Abduction

satisfies a set of integrity constraints I C, then there exists a least model of the weak completion of P that satisfies I C.

Proof.

This follows immediately from the fact that the model intersection property holds for logic programs under their weak completion under three-valued Lukasiewicz semantics.

2.5 Abduction

We will mainly focus on three-valued abduction and briefly show the correspondence to two-valued abduction. Based on (Kakas, Kowalski and Toni 1993), an abductive framework is a quadruplehP,A,I C,|=i, consisting of a programP as knowledge base, a finite set of abducibles A_P, a finite set of integrity constraintsI C, and a consequence relation |=. A two-valued abductive framework is the quadruple hP,A_2,P,I C,|=TPi, whereP is definite,|=T_P is the consequence relation with respectTP and A_2,P is defined as

{A← > |A∈undef(P)}.

Clauses inI C are of the form⊥ ←body. Observation Ois a non-empty set of literals.

Definition 2.1. Let hP,A_2,P,I C,|=TPi be a two-valued abductive framework where P satisfies I C, E ⊆ A_2,P and O is an observation.

O is two-valued explained by E given P and I C iff P∪E |=T_P O and P∪E |=T_P I C. O is two-valued explainable given P and I C iff there exists an E

such that O is two-valued explained by E given P and I C.

Normally, only set inclusion minimal (or otherwise preferred) explanations are con-sidered. We assume henceforth that explanations are minimal, that means, there ex-ists no other explanation E⁰ ⊂ E for O. Someone might possibly think of some other preference criterion instead. Note that if P |=T_P O thenE is empty.

Similarly, for the three-valued semantics considered in this thesis, we define a three-valued abductive framework as a quadruple hP,A,I C,|=wcsi, consisting of a programP as knowledge base, a set of abducibles A, a set of integrity constraints I C, and the logical consequence relation |=_wcs. Observation O is a non-empty set of literals. As we are employing the Weak Completion Semantics, abducibles may now not only be facts, but can also take the form of assumptions, otherwise they remain unknown. Therefore, the set of abducibles A_P available for the three-valued abduction is extended with the corresponding assumptions.

A_P = {A← > |A∈undef(P)} ∪ {A← ⊥ |A∈undef(P)}

Proposition 2.3. Given a definite program P, the following holds:

If {A← >} ⊆A_2,P then {A← >, A← ⊥} ⊆A_P. Proof.

This follows immediately from the definitions forA_2,P and A_P.

Definition 2.2. Let hP,A_P,I C,|=_wcsi be a three-valued abductive framework where P satisfiesI C,E ⊆ A_P and O is an observation.

O isthree-valued explained by E given P andI C iffP∪E |=wcs O and P∪E |=wcs I C. O is three-valued explainable given P and I C iff there exists anE

such that O is three-valued explained by E given P and I C.

In abduction, we distinguish betweencredulousandskeptical reasoning. Credulous reaso-ning means that there exists at least one model which entails the observation to be explained. Skeptical reasoning demands that every model of the program entails the observation.

F follows skeptically from P, I C and Oiff O can be three-valued explainable given P and I C, and for allE forOit holds that P∪E |=wcs F.

F follows credulously fromP, I C and Oiff there exists a E forO and it holds that P∪E |=wcs F.

Three-valued abduction is illustrated in Example 2.8. P,O |=^s_wcs F denotes that F follows skeptically fromP andO. P,O|=^c_wcs F denotes thatF follows credulously from P andO. Note that in the case the abducibles are not abduced as facts or assumptions, they stay unknown in the least model of the weak completion. If we do not want to allow each undefined atom to be an abducible, i.e. if we want to allow for unknown and non-abducible knowledge, we can simply add the clauseA←A for any such atomA.

Proposition 2.4. Given a two-valued abductive framework hP,A_2,P,I C,|=TPi, a three-valued abductive frameworkhP,A_P,I C,|=_wcsi, where P is definite, E ⊆ A_2,P and observation O is a non-empty set of literals. The following holds:

1. If E is a two-valued explanation for O given P and I C then E is an explanation forO givenP and I C. 2. If O is two-valued explained given P and I C

then O is three-valued explained given P and I C. Proof.

(2) follows from (1), so we show that (1) holds. Let us assume thatE is a two-valued ex-planation for O given P and I C, then P ∪E |=TP O and P ∪E |=TP I C. To show:

P∪E |=_wcs Oand P∪E |=_wcs I C.

2.5. Abduction 1. P∪E |=_wcs O follows fromP ∪E |=T_P O and Proposition 2.1.

2. P ∪E |=wcs I C means thatlm2(P ∪E∪I C) is satisfiable. This implies that the body of all clauses in I C is mapped to false in lm₂(P∪E). If they were true in lm wc(P∪E), then, according to Proposition 2.1, they would also have to be true in lm2(P ∪E). Therefore, the body of all clauses in I C is false in lm wc(P ∪E).

Thus,P∪E |=_wcs I C.

The other direction does not hold. Consider program P, which consists of one clause:

p←q.

Given O = {¬p}, the only three-valued explanation is E = {q ← ⊥}, where E ∈ A_P. However, E 6∈A_2,P and therefore E cannot be a two-valued explanation forO.

As in the following we will mainly consider three-valued semantics, we implicitly assume all the abductive frameworks and explanations to be three-valued, if not explicitly stated otherwise.³ The entailment relations |=^s_wcs and |=^c_wcs are abbreviations for expressing that a formula follows skeptically or credulously, respectively.

3Lu´ıs Moniz Pereira observed that, different to the classical TP based definition, under the Weak Completion Semantics we can also allow for both facts and assumptions in two-valued abduction. (per-sonal communication, February 10, 2016)

Example 2.8. Consider program P consisting of the following three clauses:

p(X) ← ¬q(X)∧r(X)∧t(X).

p(X) ← ¬s(X)∧r(X).

t(a) ← >.

Assume thatI C =∅and thatO={p(a)}. gP consists of the following three clauses:

p(a) ← ¬q(a)∧r(a)∧t(a).

p(a) ← ¬s(a)∧r(a).

t(a) ← >.

Let us consider this observation in the three-valued abductive framework hP,A_P,I C,|=wcsi, where the set of abducibles A_P consists of the following facts and assumptions:

q(a) ← >.

q(a) ← ⊥.

r(a) ← >.

r(a) ← ⊥.

s(a) ← >.

s(a) ← ⊥.

There are two (minimal) explanations forO:

E_rq = { r(a)← >, q(a)← ⊥ } and E_sr = { s(a)← ⊥, r(a)← > }.

Asr(a) follows from all (minimal) explanations, it follows skeptically fromP andO, whereas¬q(a) and ¬s(a) only follow credulously.