Dixin's Blog

I uploaded a NuGet package of Microsoft Concurrency Visualizer SDK: ConcurrencyVisualizer. Microsoft Concurrency Visualizer is an extension tool for Visual Studio. It is a great tool for performance profiling and multithreading execution visualization. It also has a SDK library to be invoked by code and draw markers and spans in the timeline. I used it to visualize sequential LINQ and Parallel LINQ (PLINQ) execution in my Functional Programming and LINQ tutorials. For example, array.Where(…).Select(…) sequential LINQ query and array.AsParallel().Where(…).Select(…) Parallel LINQ query can be visualized as following spans:

TransientFaultHandling.Core is retry library for transient error handling. It is ported from Microsoft Enterprise Library’s TransientFaultHandling library, a library widely used with .NET Framework. The retry pattern APIs are ported to .NET Core/.NET Standard, with outdated configuration API updated, and new retry APIs added for convenience.

Monad is a powerful structure, with the LINQ support in C# language, monad enables chaining operations to build fluent workflow, which can be pure. With these features, monad can be used to manage I/O, state changes, exception handling, shared environment, logging/tracing, and continuation, etc., in the functional paradigm.

As fore mentioned endofunctor category can be monoidal (the entire category. Actually, an endofunctor In the endofunctor category can be monoidal too. This kind of endofunctor is called monad. Monad is another important algebraic structure in category theory and LINQ. Formally, monad is an endofunctor equipped with 2 natural transformations:

Given monoidal categories (C, ⊗, I_C) and (D, ⊛, I_D), a strong lax monoidal functor is a functor F: C → D equipped with:

A functor is the mapping from 1 object to another object, with a “Select” ability to map 1 morphism to another morphism. A bifunctor (binary functor), as the name implies, is the mapping from 2 objects and from 2 morphisms. Giving category C, D and E, bifunctor F from category C, D to E is a structure-preserving morphism from C, D to E, denoted F: C × D → E:

If F: C → D and G: C → D are both functors from categories C to category D, the mapping from F to G is called natural transformation and denoted α: F ⇒ G. α: F ⇒ G is actually family of morphisms from F to G, For each object X in category C, there is a specific morphism α_X: F(X) → G(X) in category D, called the component of α at X. For each morphism m: X → Y in category C and 2 functors F: C → D, G: C → D, there is a naturality square in D:

In category theory, functor is a mapping from category to category. Giving category C and D, functor F from category C to D is a structure-preserving morphism from C to D, denoted F: C → D:

Monoid is an important algebraic structure in category theory. A monoid M is a set M equipped with a binary operation ⊙ and a special element I, denoted 3-tuple (M, ⊙, I), where

Category theory is a theoretical framework to describe abstract structures and relations in mathematics, first introduced by Samuel Eilenberg and Saunders Mac Lane in 1940s. It examines mathematical concepts and properties in an abstract way, by formalizing them as collections of items and their relations. Category theory is abstract, and called "general abstract nonsense" by Norman Steenrod; It is also general, therefore widely applied in many areas in mathematics, physics, and computer science, etc. For programming, category theory is the algebraic theory of types and functions, and also the rationale and foundation of LINQ and any functional programming. This chapter discusses category theory and its important concepts, including category, morphism, natural transform, monoid, functor, and monad, etc. These general abstract concepts will be demonstrated with intuitive diagrams and specific C# and LINQ examples. These knowledge also helps building a deep understanding of functional programming in C# or other languages, since any language with types and functions is a category-theoretic structure.

All the previous parts demonstrated what lambda calculus can do – defining functions to model the computing, applying functions to execute the computing, implementing recursion, encoding data types and data structures, etc. Lambda calculus is a powerful tool, and it is Turing complete. This part discuss some interesting problem that cannot be done with lambda calculus – asserting whether 2 lambda expressions are equivalent.

p is the fixed point (aka invariant point) of function f if and only if:

In lambda calculus, the primitive is function, which can have free variables and bound variables. Combinatory logic was introduced by Moses Schönfinkel and Haskell Curry in 1920s. It is equivalent variant lambda calculus, with combinator as primitive. A combinator can be viewed as an expression with no free variables in its body.

In lambda calculus and Church encoding, there are various ways to represent a list with anonymous functions.

Besides modeling values like Boolean and numeral, anonymous function can also model data structures. In Church encoding, Church pair is an approach to use functions to represent a tuple of 2 items.

Anonymous functions can also model numerals and their arithmetic. In Church encoding, a natural number n is represented by a function that calls a given function for n times. This representation is called Church Numeral.