# An Introduction to Functional Programming Through Lambda Calculus (Dover Books on Mathematics)

## Greg Michaelson

Language: English

Pages: 336

ISBN: 0486478831

Format: PDF / Kindle (mobi) / ePub

Functional programming is rooted in lambda calculus, which constitutes the world's smallest programming language. This well-respected text offers an accessible introduction to functional programming concepts and techniques for students of mathematics and computer science. The treatment is as nontechnical as possible, and it assumes no prior knowledge of mathematics or functional programming. Cogent examples illuminate the central ideas, and numerous exercises appear throughout the text, offering reinforcement of key concepts. All problems feature complete solutions.

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The numbers after decrementing both. The difference between a number and zero is the number: For example: Notice that this version of subtraction will return zero if the second number is larger than the first, for example: This is because pred returns zero from decrementing zero. This form of subtraction is known as natural subtraction. 4.7.3 Comparison There are a number of ways of defining equality between numbers. One approach is to notice that the difference between two equal.

Items. For example, for a worst case sort of 1000 items, it is necessary to skip 500 000 items. Remember, for searching and sorting in a linear list, each skip involves a comparison between a list item and a required or new item. If the items are strings, then comparison is character by character, so the number of comparisons can be rather big for relatively short lists. Note that we have been considering naive linear list algorithms. For particular problems, if there is a known ordering on a.

Solving a problem involves carrying out operations on values. Different values are used to solve different instances of a problem. If the values for a particular instance were built into the program, then they would have to be changed when the program was used to solve a different instance. A fruitful approach to problem analysis is to try to identify a general case of the problem. Programming languages enable the implementation of general case solutions through the use of names to stand for.

Bound variable of the innermost enclosing function to define it. Consider the following contrived example. We might define: This is a function with bound variable: and body: Thus, in the expression: the xs correspond to the second rather than the first bound variable. We would normally avoid any confusion by renaming: For lexical scope, the bound variable corresponding to a name in an expression is determined by their relative positions in that expression, before the expression is.

Programming languages will also be considered. Next, we will take an overview of λ calculus as a system for abstractions based on functions and function applications. The rules for constructing and evaluating λ expressions will be discussed informally in some detail. A notation for defining named functions will be introduced, and we will look at how functions may be constructed from other functions. We will also consider functions for manipulating pairs of values. These will be used as building.