In one of my previous post I showed that any theory featuring general recursion is inconsistent when viewed as a logical system which inevitably leads to the idea that all definable functions in such a theory should be total (or productive). However, losing Turing-completeness could be somewhat problematic for some, but it can be addressed in several ways. One of these, an possibly the most popular, is to isolate recursion into a monad, effectively regarding recursion as an effect. We discuss several lifting monads which are fit for purpose.
Strong bisimulation for CCS is the preferred equivalence method in concurrency because it relates less programs than trace equality. However, the reality is that is strong bisimulation and trace equality ought to be regarded as equivalent. This is the essence behind proof assistant’s like (e.g.) Isabelle. So what is going here?
The Axiom of Choice (AC) is an axiom that states that the product of a family of non-empty sets is itself non-empty. This is a rather controversial axiom amongst mathematicians but in type theory this axiom is provable within the logic.
Any Cartesian Closed Category (CCC) with an initial object and a fixed-point operator is trivial. Essentially this means that in languages like (e.g.) Haskell the empty type is not actually empty as it contains the non-terminating computation. Perhaps this is obvious, but here’s the categorical explanation.
