Effective AI Makes Peace with the Paradox
“This sentence is false.” Is the statement true? If it’s true, then the sentence is false. If the statement is false, then the sentence must be true. Both are contradictions. This is referred to as the Liar’s Paradox, and you may already be familiar with it in different forms.
A fun example comes from the original Star Trek TV series in which the Liar’s Paradox is presented to outwit an android. The endless hall-of-mirrors of contradictions that emerge result in smoke pouring from the hapless android’s ears. Artificial Intelligence (AI)’s initially threatening presence is revealed to be an artifice over a brittle core—humanity reigns supreme.
As humans, contradictions aren’t deal-breakers. We can brush our concerns under the rug and “get on with life.” But real-world autonomous machines cannot. The black-and-white absolutes of logic lead to suboptimal performances of real-world autonomous machines.
Watch the blooper-reel below of the 2015 DARPA humanoid robot challenge. When a robot attempts to open a door or valve without properly aligning itself to the handle before attempting to turn, the intelligent machine becomes a crash-test dummy. This is an example of an “open-loop” control system. A system commits absolutely to its current world model without the ability to adapt to changed conditions. It fails to admit confusion and thereby fails. We need to engineer systems that can gracefully handle confusion and uncertainty in order to successfully adapt. It’s something I like to call “salsa dancing with confusion,” in reference to a moment from the film Waking Life.
Statisticians and control theorists will likely argue that autonomous systems do indeed accommodate error and uncertainty as first principles. Closed-loop control explicitly minimizes and rejects errors in a system state relative to an ideal state. They use probability as a model for uncertainty to achieve success in the presence of noise.
But this argument has a deeply hidden flaw. Statistics and probability theory, like much of modern mathematics, is based on Zermelo-Fraenkel set theory. This was developed with the explicit intent to eliminate the Liar’s Paradox—confusion and contradiction—and create a walled-garden of absolutes and certainty. For example, a flipped coin may land heads or tails. The probability of the event that a coin is simultaneously both heads and tails is 0—that outcome is rejected as impossible.
To create effective AI technology, we must begin to reject these absolutes and embrace ambiguous outcomes. We should “salsa” with confusion—accept that something can be both true and untrue, to some degree, at the same time.
Foundations of AI: Perception & Measurement
Thus, the grand challenge of robotics and artificial intelligence systems is for them to operate successfully in an open world. This world doesn’t have absolutes. It is unbounded, unmodeled, and uncertain. Success in an environment beyond the scope of explicit mathematical models depends on perception, measurement, and accurately understanding environment and operational context.
Consider a self-driving car approaching an intersection with a traffic light. Safe driving depends on its ability to distinguish between the colors yellow and red. Using Bayes rule, we can infer the probability of the light being red or yellow given the color that is perceived—the perceptual stimulus. Driving decisions can be tied to the “maximum likelihood” light color.
Again, I argue that this method is flawed. Do we really believe that the redness of a light is a random variable conditioned only our observations? The world is ambiguous, and models must not only reflect that, but take the level of ambiguity present into account. It is meaningless to consider the probability of a light being red given our observations. What happens if the probability model is wrong? For example, if the sun is setting behind the traffic light an atypical perceptual stimulus is created. This could invalidate the probability model and perhaps making a decision based on yellow being most likely when it is truly red, so we best make sure we know when we aren’t certain before taking decisive action.
In the context of machine learning (ML), we can reflect on the success of regularization, which improves performance on inputs for which the system was not trained. It contradicts overfitting—a model that internalizes so much of the controlled experimental inputs and outputs that there is no room for deviation from the specifics of past observations. Regularization discards the drive for an exact, complete, and correct model. It instead favors an approximation which is more effective than the “exact” model.
Here’s a provocative alternative—rather than modeling the objective, external world, let’s consider the subjective experiences of autonomous systems when designing real-world systems. Rather than taking action probability of the “redness” or “yellowness” of a light, a vehicle can take action based on its internal belief of the state of the light. Control decisions could be taken based on levels of uncertainty and towards reducing uncertainty—leading to much more graceful and adaptive systems. However, to get there, we require special tools and careful reconsiderations of machine learning fundamentals.
Machine Learning’s Fuzzy Logic
In order to fully embrace uncertainty, belief, and disbelief, we need a toolkit that models ambiguity from its very core. Probability theory isn’t sufficient because of its baked-in reliance on Zermelo-Fraenkel set theory. There are rich, formal logics called “fuzzy logic” or “multi-valued logic” that can model degrees of belief in logical propositions. They allow us to reasonably state “I can both believe the traffic light is red and that the traffic light is yellow, to degrees. With more evidence, perhaps I can be more certain.”
Similar to probability, belief in the truth and falsehood of a logical proposition is modeled in the range . It defines a set of operations on these values that is distinct from probability theory. Unlike traditional logical operations, however, that may be defined algebraically including Union, Intersection, and Implication, fuzzy logic values are computed arithmetically over .
Inference is handled through modus ponens, where combining with allows us to deduce belief in the truth of from our belief in . Think about how intuitive this inferential form is as compared with Bayes rule. There are no priors. There is no conditioning process. There is only our subjective belief and a rigorous algebraic system in which to combine evidence. I think I have seen “X” and I know “If X then Y,” so I can be fairly certain “Y” is true.
One type of fuzzy logic called Łukasiewicz logic defines the following operations:
- Strong Conjunction (Intersection, multiplication):
- Strong Disjunction (Union, addition):
- Weak Conjunction (Meet):
- Weak Disjunction (Join):
These operations allow us to arithmetically combine and evaluate logical formulae. Logical relations such as De Morgan’s laws hold.
This sort of reasoning enables a far more efficient AI in an open world—a key that will unlock a new wave of reliable autonomous systems.
Ambiguity as an Asset in AI
ML and AI systems can be brittle and unreliable. We have come to expect computerized systems to be rigid in their interpretations and unable to break out of their pre-programmed assumptions. We expect systems to be logical, to a fault. However, new directions in ML and AI point towards systems that can handle uncertainty with grace—systems that can “salsa with their confusion” rather than fail and fall flat. The flexibility allowed by fuzzy logic means that there is no paradox, there is only ambiguity. And that ambiguity is itself a useful measurement.
These concepts have sociopolitical ramifications too. Can we learn from fuzzy logic ourselves, and hold ideas apparently in conflict in some balance of truth and falsehood? Can we break away from the rigid cultural binaries we’ve imposed upon ourselves to find a more graceful and nuanced coexistence? Can we rethink not only machine learning but human learning and understanding?