[PPT] - A Standard Theory for the Pure Lambda-Value Calculus lvaro PowerPoint Presentation

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A Standard Theory for the Pure Lambda-Value Calculus

Álvaro García-Pérez Pablo Nogueira Paris, September 2014

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Standard theory in the classical lambda calculus (a.k.a. Barendregt’s theme)

◮ Pure untyped lambda calculus (λK) with reduction (→∗ β) and

conversion (=β) theories.

◮ Solvability:

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =β I. [Barendregt, 1984]

◮ Unsolvable terms denote ⊥ in D∞ [Wadsworth, 1976]. ◮ H = {M =β N | M, N ∈ Λ0 unsolvables}+ is a lambda theory

[Barendregt, 1984].

◮ D∞ satifies H, i.e., D∞ is sensible [Barendregt, 1984]. ◮ Quasi-leftmost reduction characterises complete strategies

[Hindley and Seldin, 2008].

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Solvability and transformability in λK

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =β I. [Barendregt, 1984] M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =β N ∈ NF. [Barendregt, 1971]

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Solvability and transformability in λK

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =β I. [Barendregt, 1984]

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that

Ch[M] =β N ∈ NF. [Barendregt, 1971]

Theorem ([Wadsworth, 1976])

Let N ∈ NF. For every M, there exists Ch[ ] such that Ch[N] →∗

β M.

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Solvability and transformability in λK

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =β I. [Barendregt, 1984]

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that

Ch[M] =β N ∈ NF. [Barendregt, 1971]

Theorem ([Wadsworth, 1976])

Let N ∈ NF. For every M, there exists Ch[ ] such that Ch[N] →∗

β M.

M is solvable iff it has a head normal form.

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Definition (Needed Redex in λK [Barendregt et al., 1987])

A redex R in M ≡ C[R] is needed iff R (or some residual of it) is contracted in every reduction sequence ending in normal form: M ≡ C[R] →β . . . →β N ∈ NF

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Polarity in λK

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) N(N))+ N(x) = x− N(λx.B) = (λx.N(B))− N(M N) = (N(M) N(N))−

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Polarity in λK

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) N(N))+ N(x) = x− N(λx.B) = (λx.N(B))− N(M N) = (N(M) N(N))− λx.(λy.y (λz.M1))(x M2) λx

λy
y

λz M1

x

λt M2

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Polarity in λK

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) N(N))+ N(x) = x− N(λx.B) = (λx.N(B))− N(M N) = (N(M) N(N))− (λx.((λy.y + (λz.M−

1 )−)+(x− M− 2 )−)+)+

λx+

+

λy +

+

y + λz− M−

1

−

x− λt− M−

2

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Polarity in λK (a.k.a. head spine [Barendregt et al., 1987])

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) N(N))+ N(x) = x− N(λx.B) = (λx.N(B))− N(M N) = (N(M) N(N))− hs(x) = x hs(λx.B) = λx.hs(B) hs(M N) = hs(M)N bn(x) = x bn(λx.B) = λx.B bn(M N) = bn(M)N λx.(λy.y (λz.M1))(x M2) λx

λy
y

λz M1

x

λt M2

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Polarity in λK (head reduction)

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) N(N))+ N(x) = x− N(λx.B) = (λx.N(B))− N(M N) = (N(M) N(N))− hr(x) = x hr(λx.B) = λx.hr(B) hr(M N) = bn(M)N bn(x) = x bn(λx.B) = λx.B bn(M N) = bn(M)N λx.(λy.y (λz.M1))(x M2) λx

λy
y

λz M1

x

λt M2

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Head normal forms

HNF ::= x | λx.HNF | HNF Λ λx1 . . . xn.y M1 . . . Mm λx1 λxn

y

M1 Mm

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Pure lambda-value calculus (λV) [Egidi et al., 1991]

◮ λV in [Plotkin, 1975] without primitive constants and δ-rules.

N ∈ Val (λx.B)N =βV [N/x]B

(βV )

Val ::= x | λx.Λ

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Pure lambda-value calculus (λV) [Egidi et al., 1991]

◮ λV in [Plotkin, 1975] without primitive constants and δ-rules.

N ∈ Val (λx.B)N =βV [N/x]B

(βV )

Val ::= x | λx.Λ

◮ Why values as non applications?

An irreducible application may become a divergent term when the free variables in it are substituted by other terms [Plotkin, 1975].

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Pure lambda-value calculus (λV) [Egidi et al., 1991]

◮ λV in [Plotkin, 1975] without primitive constants and δ-rules.

N ∈ Val (λx.B)N =βV [N/x]B

(βV )

Val ::= x | λx.Λ

◮ Why values as non applications?

An irreducible application may become a divergent term when the free variables in it are substituted by other terms [Plotkin, 1975]. “Preserving confluence by preserving potential divergence”.

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Is there a standard theory for λV?

◮ Call-by-value solvability [Paolini and Ronchi Della Rocca, 1999]:

M is v-solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =βV I. The order n of a v-unsolvable M informs about the number of preceding lambdas in the term, i.e., M =βV λx1 . . . λxn.B.

◮ The v-unsolvables of order 0 can be equated in a consistent way

[Paolini and Ronchi Della Rocca, 1999].

◮ Domain H constructed from canonical solution of equation

D ∼ = [D →⊥ D]⊥ [Egidi et al., 1991]. The v-unsolvable terms of order 0 denote ⊥ in H.

◮ Standard reduction in λV [Plotkin, 1975] is complete.

Does ‘complete’ imply ‘standard’?

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent.

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . .

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V →βV (λy.∆)(V I)∆

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V →βV (λy.∆)(V I)∆

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V →βV (λy.∆)(V I)∆ →∗

βV . . .

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V →βV (λy.∆)(V I)∆ →∗

βV . . .

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V →βV (λy.∆)(V I)∆ →∗

βV (λy.∆)V ′ ∆

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V →βV (λy.∆)(V I)∆ →∗

βV (λy.∆)V ′ ∆

→βV ∆ ∆ ≡ Ω

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V →βV (λy.∆)(V I)∆ →∗

βV (λy.∆)V ′ ∆

→βV ∆ ∆ ≡ Ω →βV . . .

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Objection on λV normal forms in [Paolini and Ronchi Della Rocca, 1999]

Ω and U ≡ (λy.∆)(x I)∆ are observationally equivalent. Consider the context (λx.[ ])V (λx.Ω)V →βV Ω →βV . . . (λx.(λy.∆)(x I)∆)V →βV (λy.∆)(V I)∆ →∗

βV (λy.∆)V ′ ∆

→βV ∆ ∆ ≡ Ω →βV . . .

But. . .

◮ Differences in sequentiality.

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Solvability and transformability in λV

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =βV I. M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =βV N ∈ NFV .

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Solvability and transformability in λV

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =βV I.

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that

Ch[M] =βV N ∈ NFV .

Theorem (Transformability for λV)

. . .

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Solvability and transformability in λV

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =βV I.

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that

Ch[M] =βV N ∈ NFV .

Theorem (Transformability for λV)

. . . The U ≡ (λy.∆)(x I)∆ is a counterexample which is not transformable.

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Solvability and transformability in λV

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =βV I [Paolini and Ronchi Della Rocca, 1999].

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that

Ch[M] =βV N ∈ NFV [our research].

Theorem (Transformability for λV)

. . . The U ≡ (λy.∆)(x I)∆ is a counterexample which is not transformable.

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Our research

◮ Reconsider definition of solvability in λV :

M is solvable iff there exists Ch[ ] ≡ (λx1 . . . xm[ ])N1 . . . Nn such that Ch[M] =βV N ∈ NFV . solvable = transformable + freezable A term which is freezable into a NFV is also operational relevant!

◮ Generalisation of needed reduction [Barendregt et al., 1987] to λV . ◮ HV = {M =βV N | M, N ∈ Λ0 unsolvables of the same order}+V is

a consistent theory. E.g., HV ⊢ λx.Ω =βv λx.(λy.∆)(x I)∆.

◮ A theory or a model is ω-sensible if it satisfies HV . ◮ Hereditary quasi-chest reduction (see next slides) characterises

complete strategies. Now ‘complete’ implies ‘hereditary quasi-chest’!

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Definition (Needed Redex in λV [Our research])

M ≡ C[R] is needed iff R (or some residual of it) is contracted in every reduction sequence ending in λV normal form: M ≡ C[R] →βV . . . →βV N ∈ NFV

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Polarity in λV

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) N(N))+ N(x) = x− N(λx.B) = (λx.N(B))− N(M N) = (N(M) N(N))−

λx.(λy.y(λz.M1))(x(λt.M2)) λx

λy
y

λz M1

x

λt M2

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Polarity in λV

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) A(N))+ A(x) = x− A(λx.B) = (λx.B(B))− A(M N) = (A(M) A(N))− B(x) = x− B(λx.B) = (λx.B(B))− B(M N) = (B(M) B(N))−

λx.(λy.y(λz.M1))(x(λt.M2)) λx

λy
y

λz M1

x

λt M2

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Polarity in λV

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) A(N))+ A(x) = x− A(λx.B) = (λx.B(B))− A(M N) = (A(M) A(N))− B(x) = x− B(λx.B) = (λx.B(B))− B(M N) = (B(M) B(N))−

λx.(λy.y(λz.M1))(x(λt.M2)) λx

λy
y

λz M1

x

λt M2

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Polarity in λV

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) A(N))+ A(x) = x− A(λx.B) = (λx.B(B))− A(M N) = (A(M) A(N))− B(x) = x− B(λx.B) = (λx.B(B))− B(M N) = (B(M) B(N))−

(λx.(λy.y +(λz.M−

1 )−)+(x−(λt.M− 2 )−)−)+

λx+

+

λy +

+

y + λz− M−

1

−

x− λt− M−

2

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Polarity in λV (ribcage reduction)

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) A(N))+ A(x) = x− A(λx.B) = (λx.B(B))− A(M N) = (A(M) A(N))− B(x) = x− B(λx.B) = (λx.B(B))− B(M N) = (B(M) B(N))− rc(x) = x rc(λx.B) = λx.rc(B) rc(M N) = rc(M)bv(N) bv(x) = x bv(λx.B) = λx.B bv(M N) = bv(M)bv(N)

λx.(λy.y(λz.M1))(x(λt.M2)) λx

λy
y

λz M1

x

λt M2

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Polarity in λV (chest reduction)

P(x) = x+ P(λx.B) = (λx.P(B))+ P(M N) = (P(M) A(N))+ A(x) = x− A(λx.B) = (λx.B(B))− A(M N) = (A(M) A(N))− B(x) = x− B(λx.B) = (λx.B(B))− B(M N) = (B(M) B(N))− ch(x) = x ch(λx.B) = λx.ch(B) ch(M N) = bv(M)bv(N) bv(x) = x bv(λx.B) = λx.B bv(M N) = bv(M)bv(N)

λx.(λy.y(λz.M1))(x(λt.M2)) λx

λy
y

λz M1

x

λt M2

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Chest normal forms

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Chest normal forms

1 . . . W 0 m0)W 1 1 . . . W 1 m1)

. . . )W p

1 . . . Mp mp

λx1 λxn

λyp

Bp

λy1

B1

z

W 0

1

W 0

m0

W 1

1

W 1

m1

W p

1

W p

mp 14 / 18

SLIDE 44

Semi-decision procedure for λV solvability

(λx.x ∆)(x I)∆ (λx.Ω)

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Semi-decision procedure for λV solvability

(λx.x ∆)(x I)∆ (λx.Ω)

◮ Mark-test-contract algorithm

◮ Positive subterms checked for solvability, negative subterms checked

nly for valuability.

◮ Stops anytime a (sub)term is freezable, avoiding to diverge when it is

not transformable.

◮ On-line implementation at

http://babel.ls.fi.upm.es/~agarcia/talks/DomainsXI

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To summarise

◮ Reconsider the definition of solvability in λV . ◮ The λv normal forms informs about sequentiality. ◮ Generalisation of needed reduction to λV . ◮ Hereditary chest reduction characterises complete strategies in λV . ◮ Semi-decision procedure for solvability in λV . ◮ Consistent theory HV equates unsolvables of the same order. ◮ A theory or model is ω-sensible iff it satisfies HV .

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Full abstraction

Questions:

◮ Fully abstract model w.r.t. HV ? ◮ Böhm trees for λV ?

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SLIDE 48

Full abstraction

Questions:

◮ Fully abstract model w.r.t. HV ? ◮ Böhm trees for λV ?

⊥ω ⊥1 ⊥0

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SLIDE 49

Full abstraction

Questions:

◮ Fully abstract model w.r.t. HV ? ◮ Böhm trees for λV ?

Thanks!

⊥ω ⊥1 ⊥0

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SLIDE 50

References I

Abramsky, S. (1990). The lazy lambda calculus. In Turner, D. A., editor, Research Topics in Functional Programming, pages 65–116. Addison-Welsey, Reading, MA. Barendregt, Kennaway, Klop, and Sleep (1987). Needed reduction and spine strategies for the lambda calculus. Information and Computation, 75(3):191–231. Barendregt, H. (1984). The Lambda Calculus, Its Syntax and Semantics. North Holland. Barendregt, H. P. (1971). Some Extensional Term Models for Combinatory Logics and Lambda Calculi. Ph.D. thesis, University of Utrecht, Utrecht, NL.

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References II

Egidi, L., Honsell, F., and Ronchi della Rocca, S. (1991). The lazy call–by–value λ–calculus. In Tarlecki, A., editor, Proceedings of Mathematical Foundations of Computer Science. (MFCS ’91), volume 520 of LNCS, pages 161–169, Berlin, Germany. Springer. Espírito Santo, J. (2007). Completing Herbelin’s programme. In Typed Lambda Calculi and Applications, 8th International Conference, TLCA 2007, Paris. Hindley, J. and Seldin, J. (2008). Lambda-calculus and combinators, an introduction. Cambridge University Press. Milner, R. (1990). Functions as processes. Research Report 1154, INRIA.

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References III

Paolini, L. and Ronchi Della Rocca, S. (1999). Call-by-value solvability. ITA, 33(6):507–534. Plotkin, G. (1975). Call-by-name, call-by-value and the lambda calculus. Theoretical Computer Science, 1:125–159. Wadsworth, C. (1976). The relation between computational and denotational properties for Scott’s D∞ models. Siam J. Comput., 5(3):488–521.

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