![]() So according to the intensional definition of equality, the expressions are not equal. Still they are not alpha convertible, or even eta convertible (the latter follows because both terms are already in eta-long form). The two terms beta reduce to similar expressions. Also, for some examples, the let expression will be used. So multiplication will be represented by a dot. ![]() The purpose is to see what problems arise from this definition.įunction application will be represented using the lambda calculus syntax. Mathematical equality will be applied to these domains. First, lets consider how to encode a conditional expression of the form: if P then A else B (i.e., the value of the. The usual domains, such as Boolean and real will be available. Encoding Conditionals and Boolean Values. Being aware of the problems allows them to be avoided in some cases.įor this discussion, the lambda abstraction is added as an extra operator in mathematics. The problems arise with the interaction of lambda calculus with other mathematical systems. This is not a criticism of pure lambda calculus, and lambda calculus as a pure system is not the primary topic here. This article describes these problems and how they arise. The problems are related to the definition of the lambda abstraction and the definition and use of functions as the basic type in lambda calculus. The use of lambda abstractions, which are then embedded into other mathematical systems, and used as a deductive system, leads to a number of problems, such as Curry's paradox. These languages implement the lambda abstraction, and use it in conjunction with application of functions, and types. Boolean Logic in the Lambda Calculus Logic operators can be expressed with if-then-else: and p. Lambda calculus is the model and inspiration for the development of functional programming languages. Later the lambda calculus was resurrected as a definition of a programming language. Combinatory logic is closely related to lambda calculus, and the same paradoxes exist in each. Haskell Curry studied of illative (deductive) combinatory logic in 1941. The existence of these paradoxes meant that the lambda calculus could not be both consistent and complete as a deductive system. Haskell Curry found that the key step in this paradox could be used to implement the simpler Curry's paradox. However soon after inventing it major logic problems were identified with the definition of the lambda abstraction: The Kleene–Rosser paradox is an implementation of Richard's paradox in the lambda calculus. ![]() ![]() The expression would equal the reduction of the expression.Īlonzo Church invented the lambda calculus in the 1930s, originally to provide a new and simpler basis for mathematics. Considered as a mathematical deductive system, each reduction would not alter the value of the expression. In this interpretation, if the expression never reduces to normal form then the program never terminates, and the value is undefined. One interpretation of the untyped lambda calculus is as a programming language where evaluation proceeds by performing reductions on an expression until it is in normal form. Lambda expressions come in four varieties: Variables, which are usually taken to be any lowercase letters. In order to see if the definitions of true and false you mention are adequate, try to define the other logical operators to work with them and see if the whole picture is better suited for your purposes.Deductive lambda calculus considers what happens when lambda terms are regarded as mathematical expressions. The lambda calculus derives its usefulness from having a sparse syntax anda simple semantics, and yet it retains sufcient power to represent all com-putable functions. I don't think it makes sense to say that some definition is outright wrong maybe you can say it is not interesting for it lacks some important property. Even so that there are two possible definitions for not, depending on the evaluation strategy used for the lambda terms (as you can see in the link). Observe that for this to work, lambda terms must be considered equivalent under certain reduction procedures. If you swap the definitions for $true$ and $false$ you need to do as well for, e.g., the definitions of $or$ and $and$, as already pointed out. Roughly speaking, our lambda calculus (extended with booleans) has two kinds of values: bool s ( true or false) and abstractions. In this sense they are somewhat arbitrary. The definitions of true as $\lambda xy.x$ and of false as $\lambda xy.y$ were given in order to be used together with, e.g., the definitions of and as $\lambda x y. You're asking about what is known as Church Encoding.
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