Changes

Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

Revision as of 18:08, 19 June 2009

98 bytes added , 18:08, 19 June 2009

Line 6: Line 6:

==Note 1==

−

One of the first things that you can do, once you have a moderately ~~functional~~ calculus for boolean functions or propositional logic, whatever you choose to call it, is to start thinking about, and even start computing, the differentials of these functions or propositions.

+

One of the first things that you can do, once you have a moderately efficient calculus for boolean functions or propositional logic, whatever you choose to call it, is to start thinking about, and even start computing, the differentials of these functions or propositions.

Let us start with a proposition of the form <math>p ~\operatorname{and}~ q</math> that is graphed as two labels attached to a root node:

Line 34: Line 34:

|}

−

Now ask yourself: What is the value of the proposition <math>pq\!</math> at a distance of <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> from the cell <math>pq\!</math> where you are standing?

+

Now ask yourself: What is the value of the proposition <math>pq\!</math> at a distance <math>\operatorname{d}p</math> and <math>\operatorname{d}q</math> from the cell <math>pq\!</math> where you are standing?

Don't think about it — just compute:

Line 42: Line 42:

|}

−

~~This expression~~ arises by substituting <math>p + \operatorname{d}p</math> for <math>p\!</math> and <math>q + \operatorname{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and ~~expressing~~ the result as a cactus graph. This follows from the fact that the boolean sum <math>p + \operatorname{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:

+

The formula shown in the Figure arises by substituting <math>p + \operatorname{d}p</math> for <math>p\!</math> and <math>q + \operatorname{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result as a cactus graph. This follows from the fact that the boolean sum <math>p + \operatorname{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:

{| align="center" cellpadding="10"

Line 49: Line 49:

Next question: What is the difference between the value of the

−

proposition <math>pq\!</math> "over there" and the value of the proposition <math>pq\!</math> where you are, all expressed in the form of

+

proposition <math>pq\!</math> over there, at a distance <math>\operatorname{d}p</math> and <math>\operatorname{d}q,</math> and the value of the proposition <math>pq\!</math> where you are satnding, all expressed in the form of a general formula, of course? Here is the appropriate formulation:

−

a general formula, of course? Here is the appropriate formulation:

{| align="center" cellpadding="6" width="90%"

Jon Awbrey

12,080

edits