Overview of Previous Work on Causal Logic

This article discusses work that can be considered as preliminary to the theory presented here. It is important to know the nature of the problem being addressed, the outcome of previous attempts, and the stature of the researchers that tried.

There really isn't any preliminary work by other authors conducive to Causal Logic (CL). This fact is consistent with the fact that CL is an experimental discovery, not a conclusion from previous research. There is, instead, a very large body of work by authors who tried more traditional approaches to some CL-related problems and obtained very important but mostly negative results. Some highlights of that work are covered in this Section.

One of the lines of thought followed by Boole, Frege, Gödel,
Church and Turing, focused on the limits of *mathematical deduction*.
We now know that limits to deduction do exist and that not all functions can be
computed, and we generally
accept the definition of computation as anything that the Turing machine can
compute. An overview and references can be found in Section 1.2.2 of
Artificial Intelligence
(2010).

Another line of thought focused on the *binding problem*. A quick way to
introduce the binding problem to those unfamiliar with it, is the following
statement written by Douglas Hofstadter in his characteristic style, page 633 of
Metamagical Themas:

*The major question of AI is this: What in the world is going on to enable you
to convert 100,000,000 retinal dots into one single word `mother' in one tenth
of a second?*

What's going on, is known as a process of *binding*. I propose that CL is
the process that binds the retinal dots and makes sense of them.

Ca. 1850, physicist and physiologist Hermann von Helmholtz, in his studies of
vision, concluded that a process of inference was needed to explain the
processing of visual images in the brain. He named that process
"unconscious inference," because we are not aware that it is happening. He
appears to have been the first person who correctly identified what was later
called the *binding problem*. I propose that CL is the same as unconscious
inference.

In 1912, Bertrand Russell, who wanted to build up knowledge from experience and hoped to construct the mathematical world from first-order predicate calculus, confronted the binding problem and initiated a line of thought that was pursued by mathematical logicians such as Wittgenstein and the later Wittgenstein for decades, and finally proved to have no solution in the context of that calculus. A more detailed account of the history of the binding problem and the notion of mind as machine in the 20th century, including references, is found in Sections 1, 2, and 3 of a 2011 paper by evolutionary biologist Stuart Kauffman.

The modern view of the same old binding problem is the phenomenon known as *
self-organization*, routinely observed in all kinds of dissipative physical
systems and the object of intense scrutiny in Complex Systems Science (CSS). A
dynamical system is said to self-organize when it suddenly and unexpectedly
evolves towards an organized state where recognizable structures and behaviors
are present. But self-organization, too, remains unexplained. Mathematician and
philosopher S. Barry
Cooper refers to it as "a case of complexity outstripping computer resources
and human ingenuity." Cooper focuses on computability and definability, and
writes about "the smallness of the computable world in relation with what we can
describe." Authors also refer to self-organization as a case where "you get
something for nothing," or even as "magic." The fact is, the binding problem
remains unsolved to these days. I propose that CL has finally solved the binding
problem.

Physics has many laws, but the supreme law of nature, the law of laws, is the
principle of symmetry. The term symmetry refers to transformations of a physical
system that leave the system unchanged. For example, the laws of motion of a
billiard ball are the same in America or in Antarctica. In short, the principle
states that symmetries in physical systems lead to conservation laws and the
existence of *conserved quantities*, which appear as functions or
structures and have the property of remaining invariant under the
transformations. This fundamental principle is today of critical importance in
modern theoretical Physics.

In 1918, Emmy Noether
formalized the principle and published her now famous theorem, where she
followed a constructivist approach and proved the existence of conserved
quantities by actually *calculating* them. Her theorem has since played and
still plays a fundamental role in the development of modern theoretical Physics,
but has been largely ignored in other sciences because it applies only to a
certain class of differential equations that appear in the variational analysis
of Lagrangian systems.

But differential equations can be discretized and expressed as infinitesimal causal sets (Section 3.4 of the main paper), from which the original equations can be recovered as a limiting case. Noether's theorem is actually one of causality, a consequence of CL. I believe it can be proved from CL in its entirety, and I have actually proved one simple subcase of it in Section 2.6 of the main paper. I propose that CL generalizes Noether's theorem.

Noether's theorem is also one of self-organization, because conserved quantities
are self-organized structures. And it actually solves the binding problem in one
case, a fact that few people have realized. The theorem applies only to *
conservative systems*, but not to *dissipative systems*, a fact that has
profound consequences in AGI.

A physical system that is dissipative and has excess energy will dissipate its
energy and settle into a stationary state, also known as an *attractor*. At
that point, dissipation stops, simply because there is no more free energy, and
the system becomes conservative. At that point, Noether's theorem kicks in, and,
if symmetries exist, self-organized, conserved structures make their unexpected
appearance.

But symmetries always exist in a causal set. If a differential equation is discretized as a causal set, the causal set will always have symmetries and will always give rise to self-organized, conserved structures. All of which, even the convergence to attractors, is predicted by CL. I propose CL as the solution for both the binding problem and the self-organization problem.

In Computer Science, most of the attention has focused on algorithms, their
development, and their complexity. But the transformations of algorithms such as
*refactoring* and the object-oriented analysis of software, where binding
takes place and conserved quantities are obtained, have received relatively
little theoretical attention, and are considered, even today, as something that
human developers do better than machines. Tools have been developed, of course,
but they must be used under human control.

Still in Computer Science, it is useful to mention *Solomonoff's paradox*,
which is very much relevant in self-programming and AGI in general. In his
studies of machine learning,
Solomonof(2010),
who did not intend to produce a paradox, writes:

*A heuristic programmer would try to discover how he himself would solve (some
problem) - then write a program to simulate himself.*

When Solomonoff writes "would try to discover," he is referring to a certain
process that takes place in the programmer's brain - perhaps with the help of
some notes on paper. When he writes "`how he himself would solve (the problem),"
he is referring to the result or output from that process. And when he refers to
"a program to simulate himself," he is referring to a different process, the
program. A program is an algorithm, a set of rules that explain how to solve the
problem, but it is not the process that found those rules. Solomonoff clearly
identifies two different processes, the process of discovery of the process that
solves the problem, which in this theory corresponds to CL, and the process that
actually solves the problem, the program, which in the theory corresponds to the
algorithm inferred by CL. When the programmer writes the program, she is only
copying the algorithm from her brain to the computer. The process that
discovered the algorithm remains hidden in her brain and is not in the program.
CL is not in the algorithm. Paradoxically, the more the heuristic programmer
endeavors to program her intelligence, the farther away she gets from that goal.
Systems that display this types of behavior, where one process controls another,
are known as *host-guest systems*, see
Pissanetzky(2010a).

The effect of causal inference (CI) on a sparse canonical matrix representing a
causal set, is to *block-diagonalize* it. Techniques and approximations for
block-diagonalizing sparse matrices are well-known. Several of them are
discussed in
Pissanetzky(1984). One of the them is *reduction* (Section 5.3 ibidem),
where the associated digraph is partitioned into strong components, and the
associated causal set into blocks. Another is the Cuthill-McKey algorithm
(Section 4.6 ibidem), which in its reverse form can be used to reduce the *
profile* of the sparse matrix. Both algorithms result in a compaction of most
elements of the matrix near the diagonal, and could be contemplated as processes
of self-organization and binding. They may find an application as preprocessors
for CI. However, there are two problems. First, the profile is a global function
of the entire matrix, unlike the functional proposed here, which is local and
can be easily implemented on a massively parallel computer. Second, invariance
of the blocks under transformations is not proved, because the set of
permutations that define the symmetry and the transformation is never obtained.