Algebraic Information Theory For Binary Channels

We study the algebraic structure of the monoid of binary channels and show that it is dually isomorphic to the interval domain over the unit interval with the operation from [K. Martin. Entropy as a fixed point. ICALP 2004. Theoretical Computer Science, Volume 350, Issues 2–3, p. 292–324, 2006]. We show that the capacity of a binary channel is Scott continuous as a map on the interval domain and that its restriction to any maximally commutative submonoid of binary channels is an order isomorphism onto the unit interval. These results allows us to solve an important open problem in the analysis of covert channels: a provably correct method for injecting noise into a covert channel which will reduce its capacity to any level desired in such a way that the practitioner is free to insert the noise at any point in the system.