Formally, beta reduction (also written \(\beta\)-reduction) is the replacement of a bound variable in a function body with a function argument. The purpose of \(\beta\)-reduction is to compute the result of a function by function application using specific rules. More formally, the beta reduction rule states that a function application of the form \((\lambda x.t)s\) reduces to the term \(t[x := s]\). The term \(t[x := s]\) means that all instances of \(x\) in \(t\) are replaced with \(s\). The \(\rightarrow\) syntax is used as a shorthand for beta reduction. We can specify beta-reduction explicitly using the notation \((\lambda x.t)s \rightarrow t[x := s]\), which means that the beta reduction of \((\lambda x.t)s)\) is \(t[x := s]\). The beta reduction removes the \(\lambda\) symbol and resolves to the function’s body with the argument \(s\) substituted into the body. ...

The Kubernetes sweet-spot is running stateless microservices that can scale horizontally. By keeping state out of your application, Kubernetes can seamlessly add, remove, or restart pods to keep your service healthy and scalable. Developing a stateless application is, without question, the easiest way to ensure that your app can scale with Kubernetes. However, there are some workloads that do not run effectively in a stateless way, and for that, Kubernetes offers a few tools for developing stateful applications: leader election, StatefulSets and session affinity. ...

The purpose of eta reduction (also written \(\eta\)-reduction) is to drop an abstraction over a function to simplify it. This is possible when there is nothing more that a function can do to its argument. For example, imagine that we have a simple function \( f\ x = g\ x \). Both \(g\) and \(f\) take the same argument, \(x\), and the function application function results in the same value (specified by the equality symbol). Since both \(f\) and \(g\) take the same argument and produce the same result, we can simplify the equation to just \(f = g\). In lambda calculus, this simplification is called \(\eta\)-reduction. ...

Complex systems exhibit unexpected behavior. — John Gall, The Systems Bible One reaction to the myriad failure cases we encounter with distributed systems is to add more testing. Unfortunately, testing is a best-effort verification of system correctness — we simply cannot predict the failure cases that will happen in production. What’s more, any environment that we use to verify system behaviour is — at best — a pale imitation of our production environment. An alternative to adding more tests is to add more monitoring and alerting so we know as soon as possible that a system shows signs of degradation. Unfortunately, monitoring and alerting suffers from the same faults as testing — we monitor for behaviour that is predictable in nature or that we have experienced in the past. Together, monitoring and testing try to enumerate all possible permutations of partial and total system failure. Unfortunately, they only provide a simulation of how you expect a system to actually function in production. ...

A lambda expression is said to be in normal form if it cannot be reduced any further, meaning that the expression no longer contains any function applications. More formally, a reducible expression is called a redex, and a lambda expression is in normal form when it contains no more redexes. Redex A reducible function expression. Normal Form A lambda expression that contains no redexes. Given a lambda expression, there are two primary strategies for reducing it to normal form: normal-order evaluation, or applicative-order evaluation. This article discusses both methods, their pros and cons, and an alternative evaluation strategy that combines the pros from both — lazy evaluation. ...

Having covered types, let’s now turn our attention to lists. Lists are general purpose data structures for storing sequences of items. In lambda calculus, lists are represented using pairs, with the first item of the pair representing the head of the list, and the second item representing the rest of the list. A special value, nil, at as the second item of the pair terminates the list. Pairs Let’s start by focusing on pairs (or tuples). A pair is built from two arguments, \(a\) and \(b\), and returns a function \(f\) enclosing those two arguments: ...

Lambda calculus is a very simple language. If you take away any syntactic sugar, all you are left with is functions that take arguments and return results. You can use these simple building blocks to construct functions that represent numbers and arithmetic, but there is no way to restrict, for example, arithmetic functions to require numeric operands. This is where types come in. Generally speaking, types allow you to control the use of functions so that only meaningful combinations of inputs and outputs are used. ...

Kubernetes was built to run distributed systems over a cluster of machines. The very nature of distributed systems makes networking a central and necessary component of Kubernetes deployment, and understanding the Kubernetes networking model will allow you to correctly run, monitor and troubleshoot your applications running on Kubernetes. Networking is a vast space with a lot of mature technologies. For people unfamiliar with the landscape, this can be uncomfortable because most people have existing preconceived notions about networking, and there are a lot of both new and old concepts to understand and fit together into a coherent whole. A non-exhaustive list might include technologies like network namespaces, virtual interfaces, IP forwarding, and network address translation. This guide intends to demystify Kubernetes networking by discussing each of Kubernetes dependent technologies along with descriptions on how those technologies are used to enable the Kubernetes networking model. ...

Early computers were described using mathematical notation and theoretical constructions, which were then translated by enterprising machinists into custom built calculation engines. The first computer programmers using these machines built applications directly in hardware by plugging together wires and sockets in varying configurations. Even with these short-comings, the value of general purpose computing machines was evident. As companies began to realize the utility of computing, they saw computing as a means for differentiating themselves from their competitors. ...

In a purely functional language — like lambda calculus — programs are expressed as nested function calls. Repetition in such an environment requires that nesting of function calls continues until some condition is met. During the repetition, each function passes its result to the next function in the nested chain and this repetition is completed when a test for some condition passes. The repetitive behaviour I’ve just described is recursion: ...