Partial Abductive Explanations

Luciano Caroprese1, Irina Trubitsyna1, Mirosław Truszczynski2, and Ester Zumpano1

1 DEIS, Universita della Calabria, 87030 Rende, Italycaroprese|irina|zumpano@deis.unical.it

2 Department of Computer Science, University of Kentucky, Lexington, KY 40506, USAmirek@cs.uky.edu

Abstract. This paper stems from the work in [3, 4] in which a new mea-sure of the simplicity of an explanation based on its degree of arbitrari-ness is proposed: the more arbitrary the explanation, the less appealingit is, with explanations having no arbitrariness — called constrained— being the preferred ones. In that previous work, as commonly donein the literature of abductive logic programming, a set of hypothesisis not an explanation, unless it is complete, i.e. it explains all the databelonging to the observation. In this paper we follow a different per-spective. We recognize that completeness is a laudable goal, but alsoobserve that it is rarely accomplish in realistic situations. In this paperwe propose an extension of [3, 4] by defining the concept of partial con-strained explanations, i.e. constrained explanations that do not followthe completeness requirement, but admit some indefiniteness.A partial constrained explanation captures the intuition of the existenceof an explanation (partial explanation) that would best explain the givenevidence, while no longer making arbitrary choices (constrained expla-nation).

1 Introduction

Abductive reasoning describes the process of discovering explanations that would entaila given observation. It basically consists of inferring updates to the set of abducibles –facts that can be confirmed or negated – so that the updated knowledge base, the logicprogram, the integrity constraints and the observation “agree.” Each update that yieldsan agreement constitutes a possible explanation of the observation.

A general characteristic of abductive reasoning is the existence of multiple abduc-tive explanations, which are typically not equally compelling. Therefore, identifying asubclass, possibly narrow, of “preferred explanations” is an important problem. Follow-ing the Occam’s principle, a typical approach is to identify as “preferred” those expla-nations that are, in some sense, simple. Several concepts of simplicity were consideredin the literature. They range from the standard one based on the subset minimality,to its versions and refinements that require minimum cardinality, minimum weight, orminimality under prioritization of individual hypotheses [8].

This paper stems from the work in [3, 4] in which a new measure of the simplicityof an explanation based on its degree of arbitrariness is proposed: the more arbitrarythe explanation, the less appealing it is, with explanations having no arbitrariness —

Fig. 1. A target practice game.

called constrained — being the preferred ones. A constrained explanation connectsthe structural information present in the theory and the knowledge embedded in theobservation in a non-arbitrary (constrained) way – it “invents” no new entities andmakes no arbitrary assumptions. Constrained explanations change the knowledge base“minimally” and avoid making arbitrary commitments.

In our previous work in [3, 4], as commonly done in the literature of abductive logicprogramming, a set of hypothesis is not taken as an explanation unless it is complete.In this paper we follow a different perspective. We recognize that completeness is alaudable goal, but also observe that it is rarely accomplish in realistic situations: frommedicine to science it is not typically the case that everything is known in the causesof an observation. This paper proposes an extension of the approach from [3, 4] aim-ing to allow partial explanations, i.e. explanations that do not embed the completenessrequirement, but admit some indefiniteness. The proposal of partial constrained expla-nations can be thought of as a means to capture the intuition of the existence of anexplanation (partial explanation) that would best explain the given evidence, while nolonger making arbitrary choices (constrained explanation).

Let’s now introduce a running example that intuitively presents the basic conceptsof partial constrained explanations.

Example 1. Let us consider a target practice game in which we have a target, a player,a rifle and a judge (see Figure1). The practice field is a square ABCD (whose edges areAB, BC, CD and DA). The length of each edge is 5, and the value of the distance ofthe target from the edge DA is 2. The target is along the edge AB, the player and the

rifle are along the edge CD and the judge is along the adge BC. The positions of thetarget, the player and the rifle are expressed by means of three real values that representthe distances from the edge DA. Similarly, the positions of the judge is expressed by areal value that represent the distances from the edge CD. It is imposed the uniquenessof the target, the player, the rifle and the judge, i.e. each of them can be in just oneposition. If we observe a shot, then the player and the rifle are in the same position(shot← player(X),ri f le(X)). If we observe an alignment, then the rifle and the targetare at the same distance from the edge DA (alignment← ri f le(X), target(X)). A victoryis sanctioned by the presence of the judge. Therefore, if we observe a victory, then wehave a shot, an alignment, and the judge in some position along the edge (victory←shot,alignment, judge(X)). Given an observation we have to abduce a set of updatesof the abducibles, facts whose predicate symbol is player,ri f le and judge (abduciblepredicates)1 such that the updated theory and the observation agree.

Even assuming that the target cannot be moved (as it does not belong to the set ofadbucibles) we may have multiple explanations:

– A set of explanations of the form E i = {player(2),ri f le(2), judge(i)}, each ofthem assigning to the judge an established position, i.

– An explanation E ⊥ = {player(2),ri f le(2), judge(⊥)}.

Let us look more carefully at these two types of explanations. Each explanation E iselects an arbitrary position in the domain, with no particular reason to choose oneover another. This simple observation is at the basis of the intuition that each of theseexplanation is arbitrary. On the other hand, the explanation E ⊥ connects the structuralinformation present in the program and the knowledge provided by the observation in anon-arbitrary (constrained) way — it “invents” no new entities and makes no arbitraryassumptions. This explanation (together with the description of the situation), entailsthe observation and intuitively represents a constrained explanation. Therefore, in thespecific case the only way to obtain a constrained explanation consists in allowing apartial explanation containing the indefinite fact judge(⊥). �

In this paper we study the framework of abductive logic programming extendedwith integrity constraints in the setting in which both the initial knowledge base and theabductive explanations are indefinite (may contain occurrences of null values) and thedomain is possibly infinite. We denote the null value by⊥ and interpret it as expressingthe existence of unknown values for each attribute it is used for [19]. The benefit of theintroduction of partial explanations is evident in all cases in which it is sufficient to besure of the existence of a value, but is not relevant to know it. The present proposal alsoprovides interesting results on the complexity of problems concerning the existence ofpartial constrained explanations.

1 Note that, for the sake of simplicity of presentation (in the sequel we will remove this hypothe-sis) we do not allow target among the abducible predicates. This simply means that we cannotmove the target, i.e. that its position is fixed at 2.

2 Basic concepts and notation

We consider a finite set Π of relation symbols and fix a set D of constants that includea designated element ⊥, called the null value. We write L for the resulting first-orderlanguage. Unless explicitly stated otherwise, we assume that D is an infinite set. How-ever, in several examples, for the sake of simplicity, we assume that D is a finite set.Some predicates in Π are designated as base (or extensional) predicates and all the re-maining ones are understood as derived (or intensional) predicates. We denote the setsof base and derived predicates by Π b and Π d , respectively.

A term is a constant from D or a variable. An atom is an expression p(t1, . . . , tk),where p∈Π is a predicate symbol of arity k and ti’s are terms. If p∈Π b then p(t1, . . . , tk)is a base atom; otherwise it is a derived atom. An atom is ground if it does not containvariables. We refer to ground atoms as facts. We denote the set of all facts by At. Wewrite Atb and Atd for the sets of facts defined in terms of base and derived predicates,respectively. Clearly, At = Atb∪Atd .

Facts may contain occurrences of ⊥. A fact (ground atom) is definite if it doesnot contain any occurrences of ⊥. Otherwise, a fact is indefinite. We denote the set ofdefinite facts by Atdef .

Let a = p(t1, . . . , tk) and b = p(t ′1, . . . , t′k) be facts. We say that b is at least as indef-

inite as a, denoted b ≽ a, if for every i, 1 ≤ i ≤ k, t ′i = ti or t ′i =⊥. If b ≽ a and b = a,then b is more indefinite than a, denoted b≻ a. Finally, if there is a definite fact c suchthat a ≽ c and b ≽ c, then a and b are compatible. We denote that by a ≈ b. For a setI ⊆ At, we define

I ⇑ = {a ∈ At | there is b ∈I s.t. a≽ b}I ⇓ = {a ∈ At | there is b ∈I s.t. b≽ a}I ↑ = {a ∈ At | there is b ∈I s.t. a≻ b}I ↓ = {a ∈ At | there is b ∈I s.t. b≻ a}I ≈ = {a ∈ At | there is b ∈I s.t. b≈ a}.

To illustrate these notions, let us assume D = {⊥,1,2}. We then have{q(1,⊥)}⇑ = {q(1,⊥),q(⊥,⊥)}{q(1,⊥)}⇓ = {q(1,⊥),q(1,1),q(1,2)}{q(1,⊥)}≈ = {q(1,⊥),q(1,1),q(1,2),q(⊥,1),q(⊥,2),q(⊥,⊥)}[{q(1,⊥)}⇓]⇑ = {q(1,⊥),q(1,1),q(1,2),q(⊥,1),q(⊥,2),q(⊥,⊥)}.

In particular, [{q(1,⊥)}⇓]⇑ = {q(1,⊥)}≈. It is not a coincidence. One can show that,in general, I ≈ = [I ⇓]⇑.

Following Ullman [27], we use the standard terminology and talk about Datalog¬

programs, rules, bodies and heads of rules. A rule is safe if each variable occurring inthe head or in a negative literal in the body also occurs in a positive literal in the body. ADatalog¬ program is safe if each rule is safe. We assume that programs do not containoccurrences of⊥. The semantics of Datalog¬ programs is given in terms of answer sets[6, 7].

Integrity constraints are an essential component of knowledge bases. We will con-sider integrity constraints of the form ∀X [A1∧ . . .∧Ak→∃Y (B1∨ . . .∨Bm)], where Aiand Bi are atoms with no occurrences of⊥, built of base and built-in predicates (if any),and where every variable occurring in the constraint belongs to X ∪Y and occurs insome atom Ai built of a base predicate.

3 Indefinite Knowledge Base

In the most common case, (extensional) knowledge bases are finite subsets of At thatcontain definite facts only. The semantics of such knowledge base is given by the closed-world assumption [24]: a definite fact q is true in a knowledge base I if q ∈I . Other-wise, q is false in I . We are, however, interested in a more general setting of knowledgebases containing indefinite facts, too.

Definition 1. [INDEFINITE KNOWLEDGE BASE] An indefinite knowledge base (know-ledge base, for short) is a finite subset of At such that there are no atoms a,b in theknowledge base with a≻ b (elements of I are ≻-incomparable).

Indefinite facts are meant to represent the existence of an object, without fully de-scribing its attributes. Having the definite fact employee(Dan,Paris) in a companyknowledge base tells us that the company has an employee named Dan, who worksin the branch of the company located in Paris. On the other hand, the indefinite factemployee(Dan,⊥) tells us only that the company employs Dan and that he is assignedto one of its branches. Intuitively, if a ∈ At is true then each fact b at least as indefiniteas a (b≽ a) is true as well. For instance, if employee(Dan,Paris) is true, all indefinitefacts employee(Dan,⊥), employee(⊥,Paris) and employee(⊥,⊥) are true, as well.In the other direction, only a weaker inference is possible. For example, if we knowemployee(Ann,⊥) is true, then we know that Ann is employed, but we do not knowwhere. Thus, we treat each definite fact employee(Ann,c) as unknown.

The role of the ≻-incomparability requirement in Definition 1 is to eliminate re-dundancy. If a knowledge base contains employee(Dan,Paris), then it implies thatemployee(Dan,⊥) holds and so, there is no need to store the latter.

We will now formally define the truth value of a fact wrt a knowledge base.

Definition 2 (TRUTH VALUE OF A FACT WRT A KNOWLEDGE BASE). Let I be aknowledge base, possibly containing indefinite facts, and let q ∈ At be a fact also pos-sibly indefinite.

1. q is true in I , denoted by I |= a, if q ∈I ⇑ (that is, q is in I or is more indefinitethan an element of I ).

2. q is unknown in I , if q∈I ≈ \I t (that is, if q is compatible with a fact mentionedin I but is not true).

3. q is false in I , denoted I |= not a, if q ∈ At\I ≈ (that is, q is not compatible withany element in I ).

The sets of all facts that are true, unknown and false in I are denoted by I t ,I u andI f , respectively. �

From the above definition, we have I t = I ⇑ and I u = I ≈ \I ⇑.Example 2. Let us assume that D = {1,2,⊥} and I = {p(1,⊥), p(⊥,1)}. Then, I isa knowledge base as p(1,⊥) and p(⊥,1) are ≻-incomparable. We have

I t = {p(⊥,⊥), p(1,⊥), p(⊥,1)}

I ≈ = {p(⊥,⊥), p(1,⊥), p(1,1), p(1,2), p(⊥,1), p(2,1), p(2,⊥), p(⊥,2)}.

Thus, p(⊥,⊥), p(1,⊥) and p(⊥,1) are true in I while p(1,1), p(1,2), p(2,1), p(2,⊥)and p(⊥,2) are unknown in I . The only remaining fact, p(2,2), is false in I . �

Next, we define a possible world, that is, a possible two-valued realization of athree-valued knowledge base I .

Definition 3. [POSSIBLE WORLD] A set W of definite facts is a possible world fora knowledge base I if I t ⊆W⇑ (W “explains” all facts that are true in I ), andW ⊆ I ⇓ (only definite facts that are compatible with I can appear in a possibleworld, as all other definite facts are false in I ). �

Indefinite knowledge bases represent sets of possible worlds. However, if a know-ledge base contains no indefinite facts, it has only one possible world, itself. For aknowledge base I , we write W (I ) for the family of all possible worlds for I .

Since a possible world W consists of definite atoms only, it can be regarded asa knowledge base and we can talk about truth values with respect to W . Due to theabsence of indefinite facts in W , each atom is either true (if it belongs to W⇑) or false(otherwise). Extending the notation we introduced earlier, for a possible-world W anda ∈ At we write W |= a if a ∈W⇑ and W |= not a, otherwise.

The following proposition shows that it is possible to characterize the semantics ofa knowledge base in terms of its possible worlds.

Proposition 1. Let I be a knowledge base and q a fact.

1. q ∈I t if and only if W |= q, for every W ∈W (I )2. q ∈I f if and only if W |= q, for every W ∈W (I ) �

4 Partial Constrained Abductive Explanations

Definition 4 (ABDUCTIVE THEORY). An abductive theory T over a vocabulary Π isa tuple ⟨P,I ,A ,C ⟩, where

– P is a finite logic program over Π– I is an indefinite knowledge base over Πb– A ⊆Πb is a finite set of base predicate names called the abducible predicates– C is a finite set of integrity constraints over Π �

Informally, the program P and the integrity constraints C provide a model (de-scription) of the problem domain while I is the initial knowledge base. The integrityconstraints in C impose domain constraints on abducible and non-abducible predicatesin the language. According to the definition, information about abducible predicatesis given in terms of ground facts. They explicitly specify the extensions of abduciblepredicates. We refer to ground atoms based on abducible predicates as abducibles. Anobservation is a set of ground facts. An observation may “agree” with the theory T ,but if it does not, we assume that this “disagreement” is caused by the incorrect infor-mation about the properties modeled by the abducible predicates. Abductive reasoningconsists of inferring updates to the set of abducibles (removal of some and inclusionof some new ones) so that the updated knowledge base, the program, the integrity con-straints and the observation “agree.” Each update that yields an agreement constitutes apossible explanation of the observation.

Example 3. The target practice game informally described in Example 1 can be mod-eled by the abductive theory T = ⟨P,I ,A ,C ⟩, whereP is the logic program

shot ← player(X),ri f le(X)alignment ← ri f le(X), target(X)victory ← shot,alignment, judge(X)

I is a set containing only the fact target(2)A is a set containing the abducible predicates player, ri f le, judge and target.2

C (that impose the uniqueness of the target, the player, the rifle and the judge) containsthe integrity constraints

∀XY (target(X), target(Y )→ X = Y )∀XY (player(X), player(Y )→ X = Y )∀XY (ri f le(X),ri f le(Y )→ X = Y )∀XY ( judge(X), judge(Y )→ X = Y )

We here recall that the domain of our system is the interval [0..5]. Observe that it isinfinite. �

Definition 5 (CONSISTENT ABDUCTIVE THEORY). An abductive theory T = ⟨P,I ,A ,C ⟩ is consistent if there is at least one a possible world W for I s.t. W |= C andthe program P ∪W has answer sets. We denote the family of all those answer sets byW (T ). We call elements of W (T ) possible worlds of T .

Definition 6 (OBSERVATION). An observation is a pair O = (Ot ,O f ), where Ot andO f are disjoint sets of facts observed to be true and false, respectively. �

We define the concept of truth value of a fact wrt a theory.

Definition 7. [TRUTH VALUE OF A FACT WRT A THEORY] A fact a ∈ At is confirmedby a theory T (or a is true in T ), denoted by T |= a, if for every possible worldW ∈ W (T ), W |= a; a is negated by a theory T (or a is false in T ), denoted byT |= ¬a, if for every W ∈W (T ), W |= ¬a; a is not explained in T (or a is unknownin T ), otherwise. Given an observation O = (Ot ,O f ) we write T |= O if T confirmseach atom in Ot and negates each fact in O f . We denote the truth value of a in T byvT (a). �

Example 4. Consider the abductive theory in Example 3 modeling the target practicegame informally described in Example 1. Possible observations are:- O = ({shot}, /0), stating that the fact shot is observed as true.- O = ({shot},{target(3)}), stating that the fact shot is observed as true, whereas thefact target(3) is observed as false.- O = ({victory}, /0), stating that the fact victory is observed as true. �

2 The set of abducibles coincides with the base predicates. We have removed the restriction ofExample 1 that constraints the target to remain in its starting position 2.

Now we define the notion of update of an abductive theory. It consists in updatingthe set of abducible in the knowledge base, by removing some and including some newones, such that the updated theory agrees with the observation.

Definition 8 (THEORY UPDATE). Let T = ⟨P,I ,A ,C ⟩ be an abductive theory. Anupdate for T is ∆ = (E,F), where E and F are disjoint finite sets of abducibles andF ⊆I . We denotes as T ◦∆ the updated theory ⟨P,(I ∪E)\F,A ,C ⟩.

Example 5. Consider the abductive theory in Example 3 modeling the target practicegame informally described in Example 1. The updates:- ∆1 =({player(4), target(4)}, /0), suggests the insertion of the set of abducibles {player(4), target(4)}. The updated theory is ⟨P,I ∪{player(4), target(4)},A ,C ⟩. It is in-consistent as it does not satisfy the uniqueness of the target.- ∆2 = ({player(4), target(4)},{target(2)}), suggests the insertion of the the set of ab-ducibles {player(4), target(4)} and the removal of the the set of abducibles {target(2)},therefore it suggest to move the target from the position 2 to the position 4. The updatedtheory is ⟨P,(I ∪{player(4), target(4)})\{target(2)},A ,C ⟩. �

Each updates that yields an agreement constitutes a possible explanation of an ob-servation. In the following we formally define the basic concept of explanation avoidingembedding the completeness requirement, i.e. we define the concept of partial explana-tion.

Definition 9 (PARTIAL EXPLANATION). Let T = ⟨P,I ,A ,C ⟩ be an abductive the-ory and O = (Ot ,O f ) an observation. An update ∆ = (E,F) for T is a partial expla-nation for O wrt T if T ◦∆ |= O.

The final step consists in modeling a constrained partial explanation. In order to dothis we need some preliminary definitions.

Definition 10. Let T = ⟨P,I ,A ,C ⟩ be an abductive theory and O = (Ot ,O f ) anobservation. A constant is relevant with respect to T and O if it occurs in T , or O, orif it is ⊥. A predicate is relevant with respect to T and O if it occurs in T or in O. �

Definition 11 (PARTIAL CONSTRAINED EXPLANATION). Let T = ⟨P,I ,A ,C ⟩ bean abductive theory and O= (Ot ,O f ) an observation. A partial explanation ∆ = (E,F)of O wrt T is constrained if (i) every constant and predicate occurring in ∆ is relevantand (ii) there is no non-nullary constant a in E such that replacing some occurrencesof a with a constant b = a (b might be ⊥), results in a partial explanation of O wrt T .If ∆ is not constrained, it is arbitrary.

Intuitively, a partial constrained explanation can be thought of as a mean to cap-ture the intuition of the existence of an explanation (partial explanation) that would bestexplain the given evidence, while no longer making arbitrary choices (constrained ex-planation). Constrained explanations are non-drastic, as they minimizes the change itincurs [26]. Specifically, in Definition 11 constrained explanation are those satisfyingtwo requirements to the minimality of change: The first concern is to perform updatesthat do not introduce new predicate or constant symbols.Following the Occam’s Razor

principle to avoid introducing new entities unless necessary, we take the minimality ofthe set of new symbols as a primary consideration. Anyhow, this is not sufficient as itcould be the case that some occurrences of definite constants in a partial explanationmay still be “ungrounded” or “arbitrary,” that is, replacing them with another constantresults in a partial explanation. To avoid this, the second concern, is to perform updatesthat connect the structural information present in the theory and the knowledge em-bedded in the observation in a non-arbitrary (constrained) way — it “invents” no newentities and makes no arbitrary assumptions.

Example 6. Consider the abductive theory in Example 3 modeling the target practicegame informally described in Example 1.

If the observation is O=({shot}, /0), i.e. we observe a shot, we have an infinite num-ber of partial explanations of type E i = ({player(i),ri f le(i)}, /0) where i ∈ [0..5]. Eachof them is arbitrary because the choice of the constant i has not been constrained bythe theory and the observation. In fact, the occurrences of i can be replaced with a freeconstant ξ , and, consequently, with any other admissible constant, obtaining anotherexplanation. No constrained explanation exists in this case.

If the observation is O=({victory}, /0), then E ⊥=({player(2),ri f le(2), judge(⊥)},/0) is the unique constrained partial explanation. In fact, the value 2 is constrained by thedata already present in the knowledge base and it is used to satisfy a more complex re-quest. Observe that, any explanation of the form E i = ({player(2),ri f le(2), judge(i)},/0), meaning that the judge is in an established position, is arbitrary. At the same time,note that all the explanations of the form E i, j =({player( j),ri f le( j), target( j), judge(i)},{target(2)}) where i, j ∈ [0..5] and j = 2 are arbitrary. These set of explanations movesthe target from its initial position 2 to a new (arbitrary) position j, set both the playerand the rifle at the same position j and set a position i for the judge. Explanations inE i, j are arbitrary.

In some sense we may say that the explanation E ⊥ simply points to the existenceof a judge along the edge that sanctioned the victory. It represents an infinite set ofpossible worlds that are collapse into a single explanation, i.e. E ⊥ models existentiallyquantified formulas (along the lines of an early work by Pople [23]). However, such“existential” explanations while no longer making arbitrary choices lack specificity. �

5 Complexity

In this section, we discuss the complexity of problems concerning (constrained) partialexplanations.

We assume that the sets of base and derived predicate symbols, the program P ,the set of abducible predicate and the set of integrity constraints C are fixed. The onlyvarying parts in the problems are a knowledge base I and the observation O. That is,we consider the data complexity setting. Moreover, we assume that D = {⊥,1,2, . . .},and take = and≤, both with the standard interpretation on {1,2, . . .}, as the only built-inrelations. We restrict integrity constraints to expressions of the form:

∃X(∀Y (A1∧ . . .∧Ak→ B1∨ . . .∨Bm)) (1)

where Ai and Bi are atoms with no occurrences of ⊥ constructed of base and built-inpredicates, and where every variable occurring in the constraint belongs to X ∪Y , andoccurs in some atom Ai built of a base predicate.

We start by stating the result on the complexity of deciding the consistency of anindefinite knowledge base wrt a set of integrity constraints.

Proposition 2. Let C be a set of integrity constraints (of the form specified above). Forevery knowledge base I , I is consistent wrt C if and only if it has a possible world ofthe size bounded by a polynomial in the size of I . �

Proof. We can eliminate indefinite atoms by introducing fresh Skolem constants forevery occurrence of⊥ in I (in other words, we replace each atom p(t)∈I , where t isa sequence of elements from Dom, with p(t ′), where t ′ is obtained from t by replacingeach occurrence of ⊥ in t with a new Skolem constant). In other words we performthe standard process of Skolemization. Similarly, we replace each integrity constraint∃X∀YC with ∀YC′, where C′ is obtained from C by selecting a new Skolem constantfor each variable x ∈ X , and by replacing all occurrences of variables from X ∈C withthe appropriate Skolem constants. The number of Skolem constants we introduce in theprocess is O(|I |+ |C |). The resulting theory, say T h, has a model M with the domainDomd = Dom\{⊥} (and the natural interpretation of constants from Domd) if and onlyif I has possible worlds consistent wrt C . Let us consider a model of T h. Let D be theset of all elements from Domd that occur in I and C or that are assigned in M to theSkolem constants of T h. Clearly, by restricting the domain of M to D , keeping the wayconstants are interpreted, and restricting all the relations in M to D , we obtain a modelof T h with the domain of size O(|I |+ |C |) (in other words, a possible world of sizepolynomial in |I |+ |C |. We can also assume that all integers interpreting new Skolemconstants are not larger than Max+O(|I |+ |C |), where Max is the largest elementfrom Domd occurring in I or C . (What matters is only how those integers relate toeach other with respect to = and ≤. Thus, if there are gaps among those integers in Dthat are larger than Max, the numbers can be compressed by shifting them to the left.)�

This result allows us to show that the problem to decide whether a knowledge baseI is consistent wrt a set of integrity constraints C is in NP. A standard reduction fromSAT shows the NP-completeness.

Proposition 3. Let C be a finite set of integrity constraints. The following problem isNP-complete: given a knowledge base I , decide whether I is consistent wrt C . �

Proof. We note that ground(C ) is a collection of propositional constraints with the sizethat is polynomial in the size of I . Once we guess a set of facts W , one can checkin polynomial time that W ∈W (I ) and that W satisfies all constraints in ground(C ).Thus, the problem is in the class NP.

The NP-hardness follows by a reduction from the 3-SAT problem. It consists todecide whether a formula Φ = ∃X φ is true, where X is a set of propositional atomsand φ is a CNF formula over the set of atoms in X s.t. each clause is a disjunction of 3literals.

Given such a formula Φ , we set Π b = {truthValue,valueX ,sat,occurrence}. Thepredicate truthValue is unary and stores the truth values true and f alse. The predi-cates valueX and sat are binary and the predicate occurrence has four attributes. Weuse valueX to specify the truth value of atoms in X and sat to specify the truth value ofclauses in φ . Further, if an atom x occurs in a clause c positively (negatively, respec-tively) in the position p ∈ [1..3], we represent that by an atom occurrence(c, p,x, true)(occurrence(c, p,x, f alse), respectively). All these atoms form a representation of Φ .We define Cl as the set of clauses in φ and I to contain the following facts:

1. truthValue(true)2. truthValue( f alse)3. valueX (x,⊥) for each x ∈ X4. sat(c,⊥), c ∈Cl5. occurrence(c, p,x, true), if v occurs positively in c in position p ∈ [1..3]6. occurrence(c, p,x, f alse), if v occurs negatively in c in position p ∈ [1..3]

Next, we define C to consist of the constraints:

1. ⊥← valueX (X ,V ),not truthValue(V )2. ⊥← sat(C,V ),not truthValue(V )3. ⊥← sat(C, f alse),

occurrence(C,P,X ,V ),valueX (X ,V )4. ⊥← sat(C, true),

occurrence(C,1,X1,V1),not valueX (X1,V1),occurrence(C,2,X2,V2),not valueX (X2,V2),occurrence(C,3,X3,V3),not valueX (X3,V3),

5. ⊥← sat(C, f alse)

It can be proved that Φ is true iff I is consistent wrt C . �

We restrict our attention to stratified programs, which represent the most importantcase in practice. In this case, Proposition 2 and 4 imply the following result.

Corollary 1. Let C be a set of integrity constraints (of the form specified above) andP be stratified program. For every knowledge base I , the theory T = ⟨P,I ,A ,C ⟩has a possible world if and only if it has a possible world of the size bounded by apolynomial in the size of I . �

Thus, the problem of the consistency of a theory with a stratified program is in theclass NP. As the empty program is trivially stratified, the result mentioned above impliesNP-completeness.

Corollary 2. Let C be a finite set of integrity constraints, P a finite stratified program.The following problem is NP-complete: given a knowledge base I , decide whether thetheory T = ⟨P,I ,A ,C ⟩ is consistent. �

Now let us consider integrity constraints of the form

∀Y (A1∧ . . .∧Ak→ B1∨B2) (2)

.

Proposition 4. Let C be a finite set of integrity constraints of the form (2). The follow-ing problem is NL-complete: given a knowledge base I , decide whether I is consis-tent wrt C . �Therefore, given a knowledge base I its consistency wrt a set of integrity constraintsC of the form (2) can be checked in polynomial time.

Proposition 5. Let T = ⟨P,I ,A ,C ⟩, where P is a stratified program3, and p(X)an atom (eventually obtained by replacing the null values with distinct variables).

– p(X) is confirmed by T iff the theory ⟨P∪{µ← p(X);σ← not σ ,not µ},I ,A ,C ⟩,where µ and σ are propositional atoms, is consistent.

– p(X) is false iff the database ⟨P ∪{µ ← p(X);σ ← not σ ,µ},I ,A ,C ⟩, whereµ and σ are propositional atoms, is consistent. �It seems that the complexity does not increase. We know that the problem of verify

whether I is consistent wrt C , where C is a set of integrity constraint of the form (1)(resp. (2)) is NP-complete (resp. NL-complete). Now, having T = ⟨P,I ,A ,C ,⟩, foreach possible world W of I consistent wrt C , we have to verify that W ∪P ∪{µ ←p(X);σ ← not σ ,not µ} (or W ∪P ∪{µ ← p(X);σ ← not σ ,µ}) has a model. Thistask can be performed in polynomial time because P is stratified. It follows that check-ing whether an atom is true or false wrt D is a coNP-complete (resp. coNL-complete)problem.

Now we derive results about partial explanations.

Proposition 6. Let I be a knowledge base and C a set of integrity constraint of theform (1) (resp. (2)), E an update and O = (Ot ,O f ) an observation. The problem ofchecking whether E is a partial explanation for O wrt T is ∆ P

2 -complete (resp. NL-complete). �Proof. Let T = ⟨P,I ,A ,C ⟩ and let T ′ = T ◦E , where. In order to test that E is apartial explanation for O wrt T we need to check that T ′ is consistent and T ′ |=O. Thefirst problem is NP-complete (resp. NL-complete). The second one is coNP-complete(resp. coNL-complete - we recall that NL=coNL). �Proposition 7. The problem of checking whether an update E = (E,F) is an arbitrarypartial explanation for an observation O wrt a theory T is in Σ P

2 . �Proof. In order to test that E is an arbitrary partial explanation we need to check that1) E is a partial explanation 2) E is arbitrary. The first problem is NP-complete (seeProposition 6). The test of the second condition consists in: 2.1) verifying whetherthere is an arbitrary constant or predicate symbol in E (this task can be performed inpolynomial time) and, if no arbitrary constant or predicate symbol is found, 2.2) inguessing a non-nullary constant a presented in E and the subset of its occurrences in Eand check that replacing these occurrences of a with ξ results in a partial explanationfor O wrt T (NP-complete problem). �Corollary 3. The problem of checking whether an update E = (E,F) is a constrainedpartial explanation for an observation O wrt a theory T is in Π P

2 . �3 if P is not stratified the proposition does not hold.

6 Discussion and Conclusions

Related Works. The abduction reasoning formalism we study in the paper uses logicprograms to represent background knowledge in abductive theories. It is referred to asabductive logic programming [7, 9, 16]. Abductive Logic Programming (ALP) is recog-nized as a high level knowledge representation framework that enable to solve problemsdeclaratively based on abductive reasoning. See [5] for a nice survey.

Abduction was introduced to artificial intelligence in early 1970s by Harry Pople Jr.[23], where it is now commonly understood as the inference to the best explanation [14].Abductive reasoning has grown to be a well-established area of artificial intelligence.

Besides the use of abduction for elegantly solving a broad collection of problems,in a more general perspective it has been shown first by Eshghi and Kowalski in [9] andthen by Kakas and Mancarella in [15] and Dung in [7] that abduction can be recognizedas a semantical device to describe the non-monotonic semantics of Logic Programming.Moreover, it has been shown in [6] that abduction is a powerful tool in the field ofknowledge representation as it is suitable to represent and reasoning on incompleteinformation.

For the above mentioned reasons ALP is actually recognized as a computationalparadigm that can solve many limitations of logic programming with respect to higherlevel knowledge representation and reasoning tasks [1, 6]. ALP frameworks use abduc-tion as theory formation and differ each other with respect syntactic restrictions and onthe form of the resultant best explanation. This variety, obviously, reflects in the manykind of the abductive systems developed in the literature, in which specific algorithmsimplementing specific abductive solutions are proposed. The importance of abductivelogic programming to knowledge representation was argued by Denecker and Schreye[6]. The complexity of abductive reasoning was first studied by Bylander et al. [2]. Eiteret al. [8] studied the complexity of reasoning tasks in the abductive logic programmingsetting.

The profusion of abductive explanations was explicitly noted by Maher [20] in hiswork on constraint abduction. Maher considers a different setting and handles the prob-lem by different techniques. The key similarity is that in Maher’s setting, as in ours,symbols from the vocabulary not present in the theory can give rise to alternative ex-planations.

Besides these, a more general comparison can be done by considering the differentsemantics used to abduce solutions. In the seminal work by [9] it is proposed an ab-ductive semantics in the presence of negation as failure, equivalent to the Stable Modelsemantics [10], in which negative literals are regarded as abducibles assumptions.

Abductive explanations which allow the removal of hypotheses are first introducedby Inoue and Sakama [13].

Starting from the first work by Eshghi and Kowalski [9] different extensions sup-porting more general forms of abduction have been proposed. The need for an extensionof stable models mainly comes from the fact that, as notorious, several programs havean evident intended meaning, but they do not have any stable model. Generalizationof the Stable Model semantics of Gelfond and Lifschitz, applicable to any logic pro-gram, have been proposed in [7, 15, 17]. In [22] hypothetical reasoning is representedby the well founded semantics [28] that has the nice property of being defined for any

program, even if it only allows skeptical forms of reasoning. A declarative semantics,called Preferential Semantics, for abduction in which negation is treated as a form ofhypothesis is proposed in [7]. Dung showed that the set of complete scenario forms asemi-lattice with respect to set inclusion. Stable models correspond to maximal com-plete scenario (even if not not every maximal complete scenario represents a stablemodel), whereas the least complete scenario is captured by the well founded model.Different proposals have been provided in the literature in order to explicitly allow thepossibility of encoding disjunctive information in the consequents. In [11] it is proposedan abductive framework within extended disjunctive programs that allows disjunctionin the head and both classical negation and negation as failure. A new fixpoint seman-tics for abductive logic programs is provided in [12] in which the generalized stablemodels are characterized as the fixpoint of a disjunctive program obtained by a suit-able program transformation. The proposed technique also allows, for the first time,the removal of hypothesis.Later on a variety of approaches have been proposed withdifferent optimality (or preference criteria) criteria to identify the best solutions or ex-planations in accordance to Occam’s razor principle. The best solutions are obtained byimposing some minimality criterion on abductive explanations. Each minimality crite-rion determines a preference relation on the solution space, therefore it can be regardedas a qualitative version of probability such that the minimal explanations correspond tothe most likely ones. Different minimality criteria can be used. See [8] for a very inter-esting survey. The most commonly applied criterion is the subset-minimality (⊆) [2, 18,25]. Besides subset minimality smallest size of solutions (≤) [21] is also a commonlyapplied criterion. It states that a solution A is preferable to a solution B if |A|< |B|. Al-ternative criteria are the method of priorities that establishes priority levels over the setof solutions and the method of penalties that allows to assign a weight (penalty) to eachhypothesis and look for a solution with minimum total weight. In [3, 4] it is proposeda framework of abductive logic programming extended with integrity constraints. Forthis framework, we introduce a new measure of the goodness of an explanation basedon its degree of arbitrariness: the more arbitrary the explanation, the less appealing itis, with explanations having no arbitrariness — they are called constrained — beingthe preferred ones. As for a comparison, we point out that none of the earlier works onabduction considered the concept partial explanations.

Concluding Remarks and Directions for Further Research. In this paper we studythe framework of abductive logic programming extended with integrity constraints inthe setting in which both the original and the abductive explanations are indefinite (maycontain occurrences of null values) and the knowledge base domain is possibly infinite.We denote the null value by ⊥ and interpret it as expressing the existence of unknownvalues for each attribute it is used for [19]. The benefit of the introduction of partialexplanations is evident in all such cases in which it is sufficient to be sure of the exis-tence of a value, but is not relevant to know it. This proposal stems from the work in[3, 4] in which a new measure of the simplicity of an explanation based on its degreeof arbitrariness is proposed: the more arbitrary the explanation, the less appealing it is,with explanations having no arbitrariness — called constrained — being the preferredones. Nevertheless, in our previous work, as commonly done in the literature of abduc-tive logic programming, a set of hypothesis does not concur to become an explanation,

unless it is complete, i.e. it explains all the data belonging to the observation. This workfollows a different perspective, we recognize that completeness is a laudable goal, butalso observe that it is rarely accomplish in realistic situations, from medicine to sci-ence, it is unrealistic, even for the ’best theory’, to explain everything given the knowncauses. As an example in the medicine field doctors often need to made diagnoses evenwhen some symptoms remain enigmatic. In this paper we propose an extension of thepaper in [3, 4] in order to admit partial explanations, i.e. explanations that do not embedthe completeness requirement, but admit some indefiniteness. The proposal of partialconstrained explanations can be thought of as a mean to capture the intuition of theexistence of an explanation (partial explanation) that would best explain the given ev-idence, while no longer making arbitrary choices (constrained explanation). Importantissues, that could actually translate our research into practical applications, are left forfurther research. First, the study of the proposed framework in the presence of particu-lar types of constraints, those implemented and maintained in commercial DBMS, suchas primary keys and foreign key constraints, for which the complexity of computingconstrained explanations is expected to reduce. Moreover, further research is plannedto investigate our abduction approach in the context of policy analysis and authorizationpolicies in order to detect unauthorized access or compromized security.

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