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The formal system which is written in the form of λ-calculus is the mathematical logic in the motive of conveying computation dependent on the functions, application and abstraction by utilizing the variable connecting and replacing the equation. The general model regarding computation which can be utilized for stimulating Turning machine. The concept of Lambda calculus reflects the concept regarding Turning complete, which is a universal model framework of computation. The letter λ, Lambda is utilized for the lambda expressions and the terms of lambda depicts the binding feature of a function. The depiction of Lambda calculus may be typed or untyped. The typed form of lambda calculus reflects the application of lambda with the ability of accommodating with the existent input’s ‘kind’ of data. The typed form of lambda calculus, untyped lambda calculus is not reflected. However, the typed form of lambda calculus allows numerous items for proof. Lambda calculus, also denoted as λ-calculus is considered to be the simple notation in application and functions. The main purpose of lambda calculus is to apply function in argument by establishing functions through abstraction. Lambda calculus or λ-calculus represents an intensive notation in the function of demonstrating functions due to the composition of basic Lambda calculus. Outcome of the application of Lambda calculus is non-extensional concept regarding the functions due to the structure and rules related to computation in comparing with extensional theory in the form of sets in ordered pairs. Lambda terms is used in the form of language for lambda calculus. The lambda terms is considered as the formal syntax and group of the transformation rules that enables the manipulation of lambda terms. The depiction of transformation rules is depicted in the form of operational definition or operational definition.

The introduction and application of Lambda calculus was first established during 1930s by Alonzo Church, a mathematician in the form of a research regarding foundations in mathematics. The complete system and representation depicted inconsistency in logic presentation in the year 1935 during the development of Kleene-Rosser paradox by J.B. Rosser and Stephen Kleene. Consequently after the discovery of Kleene-Rosser paradox, the church published a portion applicable for computation, commonly known as untyped lambda calculus. During the year 1940, the concept of simply typed lambda calculus was introduced with the feature of logical consistency. Richard Montague played a major role in diffusing the significance of Lambda calculus in computer science and linguistics.

The concept of Lambda Calculus is used in the field computer science and linguistics. The function of Lambda Calculus is needed during the depiction of lambda function in short period. The need of Lambda Calculus can be reflected in the procedure of presenting high-order function in the argument for Python. The functions of Lambda are utilized in association with functions that are built-in like map and filter.

The application of Lambda Calculus can be reflected in various regions of linguistics, computer science, philosophy and mathematics. The concept and application of Lambda Calculus plays a significant role in developing theory of the programming languages. The application of Lambda Calculus can also be reflected in functional programming languages. Category theory contains the concept of Lambda calculus as recent research topic.

The application of Lambda Calculus is perceived to be inconsistent and with discrepancies that diverts the application from logic. For incorporating the factor of logic in the concept, the mathematicians and researchers introduced the concept of untyped lambda calculus.

The concept of Reduction is related to Lambda Calculus, the reduced form of the expressions reflects the concept of lambda expressions. Three forms of the reduction includes, alpha reduction, eta reduction and beta reduction. For reflecting improved logic in the concept of Lambda Calculus, untyped lambda was used where the beta reduction in the form of rewriting rule is neither weakly normalized nor strongly.

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