Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

1. INTRODUCTION

An R-function, or Rvachev function, is a real-valued function whose sign does not change if none of the signs of its arguments change; that is, its sign is determined solely by the signs of its arguments. Interpreting positive values as true and negative values as false, an R-function is transformed into a “companion” Boolean function (the two functions are called friends). R-functions are used in computer graphics and geometric modeling in the context of implicit surfaces and function representations. They also appear in certain boundary-valued problems, and are also popular in certain artificial intelligence applications, where they are used in pattern recognition. The Rvachev’s method implies an ability to represent a geometric object “implicitly” by a property Q(x) where x∈Rn, as Ω={x : Qx is true}.

R-function is a real-valued function of real variables having the property that their signs are completely determined by the signs of their arguments, and are independent of the magnitude of the arguments. For example, the following functions satisfy this property:

• •

  W1=xyz

• •

  W2=x+y+xy+x2+y2

• •

  W3=2+x2 + y2+22

The main idea is to find corresponding R-functions f : Rⁿ → R for some Boolean function φ : {0,1}ⁿ → {0,1}. Roughly speaking, the Boolean functions are usually defined using logic operations ∧ (and; minimum of the two arguments), ∨ (or; maximum of two arguments), and ¬ (negation; 1−x) on n logic variables. The Boolean function φ in the above definition is called the companion function of the R-function f. Informally, a real function f is an R-function if it can change its property (sign) only when some of its arguments change the same property (sign). The notion of R-functions is a special case of a more general concept of R-mapping that is associated with qualitative k-partitions of arbitrary domains and multi-valued logic functions [1]. We follow this idea and have proposed multiple-valued logic, namely 3-valued Lukasiewicz logic [2].

The set of all R-functions that have the same logic companion function is called a branch of the set of R-functions. Since the number of distinct logic functions of n arguments is 22n, there are also 22n distinct branches of R-functions of n arguments.

The set of R-functions is infinite. However, for applications, it is not necessary to know all R-functions; one needs only to be able to construct R-functions that belong to a specified branch. The recipes for such constructions are implied by the general properties of R-functions that follow almost immediately from their definition. Complete proofs, as well as many additional properties, can be found in [1,3].

1. 1.

   The set of R-functions is closed under composition. In other words, any function obtained by composition of R-functions is also an R-function.

2. 2.

   If a continuous function fx1,...,xn has zeros only on coordinate hyperplanes (i.e. f = 0 implies that one or more xj= 0, j = 1,2,...,n), then f is an R-function.

3. 3.

   The product of R-functions is an R-function. If the R-function fx1,...,xn belongs to some branch, and gx1,...,xn> 0 is an arbitrary function, then the function fg also belongs to the same branch.

4. 4.

   If f1 and f2 are R-functions from the same branch, then the sum f1+f2 is an R-function belonging to the same branch.

5. 5.

   If f_φ is an R-function whose logic companion function is φ, and C is some constant, then Cf_φ is also an R-function. The logic companion function of Cf is φ if C > 0, or ¬φ if C < 0.

6. 6.

   If f_φ(x₁,...,x_n) is an R-function whose logic companion function is φ(X₁,...,X_n) and f can be integrated with respect to x_i, then the function ∫0xifx1,...,xndxi is an R-function whose logic companion function is X_i ⇔ φ(X₁,...,X_n).

The above list of properties is not exhaustive, but it is enough to suggest that more complex R-functions may be constructed from simpler functions. In particular, the closure under composition leads to the notion of sufficiently complete systems of R-functions, i.e. collections of R-functions that can be composed in order to obtain an R-function from any branch.

2. THREE-VALUED GÖDEL LOGIC WITH CONSTANTS AND INVOLUTION

Partition −∞,+∞ into three sets: −∞,0,0,+∞).

Let G3=−1,0,1,∨,∧,→,∼,−1,0,1 be the algebra of type 2,2,2,1,0,0,0, where x∨y=maxx,y, x∧y=minx,y, ∼ is changing of sign and x→y is given in Table 1.

Table 1

The → operation.

For the R-function x·y we can take the logic companion φp,q=p↔q∧p∨q∨ ∼p∧q, the logical function of which in −1,0,1 is given in Table 2 with the following correspondence: + ↔1, 0, − ↔−1.

Table 2

Logic companion for the R-function x · y.

For the R-function x·y please refer to Table 3.

Table 3

The R-function x · y.

R-function −x and its Gödel companion is changing sign.

R-function c ∈ 0,+∞ and its Gödel companion is 1.

R-function c ∈ −∞,0 and its Gödel companion is −1.

R-function 0 and its Gödel companion is 0.

R-function maxx,y and its Gödel companion is disjunction ∨.

R-function minx,y and its Gödel companion is conjunction ∧.

In the study of many-valued logics, one is led to consider a finite algebra (A, o₁, ..., o_n) that generate by composition all functions in AAm for each m ∈ Z+. Such algebras are called primal. For example, Boolean algebra 0,1,∨,∧,−.0,1 and, more generally, Post algebras 0,...,n − l,minx,y,x+lmod n are primal algebras.

For primal algebras the following theorem holds:

Figure 1

Graphical representation of the algebra G33.

In the same manner the following is proven:

From this theorem holds:

From the above mentioned we conclude:

REFERENCES