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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).
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What is the difference between
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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?
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I have managed to teach the Yoneda lemma to students who knew very little about category theory, here’s how you do it.
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Cartesian closed categories are not regarded as complete models of the Simply Typed \(\lambda\)-calculus in the traditional sense. Let’s see why.
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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.
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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.