Reliable Approximated Number System with Exact Bounds and Three-valued Logic

Abstract

Many programming languages provides mechanism to guarantee the error ranges of exact numbers and intervals. However, when they are integrated with unreliable approximated numbers, we cannot rely on the error-ranges anymore. Such unreliable error-ranges may cause serious errors in programs, and especially in safety critical systems they cost us huge amount of money and/or threaten human's life. Hence, in this paper, we propose a novel number system to safely perform arithmetic operations with guaranteed error ranges. In the number system, exact numbers are separated from approximated numbers, and approximated numbers with strictly guaranteed error-ranges are again separated from unwarranted numbers such as floating-point numbers. A three-valued logic is also shipped with our number system to appropriately deal with uncertainties due to approximations. A prototype implementation of our number system in Python is demonstrated. With this module, we can more reliably execute operations on numbers and make judgments on the conditions involving numbers