Deriving Modular Recursion Schemes from Tree Automata Patrick Bahr - - PowerPoint PPT Presentation

Deriving Modular Recursion Schemes from Tree Automata

Patrick Bahr

University of Copenhagen, Department of Computer Science paba@diku.dk

Computing Science Colloquium, Utrecht University, April 12th, 2012

Outline

1

Tree Automata Bottom-Up Tree Acceptors Bottom-Up Tree Transducers

2

Introducing Modularity Composing State Spaces Compositional Signatures Decomposing Tree Transducers

3

Other Automata

2

Outline

1

Tree Automata Bottom-Up Tree Acceptors Bottom-Up Tree Transducers

2

Introducing Modularity Composing State Spaces Compositional Signatures Decomposing Tree Transducers

3

Other Automata

3

Bottom-Up Tree Acceptors

f q1

• q2. . .

qn f q . . .

4

Bottom-Up Tree Acceptors

f q1

• q2. . .

qn f q . . . f (q1(x1), q2(x2), . . . , qn(xn)) → q(f (x1, x2, . . . , xn))

4

An Example

The signature F = {and/2, not/1, tt/0, ff/0} e.g.: not(and(not(ff), and(tt, ff)))

5

An Example

The signature F = {and/2, not/1, tt/0, ff/0} e.g.: not(and(not(ff), and(tt, ff))) The states set of states: Q = {q0, q1} q0 false q1 true accepting states: Qa = {q1}

5

An Example

The signature F = {and/2, not/1, tt/0, ff/0} e.g.: not(and(not(ff), and(tt, ff))) The states set of states: Q = {q0, q1} q0 false q1 true accepting states: Qa = {q1} The rules of the automaton ff → q0(ff) tt → q1(tt) not(q0(x)) → q1(not(x)) not(q1(x)) → q0(not(x)) and(q1(x), q1(y)) → q1(and(x, y)) and(q0(x), q1(y)) → q0(and(x, y)) and(q1(x), q0(y)) → q0(and(x, y)) and(q0(x), q0(y)) → q0(and(x, y))

5

A Run of a Bottom-Up Tree Acceptor

not and not ff and tt ff

6

A Run of a Bottom-Up Tree Acceptor

not and not ff and tt ff not and not q0 ff and q1 tt q0 ff

3

ff → q0(ff) tt → q1(tt)

6

A Run of a Bottom-Up Tree Acceptor

not and not ff and tt ff not and not q0 ff and q1 tt q0 ff not and q1 not ff q0 and tt ff

3 2

not(q0(x)) → q1(not(x)) and(q1(x), q0(y)) → q0(and(x, y))

6

A Run of a Bottom-Up Tree Acceptor

not and not ff and tt ff not and not q0 ff and q1 tt q0 ff not and q1 not ff q0 and tt ff not q0 and not ff and tt ff

3 2

not(q0(x)) → q1(not(x)) and(q1(x), q0(y)) → q0(and(x, y))

6

A Run of a Bottom-Up Tree Acceptor

not and not ff and tt ff not and not q0 ff and q1 tt q0 ff not and q1 not ff q0 and tt ff not q0 and not ff and tt ff q1 not and not ff and tt ff

3 2

not(q0(x)) → q1(not(x)) and(q1(x), q0(y)) → q0(and(x, y))

6

Terms in Haskell

Data types as fixed points of functors data Term f = In (f (Term f ))

7

Terms in Haskell

Data types as fixed points of functors data Term f = In (f (Term f )) Functors class Functor f where fmap :: (a → b) → f a → f b

7

Terms in Haskell

Data types as fixed points of functors data Term f = In (f (Term f )) Functors class Functor f where fmap :: (a → b) → f a → f b Example data F a = And a a | Not a | TT | FF

7

Terms in Haskell

Data types as fixed points of functors data Term f = In (f (Term f )) Functors class Functor f where fmap :: (a → b) → f a → f b Example data F a = And a a | Not a | TT | FF instance Functor F where fmap f TT = TT fmap f (And x y) = And (f x) (f y) . . .

7

Terms in Haskell

Data types as fixed points of functors data Term f = In (f (Term f )) Functors class Functor f where fmap :: (a → b) → f a → f b Example data F a = And a a | Not a | TT | FF deriving Functor instance Functor F where fmap f TT = TT fmap f (And x y) = And (f x) (f y) . . .

7

Bottom-Up State Transitions in Haskell

f q1

• q2. . .

qn f q . . .

8

Bottom-Up State Transitions in Haskell

f q1

• q2. . .

qn q

8

Bottom-Up State Transitions in Haskell

f q1

• q2. . .

qn q1 q2 qn q

8

Bottom-Up State Transitions in Haskell

f q1

• q2. . .

qn q1 q2 qn q Bottom-up state transition rules as algebras type UpState f q = f q → q

8

Bottom-Up State Transitions in Haskell

f q1

• q2. . .

qn q1 q2 qn q Bottom-up state transition rules as algebras type UpState f q = f q → q runUpState :: Functor f ⇒ UpState f q → Term f → q runUpState φ (In t) = φ (fmap (runUpState φ) t)

8

Bottom-Up State Transitions in Haskell

f q1

• q2. . .

qn q1 q2 qn q Bottom-up state transition rules as algebras type UpState f q = f q → q runUpState :: Functor f ⇒ UpState f q → Term f → q runUpState φ (In t) = φ (fmap (runUpState φ) t) a.k.a. catamorphism / fold

8

An Example

Signature data F a = And a a | Not a | TT | FF

9

An Example

Signature data F a = And a a | Not a | TT | FF States data Q = Q0 | Q1 Accepting states acc :: Q → Bool acc Q1 = True acc Q0 = False

9

An Example

Signature data F a = And a a | Not a | TT | FF States data Q = Q0 | Q1 Accepting states acc :: Q → Bool acc Q1 = True acc Q0 = False State transition function trans :: F Q → Q trans FF = Q0 trans TT = Q1 trans (Not Q0) = Q1 trans (Not Q1) = Q0 trans (And Q1 Q1) = Q1 trans (And ) = Q0

9

Bottom-Up Tree Transducers

f q1

• q2. . .

qn f . . . q

10

Bottom-Up Tree Transducers

f q1

• q2. . .

qn q

10

Bottom-Up Tree Transducers

f q1

• q2. . .

qn q f (q1(x1), q2(x2), . . . , qn(xn)) − → q(t) f ∈ F t ∈ T (G, X) X = {x1, x2, . . . , xn}

10

An Example

The signature F = {and/2, not/1, ff/0, tt/0, b/0}

11

An Example

The signature F = {and/2, not/1, ff/0, tt/0, b/0} The states q0 false q1 true q2 don’t know

11

An Example

The signature F = {and/2, not/1, ff/0, tt/0, b/0} The states q0 false q1 true q2 don’t know Transduction rules tt → q1(tt) ff → q0(ff) b → q2(b) not(q0(x)) → q1(tt) not(q1(x)) → q0(ff) not(q2(x)) → q2(not(x)) and(q(x), p(y)) → q0(ff) if q0 ∈ {p, q} and(q1(x), q1(y)) → q1(tt) and(q1(x), q2(y)) → q2(y) and(q2(x), q1(y)) → q2(x) and(q2(x), q2(y)) → q2(and(x, y))

11

A Run of a Bottom-Up Transducer

and not b not and ff b

12

A Run of a Bottom-Up Transducer

and not b not and ff b and not q2 b not and q0 ff q2 b

3

ff → q0(ff) b → q2(b)

12

A Run of a Bottom-Up Transducer

and not b not and ff b and not q2 b not and q0 ff q2 b and q2 not b not q0 ff

3 2

not(q2(x)) → q2(not(x)) and(q(x), p(y)) → q0(ff) if q0 ∈ {p, q}

12

A Run of a Bottom-Up Transducer

and not b not and ff b and not q2 b not and q0 ff q2 b and q2 not b not q0 ff and q2 not b q1 tt

3 2

not(q0(x)) → q1(tt)

12

A Run of a Bottom-Up Transducer

and not b not and ff b and not q2 b not and q0 ff q2 b and q2 not b not q0 ff and q2 not b q1 tt q2 not b

3 2

and(q2(x), q1(y)) → q2(x)

12

Bottom-Up Tree Transducers

f q1

• q2. . .

qn f q

13

Bottom-Up Tree Transducers

f q1

• q2. . .

qn f q ∈ T (G, X)

13

Bottom-Up Tree Transducers

f q1

• q2. . .

qn f q ∈ T (G, X) From terms to contexts data Term f = In (f (Term f )) data Context f a = In (f (Context f a)) | Hole a

13

Bottom-Up Tree Transducers

f q1

• q2. . .

qn f q ∈ T (G, X) From terms to contexts data Term f = In (f (Term f )) data Context f a = In (f (Context f a)) | Hole a type Term f = Context f Empty

13

Bottom-Up Tree Transducers

f q1

• q2. . .

qn f q ∈ T (G, X) From terms to contexts data Term f = In (f (Term f )) data Context f a = In (f (Context f a)) | Hole a Representing transduction rules, [Hasuo et al. 2007] type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a)

13

Bottom-Up Tree Transducers

f q1

• q2. . .

qn f q ∈ T (G, X) From terms to contexts data Term f = In (f (Term f )) data Context f a = In (f (Context f a)) | Hole a Representing transduction rules, [Hasuo et al. 2007] type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a)

13

Bottom-Up Tree Transducers

f q1

• q2. . .

qn f q ∈ T (G, X) f (q1(x1), q2(x2), . . . , qn(xn)) − → q(t) From terms to contexts data Term f = In (f (Term f )) data Context f a = In (f (Context f a)) | Hole a Representing transduction rules, [Hasuo et al. 2007] type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a)

13

An Example

Signature And state data F a = And a a | Not a | TT | FF | B data Q = Q0 | Q1 | Q2

14

An Example

Signature And state data F a = And a a | Not a | TT | FF | B data Q = Q0 | Q1 | Q2 type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a) data Context f a = In (f (Context f a)) | Hole a The transduction function trans :: UpTrans F Q F

14

An Example

Signature And state data F a = And a a | Not a | TT | FF | B data Q = Q0 | Q1 | Q2 type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a) data Context f a = In (f (Context f a)) | Hole a The transduction function trans :: UpTrans F Q F tt → q1(tt) trans TT = (Q1, tt )

14

An Example

Signature And state data F a = And a a | Not a | TT | FF | B data Q = Q0 | Q1 | Q2 type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a) data Context f a = In (f (Context f a)) | Hole a The transduction function trans :: UpTrans F Q F tt → q1(tt) trans TT = (Q1, tt ) tt :: Context F a tt = In TT

14

An Example

Signature And state data F a = And a a | Not a | TT | FF | B data Q = Q0 | Q1 | Q2 type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a) data Context f a = In (f (Context f a)) | Hole a The transduction function trans :: UpTrans F Q F tt → q1(tt) trans TT = (Q1, tt ) not(q2(x)) → q2(not(x)) trans (Not (Q2, x)) = (Q2, not (Hole x))

14

An Example

Signature And state data F a = And a a | Not a | TT | FF | B data Q = Q0 | Q1 | Q2 type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a) data Context f a = In (f (Context f a)) | Hole a The transduction function trans :: UpTrans F Q F tt → q1(tt) trans TT = (Q1, tt ) not(q2(x)) → q2(not(x)) trans (Not (Q2, x)) = (Q2, not (Hole x)) not :: Context F a → Context F a not = In . Not

14

An Example

Signature And state data F a = And a a | Not a | TT | FF | B data Q = Q0 | Q1 | Q2 type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a) data Context f a = In (f (Context f a)) | Hole a The transduction function trans :: UpTrans F Q F tt → q1(tt) trans TT = (Q1, tt ) not(q2(x)) → q2(not(x)) trans (Not (Q2, x)) = (Q2, not (Hole x)) and(q(x), p(y)) → q0(ff) if q0 ∈ {q, p}

14

An Example

Signature And state data F a = And a a | Not a | TT | FF | B data Q = Q0 | Q1 | Q2 type UpTrans f q g = ∀ a.f (q,a) → (q, Context g a) data Context f a = In (f (Context f a)) | Hole a The transduction function trans :: UpTrans F Q F tt → q1(tt) trans TT = (Q1, tt ) not(q2(x)) → q2(not(x)) trans (Not (Q2, x)) = (Q2, not (Hole x)) and(q(x), p(y)) → q0(ff) if q0 ∈ {q, p} trans (And (q, x) (p, y)) | q ≡ Q0 ∨ p ≡ Q0 = (Q0, ff )

14

Outline

1

Tree Automata Bottom-Up Tree Acceptors Bottom-Up Tree Transducers

2

Introducing Modularity Composing State Spaces Compositional Signatures Decomposing Tree Transducers

3

Other Automata

15

Composing State Spaces – Motivating Example

A simple expression language data Sig e = Val Int | Plus e e

16

Composing State Spaces – Motivating Example

A simple expression language data Sig e = Val Int | Plus e e Task: writing a code generator type Addr = Int data Instr = Acc Int | Load Addr | Store Addr | Add Addr type Code = [Instr ]

16

Composing State Spaces – Motivating Example

A simple expression language data Sig e = Val Int | Plus e e Task: writing a code generator type Addr = Int data Instr = Acc Int | Load Addr | Store Addr | Add Addr type Code = [Instr ] The problem codeSt :: UpState Sig Code codeSt (Val i) = [Acc i ] codeSt (Plus x y) = x + + [Store a] + + y + + [Add a] where a = . . .

16

Composing State Spaces – Motivating Example

A simple expression language data Sig e = Val Int | Plus e e Task: writing a code generator type Addr = Int data Instr = Acc Int | Load Addr | Store Addr | Add Addr type Code = [Instr ] The problem codeSt :: UpState Sig Code codeSt (Val i) = [Acc i ] codeSt (Plus x y) = x + + [Store a] + + y + + [Add a] where a = . . . Sig Code → Code

16

Tupling

Tuple the code with an address counter codeAddrSt :: UpState Sig (Code, Addr) codeAddrSt (Val i) = ([Acc i ], 0) codeAddrSt (Plus (x, a′) (y, a)) = (x + + [Store a] + + y + + [Add a], 1 + max a a′)

17

Tupling

Tuple the code with an address counter codeAddrSt :: UpState Sig (Code, Addr) codeAddrSt (Val i) = ([Acc i ], 0) codeAddrSt (Plus (x, a′) (y, a)) = (x + + [Store a] + + y + + [Add a], 1 + max a a′) Run the automaton code :: Term Sig → (Code, Addr) code = runUpState codeAddrSt

17

Tupling

Tuple the code with an address counter codeAddrSt :: UpState Sig (Code, Addr) codeAddrSt (Val i) = ([Acc i ], 0) codeAddrSt (Plus (x, a′) (y, a)) = (x + + [Store a] + + y + + [Add a], 1 + max a a′) Run the automaton code :: Term Sig → (Code, Addr) code = fst . runUpState codeAddrSt

17

Tupling

Tuple the code with an address counter codeAddrSt :: UpState Sig (Code, Addr) codeAddrSt (Val i) = ([Acc i ], 0) codeAddrSt (Plus (x, a′) (y, a)) = (x + + [Store a] + + y + + [Add a], 1 + max a a′) Run the automaton code :: Term Sig → Code code = fst . runUpState codeAddrSt

17

Product Automata

Deriving projections class a ∈ b where pr :: b → a

18

Product Automata

Deriving projections class a ∈ b where pr :: b → a a ∈ b iff b is of the form (b1, (b2, ...)) and a = bi for some i

18

Product Automata

Deriving projections class a ∈ b where pr :: b → a a ∈ b iff b is of the form (b1, (b2, ...)) and a = bi for some i For example: Addr ∈ (Code, Addr)

18

Product Automata

Deriving projections class a ∈ b where pr :: b → a a ∈ b iff b is of the form (b1, (b2, ...)) and a = bi for some i For example: Addr ∈ (Code, Addr) Dependent state transition functions type UpState f q = f q → q

18

Product Automata

Deriving projections class a ∈ b where pr :: b → a a ∈ b iff b is of the form (b1, (b2, ...)) and a = bi for some i For example: Addr ∈ (Code, Addr) Dependent state transition functions type UpState f q = f q → q type DUpState f p q = (q ∈ p) ⇒ f p → q

18

Product Automata

Deriving projections class a ∈ b where pr :: b → a a ∈ b iff b is of the form (b1, (b2, ...)) and a = bi for some i For example: Addr ∈ (Code, Addr) Dependent state transition functions type UpState f q = f q → q type DUpState f p q = (q ∈ p) ⇒ f p → q

18

Product Automata

Deriving projections class a ∈ b where pr :: b → a a ∈ b iff b is of the form (b1, (b2, ...)) and a = bi for some i For example: Addr ∈ (Code, Addr) Dependent state transition functions type UpState f q = f q → q type DUpState f p q = (q ∈ p) ⇒ f p → q

18

Product Automata

Deriving projections class a ∈ b where pr :: b → a a ∈ b iff b is of the form (b1, (b2, ...)) and a = bi for some i For example: Addr ∈ (Code, Addr) Dependent state transition functions type UpState f q = f q → q type DUpState f p q = (q ∈ p) ⇒ f p → q Product state transition (⊗) :: (p ∈ c, q ∈ c) ⇒ DUpState f c p → DUpState f c q → DUpState f c (p, q) (sp ⊗ sq) t = (sp t, sq t)

18

Running Dependent State Transition Functions

The types type UpState f q = f q → q type DUpState f p q = (q ∈ p) ⇒ f p → q

19

Running Dependent State Transition Functions

The types type UpState f q = f q → q type DUpState f p q = (q ∈ p) ⇒ f p → q From state transition to dependent state transition dUpState :: Functor f ⇒ UpState f q → DUpState f p q dUpState st = st . fmap pr

19

Running Dependent State Transition Functions

The types type UpState f q = f q → q type DUpState f p q = (q ∈ p) ⇒ f p → q From state transition to dependent state transition dUpState :: Functor f ⇒ UpState f q → DUpState f p q dUpState st = st . fmap pr Running dependent state transitions runDUpState :: Functor f ⇒ DUpState f q q → Term f → q runDUpState f = runUpState f

19

The Code Generator Example

The code generator codeSt :: (Int ∈ q) ⇒ DUpState Sig q Code codeSt (Val i) = [Acc i ] codeSt (Plus x y) = pr x + + [Store a] + + pr y + + [Add a] where a = pr y

20

The Code Generator Example

The code generator codeSt :: (Int ∈ q) ⇒ DUpState Sig q Code codeSt (Val i) = [Acc i ] codeSt (Plus x y) = pr x + + [Store a] + + pr y + + [Add a] where a = pr y Generating fresh addresses heightSt :: UpState Sig Int heightSt (Val ) = 0 heightSt (Plus x y) = 1 + max x y

20

The Code Generator Example

The code generator codeSt :: (Int ∈ q) ⇒ DUpState Sig q Code codeSt (Val i) = [Acc i ] codeSt (Plus x y) = pr x + + [Store a] + + pr y + + [Add a] where a = pr y Generating fresh addresses heightSt :: UpState Sig Int heightSt (Val ) = 0 heightSt (Plus x y) = 1 + max x y Combining the components code :: Term Sig → Code code = fst . runUpState (codeSt ⊗ dUpState heightSt)

20

Outline

1

Tree Automata Bottom-Up Tree Acceptors Bottom-Up Tree Transducers

2

Introducing Modularity Composing State Spaces Compositional Signatures Decomposing Tree Transducers

3

Other Automata

21

Combining Signatures

Coproduct of signatures data (f ⊕ g) e = Inl (f e) | Inr (g e) f ⊕ g is the sum of the signatures f and g

22

Combining Signatures

Coproduct of signatures data (f ⊕ g) e = Inl (f e) | Inr (g e) f ⊕ g is the sum of the signatures f and g Example data Inc e = Inc e type Sig′ = Inc ⊕ Sig

22

Combining Automata

Making the height compositional class HeightSt f where heightSt :: DUpState f q Int instance (HeightSt f , HeightSt g) ⇒ HeightSt (f ⊕ g) where heightSt (Inl x) = heightSt x heightSt (Inr x) = heightSt x

23

Combining Automata

Making the height compositional class HeightSt f where heightSt :: DUpState f q Int instance (HeightSt f , HeightSt g) ⇒ HeightSt (f ⊕ g) where heightSt (Inl x) = heightSt x heightSt (Inr x) = heightSt x Defining the height on Sig instance HeightSt Sig where heightSt (Val ) = 0 heightSt (Plus x y) = 1 + max x y

23

Combining Automata

Making the height compositional class HeightSt f where heightSt :: DUpState f q Int instance (HeightSt f , HeightSt g) ⇒ HeightSt (f ⊕ g) where heightSt (Inl x) = heightSt x heightSt (Inr x) = heightSt x Defining the height on Sig instance HeightSt Sig where heightSt (Val ) = 0 heightSt (Plus x y) = 1 + max x y Defining the height on Inc instance HeightSt Inc where heightSt (Inc x) = 1 + x

23

Subsignatures

Subsignature type class class f g where inj :: f a → g a

24

Subsignatures

Subsignature type class class f g where inj :: f a → g a f g iff g = g1 ⊕ g2 ⊕ ... ⊕ g n and f = gi, 0 < i ≤ n

24

Subsignatures

Subsignature type class class f g where inj :: f a → g a For example: Inc Inc ⊕ Sig

• Sig′

f g iff g = g1 ⊕ g2 ⊕ ... ⊕ g n and f = gi, 0 < i ≤ n

24

Subsignatures

Subsignature type class class f g where inj :: f a → g a For example: Inc Inc ⊕ Sig

• Sig′

f g iff g = g1 ⊕ g2 ⊕ ... ⊕ g n and f = gi, 0 < i ≤ n Injection and projection functions inject :: (g f ) ⇒ g (Context f a) → Context f a inject = In . inj

24

Outline

1

Tree Automata Bottom-Up Tree Acceptors Bottom-Up Tree Transducers

2

Introducing Modularity Composing State Spaces Compositional Signatures Decomposing Tree Transducers

3

Other Automata

25

Tree Homomorphisms

type UpTrans f q g = ∀ a . f (q, a) → (q, Context g a)

26

Tree Homomorphisms

type UpTrans f g = ∀ a . f a → Context g a

26

Tree Homomorphisms

type Hom f g = ∀ a . f a → Context g a

26

Tree Homomorphisms

type Hom f g = ∀ a . f a → Context g a Example (Desugaring) class DesugHom f g where desugHom :: Hom f g desugar :: (Functor f , Functor g, DesugHom f g) ⇒ Term f → Term g desugar = runHom desugHom

26

Tree Homomorphisms

type Hom f g = ∀ a . f a → Context g a Example (Desugaring) class DesugHom f g where desugHom :: Hom f g desugar :: (Functor f , Functor g, DesugHom f g) ⇒ Term f → Term g desugar = runHom desugHom instance (Sig g) ⇒ DesugHom Inc g where desugHom (Inc x) = Hole x ‘plus‘ val 1 instance (Functor g, f g) ⇒ DesugHom f g where desugHom = simpCxt . inj

26

Tree Homomorphisms

type Hom f g = ∀ a . f a → Context g a Example (Desugaring) class DesugHom f g where desugHom :: Hom f g desugar :: (Functor f , Functor g, DesugHom f g) ⇒ Term f → Term g desugar = runHom desugHom instance (Sig g) ⇒ DesugHom Inc g where desugHom (Inc x) = Hole x ‘plus‘ val 1 instance (Functor g, f g) ⇒ DesugHom f g where desugHom = simpCxt . inj simpCxt :: Functor g ⇒ g a → Context g a simpCxt t = In (fmap Hole t)

26

Stateful Tree Homomorphisms

Decomposing tree transducers type Hom f g = ∀ a . f a → Context g a type UpState f q = f q → q type UpTrans f q g = ∀ a . f (q, a) → (q, Context g a)

27

Stateful Tree Homomorphisms

Decomposing tree transducers type Hom f g = ∀ a . f a → Context g a type UpState f q = f q → q type UpTrans f q g = ∀ a . f (q, a) → (q, Context g a) Making homomorphisms dependent on a state type QHom f q g = ∀ a. f a → Context g a

27

Stateful Tree Homomorphisms

Decomposing tree transducers type Hom f g = ∀ a . f a → Context g a type UpState f q = f q → q type UpTrans f q g = ∀ a . f (q, a) → (q, Context g a) Making homomorphisms dependent on a state type QHom f q g = ∀ a. f (q, a) → Context g a

27

Stateful Tree Homomorphisms

Decomposing tree transducers type Hom f g = ∀ a . f a → Context g a type UpState f q = f q → q type UpTrans f q g = ∀ a . f (q, a) → (q, Context g a) Making homomorphisms dependent on a state type QHom f q g = ∀ a. (a → q) → f a → Context g a

27

Stateful Tree Homomorphisms

Decomposing tree transducers type Hom f g = ∀ a . f a → Context g a type UpState f q = f q → q type UpTrans f q g = ∀ a . f (q, a) → (q, Context g a) Making homomorphisms dependent on a state type QHom f q g = ∀ a. q → (a → q) → f a → Context g a

27

Stateful Tree Homomorphisms

Decomposing tree transducers type Hom f g = ∀ a . f a → Context g a type UpState f q = f q → q type UpTrans f q g = ∀ a . f (q, a) → (q, Context g a) Making homomorphisms dependent on a state type QHom f q g = ∀ a. q → (a → q) → f a → Context g a Using implicit parameters

type QHom f q g = ∀ a . (?above :: q, ?below :: a → q) ⇒ f a → Context g a

27

An Example

Extending the signature with let bindings type Name = String data Let e = LetIn Name e e | Var Name type LetSig = Let ⊕ Sig

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An Example

Extending the signature with let bindings type Name = String data Let e = LetIn Name e e | Var Name type LetSig = Let ⊕ Sig

type Vars = Set Name class FreeVarsSt f where freeVarsSt :: UpState f Vars

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An Example

Extending the signature with let bindings type Name = String data Let e = LetIn Name e e | Var Name type LetSig = Let ⊕ Sig

type Vars = Set Name class FreeVarsSt f where freeVarsSt :: UpState f Vars instance FreeVarsSt Sig where freeVarsSt (Plus x y) = x ‘union‘ y freeVarsSt (Val ) = empty instance FreeVarsSt Let where freeVarsSt (Var v) = singleton v freeVarsSt (LetIn v e s) = if v ‘member‘ s then e ‘union‘ delete v s else s

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An Example (Cont’d)

class RemLetHom f q g where remLetHom :: QHom f q g instance (Vars ∈ q, Let g, Functor g) ⇒ RemLetHom Let q g where remLetHom (LetIn v s) | ¬ (v ‘member‘ below s) = Hole s remLetHom t = simpCxt (inj t) instance (Functor f , Functor g, f g) ⇒ RemLetHom f q g where remLetHom = simpCxt . inj

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An Example (Cont’d)

class RemLetHom f q g where remLetHom :: QHom f q g instance (Vars ∈ q, Let g, Functor g) ⇒ RemLetHom Let q g where remLetHom (LetIn v s) | ¬ (v ‘member‘ below s) = Hole s remLetHom t = simpCxt (inj t) instance (Functor f , Functor g, f g) ⇒ RemLetHom f q g where remLetHom = simpCxt . inj Combining state transition and homomorphism remLet :: (Functor f , FreeVarsSt f , RemLetHom f Vars f ) ⇒ Term f → (Vars, Term f ) remLet = runUpHom freeVarsSt remLetHom

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An Example (Cont’d)

class RemLetHom f q g where remLetHom :: QHom f q g instance (Vars ∈ q, Let g, Functor g) ⇒ RemLetHom Let q g where remLetHom (LetIn v s) | ¬ (v ‘member‘ below s) = Hole s remLetHom t = simpCxt (inj t) instance (Functor f , Functor g, f g) ⇒ RemLetHom f q g where remLetHom = simpCxt . inj Combining state transition and homomorphism remLet :: (Functor f , FreeVarsSt f , RemLetHom f Vars f ) ⇒ Term f → (Vars, Term f ) remLet = runUpHom freeVarsSt remLetHom runUpHom st hom = runUpTrans (upTrans st hom)

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An Example (Cont’d)

class RemLetHom f q g where remLetHom :: QHom f q g instance (Vars ∈ q, Let g, Functor g) ⇒ RemLetHom Let q g where remLetHom (LetIn v s) | ¬ (v ‘member‘ below s) = Hole s remLetHom t = simpCxt (inj t) instance (Functor f , Functor g, f g) ⇒ RemLetHom f q g where remLetHom = simpCxt . inj Combining state transition and homomorphism remLet :: (Functor f , FreeVarsSt f , RemLetHom f Vars f ) ⇒ Term f → (Vars, Term f ) remLet = runUpHom freeVarsSt remLetHom remLet :: Term LetSig → Term LetSig remLet :: Term (Inc ⊕ LetSig) → Term (Inc ⊕ LetSig)

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Outline

1

Tree Automata Bottom-Up Tree Acceptors Bottom-Up Tree Transducers

2

Introducing Modularity Composing State Spaces Compositional Signatures Decomposing Tree Transducers

3

Other Automata

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Top-Down Tree Transducers

f q1

• q2. . .

qn f q1 q2 q q

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Top-Down Tree Transducers

f q1

• q2. . .

qn f q1 q2 q q

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Top-Down Tree Transducers

f q1

• q2. . .

qn f q1 q2 q q q(f (x1, x2, . . . , xn)) − → t f ∈ F t ∈ T (G, Q(X)) Q(X) = {p(xi) | p ∈ Q, 1 ≤ i ≤ n}

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Top-Down Tree Transducers

f q1

• q2. . .

qn f q1 q2 q q q(f (x1, x2, . . . , xn)) − → t f ∈ F t ∈ T (G, Q(X)) Q(X) = {p(xi) | p ∈ Q, 1 ≤ i ≤ n} Representation in Haskell type DownTrans f q g = ∀ a . (q, f a) → Context g (q, a)

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Decomposing Top-Down Tree Transducers

State transition depends on transformation Successor states are assigned to variable occurrences on right-hand side. q0(f (x)) → g(q1(x), q2(x))

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Decomposing Top-Down Tree Transducers

State transition depends on transformation Successor states are assigned to variable occurrences on right-hand side. q0(f (x)) → g(q1(x), q2(x)) Assignment to variables instead restriction: if q(x) and p(x) occur on right-hand side, then p = q.

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Decomposing Top-Down Tree Transducers

State transition depends on transformation Successor states are assigned to variable occurrences on right-hand side. q0(f (x)) → g(q1(x), q2(x)) Assignment to variables instead restriction: if q(x) and p(x) occur on right-hand side, then p = q. How to represent top-down state transformations? first try: type DownState f q = ∀ a . (q, f a) → f q

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Decomposing Top-Down Tree Transducers

State transition depends on transformation Successor states are assigned to variable occurrences on right-hand side. q0(f (x)) → g(q1(x), q2(x)) Assignment to variables instead restriction: if q(x) and p(x) occur on right-hand side, then p = q. How to represent top-down state transformations? first try: type DownState f q = ∀ a . (q, f a) → f q permits changing the shape of the input: bad :: ∀ a . (q, Sig a) → Sig q bad (q, Plus x y) = Val 1 bad (q, Val i) = Val 1

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Top-Down State Transitions

Using explicit placeholders type DownState f q = ∀ i . Ord i ⇒ (q, f i) → Map i q

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Top-Down State Transitions

Using explicit placeholders type DownState f q = ∀ i . Ord i ⇒ (q, f i) → Map i q construct function of type ∀ a . (q, f a) → f q that preserves the shape

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Top-Down State Transitions

Using explicit placeholders type DownState f q = ∀ i . Ord i ⇒ (q, f i) → Map i q construct function of type ∀ a . (q, f a) → f q that preserves the shape Combining with stateful tree homomorphisms

type QHom f q g = ∀ a . (?above :: q, ?below :: a → q) ⇒ f a → Context g a

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Top-Down State Transitions

Using explicit placeholders type DownState f q = ∀ i . Ord i ⇒ (q, f i) → Map i q construct function of type ∀ a . (q, f a) → f q that preserves the shape Combining with stateful tree homomorphisms

type QHom f q g = ∀ a . (?above :: q, ?below :: a → q) ⇒ f a → Context g a type DownTrans f q g = ∀ a . (q, f a) → Context g (q, a)

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Why Tree Transducers

More structure, more flexibility Tree transducers can manipulated more easily

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Why Tree Transducers

More structure, more flexibility Tree transducers can manipulated more easily, e.g. data (f :&: a) e = f e :&: a

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Why Tree Transducers

More structure, more flexibility Tree transducers can manipulated more easily, e.g. data (f :&: a) e = f e :&: a lift :: UpTrans f q g → UpTrans (f :&: a) q (g :&: a)

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Why Tree Transducers

More structure, more flexibility Tree transducers can manipulated more easily, e.g. data (f :&: a) e = f e :&: a lift :: UpTrans f q g → UpTrans (f :&: a) q (g :&: a) Tree transducers compose We may leverage composition theorems for tree transducers. comp :: DownTrans g p h → DownTrans f q g → DownTrans f (q, p) h

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Why Tree Transducers

More structure, more flexibility Tree transducers can manipulated more easily, e.g. data (f :&: a) e = f e :&: a lift :: UpTrans f q g → UpTrans (f :&: a) q (g :&: a) Tree transducers compose We may leverage composition theorems for tree transducers. comp :: DownTrans g p h → DownTrans f q g → DownTrans f (q, p) h potential for fusion

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Beyond Tree Transducers

Bidirectional state transitions bottom-up + top-down state transition bottom-up + top-down state transition + stateful homomorphism

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Beyond Tree Transducers

Bidirectional state transitions bottom-up + top-down state transition bottom-up + top-down state transition + stateful homomorphism Macro Tree Transducers states may have arguments taken from the term necessary for ’non-local’ transformations, e.g. substitution, inlining

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Beyond Tree Transducers

Bidirectional state transitions bottom-up + top-down state transition bottom-up + top-down state transition + stateful homomorphism Macro Tree Transducers states may have arguments taken from the term necessary for ’non-local’ transformations, e.g. substitution, inlining

type UpTrans f q g = ∀ a . f (q , a) → (q , Context g a)

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Beyond Tree Transducers

Bidirectional state transitions bottom-up + top-down state transition bottom-up + top-down state transition + stateful homomorphism Macro Tree Transducers states may have arguments taken from the term necessary for ’non-local’ transformations, e.g. substitution, inlining

type UpTrans’ f q g = ∀ a . f (q a, a) → (q (Context g a), Context g a)

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Beyond Tree Transducers

Bidirectional state transitions bottom-up + top-down state transition bottom-up + top-down state transition + stateful homomorphism Macro Tree Transducers states may have arguments taken from the term necessary for ’non-local’ transformations, e.g. substitution, inlining

type UpTrans’ f q g = ∀ a . f (q a, a) → (q (Context g a), Context g a) type DownTrans f q g = ∀ a . (q , f a) → Context g (q , a)

35

Beyond Tree Transducers

Bidirectional state transitions bottom-up + top-down state transition bottom-up + top-down state transition + stateful homomorphism Macro Tree Transducers states may have arguments taken from the term necessary for ’non-local’ transformations, e.g. substitution, inlining

type UpTrans’ f q g = ∀ a . f (q a, a) → (q (Context g a), Context g a) type DownTrans’ f q g = ∀ a . (q a, f a) → Context g (q (Context g a), a)

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Beyond Tree Transducers

Bidirectional state transitions bottom-up + top-down state transition bottom-up + top-down state transition + stateful homomorphism Macro Tree Transducers states may have arguments taken from the term necessary for ’non-local’ transformations, e.g. substitution, inlining

type UpTrans’ f q g = ∀ a . f (q a, a) → (q (Context g a), Context g a) type DownTrans’ f q g = ∀ a . (q a, f a) → Context g (q (Context g a), a) e.g. for substitutions: q = Map Var

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Beyond Tree Transducers

Bidirectional state transitions bottom-up + top-down state transition bottom-up + top-down state transition + stateful homomorphism Macro Tree Transducers states may have arguments taken from the term necessary for ’non-local’ transformations, e.g. substitution, inlining

type UpTrans’ f q g = ∀ a . f (q a, a) → (q (Context g a), Context g a) type DownTrans’ f q g = ∀ a . (q a, f a) → Context g (q (Context g a), a) e.g. for substitutions: q = Map Var

generic programming

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