A Nonstandard Standardization Theorem Beniamino Accattoli1 Eduardo Bonelli2 Carlos Lombardi 2,3 1 Carnegie

Delia Kesner3

Mellon University and Universit` a di Bologna

2 Univ.

Nac. de Quilmes and CONICET

3 Universit´ e

Paris-Diderot and CNRS

Outline

Standardization Explicit Substitutions Orthogonality and the Linear Substitution Calculus Proof Nets and Partial Standardization Orders Advertisement and Conclusions

Sorting Pick any vector of integers: 3, 1, 8, 6, 12, 4 It can always be sorted by swapping elements: 3, 1, 8, 6, 12, 4 1, 3, 8, 6, 12, 4 ... 1, 3, 4, 6, 8, 12 Remark that 1. Any two elements are comparable. 2. Swapping is linear and does not change the elements.

What is standardization about? I

Rough Idea: Standardization = Sorting redexes. R R

R

t →1 →2 . . . →k s ... Rσ(1) Rσ(2) Rσ(k) t → → ... → s with respect to an order ≺ (typically the left-to-right order). I

Standard derivation = ≺-ordered derivation.

I

Way subtler than sorting: 1. Redex creation ⇒ elements may not be comparable. 2. Swapping is non-linear: it may duplicate/erase redexes. 3. Swapping changes the elements.

Standardization and λ-calculus I

β-reduction: (λx.t)s →β t{x/s}.

I

λ-calculus does not come with an evaluation strategy.

I

Non-deterministic but confluent (unique result, if any).

Theorem (Standardization) Pick your favourite, arbitrary derivation d : t →∗β s. 1. Existence: it can be ordered wrt the left-to-right order. 2. Uniqueness: any sequence of swaps gives the same result. I

Completeness: non-standard derivations can be discarded.

I

Existence = Termination, Uniqueness = Confluence.

Leftmost Reduction, Normalization, and PL I

A further concept: Leftmost Reduction Strategy →leftmost : Reduce the leftmost redex at each step

I

More abstractly: leftmost = ≺-minimum.

I

→leftmost is standard (by minimality).

I

Normalization Thm: →leftmost reaches a normal form, if any.

I

More abstractly: ≺-minimum strategies are normalizing.

I

Plotkin’s Approach: 1. Standardization: calculus

programming language.

2. Idea: Abstract machines implement ≺-minimum strategies. 3. Justifications: completeness and normalization.

Outline

Standardization Explicit Substitutions Orthogonality and the Linear Substitution Calculus Proof Nets and Partial Standardization Orders Advertisement and Conclusions

Explicit Substitutions

I

λ-calculus is based on substitution (noted t{x/s}).

I

Implementations delay it, doing it lazily and partially.

I

Explicit Substitutions tried to close the gap: 1. Statically: a new constructor t[x/s] for delayed subst. 2. Dynamically: small-step operational semantics (λx.t)s →β t{x/s} becomes (λx.t)s

→B

t[x/s] →∗s t{x/s}

Ordinary Explicit Substitutions

(λx.t)s

→B

(tu)[x/s] →@ (λx.t)[x/s] →λ x[x/s] →var y [x/s] →gc

t[x/s] t[x/s]u[x/s] λx.t[x/s] s y

I

Substitutions percolate through the term structure;

I

Commuting with applications and abstractions;

I

Substituting only when facing variables.

This Paper

I

Natural expectation: Abstract Machines = ≺-minimum Strategies for ES.

I

No such Plotkin’s-style-theorem in the literature.

I

ES calculi are complex, non-orthogonal systems.

I

Standardization is out of scope.

I

This paper: a standardization theory for ES.

Outline

Standardization Explicit Substitutions Orthogonality and the Linear Substitution Calculus Proof Nets and Partial Standardization Orders Advertisement and Conclusions

Syntactic Orthogonality I

Syntactic Orthogonality: redexes cannot overlap.

I

Most standardization proofs rely on synt. orthogonality.

I

No ES calculus is syntactically orthogonal.

I

Example, consider the following reduction rules: (λx.t)s →B t[x/s] (tu)[x/s] →@ t[x/s]u[x/s] They induce a critical pair: (λx.t)[y /u]s[y /u]

@←

( (λx.t)s )[y /u] →B t[x/s][y /u]

Semantic Orthogonality I

Syntactic Orthogonality implies Semantical Orthogonality: s

P

s1 R/P

R

s2

P/R

t

aka parallel moves, or cube property, or residual system. I

Abstract Standardization relies on semantical orthogonality. [Gonthier, L´evy, and Melli`es, LICS ’91] [Melli` es PhD thesis, ’96]

I

No ordinary ES calculus is semantically orthogonal either.

A Nonstandard Calculus of ES

I

Linear logic and ES are related decompositions of λ-calculus. [Kesner and co-authors, LICS ’97, FOSSACS 2000,...]

I

Research culminated in the Linear Substitution Calculus. (next slide)

I

It is a calculus at a distance, where Distance = Use of contexts

I

No commutative rule, way simpler meta-theory.

I

Variation over calculi by Milner, De Bruijn, and Nederpelt.

Linear Substitution Calculus (LSC) I

Language: t, s, u

I

x

|

λx.t

|

ts

|

t[x/s]

One-Hole Contexts and Lists of substitutions contexts: C L

I

:=

:= :=

[·] | Ct | tC [·] | L[x/t]

| λx.C

| C [x/t] | t[x/C ]

Rewriting rules (closed by contexts C [·]): Beta at a Distance L[λx.t]s →dB L[t[x/s]] Linear Substitution C [x][x/s] →ls C [s][x/s] Garbage Collection t[x/s] →gc t

if x ∈ / fv(t)

Distance 1 I

Rule at a distance: L[λx.t]s →dB L[t[x/s]] (λx.t)[·/·] . . . [·/·] s →dB t[x/s][·/·] . . . [·/·]

I

Traditionally a configuration like: (λx.t)[y /s] u is not a redex, as it is blocked by [y /s].

I

Here, instead, it is a redex.

Distance 2 I

The substitution rules at a distance: C [x][x/s] →ls C [s][x/s] t[x/s] →gc t are the ordinary rules: x[x/s] →var y [x/s] →gc

if x ∈ / fv(t)

s y

integrating the commutative rules via contexts: (tu)[x/s] →@ t[x/s]u[x/s] (λx.t)[x/s] →λ λx.t[x/s] I

Distance

simpler proof of SN for LL [A., RTA 2013].

I

Implementable using pointers (vs de Bruijn indexes).

Features The Linear Substitution Calculus has many nice properties: I

Bisimilar to linear logic proof nets;

I

Expresses naturally 1. Linear head reduction (LHR) = Game semantics eval. 2. Weak LHR = π-calculus evaluation = KAM

I

...

I

As any other ES-calculus it is not syntactically orthogonal...

I

...but it is the first semantically orthogonal ES-calculus.

Left-to-Right Standardization For the LSC we provide: I

A left-to-right standardization theorem.

I

Proved by means of Melli` es axiomatics.

I

Simpler approaches as Plotkin’s or Takahashi’s do not work.

I

A notion of leftmost reduction →leftmost .

I

A proof that →leftmost is standard and normalising.

I

Weak linear head reduction (π and KAM) is standard.

I

Plotkin’s approach successfully lifted to ES.

Outline

Standardization Explicit Substitutions Orthogonality and the Linear Substitution Calculus Proof Nets and Partial Standardization Orders Advertisement and Conclusions

Inception of the LSC

I

LSC = algebraic copy of linear logic proof nets.

I

Every term t translates to a proof net Pt .

I

Every proof net P can be read back as a term tP .

I

Different terms may translate to the same proof net.

I

Proof nets are a syntactic quotient.

I

Dynamically: PN and LSC are behaviourally isomorphic.

Proof Nets Strong Bisimulation LSC terms and proof nets satisfy: s

t

t

s

Pt

Ps

⇒ Pt

and t

t

s

Pt

P0

⇒ ∃s s.t. Pt

P0

Strong Bisimulation: one step to one step, and viceversa.

The Graphical Equivalence I

The quotient is given by the following equivalence: t[x/s][y /u] ∼ t[y /u][x/s] (λx.t)[y /s] ∼ λx.t[y /s] (ts)[x/u] ∼ t[x/u]s

if x ∈ / fv(u) and y ∈ / fv(s) if x ∈ / fv(s) if x ∈ / fv(s)

I

Proof Nets = LSC/∼.

I

∼ preserves redexes and residuals.

I

∼ is a behavioural isomorphism of the LSC with itself.

I

The problem is that: t[x/s][y /u] ∼ t[y /u][x/s] does not preserve left-to-right derivations.

Partial Standardization Orders I

A standardization order ≺ validating: t[x/s][y /u] ∼ t[y /u][x/s] is necessarily partial.

I

Abstract standardization for partial orders exists (Melli` es).

I

But is very delicate and complex.

I

Standard derivations are up to swaps of incomparable elem.

I

Total Orders: Standardization = Term. & Conf. of swaps.

I

Partial Orders: Standardization = Term. & Conf. modulo.

The Nonstandard Standardization Theorem I

We define a partial box order validating ∼.

I

Based on the notion of explicit box for ! in linear logic.

I

The order satisfies only Melli`es’ axioms for existence.

I

We prove uniqueness in a novel abstract way.

I

Argument: the box embeds into the left-to-right order.

I

Main Result:

Theorem If t →∗∼ s then there is a box standard derivation from t to s that is unique up to swaps of incomparable elements. I

Note: modulo at two levels, Reduction (→∼ ) and Standardization (swaps).

Linear Head Reduction

I

LHR is non-deterministic, but it has the diamond property: s

s1

s2

t

Thus its non-determinism is harmless. I

It is not left-to-right standard.

I

We provide an original proof that it is box standard (next slide).

Co-Inductive External Redexes I

LHR picks ≺-minimal redexes.

I

Total order ⇒ ≺-minimum strategy is standard.

I

Partial order ⇒ ≺-minimal strategy may not be standard.

I

One has to pick special ≺-minimal redexes, called external.

I

Melli`es definition of external redexes is complex.

I

We provide a new co-inductive characterisation. A redex is external if 1) it is ≺-minimal and 2) its eventual residual is external

Outline

Standardization Explicit Substitutions Orthogonality and the Linear Substitution Calculus Proof Nets and Partial Standardization Orders Advertisement and Conclusions

Advertisement I

Reviewers: what is the connection with PL?

I

Abstract machines for call-by-name/value/need: B. Accattoli, P. Barenbaum, D. Mazza. Distilling Abstract Machines.

I

Another work: B. Accattoli, U. Dal Lago. Beta Reduction is Invariant, Indeed. solves a long-standing open problem using standardization.

I

Leftmost reduction is invariant wrt Turing machines.

I

Proved via leftmost reduction for the LSC.

Conclusions

I

A semantically orthogonal ES calculus.

I

A left-to-right stand. thm, lifting Plotkin’s approach to ES.

I

Standardization for a calculus modulo, via a partial order.

I

New light on the abstract theory.

I

General Perspective: Standard Derivations = Abstract Machine Hidden Inside the Calculus Itself

THANKS!