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No solvable lambda-value term left behind erez and Pablo Nogueira Alvaro Garc a P Reykjavik University IMDEA Software Institute ICE-TCS Theory Week, May 27th 2016 1 / 13 Computable functions and pure lambda calculus f N N f = {

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SLIDE 1

No solvable lambda-value term left behind

Alvaro Garc´ ıa P´ erez and Pablo Nogueira

Reykjavik University IMDEA Software Institute

ICE-TCS Theory Week, May 27th 2016

1 / 13

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SLIDE 2

Computable functions and pure lambda calculus

f ∈ N → N f = { (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), . . . } f = { (x, y) | y = x + x } f (x) = x + x

2 / 13

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SLIDE 3

Computable functions and pure lambda calculus

M, N ::= x | λx.M | M N f ∈ N → N f = { (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), . . . } f = { (x, y) | y = x + x } f (x) = x + x f ≡ λx. ADD x x

2 / 13

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SLIDE 4

Computable functions and pure lambda calculus

M, N ::= x | λx.M | M N f ∈ N → N f = { (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), . . . } f = { (x, y) | y = x + x } f (x) = x + x f ≡ λx. ADD x x ((ADD x1) ≡ g where g(x2) = x1 + x2)

2 / 13

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SLIDE 5

Computable functions and pure lambda calculus

M, N ::= x | λx.M | M N f ∈ N → N f = { (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), . . . } f = { (x, y) | y = x + x } f (x) = x + x f ≡ λx. ADD x x ((ADD x1) ≡ g where g(x2) = x1 + x2) f (2) ≡ (λx.ADD x x) TWO − → ADD TWO TWO − → . . .

2 / 13

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SLIDE 6

Computable functions and pure lambda calculus

M, N ::= x | λx.M | M N f ∈ N → N f = { (0, 0), (1, 2), (2, 4), (3, 6), (4, 8), . . . } f = { (x, y) | y = x + x } f (x) = x + x f ≡ λx. ADD x x ((ADD x1) ≡ g where g(x2) = x1 + x2) f ≡ λx. MULT x TWO f ≡ λx. SUBS (MULT (ADD x ONE) TWO) TWO f (2) ≡ (λx.ADD x x) TWO − → ADD TWO TWO − → . . .

2 / 13

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SLIDE 7

I is λx. x I M − → M K is λx. λy. x K M N − → (λy.M)N − → M ∆ is λx. x x ∆ M − → M M Ω is ∆ ∆ is (λx. x x)(λx. x x) Ω − → Ω − → Ω − → . . .

3 / 13

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SLIDE 8

Reduction in λ

(λx.B) N − → [N/x]B

(β)

4 / 13

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SLIDE 9

Reduction in λ

(λx.B) N − → [N/x]B

(β)

M − → M′ M N − → M′ N N − → N′ M N − → M N′ B − → B′ λx.B − → λx.B′

4 / 13

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SLIDE 10

Reduction in λ

(λx.B) N − → [N/x]B

(β)

M − → M′ M N − → M′ N N − → N′ M N − → M N′ B − → B′ λx.B − → λx.B′ M − → M M − → N N − → P M − → P

4 / 13

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SLIDE 11

Reduction in λ

(λx.B) N − → [N/x]B

(β)

M − → M′ M N − → M′ N N − → N′ M N − → M N′ B − → B′ λx.B − → λx.B′ M − → M M − → N N − → P M − → P M is a normal form : no (λx.B)N subterm in M M has normal form : M − → Z and Z is a normal form

4 / 13

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SLIDE 12

Reduction in λ

(λx.B) N − → [N/x]B

(β)

M − → M′ M N − → M′ N N − → N′ M N − → M N′ B − → B′ λx.B − → λx.B′ M − → M M − → N N − → P M − → P M is a normal form : no (λx.B)N subterm in M M has normal form : M − → Z and Z is a normal form Z unique

4 / 13

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Conversion in λ

(λx.B) N = [N/x]B

(β)

M = M′ M N = M′ N N = N′ M N = M N′ B = B′ λx.B = λx.B′ M = M M = N N = P M = P M = N N = M M is a normal form : no (λx.B)N subterm in M M has normal form : M = Z and Z is a normal form Z unique : consistent proof theory

4 / 13

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SLIDE 14

Can all terms without normal form be considered equal?

Ω = λx.x Ω iff Ω (K I) = (λx.x Ω)(K I) iff Ω (K I) = K I Ω iff Ω (K I) = I

5 / 13

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SLIDE 15

Can all terms without normal form be considered equal?

Ω = λx.x Ω iff Ω (K I) = (λx.x Ω)(K I) iff Ω (K I) = K I Ω iff Ω (K I) = I Adding axioms that equate terms without normal form: inconsistent proof theory!

5 / 13

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SLIDE 16

Solvability

Definition (Barendregt ’71)

M closed is solvable iff there exist closed operands N1, . . . , Nn with n ≥ 0 s.t. M N1 · · · Nn = Z a normal form.

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SLIDE 17

Solvability

Definition (Barendregt ’71)

M closed is solvable iff there exist closed operands N1, . . . , Nn with n ≥ 0 s.t. M N1 · · · Nn = Z a normal form. Ω is unsolvable λx.x Ω is solvable

6 / 13

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SLIDE 18

Solvability

Definition (Barendregt ’71)

M closed is solvable iff there exist closed operands N1, . . . , Nn with n ≥ 0 s.t. M N1 · · · Nn = Z a normal form.

Definition (Wadsworth ’78)

M closed is solvable iff for every term X there exist closed

  • perands N1, . . . , Nn with n ≥ 0 s.t. M N1 · · · Nn = X.

Lemma (Wadsworth ’78)

If T closed has normal form then for every term X there exist X1, . . . , Xk s.t. T X1 · · · Xk = X.

6 / 13

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SLIDE 19

Solvability

Definition (Barendregt ’71)

M closed is solvable iff there exist closed operands N1, . . . , Nn with n ≥ 0 s.t. M N1 · · · Nn = Z a normal form.

Definition (Wadsworth ’78)

M closed is solvable iff for every term X there exist closed

  • perands N1, . . . , Nn with n ≥ 0 s.t. M N1 · · · Nn = X.

Lemma (Wadsworth ’78)

If T closed has normal form then for every term X there exist X1, . . . , Xk s.t. T X1 · · · Xk = X.

Definition (Barendregt ’84)

M closed is solvable iff there exist closed operands N1, . . . , Nn with n ≥ 0 s.t. M N1 · · · Nn = I.

6 / 13

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Solvability (open terms)

Definition (Wadsworth ’78)

M arbitrary is solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn = Z a normal form.

7 / 13

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SLIDE 21

Solvability (open terms)

Definition (Wadsworth ’78)

M arbitrary is solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn = Z a normal form. x Ω

7 / 13

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SLIDE 22

Solvability (open terms)

Definition (Wadsworth ’78)

M arbitrary is solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn = Z a normal form. (λx.[x Ω])(K I)

7 / 13

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SLIDE 23

Solvability (open terms)

Definition (Wadsworth ’78)

M arbitrary is solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn = Z a normal form. (λx.[x Ω])(K I) − → K I Ω − → I

7 / 13

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SLIDE 24

Solvability (open terms)

Definition (Wadsworth ’78)

M arbitrary is solvable iff for all X there exists (λx1 . . . xm.[ ])N1 · · · Nn with m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn = X. (λx.[x Ω])(K I) − → K I Ω − → I

7 / 13

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SLIDE 25

Solvability (open terms)

Definition (Wadsworth ’78)

M arbitrary is solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn = I. (λx.[x Ω])(K I) − → K I Ω − → I

7 / 13

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SLIDE 26

Operational relevance and effective use

Genericity Lemma (Barendregt ’84)

M unsolvable then ∀C[ ]. C[M] = Z a normal form ⇒ ∀X. C[X] = Z.

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Operational relevance and effective use

Genericity Lemma (Barendregt ’84)

M unsolvable iff ∀C[ ]. C[M] = Z a normal form ⇒ ∀X. C[X] = Z.

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SLIDE 28

Operational relevance and effective use

Genericity Lemma (Barendregt ’84)

M is solvable iff ∃C[ ]. C[M] = Z a normal form ∧ ¬(∀X. C[X] = Z).

8 / 13

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SLIDE 29

Operational relevance and effective use

Genericity Lemma (Barendregt ’84)

M is solvable iff ∃C[ ]. C[M] = Z a normal form ∧ ¬(∀X. C[X] = Z). normal forms ⊂ solvables

8 / 13

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SLIDE 30

Operational relevance and effective use

Genericity Lemma (Barendregt ’84)

M is solvable iff ∃C[ ]. C[M] = Z a normal form ∧ ¬(∀X. C[X] = Z). normal forms ⊂ solvables Adding axioms that equate unsolvable terms: consistent proof theory! Full reduction and open terms: suitable for metaprogramming!

8 / 13

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Conversion in the lambda-value calculus (λ

V) (Plotkin ’75) N ∈ Val ::= x | λx.M (λx.B) N =V [N/x]B

(βV )

M =V M′ M N =V M′ N N =V N′ M N =V M N′ B =V B′ λx.B =V λx.B′ M =V M M =V N N =V P M =V P M =V N N =V M

9 / 13

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SLIDE 32

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆

10 / 13

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SLIDE 33

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆

10 / 13

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SLIDE 34

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆

10 / 13

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SLIDE 35

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆

10 / 13

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SLIDE 36

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆ − →V I

10 / 13

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SLIDE 37

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆ − →V I (λx.(λy.I)(x ∆))∆

10 / 13

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SLIDE 38

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆ − →V I (λx.(λy.I)(x ∆))∆ − →V (λy.I)(∆ ∆)

10 / 13

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SLIDE 39

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆ − →V I (λx.(λy.I)(x ∆))∆ − →V (λy.I)Ω

10 / 13

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SLIDE 40

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆ − →V I (λx.(λy.I)(x ∆))∆ − →V (λy.I)Ω − →V . . .

10 / 13

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SLIDE 41

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆ − →V I (λx.(λy.I)(x ∆))∆ − →V (λy.I)Ω − →V . . .

10 / 13

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SLIDE 42

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆ − →V I (λx.(λy.I)(x ∆))∆ − →V (λy.I)Ω − →V . . . λ

V -normal-forms don’t have subterms (λx.B)N with N ∈ Val.

(λy.I)(x ∆) is a λ

V -normal-form

10 / 13

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SLIDE 43

Why values instead of normal forms?

Preserving confluence by preserving potential divergence. (λx.(λy.I)(x ∆))∆ − →V (λx.I)∆ − →V I (λx.(λy.I)(x ∆))∆ − →V (λy.I)Ω − →V . . . λ

V -normal-forms don’t have subterms (λx.B)N with N ∈ Val.

(λy.I)(x ∆) is a λ

V -normal-form

Sequential character of λ

V -normal-forms.

(x M)(y N) (λz.z(y N))(x M)

10 / 13

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SLIDE 44

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

11 / 13

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SLIDE 45

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X.

11 / 13

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SLIDE 46

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

λx.(λy.∆)(x I)∆

11 / 13

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SLIDE 47

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1

11 / 13

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SLIDE 48

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1 − →V (λy.∆)(V1 I)∆

11 / 13

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SLIDE 49

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1 − →V (λy.∆)(V1 I)∆

11 / 13

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SLIDE 50

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1 − →V (λy.∆)(V1 I)∆ − → . . .

11 / 13

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SLIDE 51

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1 − →V (λy.∆)(V1 I)∆

11 / 13

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SLIDE 52

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1 − →V (λy.∆)(V1 I)∆ − → (λy.∆)∆

11 / 13

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SLIDE 53

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1 − →V (λy.∆)(V1 I)∆ − → (λy.∆)∆ − → ∆∆

11 / 13

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SLIDE 54

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1 − →V (λy.∆)(V1 I)∆ − → (λy.∆)∆ − → Ω − → . . .

11 / 13

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SLIDE 55

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I.

Lemma (analogous to Wadsworth ’78)

If T closed has λ

V -normal-form then for every term X there

exist values V1, . . . , Vk s.t. T V1 · · · Vk =V X. Some λ

V -normal-forms are v-unsolvable!

(λx.(λy.∆)(x I)∆)V1 − →V (λy.∆)(V1 I)∆ − → (λy.∆)∆ − → Ω − → . . . Does not capture operational relevance! Adding axioms that equate v-unsolvables: inconsistent proof theory!

11 / 13

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SLIDE 56

v-solvability

Definition (Paolini & Ronchi della Rocca ’99)

M closed is v-solvable iff there exist closed values V1, . . . , Vn with n ≥ 0 s.t. M V1 · · · Vn =V I. SECD Machine (Landin ’64). Lazy call-by-value evaluation (Egidi, Honsell, Ronchi della Rocca ’91). Only weak reduction and closed terms: not suitable for metaprogramming!

11 / 13

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SLIDE 57

Start from scratch

Definition (Garc´ ıa-P´ erez & Nogueira 2016)

M is λ

V -solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with

m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn has λ

V -normal-form.

12 / 13

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SLIDE 58

Start from scratch

Definition (Garc´ ıa-P´ erez & Nogueira 2016)

M is λ

V -solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with

m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn has λ

V -normal-form.

The λ

V -normal forms are λ V -solvable!

Adding axioms equating λ

V -unsolvables of the same order:

consistent proof theory! Full reduction and open terms: suitable for metaprogramming!

12 / 13

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SLIDE 59

Start from scratch

Definition (Garc´ ıa-P´ erez & Nogueira 2016)

M is λ

V -solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with

m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn has λ

V -normal-form.

The λ

V -normal forms are λ V -solvable!

Adding axioms equating λ

V -unsolvables of the same order:

consistent proof theory! Full reduction and open terms: suitable for metaprogramming! λ

V -solvability captures operational relevance: ◮ Transformability. Ability to send a term to a value of your

choice (captured by v-solvability too).

◮ Freezability. Ability to send a term to a λ V -normal-form,

albeit not of your choice (captured by λ

V -solvability only).

12 / 13

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SLIDE 60

Start from scratch

Definition (Garc´ ıa-P´ erez & Nogueira 2016)

M is λ

V -solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with

m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn has λ

V -normal-form.

The λ

V -normal forms are λ V -solvable!

Adding axioms equating λ

V -unsolvables of the same order:

consistent proof theory! Full reduction and open terms: suitable for metaprogramming! λ

V -solvability captures operational relevance: ◮ Transformability. Ability to send a term to a value of your

choice (captured by v-solvability too).

◮ Freezability. Ability to send a term to a λ V -normal-form,

albeit not of your choice (captured by λ

V -solvability only).

λx.(λy.∆)(x I)∆

12 / 13

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SLIDE 61

Start from scratch

Definition (Garc´ ıa-P´ erez & Nogueira 2016)

M is λ

V -solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with

m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn has λ

V -normal-form.

The λ

V -normal forms are λ V -solvable!

Adding axioms equating λ

V -unsolvables of the same order:

consistent proof theory! Full reduction and open terms: suitable for metaprogramming! λ

V -solvability captures operational relevance: ◮ Transformability. Ability to send a term to a value of your

choice (captured by v-solvability too).

◮ Freezability. Ability to send a term to a λ V -normal-form,

albeit not of your choice (captured by λ

V -solvability only).

(λx.(λy.∆)(x I)∆)(λx.z K)

12 / 13

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SLIDE 62

Start from scratch

Definition (Garc´ ıa-P´ erez & Nogueira 2016)

M is λ

V -solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with

m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn has λ

V -normal-form.

The λ

V -normal forms are λ V -solvable!

Adding axioms equating λ

V -unsolvables of the same order:

consistent proof theory! Full reduction and open terms: suitable for metaprogramming! λ

V -solvability captures operational relevance: ◮ Transformability. Ability to send a term to a value of your

choice (captured by v-solvability too).

◮ Freezability. Ability to send a term to a λ V -normal-form,

albeit not of your choice (captured by λ

V -solvability only).

(λx.(λy.∆)(x I)∆)(λx.z K) − → (λy.∆)((λx.z K)I)∆

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slide-63
SLIDE 63

Start from scratch

Definition (Garc´ ıa-P´ erez & Nogueira 2016)

M is λ

V -solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with

m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn has λ

V -normal-form.

The λ

V -normal forms are λ V -solvable!

Adding axioms equating λ

V -unsolvables of the same order:

consistent proof theory! Full reduction and open terms: suitable for metaprogramming! λ

V -solvability captures operational relevance: ◮ Transformability. Ability to send a term to a value of your

choice (captured by v-solvability too).

◮ Freezability. Ability to send a term to a λ V -normal-form,

albeit not of your choice (captured by λ

V -solvability only).

(λx.(λy.∆)(x I)∆)(λx.z K) − → (λy.∆)((λx.z K)I)∆ − → (λy.∆)(z K)∆

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slide-64
SLIDE 64

Start from scratch

Definition (Garc´ ıa-P´ erez & Nogueira 2016)

M is λ

V -solvable iff there exists (λx1 . . . xm.[ ])N1 · · · Nn with

m, n ≥ 0 s.t. (λx1 . . . xm.[M])N1 · · · Nn has λ

V -normal-form.

The λ

V -normal forms are λ V -solvable!

Adding axioms equating λ

V -unsolvables of the same order:

consistent proof theory! Full reduction and open terms: suitable for metaprogramming! λ

V -solvability captures operational relevance: ◮ Transformability. Ability to send a term to a value of your

choice (captured by v-solvability too).

◮ Freezability. Ability to send a term to a λ V -normal-form,

albeit not of your choice (captured by λ

V -solvability only).

(λx.(λy.∆)(x I)∆)(λx.z K) − → (λy.∆)((λx.z K)I)∆ − → (λy.∆)(z K)∆

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slide-65
SLIDE 65

Our contributions

◮ Definition of λ V -solvability that captures operational

relevance of arbitrary terms (not necessarily closed).

◮ Full reduction and open terms. ◮ Sequentiality of terms matters. ◮ Consistent proof theory. ◮ Partial Genericity Lemma (order of a term matters). ◮ Characterisation of complete reduction strategies with

respect to λ

V -normal-form. ◮ Do models for sequentiality (Berry & Curien ’82) satisfy

  • ur theory?

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slide-66
SLIDE 66

Our contributions

◮ Definition of λ V -solvability that captures operational

relevance of arbitrary terms (not necessarily closed).

◮ Full reduction and open terms. ◮ Sequentiality of terms matters. ◮ Consistent proof theory. ◮ Partial Genericity Lemma (order of a term matters). ◮ Characterisation of complete reduction strategies with

respect to λ

V -normal-form. ◮ Do models for sequentiality (Berry & Curien ’82) satisfy

  • ur theory?

Thanks!

13 / 13