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

Three-Valued Gödel Logic With Constants and Involution for Application to R-Functions
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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 xRn, as ={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 : RnR 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 fφ is an R-function whose logic companion function is φ, and C is some constant, then Cfφ is also an R-function. The logic companion function of Cf is φ if C > 0, or ¬φ if C < 0.

  6. 6.

    If fφ(x1,...,xn) is an R-function whose logic companion function is φ(X1,...,Xn) and f can be integrated with respect to xi, then the function 0xifx1,...,xndxi is an R-function whose logic companion function is Xiφ(X1,...,Xn).

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.

Theorem 1.

Let H be some system of R-functions and G be the corresponding system of the logic companion functions. The system H is sufficiently complete if the system G is complete. [2]

It is easy to check that the following functions are R-functions (their logic companion function in parentheses):

  • Cconst                              (logical 1)

  • x¯  x                                     (logical negation ¬)

  • x1 1x1  minx1,x2       (logical conjunction ∧)

  • x1 1 x1  maxx1,x2 (logical disjunction ∨)

Theorem 1 states that an R-function from any branch can be defined using composition of just these functions. But these functions are not differentiable. For applications where differentiability is important, for example in solutions of boundary value problems, another system is needed. For this one we need suitable R-conjunction and R-disjunction. Let us consider a triangle with two sides of length x1 and x2. The square of the third side is determined by the law of cosines as x12+x22  2αx1x2, where α is the cosine of the angle between the two sides. It is easy to see that the function:

f=x1+x2  x12+x22  2αx1x2
satisfies the desired properties. Moreover, the R-function corresponding to logical disjunction is:
f=x1+x2 + x12+x22  2αx1x2

In this article we suggest a new logic: 3-valued Gödel Logic with constant and involution for application to R-functions that will be a new companion function for R-functions.


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 xy=maxx,y, xy=minx,y, ∼ is changing of sign and xy is given in Table 1.

Table 1

The → operation.

For the R-function x·y we can take the logic companion φp,q=pqpq pq, 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, ..., on) 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:

Theorem 2.

An algebra (A, o1, ..., on) is primal iff 2-generated free algebra is isomorphic to A|A|2. [4,5]

Let G3 be the variety generated by algebra G3 = ({−1,0,1}, ∨, ∧, →, ∼, −1,0,1). The algebra G33 is depicted in Fig. 1.

Figure 1

Graphical representation of the algebra G33.

Theorem 3.

The algebra G33 is 1-generated free algebra in the variety G3 with free generator g = (−1,0,1).

Proof. Let ¬x = x → −1. Let us consider the algebra:

G3=({ 1,0,1 },,,,,1,0,1)
and its element g=1,0,1. Now we show that g generates the algebra G3. It can be shown this fact if we obtain the elements 1,1,1, 1,1,1, 1,1,1 previously having the constant elements 1,1,1,  0,0,0,1,1,1. ¬g=1,1,1, ¬ g=1,1,1, ¬g  ¬ g=1,1,1.

From here we can obtain all elements of the algebra G3 by the lattice operations ∨ and ∧. Now we show that one-variable identity P=Q is true in the variety G3 iff the identity is true in G33 for generator g. Indeed, it is obvious that if P=Q is true in the variety G3, then it is true in the algebra G33. Let us suppose that P=Q is not true in the variety G3. Then it is not true in the algebra G3 for some element aG3. Then, we can take corresponding projection πk: G33G3. where  πkg=a. It means that P=Q is not true in G33 for the generator g. From here we conclude that G33 is 1-generated free algebra in the variety G3 with free generator g=1,0,1.

In the same manner the following is proven:

Theorem 4.

The algebra G332 is 2-generated free algebra in the variety G3 with free generators g1=1,1,1,0,0,0,1,1,1, g2=1,0,1,0,1,1,1,0,1.

From this theorem holds:

Corollary 5.

The algebra G3 is primal. In other words, the operations ∨, ∧, →, ∼, −1,0,1 of the algebra G3 generate all functions in AAm for each mZ+.

From the above mentioned we conclude:

Theorem 6.

The number of branches of n-ary R-functions is equal to 33n.


V.L. Rvachev. Geometric Applications of Logic Algebra. Kyiv: Naukova Dumka, 1967. (in Russian)
J. Gielis, R. Grigolia. Lamé Curves and Rvachev’s R-Functions. Reports of Enlarged Sessions of the Seminar of I. Vekua Institute of Applied Mathematics, 2022, 36: 27–30.
V.L. Rvachev. Theory of R-Functions and Some Applications. Kyiv: Naukova Dumka, 1982. (in Russian)
A.L. Foster. On the Finiteness of Free (Universal) Algebras. Proceedings of the American Mathematical Society, 1956, 7(6): 1011–1013.
F.M. Sioson. Free-Algebraic Characterizations of Primal and Independent Algebras. Proceedings of the American Mathematical Society, 1961, 12(3): 435–439.

Cite This Article

AU  - Revaz Grigolia
PY  - 2023
DA  - 2023/11/29
TI  - Three-Valued Gödel Logic With Constants and Involution for Application to R-Functions
BT  - Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)
PB  - Athena Publishing
SP  - 47
EP  - 51
SN  - 2949-9429
UR  -
DO  -
ID  - Grigolia2023
ER  -