Expressions -equivalence Start by defining possible expressions, - - PowerPoint PPT Presentation

Expressions

Start by defining possible expressions, whether meaningful or not. Definition The expressions of the simply-typed λ-calculus are given by the following BNF: A ::= Z | bool | A × A | A ⇒ A M ::= x | n | M op M | true | false | if M then M else M | M, M | π(M) | π′(M) | λx:A.M | MM where n is an integer, op is an arithmetic operation, and x is a variable. We call an expression A a type, and an expression M a term.

Eike Ritter Typed Lambda-Calculus MGS 2010 7

α-equivalence

Functions which differ only in the name of the argument have the same effect and should be identified. Captured by so-called α-equivalence: Definition (i) Let x be a variable. An occurrence of x in M is called bound by the binder λ in λx:A.M′ if the following conditions are satisfied: the term λx:A.M′ is a subterm of M; the occurrence of x is also an occurrence of x in M′; there exists no proper subterm N = λx:B.Q of λx:A.M′ such that the occurrence of x in M is also an occurrence

• f x in N.

(ii) An occurrence of x in M is called free if it is not a bound

• ccurrence.

Eike Ritter Typed Lambda-Calculus MGS 2010 8

Renaming

Definition Let M be a term and let x and y be variables. We define the renaming of x to y in M, which is a term written as M[y/x], by induction over the structure of M as follows: z[y/x]

def

= y if z = x z

• therwise

n[y/x]

def

= n (M op N)[y/x]

def

= M[y/x] op N[y/x] similarly for all other terms except λ (λz:A.M)[y/x]

def

=    λz:A.M if z = x λu:A.((M[u/z])[y/x]) if z = y λz:A.(M[y/x])

• therwise

Eike Ritter Typed Lambda-Calculus MGS 2010 9

Use of renaming

Using renaming, we can always ensure that each binder binds a different variable all bound variables differ from all free variables Formally: Theorem Let M be a term. Then there exists a term N which is α-equivalent to M such that if there exists an occurrence x which is bound by a binder λ all occurrences of x are bound by the same binder λ. From now on, consider terms up to α-equivalence. Hence will always make this assumption about bound and free variables.

Eike Ritter Typed Lambda-Calculus MGS 2010 10

Free variables

Can now define formally what a free variable in a term is. Definition Let M be any λ-term. Let N be the α-equivalent term given in theorem 4. Define the set of free variables of M, written FV (M), to be the set of all variables x which occur in N and are not bound variables in N. Note: Set of free variables independent of choice of N.

Eike Ritter Typed Lambda-Calculus MGS 2010 11

Contexts

Now want to formally define well-formed terms. For this, need assignment of types to variables. Definition A context Γ is a finite list of assignments of types to variables such that no variable occurs twice in Γ. We write x1:A1, . . . , xn:An for the list of assignments which indicates that the variable xi has type Ai and write for the empty context.

Eike Ritter Typed Lambda-Calculus MGS 2010 12

Well-formed terms

Now define judgement Γ ⊢ M : A, meaning “term M is well-formed given the assignment of types to the free variables of M listed in Γ and has type A.” Such a judgement depends on assignment Σ of types to constants

• f the language

Eike Ritter Typed Lambda-Calculus MGS 2010 13

Well-formed terms, continued

Definition Let Σ be a set of assignments of types. By induction over the structure of M, we define a judgement Γ ⊢ M : A as follows: Γ, x:A, Γ′ ⊢ x : A c:A is an element of Σ Γ ⊢ c : A Γ ⊢ M : bool Γ ⊢ N1 : A Γ ⊢ N2 : A Γ ⊢ if M then N1 else N2 : A Γ ⊢ M : A Γ ⊢ N : B Γ ⊢ M, N: A × B Γ ⊢ M : A × B Γ ⊢ π(M): A Γ ⊢ M : A × B Γ ⊢ π′(M): B Γ, x:A ⊢ M : B Γ ⊢ λx:A.M : A ⇒ B Γ ⊢ M : A ⇒ B Γ ⊢ N : A Γ ⊢ MN : B

Eike Ritter Typed Lambda-Calculus MGS 2010 14

Properties of well-formed terms

Theorem Assume Γ ⊢ M : A. (i) All free variables of M are contained in the context Γ. (ii) If also Γ ⊢ M : B, then A = B. (Uniqueness of types) (iii) If x does not occur in the context Γ, then also Γ, x : B ⊢ M : A. (Weakening) (iv) If Γ = Γ′, x : A and x ∈ FV (M), then also Γ′ ⊢ M : B. (Strengthening) Moreover, there is at most one derivation for the judgement Γ ⊢ M : A Derivation can be constructed bottom-up by induction over the structure of M. We say that the typing judgement is syntax-directed.

Eike Ritter Typed Lambda-Calculus MGS 2010 15

Substitution

Key problem: In M[N/x], N contains variable which occurs as bound variable in term M (Variable capture) Solution: Perform a suitable renaming before substitution Formal definition: Definition Let M be a term and x be a free variable in N. We define the term M[N/x], called the substitution of M for x in N by y[N/x]

def

= N if x = y y

• therwise

M1, M2[N/x]

def

= M1[N/x], M2[N/x] (λy:A.M)[N/x]

def

=        λy:A.M if y = x λz:A.(M[z/y])[N/x] if y ∈ FV (N) and z ∈ (FV (M) ∪ FV (N)) λy:A.M[N/x]

• therwise

Eike Ritter Typed Lambda-Calculus MGS 2010 16

Substitution lemma

Substitution preserves typing: Lemma Assume Γ, x:A is a context, Γ ⊢ N : A and and Γ, x:A ⊢ M : B. Then Γ ⊢ M[N/x]: B. Proof. Induction over the structure of M. Such a substitution lemma is typical: When you define a type-theoretic concept you need to show that substitution preserves this concept

Eike Ritter Typed Lambda-Calculus MGS 2010 17