E -Unification of Higher-order Patterns Alexandre Boudet Evelyne - PowerPoint PPT Presentation

E -Unification of Higher-order Patterns Alexandre Boudet Evelyne Contejean LRI Orsay France

Motivations • Higher-order unification is undecidable (Huet) • Unification of higher-order patterns is decidable (Miller) • Combination of algebraic and functional programming paradigms • Local confluence of HRSs ⇓ Unification of higher-order patterns modulo equational theories 2

Patterns First-Order Term Pattern General High-Order Term ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ✁ ❆ ❆ ❆ ✁ ✁ ✁ ❆ ❆ ❆ ✁ ✁ ✁ ❆ ❆ ❆ ✁ ✁ ✁ F ❆ ❆ ❆ ✁ ✁ ✁ ❆ ❆ ❆ FG H H ✁ ✁ ✁ ❆ ❆ G ❆ ✁ ✁ ✁ x y z x y z ❆ ❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ Definition Pattern: • term of the simply-typed λ -calculus in β -normal form • the arguments of a free variable are η -equivalent to distinct bound variables. Patterns Not patterns λxyz.f ( H ( x, y ) , H ( x, z )) λxy.G ( x, x, y ) λx.F ( λz.x ( z )) = η λx.F ( x ) λxy.H ( F ( x ) , y ) No equational theory, but α, β, η . Theorem (Miller) In the case of patterns, unifiability is decidable there is an algorithm for computing a mgu. 3

Patterns First-Order Term Pattern General High-Order Term ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ✁ ❆ ❆ ❆ ✁ ✁ ✁ ❆ ❆ ❆ ✁ ✁ ✁ ❆ ❆ ❆ ✁ ✁ ✁ F ❆ ❆ ❆ ✁ ✁ ✁ ❆ ❆ ❆ FG H H ✁ ✁ ✁ ❆ ❆ G ❆ ✁ ✁ ✁ x y z x y z ❆ ❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ Definition Pattern: • term of the simply-typed λ -calculus in β -normal form • the arguments of a free variable are η -equivalent to distinct bound variables. Patterns Not patterns λxyz.f ( H ( x, y ) , H ( x, z )) λxy.G ( x, x, y ) λx.F ( λz.x ( z )) = η λx.F ( x ) λxy.H ( F ( x ) , y ) No equational theory, but α, β, η . Theorem (Miller) In the case of patterns, unifiability is decidable there is an algorithm for computing a mgu. 4

E-unification of Patterns Definition E = { l 1 ≃ r 1 , . . . , l n ≃ r n } : set of First-Order axioms. Equational theory = E : least congruence containing all the l i σ ≃ r i σ (context, application and abstraction) Definition s = t , pair of patterns of the same type. Equation : Unification problem : ⊤ , ⊥ or P ≡ s 1 = t 1 ∧ · · · ∧ s n = t n . substitution σ such that ∀ i , s i σ = βηE t i σ . E -unifier of P : ⇒ u � η β = E v � η Theorem ( Tannen) ∀ u, v u = βηE v ⇐ β . 5

How to split when E = E 1 ∪ . . . ∪ E n ? Aim : unification of patterns modulo ✛ ✘ β , η ✚ ✙ ✛ ✘ E 1 . . . E i . . . E n ✚ ✙ Naive approach Counter-Example (Qian & Wang) with E = AC (+) : λxy · F ( x, y ) = λxy · F ( y, x ) has the solutions ∀ n ∈ N σ n = { F �→ λxy · G ( H 1 ( x, y ) + H 1 ( y, x ) , . . . , H n ( x, y ) + H n ( y, x )) } 6

How to split when E = E 1 ∪ . . . ∪ E n ? Aim : unification of patterns modulo ✛ ✘ β , η ✚ ✙ ✛ ✘ E 1 . . . E i . . . E n ✚ ✙ Naive approach Counter-Example (Qian & Wang) with E = AC (+) : λxy · F ( x, y ) = λxy · F ( y, x ) has the solutions ∀ n ∈ N σ n = { F �→ λxy · G ( H 1 ( x, y ) + H 1 ( y, x ) , . . . , H n ( x, y ) + H n ( y, x )) } 7

How to split when E = E 1 ∪ . . . ∪ E n ? Aim : unification of patterns modulo ✛ ✘ β , η ✚ ✙ ✛ ✘ E 1 . . . E i . . . E n ✚ ✙ Naive approach Counter-Example (Qian & Wang) with E = AC (+) : λxy · F ( x, y ) = λxy · F ( y, x ) has the solutions ∀ n ∈ N σ n = { F �→ λxy · G ( H 1 ( x, y ) + H 1 ( y, x ) , . . . , H n ( x, y ) + H n ( y, x )) } 8

How to split when E = E 1 ∪ . . . ∪ E n ? Aim : unification of patterns modulo ✛ ✘ β , η ✟ ❍ ✟✟✟✟✟✟✟✟✟✟✟ ❍ ❍ ❍ ❍ ✟ ❍ ✟✟✟✟✟✟✟✟✟✟✟✟✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ E 1 . . . E i . . . E n ❍ ❍ ✚ ✙ ❍ ❍ ❍ Realistic approach practice, ∅ , AC, ACU, ACUN, AG Counter-Example (Qian & Wang) with E = AC (+) : λxy · F ( x, y ) = λxy · F ( y, x ) has the solutions ∀ n ∈ N σ n = { F �→ λxy · G ( H 1 ( x, y ) + H 1 ( y, x ) , . . . , H n ( x, y ) + H n ( y, x )) } 9 Algorithms for patterns unification modulo the E i s are assumed to be given. In

How to split when E = E 1 ∪ . . . ∪ E n ? Aim : unification of patterns modulo ✛ ✘ β , η ✟ ❍ ✟✟✟✟✟✟✟✟✟✟✟ ❍ ❍ ❍ ❍ ✟ ❍ ✟✟✟✟✟✟✟✟✟✟✟✟✟ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ E 1 . . . E i . . . E n ❍ ❍ ✚ ✙ ❍ ❍ ❍ Realistic approach Counter-Example (Qian & Wang) with E = AC (+) : Algorithms for patterns unification modulo the E i s are assumed to be given. λxy · F ( x, y ) = λxy · F ( y, x ) has the solutions In practice, ∅ , AC, ACU, ACUN, AG ∀ n ∈ N σ n = { F �→ λxy · G ( H 1 ( x, y ) + H 1 ( y, x ) , . . . , H n ( x, y ) + H n ( y, x )) } 10

Splitting the unification problem Definition Theory of f , algebraic symbol, or of x , bound variable Th ( f ) = E i , E i such that f ∈ F i Th ( x ) = E ∅ Alien subterm u in t [ u ] p : u argument of f and Th ( f ) � = Th ( head ( u )) . VA → ∃ H λx. t [ H ( y )] p = λx. s ∧ λy. H ( y ) = λy. u λx. t [ u ] p = λx. s if u is an alien subterm of t [ u ] p , y = FV ( u ) ∩ x , and H new variable. Split ∃ F λx. F ( x ) = λx.γ ( s ) ∧ λx.γ ( s ) = λx.δ ( t ) → λx. F ( x ) = λx.δ ( t ) if γ and δ not free variables, Th ( γ ) � = Th ( δ ) , and F new variable. 11

Split unification problem A unification problem in NF wrt VA and Split : P ≡ P F ∧ P 0 ∧ P 1 ∧ · · · ∧ P n • P F contains all the Flex-Flex equations λx.F ( x ) = λx.F ( x π ) . • P 0 is pure in E 0 , with no λx.F ( x ) = λx.F ( x π ) . • P 1 is a pure unification problem in E 1 . • P n is a pure unification problem in E n . Notation λx.F ( x π ) : λx 1 . . . λx n .F ( x π (1) ,. . .,x π ( n ) ) , where π is a permutation over { 1 ,. . .,n } . 12

A combination algorithm through don’t know non-determinism Guess the actual arguments of a variable Definition Constant preserving substitution: σ = { F �→ λx.s } , λx.s in NF and every x i of x has a free occurrence in s . Projection: σ = { F �→ λx.F ′ ( y ) | { y } ⊆ { x }} Lemma σ a substitution, then σ � η β = ( πθ ) � η β with π projection and θ constant- preserving substitution. → ∃ F ′ F = λx.F ′ ( y ) ∧ P { F �→ λx.F ′ ( y ) } Project P where F ′ is a new variable and { y } ⊂ { x } 13

A combination algorithm through don’t know non-determinism Guess the flex-flex equations → F = λx.G ( x π ) ∧ P { F �→ λx.G ( x π ) } FF � = P where π is a permutation, types of F and G π are compatible, F � = G and F and G occur in P . Guess the permutations over the arguments → λx.F ( x )= λx.F ( x π ) ∧ P FF = P where F is a free variable of P , types of F and F π are compatible. 14

A combination algorithm through don’t know non-determinism Find a representative for each variable Apply as long as possible Coalesce λx.F ( y )= λx.G ( z ) ∧ P → F = λy.G ( z ) ∧ P { F �→ λy.G ( z ) } if F � = G and F, G ∈ FV ( P ) , where y is a permutation of z . Guess the theory of the representatives Guess an ordering on representatives 15

Dealing with λx.F ( x ) = λx.F ( x π ) by freezing Example (Qian & Wang) E = AC (+) : λxy · F ( x, y ) = λxy · F ( y, x ) has the solutions σ n = { F �→ λxy · G ( H 1 ( x, y )+ H 1 ( y, x ) , . . . , H n ( x, y )+ H n ( y, x )) } for all n ∈ N . In addition σ n +1 is strictly more general than σ n (nullary theory). 16

Solving the pure problems, compatiblity with frozen equations Definition Solve rule for E i : algorithm input: P i , pure problem in E i and P F frozen equations output: P ′ and P ′ iF such that i 1. P ′ i ≡ σ E i , is a solved form without flex-flex equations. 2. P ′ iF is equal to P F plus some additonnal λx.F ( x )= λx.F ( x π ) . 3. F instantiated by σ E i only if E i is the chosen theory of F 4. the value of F may contain G only if F < oc G , for the chosen ordering 5. for all the equations s = t of P i and P F , sσ E i and tσ E i can be proven E i -equal (by using the equations in P ′ iF ). 17

Example: AC(+) Input : λxy.F ( x, y ) + G ( x, y ) = λxy. 2 H ( x, y ) ∧ λxy.H ( x, y ) = λxy.H ( y, x ) Output : F = λxy.F ′ ( x, y ) + 2 F ′′ ( x, y ) G = λxy.F ′ ( x, y ) + 2 F ′′ ( y, x ) ∧ λxy.F ′ ( x, y ) = λxy.F ′ ( y, x ) H = λxy.F ′ ( x, y ) + F ′′ ( x, y ) + F ′′ ( y, x ) In order to prove that λxy.Hσ ( x, y ) = AC λxy.Hσ ( y, x ) , that is λxy.F ′ ( x, y ) + F ′′ ( x, y ) + F ′′ ( y, x ) = AC λxy.F ′ ( y, x ) + F ′′ ( y, x ) + F ′′ ( x, y ) , we need λxy.F ′ ( x, y ) = λxy.F ′ ( y, x ) . 18

Solving the pure problems, compatiblity with frozen equations Proposition s = t a pure equation in E i , and σ such that sσ = E tσ . Then there exists P perm = { λx π .F ( x )= λx ϕ .F ( x ) } , σ E i and θ such that • σ = E σ E i θ . • σ E i pure in E i , • θ E -solution of P perm . • for all pure equations s ′ = t ′ (in particular s = t ) such that s ′ σ = E t ′ σ , s ′ σ E i and t ′ σ E i can be proven E i -equal (using the equations of P perm ). 19