[PPT] - Generalising Tree Traversals to DAGs Exploiting Sharing without the PowerPoint Presentation

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Generalising Tree Traversals to DAGs

Exploiting Sharing without the Pain Patrick Bahr1 Emil Axelsson2

1University of Copenhagen

paba@diku.dk

2Chalmers University of Technology

emax@chalmers.se

PEPM 2015

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Motivation

Goal

Do stuff on acyclic graphs, but pretend they are only trees.

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Motivation

Goal

Do stuff on acyclic graphs, but pretend they are only trees.

Primary Application

Abstract Syntax Graphs/Trees:

◮ type inference ◮ program analyses ◮ program transformations ◮ . . .

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The Idea

Γ ⊢ p : τ

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The Idea

Γ ⊢ p : τ

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The Idea

Γ ⊢ p : τ

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The Idea

Γ ⊢ p : τ

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The Idea

Γ ⊢ p : τ Result

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The Idea

Γ ⊢ p : τ Result

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unravels to

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The Idea

Γ ⊢ p : τ Result

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unravels to

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The Idea

Γ ⊢ p : τ Result

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unravels to

Attribute Grammar

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The Idea

Γ ⊢ p : τ Result

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unravels to

Attribute Grammar

Why?

◮ It’s more difficult to get a traversal on graphs right. ◮ But: it’s more efficient to traverse the graph.

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What’s the Catch?

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What’s the Catch? It doesn’t work

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What’s the Catch? It doesn’t work in general.

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What’s the Catch? It doesn’t work in general. But: it does work for many cases.

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What’s the Catch? It doesn’t work in general. But: it does work for many cases.

Our Contribution

◮ Identify classes of AGs for which this approach works. ◮ Prototype implementation in Haskell. ◮ Case studies and benchmarks.

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A Toy Example

data IntTree = Leaf Int | Node IntTree IntTree leavesBelow :: Int → IntTree → Set Int leavesBelow d (Leaf i) | d 0 = Set.singleton i | otherwise = Set.empty leavesBelow d (Node t1 t2) = leavesBelow (d − 1) t1 ∪ leavesBelow (d − 1) t2

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A Toy Example

data IntTree = Leaf Int | Node IntTree IntTree leavesBelow :: Int → IntTree → Set Int leavesBelow d (Leaf i) | d 0 = Set.singleton i | otherwise = Set.empty leavesBelow d (Node t1 t2) = leavesBelow (d − 1) t1 ∪ leavesBelow (d − 1) t2 depth leaves

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A Toy Example

data IntTree = Leaf Int | Node IntTree IntTree leavesBelow :: Int → IntTree → Set Int leavesBelow d (Leaf i) | d 0 = Set.singleton i | otherwise = Set.empty leavesBelow d (Node t1 t2) = leavesBelow (d − 1) t1 ∪ leavesBelow (d − 1) t2 depth leaves

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Traversal on Graphs

A B C depth leaves

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Traversal on Graphs

A B C depth leaves d1 d2

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Traversal on Graphs

A B C depth leaves d′

1

d′

2

d1 d2

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Traversal on Graphs

A B C depth leaves ⊕ d′

1

d′

2

d1 d2

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Traversal on Graphs

A B C depth leaves ⊕ d′

1

d′

2

d1 d2

For which traversals is this correct?

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But before that, let’s implement it!

data IntTree = Leaf Int | Node IntTree IntTree

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But before that, let’s implement it!

data IntTree a = Leaf Int | Node a a

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But before that, let’s implement it!

data IntTreeF a = Leaf Int | Node a a

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But before that, let’s implement it!

data IntTreeF a = Leaf Int | Node a a leavesBelow :: Int → Tree IntTreeF → Set Int leavesBelow = runAG leavesBelowS leavesBelowI

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But before that, let’s implement it!

data IntTreeF a = Leaf Int | Node a a leavesBelow :: Int → Tree IntTreeF → Set Int leavesBelow = runAG leavesBelowS leavesBelowI leavesBelowG :: Int → Dag IntTreeF → Set Int leavesBelowG = runAGDag min leavesBelowS leavesBelowI ⊕

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Implementing the semantic functions

leavesBelowI :: Inh IntTreeF atts Int leavesBelowI (Leaf i) = ∅ leavesBelowI (Node t1 t2) = t1 → d & t2 → d where d = above − 1

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Implementing the semantic functions

leavesBelowI :: Inh IntTreeF atts Int leavesBelowI (Leaf i) = ∅ leavesBelowI (Node t1 t2) = t1 → d & t2 → d where d = above − 1 leavesBelowS :: (Int ∈ atts) ⇒ Syn IntTreeF atts (Set Int) leavesBelowS (Leaf i) | (above :: Int) 0 = Set.singleton i | otherwise = Set.empty leavesBelowS (Node t1 t2) = below t1 ∪ below t2

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Correctness

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unravels to

Result

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Correctness

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unravels to

Result Attribute Grammar

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Correctness

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unravels to

Result & ⊕ Attribute Grammar

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Correctness

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unravels to

Result & ⊕ the same Attribute Grammar merge operator Attribute Grammar

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Correspondence Theorems

Theorem (Monotone AGs)

Let (1) G be a non-circular AG, (2) ⊕ an assoc., comm. operator on inherited attributes, and (3) such that G is monotone and ⊕ is decreasing w.r.t. . If (G, ⊕) terminates on a DAG g with result r, then G terminates on U (g) with result r′ such that r r′.

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Correspondence Theorems

Theorem (Monotone AGs)

Let (1) G be a non-circular AG, (2) ⊕ an assoc., comm. operator on inherited attributes, and (3) such that G is monotone and ⊕ is decreasing w.r.t. . If (G, ⊕) terminates on a DAG g with result r, then G terminates on U (g) with result r′ such that r r′.

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U r r′

& ⊕

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Correspondence Theorems

Theorem (Monotone AGs)

Let (1) G be a non-circular AG, (2) ⊕ an assoc., comm. operator on inherited attributes, and (3) such that G is monotone and ⊕ is decreasing w.r.t. . If (G, ⊕) terminates on a DAG g with result r, then G terminates on U (g) with result r′ such that r r′.

Example

For the leavesBelow AG, define as follows:

◮ on Int: x y ⇐

⇒ x ≤ y

◮ on Set Int: S T ⇐

⇒ S ⊇ T

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Correspondence Theorems

Theorem (Monotone AGs)

Let (1) G be a non-circular AG, (2) ⊕ an assoc., comm. operator on inherited attributes, and (3) such that G is monotone and ⊕ is decreasing w.r.t. . If (G, ⊕) terminates on a DAG g with result r, then G terminates on U (g) with result r′ such that r r′.

Example

For the leavesBelow AG, define as follows:

◮ on Int: x y ⇐

⇒ x ≤ y

◮ on Set Int: S T ⇐

⇒ S ⊇ T = ⇒ leavesBelowG d g ⊇ leavesBelow d (U (g))

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Correspondence Theorems

Theorem (Monotone AGs)

Let (1) G be a non-circular AG, (2) ⊕ an assoc., comm. operator on inherited attributes, and (3) such that G is monotone and ⊕ is decreasing w.r.t. . If (G, ⊕) terminates on a DAG g with result r, then G terminates on U (g) with result r′ such that r r′.

Example

For the leavesBelow AG, define as follows:

◮ on Int: x y ⇐

⇒ x ≤ y

◮ on Set Int: S T ⇐

⇒ S ⊇ T = ⇒ leavesBelowG d g ⊇ leavesBelow d (U (g)) for leavesBelowG d g ⊆ leavesBelow d (U (g)) see paper

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Termination

◮ We know: non-circular AGs terminate on any tree. ◮ But: non-circular AGs may diverge on DAGs.

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Termination

◮ We know: non-circular AGs terminate on any tree. ◮ But: non-circular AGs may diverge on DAGs.

Example

A B1 B2

1 2 3 4

A B

1 2 3 4

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Termination

◮ We know: non-circular AGs terminate on any tree. ◮ But: non-circular AGs may diverge on DAGs.

Example

A B1 B2

1 2 3 4

A B

1 2 3 4

Theorem (termination)

Let G, ⊕, and be as before. If is well-founded on inherited attributes, then (G, ⊕) terminates on any DAG.

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Correspondence Theorem for Copying AGs

Copying AGs

◮ inherited attributes are just propagated, not changed ◮ Example: Bird’s repmin problem.

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Correspondence Theorem for Copying AGs

Copying AGs

◮ inherited attributes are just propagated, not changed ◮ Example: Bird’s repmin problem.

Theorem (copying AGs)

Let (1) G be a copying, non-circular AG, and (2) x ⊕ y ∈ {x, y} for all x, y. Then (i) (G, ⊕) terminates on any DAG, and (ii) G, ⊕(g) = G (U (g)).

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Graph Transformations

◮ Our framework generalises to tree/DAG-transformations ◮ Idea: attributes may contain trees/DAGs.

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Graph Transformations

◮ Our framework generalises to tree/DAG-transformations ◮ Idea: attributes may contain trees/DAGs.

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unravels to

& ⊕

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Graph Transformations

◮ Our framework generalises to tree/DAG-transformations ◮ Idea: attributes may contain trees/DAGs.

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unravels to

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& ⊕

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Graph Transformations

◮ Our framework generalises to tree/DAG-transformations ◮ Idea: attributes may contain trees/DAGs.

7 11 8 10 2 9 5 3

unravels to

7 11 8 10 2 9 5 3

& ⊕

unravels to

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Example: Bird’s Repmin Problem

newtype MinS = MinS Int; newtype MinI = MinI Int minS :: Syn IntTreeF atts MinS minS (Leaf i) = MinS i minS (Node a b) = min (below a) (below b) minI :: Inh IntTreeF atts MinI minI = ∅ rep :: (MinI ∈ atts) ⇒ Rewrite IntTreeF atts IntTreeF rep (Leaf i) = let MinI i′ = above in Leaf i′ rep (Node a b) = Node a b

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Example: Bird’s Repmin Problem

newtype MinS = MinS Int; newtype MinI = MinI Int minS :: Syn IntTreeF atts MinS minS (Leaf i) = MinS i minS (Node a b) = min (below a) (below b) minI :: Inh IntTreeF atts MinI minI = ∅ rep :: (MinI ∈ atts) ⇒ Rewrite IntTreeF atts IntTreeF rep (Leaf i) = let MinI i′ = above in Leaf i′ rep (Node a b) = Node a b repmin :: Tree IntTreeF → Tree IntTreeF repmin = runRewrite minS minI rep init where init (MinS i) = MinI i

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Example: Bird’s Repmin Problem

newtype MinS = MinS Int; newtype MinI = MinI Int minS :: Syn IntTreeF atts MinS minS (Leaf i) = MinS i minS (Node a b) = min (below a) (below b) minI :: Inh IntTreeF atts MinI minI = ∅ rep :: (MinI ∈ atts) ⇒ Rewrite IntTreeF atts IntTreeF rep (Leaf i) = let MinI i′ = above in Leaf i′ rep (Node a b) = Node a b repmin :: Tree IntTreeF → Tree IntTreeF repmin = runRewrite minS minI rep init where init (MinS i) = MinI i repminG :: Dag IntTreeF → Dag IntTreeF repminG = runRewriteDag const minS minI rep init where init (MinS i) = MinI i

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Summary

Our Contributions

◮ Haskell library to run AGs on DAGs ◮ Correspondence & termination theorems to prove correctness

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Summary

Our Contributions

◮ Haskell library to run AGs on DAGs ◮ Correspondence & termination theorems to prove correctness

More in the paper

◮ Examples: type inference; circuits ◮ full theory & proofs ◮ parametric AGs (→ tech report) ◮ Benchmarks (→ tech report)

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Conclusion

Future and Ongoing Work

◮ AGs with fixpoint iteration cyclic graphs ◮ mutually recursive data types and GADTs ◮ deep pattern matching in AGs ◮ corresponding notion of non-circularity for AGs on DAGs

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Conclusion

Future and Ongoing Work

◮ AGs with fixpoint iteration cyclic graphs ◮ mutually recursive data types and GADTs ◮ deep pattern matching in AGs ◮ corresponding notion of non-circularity for AGs on DAGs

Implementation

Available from http://j.mp/AG-DAG.

◮ Haskell library source code ◮ more examples ◮ benchmarks

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Conclusion

Future and Ongoing Work

◮ AGs with fixpoint iteration cyclic graphs ◮ mutually recursive data types and GADTs ◮ deep pattern matching in AGs ◮ corresponding notion of non-circularity for AGs on DAGs

Implementation

Available from http://j.mp/AG-DAG.

◮ Haskell library source code ◮ more examples ◮ benchmarks

Try the compositional datatypes library

> cabal install compdata-dags

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Generalising Tree Traversals to DAGs

Exploiting Sharing without the Pain Patrick Bahr1 Emil Axelsson2

1University of Copenhagen

paba@diku.dk

2Chalmers University of Technology