상세정보

Many programming languages provides mechanism to guarantee the error ranges of exact numbers and intervals. However, when exact numbers and intervals are integrated with unreliable approximated numbers, the error ranges are not reliable anymore. Such unreliable error-ranges may cause serious errors in programs, and in safety critical systems they cost us huge amount of money and/or threaten human's life.

Hence, in this thesis, we propose a novel number system to safely perform arithmetic operations with guaranteed error ranges. In our number system, exact numbers are separated from approximated numbers. Among approximated numbers, numbers with strictly guaranteed error-ranges are again separated from numbers without any guarantee, such as floating-point numbers. A three-valued logic is shipped with our number system to appropriately deal with uncertainties due to approximations. We implement a prototype of our number system in Python, and demonstrate that this system provides more reliable arithmetic operations and conditional judgments involving numbers.

1 Introduction                                              1
1.1 Motivation and Goal . . . . . . . . . . . . . . . . . . 1
1.2 Related Works . . . . . . . . . . . . . . . . . . . . . 4
1.3 Backgrounds . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Organization  . . . . . . . . . . . . . . . . . . . . . 7
2 Numeric Datatypes                                         8
2.1 Categorization of Numbe . . . . . . . . . . . . . . . . 8
2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Exact numbers . . . . . . . . . . . . . . . . . . . . 9
2.2.2 Proper intervals . . . . . . . . . . . . . . . . . . 12
2.2.3 Unwarranted numbers  . . . . . . . . . . . . . . . . 12
2.3 Arithmetic operations and type conversions . . . . . . 13
2.3.1 Operations on exact numbers  . . . . . . . . . . . . 13
2.3.2 Operations on intervals and exact numbers  . . . . . 14
2.3.3 Implicit coercions for foreign datatypes . . . . . . 15
2.3.4 Explicit type conversions  . . . . . . . . . . . . . 16
3 Three-Valued Logic                                       17
3.1 TTV: The set of three truth values . . . . . . . . . . 17
3.2 Three-valued comparisons between two exact numbers . . 20
3.3 Equalities between reliable numbers  . . . . . . . . . 22
3.4 Orders between reliable numbers  . . . . . . . . . . . 24
4 Implementation in Python                                 28
4.1 Exact numbers  . . . . . . . . . . . . . . . . . . . . 28
4.2 Intervals and unwarranted numbers  . . . . . . . . . . 31
4.3 Three-valued logic . . . . . . . . . . . . . . . . . . 32
4.4 Case studies . . . . . . . . . . . . . . . . . . . . . 33
5 Conclusion                                               37
5.1 Summary  . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Future works . . . . . . . . . . . . . . . . . . . . . 37
A Python Code                                              39
A.1 tvl.py: Three-valued Logic . . . . . . . . . . . . . . 39
A.2 relnum.py: Reliable Numbers  . . . . . . . . . . . . . 41
B Bibliography                                             65