in a
subsystem. For example, biologists would likely target genetic information,
and be interested in whether such information has future causal influence on
the evolutionary or ecological properties of the broader environment. AI
researchers would instead be interested in the weights of their neural
networks and the degree to which such information can change the future
trajectory of humanity (or perhaps more humble questions).

Again, as addressed by Kolchinsky and Wolpert (2018), there may be methods to
objectively determine the choice of information to scramble, and
characteristic timescale $\tau$, and one could conceive of performing a
variational analysis over these quantities in order to discover which
subsystems contain semantic information. But I leave such questions to future
work.

In terms of applying this method in the real world, one might suggest that
knowing the exact (stochastic) dynamics of most complex systems is difficult
at best and impossible at worst. How can we realistically map out the state
space of a high-dimensional, non-linear stochastic system, predict ensembles
of future phase space trajectories and then analyse the statistics of those
distributions of trajectories? Perhaps such a fine-grained level of detail is
not strictly necessary to attain useful estimates of CLD. For many systems of
interest, we often have robust statistical data (probability distributions)
from different instantiations (equivalent to a set of interventions). In this
case we can simply apply our expression for $\chi_{LD}$ and still obtain a
dimensionless measure of the extent to which our system has diverged due to
the erasure of a subset of its information content.

Note that in some complex systems, we might expect a compression of the future
spread of phase space trajectories due to the destruction of semantic
information. For example, let us imagine that an archaeologist unearths an
ancient Egyptian text containing a formula for the elixir of life. Clearly
this text has significant semantic information, for it could usher in a new
path for the Anthroposphere where no human has to die. In fact, depending on
stochastic factors, such as whether the text gets inadvertently destroyed by
bandits soon after discovery, the future trajectories of humanity now span a
spectrum from ‘business as usual’ to ‘immortal utopia’. If we intervene on the
state of the system by destroying the ‘book of vitality’, this broad spread of
future trajectories for humanity gets immediately squished into a miniscule
bundle, most of which look fairly mundane compared to the just-erased
alternatives. Hence in this scenario, the intervened distribution of
trajectories spans a smaller cone than the un-intervened, original
distribution. However, this difference will be reflected in the high value of
$\chi_{LD}$.

The opposite extreme is when the erasure of some semantic information
destablises or ‘blows up’ the future set of trajectories in a system’s phase
space. Such would be the case if some crucial, homeostatic information was
destroyed in an intervention. For example, if we were to scramble the genes
relevant to cellular gap junctions in a eukaryotic multicellular organism,
it’s highly likely that cancer would ensue for the unfortunate individual. In
this case the intervention has caused a divergence of future trajectories
through the system’s phase space, from one where the organism maintained
cellular homeostasis and integrity (a relatively confined set of future
trajectories) to one where a whole range of future paths could now ensue, most
of them sadly pathological.

There is a conspicuous potential connection between CLD and major transitions
in evolution, including the origins of life and the AI revolution. In these
transitions, there is a significant jump in the CLD, ushering in a new epoch
for life on earth as a result. At the origins of life, molecular information
gained causal control over the future infolding of the wider (otherwise
abiotic, passive, dead) environment. As we live through the AI revolution, we
see the accumulation of semantic information in compressed, neural network
representations of human knowledge. Many of these evolutionary transitions
involved increases in the viability-based semantic information introduced by
Kolchinsky and Wolpert (2018) as well as the CLD (often they coincide).
However, in many of the major evolutionary transitions, while the viability of
the extant life forms increased in the short term, the transitions often
ushered in shifts towards novel forms of life in the long term. In the case of
AI, it is not clear whether the viability of humans or life as we know it will
ultimately improve as a result of the shift, or whether the shift will
catalyse the emergence of a new form of lyfe, endowed with the most universal
thermodynamic tool of all: CLD.

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