It is known that a consistent application of quantization leads to the emergence of all logically possible physical models. This idea -- that logic underlies physics -- is known as "it from bit". Traditional applications of this idea are based on the 2-valued logic. In this paper, we show that to get a more adequate picture, we need to use ideas behind intuitionistic fuzzy approach. In the first qualitative approximation, this approach leads to a logic that can be naturally interpreted in color terms -- and for which related computations can be performed by color optical computing. The simplest quantitative approximation to this approach leads to a possible physically and logically meaningful explanation of the fact that the smallest space-time dimension for which a consistent quantum field theory is possible is 10 or 11.

Recently, an AI helped to find a counterexample to a long-standing mathematical problem: Jacobian Conjecture about invertibility of polynomial mappings. The problem that AIs in general -- and in this case in particular -- do not provide any explanations for their answers. It is therefore desirable to provide some explanations for this counterexample. In this paper, we show that some simple properties explain the set of monomials used in this counterexample. How to explain the coefficients at these monomials is still an open problem.

Reliable uncertainty estimation is essential for trustworthy artificial intelligence, particularly in safety-critical applications such as medical imaging and clinical decision support. Interval neural networks provide mathematically guaranteed output bounds by propagating interval-valued inputs through neural network layers. However, interval propagation often suffers from severe overestimation, especially when evaluating nonlinear activation functions, causing uncertainty bounds to widen rapidly as information propagates through the network.

This work investigates certified polynomial approximations and partitioning as a mechanism for reducing interval overestimation. Given an activation function over a bounded domain, minimax polynomial approximations are generated using the \texttt{Sollya} framework together with certified uniform error bounds. These approximations produce guaranteed interval enclosures while avoiding direct interval evaluation of nonlinear activation functions.

To further improve enclosure tightness, the input domain is partitioned into local subintervals and independent certified polynomial approximations are constructed for each partition. Preliminary experiments on the $\tanh$ and sigmoid activation functions demonstrate substantially tighter interval enclosures than global polynomial approximations while preserving mathematical guarantees.\\ Although the present study focuses on activation functions, the proposed methodology constitutes a fundamental building block toward reliable uncertainty propagation in interval neural networks for trustworthy clinical artificial intelligence.

It has been recently shown that, similarly to the fact that every Boolean operation can be performed if we only have gates of a single type -- NAND or NOR -- many elementary functions can be obtained by using a constant 1 and a single function eml(x,y) = exp(x) − ln(y). This function is known as Exp-Minus-Log, eml, for short. There are several other functions with the same property, but among them, eml is the simplest. In this paper, we show that, in some reasonable sense, eml is optimal. To prove this, we use symmetry-based approach and its relation to optimization.

Data Envelopment Analysis (DEA) describes how relatively efficient a production facility can be under the most beneficial combination of prices for its resources and for what it produces. To estimate this efficiency, the original DEA uses the exact amounts of resources used and items produced -- e.g., the overall amounts during some period of time. However, the resulting efficiency value does not take into account that in real life, in general, a facility's efficiency fluctuates. To capture these fluctuations, it is desirable to produce not a single efficiency value, but rather an interval of possible values of efficiency. Such interval-valued version of DEA has indeed been proposed and used. In this paper, we argue that while in some cases, the current interval DEA adequately describes these fluctuations, in some other reasonable situations, this method overestimates the fluctuations range. To overcome this limitation, we propose a more economically adequate interval version of DEA.

In many practical problems, there is a need for faster computations. One of such problems is designing an AI that would constantly re-train itself, just like a human brain does. Because physics restricts the speed of all communications to the speed of light, to drastically speed up computations, we need to drastically decrease the size of all computational units. Up to now, most actual and proposed ways to do it was to use only a part of what formed the previous computational cell: a few molecules instead of the whole cell, and atoms, elementary particles, and quarks instead of a few molecules. However, since quarks do not have sub-parts, we cannot use this approach to go faster than quark computing. A hope to go faster comes from the well-known problem of fundamental physics: if we naively compute such simple things as the overall energy of an electron, we get meaningless infinities. Physicists overcome this problem by applying a special mathematical trick called renormalization -- but it is definitely desirable to come up with a more physically meaningful solution. In this paper, we show that a natural physical solution implies the existence of special particles whose electric charge is much smaller than the electric charge of an electron. Due to the smallness of their electric charge, they have very little effect on the usual matter -- while their mass contributes to the overall mass of the Universe. From this viewpoint, they are perfect candidates for mysterious dark matter -- and indeed, this is where they were originally proposed, under the name of millicharged particles. These particles are much smaller than quarks, so the use of these particles can potentially enable us to further speed up computations even more than quark computing.

An important part of Computer Science education is learning theoretical foundations of computing. This helps students better understand which problems are solvable and which are not, what tools they need to solve problems which are solvable, how to best implement these solutions -- and also helps them to get a general understand how computers translate our instructions -- i.e., our code -- into efficient step-by-step computations. The corresponding Automata class is often difficult to students, because many concepts, ideas, algorithms, and proofs do not come naturally from the formulation of the corresponding problems: these ideas, algorithms, and proofs have come from the insights of founders of this field. This situation is similar to the need for make AI more explainable: we have interesting and often very effective ideas and results generated by AI, but it is difficult for us to understand them -- since AI usually does not provide us with an explanation of how it came up with these ideas. In this paper, on the example of several basic Automata concepts and proofs, we show that the use of invariances can help to make many such ideas more natural.

For effective human-AI collaboration, systems need to be able to communicate effectively with their partner, including through use of appropriate prosody. While this will ultimately require detailed modeling of the user and their actions relative to shared goals, we here explored a possible shortcut: conditioning the prosody directly on an easily observable aspect of the current situation. Specifically, we investigated whether the pitch height of utterances could be predicted directly from the players' positions in a navigation-with-puzzles game. We developed a pipeline to extract, align, and visualize the relation between pitch and player position through heatmaps. There were some apparent spatial patterns. However, when we built a predictor and evaluated it in a leave-one-out experiment, spatial position alone was not a consistent predictor of pitch. This confirms that prosodic behavior is influenced by many factors beyond location.

It is well known that LLMs' hallucinations are a big problem. Usually, we detect hallucinations by noticing that the LLM's answer is inconsistent with some well-established rules or facts. From this viewpoint, a natural way to decrease hallucinations is as follows: every time we have new pairs (input, output) on which to train, we should also again train on the pairs (input, output) corresponding to well-established facts and rules.

To arrange a proper coordination between several mobile devices flying together, it makes sense to analyze how such a coordination is arranged in nature, e.g., in a bird flock. In this paper, we analyze to what extent recent discoveries about the bird flocks can be applied to mobile devices. For this purpose, it is necessary: (1) to make conclusions from these discoveries more convincing, and also (2) to explain why these conclusions can be applied to general flying devices, not only to birds. This is what we do in this paper: (1) we show that additional information from these discoveries provides an additional confirmation that birds in a flock only take into account location of nearest neighbors, and (2) we also explain why the discoveries-based conclusions are applicable not only to birds, but to mobile devices as well. We also explain the relation between these discoveries, seven plus minus two law from psychology, and color optical computing.

Experiments have shown that as we get closer and closer to an object, our perception goes through several distinct stages. In his groundbreaking book "Cybernetics", Norbert Wiener provided a theoretical explanation of this phenomenon, explanation based on the notion of invariance. In this paper, we show that similar ideas can explain empirical success of many uncertainty-related : use of polytopes and ellipsoids in describing uncertainty, topological data processing, topological quantum computing, fuzzy techniques, and color optical computing.

In several practical situations, it is effective to use color optical computing, where a signal is represented by a combination of basic colors. This technique does not require accurate maintenance of intensity and thus, require much fewer resources than other computational techniques. This makes it very appropriate for mobile devices, devices that need to operate with limited resources. However, it is somewhat surprising that this technique -- that only uses a few features of the signal -- is very empirically successful. In this paper, we provide a theoretical explanation for this empirical success, an explanation based on the general theoretical results about computational efficiency. This general explanation also helps to explain several difficult-to-explain features of human reasoning and of physics, such as seven plus minus two law and homogeneity of the Universe. The physics-related analysis bring us to back to color optical computing -- namely, how its ideas can be used in computational devices of the distant future.

When we design control systems, it is beneficial to take into account expert knowledge. An important part of expert knowledge is formulated by using imprecise ("fuzzy") words from natural language like "small". We need to translate such expert statements into computer-understandable terms. Techniques for such translation are known as fuzzy techniques. When we use these techniques to process fuzzy inputs, we get fuzzy conclusions about the recommended control. In automatic control, we need to transform this fuzzy conclusion into a single control value that the system should apply. Such a transformation is known as defuzzification. At present, the most widely used defuzzification is so-called centroid defuzzification. It has led to many successful applications, but it has a limitation. This limitation is related to the fact that experts are not perfect, they sometimes make mistakes. With centroid defizzification, even a single expert mistake can drastically affect the resulting control. It is therefore desirable to come up with a defuzzification procedure that is as robust as possible against such mistakes. In this paper, we describe such most robust defuzzification: it turns out to be a modification of centroid defuzzification that is based on the median instead of the mean.

Many traditional statistical methods implicitly assume that we know the actual probability distributions. However, based on a finite sample, we can only provide approximate values of the actual probabilities -- and, with a certain confidence level, bounds of these probabilities. In particular, instead of the actual cumulative distribution function, we only have bounds on its values. This is known as a p-box. Another important case when p-boxes appear is the case of chaotic dynamics, when we cannot exactly predict the probabilities of different future states, we can only predict ranges of these probabilities. In this paper, we show how to propagate such p-box uncertainty throught data processing algorithms.

In many practical situations, distributions are non-Gaussian. One of the most efficient ways to describe a general non-Gaussian distribution is to use the sliced-normal approach, when we find a polynomial transformation that transform the original multi-D distribution into a Gaussian one. However, this approach has a limitation. This limitations comes from the fact that in many cases, it should not matter whether we use the original quantity or its inverse. For example, when studying fluctuations of exchange rates, it should not matter whether we use dollar-to-Euro or Euro-to-dollar ratio. However, when we use the traditional sliced-normal approach, in these two cases, we get different results. In this paper, we show how to generalize the traditional sliced-normal approach so as to eliminate this difference.

For many real-life phenomena -- in particular, for many economic phenomena -- it is not possible to exactly predict neither the future sare, not the probabilities of different future states. Such situations are known as chaos. In this chapter, we analyze chaos from the practical viewpoint: what can we predict, how can we make decisions in such situations, and how can we perform computations needed for such predictions and decisions.

We show that Z-numbers -- and their natural generalizations -- form a natural foundation for type-2 and other generalizations of traditional fuzzy techniques. We also show a natural analogy between (generalized) Z-numbers and physics. This analogy provide a possible path for possible future developments in physics, developments that can potentially lead, in particular, to faster computations.

In the original fuzzy technique, the expert describes his/her degree of confidence in their imprecise ("fuzzy") statement by a number from the interval [0,1] -- so that 1 means absolute confidence, 0 means absolute lack of confidence, and intermediate values describe intermediate degrees of confidence. This technique has many useful practical application, but it has a serious limitation: just like an expert is unable to describe the exact value of the corresponding physical quantity, the same expert is unable to describe his/her degree of confidence by a exact number. A natural solution to this challenge is to allow the expert to describe his/her degree not by a number, but by a natural-language words like "very confident" or "somewhat confident". This approach is known as type-2 approach. It provides a more adequate description of expert knowledge, but it requires more computations to handle. As a result, sometimes it leads to better application results, and sometime, more traditional techniques (called type-1) work better. This raises several natural questions: why are type-2 techniques effective, when are these techniques effective, and what is the most efficient way to use these techniques. In this paper, we overview theoretical results that provide (at least partial) answers to these questions.

Computers deal with numbers that represent the values of physical quantities. What will be the numbers of the future? They need to take into account all kinds of uncertainty, they need to take into account dependence between different quantities. This paper provides a vision of how such numbers of the future will look like.

In many practical situations, a data processing algorithm uses data with known with different types of uncertainty. For some data points, we know the probability distribution of the approximation error. For other points, we only know the upper bound on the asbolute value of approximation error -- which corresponds to interval uncertainty. For yet other data point, we only have imprecise (``fuzzy") expert estimates. There are effective techniques for dealing with each type of uncertainty, but situations when different types of uncertainty are present remain a challenge. Usually, such situations are dealt with by reducing to a single type of uncertainty, but, as we show, this can lead to a drastic under- or over-estimations of the resulting uncertainty. In this paper, we show to avoid such mis-estimations and thus, to make the resulting uncertainty estimations more accurate.

Published in IEEE Systems, Man, and Cybernetics Society eNewsletter, June 2026, Vol. 89.

Notes from the IEEE Conference on AI CAI 2026, Granada, Spain, May 8-10, 2026

In a class where the students learn both theoretical and practical skills, at the end of the class, we need to combine the overall grades for these two skills into a single final grade. Usually, instructors use weighted average of these grades -- e.g., a simple arithmetic average. However, this approach has a problem. For example, if a student got 100 for the theoretical part of a computing class and 60 for the practical part, his/her average grade 80 would provide a false impression to future employers that this student is good in programming -- while in reality, his/her programming skills are below satisfactory. To avoid this problem, some instructors use the smallest of the two grades. However, this can also be misleading. For example, the above 100-60 student would get a not-good overall grade of 60, so future employers will consider hum/her useless for computing-related tasks -- while in reality, he/she can help with theoretical aspects of the computing tasks. Recently, a new heuristic formula was proposed for such a combination that seems to avoid the limitations of the weighted average and minimum approaches. In many situations when we value both components equally, this formula leads to a more adequate description of the student skills. In this paper, we provide a theoretical explanation for this formula's success. We also use the ideas behind this theoretical explanation to show how this formula can be naturally extended to situations when one of the class components is valued more -- and also to situations when the class has more than two natural components.

To appear in: Hiram Ponce, Gilberto Ochoa-Ruiz, and Iris Mendez (eds.), Proceedings of the 25th Mexican International Conference on Artificial Intelligence MICAI 2026, Chihuahua, Mexico, November 2-6, 2026, Springer Lecture Notes in AI.

In the traditional neural networks, for each neuron, the activation function is fixed, and learning means finding the appropriate weights. It has been recently shown that in some cases, we can enhance the effectiveness of a neural network if we also allow it to select appropriate activation function for each neuron. Such networks are known as Kolmorogov-Arnold networks (KAN) -- after researchers who showed, in effect, that such networks can, in principle, exactly represent any continuous function on a bounded domain. Empirical results have shows that the best results are obtained when we use piecewise linear activation functions and when we use either the standard addition or multiplication to combine the inputs. In this paper, we provide semi-heuristic ideas that may help to understand both empirical observations.

To appear in Notes on Intuitionistic Fuzzy Sets

While modern AI techniques are very successful, they are not as reliable as we would like them to be: in about 5\% of the cases, they provide wrong answers. In this paper, we show that the problem facing modern AI tools is similar to the problem that, in the past, faced the original fuzzy logic -- and that led to the development of intuitionistic fuzzy techniques: the need to consider the "don't know" option in addition to the usual "yes" and "no" answers to a binary ("yes"-"no") question. Based on this analogy, we provide recommendations on how to make AI tools more reliable, and we analyze advantages and limitations on these recommendations.

To appear in: Laxminarayan Sahoo, Tapan Senapati, and Radko Mesiar (eds.), Advanced Decision Making Under Uncertainty: Fuzzy Logic, Data Science, and Machine Learning, Springer, Cham., Switzerland.

Many decisions -- in particular, medical decisions -- are based on a discrete (crisp) threshold, such as 100 F threshold for body temperature. As a result, someone whose temperature is 100.0 gets a treatment, but a person with similar symptoms whose body temperature is 99.9 -- practically the same -- has to wait. This does not seem fair. At first glance, it may sound like such unfairness is inevitable. However, inspired by one of Zadeh's main ideas -- that everything is a matter of degree -- we started looking for a solution. And indeed, it turned out that by using appropriate recommendations of decision theory, we can avoid such unfair discreteness in decision making.

In this article, we briefly describe what we learned from our teacher and friend Lotfi Zadeh -- and what others can learn from our experience.

Many physical phenomena are described by complex systems of partial different equations. Even on high-performance parallel computers, numerical methods for solving these equations often require a large amount of computation time. Lately, AI techniques -- namely, neural networks (NNs) -- have been successfully used to speed up these computations: a NN is trained on several examples and, once trained, generates solutions for new initial and/or boundary conditions. These computations are faster, but still require a lot of computation resources: time, memory, and energy. It is known that in AI computations, we can often use fewer resources by using limited precision when computing and processing the weights of a NN. To utilize this idea, we need to be able to find out how bounded precision in computing weights affects the AI computation results, and -- ideally -- what is the optimal allocation of weight precisions. In this paper, we describe general algorithms for solving these two problems. As usual with algorithms, a lot of additional work is needed to make these algorithms practically efficient and easy to use -- but what we show is that this is all doable.

To appear in: Proceedings of the 11th International Workshop on Reliable Engineering Computing REC 2026, Dortmund, Germany, September 2-5, 2026.

Complex computations often contain difficult-to-detect mistakes. This problem is very acute for AI-based computations -- where 5% of the corresponding answers are wrong, but this happens in more traditional computations as well. A natural way to detect such mistakes is to supplement the actual computation results with some easy-to-understand explanations. This is what researchers are trying to do for AI for make its results more reliable, and this is what we propose to do for computations in general. In this paper, we illustrate this idea on the example of reliable engineering computing, where it is important not only to get an estimate, but to also inform the user how accurate is the provided estimate.

It was recently proven that any countable partially ordered set with no maximal elements can be divided into disjoint omega-chains, i.e., ordered subsets isomorphic to the set of natural numbers. In this paper, we prove that this division can be done algorithmically. We also discuss possible applications to the analysis of multiple personality disorder.

It is known that the famous Catalan architect Antoni Gaudi had arches in many of his buildings. A recent book has shown that practically all his arches have one of the following two shapes; they are either parabolic arches, in which the y-coordinate is a quadratic function of x, or so-called catenary arches. In this paper, we provide a possible mathematical explanation of why Gaudi used only these two types of arches.

A recent paper has shown that dolphins living in small social groups age slower than solitary dolphins, but dolphins living in larger social groups age faster than solitary dolphins. That paper provided explanations based, to some extent, on the specifics of the social life, with its mutual help and -- at the same time -- stressful conflicts. The current paper intends to provide a more general explanation of the newly observed phenomena, an explanation based on the ideas of the general decision theory.

While convolutional neural networks (CNNs) are very effective in image processing, they are not robust: a minor change in a few pixels can drastically change the image processing result -- and thus, to a misclassification of the image. A recent paper has shown that CNNs can be made more robust if instead of the usual max-neurons that return the largest of the inputs, we use neurons that return the second largest of the inputs. Such neurons are known as drop-max neurons. In this paper, we prove that a natural robustness requirement uniquely determines the use of drop-max neurons. We also describe what type of neurons we should use if we want to achieve a stronger robustness.

One of the main direction in modern pedagogy is constructivism, when instead of explicitly teaching general rules and algorithms, the instructor provides the students with a well-design sequence of examples, based on which the students can easily reconstruct the general rules. This direction has been very successful -- and its success seems to be confirmed by spectacular successes of modern AI, successes based on a similar idea -- that teaching computer examples from which the computer can reconstruct the rules is much more productive than explicitly teaching the rules. However, our experience of teaching complex rules and algorithms shows that sometimes, teaching rules first leads to better results, In this paper, we show that several recent machine learning results show a similar tendency -- that for complex rules and algorithms, it is sometimes beneficial to explicitly teach computer the rules.

In many practical situations - in particular, in many medical problems - it is important to find the coefficients of linear regression based on the empirical data. In many such situations, we only know the upper bound on the absolute value of the measurement error - i.e., in effect, we only know intervals containing the actual values. When we know that the dependence is exactly linear, finding the exact ranges of possible values of the regression coefficients is NP-hard -- meaning that, in general (unless P = NP), the exact computation of these ranges is not practically feasible. However, in many practical cases - in particular, in many medical applications - linear regression is only an approximate model, obtained by ignoring quadratic and higher order terms. In such cases, it is reasonable to also ignore quadratic order terms in our estimation of the ranges of regression coefficients. We show that this natural idea enables us to design efficient algorithms for estimating these ranges. Specifically, we present a polynomial-time algorithm.

To appear in Proceedings of the 2026 Annual Conference of North American Fuzzy Information Processing Society NAFIPS 2026, El Paso, Texas, March 14-16, 2026.

The reason why we have seasons is that the Earth's rotation axis is tilted. An interesting fact that the sine of the tilt is almost exactly 2/5. This fact leads to a natural question: is this an indication of a physical resonance -- or is this a random coincidence? In this paper, we show that this is an accidental coincidence.

To appear in Proceedings of the 2026 Annual Conference of North American Fuzzy Information Processing Society NAFIPS 2026, El Paso, Texas, March 14-16, 2026.

In a recent book, two veteran Air Force leaders provide general advice on how to deal with real-life challenges. In this paper, we summarize this advice in precise terms, and explain that this advice fits with common sense.

To appear in Proceedings of the 2026 Annual Conference of North American Fuzzy Information Processing Society NAFIPS 2026, El Paso, Texas, March 14-16, 2026.

Many Gulag memoirs mention that to avoid starvation, smart team leaders "cheated" -- fictitiously redistributed the overall production between team members, as a result of which the overall award increased. This practice leads to a natural question: which award system prevents such cheating? In which award system such a fictitious redistribution will not change the overall team award? In this paper, we show that the only award system that prevents such cheating is linear, when the award is a linear function of productivity.