2017q3 Homework1 (ternary)
===
contributed by <`ZixinYang`>

---
### Review by `williamchangTW`

- 根據作者的內容，對數位邏輯的部份做了詳細的解釋，讓我有了初步的了解，或許可以搭配電路圖的呈現可以更加了解其中的運作
- 作者有提到 IOTA 其中的一些重點，詳細分析下去應該可以得到不錯的結論
>小弟坐井觀天，如有冒犯請見諒[name="williamchangTW"]

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## Balanced Ternary 簡介

### 原理
* 3為基數,每一位元用-1(T),0,1表示
* 因不需額外符號表示負號,故在加減法及乘法上效率較二進位高
* [Ternary Adder](http://homepage.divms.uiowa.edu/~jones/ternary/arith.shtml)
* Ternary Multiplier
* Truth Table
* K-map
* Implementation
* Output equations
* F~prod~ = X~1~^1^*X~2~^2^ + X~1~^2^*X~2~^1^ + [X~1~^1^*X~2~^1^ + X~1~^2^*X~1~^2^*X~2~^2^]
* F~carry~ = X~1~^2^*X~2~^2^
* 出處：[Ternary Digital System: Concepts and Applications](https://www.researchgate.net/publication/266477093_Ternary_Digital_System_Concepts_and_Applications) P.87

### 應用案例
* [IOTA](https://iota.readme.io/docs)
* [Introduction to IOTA](https://www.youtube.com/watch?v=yj9j_a_ACB4)
* Utilize tangle structure instead of blockchain
* Directed acyclic graph
* No blocks, the structure is very light
* Infinitely scalable
* No transaction fee and avoid waiting for verification for several days
* Why ternary applied on IOT
> These 3 states perform transaction very balanced, which is quite helpful to build a self-organizing and self-sustaining network

## 測試 Balanced Ternary
* 此表示法為 8 digits 的表示法,範圍為 -3280 ~ +3280

┌───┐    ┌───┐    ┌───┐    ├───┐    ┌─┴─┐    ┌───┤    ┌───┐    ┌───┐
│   │    │   │    ┤   │    │   │    │   │    │   │    │   ├    │   │
└─┬─┘    ├───┘    └───┘    └───┘    └───┘    └───┘    └───┘    └───┤
 (1)      (3)      (9)      (27)     (81)    (243)    (729)    (2187)
┌───┐    ┌───┐    ┌───┐    ┌┬──┐    ┌─┬─┐    ┌──┬┐    ┌───┐    ┌───┐
│   │    │   │    ├   │    │   │    │   │    │   │    │   ┤    │   │
└─┴─┘    └┴──┘    └───┘    └───┘    └───┘    └───┘    └───┘    └──┴┘
 (-1)     (-3)     (-9)     (-27)    (-81)   (-243)   (-729)   (-2187)

* [Box-Drawing Character](https://en.wikipedia.org/wiki/Box-drawing_character)

## Balanced Ternary 教學影片重點整理
* Number Systems: Ternary
* 十進位轉三進位

* Non-Binary Computing
* History
* Thomas Fowler Machine (1838)
* Calculate up to 87,000
* Do a multiplication/division for doing logarithms
* Setun
* Composed of 18 ternary digits, giving the machine numerical range of 387 million. A binary computer would need 29 digits to reach the capacity.
* As soon as the transistor came out, it was abandoned in favor of binary machine
* Conversion
* Three trips number system
* Positive number: 5 = + - - (+3^2^-3^1^-3^0^)
* Negative number: -5 = - + + (-3^2^+3^1^+3^0^)
* Range from -13 to +13

             -13 ---     +13 +++
             -12 --0     +12 ++0
             -11 --+     +11 ++-
             -10 -0-     +10 +0+
             -9  -00     +9  +00
             -8  -0+     +8  +0-
             -7  -+-     +7  +-+
             -6  -+0     +6  +-0
             -5  -++     +5  +--
             -4  0--     +4  0++
             -3  0-0     +3  0+0
             -2  0-+     +2  0+-
             -1  00-     +1  00+
             0   000

* Logic Table
Sum | - | 0 | + |
| :---: | :---: | :---: | :---: |
| - | + | - | 0 |
| 0 | - | 0 | + |
| + | 0 | + | - |

| Carry | - | 0 | + |
| :---: | :---: | :---: | :---: |
| - | - | 0 | 0 |
| 0 | 0 | 0 | 0 |
| + | 0 | 0 | + |

| Multiply | - | 0 | + |
| :---: | :---: | :---: | :---: |
| - | + | 0 | - |
| 0 | 0 | 0 | 0 |
| + | - | 0 | 0 |

| Divide| - | 0 | + |
| :---: | :---: | :---: | :---: |
| - | + |-∞ | - |
| 0 | 0 | NAN | 0 |
| + | - | ∞ | + |

> 嘗試用 Markdown 和 HackMD 擴充語法來處理文字為主的表示法 [name="jserv"][color=red]

* Full Adder
* Implementation of a full ternary adder (A+B+C=S with C^'^)
* Modulo 3 Sum and Product Functions
* Ripple Adder in Binary and Ternary Logic
* Concepts Review
* Half Adder: two inputs (addition for two 1-digit varaibles), two outputs (sum & c_out)
* Full Adder: three inputs (two 1-digit variables & carryout from previous adder), two outputs (sum & c_out)
* Ripple Adder: composed of full adders to construct an N-digit adder
* Binary Ripple Adder
* Composed of full adders which are composed of XOR and AND gates
* Ternary Ripple Adder
* Requires fewer rounds of addition and fewer circuits

## 参考資料
* [Tangle V.S. Blockchain](https://www.youtube.com/watch?v=Mlbz6vDMnZo)
* [The Balanced Ternary Machines of Soviet Russia](https://dev.to/buntine/the-balanced-ternary-machines-of-soviet-russia)
* [Why Not Ternary Computers?](https://www.techopedia.com/why-not-ternary-computers/2/32427)
* [The tangle (White paper)](http://iotatoken.com/IOTA_Whitepaper.pdf)