This document describes the implementation of a gradual polymorphic type system with standard subtyping for the logic programming language Prolog. It introduces the concept of adding types to Prolog in a gradual way, beginning with untyped programs and incrementally adding more types through type inference and user-added type signatures. The document discusses previous work on adding types to Prolog, including the Mycroft-O'Keefe type system, type inference through abstract interpretation, and gradually applying types in other languages like ActionScript and Erlang. It proposes extensions to the type system such as a subtyping hierarchy.

1. Introduction Background Extensions Type Inference Implementation Results Conclusion
.
Implementation of a Gradual
Polymorphic Type System with
.
Standard Subtyping for Prolog
Spyros Hadjichristodoulou
David S. Warren1
Stony Brook University
Computer Science Department
WLPE 2012
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 1 / 23

2. Introduction Background Extensions Type Inference Implementation Results Conclusion
Types in Prolog?
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 2 / 23

3. Introduction Background Extensions Type Inference Implementation Results Conclusion
Types in Prolog?
.
Theorem (Most Prolog Books)
..
Prolog is an untyped language...
.
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 2 / 23

4. Introduction Background Extensions Type Inference Implementation Results Conclusion
Types in Prolog?
.
Theorem (Most Prolog Books)
..
Prolog is an untyped language...
.
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 2 / 23

5. Introduction Background Extensions Type Inference Implementation Results Conclusion
Types in Prolog?
.
Theorem (Most Prolog Books)
..
Prolog is an untyped language...
.
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 2 / 23

6. Introduction Background Extensions Type Inference Implementation Results Conclusion
Previous works
Mycroft-O’Keefe’s type system (DEC-10)
Mercury, Visual Prolog
Type inference by abstract interpretation (Barbuti,Giacobazzi
- 1990s)
Towards typed Prolog (Schrijvers, Santos Costa, Wielemaker,
Demoen - 2009)
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 3 / 23

7. Introduction Background Extensions Type Inference Implementation Results Conclusion
Previous works
Mycroft-O’Keefe’s type system (DEC-10)
Mercury, Visual Prolog
Type inference by abstract interpretation (Barbuti,Giacobazzi
- 1990s)
Towards typed Prolog (Schrijvers, Santos Costa, Wielemaker,
Demoen - 2009)
Why do Prolog vendors still keep it type-free?
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 3 / 23

8. Introduction Background Extensions Type Inference Implementation Results Conclusion
Our vision
A Gradual Polymorphic type system with Subtyping
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 4 / 23

9. Introduction Background Extensions Type Inference Implementation Results Conclusion
Our vision
A Gradual Polymorphic type system with Subtyping
T
. ype
I
.nference
M
. ore G
. radual,
U .
. ntyped T
. ypes T
. ype I
.ncremental
P
. rogram A
. dded S
. afe P
. rocess
P
. ossibly
U
. nsafe T
. ype
C
. hecking
(
. user)
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 4 / 23

10. Introduction Background Extensions Type Inference Implementation Results Conclusion
The Mycroft-O’Keefe Type System
.
Example 1
..
:- type list(A) ---> [] ; [A|list(A)].
:- pred append(list(A),list(A),list(A)).
append([],L,L).
. append([X|L1],L2,[X|L3]) :- append(L1,L2,L3).
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 5 / 23

11. Introduction Background Extensions Type Inference Implementation Results Conclusion
The Mycroft-O’Keefe Type System
.
Example 1
..
:- type list(A) ---> [] ; [A|list(A)].
:- pred append(list(A),list(A),list(A)).
append([],L,L).
. append([X|L1],L2,[X|L3]) :- append(L1,L2,L3).
Prolog code
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 5 / 23

12. Introduction Background Extensions Type Inference Implementation Results Conclusion
The Mycroft-O’Keefe Type System
.
Example 1
..
:- type list(A) ---> [] ; [A|list(A)].
:- pred append(list(A),list(A),list(A)).
append([],L,L).
. append([X|L1],L2,[X|L3]) :- append(L1,L2,L3).
Prolog code Type constructor
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 5 / 23

13. Introduction Background Extensions Type Inference Implementation Results Conclusion
The Mycroft-O’Keefe Type System
.
Example 1
..
:- type list(A) ---> [] ; [A|list(A)].
:- pred append(list(A),list(A),list(A)).
append([],L,L).
. append([X|L1],L2,[X|L3]) :- append(L1,L2,L3).
Prolog code Type constructor Type signature
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 5 / 23

14. Introduction Background Extensions Type Inference Implementation Results Conclusion
Towards Typed Prolog (2009)
Implementation of the Mycroft-O’Keefe type system
For SWI-Prolog and YAProlog
Type checking
Interaction between typed and untyped code
Typed Prolog vs untyped Prolog
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 6 / 23

15. Introduction Background Extensions Type Inference Implementation Results Conclusion
Type inference by abstract interpretation
(1990s)
A
. bstract
P
. redicate A
. bstraction
R
. epresentation
fi
.x
Extension with type union
Able to type metapredicates (univ/2)
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 7 / 23

16. Introduction Background Extensions Type Inference Implementation Results Conclusion
ActionScript and Erlang
ActionScript:
Gradual type systems: mixing static and dynamic type
checking
User has control over portion of program statically checked
Recently added type inference
Erlang:
Dialyzer tool for static analysis
Gradually applied Dialyzer to Wrangler
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 8 / 23

17. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (1)
⊤
integer float
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 9 / 23

18. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (1)
⊤
c
number
t
t
t
) t
”
integer float
t
t
t
t
”
)
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 9 / 23

19. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (1)
⊤
c
number
t .
t Example 2
t ..
) t
” foo(42).
integer float . foo(4.2).
t
t
t
t
”
)
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 9 / 23

20. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (1)
⊤
c
number
t .
t Example 2
t ..
) t
” foo(integer).
integer float . foo(float).
t
t
t
t
”
)
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 9 / 23

21. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (1)
⊤
c
number
t .
t Example 2
t ..
) t
”
foo(number).
integer float .
t
t
t
t
”
)
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 9 / 23

22. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (2)
⊤
number atom
t £
t £
) t
”
integer float £
£
t £
t £
t c£
”
t
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 10 / 23

23. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (2)
⊤
c
atomic
t
t
)
t
”
number atom
t £
t £
) t
”
integer float £
£
t £
t £
t c£
”
t
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 10 / 23

24. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (2)
⊤
c
atomic
t
t .
)
t
” Example 3
number atom ..
t £ foo(42).
t £ foo(4.2).
) t
”
integer float £ . foo(bar).
£
t £
t £
t c£
”
t
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 10 / 23

25. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (2)
⊤
c
atomic
t
t .
)
t
” Example 3
number atom ..
t £ foo(integer).
t £ foo(float).
) t
”
integer float £ . foo(atom).
£
t £
t £
t c£
”
t
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 10 / 23

26. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (2)
⊤
c
atomic
t
t .
)
t
” Example 3
number atom ..
t £ foo(number).
t £
) t
”
integer float £ . foo(atom).
£
t £
t £
t c£
”
t
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 10 / 23

27. Introduction Background Extensions Type Inference Implementation Results Conclusion
Extensions (2)
⊤
c
atomic
t
t .
)
t
” Example 3
number atom ..
t £ foo(atomic).
t £
) t
”
integer float £ .
£
t £
t £
t c£
”
t
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 10 / 23

28. Introduction Background Extensions Type Inference Implementation Results Conclusion
The struct type
.
lookup/3
..
.
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 11 / 23

29. Introduction Background Extensions Type Inference Implementation Results Conclusion
The struct type
.
lookup/3
..
:- type list(A) --- [] ; [A|list(A)].
.
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 11 / 23

30. Introduction Background Extensions Type Inference Implementation Results Conclusion
The struct type
.
lookup/3
..
:- type list(A) --- [] ; [A|list(A)].
lookup([], ,0).
. lookup([X’=’Y|T],X,Y) :- atom(X),integer(Y),lookup(T, , ).
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 11 / 23

31. Introduction Background Extensions Type Inference Implementation Results Conclusion
The struct type
.
lookup/3
..
:- type list(A) --- [] ; [A|list(A)].
:- pred lookup(list(struct),atom,integer).
lookup([], ,0).
. lookup([X’=’Y|T],X,Y) :- atom(X),integer(Y),lookup(T, , ).
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 11 / 23

32. Introduction Background Extensions Type Inference Implementation Results Conclusion
The struct type
.
lookup/3
..
:- type list(A) --- [] ; [A|list(A)].
:- type pair(A,B) --- A (=) B.
:- pred lookup(list(struct),atom,integer).
lookup([], ,0).
. lookup([X’=’Y|T],X,Y) :- atom(X),integer(Y),lookup(T, , ).
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 11 / 23

33. Introduction Background Extensions Type Inference Implementation Results Conclusion
The struct type
.
lookup/3
..
:- type list(A) --- [] ; [A|list(A)].
:- type pair(A,B) --- A (=) B.
:- pred lookup(list(pair(atom,integer),atom,integer)).
lookup([], ,0).
. lookup([X’=’Y|T],X,Y) :- atom(X),integer(Y),lookup(T, , ).
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 11 / 23

34. Introduction Background Extensions Type Inference Implementation Results Conclusion
Subtyping relation
⊤
d
© ‚
d
atomic struct
t ¢
t ¢
) t
” ¢
number atom
¢
t ¢
t ¢
)
t
” ¢
integer float
¢
 t ¢
 t
 ¢
 ”
t c ¢

~

¢
⊥
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 12 / 23

35. Introduction Background Extensions Type Inference Implementation Results Conclusion
The cult of the bound variable
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 13 / 23

36. Introduction Background Extensions Type Inference Implementation Results Conclusion
The cult of the bound variable
Why even bother doing type inference?
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 13 / 23

37. Introduction Background Extensions Type Inference Implementation Results Conclusion
The cult of the bound variable
Why even bother doing type inference?
Similar method to “Type inference by abstract interpretation”
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 13 / 23

38. Introduction Background Extensions Type Inference Implementation Results Conclusion
The cult of the bound variable
Why even bother doing type inference?
Similar method to “Type inference by abstract interpretation”
Infrastructure taken by “Towards Typed Prolog”
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 13 / 23

39. Introduction Background Extensions Type Inference Implementation Results Conclusion
The cult of the bound variable
Why even bother doing type inference?
Similar method to “Type inference by abstract interpretation”
Infrastructure taken by “Towards Typed Prolog”
Type variables vs Prolog variables
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 13 / 23

40. Introduction Background Extensions Type Inference Implementation Results Conclusion
The cult of the bound variable
Why even bother doing type inference?
Similar method to “Type inference by abstract interpretation”
Infrastructure taken by “Towards Typed Prolog”
Type variables vs Prolog variables
A
. 2 T
.2
. .
A
. 1 T
.1
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 13 / 23

41. Introduction Background Extensions Type Inference Implementation Results Conclusion
The idea
.
General Idea
..
repeat
For a predicate H, let {H1 , . . . , Hn } be the heads of the
clauses that define it
for all Hi ∈ {H1 , . . . , Hn } do
Let Hi : −Bi be the respective clause of H
for all B ∈ Bi do
Find the type of B and accumulate variable-type bindings
for the entire clause
end for
The new type for H is the intersection of the types found
for each Hi
end for
. until No change is made to the inferred type for H
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 14 / 23

42. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Type
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

43. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 1
Type
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

44. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 1
Type
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

45. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 1
member(A,list(A))
Type
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

46. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 1
member(A,list(A))
Type
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

47. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 1
member(A,list(A)) member(B,list(C))
Type
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

48. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 1
member(A,list(A)) member(B,list(C))
Type member(A,list(A))
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

49. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 1
Type member(A,list(A))
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

50. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 2
Type member(A,list(A))
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

51. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 2
Type member(A,list(A))
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

52. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 2
member(A,list(A))
Type member(A,list(A))
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

53. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 2
member(A,list(A))
Type member(A,list(A))
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

54. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 2
member(A,list(A)) member(B,list(B))
Type member(A,list(A))
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

55. Introduction Background Extensions Type Inference Implementation Results Conclusion
An example
.
Example 4: member/2
..
:- type list(A) --- [] ; [A|list(A)].
member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).
Iteration 2
member(A,list(A)) member(B,list(B))
Type member(B,list(B))
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 15 / 23

56. Introduction Background Extensions Type Inference Implementation Results Conclusion
Overview
Compile Load Execute
XSB
Disk
.P
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 16 / 23

57. Introduction Background Extensions Type Inference Implementation Results Conclusion
Overview
User
1
c
Compile Load Execute
XSB
Disk
.P
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 16 / 23

58. Introduction Background Extensions Type Inference Implementation Results Conclusion
Overview
User
1
c
Compile Load Execute
XSB
Disk
Type 2
Analysis ' .P
XSB
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 16 / 23

59. Introduction Background Extensions Type Inference Implementation Results Conclusion
Overview
User
1
c
Compile Load Execute
XSB T
3 Disk
Type 2
Analysis ' .P
XSB
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 16 / 23

60. Introduction Background Extensions Type Inference Implementation Results Conclusion
Overview
User
1
c
Compile Load Execute
XSB T 4
3 Disk
Type 2
Analysis ' .P
XSB E .xwam
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 16 / 23

61. Introduction Background Extensions Type Inference Implementation Results Conclusion
Overview
User
1
c
Compile Load Execute
XSB T 4 T
3 Disk
Type 2
Analysis ' .P 5
XSB E .xwam
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 16 / 23

62. Introduction Background Extensions Type Inference Implementation Results Conclusion
Overview
User
1
c
Compile Load 6 E Execute
XSB T 4 T
3 Disk
Type 2
Analysis ' .P 5
XSB E .xwam
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 16 / 23

63. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling
.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 17 / 23

64. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling
.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f
. (21).
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 17 / 23

65. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling
.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(21).
f(42).
.
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 17 / 23

66. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling
.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(21).
f(42).
f(84).
.
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 17 / 23

67. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling with Answer Subsumption
Answer Subsumption
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 18 / 23

68. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling with Answer Subsumption
.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
Answer Subsumption
Lattice operation: max/3
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 18 / 23

69. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling with Answer Subsumption
.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(21).
. Answer Subsumption
Lattice operation: max/3
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 18 / 23

70. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling with Answer Subsumption
.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(42).
. Answer Subsumption
Lattice operation: max/3
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 18 / 23

71. Introduction Background Extensions Type Inference Implementation Results Conclusion
Tabling with Answer Subsumption
.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(84).
. Answer Subsumption
Lattice operation: max/3
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 18 / 23

72. Introduction Background Extensions Type Inference Implementation Results Conclusion
The code (1)
.
Type inference: tabling with answer subsumption
..
type inference n(Head,Type) :-
copy term1(Head,Type),
numbervars(Head,0, ).
type inference n(Head,Type) :-
copy term(Head,Head1),
clause(Head1,Body),
get env(Body,Env),
compute type(Env,Head1,Type1),
copy term(Type1,Type),
. numbervars(Type,0, ).
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 19 / 23

73. Introduction Background Extensions Type Inference Implementation Results Conclusion
The code (2)
.
Lattice operation: unification
..
type unify(X,Y,X1) :-
unnumbervars(X,0,X1),
unnumbervars(Y,0,Y1),
X1 = Y1,
. numbervars(X1,0, ).
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 20 / 23

74. Introduction Background Extensions Type Inference Implementation Results Conclusion
Results (1)
Name LOC # Def # Inf T(s)
lib/pairlist.P 125 10 10 0.006
lib/lists.P 515 59 59 0.022
lib/ugraphs.P 850 90 90 0.102
lib/assoc xsb.P 566 49 48 0.029
lib/ordsets.P 422 39 39 0.023
examples/tree1k.P 2046 1 1 0.105
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 21 / 23

75. Introduction Background Extensions Type Inference Implementation Results Conclusion
What’s next?
D
. eclaration Free
T .
. est/Debug
M
. odules
M
. etapredicates
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 22 / 23

76. Introduction Background Extensions Type Inference Implementation Results Conclusion
Thanks for your attention :-)
Spyros Hadjichristodoulou David S. Warren Impl. of a Gradual Pol. Type System with St. Subtyping for Prolog 23 / 23