[PPT] - Conditions for Efficiency Improvement by Tree Transducer Composition PowerPoint Presentation

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RTA, July 23, 2002

Conditions for Efficiency Improvement by Tree Transducer Composition

Janis Voigtl¨ ander

Dresden University of Technology voigt@tcs.inf.tu-dresden.de

Supported by the “Deutsche Forschungsgemeinschaft” under grant KU 1290/2-1 and by the “European Commission – DG Information Society” with a FLoC’02 travel grant.

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1

Macro Tree Transducers [Eng80]

data Term = Term ⊗ Term | Term ⊕ Term | A | B data Ins = Mul Ins | Add Ins | LoadA Ins | LoadB Ins | Nil data Nat = Succ Nat | Zero pre :: Term → Ins → Ins pre (x1 ⊗ x2) y = Mul (pre x1 (pre x2 y)) pre (x1 ⊕ x2) y = Add (pre x1 (pre x2 y)) pre A y = LoadA y pre B y = LoadB y pre ⊗ A ⊕ B A Nil ⇒ Mul pre A pre ⊕ B A Nil ⇒ Mul LoadA pre ⊕ B A Nil ⇒3 Mul LoadA Add LoadB LoadA Nil

ps :: Ins → Nat
ps

(Mul x) = Succ (ops x)

ps (Add x) = Succ (ops x)
ps (LoadA x) = ops x
ps (LoadB x) = ops x
ps

Nil = Zero

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2

Intermediate Results

ps

pre ⊗ A ⊕ B A Nil ⇒5

✁ ✂

ps

Mul LoadA Add LoadB LoadA Nil ⇒6

✄ ☎

Succ Succ Zero

Inefficient !

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3

Tree Transducer Composition [EV85]: Example

preops ⊗ x1 x2 yops =

ps

Mul pre x1 pre x2 y ⇒ Succ

ps

pre x1 pre x2 y ⇒ Succ preops x1

ps

pre x2 y ⇒ Succ preops x1 preops x2

ps

y ⇒ Succ preops x1 preops x2 yops Replace occurrences of (ops (pre t Nil)) by (preops t (ops Nil)).

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4

Transformed Program:

preops :: Term → Nat → Nat preops (x1 ⊗ x2) yops = Succ (preops x1 (preops x2 yops)) preops (x1 ⊕ x2) yops = Succ (preops x1 (preops x2 yops)) preops A yops = yops preops B yops = yops preops ⊗ A ⊕ B A Zero ⇒ Succ preops A preops ⊕ B A Zero ⇒ Succ preops ⊕ B A Zero ⇒3 Succ Succ Zero

No intermediate result is produced ! Transformed program requires fewer reduction steps !

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5

Formal Efficiency Analysis

should be:

with respect to call-by-need reduction steps
input-independent
based on original program before transformation

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6

Ticking of Producer:

pre

(x1 ⊗ x2) y = ⋄ (Mul (pre x1 (pre x2 y)))

pre

(x1 ⊕ x2) y = ⋄ (Add (pre x1 (pre x2 y)))

pre

A

y = ⋄ (LoadA y) pre

B

y = ⋄ (LoadB y)

pre

✁

⊗ A ⊕ B A Nil ⇒

⋄

Mul pre

✁

A pre

✁

⊕ B A Nil ⇒

⋄

Mul

⋄

LoadA pre

✁

⊕ B A Nil ⇒3

⋄

Mul

⋄

LoadA

⋄

Add

⋄

LoadB

⋄

LoadA Nil

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7

Ticking of Consumer:

ps

(Mul x)

= • (Succ (ops

x))

ps

(Add x)

= • (Succ (ops

x))

ps

(LoadA x) = • (ops x)

ps

(LoadB x) = • (ops x)

ps

Nil

= • Zero

ps

(⋄ x1)

= • (ops

x1)

ps
⋄

Mul

⋄

LoadA

⋄

Add

⋄

LoadB

⋄

LoadA Nil ⇒

ps
Mul

⋄

LoadA

⋄

Add

⋄

LoadB

⋄

LoadA Nil ⇒

Succ
ps
⋄

LoadA

⋄

Add

⋄

LoadB

⋄

LoadA Nil ⇒9

Succ
Succ
Zero

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8

Steps of Original Program Reflected in Output (Lemma 2)

The number of •-symbols in the reduction result of:

ps
pre
⊗

A ⊕ B A Nil

is equal to the number of call-by-need reduction steps of:

ps

pre ⊗ A ⊕ B A Nil ⇒

✁

Succ Succ Zero

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9

Ticking of Composed Program:

preops

(x1 ⊗ x2) yops = ◦ (Succ (preops
x1 (preops
x2 yops)))

preops

(x1 ⊕ x2) yops = ◦ (Succ (preops
x1 (preops
x2 yops)))

preops

A

yops = ◦ yops preops

B

yops = ◦ yops preops

⊗

A ⊕ B A Zero ⇒

Succ

preops

A

preops

⊕

B A Zero ⇒

Succ
preops
⊕

B A Zero ⇒3

Succ
Succ
Zero

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10

Annotation through Composition (Lemma 4)

pre

(x1 ⊗ x2) y = ⋄ (Mul (pre
x1 (pre
x2 y)))

pre

(x1 ⊕ x2) y = ⋄ (Add (pre
x1 (pre
x2 y)))

pre

A

y = ⋄ (LoadA y) pre

B

y = ⋄ (LoadB y)

ps
(Mul x)

= Succ (ops

x)
ps
(Add x) = Succ (ops
x)
ps
(LoadA x) = ops
x
ps
(LoadB x) = ops
x
ps
Nil

= Zero

ps
(⋄ x1)

= ◦ (ops

x1)

composed into: pre

ps
(x1 ⊗ x2) yops
= ◦ (Succ (pre
ps
x1 (pre
ps
x2 yops
)))

pre

ps
(x1 ⊕ x2) yops
= ◦ (Succ (pre
ps
x1 (pre
ps
x2 yops
)))

pre

ps
A

yops

= ◦ yops
pre
ps
B

yops

= ◦ yops

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11

Steps of Composed Program Reflected in Output (Lemma 5)

The number of ◦-symbols in the reduction result of:

ps
pre
⊗

A ⊕ B A Nil

is greater or equal to the number of call-by-need reduction steps

f:

preops ⊗ A ⊕ B A Zero ⇒

✁

Succ Succ Zero

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12

Compare Annotated Programs:

pre

(x1 ⊗ x2) y = ⋄ (Mul (pre
x1 (pre
x2 y)))

pre

(x1 ⊕ x2) y = ⋄ (Add (pre
x1 (pre
x2 y)))

pre

A

y = ⋄ (LoadA y) pre

B

y = ⋄ (LoadB y)

ps
(Mul x)

= • (Succ (ops

x))
ps
(Add x) = • (Succ (ops
x))
ps
(LoadA x) = • (ops
x)
ps
(LoadB x) = • (ops
x)
ps
Nil

= • Zero

ps
(⋄ x1)

= • (ops

x1)
ps
(Mul x) = Succ (ops
x)
ps
(Add x) = Succ (ops
x)
ps
(LoadA x) = ops
x
ps
(LoadB x) = ops
x
ps
Nil

= Zero

ps
(⋄ x1)

= ◦ (ops

x1)

Always more •- than ◦-symbols !

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Abstracting from the Example (Theorem 1)

The composed program is at least as efficient as the original program, provided that:

1. the producer is context-linear or basic
2. the consumer is recursion-linear
3. the consumer is context-linear or basic

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Further Results:

Weaker pre-conditions by counting only steps of the con-

sumer (Lemma 3, Lemma 6, Lemma 7, Theorem 2)

Application to special cases of classical deforestation [Wad90]

(Corollary 1)

Analysis technique scales for the case that both involved

transducers use context parameters [VK01]; work in progress

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References

[Eng80]

J. Engelfriet. Some open questions and recent results on tree transducers and

tree languages. In R.V. Book, editor, Formal language theory; perspectives and

pen problems, pages 241–286. New York, Academic Press, 1980.

[EV85]

J. Engelfriet and H. Vogler. Macro tree transducers. J. Comput. Syst. Sci.,

31:71–145, 1985. [VK01]

J. Voigtl¨

ander and A. K¨

uhnemann. Composition of functions with accumulat-

ing parameters. Technical Report TUD-FI01-08, Dresden University of Tech- nology, August 2001. http://wwwtcs.inf.tu-dresden.de/∼voigt/TUD-FI01-08.ps.gz. [Wad90] P. Wadler. Deforestation: Transforming programs to eliminate trees. Theoret.

Comput. Sci., 73:231–248, 1990.