Open Call-by-Value Beniamino Accattoli1 1 2

Giulio Guerrieri2

´ INRIA, LIX, Ecole Polytechnique

I2M, Aix-Marseille Universit´ e & Dipartimento di Matematica e Fisica, Universit` a Roma Tre

APLAS 2016

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Call-by-Value

Plotkin’s Call-by-Value (CbV) λ-Calculus models: Programming Languages such as Ocaml or Standard ML; Proof assitants such as Coq. The two settings rely on different hypotheses.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Flavors of Call-by-Value

Programming Languages rely on Closed CbV, that is: Weak Evaluation, i.e. function bodies are not evaluated; Closed Terms. Proof assitants rely on Strong CbV, that is: Strong Evaluation, i.e. function bodies are evaluated; Open Terms.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Closed vs Strong CbV

Closed CbV is canonical and much simpler than Strong CbV. 2 Problems: Strong CbV 6= Closed CbV iterated into function bodies; No canonical presentation of Strong CbV.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Open CbV

Open CbV is halfway Closed and Strong CbV: Weak Evaluation (as in Closed CbV); Open Terms (as in Strong CbV). Open terms ⇒ Open CbV can be iterated into function bodies. Then, Strong CbV = Iterated Open CbV. No canonical presentation of Open CbV, but many equivalent ones.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Open CbV—Results We compare 4 presentations of Open CbV arising in different contexts: Abstract Machines; Linear Logic Proof Nets; Denotational Semantics; Sequent Calculus. We prove them equivalent with respect to termination. We establish precise quantitative relationships, relating the cost models. Summing Up: many presentations defining the same framework, Open CbV.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Outline 1

Closed CbV

2

Open CbV 0: Naive Open CbV

3

Open CbV 1: The Fireball Calculus

4

Open CbV 2: The Value Substitution Calculus

5

Open CbV 3: The Shuffling Calculus

6

Open CbV 4: The Value Sequent Calculus

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

7 / 39

Plotkin’s Closed CbV Language: Terms Values

t, u, s v

:= :=

v x

| |

tu λx.t

Weak evaluation contexts: E

:=

h · i | Et

| tE

Small-step Evaluation: Rule at Top Level (λx.t)v 7→βv t{x v }

B. Accattoli, G.Guerrieri

Contextual closure E hti →βv E hui iff t 7→βv u

Open CBV

APLAS 2016

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Harmony

Closed CbV harmony property: A closed term t is a →βv -normal forms iff t is a value. Harmony = Internal completeness wrt unconstrained β-reduction. Closed CbV never gets stuck: Every β-redex eventually becomes a βv -redex, unless evaluation diverges.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

9 / 39

Time Cost Model

The study of cost models for λ-calculi is a delicate subject. In Closed CbV the number of βv -steps is a reasonable time cost model. Reasonable = polynomially related to Turing machines. Roughly, one can count 1 for every βv -step. Result first proved by Blelloch and Greiner (1996).

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

10 / 39

Outline 1

Closed CbV

2

Open CbV 0: Naive Open CbV

3

Open CbV 1: The Fireball Calculus

4

Open CbV 2: The Value Substitution Calculus

5

Open CbV 3: The Shuffling Calculus

6

Open CbV 4: The Value Sequent Calculus

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

11 / 39

Adding Open Terms Naively

Open terms break harmony. They introduce stuck β-redexes. Consider t := (λx.u)(yz). t is stuck, it will never reduce to a value nor diverge.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Adding Open Terms Naively

Stuck β-redexes provide prematures βv -normal forms, e.g. t := (λy .δ)(zz)δ

u := δ((λy .δ)(zz))

where δ := λx.xx

t and u should behave as δδ, i.e. they should diverge: for semantical reasons (empty denotational semantics) for logical reasons (evaluation into linear logic proof-nets) ⇒ Open terms impact on termination and observational equivalences.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

13 / 39

Improving CbV From the start it was clear that something is wrong with CbV. Already Plotkin noted that CPS is sound but not complete for CbV. Many proposals for a better CbV in the literature. Moggi (1989), Sabry & Felleisen (1993), Sabry & Wadler (1997), Maraist et al. (1999), Curien & Herbelin (2000), Dychoff & Lengrand (2007), Herbelin & Zimmermann (2009), ...

They all focus on Strong CbV. We consider 4 representative cases, but on Open CbV.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Outline 1

Closed CbV

2

Open CbV 0: Naive Open CbV

3

Open CbV 1: The Fireball Calculus

4

Open CbV 2: The Value Substitution Calculus

5

Open CbV 3: The Shuffling Calculus

6

Open CbV 4: The Value Sequent Calculus

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

15 / 39

Fireball Calculus Statically: values are generalized to Fireballs. Dynamically: Call-by-Value ⇒ Call-by-Fireball. History: The Fireball Calculus λfire was introduced as a technical tool. (Paolini and Ronchi della Rocca 1999)

Rediscovered to improve the implementation of Coq. (Gr´ egoire and Leroy 2002)

Rediscovered (and named) to study cost models for CbV. (Accattoli and Sacerdoti Coen 2015)

Fireball is a pun on fire-able term.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Fireball Calculus Language: Terms Values Fireballs

t, u, s v f

:= := :=

v x v

| | |

tu λx.t xf 1 . . . f n | {z }

n≥1

inert terms i

Weak evaluation contexts (exactly as before): E

:=

h · i | Et

| tE

Small-step Call-by-Fireball Evaluation: Rule at Top Level (λx.t)f 7→βf t{x f }

B. Accattoli, G.Guerrieri

Contextual closure E hti →βf E hui iff t 7→βf u

Open CBV

APLAS 2016

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Properties of Fireballs 1

Open Harmony: a term t is a →βf -normal forms iff t is a fireball.

2

Conservative Extension of Closed CbV: if t is closed then t →βv u iff t →βf u.

3

Reasonable Cost Model (Accattoli and Sacerdoti Coen 2015): The number of →βf steps is a reasonable cost model. B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Properties of Fireballs

1

The calculus is strongly confluent (diamond property).

2

Fireballs remove stuck β-redexes:

3

t := (λy .δ)(zz)δ

→βf

δδ

→βf

...

u := δ((λy .δ)(zz))

→βf

δδ

→βf

...

Problem: not confluent in its strong version.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Outline 1

Closed CbV

2

Open CbV 0: Naive Open CbV

3

Open CbV 1: The Fireball Calculus

4

Open CbV 2: The Value Substitution Calculus

5

Open CbV 3: The Shuffling Calculus

6

Open CbV 4: The Value Sequent Calculus

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

20 / 39

The Value Substitution Calculus

The Value Substitution Calculus λvsub was introduced by Accattoli and Paolini. (2012) It is a λ-calculus with explicit substitutions. Modern approach to explicit substitutions, called at a distance. Grounded in linear logic.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Value Substitution Calculus Language: t, u, s

:=

x

|

λx.t

|

|

tu

t[x u]

Evaluation contexts and Lists of substitutions contexts: :=

h·i

|

Et

L :=

h·i

|

L[x t]

E

| tE

| E [x t]

| t[x E ]

Rewriting rules (closed by evaluation contexts E h · i): Lhλx.tiu

→m

Lht[x u]i

exponential t[x Lhv i]

→e

Lht{x v }i

multiplicative

Corresponding to principal (or key) cut-elimination cases. B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Value Substitution Calculus vs Stuck β-redexes Lists of substitutions contexts: L :=

h·i

|

L[x t]

Rewriting rules: Lhλx.tiu

→m

Lht[x u]i

exponential t[x Lhv i]

→e

Lht{x v }i

multiplicative

Example: (λy .δ)(zz)δ

→m

δ[y zz]δ

→m

(xx)[x δ][y zz] →e

(δδ)[y zz] . . .

(Remember δ := λx.xx). B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Value Substitution Calculus—Key Properties

1

Elegant and simple rewriting theory: Strongly confluent; The two evaluation rules commute and terminate (separately);

2

Isomorphic to linear logic proof nets.

3

Moggi’s equational theory is contained in the one of λvsub .

4

Our termination equivalences are all proved using λvsub .

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

24 / 39

Fireball and Value Substitutins Termination equivalence: A λ-term t terminates in λfire iff t terminates in λvsub . Moreover: evaluation takes the same number of →βf and →m steps. ⇒ the cost model transfer from λfire to λvsub . The result is trickier than it seems: multiplicative

Lhλx.tiu

→m

Lht[x u]i

Fireball β

(λx.t)f

→βf

t{x f }

→βf requires arguments to be fireballs while →m does not!

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

25 / 39

Outline 1

Closed CbV

2

Open CbV 0: Naive Open CbV

3

Open CbV 1: The Fireball Calculus

4

Open CbV 2: The Value Substitution Calculus

5

Open CbV 3: The Shuffling Calculus

6

Open CbV 4: The Value Sequent Calculus

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

26 / 39

The Shuffling Calculus

The Shuffling Calculus λshuf was introduced by Carraro ad Guerrieri in (2014). It solves the mismatch between denotational semantics and Naive Open CbV. It refines Plotkin’s calculus with rules shuffling constructors. The shuffling rules are CbV variants of Regnier’s σ-rules (for call-by-name).

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

27 / 39

The Shuffling Calculus Language: t, u, s

:=

x

|

λx.t

|

tu

Balanced Evaluation contexts: B ::= h·i | tB | Bt | (λx.B)t Rewriting rules (closed by balanced evaluation contexts Bh · i):

B. Accattoli, G.Guerrieri

CbV β

(λx.t)v

→βv[

t{x v }

Shuffling 1

((λx.t)u)s

→σ1[

(λx.ts)u

Shuffling 3

v ((λx.s)u)

→σ3[

(λx.v s)u

Open CBV

APLAS 2016

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Shuffling Calculus Moves Stuck β-redexes Balanced Evaluation contexts: B ::= h·i | tB | Bt | (λx.B)t Rewriting rules: CbV β

(λx.t)v

→βv[

t{x v }

Shuffling 1

((λx.t)u)s

→σ1[

(λx.ts)u

Example: stuck β-redexes are moved by shuffling rules: (λy .δ)(zz)δ

B. Accattoli, G.Guerrieri

→σ1[

(λy .δδ)(zz) →βv[

Open CBV

(λy .δδ)(zz) →βv[ . . .

APLAS 2016

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Shuffling Calculus Key Properties

Carraro and Guerrieri show that λshuf is adequate wrt denotational semantics. Denotational semantics = relational semantics induced by linear logic. Adequacy: a term t terminates iff its interpretation is non-empty.

λshuf is not strongly confluent but SN and WN coincide.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

30 / 39

Shuffling vs Value Substitutions

We show how to simulate λshuf in λvsub , obtaining termination equivalence. Quantitatively: the number of →βv[ steps coincide with the number of →e steps. (λx.t)v

→βv[

t{x v }

exponential t[x Lhv i]

→e

Lht{x v }i

CbV β

This result does not allow to transfer the cost model. Termination equivalence ⇒ adequacy transfers to all incarnations of Open CbV.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Outline 1

Closed CbV

2

Open CbV 0: Naive Open CbV

3

Open CbV 1: The Fireball Calculus

4

Open CbV 2: The Value Substitution Calculus

5

Open CbV 3: The Shuffling Calculus

6

Open CbV 4: The Value Sequent Calculus

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

32 / 39

The Value Sequent Calculus

The Value Sequent Calculus λvseq is a term syntax for sequent calculus. It is a fragment of Curien and Herbelin’s λµ˜ µ. ¯ µ (the CbV variant of λµ˜ Precisely, it is the intuitionistic restriction of λ˜ µ). Key idea: changing the applicative structure of λ-terms. Sequence of applications are looked at from the head value.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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The Value Sequent Calculus

Language: Commands Values Environments

c v e

::= hv | ei ::= x | λx.c ::=  | µ ˜x.c | v ·e

Rewriting rules: Sequent β

hλx.c | v ·ei

→λ¯

hv | (˜ µx.c)@ei

µ ˜-rule

hv | µ ˜x.ci

→µ˜

c{x v }

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Value Sequent Calculus—Key Properties

Strongly confluent. No need of contextual rewriting rules (rules at a distance). No need of shuffling rules. Drawback: drastic change of syntax.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Sequent Calculus vs Value Substitution

λvseq is isomorphic to a sub-language of λvsub . The kernel λvsubk of λvsubk is obtained by restricting the application: t, u, s

:=

x

|

λx.t

|

tv

|

t[x u]

Everything else is unchanged. This can be seen as the language of administrivative normal forms of λvsub .

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Sequent Calculus vs Value Substitution

Termination equivalence of λvsub and λvseq is proved in 2 steps: 1

Showing that λvsub is termination equivalent to its kernel λvsubk ;

2

Showing that the kernel λvsubk is isomorphic to λvseq .

The number of →m steps is preserved (becoming →λ¯ steps). ⇒ the cost model transfers to the value sequent calculus λvseq .

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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Conclusions

4 calculi arising from different perspectives. All termination equivalent. They can be seen as different incarnations of Open CbV. The cost model is stable in 3 out of 4. Adequacy of denotational semantics. Linear logic foundation. Sequent calculus interpretation.

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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THANKS!

B. Accattoli, G.Guerrieri

Open CBV

APLAS 2016

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