AC-Unification of Higher-order Patterns Alexandre Boudet Evelyne - PDF document

CP’97 1 ✬ ✩ AC-Unification of Higher-order Patterns Alexandre Boudet Evelyne Contejean LRI Orsay France ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 2 ✬ ✩ Motivations ∗ Higher-order unification is undecidable ∗ Unification of higher-order patterns is decidable ∗ Combination of algebraic and functional programming paradigms ∗ Local confluence of HRSs ⇓ Unification of higher-order patterns modulo equational theories ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 3 ✬ ✩ Patterns Definition A pattern is ∗ a term of the simply-typed λ -calculus in β -normal form ∗ in which the arguments of a free variable are η -equivalent to distinct bound variables.  λxyz.f ( H ( x, y ) , H ( x, z ))  are patterns λx.F ( λz.x ( z )) = η λx.F ( x )   λxy.G ( x, x, y )  are not patterns λxy.H ( F ( x ) , y )  No equational theory, but α, β, η . Theorem (Miller) The unifiability of patterns is decidable and if two patterns are unifiable, there is an algorithm com- puting a unique most general unifier. Notation λx.F ( x π ) denotes λx 1 . . . λx n .F ( x π (1) , . . . , x π ( n ) ), where π is a permutation over { 1 , . . . , n } . ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 4 ✬ ✩ AC-unification of Patterns Definition Let E = { l 1 ≃ r 1 , . . . , l n ≃ r n } a set of axioms such that l i and r i of the same type, for 1 ≤ i ≤ n . The equational theory = E is the least congruence (compatible also with application and abstraction) containing all the instances of the axioms of E . Definition An equation s = t is a pair of patterns of the same type.  − ⊤    A unification problem is − ⊥   − P ≡ s 1 = t 1 ∧ · · · ∧ s n = t n .  A substitution σ is an E -unifier of P if ∀ i , s i = βηE t i . ⇒ u � η β = E v � η Theorem (Tannen) ∀ u, v u = βηE v ⇐ β . We consider the case where E is � { x + y = y + x, x + ( y + z ) = ( x + y ) + z } + ∈ AC The type of + is α → α → α , where α is a base type. ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 5 ✬ ✩ Alien subterms and Purification Definition u = t | p · i is an alien subterm of t ∗ if t ( p ) ∈ AC and Head( u ) / ∈ FV , ∗ or t ( p ) ∈ FC and Head( u ) ∈ AC. VA → 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 at position p , and y = FV ( u ) ∩ x , where H is a new variable. A unification problem in normal form wrt VA has the form P ≡ P F ∧ P 0 ∧ P 1 ∧ · · · ∧ P n ∗ P 0 is pure in the free theory, with no λx.F ( x ) = λx.F ( x π ). ∗ P F contains all the Flex-Flex equations λx.F ( x ) = λx.F ( x π ). ∗ P i is a pure unification problem in the AC theory of + i , the arguments of + i being of the form F ( x ) where F is a free variable. ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 6 ✬ ✩ Why Frozen Equations ? Example (Qian & Wang) 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). ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 7 ✬ ✩ Free: a subset of Nipkow’s rules for pattern unification Dec-free λx.a ( s 1 , . . . , s n ) = λx.a ( t 1 , . . . , t n ) ∧ P 0 → λx.s 1 = λx.t 1 ∧ · · · ∧ λx.s n = λx.t n ∧ P 0 if a is a free constant symbol or a bound variable of x . FR-free λx.F ( y ) = λx.a ( s 1 , . . . , s m ) ∧ P 0 → λy.H 1 ( y ) = λy.s 1 ∧ · · · ∧ λy.H m ( y ) = λy.s m ∧ P 0 { F �→ λx.a ( H 1 ( y ) , . . . , H m ( y )) } ∧ F = λx.a ( H 1 ( y ) , . . . , H m ( y )) } If F is a free variable, a a free constant and F / ∈ FV ( s i ) for 1 ≤ i ≤ m , where H 1 , . . . , H m are new variables. FF � = λx.F ( y ) = λx.G ( z ) ∧ P 0 → P 0 { F �→ λy.H ( v ) , G �→ λz.H ( v ) } ∧ F = λy.H ( v ) ∧ G = λz.H ( v ) if F and G are different free variables where v = y ∩ z . Fail1 → ⊥ λx.a ( s ) = λx.b ( t ) if a and b are constants or bound variables and a � = b . Fail2 λx.F ( y ) = λx.a ( s ) → ⊥ if F is free in s or a ∈ x \ y . ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 8 ✬ ✩ Freeze P F ∧ ( λx.F ( x ) = λx.F ( x π ) ∧ P 0 ) ∧ P 1 ∧ · · · ∧ P n → ( λx.F ( x ) = λx.F ( x π ) ∧ P F ) ∧ P 0 ∧ P 1 ∧ · · · ∧ P n if F is a free variable. Normal Forms A unification problem (with no alien subterms) in normal form wrt ( Free + Freeze ) has the form P F ∧ P 0 ∧ P 1 ∧ · · · ∧ P n where ∗ P F contains only λx.F ( x ) = λx.F ( x π ), ∗ P 0 is solved, ∗ P i is a pure problem in the AC -theory of + i . ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 9 ✬ ✩ Pure AC Patterns: An Example From AC Problems to Diophantine Problems λxyz. 2 F ( x, y, z ) + F ( y, z, x ) = λxyz. 2 G ( x, y, z ) Let σ be a solution of the above equation: ∗ σ may introduce t ( x, y, z ) such that t ( x, y, z ) = t ( y, z, x ). λxyz. 2 F ( x, y, z )+ F ( y, z, x )= λxyz. 2 G ( x, y, z ) # of t 2 α + α = 2 β ( α, β ) is a positive solution of 3 n = 2 m . ∗ σ may introduce t ( x, y, z ) such that t ( x, y, z ) � = t ( y, z, x ). λxyz. 2 F ( x, y, z )+ F ( y, z, x )= λxyz. 2 G ( x, y, z ) # of t ( x, y, z ) 2 α 1 + α 3 = 2 β 1 # of t ( y, z, x ) 2 α 2 + α 1 = 2 β 2 # of t ( z, x, y ) 2 α 3 + α 2 = 2 β 3 ( α 1 , α 2 , α 3 , β 1 , β 2 , β 3 ) are positive solutions of the system 2 n 1 + n 3 = 2 m 1 2 n 2 + n 1 = 2 m 2 2 n 3 + n 2 = 2 m 3 ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 10 ✬ ✩ Pure AC Patterns: An Example (Continued) Recombination + Constraints The set of minimal solutions of 3 n = 2 m is { (2 , 3) } The set of minimal solutions of 2 n 1 + n 3 = 2 m 1 2 n 2 + n 1 = 2 m 2 2 n 3 + n 2 = 2 m 3 is { (2 , 0 , 0 , 2 , 1 , 0) , (0 , 2 , 0 , 0 , 2 , 1) , (0 , 0 , 2 , 1 , 0 , 2) } . F(x,y,z) G(x,y,z) L 1 2 3 L 2 2 0 0 2 1 0 L 3 0 2 0 0 2 1 L 4 0 0 2 1 0 2 (x,y,z) (y,z,x) (z,x,y) (x,y,z) (y,z,x) (z,x,y) ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 11 ✬ ✩ Pure AC Patterns: An Example (End) A solution of the original problem is { F �→ λxyz. 2 L 1 ( x, y, z ) θ + 2 L 2 ( x, y, z ) + 2 L 3 ( y, z, x ) + 2 L G �→ λxyz. 3 L 1 ( x, y, z ) θ + 2 L 2 ( x, y, z ) + L 2 ( y, z, x ) + 2 L 3 L 4 ( x, y, z ) + 2 L 4 ( z, x, y ) } provided that L 1 ( x, y, z ) θ = L 1 ( y, z, x ) θ = L 1 ( z, x, y ) θ . ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 12 ✬ ✩ Pure AC Patterns: Another Example λxyz.G ( x, y, z ) = λxyz.G ( y, z, x ) ∧ λxyz. 2 F ( x, y, z ) + F ( y, z, x ) = λxyz. 2 G ( x, y, z ) Let σ be a solution of the above problem: σ may introduce t ( x, y, z ) such that t ( x, y, z ) � = t ( y, z, x ). λxyz. 2 F ( x, y, z )+ F ( y, z, x )= λxyz. 2 G ( x, y, z ) # of t ( x, y, z ) 2 α 1 + α 3 = 2 β 1 # of t ( y, z, x ) 2 α 2 + α 1 = 2 β 2 # of t ( z, x, y ) 2 α 3 + α 2 = 2 β 3 λxyz.G ( x, y, z ) = λxyz.G ( x, y, z ) # of t ( x, y, z ) β 1 = β 3 # of t ( y, z, x ) β 2 = β 1 # of t ( z, x, y ) β 3 = β 2 ( α 1 , α 2 , α 3 , β 1 , β 2 , β 3 ) are positive solutions of the system 2 n 1 + n 3 = 2 m 1 2 n 2 + n 1 = 2 m 2 2 n 3 + n 2 = 2 m 3 m 1 = m 3 m 2 = m 1 m 3 = m 2 ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 13 ✬ ✩ Pure AC Patterns: Another Example (End) The set of minimal solutions of 3 n = 2 m is { (2 , 3) } The set of minimal solutions of P Dioph is { (2 , 2 , 2 , 3 , 3 , 3) } .  2 n 1 + n 3 = 2 m 1      2 n 2 + n 1 = 2 m 2      2 n 3 + n 2 = 2 m 3   P Dioph ≡ m 1 = m 3      m 2 = m 1      m 3 = m 2   F(x,y,z) G(x,y,z) L 1 2 3 L 2 2 2 2 3 3 3 (x,y,z) (y,z,x) (z,x,y) (x,y,z) (y,z,x) (z,x,y) A solution of the original problem is { F �→ λxyz. 2 L 1 ( x, y, z ) θ +2 L 2 ( x, y, z )+2 L 2 ( y, z, x )+2 L 2 ( z, x, y ) , G �→ λxyz. 3 L 1 ( x, y, z ) θ +3 L 2 ( x, y, z )+3 L 2 ( y, z, x )+3 L 2 ( z, x, y ) } provided that L 1 ( x, y, z ) θ = L 1 ( y, z, x ) θ = L 1 ( z, x, y ) θ . ✫ ✪ Linz, Austria. Oct 30th, 1997

CP’97 14 ✬ ✩ Pure AC Patterns: General Case n 1 n 2 � � � � a i,π F i ( x π ) = λx · a i,π F i ( x π ) E ≡ λx · i =1 π ∈ Π i = n 1 +1 π ∈ Π Π is the group of permutations over all the variables of x . n 2 � � y i,π 1 = y i,π 2 ∧ i =1 E Π 0 ( π 1 ,π 2 ) | Π 0 ◦ π 1 =Π 0 ◦ π 2 Dioph ≡ n 1 n 2 � � � � � a i,π − 1 ◦ π ′ y i,π = a i,π − 1 ◦ π ′ y i,π π ′ ∈ Π i =1 π ∈ Π i = n 1 +1 π ∈ Π Π 0 subgroup of Π E Π 0 E Dioph is the disjunction � Dioph . Proposition Let P be any subset of the minimal solutions of P Dioph , which is great enough. Then σ P is a solution of E : � � � m ( i, π ′ ) L m ( x π ′ ) } σ P = { F i �→ λx. Π 0 m ∈P Π0 π ′ ∈ Π / Π 0 where L m , m ∈ P Π 0 is a new variable constrained by ∀ π ∈ Π 0 L m ( x π ) = L m ( x ). Proposition { σ P | P is great enough } is a complete set of solutions for P . ✫ ✪ Linz, Austria. Oct 30th, 1997