- #ALPHA REDUCTION IN LAMBDA CALCULUS CODE#
- #ALPHA REDUCTION IN LAMBDA CALCULUS SERIES#
- #ALPHA REDUCTION IN LAMBDA CALCULUS FREE#
Bound variables refer to things within the lambda term, whereas free variables refer to values that aren't already defined within the expression. In the expression "λ x.E", any occurrence of x in E is bound, while any other variable is free (unless bound by another lambda expression, like the y in "λ x.λ y.xy"). That is, "E1 E2" is an example of calling E1 with E2 as its argument. Application is jargon for calling a function, and is conventionally written without any brackets. A lambda expression is either a variable (like the x in the above expression), a function in the "λ x.E" form, or an application E1E2. For instance, λ x.x is a function that takes one argument x and immediately returns that argument. The intuition is that λ x.E is a function that takes one parameter, x, and returns E. The following paragraphs give an informal introduction to lambda calculus - for a formal description of lambda calculus see under "External resources."Ī function in lambda calculus is written in the form "λ x.E", where x is the function's parameter and E is a lambda expression constituting the function body. Alan Turing 3 proved that the set of all functions definable in the lambda calculus is equivalent to the Turing-machine-computable functions (so lambda calculus is Turing-complete), thereby lending credence to the thesis - now called the Church-Turing thesis - that it's these functions, and these functions only, that one would naturally or intuitively regard as "effectively computable." In lambda calculus a function does not technically return a result based on its parameters - instead the function and its parameters are reduced to give an answer, which mathematically is equivalent to the question.
The result is the same: applying a function to its input arguments turns into the value it outputs, and these functions are strung together so that the return value of one becomes the argument of another.
#ALPHA REDUCTION IN LAMBDA CALCULUS SERIES#
The difference is that functions in imperative languages arrive at a result by executing a series of statements that correspond to an algorithm, whereas we tend to imagine lambda calculus function applications as using a process reminiscent of repeatedly replacing wildcards within a function body. The lambda calculus tries to capture the nested structure of functions within functions.
#ALPHA REDUCTION IN LAMBDA CALCULUS CODE#
The first step in an imperative language would be to replace that code with add(1, 4) after completing the multiply operation. An example is writing 1 + 2 * 2 as add(1, multiply(2, 2)). Programmers are used to calling functions and sending the values they return to other functions. Fortunately, its power and complexity is built on simple rules and intuitions, ones which the average programmer has already been exposed to.
The lambda calculus is a powerful tool, but tends to frighten away novices because of the word 'calculus' and the seemingly incomprehensible notation. That system turned out to be inconsistent, but Church salvaged and published in 1936 1 just the portion relevant to computation - what is now called the lambda calculus - and this was proved 2 to be consistent that same year.
The lambda calculus was invented by Alonzo Church in the 1930s as part of a broader attempt to formalise the foundations of mathematics.
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