Classes and subclasses of entities in the Physics Derivation Graph:
- derivations = an ordered set of two or more steps
- steps = a set of one or more statements related by an inference rule
- inference rule = identifies the relation of a set of one or more statements
- statement = two or more expressions (LHS and RHS) and a relational operator
- expressions = an ordered set of symbols
- symbols = a token
- operator = applies to one or more values (aka operands). Property: number of expected values
- value. Property: categorized as "variable" xor "constant"
- integer = one or more digits. The set of digits depends on the base
- float
- complex
- unit. Examples: "m" for meter, "kg" for kilogram
Some aspects of expressions and derivations I don't have names for yet:
- binary operators {"where", "for all", "when", "for"} used two relate two expressions, the "primary expression" on the left and one or more "scope"/"definition"/"constraint" (equation/inequality)
Some aspects of expressions and derivations I don't need to label in the PDG:
- terms = parts of the expression that are connected with addition and subtraction
- factors = parts of the expression that are connected by multiplication
- coefficients = a number that is multiplied by a variable in a mathematical expression.
- power, base, exponent
- base (as in decimal vs hexadecimal, etc)
- formula
- function
An equation is two expressions linked with an equal sign.
What is the superclass above "equation" and "inequality"?
So far I'm settling on "statement".
I am intentionally staying out of the realm of {proofs, theorems, axioms} both because that is outside the scope of the Physics Derivation Graph and because the topic is already addressed by OMDoc.
Suppose we have a statement like
y = x^2 + b where x = {5, 3, 1}
In that statement,
- "y = x^2 + b" is an equation
- "x^2 + b" is an expression and is related to the expression "y" by equality.
- "x^2" is a term in the RHS expression
- "x = {5, 3, 1}" is an equation that provides scope for the primary equation.
What is the "where" relation in the statement? The "where" is a binary operator that relates two equations. There are other "statement operators" to relate equations, like "for all"; see the statement
a + c = 2*g + k for all g \in \Re
In that statement, "g \in \Re" is (an equation?) serving as a scope for the primary equation.
All statements have supplemental scope/definition equations that are usually left as implicit. The reader is expected to deduce the scope of the statement from the surrounding context.
The supplemental scope/definition equations describe both per-variable and inter-variable constraints. For example,
x*y + 3 = 94 where ((x \in \Re) AND (y \in \Re) AND (x<y))
More complicated statement:
f(x) = { 0 for x<0
{ 1 for 0<=x<=1
{ 0 for x>1
Here the LHS is a function and the RHS is an integer, but the value of the integer depends on x.
Note that the "0<=x<=1" can be separated into "0<=x AND x<=1". Expanding this even more,
(f(x) = 0 for x<0) AND (f(x) = 1 for (0<=x AND x<=1)) AND (f(x) = 0 for x>1)
