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1 Based on slides by Mattox Beckman, as updated by Vikram
Adve, Gul Agha, Elsa Gunter, and Mark Hills
Introduction to Lambda Calculus^1 CS421 Programming Languages and Compilers http://www.cs.uiuc.edu/class/cs Dongyun Jin ([email protected])
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Outline
- Motivation
- λ-Calculus
- Syntax
- Features and Facts
- Terminology
- Computational Model
- Evaluation Rules
- Order of Evaluation
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Motivation
- λ-Calculus
- Computation
- All sequential programs may be viewed as functions from input (initial state and input values) to output (resulting state and output values)
- “Everything is Really a Function” Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ★ ☆
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Motivation
- λ-Calculus
- A Theory of Computation
- A Mathematical Formalism of Functions and Functional Computations
- Purpose
- To capture the essence of functions, function applications, and evaluation Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ λ-Calculus: Syntax of λ-Calculus
- Three kinds of expressions
- Variables: x, y, z, w, …
- Abstraction: λ x.exp Function Creation, like fun x -> exp
- Application:exp 1 exp (^2) Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ☆ ☆ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ λ-Calculus: Syntax of λ-Calculus
- Formal BNF Grammar ::= | | | ( ) ::= λ. ::= Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ☆ ☆ Motivation ★ ★
λ-Calculus: Features and Facts
- Computing Power
- Turing Complete
- Can express any sequential computation
- Integers, Booleans?
- Recursion?
- Constants?
- Other statements, i.e. if_then_else_?
- Syntactic sugar
λ-Calculus: Features and Facts
- Typed vs. Untyped λ-Calculus
- λ-Calculus has no notion of type
- Types restrict what function can be applied to what arguments
- Types are not just syntactic sugar: some terms are illegal in the typed calculus
- The simply typed λ-calculus is less powerful than the untyped λ-calculus – it is NOT Turing complete, since it lacks recursion
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ λ-Calculus: Features and Facts
- Use of λ-Calculus
- The typed and untyped λ-Calculus enable the theoretical study of sequential programming languages.
- A functional language is essentially the λ- Calculus extended with predefined constants, syntactic constructs, and types.
- Ocaml is close to the λ-Calculus:
- fun x -> exp λx. exp
- let x = e 1 in e 2 (λx. e 2 ) e (^1) Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ☆ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ λ-Calculus: Terminology
- Occurrence
- A location of a subterm in a term
- Variable Binding
- Variable x is said to be bound in λx.e
- Scope
- The scope of this binding is all of e except any terms within e that are of the form λ x.e’
- Bound Occurrence
- An occurrence of x in e is a bound occurrence if it is within the scope of some binding for x
- Free Occurrence
- Otherwise, an occurrence is said to be a free occurrence (aka free variable) Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ λ-Calculus: Terminology
- Variable Binding
- Variable x is said to be bound in λx.e (λx.λy.y (λx.x y)) x Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ λ-Calculus: Terminology
- Scope
- The scope of this binding is all of e except any terms within e that are of the form λ x.e’ (λx.λy.y (λx.x y)) x Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model: α-Equivalence
- α-Equivalence is the smallest congruence containing α-Conversion
- Generally, α-Equivalent terms are considered equal – an individual term is a representative of an equivalence class of terms.
Computational Model: α-Equivalence Example
- Show: λx.(λy.y x) x ∼α^ λy.(λx.x y) y
- λx.(λy.y x) x α^ λz.(λy.y z) z so λx.(λy.y x) x ∼α^ λz.(λy.y z) z
- (λy.y z) α^ (λx.x z) so λz.(λy.y z) z ∼ α^ λz.(λx.x z) z
- λz.(λx.x z) z α^ λy.(λx.x y) y so λz.(λy.y z) z ∼α^ λy.(λx.x y) y
- Thus, λx.(λy.y x) x ∼α^ λy.(λx.x y) y
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Computational Model: η-Reduction
- η-Reduction is defined as λx.f x η^ f If x is not free in f
- Not valid in OCaml
- Not equivalent to (λx.f) x (this is function application)
- Example: λx.(λy.y) x η^ λy.y Computational Model ★ ★ ★ ★ ★ ★ ★ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Computational Model: Substitution
- Substitution is defined over α-Equivalence classes of terms. P [N/x] means replace every free occurrence of x in P by N - Requirement: No free variable in P should become bound in P [N/x] – hence definition over α-Equivalence classes - Renaming/α-Conversion used to avoid name capture Computational Model ★ ★ ★ ★ ★ ★ ★ ★ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Computational Model: Substitution Rules
- Substitution defined inductively over the structure of λ terms - x[N/x] = N - y[N/x] = y if y ≠ x - (e 1 e 2 )[N/x] = (e 1 [N/x]) (e 2 [N/x]) - (λx.e)[N/x] = (λx.e) - (λy.e)[N/x] = λy.(e[N/x]) if y ≠ x and y not free in N (rename y if needed to ensure) Computational Model ★ ★ ★ ★ ★ ★ ★ ★ ★ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Computational Model: Substitution Example
- (λy.y z)[(λx.x y)/z] =?
- Any problems?
- z in redex (term being substituted into) in scope of y binding
- y if free in the residue (term being substituted) (λy.y z)[(λx.x y)/z] α (λw.w z)[(λx.x y)/z] = λw.w (λx.x y) Computational Model ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model: Substitution Example 2
- (λy.y z (λz.z))[(λx.x)/z] =?
- How do we do substitution here?
- Is this correct? (λy.y (λx.x) (λz.(λx.x)))
- No, this isn’t, since we only replace free occurrences. λy.y (λx.x) (λz.z)
Computational Model: β-Reduction
- β-Reduction is the essence of computation in the λ-Calculus β rule: (λx.P) N β^ P[N/x]
- Usually defined on α-Equivalence classes of terms
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Computational Model: β-Reduction Example
- (λz.(λx.x y) z)(λy.y z) β (λx.x y) (λy.y z) β (λy.y z) y β y z
- (λx.x x)(λx.x x) β (λx.x x)(λx.x x) β (λx.x x)(λx.x x) β …… Computational Model ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ☆ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Computational Model: αβ-Equivalence
- αβ-Equivalence is the smallest congruence containing α-Equivalence and β-Reduction
- Definition: A term is in normal form if no subterm is α-Equivalent to a term that can be β-reduced – i.e. there is no way to reduce the term further
- Hard fact (Church-Rosser): if e 1 and e 2 are αβ-Equivalent and both are normal forms, then they are α-Equivalent Computational Model ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ☆ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Computational Model: Order of Evaluation
- Order of Evaluation refers to the order of operations we choose to perform when reducing λ terms. For instance, should we substitute a λ term before or after reducing it? With this, there are two important points: - Not all terms reduce to normal forms - Not all reduction strategies will produce a normal form if one exists Computational Model ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ☆ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
Computational Model ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ ☆ λ-Calculus ☆ ☆ ☆ Motivation ☆ ☆ Computational Model: Lazy Evaluation
- Always reduce the left-most, top-most application
- Stop when left-most term is not an application of an abstraction to a term
- Example: (λz. (λx. x)) ((λy. y y) (λy. y y)) β^ (λx. x) Computational Model ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ☆ ☆ λ-Calculus ★ ★ ★ Motivation ★ ★
