Parametric Lambda Calculus - UniTo

“Parametric Lambda Calculus” Simona Ronchi Della Rocca Luca Paolini [email protected]

Universita` di Torino Dipartimento di Informatica

FOLLIA - Torino, 2006 – p. 1/56

Outline Historical Introduction


Outline Historical Introduction The syntax of the parametric λ calculus


Outline Historical Introduction The syntax of the parametric λ calculus The fundamental properties: Confluence and Standardization


Outline Historical Introduction The syntax of the parametric λ calculus The fundamental properties: Confluence and Standardization The Universal Reduction Machine


Outline Historical Introduction The syntax of the parametric λ calculus The fundamental properties: Confluence and Standardization The Universal Reduction Machine The Operational Semantics


Outline Historical Introduction The syntax of the parametric λ calculus The fundamental properties: Confluence and Standardization The Universal Reduction Machine The Operational Semantics Crossing Call-by-name and Call-by-value


A bit of history ∼ 1930 The λ-calculus as logical foundation of mathematics (Curry) The λ-calculus as intensional language for the computable functions (Curry, Kleene, Turing)


A bit of history ∼ 1930 The λ-calculus as logical foundation of mathematics (Curry) The λ-calculus as intensional language for the computable functions (Curry, Kleene, Turing) ∼ 1960 LISP: the first programming language inspired by λ-calculus (MC Carthy) The translation of Algol 60 into λ-calculus (Landin) The language CUCH (Böhm and Gross) The Separability Theorem (Böhm)


A bit of history ∼ 1930 The λ-calculus as logical foundation of mathematics (Curry) The λ-calculus as intensional language for the computable functions (Curry, Kleene, Turing) ∼ 1960 LISP: the first programming language inspired by λ-calculus (MC Carthy) The translation of Algol 60 into λ-calculus (Landin) The language CUCH (Böhm and Gross) The Separability Theorem (Böhm) ∼ 1970 The denotational semantics of λ-calculus (Scott) The λ-βv -calculus (Plotkin)


A bit of history 1980 − 90 The incompleteness of λ-calculus (Ronchi, Honsell) The full-abstraction problem (Plotkin, Berry) The lazy λ-calculus (Abramsky) Other λ-calculi....


Some λ-calculi The classical λ-calculus: a paradigmatic language for the call-by-name computations.


Some λ-calculi The classical λ-calculus: a paradigmatic language for the call-by-name computations. The λ-βv -calculus: a paradigmatic language for the lazy-call-by-value computations.


“Historia magistra vitae” question: Is there a unique paradigmatic language, that can model all the computations listed before (and may be that can suggest new computations)?


“Historia magistra vitae” question: Is there a unique paradigmatic language, that can model all the computations listed before (and may be that can suggest new computations)? answer: The parametric λ-calculus.


The parametric λ-calculus The parametric λ-calculus is an abstract calculus with some basic requirements, parametric with respect to a set of terms. Different instantiations of the parameter produce a different λ-calculus.


The parametric λ-calculus The parametric λ-calculus is an abstract calculus with some basic requirements, parametric with respect to a set of terms. Different instantiations of the parameter produce a different λ-calculus. It allows to generalize definitions, properties and proofs


The parametric λ-calculus The parametric λ-calculus is an abstract calculus with some basic requirements, parametric with respect to a set of terms. Different instantiations of the parameter produce a different λ-calculus. It allows to generalize definitions, properties and proofs It is a natural starting point for the study of new calculi


The parametric λ-calculus The parametric λ-calculus is an abstract calculus with some basic requirements, parametric with respect to a set of terms. Different instantiations of the parameter produce a different λ-calculus. It allows to generalize definitions, properties and proofs It is a natural starting point for the study of new calculi It helps the investigations of the relations between different calculi


The Syntax The set Λ is produced by M ::= x | M M | λx.M


The Syntax The set Λ is produced by M ::= x | M M | λx.M

Let ∆ ⊆ Λ. The ∆-reduction (→∆ ) is the contextual closure of the following rule: (λx.M )N →∆ M [N/x] if and only if N ∈ ∆


Set of Input Values Let ∆ ⊆ Λ.

∆ is a set of input values, when:



∆ is a set of input values, when: Var ⊆ ∆

(variable closure)



∆ is a set of input values, when: Var ⊆ ∆ P, Q ∈ ∆ implies P [Q/x] ∈ ∆,

(variable closure) (substitution closure)




(variable closure)

(substitution closure) M ∈ ∆ and M →∆ N imply N ∈ ∆ (reduction closure)




(variable closure)

(substitution closure) M ∈ ∆ and M →∆ N imply N ∈ ∆ (reduction closure)

The above requirements assure that the status of being an input value is preserved during the evalution process.


Some λ∆-calculi Λ is a set of input values (the λΛ-calculus is the classical λ-calculus)


Some λ∆-calculi Λ is a set of input values (the λΛ-calculus is the classical λ-calculus) Γ = V ar ∪ {λx.M | M ∈ Λ} is a set of input values (the λΓ-calculus is the Plotkin’s λβv -calculus)


Some λ∆-calculi Λ is a set of input values (the λΛ-calculus is the classical λ-calculus) Γ = V ar ∪ {λx.M | M ∈ Λ} is a set of input values (the λΓ-calculus is the Plotkin’s λβv -calculus) Ξ = V ar ∪ {M | M is a closed β normal form } is a set of input value


Some λ∆-calculi Λ is a set of input values (the λΛ-calculus is the classical λ-calculus) Γ = V ar ∪ {λx.M | M ∈ Λ} is a set of input values (the λΓ-calculus is the Plotkin’s λβv -calculus) Ξ = V ar ∪ {M | M is a closed β normal form } is a set of input value ΛI , defined by the following grammar: M ::= x | M M | λx.M (x ∈ F V (M ))

is a set of input values


Non Input Values Λ-normal forms are not input values


Non Input Values Λ-normal forms are not input values Λ-head normal forms are not input values


Non Input Values Λ-normal forms are not input values Λ-head normal forms are not input values Γ-normal forms are not input values


Confluence M →∗∆ N1 and M →∗∆ N2 imply there is N3 such that both N1 →∗∆ N3 and N2 →∗∆ N3 .



The closure conditions on the set ∆ of input values assure that the corresponding λ-∆-calculus enjoys the confluence property. The closure conditions are sufficient but not necessary for garantee the confluence property (instead of the reduction closure it would be sufficent to have: M ∈ ∆ and M →∗∆ N imply there is P ∈ ∆ such that N →∗∆ P . )



The closure conditions on the set ∆ of input values assure that the corresponding λ-∆-calculus enjoys the confluence property. The closure conditions are sufficient but not necessary for garantee the confluence property (instead of the reduction closure it would be sufficent to have: M ∈ ∆ and M →∗∆ N imply there is P ∈ ∆ such that N →∗∆ P . )


A counterexample Let us take Λ-normal forms as input values. Then the confluence fails, in fact: (λx.(λy.z)(x(λx.xx)))(λx.xx) reduces to both z and (λy.z)((λx.xx)(λx.xx)), which do not have a common reduct.


Standardization M →∗∆ N implies there is a standard reduction sequence such that M →∗∆ N (M →s∆ N ) .


Standardization M →∗∆ N implies there is a standard reduction sequence such that M →∗∆ N (M →s∆ N ) . In case of λΛ-calculus a standard reduction sequence is a strategy choosing redexes from left to right. It is not easy to formalize such a strategy for other sets of input values.


Standardization M →∗∆ N implies there is a standard reduction sequence such that M →∗∆ N (M →s∆ N ) . In case of λΛ-calculus a standard reduction sequence is a strategy choosing redexes from left to right. It is not easy to formalize such a strategy for other sets of input values. (λx.x(KI))(II) →Λ II(KI) →Λ I(KI) →Λ I(λy.I)

is a standard reduction sequence in the λΛ-calculus.


Standardization M →∗∆ N implies there is a standard reduction sequence such that M →∗∆ N (M →s∆ N ) . In case of λΛ-calculus a standard reduction sequence is a strategy choosing redexes from left to right. It is not easy to formalize such a strategy for other sets of input values. (λx.x(KI))(II) →Λ II(KI) →Λ I(KI) →Λ I(λy.I)

is a standard reduction sequence in the λΛ-calculus.

The reduction sequence reducing the same term from left to right in the λΓ-calculus: (λx.x(KI))(II) →Γ (λx.x(λy.I))(II) →Γ (λx.x(λy.I))I →Γ I(λy.I)

is not a standard reduction sequence! FOLLIA - Torino, 2006 – p. 14/56

Sequentialization The ∆-sequentialization (M )◦ of a term M is a function from Λ to Λ defined as follows: (xM1 ...Mm )◦ = x(M1 )◦ ...(Mm )◦ ; ((λx.P )QM1 ...Mm )◦ = (λx.P )◦ (Q)◦ (M1 )◦ ...(Mm )◦ if Q ∈ ∆; ((λx.P )QM1 ...Mm )◦ = (Q)◦ (λx.P )◦ (M1 )◦ ...(Mm )◦ if Q 6∈ ∆; (λx.P )◦ = λx.(P )◦ .


Sequentialization The ∆-sequentialization (M )◦ of a term M is a function from Λ to Λ defined as follows: (xM1 ...Mm )◦ = x(M1 )◦ ...(Mm )◦ ; ((λx.P )QM1 ...Mm )◦ = (λx.P )◦ (Q)◦ (M1 )◦ ...(Mm )◦ if Q ∈ ∆; ((λx.P )QM1 ...Mm )◦ = (Q)◦ (λx.P )◦ (M1 )◦ ...(Mm )◦ if Q 6∈ ∆; (λx.P )◦ = λx.(P )◦ . The degree of a redex R in M is the numbers of λ’s which both are active in M and occur on the left of (R)◦ in (M )◦ . (A symbol λ in a term M is active if and only if it is the first symbol of a ∆-redex of M ).


Sequentialization A sequence M ≡ P0 →∆ P1 →∆ ... →∆ Pn →∆ N is standard if and only if the degree of the redex contracted in Pi is less than or equal to the degree of the redex contracted in Pi+1 , for every i < n.


Sequentialization A sequence M ≡ P0 →∆ P1 →∆ ... →∆ Pn →∆ N is standard if and only if the degree of the redex contracted in Pi is less than or equal to the degree of the redex contracted in Pi+1 , for every i < n. A set ∆ of input values is standard if and only if M 6∈ ∆ and M →∗∆ N by reducing at every step a not principal redex imply N 6∈ ∆. (The principal redex is the redex of degree 0 corresponding to the head redex in the λΛ-calculus )


Sequentialization A sequence M ≡ P0 →∆ P1 →∆ ... →∆ Pn →∆ N is standard if and only if the degree of the redex contracted in Pi is less than or equal to the degree of the redex contracted in Pi+1 , for every i < n. A set ∆ of input values is standard if and only if M 6∈ ∆ and M →∗∆ N by reducing at every step a not principal redex imply N 6∈ ∆. (The principal redex is the redex of degree 0 corresponding to the head redex in the λΛ-calculus ) A λ∆-calculus is standard if and only if the set ∆ is standard


Sequentialization A sequence M ≡ P0 →∆ P1 →∆ ... →∆ Pn →∆ N is standard if and only if the degree of the redex contracted in Pi is less than or equal to the degree of the redex contracted in Pi+1 , for every i < n. A set ∆ of input values is standard if and only if M 6∈ ∆ and M →∗∆ N by reducing at every step a not principal redex imply N 6∈ ∆. (The principal redex is the redex of degree 0 corresponding to the head redex in the λΛ-calculus ) A λ∆-calculus is standard if and only if the set ∆ is standard To be standard is a necessary and sufficent condition for enjoying the standardization property. FOLLIA - Torino, 2006 – p. 16/56

Standardization (examples) (λx.x(KI))(II) →Γ (λx.x(KI))I →Γ I(KI) →Γ I(λy.I)

is a standard reduction sequence in the λΓ-calculus.



is a standard reduction sequence in the λΓ-calculus. The set of input values Λ, Γ, Ξ are standard.



is a standard reduction sequence in the λΓ-calculus. The set of input values Λ, Γ, Ξ are standard. The set of input values ΛI is not standard.


Evaluation The principal evaluation is a sequence of reduction performing at every step the principal redex.


Evaluation The principal evaluation is a sequence of reduction performing at every step the principal redex. The principal evaluation is normalizing.


Evaluation The principal evaluation is a sequence of reduction performing at every step the principal redex. The principal evaluation is normalizing. There is a Parametric Principal Reduction Machine, parametric with respect to the set of input values ∆, reducing a term according to the principal evaluation.


The PPR machine M →p∆ N λx.M →p∆ λx.N

p1

i = min{j ≤ m|Mi 6∈ ∆-nf} Mi →p∆ Ni xM1 ...Mm →p∆ xM1 ...Ni ...Mm

Q∈∆ (λx.P )QM1 ...Mm →p∆ P [Q/x]M1 ...Mm

p3

Q 6∈ ∆ Q 6∈ ∆-nf Q →p∆ Q′ (λx.P )QM1 ...Mm →p∆ (λx.P )Q′ M1 ...Mm

p4

Q 6∈ ∆ Q ∈ ∆-nf P ∈ 6 ∆-nf P →p∆ P ′ (λx.P )QM1 ...Mm →p∆ (λx.P ′ )QM1 ...Mm Q 6∈ ∆

p2

p5

P, Q ∈ ∆-nf i = min{j ≤ m|Mi 6∈ ∆-nf} Mi →p∆ Ni (λx.P )QM1 ...Mm →p∆ (λx.P )QM1 ...Ni ...Mm

p6


Output values Let ∆ be a set of input values. A set of output values with respect to ∆ is any set Θ ⊆ Λ such that: Θ contains all the ∆-normal forms; If M =∆ N and N ∈ Θ then there is P ∈ Θ such that M →∗p ∆ P (→∗p denotes the reduction sequence choosing at every step the principal redex )


Examples of Output Values ∆-normal forms is a set of output values w.r.t ∆, for all ∆.


Examples of Output Values ∆-normal forms is a set of output values w.r.t ∆, for all ∆. Λ-head normal forms is a set of output values w.r.t Λ.


Examples of Output Values ∆-normal forms is a set of output values w.r.t ∆, for all ∆. Λ-head normal forms is a set of output values w.r.t Λ. Λ-weak head normal forms is a set of output values w.r.t Λ.


Examples of Output Values ∆-normal forms is a set of output values w.r.t ∆, for all ∆. Λ-head normal forms is a set of output values w.r.t Λ. Λ-weak head normal forms is a set of output values w.r.t Λ. Γ together with Γ-normal forms are a set of output values w.r.t Γ.


The Operational Semantics Let Θ be a set of output values with respect to the set of input value ∆. ⇓∆,Θ is the evaluation relation defined through the following rules: M ∈Θ M ⇓∆,Θ M

(axiom)

M →p∆ P P ⇓∆,Θ N M ⇓∆,Θ N

(eval)



(axiom)

M →p∆ P P ⇓∆,Θ N M ⇓∆,Θ N

(eval)

Operational pre-order: M ∆,Θ N if and only if, for all contexts C[.] such that 0 C[M ], C[N ] ∈ Λ , C[M ] ⇓∆,Θ implies C[N ] ⇓∆,Θ .



(axiom)

M →p∆ P P ⇓∆,Θ N M ⇓∆,Θ N

(eval)

Operational pre-order: M ∆,Θ N if and only if, for all contexts C[.] such that 0 C[M ], C[N ] ∈ Λ , C[M ] ⇓∆,Θ implies C[N ] ⇓∆,Θ . Operational Equivalence: M ≈∆,Θ N if and only if M ∆,Θ N and N ∆,Θ M .


Normal forms: an apparent paradox In the λΛ-calculus, the set of normal forms: M ::= x | λ~x.xM...M

is semantically meaningful. The Böhm separability property holds, so two βη -different normal forms cannot be equated.


Normal forms: an apparent paradox In the λΛ-calculus, the set of normal forms: M ::= x | λ~x.xM...M

is semantically meaningful. The Böhm separability property holds, so two βη -different normal forms cannot be equated. In a generic λ∆-calculus, the set of normal forms: M ::= x | λ~x.xM...M | λ~x.(λx.P )QM..M

has not good semantical properties.

(Q 6∈ ∆)


Normal forms in λΓ In the λΓ-calculus, two Γ-normal forms can be consistently equated, for example it is consistent to put: λx.I(xx) = λx.xx.



Let D ≡ λx.xx. (λy.D)(xI)D is a Γ-nf. It is consistent to put: (λy.D)(xI)D = DD

(DD is not a value!)





A Separability Theorem holds, saying that two different Λ-normal forms can be Γ-separated, so they are different in every Γ model.





A Separability Theorem holds, saying that two different Λ-normal forms can be Γ-separated, so they are different in every Γ model. We conjecture that the same holds for all meaningful ∆. FOLLIA - Torino, 2006 – p. 24/56

Solvability & Valuability A term M is valuable iff M →∗∆ N ∈ ∆.



A term M is potentially ∆-valuable iff there is a substitution s replacing variables by closed values, s.t. s(M ) is ∆-valuable.



A term M is potentially ∆-valuable iff there is a substitution s replacing variables by closed values, s.t. s(M ) is ∆-valuable. A term M is ∆-solvable if and only if there is a head ~ such that: ∆-context C[.] ≡ (λ~x.[.])N ~x sequentialize variables in FV(M ) ~ ⊆∆ N and C[M ] =∆ I


Γ-Solvability, Γ and Γ-normal forms We already showed that the term: (λy.D)(xI)D

where D ≡ λx.xx is a Γ-normal form, which is not a Γ-input value, and which is operationally equivalent to DD



where D ≡ λx.xx is a Γ-normal form, which is not a Γ-input value, and which is operationally equivalent to DD DD is Γ-unsolvable



where D ≡ λx.xx is a Γ-normal form, which is not a Γ-input value, and which is operationally equivalent to DD DD is Γ-unsolvable (λy.D)(xI)D is Γ-unsolvable!



where D ≡ λx.xx is a Γ-normal form, which is not a Γ-input value, and which is operationally equivalent to DD DD is Γ-unsolvable (λy.D)(xI)D is Γ-unsolvable! λx.(λy.D)(xI)D ∈ Γ!



where D ≡ λx.xx is a Γ-normal form, which is not a Γ-input value, and which is operationally equivalent to DD DD is Γ-unsolvable (λy.D)(xI)D is Γ-unsolvable! λx.(λy.D)(xI)D ∈ Γ!

So neither the set of Γ-normal forms nor the set Γ is not a proper subset of the set of Γ-solvable terms!


Call-by-name, call-by-value relations Some interesting sets of terms in call-by-value calculi correspond to semantically relevant set of the classical (call-by-name) calculus. The set of potentially Γ-valuable terms coincides with the set of lazy Λ-strongly normalizing terms.


Call-by-name, call-by-value relations Some interesting sets of terms in call-by-value calculi correspond to semantically relevant set of the classical (call-by-name) calculus. The set of potentially Γ-valuable terms coincides with the set of lazy Λ-strongly normalizing terms. The set of Φ-solvable terms coincides with the set of Λ-strongly normalizing (The λΦ-calculus is a not lazy call-by-value calculus.)


Lazy Reductions Let ∆ ⊆ Λ be a set of terms.

The lazy ∆-reduction (→∆ℓ ) is the applicative closure of the following rule: (λx.M )N → M [N/x] if and only if N ∈ ∆


Lazy Reductions Let ∆ ⊆ Λ be a set of terms.

The lazy ∆-reduction (→∆ℓ ) is the applicative closure of the following rule: (λx.M )N → M [N/x] if and only if N ∈ ∆ → Λ λx.I Let I ≡ λx.x; then λx.II→ 6 Λℓ λx.I while λx.II6→


Γ-pot. valuability and Λℓ-strongly norm.

M is Λℓ-strongly norm.

B ⊢ν M : σ M is pot. Γ-valuable










Γ-solvability and Λℓ-strong norm. The set of Γ-solvable terms is a proper subset of the set of Γ-potentially valuable terms, so it is a proper subset of the set of Λℓ-strongly normalizing terms.


Γ-solvability and Λℓ-strong norm. The set of Γ-solvable terms is a proper subset of the set of Γ-potentially valuable terms, so it is a proper subset of the set of Λℓ-strongly normalizing terms. The Γ-normal form (λy.D)(xI)D is Γ-unsolvable!


Γ-solvability and Λℓ-strong norm. The set of Γ-solvable terms is a proper subset of the set of Γ-potentially valuable terms, so it is a proper subset of the set of Λℓ-strongly normalizing terms. The Γ-normal form (λy.D)(xI)D is Γ-unsolvable!


A non lazy Call-By-Value The set SNΛ of Λℓ-strongly normalizing terms is a set of input values.


A non lazy Call-By-Value The set SNΛ of Λℓ-strongly normalizing terms is a set of input values. The set of SNΛ -solvable terms coincides with the set of SNΛ -potentially valuable terms.


A non lazy Call-By-Value The set SNΛ of Λℓ-strongly normalizing terms is a set of input values. The set of SNΛ -solvable terms coincides with the set of SNΛ -potentially valuable terms. The set of SNΛ -solvable terms coincides with SNΛ .


A non lazy Call-By-Value The set SNΛ of Λℓ-strongly normalizing terms is a set of input values. The set of SNΛ -solvable terms coincides with the set of SNΛ -potentially valuable terms. The set of SNΛ -solvable terms coincides with SNΛ . SNΛ is not the minimal set of input values satisfying these properties.


A non lazy Call-By-Value Υi , Φi are defined as follows: Υ0 = Var

Φi = Var ∪ (Υi )0

Υi+1 =Var ∪{xM1 ...Mn | Mk ∈ Υi (1 ≤ k ≤ n)} ∪{λ~x.M | M ∈ Υi } ∪{(λx.P )QM1 ...Mn | Q ∈ Υi − (Λ0 ∪ Var), P [Q/x]M1 . . . Mn →∗Φi R ∈ Υi }


A non lazy Call-By-Value Υi , Φi are defined as follows: Υ0 = Var

Φi = Var ∪ (Υi )0

Υi+1 =Var ∪{xM1 ...Mn | Mk ∈ Υi (1 ≤ k ≤ n)} ∪{λ~x.M | M ∈ Υi } ∪{(λx.P )QM1 ...Mn | Q ∈ Υi − (Λ0 ∪ Var), P [Q/x]M1 . . . Mn →∗Φi R ∈ Υi }

Let Υ = ∪i Υi and Φ = Var ∪ (Υ)0


Some Properties of λΦ-calculus

M is Λ-strongly norm.

B⊢M :σ

M is Φ-solvable

M is pot. Φ-valuable

M →∗Φ N ∈ Υ




B⊢M :σ

M is Φ-solvable


M →∗Φ N ∈ Υ




B⊢M :σ

M is Φ-solvable


M →∗Φ N ∈ Υ




B⊢M :σ

M is Φ-solvable


M →∗Φ N ∈ Υ




B⊢M :σ

M is Φ-solvable


M →∗Φ N ∈ Υ


λ∆-model λ∆-model is a quadruple < D, I, ◦, [[.]] > where D is a set

◦ is an operation on D I is a subset of D

if ρ : Var → I then [[.]]ρ maps terms in D


λ∆-model λ∆-model is a quadruple < D, I, ◦, [[.]] > s.t.: [[x]]ρ = ρ(x)


λ∆-model λ∆-model is a quadruple < D, I, ◦, [[.]] > s.t.: [[x]]ρ = ρ(x) [[M N ]]ρ = [[M ]]ρ ◦ [[N ]]ρ



[[λx.M ]]ρ ◦ d = [[M ]]ρ[d/x]

if d ∈ I



[[λx.M ]]ρ ◦ d = [[M ]]ρ[d/x]

if d ∈ I

if [[M ]]ρ[d/x] = [[M ′ ]]ρ′ [d/y] for each d ∈ I,

then [[λx.M ]]ρ = [[λy.M ′ ]]ρ′



[[λx.M ]]ρ ◦ d = [[M ]]ρ[d/x]

if d ∈ I

if [[M ]]ρ[d/x] = [[M ′ ]]ρ′ [d/y] for each d ∈ I,

then [[λx.M ]]ρ = [[λy.M ′ ]]ρ′

M ∈ ∆ implies ∀ρ.[[M ]]ρ ∈ I


Intersection Types Let C be a countable set of type-constants, containing at least the constant ω .


Intersection Types Let C be a countable set of type-constants, containing at least the constant ω . The set T (C) of types is inductively defined as follows: σ ::= α | (σ→τ ) | (σ∧τ )

where α ∈ C .


Intersection Type Theory A intersection relation ≤ is a preorder relation on T (C), closed under the following rules: σ≤ω

(a)

σ≤σ ∧ σ

σ≤σ (b)

σ ∧ τ ≤σ

σ≤ρ, ρ≤τ σ≤ρ

(r)

(c)

σ ∧ τ ≤τ

(σ → τ ) ∧ (σ → π)≤σ → (τ ∧ π) σ ′ ≤σ, τ ≤τ ′ σ → τ ≤σ ′ → τ ′

(f )

(d)

(t)

(c′ )

σ≤σ ′ , τ ≤τ ′ σ ∧ τ ≤σ ′ ∧ τ ′

σ → ω≤ω → ω

(e)

(g)

≤ induce a type theory ≃ .


Intersection Type System A type system ∇ is a triple < C, ≤∇ , I(C) >: C is a set of type constants



≤∇ is an intersection relation on T (C)




I(C) ⊆ T (C) is a set of input types s.t. I(C) is not empty




I(C) ⊆ T (C) is a set of input types s.t. I(C) is not empty σ ∈ I(C) and σ ≃∇ τ imply τ ∈ I(C);




I(C) ⊆ T (C) is a set of input types s.t. I(C) is not empty σ ∈ I(C) and σ ≃∇ τ imply τ ∈ I(C); σ ∈ I(C) and τ 6∈ I(C) imply σ ≤∇ τ .


Intersection Type Assignment System Let ∇ be a type system B[σ/x] ⊢∇ x : σ B[σ/x] ⊢∇ M : τ B ⊢∇ λx.M : σ → τ B ⊢∇ M : σ B ⊢∇ M : τ B ⊢∇ M : σ ∧ τ B ⊢∇ M : σ ∧ τ B ⊢∇ M : σ

(var)

(→I)

(∧I)

(∧El )

B ⊢∇ M : ω σ ∈ I(C)

(ω)

B ⊢∇ M : σ → τ B ⊢∇ N : σ B ⊢∇ M N : τ

(→E)

B ⊢∇ M : σ σ ≤∇ τ B ⊢∇ M : τ

(≤∇ )

B ⊢∇ M : σ ∧ τ B ⊢∇ M : τ

(∧Er )


Intersection Type Assignment System Let ∇ be a type system B[σ/x] ⊢∇ x : σ B[σ/x] ⊢∇ M : τ B ⊢∇ λx.M : σ → τ B ⊢∇ M : σ B ⊢∇ M : τ B ⊢∇ M : σ ∧ τ B ⊢∇ M : σ ∧ τ B ⊢∇ M : σ

(var)

(→I)

(∧I)

(∧El )

B ⊢∇ M : ω σ ∈ I(C)

B ⊢∇ M : σ → τ B ⊢∇ N : σ B ⊢∇ M N : τ B ⊢∇ M : σ σ ≤∇ τ B ⊢∇ M : τ B ⊢∇ M : σ ∧ τ B ⊢∇ M : τ

(ω)

(→E)

(≤∇ )

(∧Er )


Filters Let ∇ be the type system < C, ≤∇ , I(C) >. Let F (∇) be the set of all filters.


Filters Let ∇ be the type system < C, ≤∇ , I(C) >. Let F (∇) be the set of all filters. Let I(∇) be the set of input filters containing at least one type belonging to I(C).


Filters Let ∇ be the type system < C, ≤∇ , I(C) >. Let F (∇) be the set of all filters. Let I(∇) be the set of input filters containing at least one type belonging to I(C). f1 ◦∇ f2 =↑ {ω}∪

{τ | σ → τ ∈ f1 and σ ∈ f2 and σ ∈ I(C) }.


Filters Let ∇ be the type system < C, ≤∇ , I(C) >. Let F (∇) be the set of all filters. Let I(∇) be the set of input filters containing at least one type belonging to I(C). f1 ◦∇ f2 =↑ {ω}∪ F(∇)

[[M ]]ρ

{τ | σ → τ ∈ f1 and σ ∈ f2 and σ ∈ I(C) }.

= {σ ∈ T (C) | ∃B ∝ ρ such that B ⊢∇ M : σ}

where B ∝ ρ when B(z) ∈ ρ(z), for all z.


Legal Type Theory Let ∇ be the type system < C, ≤∇ , I(C) >. ∇ is legal whenever, for all σ ∈ I(C) and τ 6≃∇ ω : (σ1 → τ1 ) ∧ ... ∧ (σn → τn )≤∇ σ → τ


Legal Type Theory Let ∇ be the type system < C, ≤∇ , I(C) >. ∇ is legal whenever, for all σ ∈ I(C) and τ 6≃∇ ω : (σ1 → τ1 ) ∧ ... ∧ (σn → τn )≤∇ σ → τ

implies ∃{i1 , ..., ik } ⊆ {1, ..., n} s.t. (σi1 ∧ ... ∧ σik )≥∇ σ and (τi1 ∧ ... ∧ τik )≤∇ τ .


Filters models(Coppo-Dezani) Let ∇ =< C, ≤∇ , I(C) > be legal and such that M ∈ ∆ imply ∀ρ [[M ]]ρ ∈ I(∇).

Then < F(∇), I(∇), ◦∇ , [[.]]F (∇) > is a λ∆-model.




M ⊑F N

means that

F ∀ρ, [[M ]]F ⊆ [[N ]] ρ ρ




M ⊑F N

means that

Correctness:

F ∀ρ, [[M ]]F ⊆ [[N ]] ρ ρ

⊑F ⊆ O




M ⊑F N

means that

Correctness: Completeness:

F ∀ρ, [[M ]]F ⊆ [[N ]] ρ ρ

⊑F ⊆ O

O ⊆ ⊑F




M ⊑F N

means that

Correctness: Completeness:

F ∀ρ, [[M ]]F ⊆ [[N ]] ρ ρ

⊑F ⊆ O

Full abstraction:

O ⊆ ⊑F

⊑F = O


Using Parametric Models In literature standard topological models for lazy evaluations of λ-calculus have been considered.


Using Parametric Models In literature standard topological models for lazy evaluations of λ-calculus have been considered. Abramsky and Ong (1993) have studied the call by name parameter passing


Using Parametric Models In literature standard topological models for lazy evaluations of λ-calculus have been considered. Abramsky and Ong (1993) have studied the call by name parameter passing Egidi, Honsell and Ronchi Della Rocca (1992) have studied the call by value parameter passing


Using Parametric Models In literature standard topological models for lazy evaluations of λ-calculus have been considered. Both models are correct, but not complete!


Using Parametric Models In literature standard topological models for lazy evaluations of λ-calculus have been considered. Both models are correct, but not complete! A pre-order relation on terms, stratified on intersection types which grasps exactly the two operational semantics we want to model. Such a relation can be used for building two fully abstract models starting from a logical descriptions of the two previous models.


Using Parametric Models In literature standard topological models for lazy evaluations of λ-calculus have been considered. Both models are correct, but not complete! A pre-order relation on terms, stratified on intersection types which grasps exactly the two operational semantics we want to model. Such a relation can be used for building two fully abstract models starting from a logical descriptions of the two previous models.


Lazy Evaluations (Plotkin 1975) Call by name (said simply lazy) λx.M ⇓L λx.M

(lazy)

P [Q/x]M1 . . .Mm ⇓L N (λx.P )QM1 . . .Mm ⇓L N

(head)

Call by value λx.M ⇓V λx.M

(lazy)

Q ⇓V Q′ P [Q′ /x]M1 . . .Mm ⇓V N (λx.P )QM1 . . .Mm ⇓V N

(head)


Operational Semantics Let M, N ∈ Λ0 M ⇓L if and only if M Λ-reduces to an abstr.


Operational Semantics Let M, N ∈ Λ0 M ⇓V if and only if M Γ-reduces to an abstr.



Let O ∈ {L, V}.



Let O ∈ {L, V}. M O N if and only if, for all context C[.] such that C[M ], C[N ] ∈ Λ0 , C[M ] ⇓O implies C[N ] ⇓O .



Let O ∈ {L, V}. M O N if and only if, for all context C[.] such that C[M ], C[N ] ∈ Λ0 , C[M ] ⇓O implies C[N ] ⇓O . M ≈O N if and only if M O N and N O M .


L-model (Abramsky-Ong) ∠ is the type system < C∠ , ≤∠ , I(C∠ ) > where ≤∠ is the least intersection relation, C∠ = {ω} and I(C∠ ) = T (C∠ ).


L-model (Abramsky-Ong) ∠ is the type system < C∠ , ≤∠ , I(C∠ ) > where ≤∠ is the least intersection relation, C∠ = {ω} and I(C∠ ) = T (C∠ ). L is the model < F(∠), F (∠), ◦∠ , [[.]]F (∠) >. L is isomorphic to the topological λ-model, initial solution of: X = [X → X]⊥


L-model (Abramsky-Ong) ∠ is the type system < C∠ , ≤∠ , I(C∠ ) > where ≤∠ is the least intersection relation, C∠ = {ω} and I(C∠ ) = T (C∠ ). L is the model < F(∠), F (∠), ◦∠ , [[.]]F (∠) >. L is isomorphic to the topological λ-model, initial solution of: X = [X → X]⊥ L is correct but not complete with respect to L .


V-model (Egidi et al.) √

is the type system < C√ , ≤√ , I(C√ ) > where C√ = {ω}, I(C√ ) = {σ | σ 6≃V ω} and ≤√ is the least intersection relation satisfying: (ω → ω) →

τ ≤√ ω

→τ

(v)




τ ≤√ ω

→τ

(v)

√ √ √ √ F ( ) >. V is the model < F( ), I( ), ◦ , [[.]]

V is isomorphic to the topological λ-model, initial solution of: X = [X →⊥ X]⊥ .




τ ≤√ ω

→τ

(v)

√ √ √ √ F ( ) >. V is the model < F( ), I( ), ◦ , [[.]]

V is isomorphic to the topological λ-model, initial solution of: X = [X →⊥ X]⊥ . V is correct but not complete with respect to V .


L-Characterizations B ⊢∠ M : ω → ω if and only if M ⇓L .



If σ6≃∠ ω then σ≃∠ σ0 ∧ ... ∧ σn such that i → ω → ω. ∀i ≤ n, σi ≃∠ τ1i → ... → τm i



If σ6≃∠ ω then σ≃∠ σ0 ∧ ... ∧ σn such that i → ω → ω. ∀i ≤ n, σi ≃∠ τ1i → ... → τm i σ≃∠ ω if and only if σ ≡ ω | ∧ ..... {z ∧ ω}. n


V-Characterizations B ⊢√ M : ω → ω if and only if M ⇓V .

If σ6≃√ ω then σ≃√ σ0 ∧ ... ∧ σn such that i → ω → ω. ∀i ≤ n, σi ≃√ τ1i → ... → τm i σ≃√ ω if and only if σ ≡ ω | ∧ ..... {z ∧ ω}. n


L-Typed Relation

0 defined as follows: EL is a relation on Λ σ

M EL ω N is true;


L-Typed Relation


M EL ω N is true; M EL σ→τ N where τ ≃∠ ω , if and only if B ⊢∠ M : ω → ω implies B ⊢∠ N : ω → ω ;


L-Typed Relation


M EL ω N is true; M EL σ→τ N where τ ≃∠ ω , if and only if B ⊢∠ M : ω → ω implies B ⊢∠ N : ω → ω ; M EL σ→τ N where τ 6≃∠ ω , if and only if P ∈ Λ0 and B ⊢∠ P : σ imply M P EL τ NP;


L-Typed Relation


M EL ω N is true; M EL σ→τ N where τ ≃∠ ω , if and only if B ⊢∠ M : ω → ω implies B ⊢∠ N : ω → ω ; M EL σ→τ N where τ 6≃∠ ω , if and only if P ∈ Λ0 and B ⊢∠ P : σ imply M P EL τ NP; L N and M EL N . M EL N if and only if M E σ∧τ σ τ


L-Typed Relation



M EL N if and only if M EL σ N , for all σ .


L-Typed Relation



M EL N if and only if M EL σ N , for all σ .


V-Typed Relation

EVσ is a relation on Λ0 defined as follows: M EVω N is true;

M EVσ→τ N where τ ≃√ ω , if and only if B ⊢√ M : ω → ω implies B ⊢√ N : ω → ω ; M EVσ→τ N where τ 6≃√ ω , if and only if P ∈ Γ0 -val. and B ⊢√ P : σ imply M P EVτ N P ; M EVσ∧τ N if and only if M EVσ N and M EVτ N . M EV N if and only if M EVσ N , for all σ .


L

E -Properties Let M, N ∈ Λ0 .

EL is a preorder relation


L


EL is a preorder relation M ⊑L N implies M EL N


L



M EL N if and only if M L N


L



M EL N if and only if M L N M ,N if and only if M EL N and N EL M .


V


EV is a preorder relation M ⊑V N implies M EV N

M EV N if and only if M V N M ,N if and only if M EV N and N EV M .


LL-model

, induces a equiv. relation on F 0 (∠), the set of filters being interpretations of closed terms. [f ] will be the equivalence class of f w.r.t. ,, while F,0

will be the set of all such classes.


LL-model


will be the set of all such classes. 0 I,

= {[f ] ∈

F,0

| ∃M ∈

Λ0

s.t.

F(∠) [[M ]]ρ

= f }.


LL-model



= {[f ] ∈

F,0

| ∃M ∈

Λ0

s.t.

F(∠) [[M ]]ρ

= f }.

[f ]◦, [g] = [f ◦∠ g], for all [f ], [g] ∈ F,0 .


LL-model



= {[f ] ∈

F,0

| ∃M ∈

Λ0

s.t.

F(∠) [[M ]]ρ

= f }.

[f ]◦, [g] = [f ◦∠ g], for all [f ], [g] ∈ F,0 . [[M ]]LL ζ

=

F(∠) [[[M ]]ρ ],

where ρ(x) ∈ ζ(x).


LL-model



= {[f ] ∈

F,0

| ∃M ∈

Λ0

s.t.

F(∠) [[M ]]ρ

= f }.

[f ]◦, [g] = [f ◦∠ g], for all [f ], [g] ∈ F,0 . [[M ]]LL ζ

=

F(∠) [[[M ]]ρ ],


Let LL be the quadruple: < F,0 , I,0 , ◦, [[.]]LL >.


VV-model √ 0 F ( ),

, induces a equiv. relation on interpretations of closed terms.

the set of filters being

[f ] will be the equivalence class of f w.r.t. ,, while F,0

will be the set of all such classes. 0 = {[f ] ∈ F 0 | ∃M ∈ Γ0 s.t. I, ,

√ F( ) [[M ]]ρ

= f }.

[f ]◦, [g] = [f ◦√ g], for all [f ], [g] ∈ F,0 . [[M ]]VV ζ =

√ F( ) [[[M ]]ρ ],


Let VV be the quadruple: < F,0 , I,0 , ◦, [[.]]VV >.


Full Abstraction LL is a λΛ-model. VV is a λΓ-model.


Full Abstraction LL is a λΛ-model. VV is a λΓ-model.

The model LL is fully abstract with respect to the L-operational semantics. The model VV is fully abstract with respect to the V-operational semantics.
