Programs _______ their inputs
interpret
Define pure functions
- You know exactly what the programming is doing from the code (no hidden state)
- Deterministic (same inputs always give same output)
- Output doesn't vary based on local static variables, non-local variables, etc.
- No side effects (no mutation of local static variables, non-local variables, mutable reference arguments or input/output streams)
Is this function pure?
No
Is this function pure?
Yes
Total functional program definition
Everything terminates!
Total functions always halt
"Well typed programs cannot _________"
Go wrong
Type system purpose/function
Type systems rule out certain kinds of error that might happen at runtime
They do their job before runtime even starts
Sound type system meaning
if the type system accepts the program, the program will not “go wrong”...
Type systems are usually _______ but _________
sound, incomplete
Incomplete type system meaning
sometimes “good” programs are rejected by the type system
Define compositional reasoning
the meaning or behavior of a composite structure (the "whole") can be entirely determined by the meanings or behaviors of its parts and how they are combined. meaning(whole)=f(meaning(parts)) (with the f being a composite function)
Define continuation
A continuation is a higher-order function that takes the result of the current computation as input and describes the "next step."
Functional core, imperative shell definition
The functional bit:
Easy unit testing, easy reasoning about correctness
Fearless concurrency
Build up programs from small pieces
The imperative bit:
Easy integration testing
Modular connections to the outside world
Atomic data examples
Numbers, strings, images (!), symbols, booleans, ...
6 steps in design recipe
Problem analysis + data definitions
Signature + purpose statement + header
Worked functional examples
Choose a function template
Function definition
Testing
Template definition
An abstraction describing a pattern of code used to analyze a value conforming to a particular data definition
Templates have _______ (the “...” parts), and an ________ (pieces of the inputs that are likely to be useful)
holes, inventory
Sum types vs product types
Sum types: A or B or C
Called a "sum" type because or behaves algebraically like addition.
HtDP Enumerations and intervals are sum types
Product types: A and B and C
Eg tuples, records, structures
Called a "product" type because and behaves algebraically like multiplication.
HtDP Structures are product types
Enumeration definition/example
A finite number of specific values to choose from.
Template uses cond or match.
; A TrafficLight is one of the following Strings:
; – "red" ; – "green" ; – "yellow"
; interpretation the three strings represent the three
; possible states that a traffic light may assume
(define (F traffic-light)
(cond [(string=? traffic-light "red") ...]
[(string=? traffic-light "green") ...]
[(string=? traffic-light "yellow") ...]))
Interval definition/example
Groups of elements that satisfy some specific property.
Template uses cond with predicates.
;; A Temperature can be split into bands corresponding to comfort levels: ;; - less than 0 °C ;; - at or more than 0 °C, but less than 10 °C ;; - at or more than 10 °C, but less than 22 °C ;; - at or more than 22 °C ;; F : Temperature -> ? (define (F temp) (cond [(< temp 0) ... temp ...] [(and (>= temp 0) (< temp 10)) ... temp ...] [(and (>= temp 10) (< temp 22)) ... temp ...] [(>= temp 22) ... temp ...]))Itemization definition/example
Itemizations generalize Intervals and Enumerations.
Template uses cond or match.
;; A MissionTime is one of ;; - 'preflight, denoting the time up to launch ;; - a NonNegativeNumber, denoting seconds since launch (define (F mission-time) (cond [(equal? mission-time 'preflight) ...] [(number? mission-time) ... mission-time ...])) (define (F mission-time) (match mission-time ['preflight ...] [seconds ... seconds ...]))Structure definition/example
A named collection of (sub-)values called fields.
Template uses field accessor functions e.g. movie-title, ...
Structure predicate also available e.g. movie?
;; A Movie is a (make-movie String String Number), representing
;; an entry in a movie database, with fields
;; - title, a String, the movie's title; ;; - director, a String, the name of its director; ;; - year, a Number, the year of its release.
(define-struct movie [title director year])
;; Example: (make-movie "The Fellowship of the Ring" "Peter Jackson" 2001)
Itemization of structure: definition/example
In algebra-speak, sums of products; an “or” of “ands”.
Many different alternatives, each of which is (in general) a structure.
;; A DisplayShape represents a shape to draw on the screen, and is one of
;; - a (make-rectangle Size Size), a rectangle of given width and height;
;; - a (make-circle Size), a circle of given radius; or
;; - a (make-line Size Angle), a line of the given length and angle.
(define-struct rectangle (width height))
(define-struct circle (radius))
(define-struct line (size angle))
;; A Size is a Number representing a pixel count.
;; An Angle is a Number representing an angle in degrees.
;; F : DisplayShape -> ?
(define (F ds)
(cond [(rectangle? ds) ... (rectangle-width ds) ... (rectangle-height ds) ...]
[(circle? ds) ... (circle-radius ds) ...]
[(line? ds) ... (line-size ds) ... (line-angle ds) ...]))
Self-referential data: definition/explanation
Everywhere that a data definition has self-reference,
the template (and your function) has recursion,
and you prove correctness by induction.
Natural recursion definition
Recursion exactly where self-reference occurs
Lists, self referential data definition+template in racket
;; A ListOf<X> is one of
;; - '(), the empty list
;; - (cons X ListOf<X>)
;; TEMPLATE (define (F xs)
(cond
[(null? xs) ...]
[(cons? xs) ... (first xs) ... (F (rest xs)) ...]))
;; where: ;; first : (cons X ListOf<X>) -> X
;; rest : (cons X ListOf<X>) -> ListOf<X>
Racket offers an abbreviation for uses of cons and '():
(list a b ... z) = (cons a (cons b ... (cons z '())...))
Generative recursion definition/function
Fully general recursion that doesn't have to fit the very specific recursion patterns of structural/natural recursion.
Generate subproblems that are still smaller but are not an immediate structural subproblem
Requirement: termination argument
Natural recursion is also called ________ recursion
Structural
What is the time complexity of the following function:
;; reverse : ListOf<X> -> ListOf<X>
;; Answers items from `input`, in reverse order.
(define (reverse xs)
(cond
[(null? xs) '()]
[(cons? xs) (append (reverse (rest xs)) (cons (first xs) '()))]))
O(n^2), because append() is O(n)
The following method uses an __________ to obtain O(n) time complexity, as using append() instead would result in O(n^2)
;; reverse : ListOf<X> -> ListOf<X>
;; Answers items from `input`, in reverse order.
(define (reverse xs)
(local [
;; helper : ListOf<X> ListOf<X> -> ListOf<X>
;; Prepends `remaining` in reverse order onto `reversed`. 👀
;; INVARIANT: (append (reverse remaining) reversed) === (reverse xs)
(define (helper remaining reversed)
(cond
[(null? remaining) reversed]
[(cons? remaining) (helper (rest remaining)
(cons (first remaining) reversed))]))]
(helper xs '())))
Accumulator
Differences between natural/structural and generative recursion
In structural recursion, the subproblem is directly tied to a smaller structural piece of the input. Eg. data definition has a self reference
In generative recursion, the subproblem is created through computation and may not align with the input's structure.
Eg. quicksort: The key is that the two sub lists are generated by the algorithm
rather than being directly derived from the structure of the list (e.g., first and
rest). The subproblems aren't pre-determined by the data's construction but
are dynamically computed.
Examples function values
;; A Predicate<X> is a (X -> Boolean).
;; A BinaryOperator is a (Number Number -> Number).
Do function values have applicable templates?
No
What is the '<X>' called in a function value such as Predicate<X> or List<X>?
A generic. It means the function works with generic/non-specified type(s).
Higher-Order Utilities: definition
Functions that take other functions as input or return functions as output
What is the return type of :
(define (compose g f)
(lambda (a) (g (f a))))
Function (that takes 'a' as an input parameter)
Fill in the blank:
;; filter : Predicate<X> ListOf<X> -> ListOf<X> \
(check-expect (filter (lambda (n) (< n 5)) (list 1 7 4 9 8 2 6 5 3))
???????)
(list 1 4 2 3)
Fill in the blank:
(check-expect (map number->string (list 1 2 3 4 5))
????????)
(list "1" "2" "3" "4" "5")
Fill in the blank:
(check-expect (foldr + 0 (list 1 2 3 4 5))
????????)
15
foldr general signature
(X Y -> Y) is a combiner function: It takes an element of the list (X) and an accumulated result (Y) and returns a new result (Y).
Y is the initial value for the accumulation (also called the base case).
ListOf<X> is the input list.
Returns a final value of type Y.
foldr starts at the _____ of the input list, while foldl starts at the _____
end, start
In Java, a simple “structure” data definition turns into a standalone _____ _____
record class
Is interacting with an external file a side-effect?
Yes, it breaks the purity of a function
Total function definition
Function that is defined/terminates for all possible inputs mentioned in its signature
Key advantage of functional programming
Less unpredictability
Interpreter general signature
;; interp : Expression -> Value
Define context
Refers to the environment or scope in which an expression or function is evaluated.
Includes bindings of variables to their values or functions.
Dynamic vs static/lexical scope
Static: the binding of a variable (i.e., what the variable refers to) is determined by the program's structure, specifically by where the variable is declared in the source code. This decision is made at compile time.
Dynamic: the binding of a variable is determined by the program's execution context (runtime stack) rather than its syntactic structure.
Define soundness
Every program that passes the system's checks is guaranteed to be free of certain kinds of errors (e.g., type errors, logical contradictions).
'No false negatives': the system never incorrectly claims that something is error-free when it isn't.
Define completeness
Every program or statement that is actually correct passes the system's checks.
'No false positives'
Readable is usually covariant, writeable is usually contravariant, readable and writeable is usually invariant
FILL IN
Define closure
A pairing of a function and the environment needed to execute/complete it
In Java, each “itemization” turns into a ______ ________, with one implementing _______ type per branch.
sealed interface, record
Itemization in Java: example
A Maybe<X> represents an optional X. It is one of a None(), representing absence, or a Some(X value), representing presence.
/** A Maybe<X> represents ... */ public sealed interface Maybe<X> permits None, Some {} /** A None<X> represents an absent X. See Maybe<X> */ public record None<X>() implements Maybe<X> {} /** A Some<X> represents a present X. See Maybe<X> */ public record Some<X>(X value) implements Maybe<X> {}Notation for the Temperature data definition example, in Java
// A Temperature is a `double` representing temperature in degrees Celsius (°C). ...
/** Get the active Temperature. */
public static double /*Temperature*/ activeTemperature();
Methods instead of functions: features
Use dynamic dispatch in place of the outermost layer of pattern-matching (instead of a switch case to determine type, use polymorphism)
Signature can make use of keyword this and the implicit self type
Purpose statement goes on the declaration in the interface
One implementation method in each of the classes in the interface's permits clause
Eg.
public sealed interface Maybe<X> permits None, Some {
X valueOrDefault(X d);
}
public record None<X>() implements Maybe<X> {
public X valueOrDefault(X d) { return d; }
}
public record Some<X>(X value) implements Maybe<X> {
public X valueOrDefault(X d) { return this.value; }
}
In Java, often dynamic dispatch is used instead of ______ _______
pattern matching
What should you do any time a function involves an externally observable side effect?
Add a comment about it in its signature.
Eg:
/** Append items in `this` to `xs`. SIDE EFFECT: mutates `xs`. */
void appendTo(java.util.List<X> xs);
Java's “lambda expressions” are automatically converted to _________ _______ _________ by the compiler
Anonymous inner classes
Java “lambda expressions” examples
@Test void test_simple_functions() {
java.util.function.Function<String, String> f = (String s) -> "hello, " + s;
java.util.function.Function<String, String> g = (s) -> "hello, " + s;
java.util.function.Function<String, Integer> h = (s) -> s.length();
assertEquals(f.apply("world"), "hello, world");
assertEquals(g.apply("world"), "hello, world");
assertEquals(h.apply("world").intValue(), 5);
}
public class Main { public static void main(String[] args) { ArrayList<Integer> numbers = new ArrayList<Integer>(); numbers.add(5); numbers.add(9); numbers.add(8); numbers.add(1); Consumer<Integer> method = (n) -> { System.out.println(n); }; numbers.forEach( method ); } }Java: a lambda expression can take on any interface or class type with a _______ _________ _________
single abstract method
Eg:
public interface MyFunction { int stringToInt(String s); }public class MainClass { public static void main(String[] args) { MyFunction f = (s) -> s.length(); String s = "test"; System.out.println(f.stringToInt(s)); } }Syntax, semantics and pragmatics definition
Syntax: how to write a valid program
Semantics: what a program means
Pragmatics: the human context in which a language and its programs exists
Tooling: IDEs, foreign-function interfaces, debuggers, syntax highlighting, linters, package management tools...
Context of Application: Who uses it? What for? which domain (medicine, engineering, graphic design, accountancy...)? Is it usable? Is it ergonomic? Is it error-prone? Is it safe?
Abstract syntax definition
a data definition describing "valid" programs
Concrete syntax definition
a grammar mapping text to abstract syntax
Two (sub)programs, P and Q, are equivalent iff
For all contexts C[·] (a context is a program with a “hole”),
eval(C[P]) = eval(C[Q])
This says: in every context I could place P in, I will get exactly the same result and exactly the same side effects as if I put Q in that context.
Syntactic equivalence vs semantic equivalence
Two statements are syntactically equivalent if we can reach either one from the other using only mathematical rules and not interpreting the statements at any point. Parallelly, if two statements are semantically equivalent, we can reach either one from the other by interpreting their meanings.
Concrete syntax is _______ into abstract syntax, which is then _______ into values
parsed, interpreted
Notes how to design interpreters
;; A Program is an Expression.
;; An Expression is one of:
;; - an (app Expression Expression), a function call
;; - a (var Symbol), a variable reference
;; - a (fun Symbol Expression), a function definition
;; interp : Expression -> Value
Tests for partial functions must trigger the documented ____________
exceptions
Notes dynamic scope
The decision to let control flow determine binding is called dynamic scope. It is the one unambiguously wrong design decision in programming languages
We can't be sure what a variable name means!
Neither can our IDE
Neither can our compiler - noncompositional
All bets are off
Tremendously error prone
SMoL and SImPl
Standard Model of Languages
Standard Implementation Plan (interpreters)
(foldl (lambda (x n) (+ 1 n)) 0 (1, 2, 3)):
0 is the ____________ ////
x is the ____________ ////
n is the ____________ ////
initial value of the accumulator
item from the list input into the function on each iteration
value of the accumulator
For correct functions, the interpreter saves the ________ on _________ creation
environment, function
Notes primitives
Programs start with an environment that contains the primitives, instead of an empty one
Does java use: static dispatch? or dynamic dispatch?
dynamic dispatch (not scope)
Dynamic dispatch definition
selecting which implementation of a polymorphic operation (method or function) to call at run time (as opposed to compile time, which is called static dispatch)
Delegation definition
an object forwards a method call or behavior request to another object, called its delegate, typically used to reuse behavior or share functionality dynamically.
Example: In prototype-based programming, an object inherits methods from its prototype through delegation.
Define orthogonal features
Features that don't overlap with each other
Define implementing language
The programming language used to build or implement another language
Define type system
a tractable syntactic method for proving the absence of certain program behaviors
Type safety definition
A type safe language is one that statically protects its own abstractions.
No untrapped errors. Prevents illegal operations.
Translate ⊢ To english
Under the given context
Translate ':' to english (type systems)
Is of type
Theorem for type soundness
If ⊢e:τ then either eval(e)=v and ⊢v:τ, or eval(e) yields an exception, or eval(e) does not halt.
If the type system checks expression
e, and shows that it has type
τ, then evaluation of
e either yields a value
v that is also of type
τ, or yields an error/exception, or never answers a value.
Soundness vs completeness
Sound: If the system claims property P, then P is really true.Complete: If P really is true, then system claims P.
Sound: proof implies truth; Complete: truth implies proof
Non-theorem: completeness
If eval(e)=v and ⊢v:τ, then ⊢e:τ.
Run-time checks fill in ________ _______ not covered or coverable by the type system.
Safety gaps
RTTI definition
RTTI = Run-time “type” information: not a “type” in the type-system sense — it's metadata about a value: a label, available at runtime, that describes some aspects of the value
Are run-time type checks sound? Complete?
Both
A judgement is notation for ...
An n-ary relation.
Unary example: ⊢ E ok (E is ok)
Translate Γ to english
Context
Translate to english: Γ ⊢ E ok
the context Γ proves that E is 'ok'
Top and bottom part of inference rules: names
Antecedent, consequent
A rule with 0 antecedents is called an _______
Axiom
Translate to english: Γ ⊢ x : Γ(x)
X is of the type that the type environment maps it to