1. Modular Answer Set Programming

2. Introduction One common answer set programming (ASP) methodology is to: Encode the problem Enumerate possible solutions Test and eliminate non-solutions Size of state space of the problem can limit ASP solvers to find solutions in a reasonable amount of time. Can we somehow split the representation and solve in parts?

3. Illustration of Splitting Consider an AnsProlog program  =E 1  E 2  T 1  T 2, where E 1, E 2 are enumeration parts and T 1, T 2 are test parts. 1. Split  into  1 = T 1  E 1 and  2 = T 2  E 2. 2. Compute answer sets of  1. 3. For each answer set A i of  1, compute the answer sets of  2  A i.

4. Illustration of Splitting Assume that T 1 enumerates n binary atoms and T 2 enumerates m binary atoms, and T 1  E 1 has only a small number k of solutions A i. Finding answer sets of  could involve a search space of 2 m+n. As for splitting,  1 can involve search space of 2 n and k search spaces of 2 m for  2  A i.

5. Idea – modular ASP A declarative language that builds on top of AnsProlog. Large programs can be broken into modules. Enhance readability and project management. Communication between modules are done by “import rules”.

6. ASP modules An ASP rule is of the form: An ASP module consists of: a name, a set of parameters, a collection of ASP rules, and a collection of ASP import rules.

7. Instantiation of Modules Suppose P is the name of an ASP module and P has the following rule: date(1..N). P(N=9) is an instantiation of P which indicates that the value of N becomes 9 in module P.

8. Modular ASP by examples Suppose P 1 is the following ASP module that contains the predicate p(X,Y,Z), which is declared as for export. % module P1 p(c,d,a). p(e,f,a) :- not p(g,h,a). p(g,h,a) :- not p(e,f,a). Module P 1 has two answer sets: { p(c,d,a), p(e,f,a)} and { p(c,d,a), p(g,h,a)}

9. Import rule (a) Import predicates that exist in all answer sets of module P to module P’, based on some criteria. Example: Suppose P 2 is an ASP module that contains an import rule r of the form (a) that refers to P 1. % import rule r q(X,Y) :- P1.p(X,Y,a). % answer sets of P1 are { p(c,d,a), p(e,f,a)} and { p(c,d,a), p(g,h,a)} The predicate p(c,d,a) exists in all answer sets of P 1. Therefore, the extracted set of r is {q(c,d)}.

10. Import rule (b) Import predicates that exist in any answer sets of module P to module P’, based on some criteria. Suppose P 3 is an ASP module that contains an import rule r of the form (b) that refers to P 1. % import rule r *q(X,Y) :- P1.p(X,Y,a). % answer sets of P1 are {p(c,d,a), p(e,f,a)} and { p(c,d,a), p(g,h,a)} By filtering the answer sets of P 1 using r, the extracted set of r is { { q(c,d), q(e,f) }, { q(c,d), q(g,h) } }.

11. Import rule (c) Import predicates that exist in any answer sets of module P to module P’ based on some criteria, and numerate the predicates to form a singleton set of predicates. Suppose P 4 is an ASP module that contains an import rule r of the form (c) that refers to P 1. % import rule r q(#,X,Y) :- P1.p(X,Y,a). % answer sets of P1 are {p(c,d,a), p(e,f,a)} and { p(c,d,a), p(g,h,a)} The extracted set of r is { q(1,c,d), q(1,e,f), q(2,c,d), q(2,g,h) }.

12. ACC example Description of ACC college basketball tournament problem Nine college teams play over a period of nine weeks in the Two games per week: one on a weekday and one on a weekend On each game day: 8 teams play either at home or away Double round-robin: every team plays against every other team twice, once at home and once away. Mirroring dates: teams who play on day d will play again on day d+9. 2 9*8*9 atoms of order of schedules!

13. ACC example Attempted to encode the problem using a direct encoding approach No returned solutions by the currently most efficient AnsProlog solvers - Smodels and DLV Encode the problem using ASP modules Solutions are returned in a reasonable amount of time

14. Conclusion Modular ASP is a simple extension to AnsProlog. Can write AnsProlog modules and import from one module to another. Difficult problems can be solved with certain modularization. Future Work: incorporate different ASP solvers and other declarative languages such as Prolog to modular ASP.