Non-State Hostile Intelligence Service & Maxwell Silver Hammer

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Back In The USSR
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Maxwell Silver Hammer
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Unexplainability and Incomprehensibility of Artificial Intelligence
Roman V. Yampolskiy

Computer Engineering and Computer Science
University of Louisville
roman.ya...@louisville.edu, @romanyam
June 20, 2019

"If a lion could speak, we couldn't understand him"
Ludwig Wittgenstein

“It would be possible to describe everything scientifically, but it would make no sense. It would
be a description without meaning - as if you described a Beethoven symphony as a variation of
wave pressure.”
Albert Einstein

“Some things in life are too complicated to explain in any language. … Not just to explain to
others but to explain to yourself. Force yourself to try to explain it and you create lies.”
Haruki Murakami

“I understand that you don’t understand”
Grigori Perelman


Abstract
Explainability and comprehensibility of AI are important requirements for intelligent systems
deployed in real-world domains. Users want and frequently need to understand how decisions
impacting them are made. Similarly it is important to understand how an intelligent system
functions for safety and security reasons. In this paper, we describe two complementary
impossibility results (Unexplainability and Incomprehensibility), essentially showing that
advanced AIs would not be able to accurately explain some of their decisions and for the decisions
they could explain people would not understand some of those explanations.
Keywords: AI Safety, Black Box, Comprehensible, Explainable AI, Impossibility, Intelligible,
Interpretability, Transparency, Understandable, Unserveyability.

1. Introduction
For decades AI projects relied on human expertise, distilled by knowledge engineers, and were
both explicitly designed and easily understood by people. For example, expert systems, frequently
based on decision trees, are perfect models of human decision making and so are naturally
understandable by both developers and end-users. With paradigm shift in the leading AI
methodology, over the last decade, to machine learning systems based on Deep Neural Networks
(DNN) this natural ease of understanding got sacrificed. The current systems are seen as “black
boxes” (not to be confused with AI boxing [1, 2]), opaque to human understanding but extremely
capable both with respect to results and learning of new domains. As long as Big Data and Huge
Compute are available, zero human knowledge is required [3] to achieve superhuman [4]
performance.

With their new found capabilities DNN-based AI systems are tasked with making decisions in
employment [5], admissions [6], investing [7], matching [8], diversity [9], security [10, 11],
recommendations [12], banking [13], and countless other critical domains. As many such domains
are legally regulated, it is a desirable property and frequently a requirement [14, 15] that such
systems should be able to explain how they arrived at their decisions, particularly to show that they
are bias free [16]. Additionally, and perhaps even more importantly to make artificially intelligent
systems safe and secure [17] it is essential that we understand what they are doing and why. A
particular area of interest in AI Safety [18-25] is predicting and explaining causes of AI failures
[26].

A significant amount of research [27-41] is now being devoted to developing explainable AI. In
the next section we review some main results and general trends relevant to this paper.

2. Literature Review
Hundreds of papers have been published on eXplainable Artificial Intelligence (XAI) [42].
According to DARPA [27], XAI is supposed to “produce more explainable models, while
maintaining a high level of learning performance … and enable human users to understand,
appropriately, trust, and effectively manage the emerging generation of artificially intelligent
partners”. Detailed analysis of literature on explainability or comprehensibility is beyond the scope
of this paper, but the readers are encouraged to look at many excellent surveys of the topic [43-
45]. Miller [46] surveys social sciences to understand how people explain, in the hopes of
transferring that knowledge to XAI, but of course people often say: “I can’t explain it” or “I don’t
understand”. For example, most people are unable to explain how they recognize faces, a problem
we frequently ask computers to solve [47, 48].

Despite wealth of publications on XAI and related concepts [49-51], the subject of unexplainability
or incomprehensibility of AI is only implicitly addressed. Some limitations of explainability are
discussed: “ML algorithms intrinsically consider high-degree interactions between input features,
which make disaggregating such functions into human understandable form difficult. … While a
single linear transformation may be interpreted by looking at the weights from the input features
to each of the output classes, multiple layers with non-linear interactions at every layer imply
disentangling a super complicated nested structure which is a difficult task and potentially even a
questionable one [52]. … As mentioned before, given the complicated structure of ML models,
for the same set of input variables and prediction targets, complex machine learning algorithms
can produce multiple accurate models by taking very similar but not the same internal pathway in
the network, so details of explanations can also change across multiple accurate models. This
systematic instability makes automated generated explanations difficult.” [42].

Sutcliffe et al. talk about incomprehensible theorems [53]: “Comprehensibility estimates the effort
required for a user to understand the theorem. Theorems with many or deeply nested structures
may be considered incomprehensible.” Muggleton et al. [54] suggest “using inspection time as a
proxy for incomprehension. That is, we might expect that humans take a long time … in the case
they find the program hard to understand. As a proxy, inspection time is easier to measure than
comprehension.”

The tradeoff between explainability and comprehensibility is recognized [52], but is not taken to
its logical conclusion. “[A]ccuracy generally requires more complex prediction methods [but]
simple and interpretable functions do not make the most accurate predictors'' [55]. “Indeed, there
are algorithms that are more interpretable than others are, and there is often a tradeoff between
accuracy and interpretability: the most accurate AI/ML models usually are not very explainable
(for example, deep neural nets, boosted trees, random forests, and support vector machines), and
the most interpretable models usually are less accurate (for example, linear or logistic regression).”
[42].

Incomprehensibility is supported by well-known impossibility results. Charlesworth proved his
Comprehensibility theorem while attempting to formalize the answer to such questions as: “If [full
human-level intelligence] software can exist, could humans understand it?” [56]. While describing
implications of his theorem on AI, he writes [57]: “Comprehensibility Theorem is the first
mathematical theorem implying the impossibility of any AI agent or natural agent—including a
not-necessarily infallible human agent—satisfying a rigorous and deductive interpretation of the
self-comprehensibility challenge. … Self-comprehensibility in some form might be essential for a
kind of self-reflection useful for self-improvement that might enable some agents to increase their
success.” It is reasonable to conclude that a system which doesn’t comprehend itself would not be
able to explain itself.
Hernandez-Orallo et al. introduce the notion of K-incomprehensibility (a.k.a. K-hardness) [58].

“This will be the formal counterpart to our notion of hard-to-learn good explanations. In our sense,
a k-incomprehensible string with a high k (difficult to comprehend) is different (harder) than a k-
compressible string (difficult to learn) [59] and different from classical computational complexity
(slow to compute). Calculating the value of k for a given string is not computable in general.
Fortunately, the converse, i.e., given an arbitrary k, calculating whether a string is k-
comprehensible is computable. … Kolmogorov Complexity measures the amount of information
but not the complexity to understand them.” [58].

Yampolskiy addresses limits of understanding other agents in his work on the space of possible
minds [60]: “Each mind design corresponds to an integer and so is finite, but since the number of
minds is infinite some have a much greater number of states compared to others. This property
holds for all minds. Consequently, since a human mind has only a finite number of possible states,
there are minds which can never be fully understood by a human mind as such mind designs have
a much greater number of states, making their understanding impossible as can be demonstrated
by the pigeonhole principle.” Hibbard points out safety impact from incomprehensibility of AI:
“Given the incomprehensibility of their thoughts, we will not be able to sort out the effect of any
conflicts they have between their own interests and ours.”

We are slowly starting to realize that as AIs become more powerful, the models behind their
success will become ever less comprehensible to us [61]: “… deep learning that produces
outcomes based on so many different variables under so many different conditions being
transformed by so many layers of neural networks that humans simply cannot comprehend the
model the computer has built for itself. … Clearly our computers have surpassed us in their power
to discriminate, find patterns, and draw conclusions. That’s one reason we use them. Rather than
reducing phenomena to fit a relatively simple model, we can now let our computers make models
as big as they need to. But this also seems to mean that what we know depends upon the output of
machines the functioning of which we cannot follow, explain, or understand. … But some of the
new models are incomprehensible. They can exist only in the weights of countless digital triggers
networked together and feeding successive layers of networked, weighted triggers representing
huge quantities of variables that affect one another in ways so particular that we cannot derive
general principles from them.”

“Now our machines are letting us see that even if the rules are simple, elegant, beautiful and
rational, the domain they govern is so granular, so intricate, so interrelated, with everything
causing everything else all at once and forever, that our brains and our knowledge cannot begin to
comprehend it. … Our new reliance on inscrutable models as the source of the justification of our
beliefs puts us in an odd position. If knowledge includes the justification of our beliefs, then
knowledge cannot be a class of mental content, because the justification now consists of models
that exist in machines, models that human mentality cannot comprehend. … But the promise of
machine learning is that there are times when the machine’s inscrutable models will be far more
predictive than the manually constructed, human-intelligible ones. In those cases, our
knowledge — if we choose to use it — will depend on justifications that we simply cannot
understand. … [W]e are likely to continue to rely ever more heavily on justifications that we
simply cannot fathom. And the issue is not simply that we cannot fathom them, the way a lay
person can’t fathom a string theorist’s ideas. Rather, it’s that the nature of computer-based
justification is not at all like human justification. It is alien.” [61].

3. Unexplainability
A number of impossibility results are well-known in many areas of research [62-70] and some are
starting to be discovered in the domain of AI research, for example: Unverifiability [71],
Unpredictability1 [72] and limits on preference deduction [73] or alignment [74]. In this section
we introduce Unexplainability of AI and show that some decisions of superintelligent systems will
never be explainable, even in principle. We will concentrate on the most interesting case, a
superintelligent AI acting in novel and unrestricted domains. Simple cases of Narrow AIs making
decisions in restricted domains (Ex. Tic-Tac-Toe) are both explainable and comprehensible.
Consequently a whole spectrum of AIs can be developed from completely
explainable/comprehensible to completely unexplainable/incomprehensible. We define
Unexplainability as impossibility of providing an explanation for certain decisions made by an
intelligent system which is both 100% accurate and comprehensible.


1 Unpredictability is not the same as Unexplainability or Incomprehensibility, see ref. 72. Yampolskiy, R.V.,
Unpredictability of AI. arXiv preprint arXiv:1905.13053, 2019. for details.
Artificial Deep Neural Networks continue increasing in size and may already comprise millions
of neurons, thousands of layers and billions of connecting weights, ultimately targeting and
perhaps surpassing the size of the human brain. They are trained on Big Data from which million
feature vectors are extracted and on which decisions are based, with each feature contributing to
the decision in proportion to a set of weights. To explain such a decision, which relies on literally
billions of contributing factors, AI has to either simplify the explanation and so make the
explanation less accurate/specific/detailed or to report it exactly but such an explanation elucidates
nothing by virtue of its semantic complexity, large size and abstract data representation. Such
precise reporting is just a copy of trained DNN model.

For example, an AI utilized in the mortgage industry may look at an application to decide credit
worthiness of a person in order to approve them for a loan. For simplicity, let’s say the system
looks at only a hundred descriptors of the applicant and utilizes a neural network to arrive at a
binary approval decision. An explanation which included all hundred features and weights of the
neural network would not be very useful, so the system may instead select one of two most
important features and explain its decision with respect to just those top properties, ignoring the
rest. This highly simplified explanation would not be accurate as the other 98 features all
contributed to the decision and if only one or two top features were considered the decision could
have been different. This is similar to how Principal Component Analysis works for dimensionality
reduction [75].

Even if the agent trying to get the explanation is not a human but another AI the problem remains
as the explanation is either inaccurate or agent-encoding specific. Trained model could be copied
to another neural network, but it would likewise have a hard time explaining its decisions.
Superintelligent systems not based on DNN would face similar problems as their decision
complexity would be comparable to those based on neural networks and would not permit
production of efficient and accurate explanations. The problem persists in the case of self-
referential analysis, where a system may not understand how it is making a particular decision.
Any decision made by the AI is a function of some input data and is completely derived from the
code/model of the AI, but to make it useful an explanation has to be simpler than just presentation
of the complete model while retaining all relevant, to the decision, information. We can reduce
this problem of explaining to the problem of lossless compression [76]. Any possible decision
derived from data/model can be represented by an integer encoding such data/model combination
and it is a proven fact that some random integers can’t be compressed without loss of information
due to the Counting argument [77]. “The pigeonhole principle prohibits a bijection between the
collection of sequences of length N and any subset of the collection of sequences of length N - 1.
Therefore, it is not possible to produce a lossless algorithm that reduces the size of every possible
input sequence.”2 To avoid this problem, an AI could try to produce decisions, which it knows are
explainable/compressible, but that means that it is not making the best decision with regards to the
given problem, doing so is suboptimal and may have safety consequences and so should be
discouraged.

2 https://en.wikipedia.org/wiki/Lossless_compression

Overall, we should not be surprised by the challenges faced by Artificial Neural Networks
attempting to explain their decision, as they are modeled on Natural Neural Networks of human
beings and people are also “black boxes” as illustrated by a number of split brain experiments [78].
In such experiments it is frequently demonstrated that people simply make up explanations for
their actions after the decision has already been made. Even to ourselves, we rationalize our
decision after the fact and don’t become aware of our decisions or how we made them until after
they been made unconsciously [79]. People are notoriously bad at explaining certain decisions
such as how they recognize faces or what makes them attracted to a particular person. These
reported limitations in biological agents support idea that unexplainability is a universal
impossibility result impacting all sufficiently complex intelligences.

4. Incomprehensibility
A complimentary concept to Unexplainability, Incomprehensibility of AI address capacity of
people to completely understand an explanation provided by an AI or superintelligence. We define
Incomprehensibility as an impossibility of completely understanding any 100% - accurate
explanation for certain decisions of intelligent system, by any human.
Artificially intelligent systems are designed to make good decision in their domains of deployment.

Optimality of the decision with respect to available information and computational resources is
what we expect from a successful and highly intelligent systems. An explanation of the decision,
in its ideal form, is a proof of correctness of the decision. (For example, a superintelligent chess
playing system may explain why it sacrificed a queen by showing that it forces a checkmate in 12
moves, and by doing so proving correctness of its decision.) As decisions and their proofs can be
arbitrarily complex impossibility results native to mathematical proofs become applicable to
explanations. For example, explanations may be too long to be surveyed [80, 81]
(Unserveyability), Unverifiable [71] or too complex to be understood [82] making the explanation
incomprehensible to the user. Any AI, including black box neural networks can in principle be
converted to a large decision tree of nothing but “if” statements, but it will only make it human-
readable not human-understandable.

It is generally accepted that in order to understand certain information a person has to have a
particular level of cognitive ability. This is the reason students are required to take standardized
exams such as SAT, ACT, GRE, MCAT or LCAT, etc. and score at a particular percentile in order
to be admitted to their desired program of study at a selective university. Other, but similar tests
are given to those wishing to join the military or government service. All such exams indirectly
measure person’s IQ (Intelligence Quotient) [83, 84] but vary significantly in how closely they
correlate with standard IQ test scores (g-factor loading). The more demanding the program of
study (even at the same university), the higher cognitive ability is expected from students. For
example, average quantitative GRE score of students targeting mathematical sciences is 163, while
average quantitative score for students interested in studying history is 1483. The trend may be
reversed for verbal scores.

3 https://www.prepscholar.com/gre/blog/average-gre-scores/

People often find themselves in situations where they have to explain concepts across a significant
communication range [85] for example to children or to people with mental challenges. The only
available option in such cases is to provide a greatly oversimplified version of the explanation or
a completely irrelevant but simple explanation (a lie). In fact the situation is so common we even
have a toolbox of common “explanations” for particular situations. For example, if a five-year old
asks: “Where do babies come from?” They are likely to hear something like “A seed from the
daddy and an egg from the mommy join together in the mom's tummy”4, instead of a talk about
DNA, fertilization and womb. A younger child may learn that the “stork brings them” or “they
come from a baby store”. Alternatively, an overly technical answer could be provided to confuse
the child into thinking they got an answer, but with zero chance of them understanding such
overcomplicated response. Overall, usefulness of an explanation is relative to the person who is
trying to comprehend it. The same explanation may be comprehended by one person, and
completely misunderstood by another.

4 https://www.babycenter.com/0_how-to-talk-to-your-child-about-sex-age-5_67112.bc

There is a similar and perhaps larger intelligence gap between superintelligence and adult humans,
making the communication range unsurmountable. It is likely easier for a scientist to explain
quantum physics to a mentally challenged deaf and mute four-year-old raised by wolves then for
superintelligence to explain some of its decisions to the smartest human. We are simply not smart
enough to understand certain concepts. Yampolskiy proposed [82] a complexity measure which is
based on the minimum intelligence necessary to understand or develop a particular algorithm, and
while it takes less intelligence to just understand rather than create both requirements could be well
above IQ of the smartest human. In fact it could be very hard to explain advanced concepts to even
slightly less intelligent agents.

We can predict a certain complexity barrier to human understanding for any concept for which
relative IQ of above 250 would be necessary, as no person has ever tested so high. In practice the
barrier may be much lower, as average IQ is just 100 and additional complication from limited
memory and limited attention spans can place even relative easy concepts outside of human grasp.
To paraphrase Wittgenstein: if superintelligence explained itself we would not understand it.
Given that research on deception by AI is well established [86] it would not be difficult for
advanced AIs to provide highly believable lies to their human users. In fact such explanations can
be designed to take advantage of AI’s knowledge of the human behavior [87, 88] and mental model
[89, 90], and manipulate users beyond just convincing them that explanation is legitimate [91]. AI
would be able to target explanations to the mental capacity of particular people, perhaps taking
advantage of their inherent limitations. It would be a significant safety issue, and it is surprising to
see some proposals for using human users as targets of competing (adversarial) explanations from
AIs [92].

Incomprehensibility results are well-known for different members of Chomsky hierarchy [93] with
finite state automation unable to recognize context-free languages, pushdown automata unable to
recognize context-sensitive languages and linear-bounded non-deterministic Turing machines
unable to recognize recursively enumerable languages. Simpler machines can’t recognize
languages which more complex machines can recognize.
While people are frequently equated with unrestricted Turing machines via Church-Turing thesis
[94], Blum et al. formalize human computation, in practice, as a much more restricted class [95].
However, Turing machines are not an upper limit on what is theoretically computable as described
by different hypercomputation models [96]. Even if our advanced AIs (superintelligence), fail to
achieve true hypercomputation capacity, for all practical purposes and compared to the human
computational capabilities they would be outside of what human-equivalent agents can
recognize/comprehend.

Superintelligence would be a different type of computation, far superior to humans in practice. It
is obviously not the case that superintelligent machines would actually have infinite memories or
speeds but they would appear to act as they do to unaugmented humans. For example a machine
capable of remembering one trillion items vs seven, in short-term memory of people, would appear
to have infinite capacity to memorize. In algorithmic complexity theory some algorithms become
the most efficient for a particular problem type on inputs so large as to be unusable in practice, but
such inputs are nonetheless finite [97]. So, just like a finite state automata can’t recognize
recursively enumerable languages, so will people fail in practice to comprehend some explanations
produced by superintelligent systems, they are simply not in the same class of automata, even if
theoretically, given infinite time, they are.

Additionally, decisions made by AI could be mapped onto the space of mathematical conjectures
about the natural numbers. An explanation for why a particular mathematical conjecture is true or
false would be equivalent to a proof (for that conjecture). However, due to Gödel's First
Incompleteness Theorem we know that some true conjectures are unprovable. As we have mapped
decision on conjectures and explanations on proofs, that means that some decision made by AI are
fundamentally unexplainable/incomprehensible. Explanations as proofs would be subject to all the
other limitations known about proofs, including Unserveyability, Unverifiability and
Undefinability [98, 99]. Finally, it is important to note that we are not saying that such
decision/conjecture reduction would preserve semantics of the subject, just that it is a useful tool
for showing impossibility of explainability/comprehensibility in some cases.

5. Conclusions
The issues described in this paper can be seen as a communication problem between AI encoding
and sending information (sender) and person receiving and decoding information (receiver).
Efficient encoding and decoding of complex symbolic information is difficult enough, as described
by Shannon’s Information Theory [100], but with Explainability and Comprehensibility of AI we
also have to worry about complexity of semantic communication [101]. Explainability and
Comprehensibility are another conjugate pair [71, 102] in the domain of AI safety. The more
accurate is the explanation the less comprehensible it is, and vice versa, the more comprehensible
the explanation the less accurate it is. A non-trivial explanation can’t be both accurate and
understandable, but it can be inaccurate and comprehensible. There is a huge difference between
understanding something and almost understanding it. Incomprehensibility is a general result
applicable to many domains including science, social interactions, etc. depending on a mental
capacity of a participating person(s).

Human being are finite in our abilities. For example our short term memory is about 7 units on
average. In contrast, an AI can remember billions of items and their capacity to do so grows
exponentially, while never infinite in a true mathematical sense, machine capabilities can be
considered such in comparison to ours. This is true for memory, compute speed and
communication abilities. Hence the famous: Finitum Non Capax Infiniti (The finite cannot contain
the infinite) is highly applicable to understand the incomprehensibility of the god-like [103]
superintelligent AIs.

Shown impossibility results present a number of problems for AI Safety. Evaluation and
debugging of intelligent systems becomes much harder if their decisions are
unexplainable/incomprehensible. In particular, in case of AI failures [104] accurate explanations
are necessary to understand the problem and reduce likelihood of future accidents. If all we have
is a “black box” it is impossible to understand causes of failure and improve system safety.
Additionally, if we grow accustomed to accepting AI’s answers without an explanation, essentially
treating it as an Oracle system, we would not be able to tell if it begins providing wrong or
manipulative answers.



Acknowledgments
The author is grateful to Elon Musk and the Future of Life Institute, and to Jaan Tallinn and
Effective Altruism Ventures for partially funding his work on AI Safety. It is obvious that the
subject described in this paper is accurately explained and completely comprehensible.


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[chris rodgers]

> 104. Yampolskiy, R.V. and M. Spellchecker, Artificial Intelligence Safety and Cybersecurity: a
> Timeline of AI Failures. arXiv preprint arXiv:1610.07997, 2016.

hot rod charlie runs tomorrow
last race of his career

and joe biden should tap out
and Harris? she can sell hot dogs in front of the White House

[chris rodgers]

poor hot rod charlie got beat yesterday.
didn't even finish 'in the money'. oh well.
they had some fast horses in that race.

[LowRider44M]

> > hot rod charlie runs tomorrow
> > last race of his career
> >
> > and joe biden should tap out
> > and Harris? she can sell hot dogs in front of the White House
> poor hot rod charlie got beat yesterday.
> didn't even finish 'in the money'. oh well.
> they had some fast horses in that race.

I'd bet a 20 on this dog!
https://twitter.com/i/status/1589242912109301760