WLPE12

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


Types in Prolog?



Types in Prolog?

.
Theorem (Most Prolog Books)
..
Prolog is an untyped language...
.



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)



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?



Our vision
A Gradual Polymorphic type system with Subtyping



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)



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).




.
Example 1
..

append([],L,L).

Prolog code




.
Example 1
..

append([],L,L).

Prolog code Type constructor




.
Example 1
..

append([],L,L).

Prolog code Type constructor Type signature



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



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)



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



Extensions (1)
⊤

integer float

⊥



Extensions (1)
⊤

c
number
t
t
t

) t
”
integer float
t
t
t
t
”
)
⊥



Extensions (1)
⊤

c
number
t .
t Example 2
t ..

) t
” foo(42).
integer float . foo(4.2).
t
t
t
t
”
)
⊥



Extensions (1)
⊤

c
number
t .
t Example 2
t ..

) t
” foo(integer).
integer float . foo(float).
t
t
t
t
”
)
⊥



Extensions (1)
⊤

c
number
t .
t Example 2
t ..

) t
”
foo(number).
integer float .
t
t
t
t
”
)
⊥



Extensions (2)
⊤

number atom
t £
t £

) t
”
integer float £
£
t £
t £
t c£
”
t
⊥



Extensions (2)
⊤

c
atomic
t
t
)
t
”
number atom
t £
t £

) t
”
integer float £
£
t £
t £
t c£
”
t
⊥



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
⊥



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
⊥



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
⊥



Extensions (2)
⊤

c
atomic
t
t .
)
t
” Example 3
number atom ..
t £ foo(atomic).
t £

) t
”
integer float £ .
£
t £
t £
t c£
”
t
⊥



The struct type

.
lookup/3
..

.



The struct type

.
lookup/3
..
:- type list(A) --- [] ; [A|list(A)].

.



The struct type

.
lookup/3
..

lookup([], ,0).
. lookup([X’=’Y|T],X,Y) :- atom(X),integer(Y),lookup(T, , ).



The struct type

.
lookup/3
..

:- pred lookup(list(struct),atom,integer).

lookup([], ,0).



The struct type

.
lookup/3
..
:- type pair(A,B) --- A (=) B.
:- pred lookup(list(struct),atom,integer).

lookup([], ,0).



The struct type

.
lookup/3
..
:- type pair(A,B) --- A (=) B.
:- pred lookup(list(pair(atom,integer),atom,integer)).

lookup([], ,0).



Subtyping relation
⊤
d

© ‚
d
atomic struct
t ¢
t ¢

) t
” ¢
number atom
¢
t ¢
t ¢
)
t
” ¢
integer float
¢
t ¢
t
¢
”
t c ¢

~

¢
⊥



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




Type variables vs Prolog variables

A
. 2 T
.2
. .
A
. 1 T
.1



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



An example

.
Example 4: member/2
..

member(X,[X| ]).
. member(X,[ |T]) :- member(X,T).

Type



An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 1

Type



An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 1
member(A,list(A))
Type



An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 1
member(A,list(A)) member(B,list(C))
Type



An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 1
member(A,list(A)) member(B,list(C))
Type member(A,list(A))



An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 1




An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 2




An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 2
member(A,list(A))



An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 2
member(A,list(A)) member(B,list(B))



An example

.
Example 4: member/2
..

member(X,[X| ]).
Iteration 2
member(A,list(A)) member(B,list(B))
Type member(B,list(B))



Overview

Compile Load Execute

XSB

Disk
.P



Overview

User
1
c

XSB

Disk
.P



Overview

User
1
c

XSB

Disk
Type 2
Analysis ' .P
XSB



Overview

User
1
c

XSB T

3 Disk
Type 2
Analysis ' .P
XSB



Overview

User
1
c

XSB T 4

3 Disk
Type 2
Analysis ' .P
XSB E .xwam



Overview

User
1
c

XSB T 4 T

3 Disk
Type 2
Analysis ' .P 5
XSB E .xwam



Overview

User
1
c
Compile Load 6 E Execute

XSB T 4 T

3 Disk
Type 2
Analysis ' .P 5
XSB E .xwam



Tabling

.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..



Tabling

.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f
. (21).



Tabling

.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(21).
f(42).
.



Tabling

.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(21).
f(42).
f(84).
.



Tabling with Answer Subsumption

Answer Subsumption




.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
Answer Subsumption
Lattice operation: max/3




.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(21).
. Answer Subsumption




.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(42).




.
|?- f(X).
..
f(21).
f(42).
f(84).
.
.
Global Table
..
f(84).



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, ).



The code (2)

.
Lattice operation: uniﬁcation
..
type unify(X,Y,X1) :-
unnumbervars(X,0,X1),
unnumbervars(Y,0,Y1),
X1 = Y1,
. numbervars(X1,0, ).



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



What’s next?

D
. eclaration Free

T .
. est/Debug

M
. odules

M
. etapredicates



Thanks for your attention :-)


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