On the Relative Usefulness of Fireballs Beniamino Accattoli
Claudio Sacerdoti Coen
INRIA, PARSIFAL team Universit` a di Bologna
Fireballs are a generalisation of values (in the sense of the call-by-value λ-calculus)
Outline Background Weak Call-by-Value λ-calculus Weak Call-by-Value Invariance, Revisited Fireballs Fireball Explosion How to Stop Worrying and Love the Bomb
The Invariance Thesis
Slot and van Emde Boas’ (Weak) Invariance Thesis:
Reasonable machines can simulate each other within a polynomially bounded overhead in time.
Long-standing problem: is λ-calculus a reasonable machine?
Bird’s Eye View
λ-Calculus
?
Turing Machines
Bird’s Eye View
Linear [A. & Dal Lago, RTA ’12]
λ-Calculus
Turing Machines
Bird’s Eye View
Linear [A. & Dal Lago, RTA ’12]
Turing Machines
λ-Calculus
?
Bird’s Eye View
Linear [A. & Dal Lago, RTA ’12]
Turing Machines
λ-Calculus Size Explosion Problem
Outline Background Weak Call-by-Value λ-calculus Weak Call-by-Value Invariance, Revisited Fireballs Fireball Explosion How to Stop Worrying and Love the Bomb
The Literature 1 Weak Call-by-Value λ-calculus is invariant. Blelloch & Greiner ’96; Sands & Gustavsson & Moran ’02; Dal Lago & Martini ’06.
Proof technique: simulation in a system with sharing. (Formalisms: abstract machines or graph reduction). Sands & al.: abstract machine with bilinear overhead. Bilinear = linear in 1. the number of β-steps, and 2. the size of the initial term (i.e. the input).
Schema of the Solution
Weak Call-by-Value λ-Calculus
Linear
RAM
Abstract Machine
Bilinear
The Literature 2
Strong Call-by-name λ-calculus is invariant. Accattoli & Dal Lago, CSL-LICS ’14.
Harder case, simple sharing is not enough. Proof technique: refined simulation with useful sharing Formalism: Linear substitution calculus aka explicit substitutions. Overhead: at least quadratic.
Schema of the Solutions
Strong Call-by-Name λ-Calculus Quadratic
RAM
Linear Substitution Calculus + Useful Sharing
Polynomial (linear?)
This Work Semi-strong Call-by-value λ-calculus is invariant. Semi-strong CBV = Weak CBV with open terms. Useful sharing is required. Proof technique: LSC + useful sharing + abstract machine. Main points: 1. Bilinear implementation λ → RAM; 2. Original combination of different techniques; (Practical values, distilleries, useful sharing);
3. Constant time implementation of useful sharing.
Outline Background Weak Call-by-Value λ-calculus Weak Call-by-Value Invariance, Revisited Fireballs Fireball Explosion How to Stop Worrying and Love the Bomb
Weak Call-by-Value Invariance, Revisited
Sands & al.’s result has been revisited last year. Decomposed in two works: 1. Distilling Abstract Machines. (Accattoli, Barenbaum, Mazza, ICFP 2014)
2. On the Value of Variables. (Accattoli & Sacerdoti Coen, WoLLIC 2014)
Distilling Abstract Machines Abstract machines = LSC + search of the redex.
Theorem The search of the redex is bilinear in the # of LSC steps. (for most CBN, CBV, and CBNeed machines in the literature)
Abstract machines = LSC + bilinear overhead. Two outcomes: 1. reasoning in the simpler LSC preserves complexity; 2. Principle for abstract machines.
Sands & al. Solution
Weak Call-by-Value λ-Calculus
Linear
RAM
Abstract Machine
Bilinear
Distilling Abstract Machines, ICFP 2014
Weak Call-by-Value λ-Calculus
RAM Bilinear
Weak Call-by-Value LSC
Bilinear
Abstract Machine
On the Value of Variables
Call-by-value schizophrenic literature: 1. Theoretical values = abstractions and variables. 2. Practical values = abstractions. The overhead in Weak CBV λ-calculus → Weak CBV LSC is: 1. Quadratic with theoretical values. 2. Linear with practical values.
Practical Values + Distillation
Weak Call-by-Value λ-Calculus
Bilinear
Bilinear
Linear
Weak Call-by-Value LSC
RAM
Bilinear
Abstract Machine
GLAM Leroy’s ZINC implements right-to-left CBV. GLAM = Global Leroy Abstract Machine. Global = just one global environments. Values are practical, i.e. are abstractions. Dump
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Outline Background Weak Call-by-Value λ-calculus Weak Call-by-Value Invariance, Revisited Fireballs Fireball Explosion How to Stop Worrying and Love the Bomb
Extending CBV
Weak CBV key property: closed normal forms are values Adding free variables breaks the property. There are open normal forms that are not values, as x x. Problem: they block evaluation, as in (λy .δ) (x x) δ. Solution: generalizing values to fire-able terms, called fireballs.
The Fireball Calculus Language, extended with fireballs f : t, s, u v f
a | λx.t v |
:= := :=
x
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λx.t
af . . . f | 1 {z n}
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inert terms A
Right-to-left evaluation contexts: F
:=
h · i | Ff
| tF
Small-step Evaluation: Rule at Top Level (λx.t)f 7→f t{x f }
Contextual closure F hti →f F hsi iff t 7→f s
Key Property: →f normal forms are fireballs.
Open GLAM Open GLAM = kernel of Gr´egoire and Leroy’s machine (ICFP 2002). Open GLAM = GLAM + additional transition for free variables Dump
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Outline Background Weak Call-by-Value λ-calculus Weak Call-by-Value Invariance, Revisited Fireballs Fireball Explosion How to Stop Worrying and Love the Bomb
Size Explosion
Open terms bring a malicious behavior, called size explosion. Inert terms can grow exponentially with the number of β-steps. ⇒ A single substitution step can take exponential time. Key point: Substituting inert terms cannot create β-redexes.
Size Explosion
Fireball Calculus
Exponential!
Exponential!
Linear
Fireball LSC
RAM
Bilinear
Open GLAM
Outline Background Weak Call-by-Value λ-calculus Weak Call-by-Value Invariance, Revisited Fireballs Fireball Explosion How to Stop Worrying and Love the Bomb
Useful Sharing Remedy against size explosion: useful sharing. Complex definition (omitted). Surprisingly simple implementation. Intuitions (= the relative usefulness of fireballs): 1. Do a substitution step only when it creates a β-redex; 2. Therefore, never substitute inert terms, and 3. Substitute values only when the stack is non-empty. Complication: values have to include variables, i.e. are theoretical.
GLAMOUr The Useful Open GLAM, or GLAMOUr. Every term t in a π or E is labeled with l ∈ {λ, A}. Invariants: t λ unfolds (in E ) to an abstraction. t A unfolds (in E ) to a inert term. Dump
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Usefulness is tested in constant time, as it is coded by labels.
Semi-Strong CBV is Invariant
Fireball Calculus
Quadratic
Quadratic
Useful Fireball LSC
RAM Bilinear
Bilinear
GLAMOUr
Unchaining GLAMOUr
The quadratic overhead is due to theoretical values. Theoretical values are reintroduced by usefulness. A further unchaining optimization brings back practical values. The overhead becomes linear.
Final Result: Linear Overhead
Fireball Calculus
Bilinear
Linear
Unchaining Useful Fireball LSC
RAM Bilinear
Bilinear
Unchaining
GLAMOUr
Conclusions
Useful Sharing: Theoretically born concept, surprisingly relevant in practice. The GLAMOUr can be expon. faster than the Open GLAM. Strong improvement over Gr´ egoire and Leroy’s machine (their machine is used in the implementation of Coq). Abstract and sharp decomposition of call-by-value. Fireballs open the way to a new theory of call-by-value. (CBV Bohm theorem?).
THANKS!
