Entropy-changing actions

A terminating action on state space $X$ takes a state-before $x$ to a state-after $x'$ and so is modeled as a relation on $X$. If the action is deterministic then for each state-before there is only one state-after and so the relation is a function.

Let $(X,\leq,\sim)$ be an entropy structure. An action $A$ on $X$ is entropy decreasing iff $A$ is contained in the converse of $\leq$

$$\forall x,x' : X \spot x ~A ~x' \implies x' \leq x .$$

In other words, whenever the action $A$ can change the state of the system from an initial state to a final state, the final state must be smaller (in the given order). We refer to such an action more briefly and informally as good.

Action $A$ is entropy increasing iff it intersects $\leq$

$$\exists x,x' : X \spot x ~A ~x' \land x \leq x' .$$

In other words, it is possible to start the action $A$ in some state such that $A$ leaves the system in a larger state (in the given order). We refer to such an action informally as evil.

Finally, $A$ leaves entropy invariant iff it is contained in $\sim$

$$\forall x,x' : X \spot x ~A~ x' \implies x \sim x' .$$

In other words, action $A$ always leaves the system in an equivalent state (from the viewpoint of the given order). Such an action we refer to as benign.

The asymmetry between the definitions of entropy increasing and entropy decreasing reflects the intuition that one 'bad' instance is enough for an action to be judged 'evil'; but that to be judged 'good' all of its instances must be 'good'.

For two actions $A_0$ and $A_1$ on $X$ we say that $A_1$ is more evil than $A_0$ iff from any given state, $A_1$ results in a state with higher entropy than does $A_0$:

$$\forall x,x',x'' : X \spot (x~A_0~x' \land x~A_1~x'')\ \ \implies\ \ x' \leq x'' .$$