An Analysis of Arithmetic Constraints on Integer Intervals

Abstract

Arithmetic constraints on integer intervals are supported in many constraint programming systems. We study here a number of approaches to implement constraint propagation for these constraints. To describe them we introduce integer interval arithmetic. Each approach is explained using appropriate proof rules that reduce the variable domains. We compare these approaches using a set of benchmarks. For the most promising approach we provide results that characterize the effect of constraint propagation.

Parametric Functional Representation of Interval Number with Arithmetic Operations

“…From the above definition it is clear that [0, 0] and [1,1] are the additive and multiplicative identities respectively. Again interval arithmetic is closed, associative and commutative with respect to addition and multiplication.…”

“…Then the arithmetic sum of the two intervals S = A + B = [6,8] and the corresponding interval-valued functional form in parametric and symmetrical parametric from are given by s ( 1,3] and the interval-valued function in two different forms are specified by d (…”

“…The division A B = 4 3 , 5 2 and the interval-valued function in convex combination form of the division is f ( p) = (5 p + 4 (1 − p)) 1 2 p + 1 3 (1 − p) . The resultant interval of arithmetic operations attains its lower and upper bound for p = 0 and p = 1 as presented by Table 2 and Fig.…”

“…Several researches [1,2,9,11,15,16,32,34,35] used interval arithmetic for optimization by different techniques; interval arithmetic provides an efficient method of global optimization [10]. Interval arithmetic operations [3,[6][7][8]12,29,30] studied by several researches in various approaches and form for investigation their problem with interval in real or complex domain.…”

“…Let A =[4,5] and B =[2,3], then find the arithmetic operation using convex combination form of A + B, A − B, AB, A/B.Solution:The interval-valued function in convex combination form of A and B can be written as a ( p)= 5 p + 4 (1 − p) , b ( p) = 3 p + 2 (1 − p) .The addition A + B =[6,8] and the interval-valued function in convex combination form of the sum is s ( p) = 8 p + 6 (1 − p) .The subtraction A − B =[1,3] and convex combination form of the interval-valued function is d (p) = 5 p + 4 (1 − p) − {2 p + 3 (1 − p)} = 2 p + 1.The multiplication A × B =[8,15] and convex combination form of the interval-valued function for multiplication is m ( p) = (5 p + 4 (1 − p))(3 p …”