closed und is general recursive. 3.Church defined an idealized programming language called the lambda cal an be written as a lar a term It was proved by Church,Kleene.Rosser,and Turing that all three computational of computable fun ns Whe mode notioof comutability isaquestion tht ao benseredbecause thereisn 1.5 Connections to computer science The lambda calculus is a very idealized programming language:arguably.it is the Because or Its simpl hining and proving properties of program Many real-world programming languages can be regarded as extensions of the input/output,side effects da calculus provide a a le for stu h extensions,in isolation and jointly,to see how the a well-formed rogram will n CPeiesofpoeramaminglanguae(SLc The lambda calculus is also a tool used in compiler construction,see e.g.[8.9]. 1.6 Connections to logic In the 19th and early 20th centuries,there was a philosophical dispute among t what a proof is.The 0-9 ists such as that it is sufficient to derive a contradiction from the assumption that it doesn't exist. closed under certain operations (such as compositions and recursion). He postulated that a function is computable (in the intuitive sense) if and only if it is general recursive. 3. Church defined an idealized programming language called the lambda cal￾culus, and postulated that a function is computable (in the intuitive sense) if and only if it can be written as a lambda term. It was proved by Church, Kleene, Rosser, and Turing that all three computational models were equivalent to each other, i.e., each model defines the same class of computable functions. Whether or not they are equivalent to the “intuitive” notion of computability is a question that cannot be answered, because there is no formal definition of “intuitive computability”. The assertion that they are in fact equivalent to intuitive computability is known as the Church-Turing thesis. 1.5 Connections to computer science The lambda calculus is a very idealized programming language; arguably, it is the simplest possible programming language that is Turing complete. Because of its simplicity, it is a useful tool for defining and proving properties of programs. Many real-world programming languages can be regarded as extensions of the lambda calculus. This is true for all functional programming languages, a class that includes Lisp, Scheme, Haskell, and ML. Such languages combine the lambda calculus with additional features, such as data types, input/output, side effects, udpateable memory, object orientated features, etc. The lambda calculus provides a vehicle for studying such extensions, in isolation and jointly, to see how they will affect each other, and to prove properties of programming language (such as: a well-formed program will not crash). The lambda calculus is also a tool used in compiler construction, see e.g. [8, 9]. 1.6 Connections to logic In the 19th and early 20th centuries, there was a philosophical dispute among mathematicians about what a proof is. The so-called constructivists, such as Brower and Heyting, believed that to prove that a mathematical object exists, one must be able to construct it explicitly. Classical logicians, such as Hilbert, held that it is sufficient to derive a contradiction from the assumption that it doesn’t exist. 9