Abstract—Most modern physical theories are formulated in terms of differential equations. As a result, if we know exactly the current state of the world, then this state uniquely determines all the future events – including our own future behavior. This determination seems to contradict the intuitive notion of a free will, according to which we are free to make decisions – decisions which cannot be determined based on the past locations and velocities of the elementary particles. In quantum physics, the situation is somewhat better in the sense that we cannot determine the exact behavior, but we can still determine the quantum state, and thus, we can determine the probabilities of different behaviors – which is still inconsistent with our intuition. This inconsistency does not mean, of course, that we can practically predict our future behavior; however, in view of many physicists and philosophers, even the theoretical inconsistency is somewhat troubling. Some of these researchers feel that it is desirable to modify physical equations in such a way that such a counter-intuitive determination would no longer be possible. In this paper, we analyze the foundations for such possible theories, and show that on the level of simple mechanics, the formalization of a free will requires triple interactions – while traditional physics is based on pairwise interactions between the particles. I. FREE WILL: A NATURAL IDEA Intuitively, most of us believe that we are able to make conscientious decisions, i.e., that we have free will. If we walk to a corner, then we can turn right or cross the street. The commonsense belief is that it is not possible to predict beforehand what exactly a person will do.

Abstract. In his logical papers, Leo Esakia studied corresponding ordered topological spaces and order-preserving mappings. Similar spaces and mappings appear in many other application areas such the analysis of causality in space-time. It is known that under reasonable conditions, both the topology and the original order relation � can be uniquely reconstructed if we know the “interior ” ≺ of the order relation. It is also known that in some cases, we can uniquely reconstruct ≺ (and hence, topology) from �. In this paper, we show that, in general, under reasonable conditions, the open order ≺ (and hence, the corresponding topology) can be uniquely determined from its closure �.

Abstract. We show that in many application areas including soft constraints reasonable requirements of scale-invariance lead to polynomial (tensor-based) formulas for combining degrees (of certainty, of preference, etc.) Partial orders naturally appear in many application areas. One of the main objectives of science and engineering is to help people select decisions which are the most beneficial to them. To make these decisions, – we must know people’s preferences, – we must have the information about different events – possible consequences of different decisions, and – since information is never absolutely accurate and precise, we must also have information about the degree of certainty. All these types of information naturally lead to partial orders: – For preferences, a < b means that b is preferable to a. This relation is used in decision theory; see, e.g., [1].

In many real-life applications, we have an ordered set: a set of all space-time events, a set of all alternatives, a set of all degrees of confidence. In practice, we usually only have a partial information about an element x of this set. This partial information includes positive knowledge: that a ≤ x or x ≤ a for some known a, and negative knowledge: that a ̸ ≤ a or x ̸ ≤ a for the known a. In the case of a total order, the set of all elements satisfying this partial information is an interval. We show that in the general case of a partial order, the corresponding analogue of an interval is a convex set. We also show that in general, to describe partial knowledge, it is sufficient to have only negative information about x but it is not sufficient to have only positive information.

Abstract—We show that in many application areas including soft constraints reasonable requirements of scale-invariance lead to polynomial formulas for combining degrees (of certainty, of preference, etc.)

The study of Artificial Neural Networks started with the analysis of linear neurons. It was then discovered that networks consisting only of linear neurons cannot describe non-linear phenomena. As a result, most currently used neural networks consist of non-linear neurons. In this paper, we show that in many cases, linear neurons can still be successfully applied. This idea is illustrated by two examples: the PageRank algorithm underlying the successful Google search engine and the analysis of family happiness. 1 Linear Neural Networks: A Brief Reminder Neural networks. A general neural network consists of several neurons exchanging signals. At each moment of time, for each neuron, we need finitely many numerical parameters to describe the current state of this neuron and the signals generated by this neuron. The state of the neuron at the next moment of time and the signals generated by the neuron at the next moment of time are

For a society to function efficiently, it is desirable that all members of this society care no only about themselves, but also about the society as a whole, i.e., about all the other individuals from the society. In practice, most people are only capable of caring about a few other individuals. We analyze this problem from the viewpoint of decision theory and show that even with such imperfect individuals, it is possible to make sure that everyone’s decisions are affected by the society as a whole: namely, it is sufficient to make sure that people have emotional attachment to those few individuals who are capable of caring about the society as a whole. As a side effect, our result provides a possible explanation of why the Biblical commandment to love your God encourages ethical behavior.

Many decisions are made by voting. At first glance, the more people participate in the voting process, the more democratic – and hence, better – the decision. In this spirit, to encourage everyone’s participation, several countries make voting mandatory. But does mandatory voting really make decisions better for the society? In this paper, we show that from the viewpoint of decision making theory, it is better to allow undecided voters not to participate in the voting process. We also show that the voting process would be even better – for the society as a whole – if we allow partial votes. This provides a solid justification for a semi-heuristic “fuzzy voting ” scheme advocated by Bart Kosko. Need for democratic decision making. Often, a social entity faces a problem, and there are several alternative ways to solve this problem. For example, to build a new baseball stadium, the city can either use the existing funds or issue a bond – and hope that the future profits from this stadium will pay off

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