On the Axiomatization of Interval Arithmetic Svetoslav Markov Institute of Mathematics and Informatics Bulgarian Academy of Sciences “Acad. G. Bonchev” st., block 8, 1113 Sofia, Bulgaria [email protected] Abstract In our study we follow the algebraically natural approach of completing the set IRn up to the set IRn involving im- We investigate some abstract algebraic properties of the proper intervals. Thus, starting from the above three ba- system of intervals with respect to the arithmetic operations sic operations/relations (1)–(3), we arrive to the system and the relation inclusion and derive certain practical con- (IRn, +, R, ∗, ⊆). In the next Section 2 we briefly recall sequences from these properties. In particular, we discuss the above mentioned algebraic construction and the conse- the use of improper intervals (in addition to proper ones) quent quasivector spaces. It is to be noted that everything and of midpoint-radius presentation of intervals. This work said in this section for intervals is also true for the more gen- is a theoretical introduction to interval arithmetic involving eral case of convex bodies and briefly repeats already pub- improper intervals. We especially stress on the existence of lished materials, cf. [7], [8]. In Section 3 we concentrate special “quasi”-multiplications in interval arithmetic and on the system (IRn, +, R, ∗, ⊆) obtained by algebraic com- their role in relevant symbolic computations. pletion. Using the theoretical foundations given in Section 2 and some specific properties of intervals (distinct from those of general convex bodies) we derive formulae for the 1. Introduction operations/relations involved. We show that the familiar midpoint-radius presentation of intervals is a special case of the presentation of elements in a quasivector space. Section In this work we discuss several algebraic properties of (IR, +, ×, ⊆) the system of intervals with the arithmetic operations ad- 4 is devoted to the system involving multi- dition and multiplication and the relation inclusion. Our plication of one-dimensional intervals. In the Conclusion aim is to point out certain practical advantages of using im- we discuss various topics like computer implementation of proper intervals and midpoint-radius presentation of inter- interval arithmetic, symbolic computations, etc. vals. Denote by IR the set of all compact intervals on the real 2. Quasivector spaces line R and by IRn the set of all n-tuples of intervals. For n A, B ∈ IRn, α, β ∈ Rn, γ ∈ R, one defines addition, The system (IR , +) is a commutative monoid (semi- multiplication by scalars and inclusion, resp., by: group with null) with cancellation law. There is no opposite operator in (IRn, +). The operator multiplication by the A + B = {α + β | α ∈ A, β ∈ B}, (1) scalar −1: ¬A =(−1) ∗ A = {−α | α ∈ A},A∈ IR, γ ∗ B = {γβ | β ∈ B}, (2) briefly called negation (that may be suspected for opposite), A +(¬A)=0 A ⊆ B ⇐⇒ (α ∈ A =⇒ α ∈ B), (3) is not an opposite operator, as is violated for certain A ∈ IRn. Thus IRn is not a group; however it where all operations/relations are understood component- can be embedded in a group. The algebraic construction wise. We thus obtain the system (IRn, +, R, ∗, ⊆) to be that converts an abelian monoid with cancellation law into discussed in the sequel. a group will be further refered as embedding construction. We shall refer to the definitions of the opera- Recall that this approach is used to pass from the monoid of tions/relations (1)–(3) as set-theoretic. These definitions are nonnegative reals (R+, +) to the set of reals (R, +). Thus, not suitable for computations with intervals. Our final aim it is natural instead of the original system (IRn, +, R, ∗, ⊆) is to derive computationally efficient expressions for these to consider the extended system (IRn, +, R, ∗, ⊆) obtained operations based on the intrinsic properties of intervals. by the embedding construction. 2.1. The embedding construction γ ∗ (a + b)=γ ∗ a + γ ∗ b, (7) α ∗ (β ∗ c)=(αβ) ∗ c, (8) Every abelian monoid (M,+) with cancellation law in- 1 ∗ a = a, (9) duces an abelian group (M, +), where M = M 2/ ∼ is (α + β) ∗ c = α ∗ c + β ∗ c, αβ ≥ 0. the difference (quotient) set of M consisting of all pairs if (10) (A, B) factorized by the congruence relation ∼:(A, B) ∼ The only difference between a vector (linear) space and (C, D) iff A + D = B + C, for A, B, C, D ∈ M. a quasivector space is contained in assumption (10), where Addition in M is defined by the relation (α + β) ∗ c = α ∗ c + β ∗ c is required to hold just for αβ ≥ 0, whereas in a vector space the same (A, B)+(C, D)=(A + C, B + D). (4) relation is assumed to hold for all scalars α, β ∈ R (known The neutral (null) element of M is the class (Z, Z), Z ∈ as second distributive law). Thus a vector space is a special M. Due to the existence of null element in M,wehave quasivector space. (Z, Z) ∼ (0, 0). The opposite element to (A, B) ∈ M is Conjugate elements. From opp(a)+a =0we obtain opp(A, B)=(B,A). The mapping ϕ : M −→ M defined ¬opp(a) ¬ a =0, that is ¬opp(a)=opp(¬a). The el- for A ∈ M by ϕ(A)=(A, 0) ∈ M is an embedding of ement ¬opp(a)=opp(¬a) is further denoted by a − and monoids. We embed M in M by identifying A ∈ M with the corresponding operator is called dualization or conju- the equivalence class (A, 0) ∼ (A + X, X), X ∈ M; all gation. We say that a− is the conjugate (or dual) of a.In elements of M admitting the form (A, 0) are called proper the sequel we shall express the opposite element symboli- and the remaining (new) elements are called improper. The cally as: opp(a)=¬a−, minding that a+(¬a−)=0(to be set of all proper elements of M is ϕ(M)={(A, 0) | A ∈ briefly written as a ¬ a− =0). Using conjugate elements ∼ M} = M. the quasistributive law (10) can be written in the form Using the above construction the system (IRn, +) is em- n bedded into the group (IR , +) in a unique way. (α + β) ∗ cσ(α+β) = α ∗ cσ(α) + β ∗ cσ(β), (11) Multiplication by scalars “∗” is extended from R×IRn σ n wherein is the sign functional to R × IR by means of −,α<0; n σ(α)= γ ∗ (A, B)=(γ ∗ A, γ ∗ B),A,B∈ IR ,γ∈ R. (5) +,α≥ 0,α∈ R, ¬(A, B)=(−1) ∗ In particular, negation is extended by and the convention a+ = a has been made (for a proof (A, B)=(¬A, ¬B),A,B∈ IRn . see [8]). Expression (11) is valid for all values of α, β (not In the sequel we shall use lower case roman letters only for equally signed α, β) which allows efficient sym- to denote the elements of IRn, writing e. g. a = n bolic calculations. (A1,A2),A1,A2 ∈ IR . For example, negation is writ- ¬a =(−1) ∗ a a ¬ b a +(¬b) ten: ; below means . 2.3. A decomposition theorem Inclusion “⊆” is extended in IR by means of An element y with the property y ¬ y =0(equivalently (A, B) ⊆ (C, D) ⇐⇒ A + D ⊆ B + C, (6) y = y−) is called linear or distributive; an element z such n that ¬z = z (equivalently z + z− =0) is called centred or wherein A, B, C, D ∈ IR and inclusion of interval n- 0-symmetric. tuples is meant component-wise. As is immediately seen, under this extension the practically important properties Theorem (Decomposition theorem). (Q, +, R, ∗) is a a ⊆ b ⇐⇒ a+c ⊆ b+c for c ∈ IR and a ⊆ b ⇐⇒ γ ∗a ⊆ quasivector space. For every x ∈ Q there exist unique γ ∗b for γ ∈ R are preserved. The system (IRn, +, R, ∗, ⊆) y,z ∈ Q such that: i) x = y + z; ii) y ¬ y =0; iii) involving improper intervals is now completely defined. ¬z = z;iv)y = z =⇒ y = z =0. The proof, see [8], is based on the fact that any x ∈ Q 2.2. Quasivector space: definition can be written in the form: (IRn, +, R, ∗) The interval system is a quasi-vector x = y + z =(1/2) ∗ (x + x−)+(1/2) ∗ (x ¬ x). (12) space in the sense of the following definition [8]: Definition.Aquasi-vector space (over R), denoted Note that the first summand in (12) is linear, whereas the (Q, +, R, ∗), is an abelian group (Q, +) with a mapping second one is centred. The subset of all linear elements of Q (multiplication by scalars) “∗”: R × Q −→ Q, such that is denoted Q = {x ∈ Q | x ¬ x =0} and the subset of all for a, b, c ∈ Q, α,β,γ ∈ R: centred elements of Q is denoted Q = {x ∈ Q | x = ¬x}. Note that negation coincides with opposite in Q, and that the two vector spaces (Q, +, R, ·), (Q, +, R, ·) are negation coincides with identity in Q. m-, resp. n-dimensional and therefore they are isomorphic to Rm, resp. Rn. Hence any a ∈ Q is a direct sum of the Corollary 1. Every quasivector space Q is a direct sum m n form a =(a ; a ), a ∈ R , a ∈ R . As elements of Q of Q = {x ∈ Q | x ¬ x =0 } and Q = {x ∈ Q | x = the elements of Q are of the form (a ;0)and the elements ¬x}, symbolically Q = Q Q . of Q — of the form (0; a ), so that: (Q, +, R, ∗) Q = {x ∈ Q | x ¬ x =0} Clearly, with is a linear space. Indeed, conjugation in Q coincides with a =(a ; a )=(a ;0)+(0;a ). (15) identity.