![]() To find the difference between two numbers, find the difference between the numbers after decrementing both. While the book does not define != it should be obvious. Now using the book in the link just find all of the subordinate functions and you will have =, >, <. ![]() Less is defined in the solutions to exercises section in the back. ForĮxample, a number is greater than another if subtracting the second We can also use subtraction to define arithmetic inequalities. Similarly, if the second is greater than the first then theĭifference will be the second minus the first because the first minus The first minus the second because the second minus the first will be If theįirst is greater than the second then the absolute difference will be If they are both the same then the absolute differences will be zeroīecause the result of taking each from the other will be zero. To find the absoluteĭifference between two numbers, add the difference between the firstĪnd the second to the difference between the second and the first:ĭef abs_diff x y = add (sub x y) (sub y x) ![]() Get zero so we need to find the absolute difference between them theĭifference regardless of the order of comparison. However, if we subtract a number from a smaller number we also OneĪpproach is to notice that the difference between two equal numbers is There are a number of ways of defining equality between numbers. So according to the intensional definition of equality, the expressions are not equal.Using "An Introduction To Functional Programming Through Lambda Calculus" by Greg Michaelson 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. The usual domains, such as Boolean and real will be available. 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. 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. Deductive lambda calculus considers what happens when lambda terms are regarded as mathematical expressions.
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