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## Reliable approximated number system with exact bounds and three-valued logic

Material type
학위논문
Personal Author
차리서 車里西
Title Statement
Reliable approximated number system with exact bounds and three-valued logic / Reeseo Cha
Publication, Distribution, etc
Seoul :   Graduate School, Korea University,   2018
Physical Medium
iii, 67장 : 도표 ; 26 cm
기타형태 저록
Reliable Approximated Number System with Exact Bounds and Three-valued Logic   (DCOLL211009)000000080254
학위논문주기
학위논문(박사)-- 고려대학교 대학원: 컴퓨터·전파통신공학과, 2018. 2
학과코드
0510   6YD36   342
General Note
지도교수: 최진영
부록: Python code
Bibliography, Etc. Note
참고문헌: 장 65-67
이용가능한 다른형태자료
PDF 파일로도 이용가능;   Requires PDF file reader(application/pdf)
비통제주제어
formal methods , number system , exact number , interval arithmetic , three-valued logic,,
 000 00000nam c2200205 c 4500 001 000045932642 005 20180417131634 007 ta 008 180102s2018 ulkd bmAC 000c eng 040 ▼a 211009 ▼c 211009 ▼d 211009 085 0 ▼a 0510 ▼2 KDCP 090 ▼a 0510 ▼b 6YD36 ▼c 342 100 1 ▼a 차리서 ▼g 車里西 245 1 0 ▼a Reliable approximated number system with exact bounds and three-valued logic / ▼d Reeseo Cha 260 ▼a Seoul : ▼b Graduate School, Korea University, ▼c 2018 300 ▼a iii, 67장 : ▼b 도표 ; ▼c 26 cm 500 ▼a 지도교수: 최진영 500 ▼a 부록: Python code 502 1 ▼a 학위논문(박사)-- ▼b 고려대학교 대학원: ▼c 컴퓨터·전파통신공학과, ▼d 2018. 2 504 ▼a 참고문헌: 장 65-67 530 ▼a PDF 파일로도 이용가능; ▼c Requires PDF file reader(application/pdf) 653 ▼a formal methods ▼a number system ▼a exact number ▼a interval arithmetic ▼a three-valued logic 776 0 ▼t Reliable Approximated Number System with Exact Bounds and Three-valued Logic ▼w (DCOLL211009)000000080254 900 1 0 ▼a Cha, Ree Seo, ▼e 저 900 1 0 ▼a 최진영 ▼g 崔振榮, ▼e 지도교수 945 ▼a KLPA

### Electronic Information

No. Title Service
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Reliable approximated number system with exact bounds and three-valued logic (34회 열람)

### Holdings Information

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No. 1 Location Call Number 0510 6YD36 342 Accession No. 123058291 Availability Available Due Date Make a Reservation Service
No. 2 Location Call Number 0510 6YD36 342 Accession No. 123058292 Availability Available Due Date Make a Reservation Service

### Contents information

#### Abstract

```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```