while I think this is a great idea, let me tell you: This is not a degree for the faint-hearted. In one of my previous post I showed that any theory featuring general recursion is inconsistent when viewed as a logical system which inevitabl…
while I think this is a great idea, let me tell you: This is not a degree for the faint-hearted. A PhD (Doctor of Philosophy) is a research degree focused on creating new knowledge, not just learning existing knowledge. It is radically diff…
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)…
To keep this post as concise as humanly possible I will assume knowledge of (symmetric)monoidal categories, kan extensions and enriched categories. We show informally the following proposition. Proposition. Let \((\mathcal{C}, \otimes_{\mat…
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.
I have managed to teach the Yoneda lemma to students who knew very little about category theory, here’s how you do it. Say that you want to do denotational semantics for a simply typed calculus with a unary constructor \(\textsf{R}\) which
Cartesian closed categories are not regarded as complete models of the Simply Typed \(\lambda\)-calculus in the traditional sense. Let’s see why. Assume \(\Lambda_X\) is the set of closed well-typed STLC (Simply Typed \(\lambda\)-calculus)
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. Fir…
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 computati…