Answer Set Programming with External Sources

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Άιμον Τρικούπη
6 χρόνια πριν
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1 Answer Set Programming with External Sources Christoph Redl September 4, 2012 Redl C. (TU Vienna) HEX-Programs September 4, / 14

2 ASP-Programs ASP Solver Answer Sets Rules: a 1 a n b 1,..., b m, not b m+1,..., not b n, Redl C. (TU Vienna) HEX-Programs September 4, / 14

3 HEX-Programs HEX Answer Sets Solver Plugin Rules: External atom: a 1 a n b 1,..., b m, not b m+1,..., not b n, &p[q 1,..., q k ](t 1,..., t l ) &p[q 1,..., q k ](t 1,..., t l ) = true f &p (A, q 1,..., q k, t 1,..., t l ) = 1 Redl C. (TU Vienna) HEX-Programs September 4, / 14

4 HEX-Programs &rdf addr( /data1.rdf). addr( /data2.rdf). bel(x, Y) addr(u), &rdf [U](X, Y, Z). Redl C. (TU Vienna) HEX-Programs September 4, / 14

5 HEX-Programs &rdf &diff addr( /data1.rdf). addr( /data2.rdf). bel(x, Y) addr(u), &rdf [U](X, Y, Z). dom(x) #int(x). nsel(x) dom(x), &diff [dom, sel](x). sel(x) dom(x), &diff [dom, nsel](x). sel(x1), sel(x2), sel(x3), X1 X2, X1 X3, X2 X3. Redl C. (TU Vienna) HEX-Programs September 4, / 14

6 Evaluation Translation Π: ˆΠ: p(c 1 ). dom(c 1 ). dom(c 2 ). dom(c 3 ). p(x) dom(x), &empty[p](x). p(c 1 ). dom(c 1 ). dom(c 2 ). dom(c 3 ). p(x) dom(x), e &empty[p] (X). e &empty[p] (X) e &empty[p] (X) dom(x). 8 candidates, e.g.: {Tp(c 1 ), Tp(c 2 ), Tdom(c 1 ), Tdom(c 2 ), Tdom(c 3 ), Fe &empty[p] (c 1 ), Te &empty[p] (c 2 ), Fe &empty[p] (c 3 )} Redl C. (TU Vienna) HEX-Programs September 4, / 14

7 Evaluation Conflict-driven SAT/ASP Solving C = {C 1 : { a, b}, C 2 : { a, c, i}, C 3 : { b, c, d}, C 4 : { d, e, j}, C 5 : { d, e, k}, C 6 : { e, f }, C 7 : {a, g, l}, C 8 : {a, h}, C 9 : { g, h, m}} j=f (3) C4 b=t e=t a=t C1 C3 d=t C4 C6 û C2 C3 C5 C6 c=t f=t C2 C5 i=f (1) k=f (3) Redl C. (TU Vienna) HEX-Programs September 4, / 14

8 Evaluation Conflict-driven SAT/ASP Solving C = {C 1 : { a, b}, C 2 : { a, c, i}, C 3 : { b, c, d}, C 4 : { d, e, j}, C 5 : { d, e, k}, C 6 : { e, f }, C 7 : {a, g, l}, C 8 : {a, h}, C 9 : { g, h, m}} j=f (3) C4 b=t e=t a=t C1 C3 d=t C4 C6 û C2 C3 C5 C6 c=t f=t C2 C5 i=f (1) k=f (3) Redl C. (TU Vienna) HEX-Programs September 4, / 14

9 Evaluation Conflict-driven SAT/ASP Solving C = {C 1 : { a, b}, C 2 : { a, c, i}, C 3 : { b, c, d}, C 4 : { d, e, j}, C 5 : { d, e, k}, C 6 : { e, f }, C 7 : {a, g, l}, C 8 : {a, h}, C 9 : { g, h, m}} j=f (3) C4 b=t e=t a=t C1 C3 d=t C4 C6 û C2 C3 C5 C6 c=t f=t C2 C5 i=f (1) k=f (3) Redl C. (TU Vienna) HEX-Programs September 4, / 14

10 Evaluation Conflict-driven SAT/ASP Solving C = {C 1 : { a, b}, C 2 : { a, c, i}, C 3 : { b, c, d}, C 4 : { d, e, j}, C 5 : { d, e, k}, C 6 : { e, f }, C 7 : {a, g, l}, C 8 : {a, h}, C 9 : { g, h, m}, C 10 : { a, i, j, k}} j=f (3) C4 b=t e=t a=t C1 C3 d=t C4 C6 û C2 C3 C5 C6 c=t f=t C2 C5 i=f (1) k=f (3) Redl C. (TU Vienna) HEX-Programs September 4, / 14

11 Evaluation: External Behavior Learning (EBL) Input Output {Clause} Redl C. (TU Vienna) HEX-Programs September 4, / 14

12 Evaluation: External Behavior Learning (EBL) Input Output {Clause} General Case &diff [p, q](x), ext(p, A) = {a, b}, ext(q, A) = {a, c} p(a) p(b) p(c) q(a) q(b) q(c) e &diff [p,q] (b) { p(a), p(b), p(c), q(a), q(b), q(c), e &diff [p,q] (b)} Redl C. (TU Vienna) HEX-Programs September 4, / 14

13 Evaluation: External Behavior Learning (EBL) &diff monotonic Monotonicity &diff [p, q](x), ext(p, A) = {a, b}, ext(q, A) = {a, c} p(a) p(b) p(c) q(a) q(b) q(c) e &diff [p,q] (b) { p(a), p(b), p(c), q(a), q(b), q(c), e &diff [p,q] (b)} Functionality &concat[ab, c](x) { e &concat[ab,c] (abc), e &concat[ab,c] (ab)} Redl C. (TU Vienna) HEX-Programs September 4, / 14

14 Evaluation: External Behavior Learning (EBL) &diff monotonic Monotonicity &diff [p, q](x), ext(p, A) = {a, b}, ext(q, A) = {a, c} p(a) p(b) p(c) q(a) q(b) q(c) e &diff [p,q] (b) { p(a), p(b), p(c), q(a), q(b), q(c), e &diff [p,q] (b)} Functionality &concat[ab, c](x) { e &concat[ab,c] (abc), e &concat[ab,c] (ab)} Redl C. (TU Vienna) HEX-Programs September 4, / 14

{ p(a), p(b), q(a), q(b), q(c), e &diff [p,q] (b)} Functionality &concat[ab, c](x) { e

15 Evaluation: External Behavior Learning (EBL) &diff monotonic Monotonicity &diff [p, q](x), ext(p, A) = {a, b}, ext(q, A) = {a, c} p(a) p(b) q(a) q(b) q(c) e &diff [p,q] (b) { p(a), p(b), q(a), q(b), q(c), e &diff [p,q] (b)} Functionality &concat[ab, c](x) { e &concat[ab,c] (abc), e &concat[ab,c] (ab)} Redl C. (TU Vienna) HEX-Programs September 4, / 14

16 Evaluation: Minimality Check Example Π: &g: dom(a).dom(b). p(a) dom(a), &g[p](a). p(b) dom(b), &g[p](b). {b}, {a} {a}, {b}, {a, b} {a, b} A = {Tdom(a), Tdom(b), Tp(a)} = Π But FLP-reduct f Π A = {r Π A = B(r)}: dom(a).dom(b). p(a) dom(a), &g[p](a). A = {Tdom(a), Tdom(b)} = f Π A Redl C. (TU Vienna) HEX-Programs September 4, / 14

17 Evaluation: Minimality Check Unfounded Sets a b b a. c b. a c. c a b b a. c b. a. c Redl C. (TU Vienna) HEX-Programs September 4, / 14

Benchmarks all AS first AS n 5 6 7 8 9 10 11 12 13... 20 explicit 10.9 94.3 +EBL 4.3 34.8 266.1 UFS 0.2 0.3 0.8 1.8 4.5 11.9 32.4 92.1 273.9 +EBL 0.1 0.1 0.2 0.2 0.3 0.4 0.

47 1.13 0.70 0.62 6 4.57 2.90 1.52 1.27 7 19.99 10.50 3.64 2.77 8 80.63 39.01 9.46 6.94 9 142.95 80.66 30.12 20.97 10 240.46 122.81 107.14 63.

18 Benchmarks all AS first AS n explicit EBL UFS EBL explicit EBL UFS EBL Figure: Set Partitioning A B C D n all AS first AS explicit UFS explicit UFS A B Figure: Argumentation Redl C. (TU Vienna) HEX-Programs September 4, / 14

19 Benchmarks ASP <XML> n explicit UFS check plain +EBL plain +EBL +UFL n all AS first AS explicit UFS explicit UFS plain +EBL plain +EBL +UFL plain +EBL plain +EBL +UFL Figure: MCSs Redl C. (TU Vienna) HEX-Programs September 4, / 14

20 HEX-Programs HEX Answer Sets Solver Plugin Redl C. (TU Vienna) HEX-Programs September 4, / 14

21 References Eiter, T., Ianni, G., Schindlauer, R., and Tompits, H. (2005). A uniform integration of higher-order reasoning and external evaluations in answer-set programming. In In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI-05, pages Professional Book. URL: Faber, W. (2005). Unfounded sets for disjunctive logic programs with arbitrary aggregates. In In Logic Programming and Nonmonotonic Reasoning, 8th International Conference (LPNMR 05), 2005, pages Springer Verlag. Faber, W., Leone, N., and Pfeifer, G. (2004). Recursive aggregates in disjunctive logic programs: Semantics and complexity. In In Proceedings of European Conference on Logics in Artificial Intelligence (JELIA, pages Springer. Gelfond, M. and Lifschitz, V. (1991). Classical negation in logic programs and disjunctive databases. New Generation Computing, 9: URL: Redl C. (TU Vienna) HEX-Programs September 4, / 14

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