Chu Spaces: Towards New Foundations for Fuzzy Logic and Fuzzy Control, with Applications to Information Flow on the World Wide Web
We show that Chu spaces, a new formalism used to describe parallelism and information flow, provide uniform explanations for different choices of fuzzy methodology, such as choices of fuzzy logical operations, of membership functions, of defuzzification, etc. 1. What are Chu spaces? 1.1. World according to classical physics It is well known that measurements can change the measured object: e.g., most methods of chemical analysis destroy a part of the analyzed substance; testing a car often means damaging it, etc. However, in classical (pre-quantum) physics it was assumed that in principle, we can make this adverse influence as small as possible. Therefore, ideally, each measurement can be described as a function r(x) from the set of all objects X to the set K of all measurement results. These measurements lead to a complete knowledge in the sense that an object x can be uniquely reconstructed from the results r(x) of all such measurements. 1.2. Non-determinism in modern physics: enter Chu spaces In modern physics, starting from quantum mechanics, it was realized that ideal non-influencing measurements are impossible: the more accurately we measure, the more we change the object of measurement. As a result, it is not possible to uniquely reconstruct an object from measurement results. In other words, each measurement is a function r(x, y) of two variables: an object x and a (not completely known) measuring device y. Such a function describes a so-called Chu space (see, e.g., [1, 2, 7, 8, 16, 17, 18, 19, 20, 21, 22]). 1.3. Precise definition of a Chu space To be more precise, to define a Chu space, we must fix a set K (of possible values). Then, a K-Chu space is defined as a triple (X, r, Y ), where X and Y are sets, and r : X×Y → K is a function which maps every pair (x, y) of elements x ∈ X and y ∈ Y into an element r(x, y) ∈ K. 1.4. Back to measurements: enter automorphisms of Chu spaces The fact that x cannot be uniquely reconstructed from such measurements means that the same measurement results can be explained if we take slightly different objects (f(x) instead of x) and, correspondingly, slightly different measuring instruments (g(y) instead of y): r(x, y) = r(f(x), g(y)). This formula takes a more symmetric form if we consider, instead of g(y), an inverse function y = h(z) = g−1(z): r(x, h(z)) = r(f(x), z). (1) A pair of functions (f, h) which satisfies the property (1) for all x ∈ X and z ∈ Y is called an automorphism of a Chu space. 1.5. From physics to general problem solving A general problem is: given x, find y for which a known (easy to compute) function r(x, y) takes the desired value d (e.g., 0). A problem r is reduced to a problem s if it is possible, for each instance x of the first problem, to find the correspondence instance f(x) of the second problem, so that from each solution z of the second problem, we can compute a solution h(z) to the original problem, i.e., r(x, h(z)) = s(f(x), z). (2) (This notion is central in computational complexity theory, in the definitions of NP-hardness etc., see, e.g., [5, 14].) Such a pair (f, h) is called a morphism of Chu spaces. 1.6. Morphism of Chu spaces: precise definition In general, if we have two Chu spaces A = (X, r, Y ) and B = (X ′, r′, Y ′), the a pair of functions (f : X → X ′, g : Y ′ → Y ) is called a morphism of Chu spaces if it satisfies the property (2) for all x ∈ X and for all z ∈ Y ′. 1.7. Applications to parallelism and to information flow The notion of Chu spaces was actively used by V. Pratt (Stanford) for describing parallel problemsolving algorithms (see, e.g., [7, 8, 16, 17, 18, 19, 20, 21, 22]), and by J. Barwise (Indiana) to describe information flow in general (see, e.g., [3]). 2. Chu spaces as a uniform justification for fuzzy techniques 2.1. Fuzzy is a particular case of Chu spaces Fuzzy knowledge can be naturally described as a Chu space (X, r, Y ), where X is the set of all objects, Y is the set of all linguistic properties, and r(x, y) is a degree to which x has a property y (see, e.g., [15]). This relation was originally done in two steps: • fuzzy logic can be interpreted as a particular case of so-called linear logic (see, e.g., [6, 11, 12, 15]), and • linear logic is naturally interpreted in terms of Chu spaces. 2.2. What we are planning to do We show that Chu description leads to a uniform justification of numerous choices of fuzzy membership functions, fuzzy logic operations, defuzzification procedures, etc. This justification is in line with a general group-theoretic approach described in our 1997 Kluwer book [13] (see also [4, 10]). 3. The main technical idea behind using Chu spaces as a foundation for fuzzy theory: a simplified (non-fuzzy) illustration 3.1. Example: a simple physical problem To better present our main technical idea, we will first illustrate it on a simplified (crisp) example. Let us analyze how the period t of a pendulum depends on its length l. From the purely mathematical viewpoint, this dependency can be described by a function of one variable t = F (l), i.e., as a function from real numbers to real numbers. However, from the physical viewpoint, such a mathematical description is somewhat unnatural, for the following reason: • we really want a dependence between physical quantities t and l; • in order to describe this dependence as a dependence between real numbers, we must fix some units for measuring both length l and time t; thus, the resulting function depends on the specific choice of these units; • however, the choice of the units is a matter of convention (e.g., to describe length, we can use meters or feet without changing any physical meaning). It is therefore desirable to have a mathematical description of the dependency of t on l which would reflect the physical dependency without adding any arbitrariness. 3.2. A more adequate mathematical description of the physical problem Such a description can be obtained if we explicitly add the two measuring units ul (for length) and ut (for time) to the description of this function, i.e., if we consider the function of the type t = F (l, ul, ut), where l is a numerical value of the pendulum’s length, t is a numerical value of its period, and ul and ut are the measuring units used to describe the corresponding numerical values (described in terms of some standard measuring units). If we know the dependence t0 = F (l0) in standard units, then we can easily describe the new function: Indeed, if we use the length ul as a unit of length, then in these units, the numerical value l of length means l0 = l · ul in the original units, so in the standard units, the pendulum’s period is equal to t0 = F (l0) = F (l · ul). Hence, if we use the unit ut for measuring time intervals, then in this unit, the numerical value of the time period is equal to t = t0/ut = F (l · ul)/ut. In other words, we get F (l, ul, ut) = F (l · ul)/ut. 3.3. Mathematical model naturally reformulated as a Chu space The above physically appropriate dependence can be naturally described as a Chu space, with X being the set of all possible units of length, Y the set of all possible time units, K the set of all possible functions of one real variable, and the function r(ul, ut) defined as (r(ul, ut))(l) = F (l, ul, ut). From the mathematical viewpoint, the sets X and Y coincide with the set R of all positive real numbers. 3.4. Unit-invariance formulated in precise terms Let us now formalize the requirement that this dependence be independent on the choice of the units for measuring length l and time t. If we simply change a measuring unit for length or a measuring unit for time, then we get a different numerical dependence. However, for every change of the length unit, there is an appropriate change of a time unit after which the resulting numerical dependence stays the same. This requirement can be formulated as follows. Suppose that we use a different standard unit for measuring length. Let λ > 0 be the value of the old standard unit in terms of the new one; then, 1 old standard unit = λ new standard units, so ul old standard units = ul · λ new standard units, i.e., the measuring unit for length whose value was ul in old standard units has a new value ul = λ ·ul in new standard units. Similarly, the choice of a new standard unit for time means that we replace the original value ut by a new value ut = g(ut), where g(y) = μ · y and μ is the value of the old standard unit in terms of the new standard unit for time. In these terms, the above requirement means that for every function f : X → X of the type f(ul) = λ ·ul, there exists a function g(y) of the type g(ut) = μ·ut for which, for every x ∈ X and y ∈ Y , we have r(x, y) = r(f(x), g(y)). 3.5. Unit-invariance reformulated in terms of Chu spaces We have already mentioned that this equality describes an automorphism of the Chu space. Thus, the above requirement means that for every function f : X → X from a certain transformation class can be extended to an automorphism (f, h) of the corresponding Chu space. 3.6. The Chu-space requirements describes the desired function One can show that this condition is satisfied only by functions of the type t = A · l, with A and α arbitrary constants; the actual pendulum corresponds to α = −0.5. Thus, the Chu space requirement leads to a description of a very narrow class of functions which contain the desired one. 4. Application of our main idea to fuzzy techniques: illustration and other results