1. Treatment of Types in an Implementation of λProlog Organized around Higher-Order Pattern Unification Xiaochu Qi Department of Computer Science and Engineering University of Minnesota

2. Types in Logic Programming Languages descriptive notion: typed Prolog parametric polymorphism type nil (list A). type :: A -> (list A) -> (list A). type append (list A) -> (list A) -> (list A) -> o.

3. Types in Logic Programming Languages prescriptive notion: λProlog mixed usage of parametric and ad hoc polymorphism type print_int int -> o. type print_str string -> o. type print A -> o. print X :- print_int X. print X :- print_str X. print X.

4. The Nature of Polymorphism in λProlog a polymorphic version of the simply typed λ-calculus atomic types: including type variables the intuitive meaning of a term t :σ {t :σ ’ |σ ’ is a closed instance of σ }

5. The Roles of Types in λProlog a strongly typed language type declarations for constants inferring types for all subterms prevent run-time failures well-typedness: compile time type-checking participate in unification

6. ∀ :: ∀ nil ∀ append  F ∀ c A Prefix of Mixed Quantifiers ?- (  F)(append (c::nil) nil (F c)). universal variable existential variable constant ?- (  F) ∀ c(append (c::nil) nil (F c)). ∀ c ∀ :: ∀ nil ∀ append  F

7. Types of Constants and Variables The occurrences of a constant can have different types. The occurrences of a universal or existential variable must always have identical types. type nil (list A). type :: A -> (list A) -> (list A). (1 :: nil) (“a” :: nil) (1 :int :: :int->(list int)->(list int) nil :(list int) ) (“a” :string :: :string->(list string)->(list string) nil :(list string) )

8. Higher-Order Unification determine the identity of constants decide the structures of unifiers Types are associated with every variable and constant.

9. Higher-Order Pattern Unification determine the identity of constants a universal or existential variable: always has identical types in different occurrences can have a polymorphic type, which could be specialized during unification

10. Typed Higher-Order Pattern Unification processing terms in a top-down manner iterative usage of: term simplification phase: simply non-existential variable head applications binding phase decide the structure of bindings for existential variables invariant: Terms to be unified always have identical types.

11. Typed Higher-Order Pattern Unification term simplification phase: constant heads: examine (specialize) the types of these constants others: already have identical types binding phase: identical types of the entire terms identical types of relevant variables no comparison on constants No need for type examination

12. ∀ g  F ∀ c  X ∀ a type g A -> B. τ(c)= A->B τ(a)= C τ(X)= C->A τ(F)= (A->B)->int D (c :A->B (X :C->A a :C )), (g :int->D (F :(A->B)->int c :A->B ))> (r-r): { } int (X :C->A a :C ), F :(A->int)->int c :A->int > (r-f): { } ∀ g  H  F ∀ c  X ∀ a A a :C, H :(A->int)->A c :A->int > (f-f): {, }

13. ∀ 1 ∀ ”a” ∀ f  X  Y ∀ c type f A -> B -> C. τ(c)= A->B τ(X)= A τ(Y)= B string->B->C X :A “a” :string (c :A->B X :A ), f :int->A->B->C 1 :int Y :A (c :A->B Y :A )>

14. Maintain Types with Constants only associating types with constants Exp1 : ∀ g  F ∀ c  X ∀ a D (c :A->B (X :C->A a :C )), (g :int->D (F :(A->B)->int c :A->B ))> D (c (X a)), (g :int->D (F c))> Exp2 : ∀ 1 ∀ ”a” ∀ f  X  Y string->B->C X :A “a” :string (c :A->B X :A ), f :int->A->B->C 1 :int Y :A (c :A->B Y :A )> string->B->C X “a” (c X), f :int->A->B->C 1 Y (c Y)>

15. Maintain instances of type variables well-typedness with respect to type declarations compile time type-checking instances of type variables run time type-checking (:: :[list A] (:: :[A] X nil :[A] ) nil: [list A] ) Exp1: D (c (X a)), (g :int->D (F c))> (:: (:: X nil) nil) (:: :(list A)->(list (list A))->(list (list A) ) (:: :A->(list A)->(list A) X nil :(list A) ) nil :(list (list A)) )

16. Type Preservation Type variables occurring in target types have been checked at a upper level. Constant function symbols: type variables not occurring in target types Non-function constants: no need to maintain types type :: A -> (list A) -> (list A). type nil (list A). (:: :[list A] (:: :[A] X nil :[A] ) nil :[list A] ) (:: (:: X nil) nil) type g (A -> B) -> A. type a int -> B. (:: :[int] (g :[int,B] a :[int,B] ) nil :[int] ) (:: (g [B] a) nil)

17. Teyjus: id A (:: A X L) (:: A X K) :- id A L K. Type Processing Effort in Compiled Unification type id (list A) -> (list A) -> o. id nil nil. id (X::L) (X::K) :- id L K. ?- id (1::nil) T. Our scheme: id A (:: X L) (:: X K) :- id A L K. Teyjus: ?- id int (:: int 1 nil) T. Our scheme: ?- id int (:: 1 nil) T.

18. bind (T, (:: A S2 K)); bind (A, int); bind (S2, X); (write mode) (read mode) bind (T, (:: S2 K)); Type Processing in Compiled Unification over bind (A, int); bind (X, 1); bind (L, nil);

19. Type generality Typed Prolog : type p σ 1 -> … -> σ n -> o. p t 1 … t n :- b 1, …, b m. τ ( t 1 )= σ 1, …, τ ( t n )= σ n Every sub-term of t i is type preserving. Only type general and preserving programs are well-typed. λProlog : mixed usage of parametric and ad hoc polymorphism

20. Type generality in λProlog type print A -> o. type print_int int -> o. type print_str string -> o. print X :- print_int X. print X :- print_string X. print X. type generality and preservation property should hold on every clause head defining a predicate.

21. Type generality in λProlog type foo A -> o. foo X :- print X. type variables of the clause head should not be used in body goals.

22. Conclusion Type examination is necessary in higher-order pattern unification. Reduce type association and run-time type examination constant function symbols separate static and dynamic information take advantage of the type preservation property

23. Future work prove the correctness of the typed higher-order pattern unification scheme taking advantage of the type generality property making usage in implementations