You know, back in 1956, a machine didn't just
solve a complicated math problem faster than
a human. Right. It actually found a more beautiful,
more elegant path to the truth than two of the
greatest mathematical minds of the 20th century.
Welcome to Deep Dive. It really is an incredible
story. I mean, we're looking at a field that
sits right at this crazy intersection of pure
mathematics, philosophy, and artificial intelligence.
Exactly. And our mission today is to give you,
the listener, the ultimate shortcut to understanding
this foundational, almost science fiction level
concept in computer science, which is automated
reasoning. Yeah, it's a heavy topic, but we're
going to break it down. We really are. And just
to give you a sense of our roadmap, we are pulling
from a really comprehensive Wikipedia article
on automated reasoning today, along with several
of its deep -cut reference materials. Some really
fascinating sources in there. Oh, for sure. And
for you listening, here is exactly why you need
to care about this. We are living in a moment
right now where everyone is using these AI chatbots.
And well, we all know they hallucinate, right?
Oh, constantly. Yeah, they just constantly make
things up. Right. So understanding automa - reasoning
is the key to knowing how we are actually teaching
machines not just to guess the next word in a
sentence but to you know actually think deduce
and verify truths with absolute certainty it's
the antidote to the hallucination problem exactly
okay so Let's unpack this. We need to set the
foundation first. What exactly is this field
even trying to achieve? Well, the primary goal
of automated reasoning is to basically build
computer programs that allow machines to reason
completely automatically, or I mean, at least
as close to completely as mechanically possible.
So we're not just talking about a chatbot that
sounds smart. No, no, not at all. We are talking
about a machine that deduces what is mathematically
and logically true. The most developed areas
here are automated theorem, proving where the
machine actually discovers a proof on its own,
and automated proof checking. And proof checking
is where the human does the work first. Exactly.
A human provides the reasoning, and the machine
rigorously verifies that absolutely no logical
laws were broken. OK, so to do that, to actually
automate reasoning, we have to define what perfect
reasoning looks like, right? Which means we have
to step completely away from how the human brain
actually operates. Right, because human reasoning
is just incredibly messy. I mean, it relies heavily
on intuition. Yeah. A human mathematician will
just, you know, skip a dozen minor logical steps
and a proof because a connection just feels right
or it looks obvious based on their past experience.
You know, when I was reading the sources for
this, it made me think of walking across a frozen
lake. Oh, interesting. How so? Well, human intuition
is like looking at the ice, seeing it look solid
enough and just walk it across. You take a few
steps. You don't fall in. So you just assume
the whole lake is safe. Right. You extrapolate.
Yeah. But formal logic, on the other hand, is
like pausing to check the structural integrity
of every single individual ice crystal before
you are willing to move even an inch. That is
a great analogy. And what's fascinating here
is that a formal proof requires exactly that
microscopic level of scrutiny. Is relentless.
It really is. In a formal proof, every single
logical inference has to be traced all the way
back to the fundamental axioms of mathematics.
You absolutely cannot skip steps. No intuitive
leaps allowed. Zero. you cannot appeal to intuition.
Even if translating a specific, you know, intuitive
leap into formal logic would be totally routine
and boring for a human mathematician, the machine
demands the intermediate steps. It forces you
to show your work. Exactly. Because by removing
human intuition entirely, a formal proof strips
away the exact places where logical errors like
to hide. That makes total sense. Yeah. But logic
isn't always just proving that two plus two is
four, right? Like life and human thought involve
a lot of gray areas. Does automated reasoning
only deal with rigid black and white math? No,
it goes much deeper than that, actually. The
field also tackles how to reason by analogy.
For instance, it works with induction, which
is drawing a general rule from specific examples.
OK. And also abduction. Abduction, like an alien
abduction. Ha, no, logical abduction. It's essentially
forming the most likely explanation for a set
of observations, like a doctor diagnosing a patient
based on a list of symptoms. Oh, I see. Yeah.
And the sources also highlight something called
non -monotonic reasoning. Okay, non -monotonic.
I'm going to need you to break that down for
me. Sure. So in classical logic, once you prove
something is true, it stays true forever. Adding
new information doesn't change it. That's monotonic.
Right. Non -monotonic reasoning is closer to
real life. It's when you might draw a logical
conclusion, but then you learn a new piece of
information that actually forces you to retract
that earlier conclusion. So you basically change
your mind based on new evidence. Right. Exactly.
And a brilliant example of this from the reference
materials is John Paul. Oscar system. Oscar?
Yeah, Oscar is an automated argumentation system.
So instead of just proving a rigid math theorem,
it applies constraints of minimality and consistency
to literally argue with itself. Wait, it argues
with itself? Yeah, it builds what's called a
defeasible argument. Defeasible meaning, you
know, an argument that can be defeated. The system
generates a claim and then it systematically
searches its own database for a counter example
to destroy its own claim. That's wild. It acts
as both the prosecution and the defense, weighing
evidence under uncertain conditions. Honestly,
that sounds computationally exhausting. I mean,
checking every single mathematical ice crystal
or having a machine constantly prosecute and
defend its own thoughts. What's a massive amount
of processing? It really makes me wonder. what
forced early computer scientists to even attempt
this. Did they just like get tired of doing manual
math? Well, sort of, yeah. The sheer physical
limitation of the human brain to hold massive
logical structures without making a single mistake
was a huge driver. That makes sense. But the
actual birth of the field is heavily debated.
Some historians point back to Martin Davis in
1954. He implemented a concept called Pressberger's
Decision Procedure, and his program successfully
proved that the sum of two even numbers is even.
Okay, 1954. Right. But others argue the field
really crystallized a few years later at this
really famous Cornell summer meeting in 1957
that gathered logicians and computer scientists
together. You know, the project that really caught
my eye right in that same window was the logic
theorist program from 1956. Oh, yes. A classic.
The creators, Alan Newell, Cliff Shaw and Herbert
Simon, they pointed their machine at this massive
book called the Principia Mathematica. What was
the deal with that specific book? Well, the Principia
Mathematica was a monumental multi -volume masterwork
written between 1910 and 1913 by Alfred North
Whitehead and Bertrand Russell. OK. Their whole
goal was to derive all mathematical truths purely
in terms of symbolic logic. It was widely considered
the absolute Mount Everest of formal logic. So
naturally, you point the new computer Mount Everest.
Exactly. So Newell, Shaw, and Simon fed 52 theorems
from chapter two of this absolute beast of a
book into their logic theorist program. And the
machine? proved 38 of them. It did. But here's
the part that I just couldn't believe. The logic
theorist didn't just mimic the human proofs that
Whitehead and Russell had published. For one
of those theorems, the machine actually found
a proof that was more elegant. Yeah, that's the
real kicker. Like it required fewer steps and
was more logically beautiful than the proof provided
by two of the greatest minds of the 20th century.
It totally outlogist its human masters. It found
a cleaner, purely mechanical path to the truth.
And the funny thing is, the creators actually
had trouble getting those results published in
human mathematics journals at first. You were
probably defensive. Oh, almost certainly. But
that success deeply impacted how those scientists
viewed the future. I mean, by 1958, Newell, Shaw,
and Simon published a paper containing a quote
that is just dripping with audacity. Let's hear
it. They wrote, and I quote, There are now in
the world machines that think, that learn, and
that create. Moreover, their ability to do these
things is going to increase rapidly until the
range of problems they can handle will be coextensive
with the range to which the human mind has been
applied. Wow. They were essentially calling their
shot in 1958, predicting the entire future of
AI based on this one breakthrough. They completely
saw the horizon. But you know, here's where it
gets really interesting for me though. I have
to push back a little on what this machine is
actually doing. Okay, go for it. Aren't these
early programs, and I mean even the modern ones,
aren't they just glorified calculators? Like,
how does a program actually prove a philosophical
or mathematical concept without just crunching
numbers really fast? That's a great question.
And the distinction really lies in the difference
between arithmetic and formal logic. A calculator
takes numerical inputs and runs them through
a hard -wired arithmetic operation. It just computes.
Right. Numbers in, numbers out. Exactly. But
automated reasoning systems manipulate They take
abstract logical concepts and apply a strict
set of rules to transform those symbols into
a verified truth. They don't just calculate a
result, they generate the actual chain of reasoning.
OK, so if I am a computer scientist trying to
build one of these logic engines, what kind of
programming language am I even using to talk
to the machine? Like, it can't just be standard
code. No. You are using highly specialized languages
designed specifically for symbolic manipulation.
Let's look at a major system from the sources.
the Boyer -Moore theorem prover, also known as
NQTHM. NQTHM. Got it. This started in 1971 in
Edinburgh, and it was built entirely in Pure
Lisp, which is a very unique, highly recursive
programming language. Good. And NQTHM had this
fascinating mechanism for learning. It utilized
an induction heuristic based entirely on the
failure of its own symbolic evaluations. How
does a machine logically learn from a failure?
Well, imagine the system is trying to prove a
rule that applies to all number. It tries to
calculate the steps normally, but it hits a wall
where a variable is unknown and it just can't
proceed. Look at dead end. Right. Now a normal
program would just crash or throw an error, but
Enki THM was designed to analyze the shape of
that dead end. The shape of it. Yeah, it looked
at the exact point where the evaluation failed,
and it used the structure of that failure to
guess the underlying mathematical pattern, formulating
a totally new logical step to bypass the wall.
Oh, wow. So it literally mapped its own dead
ends to find the open door. Exactly. That is
brilliant. But let's bring this into the real
world for a second, because if we're talking
about trusting software to, I don't know, fly
a commercial airplane or manage a national power
grid, we can't just have a system making good
guesses. We need Absolute certainty. Which is
exactly where a system like ROCK comes in. ROCK,
spelled R -O -Z -Q. Right. Yes, formerly known
as COCK. This system was developed in France
and represents just a massive leap forward. ROCK
operates on something called the Calculus of
Inductive Constructions, or CIC. Okay, CIC. In
the CIC framework, a mathematical proof and a
computer program are treated as the exact same
mathematical object. Wait, hold on. A proof and
a program are the same thing. In this specific
architecture, yes. And because of this, Rock
can do something incredible. Let's say you write
a logical specification proving mathematically
that a specific sorting algorithm is flawless.
Okay. Rock doesn't just give you a checkmark
saying your math is right. It mechanically extracts
that theoretical proof and translates it directly
into actual executable source code like OCaml
or Haskell. Are you serious? So you do the pure
math and the system literally spits out the working
software? Yes. And because the software was extracted
mechanically from a formal logical proof, it
is physically impossible for that resulting code
to have a logical bug. It is strictly guaranteed
to behave exactly as specified. That completely
blows my mind. What is the limit to this? How
complex of a problem can these systems actually
handle? The milestones are pretty staggering,
honestly. Back in 1986, a researcher named Natarajan
Shankar used the Boyer -Moor system we talked
about to formalize Gödel's first incompleteness
theorem. Which is huge. Massive. And by 2004,
the rock system was used to verify the four -color
theorem. But one of the most wild examples from
the sources happened in 2016 regarding the Boolean
Pythagorean triples problem. Researchers formalized
this as a Boolean Satisfiability Problem, or
a SAT problem. I saw the term SAT problem in
the reading. What actually is that? Basically,
a SAT problem asks if there is any possible way
to assign true or false to a massive set of variables
so that the entire overall equation evaluates
to true. OK. True or false logic dates. Right.
For the Pythagorean problem, imagine you have
to color every whole number either red or blue.
The rule is that no three numbers that satisfy
the Pythagorean theorem, like three, four, and
five, can all be the same color. OK. False. So
mathematicians wanted to know if you could color
numbers this way forever without breaking the
rule. And how did the automated reasoner tackle
that? By using pure brute force logic combined
with some really smart pruning techniques, the
system checked combinations all the way up to
the number 7 ,824 and proved that, yes, you can
color them without breaking the rule. But at
the number 7 ,825, it becomes mathematically
impossible. And to prove this, the system generated
a proof file that was 200 terabytes in size.
200 terabytes. Yeah. No human being could ever
read that proof in a single lifetime. Not even.
close, but we trust the result because we mathematically
trust the logic engine that generated it. But
wait, if we had the logic theorists outperforming
humans back in 1956, and we are generating 200
terabyte proofs today, why hasn't automated reasoning
just been the dominant force in tech for the
last 70 years? I mean, the reality of technological
progress is rarely a straight line. What happened
in between? Yeah, it definitely wasn't a straight
line. The field actually hit a massive wall in
the 1980s and early 1990s. This period is widely
known as the AI winter. Right, the AI winter.
The theoretical math was totally sound, but the
hardware simply couldn't handle the combinatorial
explosion of data required to run these proofs
at scale. Funding dried up. So they just got
stuck waiting for better computers. Pretty much.
But the researchers didn't stop. They basically
went into hibernation and optimized their logic
engines. And the rebound became highly visible
by 2005. That's when Microsoft started integrating
automated verification technology directly into
their internal projects. Oh, interesting. Yeah,
they even incorporated logical checking into
the 2012 version of Visual C. So the technology
moved out of academic labs and straight into
the software compiling tools used by millions
of regular developers. So what does this all
mean for us today right in the middle of this
massive boom in generative AI? Well in the 2020s
the focus has violently shifted back to automated
reasoning as the ultimate cure for the AI hallucination
problem we talked about at the start. Bringing
it full circle. Exactly. Large language models
guess the next word based on statistical patterns.
They don't actually understand truth. So to fix
this, researchers are building reasoning language
models like DeepSeq R1, which are designed to
literally spend additional computational time
pondering a problem before they generate a single
word of output. But how does spending more time
pondering actually fix the hallucination fundamentally?
Like, isn't it just thinking about a wrong answer
longer? If we connect this to the bigger picture,
it leads us to the absolute bleeding edge of
the field. Neuro -symbolic architectures. This
is the marriage of two completely different eras
of computer science. Okay, neuro -symbolic. So
the neuro part being the modern neural networks
and the symbolic part being the formal logic
engines like rock that we just discussed. You
nailed it! It sounds kind of like pairing a wildly
creative architect who dreams up impossible gravity
-defying buildings with a ruthless safety inspector
who checks the math on every single load -bearing
beam before a single brick is laid. That is a
perfect way to look at it. The neural network
provides the intuition, the pattern recognition,
and those creative leaps. It drafts the blueprint.
Right. But before that answer is ever shown to
you, the symbolic logic engine takes over. It
attempts to formally prove the neural network's
generated answer using strict mathematical axiom.
And if it fails? If the safety inspector finds
a logical flaw, it rejects the answer and forces
the neural network to try again. The entire goal
is to give AI a built -in, mathematically flawless
lie detector. Man, it really is the invisible
safety net for our modern digital infrastructure.
Yeah. Because as we discussed, we are handing
over control of power grids, financial markets,
and aviation software to machines. If there is
a logical bug in those systems, the consequences
are just catastrophic. Absolutely. And to keep
improving that safety net, the community maintains
this massive, constantly updated library called
the TPTP, the Thousands of Problems for Theorem
Provers. Library of problems. Yeah. It provides
a standardized battleground for these systems.
There are even regular competitions held at the
KD conference where different automated theorem
provers from all around the world compete to
see which system can solve the most complex logic
puzzles from the TPTP library. It is literally
the Olympics for artificial logic. It really
is. I love that so much. An international competition
to build a sharper, faster truth machine. As
we wrap up this deep dive, want to turn it over
to you for the final word, looking at this entire
journey from the 1950s to today. What is the
biggest takeaway the listener should be chewing
on? You know, I keep going back to that initial
shock in 1956 with the logic theorist. Yeah,
that was wild. The machine found a more elegant
mathematical proof than Whitehead and Russell,
and it did it by mechanically applying logical
rules without using a single drop of human intuition.
Right. If a machine can create a more elegant
truth through sheer mechanical deduction, it
really forces us to ask a rather uncomfortable
question. Which is? Is human intuition actually
the magical, irreplaceable source of creativity
we think? it is? Or is intuition just a biological
shortcut, a messy adaptation our brains use simply
because we lack the processing power to run the
formal logic? Our greatest aha moments just workarounds
for our biological hardware limits. That is a
wild thought to leave on. Thank you so much for
joining us on this journey into the invisible
logic running our world. Keep questioning your
own assumptions. Keep checking those ice crystals.
We will catch you on the next Deep Dive.