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# Lambda Calculus
Lambda calculus is an extremely simple and low-level [mathematical](math.md) system that can describe computations with [functions](function.md), and can in fact be used to describe and perform any computation. Lambda calculus provides a theoretical basis for [functional programming languages](functional.md) and is a **[model of computation](model_of_computation.md)** similar to e.g. a [Turing machine](turing_machine.md) or [interaction nets](interaction_net.md) -- lambda calculus has actually exactly the same computational power as a Turing machine, which is the greatest possible computational power, and so it is an alternative to it. Lambda calculus can also be seen as a simple [programming language](programming_language.md), however it is so extremely simple (there are e.g. no numbers) that its pure form isn't used for practical programming, it is more of a mathematical tool for studying computers theoretically, constructing proofs etc. Nevertheless anything that can be programmed in any classic programming language can in theory be also programmed in lambda calculus.
Lambda calculus is an extremely simple, low-level [mathematical](math.md) system that can perform computation with mathematical [functions](function.md), and can in fact be used to describe and perform any conceivable computation. Lambda calculus provides a theoretical basis for [functional programming languages](functional.md) and is a **[model of computation](model_of_computation.md)** similar to e.g. a [Turing machine](turing_machine.md) or [interaction nets](interaction_net.md) -- lambda calculus has actually exactly the same computational power as a Turing machine, which is the greatest possible computational power, and so these systems are alternatives to one another. Lambda calculus can also be viewed as a primitive [programming language](programming_language.md), however its extreme simplicity (for example the lack of even such basic concepts as [numbers](number.md)) doesn't allow its pure form to be used for practical programming, it is more of a mathematical tool for studying computers theoretically, constructing proofs etc. The system is a result of searching for most minimal systems capable of computation, just like the most essential physics equations come from searching for the most basic rules of our [Universe](universe.md). Nevertheless anything that can be programmed in any classic programming language can in theory be also programmed in lambda calculus.
While Turing machines use memory cells in which computations are performed -- which is similar to how real life computers work -- lambda calculus performs computations only by simplifying an expression made of pure mathematical functions, i.e. there are no [global variables](variable.md) or [side effects](side_effect.md) (the concept of memory is basically present in the expression itself, the lambda expression is both a program and memory at the same time). It has to be stressed that the functions in question are mathematical functions, also called **pure functions**, NOT functions we know from programming (which can do all kinds of nasty stuff). A pure function cannot have any side effects such as changing global state and its result also cannot depend on any global state or randomness, the only thing a pure function can do is return a value, and this value has to always be the same if the arguments to the function are same.
While Turing machines use memory cells as the medium to carry out computation -- which closely imitates the "number crouching" of real life computers -- lambda calculus instead performs computation solely by simplifying an expression made of pure mathematical functions -- that means there are no [global variables](variable.md) or [side effects](side_effect.md) (the role of memory is essentially replaced by the expression itself, the lambda expression is both the program and its memory at the same time). It has to be stressed that the functions in question are mathematical functions, also called **pure functions**, NOT functions we know from programming (which can do all kinds of nasty stuff). A pure function cannot have any side effects such as changing global state and its result also cannot depend on any global state or randomness, the only thing a pure function can do is return a value, and this value has to always be the same if the arguments to the function are same. In addition to this the pure mathematical functions are yet much simpler than those we encounter in high school, there are no algebraic operators or numbers, just symbols.
## How It Works