Functional programming in the context of Composability

Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality.

Many areas of computer science also rely on category theory, such as functional programming and semantics.

C++ is a high-level, general-purpose programming language created by Danish computer scientist Bjarne Stroustrup. First released in 1985 as an extension of the C programming language, adding object-oriented (OOP) features, it has since expanded significantly over time adding more OOP and other features; as of 1997/C++98 standardization, C++ has added functional features, in addition to facilities for low-level memory manipulation for systems like microcomputers or to make operating systems like Linux or Windows, and even later came features like generic programming (through the use of templates). C++ is usually implemented as a compiled language, and many vendors provide C++ compilers, including the Free Software Foundation, LLVM, Microsoft, Intel, Embarcadero, Oracle, and IBM.

C++ was designed with systems programming and embedded, resource-constrained software and large systems in mind, with performance, efficiency, and flexibility of use as its design highlights. C++ has also been found useful in many other contexts, with key strengths being software infrastructure and resource-constrained applications, including desktop applications, video games, servers (e.g., e-commerce, web search, or databases), and performance-critical applications (e.g., telephone switches or space probes).

In computer programming, a conditional statement directs program control flow based on the value of a condition; a Boolean expression. A conditional expression evaluates to a value without the side-effect of changing control flow.

Many programming languages (such as C) have distinct conditional statements and expressions. In pure functional programming, a conditional expression does not have side-effects, many functional programming languages with conditional expressions (such as Lisp) support side-effects.

In object-oriented (OO) and functional programming, an immutable object (unchangeable object) is an object whose state cannot be modified after it is created. This is in contrast to a mutable object (changeable object), which can be modified after it is created. In some cases, an object is considered immutable even if some internally used attributes change, but the object's state appears unchanging from an external point of view. For example, an object that uses memoization to cache the results of expensive computations could still be considered an immutable object.

Strings and other concrete objects are typically expressed as immutable objects to improve readability and runtime efficiency in object-oriented programming. Immutable objects are also useful because they are inherently thread-safe. Other benefits are that they are simpler to understand and reason about and offer higher security than mutable objects.

Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically type-checked and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming.

Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language. Python 3.0, released in 2008, was a major revision and not completely backward-compatible with earlier versions. Beginning with Python 3.5, capabilities and keywords for typing were added to the language, allowing optional static typing. As of 2025, the Python Software Foundation supports Python 3.10, 3.11, 3.12, 3.13, and 3.14, following the projects annual release cycle and five-year support policy. Earlier versions in the 3.x series have reached end-of-life and no longer receive security updates.

Ruby is a general-purpose programming language. It was designed with an emphasis on programming productivity and simplicity. In Ruby, everything is an object, including primitive data types. It was developed in the mid-1990s by Yukihiro "Matz" Matsumoto in Japan.

Ruby is interpreted, high-level, and dynamically typed; its interpreter uses garbage collection and just-in-time compilation. It supports multiple programming paradigms, including procedural, object-oriented, and functional programming. According to the creator, Ruby was influenced by Perl, Smalltalk, Eiffel, Ada, BASIC, and Lisp.

Idempotence (UK: /ˌɪdɛmˈpoʊtəns/, US: /ˈaɪdəm-/) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency).

The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from idem + potence (same + power).

In computer science, an operation or expression is said to have a side effect if it has any observable effect other than its primary effect of reading the value of its arguments and returning a value to the invoker of the operation. Example side effects include modifying a non-local variable, a static local variable or a mutable argument passed by reference; performing I/O; or calling other functions with side-effects. In the presence of side effects, a program's behaviour may depend on history; that is, the order of evaluation matters. Understanding and debugging a function with side effects requires knowledge about the context and its possible histories.Side effects play an important role in the design and analysis of programming languages. The degree to which side effects are used depends on the programming paradigm. For example, imperative programming is commonly used to produce side effects, to update a system's state. By contrast, declarative programming is commonly used to report on the state of system, without side effects.

Functional programming aims to minimize or eliminate side effects. The lack of side effects makes it easier to do formal verification of a program. The functional language Haskell eliminates side effects such as I/O and other stateful computations by replacing them with monadic actions. Functional languages such as Standard ML, Scheme and Scala do not restrict side effects, but it is customary for programmers to avoid them.

ML (Meta Language) is the metalanguage developed for the Edinburgh LCF theorem prover in the 1970s. It is an early statically typed, functional language with polymorphic type inference in the Hindley–Milner style, and other features like exceptions and mutable variables. ML's design in LCF directly inspired the later ML family (notably Standard ML, Caml, and their derivatives) and influenced subsequent functional language development.

Common Lisp (CL) is a dialect of the Lisp programming language, published in American National Standards Institute (ANSI) standard document ANSI INCITS 226-1994 (S2018) (formerly X3.226-1994 (R1999)). The Common Lisp HyperSpec, a hyperlinked HTML version, has been derived from the ANSI Common Lisp standard.

The Common Lisp language was developed as a standardized and improved successor of Maclisp. By the early 1980s several groups were already at work on diverse successors to MacLisp: Lisp Machine Lisp (aka ZetaLisp), Spice Lisp, NIL and S-1 Lisp. Common Lisp sought to unify, standardise, and extend the features of these MacLisp dialects. Common Lisp is not an implementation, but rather a language specification. Several implementations of the Common Lisp standard are available, including free and open-source software and proprietary products.Common Lisp is a general-purpose, multi-paradigm programming language. It supports a combination of procedural, functional, and object-oriented programming paradigms. As a dynamic programming language, it facilitates evolutionary and incremental software development, with iterative compilation into efficient run-time programs. This incremental development is often done interactively without interrupting the running application.