On Abductive Equivalence

1. On Abductive Equivalence Katsumi Inoue National Institute of Informatics Chiaki Sakama Wakayama University MBR 2004. 12. 17

2. Computational issues on abductive inference • Nowadays, abduction is used in many AI applications, e.g., diagnosis, design, discovery. • Abduction is an important paradigm for problem solving, and is incorporated in programming technologies, i.e., abductive logic programming (ALP). • Automated abduction is also studied in the literature as an extension of deductive methods or a part of inductive systems.

3. Issues not fully understood yet • Evaluation of abductive power in ALP. • Understanding of the semantics of ALP with respect to modularity and contexts. • Efficiency in abductive reasoning, e.g., simplification, optimization. • Debugging and verification in ALP. • Standardization in ALP. • All these issues are related to different notions of equivalence in ALP.

4. When two ALPs are equivalent? • No definition in the literature of ALP. • No such concepts in philosophy, either? • When can we consider that an explanation E is equivalent to an explanation F for an observation? • When can we say that an observation G is equivalent to an observation H in an abductive framework? • In what circumstances, can we say that abduction by person A is equivalent to abduction by person B? • When can we regard that abduction with knowledge P is equivalent to abduction with knowledge Q?

5. Considerable parameters ... • World • background knowledge • observations • Agent who performs abduction • her logic of background knowledge • language, syntax • semantics • axioms, inference procedure • her logic of hypotheses/observation • language, syntax • logic of explanation entailment • criteria of best explanations

6. Abductive framework • (L, B, H) • L : language and logic • B : background knowledge • H : possible hypotheses • Given an observation O, E is an explanationof O in (L, B, H) iffE belongs to H and • B U E ┣L O • B U E is consistent.

7. Abductive equivalence: Definition 1 • Two abductive frameworks (L1, B1, H1) and (L2, B2, H2) are explainably equivalent if, for any observation O, there is an explanation of O in (L1, B1, H1) iff there is an explanation of O in (L2, B2, H2). • Explainable equivalence requires that two abductive frameworks have the same explainability for any observation.

8. Abductive equivalence: Definition 2 • Two abductive frameworks (L, B1, H) and (L, B2, H) are explanatorily equivalent if, for any observation O, E is an explanation of O in (L, B1, H) iff E is an explanation of O in (L, B2, H). • Explanatory equivalence assures that two abductive frameworks have the same explanation power for any observation. • Explanatory equivalence implies explainable equivalence. • Note: L and H must be common.

9. Example • A1 = (FOL, B1, {a,b}) and A2 = (FOL, B2, {a,b})where B1 : a → p B2 : b → p • A1 andA2 are explainably equivalent. • A1 andA2 are not explanatorily equivalent. • A3 = (FOL, B1, {b}) and A4 = (FOL, B2, {b}) are not explainably equivalent.

10. Example • A1=(FOL, B1, {a,b}), A2=(FOL, B2, {a,b}), A3=(FOL, B3, {a,b}), where B1 : a → p, b → a B2 : a → p, b → p, b → a B3 : b → p, b → a • A1, A2 andA3 are explainably equivalent. • A1 andA2 are explanatorily equivalent. • A1 andA3 are not explanatorily equivalent. In fact, {a} is an explanation of p in A1 but is not in A3.

11. Results in first-order logic • Theorem: (FOL, B1, H) and (FOL, B2, H) are explainably equivalent iff ∃H1 ,H2⊆H such that B1 UH1 ≡B2 U H2. • Corollary: If (B1＼B2) U(B2＼B1) ⊆H, then (FOL, B1, H) and (FOL, B2, H) are explainably equivalent. • Corollary: If B1 ≡B2 then (FOL, B1, H) and (FOL, B2, H) are explainably equivalent. • Theorem: (FOL, B1, H) and (FOL, B2, H) are explanatorily equivalent iff B1 ≡B2 .

12. Nonmonotonic effects • Addition of hypotheses may invalidate explanations of some observations if the background theory is nonmonotonic. • Nonmonotonic reasoning often appears in logic programming through negation as failure. • Abductive equivalence with nonmonotonic background theory is more complicated than in the case of FOL.

13. Answer Set Programming A logic program is regarded as the constraints to be satisfied by solutions. Each solution is obtained by computing an answer set of the program. Program：　 r　←　p r　←　q p ←not q q ←not p Answer Sets: { p , r }, { q , r }

14. Strong equivalence in ASP [Maher 88; Lifschitz, Pearce & Valverde 01] • Programs P1 and P2 are strongly equivalentif for any program R, P1 U R and P2 U R have the same answer sets. • E.g.P1: p ← q. q ← . P2: p ←. q ← . are strongly equivalent, but P3 : p ← not q. P4: p ←. are not strongly equivalent.

15. Relative strong equivalence[Inoue and Sakama, JELIA’04] • Let P1 and P2 be programs, and R be a set of rules. • P1 and P2are strongly equivalent with respect toR if AS(P1 R) = AS(P2  R)for anyR ⊆R . • Relative equivalence restricts the language of additional sets or rules as contexts.

16. Results in nonmonotonic programs • Theorem: (NMP, B1, H) and (NMP, B2, H) are explanatorily equivalent iff B1 andB2 are strongly equivalent with respect to H. • Theorem:Deciding explanatory equivalence is coNP-complete. • Further generalization of abductive equivalence in which removal of hypotheses is allowed in extended abduction [Inoue and Sakama, IJCAI’95] can be characterized by the notion of update equivalence [Inoue and Sakama, JELIA’04].

17. Discussion • Equivalence of abductive theories should be attracted more attention to. • Abductive equivalence is complicated. • Logical equivalence of background theories does not necessarily implies abductive equivalence (except for some simple cases). • In future work, further parameters should be considered.