Epistemic Logic Programs: A Study of Some Properties

Abstract

Epistemic Logic Programs (ELPs), extend Answer Set Programming (ASP) with epistemic operators. The semantics of such programs is provided in terms of world views, which are sets of belief sets. Different semantic approaches propose different characterizations of world views. Recent work has introduced semantic properties that should be met by any semantics for ELPs, like the Epistemic Splitting Property, that, if satisfied, allows to modularly compute world views in a bottom-up fashion, analogously to ‘traditional’ ASP. We analyze the possibility to change the perspective, shifting from a bottom-up to a top-down approach to splitting. Our new definition: (i) copes with concerns regarding unfoundedness of world views and subjective constraint monotonicity; (ii) is provably applicable to many of the existing semantics; (iii) operates similarly to “traditional” ASP; (iv) provably coincides with the bottom-up notion of splitting at least on the class of Epistemically Stratified Programs (which are, intuitively, those where the use of epistemic operators is stratified).

A Epistemic Logic Programs: Useful Properties

Following [1], an (abstract) semantics $\mathcal {S}$ is a function mapping each program $\varPi$ into sets of $\mathcal{{S}}$-world view of $\varPi$, i.e., sets of sets of objective literals, where if $\varPi$ is an objective program, then the unique member of $\mathcal{{S}}(\varPi )$ is the set of its stable models. Drawing inspiration from the Splitting Theorem [17], an analogous properties is defined for ELPs:

Definition 1 (Epistemic splitting set. [1, Definition 4])

A set of atoms $U \subseteq At$ is said to be an epistemic splitting set of a program $\varPi$ if for any rule r in $\varPi$ one of the following conditions hold: (i) $Atoms (r) \subseteq U$; (ii) $Body_{obj} (r) \cup Head (r))\ \cap \ U = \emptyset$. An epistemic splitting of $\varPi$ is a pair $\langle B_U(\varPi ),T_U(\varPi )\rangle$ satisfying $B_U(\varPi ) \cap T_U(\varPi ) = \emptyset$, $B_U(\varPi ) \cup T_U(\varPi ) = \varPi$, and also that all rules in $B_U(\varPi )$ satisfy (i) and all rules in $T_U(\varPi )$ satisfy (ii).

Intuitively, condition (ii) means that the top program $T_U(\varPi )$ may refer to atoms in U which occur as heads of rules in the bottom $B_U(\varPi )$, only through epistemic operators.

Epistemic splitting can be used, similarly to ‘traditional’ Lifschitz &Turner splitting, for iterative computation of world views. Indeed, Cabalar et al. [1] propose to compute first the world views of the bottom program $B_U(\varPi )$ and, for each of them, simplify the corresponding subjective literals in the top part. Given an epistemic splitting set U for $\varPi$ and a set of interpretations W, they define the subjective reduct of the top with respect to W and signature U, called $E_U(\varPi ,W)$. This operator, according to [1] considers all subjective literals L occurring in $T_U(\varPi )$, such that the atoms occurring in them belong to $B_U(\varPi )$. In particular, L will be substituted by $\top$ in $E_U(\varPi ,W)$ if $W \models L$, and by $\bot$ otherwise. So, $E_U(\varPi ,W)$ is a version of $T_U(\varPi )$ where some subjective literal, namely those referring to the bottom part of the program, have been simplified as illustrated.

Definition [1, Definition 5]

Given a semantics $\mathcal {S}$, a pair $\langle W_b,W_t \rangle$ is said to be an $\mathcal {S}$-solution of $\varPi$ with respect to an epistemic splitting set U if $W_b$ is a $\mathcal {S}$-world view of $B_U(\varPi )$ and $W_t$ is a $\mathcal {S}$-world view of $E_U(\varPi ,W_b)$.

The definition is parametric w.r.t. $\mathcal {S}$, as each different semantics $\mathcal {S}$ will define in its own way the $\mathcal {S}$-solutions for a given U and $\varPi$. So, world views of the entire program will be obtainable by suitably combining some world view of the bottom with some world view of the top, i.e., the world views of the entire program should be obtained as (where $I_b$ and $I_t$ are answer sets occurring respectively in $W_b$ and $W_t$): $W_b \sqcup W_t = \{ I_b \cup I_t | I_b \in W_b \wedge I_t \in W_t \}$. Therefore, the following property can be stated:

Property (Epistemic splitting. [1, Property 4])

A semantics $\mathcal {S}$ satisfies epistemic splitting if, for any epistemic splitting set U of any given program $\varPi$: W is an $\mathcal {S}$-world view of $\varPi$ iff there is an $\mathcal {S}$-solution $\langle W_b,W_t \rangle$ of $\varPi$ with respect to U such that $W = W_b \sqcup W_t$.

As discussed in [1], many semantics do not satisfy this property, which is satisfied only by the very first semantics of ELPs, proposed in [12] (and in some of its generalizations), and by Founded Autoepistemic Equilibrium Logic (FAEEL), defined in [2]. Epistemic splitting property implies subjective constraint monotonicity.

Another interesting property is foundedness. Again, such a notion has been extended from objective programs (see [1, Definition 15]). Intuitively, a set X of atoms is unfounded w.r.t. a (objective) program $\varPi$ and an interpretation I, if for every $A \in X$ there is no rule of r by which A might be derived, without incurring in positive circularities and without forcing the derivation of more than one atom from the head of a disjunctive rule (see, e.g., [15] for a formal definition). For ELPs one has to consider that unfoundedness can originate also from positive dependencies on positive subjective literals, like, e.g., in the program $A \leftarrow {\textbf {K}}A$. Among the existing semantics, only FAEEL satisfies foundedness.

Definition [1, Definition 6] [Epistemic Stratification]

We say that an ELP $\varPi$ is epistemically stratified if we can assign an integer mapping $\lambda : At \rightarrow N$ to each atom [occurring in the program] such that:

• $\lambda (a) = \lambda (b)$ for any rule $r \in \varPi$ and atoms $a,b \in ( Atoms (r) \setminus Body_{subj} (r))$, and

• $\lambda (a) > \lambda (b)$ for any pair of atoms a, b for which there is a rule $r \in \varPi$ with $a \in ( Head(r) \cup Body_{obj} (r))$ and $b \in Body_{subj} (r)$.