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02b What is Functional Programming?
William Farmer
Overview
This video introduces fundamental concepts in functional programming, starting with the nature of values and how they are represented in programming. It defines expressions as syntactic entities that denote values and explains atomic versus compound expressions. The concept of types is introduced as a way to categorize values, preventing type errors. Finally, the video delves into the nature of functions, explaining them as both rules and sets of ordered pairs, and discussing their properties like domain, range, totality, injectivity, and surjectivity. It highlights how Haskell handles functions, particularly its use of currying to represent n-ary functions as unary functions.
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Chapters
- Programs manipulate information, which are represented as values like booleans, integers, floating-point numbers, strings, tuples, lists, and functions.
- Expressions are syntactic constructs that represent or 'denote' a value.
- Atomic expressions (like identifiers 'x' or literals '2.3', 'cat', 'true') cannot be broken down further.
- Compound expressions are built by applying functions or operators to other expressions (e.g., 'abs 10', '1 + 2').
- Evaluating an expression determines the value it denotes.
Understanding values and expressions is crucial because they form the basic building blocks of any program, allowing us to represent and manipulate data.
The expression 'x * 2 + 7' denotes a value. If 'x' has the value 2, the expression evaluates to 11.
- Types are syntactic entities that denote collections of values with similar forms.
- Examples of types include 'bool' (true, false), 'int' (machine integers), 'float' (rational numbers), and 'integer' (all integers).
- A 'type error' occurs when an operation is applied to a value of an inappropriate type (e.g., adding a string to a number).
- Haskell allows for type checking before evaluation, catching type errors early and preventing them during program execution.
- The 'integer' type in Haskell represents an infinite set of all possible integers, unlike 'int' or 'float' which are finite due to fixed bit representations.
Types help prevent errors by ensuring that operations are only applied to compatible data, leading to more robust and reliable programs.
In Haskell, typing ':type 5 + "hello"' would reveal a type error because the addition operator expects numbers, not strings.
- Functions can be viewed as a rule that maps inputs to at most one output, or as a set of ordered pairs (input, output).
- The domain is the set of all possible inputs for which a function is defined.
- The range is the set of all outputs that a function can actually produce.
- A function is 'total' if it's defined for all possible inputs in its domain.
- A function is 'surjective' if its range covers all possible outputs, and 'injective' if different inputs always produce different outputs.
Grasping the dual nature of functions (rule vs. set) and their properties is fundamental to understanding how programs transform data and solve problems.
The function 'reciprocal(x) = 1/x' is a rule. As a set, it includes pairs like (2, 0.5), (4, 0.25), but is undefined for the input 0.
- Functions can take any number of arguments (n-ary functions).
- In Haskell, all functions are technically unary (take only one argument).
- N-ary functions are represented in Haskell using two main methods: tuple arguments or currying.
- Currying transforms an n-ary function into a sequence of unary functions, where each function takes one argument and returns another function, until the final argument is processed.
- A function that takes a tuple of arguments (e.g., f(x, y)) is distinct from a curried function that takes arguments sequentially (e.g., g(x)(y)).
Understanding currying is essential for writing and comprehending Haskell code, as it's the primary mechanism for handling functions with multiple arguments.
A function `add(x, y)` that takes two numbers can be represented in Haskell as `add x y = x + y`. Calling `add 5` returns a new function that waits for the second argument, and then `(add 5) 3` evaluates to 8.
Key takeaways
- Programs are built upon values, which are the pieces of information manipulated by code.
- Expressions provide a way to refer to and compute values within a program.
- Types categorize values, helping to prevent errors by ensuring operations are applied correctly.
- The 'integer' type in Haskell represents a theoretical infinite set of numbers, distinct from finite machine integer types.
- Functions are core concepts, acting as both computational rules and sets of input-output mappings.
- Key function properties include totality, surjectivity, and injectivity, which describe their behavior and coverage.
- Haskell uses currying to elegantly handle functions with multiple arguments by chaining unary functions.
- Distinguishing between functions that accept tuples and curried functions is important for understanding Haskell's functional nature.
Key terms
ValueExpressionLiteralIdentifierTypeType ErrorFunctionDomainRangeCurrying
Test your understanding
- What is the difference between an atomic expression and a compound expression?
- Why are types important in programming, and what is a type error?
- How can a function be viewed both as a rule and as a set of ordered pairs?
- What does it mean for a function to be total, surjective, and injective?
- How does currying allow Haskell to represent functions that appear to take multiple arguments?