Why Tensors?

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]. – For events, a < b means that a can influence b. This causality relation is used in space-time physics. – For uncertain statements, a < b means that a is less certain than b. This relation is used in logics describing uncertainty such as fuzzy logic (see, e.g., [3]) and in soft constraints. Numerical characteristics related to partial orders. While an order may be a natural way of describing a relation, orders are difficult to process, since most data processing algorithms process numbers. Because of this, in all three application areas, numerical characteristics have appeared that describe the corresponding orders: – in decision making, utility describes preferences: a < b if and only if u(a) < u(b); – in space-time physics, metric (and time coordinates) describes causality relation; – in logic and soft constraints, numbers from the interval [0, 1] are used to describe degrees of certainty; see, e.g., [3]. Need to combine numerical characteristics, and the emergence of polynomial aggregation formulas. – In decision making, we need to combine utilities u1, . . . , un of different participants. Nobelist Josh Nash showed that reasonable conditions lead to u = u1 · . . . · un; see, e.g., [1, 2]. – In space-time geometry, we need to combine coordinates xi into a metric; reasonable conditions lead to polynomial metrics such as Minkowski metric in which s = c · (x0 − x0) − (x0 − x0) − (x1 − x1) − (x2 − x2) − (x3 − x3) and of a more general Riemann metric where ds = ∑ i,j gij · dx · dx . – In fuzzy logic and soft constraints, we must combine degrees of certainty di in Ai into a degree d for A1 & A2; reasonable conditions lead to polynomial functions like d = d1 · d2. In mathematical terms, polynomial formulas are tensor-related. In mathematical terms, a general polynomial dependence f(x1, . . . , xn) = f0+ n ∑