四十岁满，
\(\{0, 1\}\) The simplest set represents the difference.
0, 1 are just symbols in the alphabet of size 2. Let 1-bit variable, hold either 0 or 1.
Operations on the variable(s)
- unary:
- NOT: represents a change, either from 0 to 1, or from 1 to 0.
- binary:
- The simplest relationship involves two variables. Let's care about the symmetrical relaitonships only.
| input | Gnd | NOR | XOR | NAND | AND | XNOR | OR | Vcc |
|---|---|---|---|---|---|---|---|---|
| 00 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 01/10 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 11 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| A, b | A->B | A<-B | A<->B |
|---|---|---|---|
| 00 | 1 | 1 | 1 |
| 01 | 1 | 0 | 0 |
| 10 | 0 | 1 | 0 |
| 11 | 1 | 1 | 1 |
With the help of NOT, we can reduce the operations into three:
- AND
- OR
- XOR: \(F_{SOP} = A\cdot\bar{B} + \bar{A}\cdot B\); \(F_{POS} = (A+B)\cdot(\bar{A} + \bar{B})\)
These operations are communitive and associative, so they can apply to multiple variables to form relationship:
- ALL
- ANY
Truth Table
| \(I_{n-1}\) | ... | \(I_1\) | \(I_0\) | - | \(O_{m-1}\) | ... | \(O_1\) | \(O_0\) |
|---|---|---|---|---|---|---|---|---|
| 0 | ... | 0 | 0 | - | 1 | ... | 0 | 1 |
| 0 | ... | 0 | 1 | - | 0 | ... | 1 | 1 |
| ... | ... | ... | ... | - | ... | ... | ... | ... |
| 1 | ... | 1 | 1 | - | 1 | ... | 1 | 0 |
Every \(O_j\) for \(j \in \{0, 1, \cdots, m-1\}\), is a function of \(\{I_{0},I_{1},\cdots,I_{n-1}\}\), which can be represented as POS or SOP.
- Think SOP as
any of the possible 1 filters work - Think POS as
all the 0 filters failed
LISP揭示了程序语言的本质，LISP程序本身就是一个AST。非叶子结点代表操作符，其叶子结点代表操作数。