On Abductive Equivalence Katsumi Inoue National Institute of Informatics Chiaki Sakama Wakayama University MBR

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.

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.

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?

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

Abductive framework (L, B, H)  L : language and logic  B : background knowledge  H : possible hypotheses Given an observation O, E is an explanation of O in (L, B, H) iff E belongs to H and  B U E ┣ L O  B U E is consistent.

Abductive equivalence: Definition 1 Two abductive frameworks (L 1, B 1, H 1 ) and (L 2, B 2, H 2 ) are explainably equivalent if, for any observation O, there is an explanation of O in (L 1, B 1, H 1 ) iff there is an explanation of O in (L 2, B 2, H 2 ). Explainable equivalence requires that two abductive frameworks have the same explainability for any observation.

Abductive equivalence: Definition 2 Two abductive frameworks (L, B 1, H) and (L, B 2, H) are explanatorily equivalent if, for any observation O, E is an explanation of O in (L, B 1, H) iff E is an explanation of O in (L, B 2, 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.

Example  A 1 = ( FOL, B 1, {a,b}) and A 2 = ( FOL, B 2, {a,b}) where B 1 : a → p B 2 : b → p  A 1 and A 2 are explainably equivalent.  A 1 and A 2 are not explanatorily equivalent.  A 3 = ( FOL, B 1, {b}) and A 4 = ( FOL, B 2, {b}) are not explainably equivalent.

Example  A 1 =( FOL, B 1, {a,b}), A 2 =( FOL, B 2, {a,b}), A 3 =( FOL, B 3, {a,b}), where B 1 : a → p, b → a B 2 : a → p, b → p, b → a B 3 : b → p, b → a  A 1, A 2 and A 3 are explainably equivalent.  A 1 and A 2 are explanatorily equivalent.  A 1 and A 3 are not explanatorily equivalent. In fact, {a} is an explanation of p in A 1 but is not in A 3.

Results in first-order logic Theorem: ( FOL, B 1, H) and ( FOL, B 2, H) are explainably equivalent iff ∃ H 1,H 2 ⊆ H such that B 1 U H 1 ≡ B 2 U H 2. Corollary: If (B 1 ＼ B 2 ) U (B 2 ＼ B 1 ) ⊆ H, then ( FOL, B 1, H) and ( FOL, B 2, H) are explainably equivalent. Corollary: If B 1 ≡ B 2 then ( FOL, B 1, H) and ( FOL, B 2, H) are explainably equivalent. Theorem: ( FOL, B 1, H) and ( FOL, B 2, H) are explanatorily equivalent iff B 1 ≡ B 2.

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.

Answer Set Programming Program ： r ← p r ← q p ← not q q ← not p Answer Sets: { p, r }, { q, r } 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.

Strong equivalence in ASP [Maher 88; Lifschitz, Pearce & Valverde 01]  Programs P 1 and P 2 are strongly equivalent if for any program R, P 1 U R and P 2 U R have the same answer sets.  E.g. P 1 : p ← q. q ←. P 2 : p ←. q ←. are strongly equivalent, but P 3 : p ← not q. P 4 : p ←. are not strongly equivalent.

Relative strong equivalence [Inoue and Sakama, JELIA’04]  Let P 1 and P 2 be programs, and R be a set of rules.  P 1 and P 2 are strongly equivalent with respect to R if AS(P 1  R) = AS(P 2  R) for any R ⊆ R.  Relative equivalence restricts the language of additional sets or rules as contexts.

Results in nonmonotonic programs Theorem: ( NMP, B 1, H) and ( NMP, B 2, H) are explanatorily equivalent iff B 1 and B 2 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].

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.