# How does Lamport imply anomalous behavior is impossible given these constraints?

Lamport's paper: Time, Clocks, and the ordering of Events in a Distributed System

Anomalous behavior occurs because Lamport's logical clocks are not enough to guarantee that the total ordering of events derived from the unique partial ordering and a non-unique total ordering is not enough to guarantee that the ordering is correct with respect to physical time.

He gives two conditions PC1 and PC2:

PC1: There exists a constant $$\kappa$$ << 1 such that for all i:

$$|\frac{dC_{i}}{dt} - 1| < \kappa$$

PC2: For all i, j:

$$|C_{i}(t) - C_{j}(t)| < \epsilon$$

PC1 says that individual clock drift in the system must be less than some constant $$\kappa$$. PC2 says that clocks must be synchronized so that the difference in time between any 2 clocks does not exceed $$\epsilon$$.

This is what I do not understand. He says that anomalous behavior is impossible if this proposition holds, deduced from PC2.

proposition: PC2 implies $$|C_{i}(t + \mu) - C_{j}(t)| > 0$$ if $$\frac{\epsilon}{1 - \kappa} \leq \mu$$.

I have no idea where he comes up with this, maybe my math isn't too great.

Note: $$\mu$$ is a constant that must be less than the shortest transmission time for interprocess messages.

PC1 implies $$|C_{i}(t + \mu) - C_{i}(t)| > (1 - \kappa)\mu$$. This is apparent because (1 - $$\kappa$$) is less than 1. But apparently he combines this with PC2 to deduce the proposition.

How does he arrive at this??

The constraint as written by Lamport is $$C_i(t+\mu)-C_j(t)\gt0$$ or $$C_i(t+\mu)\gt C_j(t)$$ which can be re-written $$C_i(t+\mu) - C_i(t)\gt C_j(t) - C_i(t) \tag{*}\label{*}$$
Given Lamport's previous deductions, we recognise the boundary conditions each side must satisfy namely: $$(1-\kappa)\mu \lt C_i(t+\mu) - C_i(t)$$ $$\epsilon \gt C_j(t) - C_i(t)$$
Now, Lamport wants to find out the range of values for 'variables' $$\epsilon,\kappa$$ ($$\mu$$ is fixed) that will ensure the strict inequality $$\eqref{*}$$. Obviously, for real numbers (Lamport's starting assumption) if $$x \lt A$$ and $$y \gt B$$, then as long as $$y \leq x$$ we know $$A\gt B$$.
In other words, as long as $$\epsilon \leq (1-\kappa)\mu$$ then $$\eqref{*}$$ is true and so $$C_i(t+\mu)-C_j(t)\gt0$$