2 Preliminaries

In this section, we briefly introduce some basics of possibilistic (normal) logic programs (poss-NLPs) and fix the notations that will be used in the paper (Nicolas et al. Reference Nicolas, Garcia, Stéphan and Lefèvre2006). For a set $O$ , $\vert O \vert$ denotes cardinality of $O$ .

2.1 Possibilistic logic programs

We first introduce the syntax of poss-NLPs in this subsection. We assume a propositional logic $\mathcal{L}$ over a finite set $\mathcal{A}$ of atoms. Literals (positive and negative), terms, clauses, models, and satisfiability are defined as usual. A possibilistic formula $\overline {\phi }$ is a pair $(\phi ,\alpha )$ with $\phi$ being a (propositional) formula of $\mathcal L$ and $\alpha$ being the weight of $\phi$ . In general, this $\alpha$ is an element of a given lattice $({\mathcal{Q}},\le )$ (Dubois and Prade Reference Dubois and Prade1998). In this paper, $({\mathcal{Q}},\le )$ is assumed to be a totally ordered set where $\mathcal{Q}$ is finite. Intuitively, every pair of weights in the finite set $\mathcal{Q}$ of a given induction task is comparable.

For example, ${\mathcal{Q}}= \{ 0.1, 0.6, 0.7, 1 \}$ and $\leq$ is the ‘less than or equal’ relation for real numbers. The supremum of $\mathcal{Q}$ in this $({\mathcal{Q}},\le )$ is $1$ . Except for a finite set of decimals in $[0, 1]$ , $\mathcal{Q}$ can also be a set of adverbs representing necessities. For instance, ${\mathcal{Q}}= \{ slightly, highly, extremely, absolutely \}$ and $slightly \leq highly \leq extremely \leq absolutely$ . The supremum of $\mathcal{Q}$ in this $({\mathcal{Q}},\le )$ is $absolutely$ . Hereafter, we use $\mu$ to denote the supremum of $\mathcal{Q}$ in $({\mathcal{Q}},\le )$ .

The possibilistic formula $(\phi ,\alpha )$ intuitively states that the ordinary formula $\phi$ is certain at least to the level $\alpha$ . If $\phi$ is an atom (resp. a literal, a clause and a term) then $(\phi ,\alpha )$ is a possibilistic atom (resp. literal, clause and term). Hereafter, we use the symbol $X$ to denote the classic projection ignoring all uncertainties from its possibilistic counterpart  $\overline {X}$ . Given a set $\mathcal{A}$ of atoms and a complete lattice $(\mathcal{Q}, \le )$ , a set $\overline {I}\subseteq {\mathcal{A}}\times {\mathcal{Q}}$ of possibilistic atoms is called a possibilistic interpretation over $\mathcal{A}$ and $(\mathcal{Q},\le )$ if $\vert \{ (x,\alpha ) \in \overline {I} \mid \alpha \in {\mathcal{Q}} \} \vert \leq 1$ for each $x \in {\mathcal{A}}$ . Thus, for a possibilistic interpretation $\overline {I}$ , $(x,\alpha _1)\in \overline {I}$ and $(x,\alpha _2)\in \overline {I}$ imply $\alpha _1=\alpha _2$ . Given a possibilistic interpretation $\overline {I}$ , we use $I$ to denote the classic interpretation $\{p\mid (p,\alpha )\in \overline {I} \}$ .

Before introducing possibilistic logic programs, we recall the set operations for finite sets of possibilistic formulas.

Let $\overline {\Gamma _1}$ and $\overline {\Gamma _2}$ be two finite sets of possibilistic formulas.

• • $\overline {\Gamma _1} \sqsubseteq \overline {\Gamma _2}$ if for each $(\phi ,\alpha )\in \overline {\Gamma _1}$ there exists $(\phi ,\beta )\in \overline {\Gamma _2}$ such that $\alpha \le \beta$ . Consequently, $\overline {\Gamma _1} \sqsubset \overline {\Gamma _2}$ if $\overline {\Gamma _1} \sqsubseteq \overline {\Gamma _2}$ and $\overline {\Gamma _1} \neq \overline {\Gamma _2}$ .

• • $ \overline {\Gamma _1}\sqcup \overline {\Gamma _2}=\{(x, \alpha )\mid (x,\alpha )\in \overline {\Gamma _1}, x \notin \Gamma _2\}\cup \{(x,\beta )\mid (x,\beta )\in \overline {\Gamma _2},x \notin \Gamma _1\}\cup \{(x,\max \{\alpha ,\beta \})\mid (x,\alpha )\in \overline {\Gamma _1}, (x,\beta )\in \overline {\Gamma _2}\}$ .

• • $ \overline {\Gamma _1}\sqcap \overline {\Gamma _2}=\{(x, \min \{\alpha ,\beta \})\mid (x,\alpha )\in \overline {\Gamma _1}, (x,\beta )\in \overline {\Gamma _2}\}$ .

A possibilistic normal logic program (poss-NLP for short) on a complete lattice $({\mathcal{Q}},\le )$ is a finite set of possibilistic normal rules (or rules) of the form $\overline {r} = (r,\alpha )$ , where

• • $\alpha \in {\mathcal{Q}}$ is the weight of $\overline {r}$ , also written as $N(\overline {r})$ , and

• • $r$ is a classic (normal) rule of the form:

  (1) \begin{align} p_0\leftarrow p_1,\ldots , p_m,\textit {not}\, p_{m+1}, \ldots , \textit {not} \, p_n \end{align}
  with $n \geq 0$ and $p_i\in {\mathcal{A}}\,(0\le i\le n)$ .

A possibilistic logic program is also referred to as a possibilistic logic knowledge base or a possibility theory in the literature.

Given a poss-NLP $\overline {P}$ , its classic counterpart, consisting of all classic rules in the poss-NLP, is denoted as $P$ . Let $r$ be a classic rule of the form (1). We denote $\textit {hd}(r)=p_0$ , ${\textit {bd}{^+}}(r)=\{p_1,\ldots , p_m\}$ , ${\textit {bd}{^-}}(r)=\{p_{m+1},\ldots , p_n\}$ and $\textit {bd}(r)={\textit {bd}{^+}}(r)\cup \textit {not} \,{\textit {bd}{^-}}(r)$ . Thus, the rule $r$ can be written as

\begin{align*} \textit {hd}(r)\leftarrow {\textit {bd}{^+}}(r),\textit {not} \,{\textit {bd}{^-}}(r) \qquad \textit {or}\qquad \textit {hd}(r)\leftarrow \textit {bd}(r) \end{align*}

where $\textit {not} \, S=\{\textit {not} \, p\mid p\in S\}$ . The rule $r$ is definite if ${\textit {bd}{^-}}(r)=\emptyset$ . For a possibilistic normal logic rule $\overline {r}$ , we also denote $\textit {hd}(\overline {r})=\textit {hd}(r)$ , ${\textit {bd}{^+}}(\overline {r})={\textit {bd}{^+}}(r)$ and ${\textit {bd}{^-}}(\overline {r})={\textit {bd}{^-}}(r)$ . A possibilistic definite (logic) program is a finite set of possibilistic definite rules.

Two classic rules $r_1$ and $r_2$ of the form (1) are identical if $\textit {hd}(r_1) = \textit {hd}(r_2)$ and $\textit {bd}(r_1) = \textit {bd}(r_2)$ . For instance, rule $p_0\leftarrow p_1, p_2,\textit {not} \, p_3, \textit {not} \, p_4$ is the same as rule $p_0\leftarrow p_2, p_1,\textit {not} \, p_4, \textit {not} \, p_3$ . Similar to the assumption in Possibilistic Logic, we assume that every classic rule occurs at most once in a poss-NLP.

Given two poss-NLPs $\overline {P_1}$ and $\overline {P_2}$ , the operations $\cup$ and $-$ over poss-NLPs can be defined as follows (Garcia et al. Reference Garcia, Lefèvre, Papini, Stéphan and Würbel2018).

• • $\overline {P_1}\sqcup \overline {P_2}=\{(r, \alpha )\mid (r,\alpha )\in \overline {P_1}, r \notin P_2\}\cup \{(r,\beta )\mid (r,\beta )\in \overline {P_2},r \notin P_1\}\cup \{(r,\max \{\alpha ,\beta \})\mid (r,\alpha )\in \overline {P_1}, (r,\beta )\in \overline {P_2}\}$ .

• • $\overline {P_1} - \overline {P_2} = \{ (r, \alpha ) \mid (r,\alpha )\in \overline {P_1}, r \notin P_2 \} \cup \{ (r, \alpha ) \mid (r,\alpha )\in \overline {P_1}, (r,\beta )\in \overline {P_2}, \alpha \gt \beta \}$ .

2.2 Possibilistic stable models

In this subsection, we introduce the semantics of poss-NLPs, that is, possibilistic stable models or poss-stable models (Nicolas et al. Reference Nicolas, Garcia, Stéphan and Lefèvre2006). To this end, we first recall basics of normal logic programs under stable models (Gelfond and Lifschitz Reference Gelfond and Lifschitz1988).

An atom set $S$ satisfies a definite logic program $P$ , written as $S \models P$ , if ${\textit {bd}{^+}}(r) \subseteq S$ implies $\textit {hd}(r) \in S$ for each $r \in P$ . $S$ is a stable model of $P$ , written as $S \in \mathit{SM}(P)$ , if $S \models P$ and there exists no $S' \subset S$ such that $S' \models P$ . In fact, a definite logic program $P$ has the unique stable model which is its least Herbrand model $\mathit{Cn}(P)$ (Lloyd Reference Lloyd2012) (the set of consequences of $P$ ). $\mathit{Cn}(P)$ is also the least fixpoint $\mathit{lfp}(T_P)$ of the immediate consequence operator $T_P:2^{\mathcal{A}} \rightarrow 2^{\mathcal{A}}$ defined by $T_P(A)=\textit {hd}(\mathit{App}(P, A))$ , where $\textit {hd}(P) = \{ \textit {hd}(r) \mid r \in P \}$ and $\mathit{App}(P, A) = \{ r \in P \mid {\textit {bd}{^+}}(r) \subseteq A \}$ . That is, for a definite logic program $P$ , we have $\mathit{Cn}(P) = \mathit{lfp}(T_P)$ .

The following proposition is useful for checking whether $S = \mathit{lfp}(T_P)$ for a set $S$ of atoms. A definite logic program $P$ is grounded if it can be ordered as a sequence $(r_1, \ldots , r_n)$ such that $ r_i \in \mathit{App}(P,\textit {hd}(\{ r_1, \ldots , r_{i-1} \}))$ for each $1 \leq i \leq n$ .

Proposition 2.1 (Proposition 1 of Nicolas et al. (Reference Nicolas, Garcia, Stéphan and Lefèvre2006)). Let $P$ be a definite logic program and $S$ be an atom set. $S$ is a least Herbrand model of $P$ if and only if

• • $S = \textit {hd}(\mathit{App}(P,S))$ , and

• • $\mathit{App}(P,S)$ is grounded.

An atom set $S$ is a stable model of normal logic program (NLP) $P$ if $S$ is the stable model of $P^S$ , the reduct of $P$ w.r.t. $S$ , where $P^S$ denotes the definite logic program $\{\textit {hd}(r) \leftarrow {\textit {bd}{^+}}(r) \mid r \in P, {\textit {bd}{^-}}(r) \cap S = \emptyset \}$ (Gelfond and Lifschitz Reference Gelfond and Lifschitz1988). A stable model of $P$ is also referred to as an answer set of $P$ . The set of all stable models for $P$ is denoted as $\mathit{SM}(P)$ .

Now we are ready to introduce the notion of poss-stable models (Dubois and Prade Reference Dubois and Prade2024), which is a generalization of the stable models for ordinary NLPs. First, we extend the definitions of reduct, rule applicability, and consequence operator to the case of poss-NLPs.

Given a possibilistic definite rule $\overline {r} = (r, \alpha )$ where $r$ is of the form $p \leftarrow \{p_1,\ldots ,p_m$ , if there exists weights $\alpha _1, \ldots , \alpha _m$ such that $(p_i,\alpha _i)\in \overline {A}$ for all $i\,(1\le i\le m)$ ( $1\le i\le m$ ), then we say that $\overline {r}$ is $\beta$ -applicable in a possibilistic atom set $\overline {A}$ .

A definite rule $r$ is applicable in an atom set $A$ if ${\textit {bd}{^+}}(r) \subseteq A$ . Correspondingly, a possibilistic definite rule $(r,\alpha )$ with ${\textit {bd}{^+}}(r)=\{p_1,\ldots ,p_m\}$ is $\beta$ -applicable in a possibilistic atom set $\overline {A}$ with $\beta =\min \{\alpha ,\alpha _1,\ldots , \alpha _m\}$ if there exists $(p_i,\alpha _i)\in \overline {A}$ for every $i\,(1\le i\le m)$ . Otherwise, it is not applicable in $\overline {A}$ . For a certain atom $q \in {\mathcal{A}}$ and a possibilistic definite program $\overline {P}$ , define

(2) \begin{align} \mathit{App}(\overline {P}, \overline {A}, q) = \{ \overline {r} \in \overline {P} \mid \textit {hd}(\overline {r}) = q,\overline {r} \mbox{is} \beta -\mbox{applicable in } \overline {A} \}. \end{align}

Additionally, $\mathit{App}(\overline {P}, \overline {A}) = \bigcup _{q \in A} \mathit{App}(\overline {P}, \overline {A}, q)$ .

Given a possibilistic definite logic program $\overline {P}$ and a possibilistic atom set $\overline {A}$ , the immediate possibilistic consequence operator ${\mathcal{T}}_{\overline {P}}$ as Definition 9 in (Nicolas et al. Reference Nicolas, Garcia, Stéphan and Lefèvre2006) maps a possibilistic atom set $\overline {A}$ to another one as follows.

(3) \begin{align} {\mathcal{T}}_{\overline {P}}(\overline {A}) = &\{(q,\delta ) \mid q \in {\mathcal{A}}, \mathit{App}(\overline {P},\overline {A},q) \neq \emptyset ,\nonumber\\ \delta =&\max \{\beta \mid \overline {r} \in \mathit{App}(\overline {P},\overline {A},q)\mbox{ is $\beta $-applicable in } \overline {A}\}\}. \end{align}

Then the iterated operator ${\mathcal{T}}_{\overline {P}}^k$ is defined by ${\mathcal{T}}_{\overline {P}}^0 = \emptyset$ and ${\mathcal{T}}_{\overline {P}}^{n+1} = {\mathcal{T}}_{\overline {P}}({\mathcal{T}}_{\overline {P}}^{n})$ for each $n \geq 0$ . For a possibilistic definite logic program $\overline {P}$ , we can compute the set $\mathit{Cn}(\overline {P})$ of possibilistic consequences via $\mathit{Cn}(\overline {P}) = \mathit{lfp}({\mathcal{T}}_{\overline {P}})$ where $\mathit{lfp}({\mathcal{T}}_{\overline {P}}) = \bigsqcup _{n\geq 0}{\mathcal{T}}_{\overline {P}}^n$ is the least fixpoint of the immediate possibilistic consequence operator ${\mathcal{T}}_{\overline {P}}$ .

The possibilistic reduct of a poss-NLP $\overline {P}$ w.r.t. an atom set $S$ is the possibilistic definite logic program

\begin{equation*} \overline {P}^S = \{(\textit {hd}(r) \leftarrow {\textit {bd}{^+}}(r), N(\overline {r})) \mid \overline {r} \in \overline {P},{\textit {bd}{^-}}(r) \cap S = \emptyset \}. \end{equation*}

A set $\overline {S}$ of possibilistic atom is a poss-stable model (or poss-stable model) of poss-NLP $\overline {P}$ if $\overline {S} = \mathit{Cn}(\overline {P}^S)$ . The set of all poss-stable models of $\overline {S}$ is denoted $\mathit{PSM}(\overline {P})$ .

The following example illustrates some of the above notions for poss-NLPs.

Example 2.1. Let ${\mathcal{A}} = \{ a,b,c \}$ , $\overline {P} = \{(a \leftarrow \textit {not} \, b,0.6),(a \leftarrow ,0.9),(b \leftarrow , 0.6),(c \leftarrow a,b,0.8)\}$ and $\overline {S} = \{ (a,0.9), (b,0.6), (c,0.6) \}$ . Then $\mathit{Cn}(\overline {P}^S) = \{ (a,0.9),(b,0.6),(c,0.6) \}$ since $\overline {P}^S = \{(a \leftarrow , 0.9),(b \leftarrow , 0.6),(c \leftarrow a,b,0.8)\}$ and

\begin{align*} {\mathcal{T}}_{\overline {P}^S}^0 = &{\mathcal{T}}_{\overline {P}^S}(\emptyset ) = \{ (a,0.9),(b,0.6) \}, \\ {\mathcal{T}}_{\overline {P}^S}^1 = &{\mathcal{T}}_{\overline {P}^S}(\{ (a,0.9),(b,0.6) \}) = \{ (a,0.9),(b,0.6),(c,0.6) \}, \\ {\mathcal{T}}_{\overline {P}^S}^2 = &{\mathcal{T}}_{\overline {P}^S}(\{ (a,0.9),(b,0.6),(c,0.6) \}) = \{ (a,0.9),(b,0.6),(c,0.6) \} = \overline {S}. \end{align*}

As a result, $\overline {S} \in \mathit{PSM}(\overline {P})$ .

The next proposition shows that there is a one-to-one correspondence between the set $\mathit{PSM}(\overline {P})$ of poss-stable models for $\overline {P}$ and the set $\mathit{SM}(P)$ of stable models for $P$ .

Proposition 2.2 (Proposition 10 of Nicolas et al. (Reference Nicolas, Garcia, Stéphan and Lefèvre2006)). Let $\overline {P}$ be a poss-NLP.

• • If $A$ is a stable model of $P$ , then $\mathit{Cn}(\overline {P}^A)$ is a poss-stable model of $\overline {P}$ .

• • If $\overline {A}$ is a poss-stable model of $\overline {P}$ , then $A$ is a stable model of $P$ .

The immediate possibilistic consequence operator introduced in Nicolas et al. (Reference Nicolas, Garcia, Stéphan and Lefèvre2006) is for possibilistic definite logic programs only, but it can be generalized to poss-NLPs as follows. Beforehand, let us define the applicability of poss-rules in a poss-NLP.

Definition 2.1 (Poss-rule applicability). Given a possibilistic interpretation $ \overline {I}$ and a weight $\beta$ , a possibilistic normal rule $(r,\alpha )$ is $\beta$ -applicable in $ \overline {I}$ if the possibilistic definite rule $({hd}(r)\leftarrow {{bd}{^+}}(r),\alpha )$ is $\beta$ -applicable in $ \overline {I}$ and, ${{bd}{^-}}(r)\cap I=\emptyset$ . Otherwise, it is not applicable in $\overline {I}$ .

With this new definition of applicability in hand, Equation (2), the definition of $\mathit{App}(\overline {P}, \overline {A}, q)$ of applicable rules over a poss-NLP $\overline {P}$ , still works for poss-NLPs. Consequently, the immediate possibilistic consequence operator over a poss-NLP, Equation 3, can be easily extended to poss-NLPs as follows.

Definition 2.2 (Immediate possibilistic consequence operator ${\mathcal{T}}_{\overline {P}}$ ). Let $\overline {P}$ be a poss-NLP and $\overline {I} \in 2^{{\mathcal{A}}\times {\mathcal{Q}}}$ be a possibilistic interpretation. We define ${\mathcal{T}}_{\overline {P}}: 2^{{\mathcal{A}}\times {\mathcal{Q}}}\rightarrow 2^{{\mathcal{A}}\times {\mathcal{Q}}}$ as follows.

\begin{align*} {\mathcal{T}}_{\overline {P}}(\overline {A}) = &\{(q,\delta ) \mid q \in {\mathcal{A}}, \mathit{App}(\overline {P},\overline {A},q) \neq \emptyset ,\\ \delta = & \max \{\beta \mid \overline {r} \in \mathit{App}(P,\overline {A},q)\mbox{ is $\beta $-applicable in } \overline {A}\}\}. \end{align*}

The following example illustrates how the reduct of a poss-NLP wrt a poss-interpretation and its least fixpoint can be computed.

Example 2.2. Let $\overline {I}=\{(q, 0.9), (s, 0.7)\}$ be a possibilistic interpretation and $\overline {P} = \{ r_1, r_2, r_3, r_4 \}$ be a poss-NLP where

\begin{align*} & r_1 = ( p \leftarrow q,s, 0.9),\\ &r_2 = ( p \leftarrow \textit {not} \, r, 0.9),\\ &r_3 = ( p \leftarrow \textit {not} \, s, 0.7),\\ &r_4 = ( p \leftarrow r, 0.7). \end{align*}

By Definition 2.1 , we have

• • $r_1$ is 0.7-applicable in $\overline {I}$ since ${\textit {bd}{^+}}(r_1) \subseteq I$ and ${\textit {bd}{^-}}(r_1) \cap I = \emptyset$ and $\min \{0.9, 0.7, 0.9\}=0.7$ ;

• • $r_2$ is 0.9-applicable in $\overline {I}$ since ${\textit {bd}{^+}}(r_2) \subseteq I$ and $\textit {bd}{^-}(r_2) \cap I = \emptyset$ and $\min \{0.9\}=0.9$ ;

• • $r_3$ is not applicable in $\overline {I}$ since ${\textit {bd}{^-}}(r_3) \cap I \neq \emptyset$ ;

• • $r_4$ is not applicable in $\overline {I}$ since $\textit {bd}{^+}(r_4) \not \subseteq I$ .

By Definition 2.2 , it follows that ${\mathcal{T}}_{\overline {P}}(\overline {I})=\{(p,0.9)\}$ as $\max \{0.9,0.7\}=0.9$ .

As the next proposition shows, the immediate consequence operator for poss-NLPs in Definition 2.2 reserves the key properties of the original immediate consequence operator for ordianry NLPs.

Proposition 2.3 (Equivalent consequence). For a given poss-NLP $\overline {P}$ and a possibilistic interpretation $\overline {I}$ , ${\mathcal{T}}_{\overline {P}^I}(\overline {I}) = {\mathcal{T}}_{\overline {P}}(\overline {I})$ .

Such an integration simplifies the reasoning and representation concerning poss-stable models. For instance, Corollary 2.1 discussed a property of a poss-stable model without directly touching upon the GL-reduct. This corollary states that a possibilistic interpretation $\overline {I}$ is still a poss-stable model of the extension of a poss-NLP $\overline {P}$ when $\overline {P}$ absorbs a new poss-NLP $\overline {B}$ such that ${\mathcal{T}}_{\overline {B}}(\overline {I}) \sqsubseteq \overline {I}$ .

Corollary 2.1 (Absorption for a poss-stable model). Given a possibilistic interpretation $\overline {I}$ and two poss-NLPs $\overline {P}$ and $\overline {B}$ , $\overline {I} \in \mathit{PSM}(\overline {P} \sqcup \overline {B})$ if $\overline {I} \in \mathit{PSM}(\overline {P})$ and ${\mathcal{T}}_{\overline {B}}(\overline {I}) \sqsubseteq \overline {I}$ .

3 Induction tasks for possibilistic logic programs

In this section, we first present the definition of induction tasks for poss-NLPs and then investigates some properties. These results will further be applied in computing (minimal) induction solutions. The necessary and sufficient condition for existing a solution for an induction task relies on the relationships such as (in)comparability and coherency among the components of the task. To solve such an induction task, we propose an algorithm ilpsm for the construction of a particular solution and further propose another algorithm ilpsmmin to identify a minimal solution. To narrow the search scope of the algorithm ilpsmmin, the concept of solution space is introduced and its properties are investigated.

Informally, assume that there is a given background knowledge base formalized as a set of possibilistic rules, and a set of (positive and negative) examples, the induction task is to extract a new poss-NLP that covers positive examples but none of the negative examples. We first formally define the notion of induction tasks for poss-NLPs in Definition 3.1.

Definition 3.1 (Induction task for poss-NLPs). Let $\mathcal{A}$ be a set of atoms and $\mathcal{Q}$ be a complete lattice. An induction task for poss-NLPs is a tuple $T = {\langle { \overline {B}, E^+, E^-}\rangle }$ , where the background knowledge $\overline {B}$ is a poss-NLP over $\mathcal{A}$ and $\mathcal{Q}$ , $E^+$ and $E^-$ are two sets of possibilistic interpretations called the set of positive examples and the set of negative examples, respectively. A hypothesis $\overline {H}$ is a NLP over $\mathcal{A}$ and $\mathcal{Q}$ . We say $\overline {H}$ is a solution of the induction task $T$ if the following two conditions are satisfied:

(G1) \begin{align} E^+\subseteq \mathit{PSM}(\overline {B} \sqcup \overline {H}), \end{align}

(G2) \begin{align} E^-\cap \mathit{PSM}(\overline {B} \sqcup \overline {H})=\emptyset . \end{align}

The set of all solutions of the induction task is denoted $\mathit{ILP_{LPoSM}}(T)$ .

For convenience, we assume that $\mathcal{A}$ and $\mathcal{Q}$ consists of the atoms and weights occurring in the induction task $T$ , receptively, unless explicitly stated otherwise.

Note that Condition G1 requires only $E^+$ is a portion of the poss-stable models of $\overline {B} \sqcup \overline {H}$ . This “partiality of examples” should not be confused with the “model partiality”. The following example shows some instances of induction tasks.

Example 3.1. For the scenario in Example 1.1 , the induction task can be represented as $T_1 = {\langle {\overline {B_{med}}, \{ \overline {A_1},\overline {A_2} \}, \{ \overline {A_3} \} }\rangle }$ where

\begin{align*} & \overline {A_1} = \{ (pregnancy,1), (vomiting,1), (medA,1), (\mathit{relief},0.7), (malnutrition,0.7) \},\\ & \overline {A_2} = \{ (pregnancy,1), (vomiting,1), (medB,1), (\mathit{relief},0.6), (malnutrition,0.1) \},\\ & \overline {A_3} = \{ (pregnancy,1), (vomiting,1), (medA,0.7), (\mathit{relief},0.7) \}. \end{align*}

Here ${\mathcal{A}} = \{ pregnancy, vomiting, medA, medB, \mathit{relief}, malnutrition \}$ and ${\mathcal{Q}} = \{ 0.1,0.6,0.7,1 \}$ and $0.1 \leq 0.6 \leq 0.7 \leq 1$ . The induction task $T_1$ has a solution $\overline {H_1} = \{ (medA \leftarrow vomiting, \textit {not} \, medB, 1) \}$ . Also, $\overline {H_2} = \overline {H_1} \cup \{ (\mathit{relief} \leftarrow vomiting, 0.6) \}$ is a solution of $T_1$ . Let $\overline {P_{med}} = \overline {B_{med}} \cup \overline {H_1}$ . Then $\overline {A_1}$ and $\overline {A_2}$ are poss-stable models of $\overline {P_{med}}$ , while $\overline {A_3}$ is not.

However, the induction task $T_2 = {\langle {\overline {B_{med}}, \{ \overline {A_1},\overline {A_2}, \{ (pregnancy, 0.6) \} \}, \{ \overline {A_3} \} }\rangle }$ has no solution, that is, $\mathit{ILP_{LPoSM}}(T_2) = \emptyset$ .

For induction task $T_3 = {\langle {\emptyset , \{ \{ (p, 0.3), (q,0.3) \} \}, \emptyset }\rangle }$ , we have ${\mathcal{A}} = \{ p, q \}$ and ${\mathcal{Q}} = \{ 0.3 \}$ . This induction task has more than one solution. For instance, $\overline {H_3}$ , $\overline {H_4}$ , $\overline {H_5}$ , $\overline {H_6}$ are all solutions of $T_3$ , that is, $\{ \overline {H_3}, \overline {H_4}, \overline {H_5}, \overline {H_6} \} \subseteq \mathit{ILP_{LPoSM}}(T_3)$ , where $\overline {H_3} = \{ (p \leftarrow , 0.3), (q \leftarrow p, 0.3) \}$ , $\overline {H_4} = \{ (p \leftarrow q, 0.3), (q \leftarrow , 0.3) \}$ , $\overline {H_5} = \{ (p \leftarrow , 0.3), (q \leftarrow , 0.3) \}$ , and $\overline {H_6} = \{ (p \leftarrow , 0.3), (q \leftarrow , 0.3), (q \leftarrow p, 0.3), (p \leftarrow q, 0.3) \}$ .

For induction task $T_4 = {\langle { \emptyset , \{ \{ (p, 0.3), (q,0.3) \} \}, \{ \{ (p, 0.3), (q,0.3) \} \}}\rangle }$ , it is obvious that $\mathit{ILP_{LPoSM}}(T_4) = \emptyset$ .

For induction task $T_5 = {\langle { \{ (p \leftarrow , 1) \}, \{\{(q,1),(p,1)\}\}, \{\{(q,1)\}\} }\rangle }$ , it can be checked that $\{ (q \leftarrow , 1) \} \subseteq \mathit{ILP_{LPoSM}}(T_5)$ .

We recall that any two stable models of a NLP are $\subseteq$ -incomparable. Namely, $\mathit{SM}(P)$ of any NLP $P$ is $\subseteq$ -incomparable. For a poss-NLP, there is a similar property. Before formally stating the property, we note that the (in)comparability of interpretations can be similarly defined for poss-interpretations.

By the minimality of stable models, two different $\subseteq$ -comparable interpretations cannot simultaneously be poss-stable models of the same poss-NLP.

Intuitively, two different comparable possibilistic interpretations cannot simultaneously appear in the set of stable models of the same poss-NLP. This result are useful for optimizing the algorithms for solving induction tasks in poss-NLPs, which actually provides two heuristics for early termination of some search paths. First, the induction task has no solution if two positive examples in the given $E^+$ are comparable. Moreover, a negative example in $E^-$ can be ignored immediately if it is comparable with a positive example in  $E^+$ .

Induction task $T = {\langle { \overline {B}, E^+, E^-}\rangle }$ of poss-NLP from poss-stable models in two aspects at least. On one hand, an induction task has no solution if two positive examples in the given $E^+$ are comparable as Example 3.2. On the other hand, a negative example $\overline {e} \in E^-$ can be ignored before the solving process if $\overline {e}$ is comparable with a positive example in $E^+$ .

In the process of extracting a poss-NLP from both positive and negative examples, we are going to obtain some other necessary conditions for a solution of the induction task, which will be useful for implementation of algorithms for induction for poss-NLPs.

By Proposition 2.2, $\mathit{PSM}(\overline {P}) = \overline {S}$ implies $\mathit{SM}(P) = S$ , but not vice versa.

From the above example, given a collection $\overline {S}$ of poss-interpretations, an ordinary NLP $H$ can be constructed from $\overline {S}$ such that $\mathit{SM}(H) = S$ , but there may be no $\overline {H}$ such that $\mathit{PSM}(\overline {H}) = \overline {S}$ . This shows that, due to the introduction of weights in rules, the problem of induction for poss-NLPs is not as easy as in the case of ordinary NLPs.

In the rest of this section, we will investigate some properties of induction tasks for poss-programs, especially, the existence of induction solutions. To this aim, we first introduce two notions that are used for characterizing rules that can cover a candidate model (supporting positive examples and blocking negative examples, respectively).

Let $\overline {S}$ be a set of possibilistic interpretations. We define

(4) \begin{equation} \mathit{PPE}(\overline {S}) = \bigsqcup _{\overline {I} \in \overline {S}} \mathit{PPE}(\overline {I}) \end{equation}

where $\mathit{PPE}(\overline {I}) = \{ (x \leftarrow \textit {not} \, ({\mathcal{A}} - I), \alpha ) \mid \text{ for all } (x,\alpha ) \in \overline {I} \}$ .

In an induction task, if $\overline {I}$ in $E^+$ , then $\mathit{PPE}(\overline {I})$ contains the possibilistic rules that potentially cover the positive example. Thus, $\mathit{PPE}(E^+)$ contains the rules that are potentially cover $E^+$ (i. e., all positive examples) if $E^+$ is incomparable.

Apart from the positive examples, the impact of negative examples on the induction solution can be also handled by constructing a special poss-NLP. Let $\overline {S_N}$ and $\overline {S_P}$ be two sets of possibilistic interpretations. We define

(5) \begin{equation} \mathit{PNE}(\overline {S_N},\overline {S_P}) = \bigsqcup _{\overline {I} \in \overline {S_N}, I \neq {\mathcal{A}}, \overline {I} \not \parallel \overline {S_P}} \mathit{PNE}(\overline {I}) \end{equation}

where $\mathit{PNE}(\overline {I}) = \{ (x_0 \leftarrow I, \textit {not} \, ({\mathcal{A}} - I), \mu ) \mid \text{ for some } x_0 \in ({\mathcal{A}} - I) \}$ . Here $x_0$ is an arbitrary element in ${\mathcal{A}} - I$ and $\mu$ is the supremum of $\mathcal{Q}$ in $({\mathcal{Q}},\le )$ . In other words, $\mathit{PNE}(\overline {I})$ contains exactly one rule that guarantees the second condition in the definition of induction tasks is satisfied.

For an induction task $T = {\langle {\overline {B}, E^+, E^-}\rangle }$ , it will be proved that $\mathit{PNE}(E^-,E^+)$ corresponds to a poss-NLP blocking some of possibilistic interpretations in $E^-$ to become the poss-stable models. Please note that $|\mathit{PPE(\overline S)}|\le \sum _{\overline I\in \overline S}|\overline I|$ and $|\mathit{PNE(\overline {S_N},\overline {S_P})}|\le |S_N|$ . Both of them can be computed in polynomial time.

By the above proposition, the poss-NLP $\mathit{PNE}(\overline {S_N},\overline {S_P})$ blocks some possibilistic interpretations in $\overline {S_N}$ to become poss-stable models. Consider the example below.

As explained, the poss-NLP $\mathit{PNE}(\overline {S_N},\overline {S_P})$ only considers a part of possibilistic interpretations in $\overline {S_N}$ . A possibilistic interpretation $\overline {G}$ such that $G = {\mathcal{A}}$ goes beyond the consideration. So we also construct another special poss-NLP $\mathit{PPE}(\overline {G})$ . As Lemma 3.1 shows, $\mathit{PPE}(\overline {G})$ can be used to generate the poss-stable model $\overline {G}$ under certain conditions. Intuitively, a possibilistic interpretation $\overline {I}$ such that $I = {\mathcal{A}}$ is so particular since it is comparable to any possibilistic interpretation. Hereafter, we denote $\overline {\mathbb{A}}=\{\overline I\mid \mbox{$\overline I$ is a possibilistic interpretation with $I=\mathcal{A}$}\}$ .

With constructions of the above special poss-NLPs, for a given poss-interpretation $\overline {I}$ , the existence of a poss-NLP $\overline {G}$ with $\overline {I}$ being a poss-stable mode of $\overline {G}$ is guaranteed by the closeness of $\overline {I}$ under the immediate consequence operator of $\overline {G}$ .

By the above proposition, we are able to state useful corollaries in the following.

The condition ${\mathcal{T}}_{\overline {B}}(\overline {I}) \sqsubseteq \overline {I}$ is important and often referred in our subsequent discussions. Given the above corollary, it makes sense to give the definition below.

Given this definition, the next corollary is straightforward.

In a special case when the set of positive examples may be empty, a necessary and sufficient condition for the existence of an induction solution can be provided as follows.

Let us look at an example.

By Proposition 3.5, if we consider only negative examples, the above three conditions provide some clue on the existence or nonexistence of induction solutions (even if an induction task contains positive examples). Thus, we have the follwoing definition.

Putting all these considerations together, we present Theorem 3.1. It reveals a necessary and sufficient condition for the existence of a solution for a general induction task. All indispensable relations in $\langle {\overline {B}, E^+, E^-}\rangle$ has been formally exhibited.

Now we are ready to present the major result in this section, which provides a necessary and sufficient condition for the existence of a solution for a general induction task.

Intuitively, $E^+$ is utilized to construct the candidate induction solution. Conversely, in a general induction task $T = {\langle {\overline {B}, E^+, E^-}\rangle }$ , $E^-$ is employed to eliminate unintended candidate solutions covering $E^+$ though. It functions akin to a hard constraint in ASP (Brewka et al. Reference Brewka, Eiter and Truszczynski2011). After a candidate solution is constructed, we check whether the candidate conflicts with $E^-$ . Then, either the candidate solution satisfy the requirements (return as a solution) or repeat this process.

In the following, we consider a special case of induction task, that is, when a fact rule is in the background knowledge.

Intuitively, for a special induction task $T = {\langle {\overline {B}, E^+, E^-}\rangle }$ whose background knowledge base contains a fact rule $(a \leftarrow , \mu )$ , the construction of an induction solution is significantly simplified.

1. (i) The induction task $T$ has no solution if we see a positive example without $(a, \mu )$ .

2. (ii) For each $\overline {e} \in E^-$ such that $(a, \mu ) \notin \overline {e}$ , $\overline {e}$ can be skipped during the construction of a solution.

3. (iii) When constructing a minimal solution of $T$ (i.e., with a minimum number of rules), we can ignore the rules whose head is $a$ .

Thus, Proposition 3.6 provides a way for computing a minimal solution (i.e., the number of rules in the induction solution is minimal).

Theorem 3.1 provides a method (Algorithm 1) for checking the existence of a (minimal) solution of an induction task. It is not difficult to verify that its time complexity is $O(n^c)$ where $n$ is the size of its input and $c$ is a constant.

Algorithm 1 Existence ${\langle {\overline {B}, E^+, E^-}\rangle }$

4 Algorithms for computing induction solutions

Theorem 3.1 not only provides a sufficient and necessary condition for the existence of an induction solution, but its proof also paves a way for constructing a particular solution. Thus, the method for computing an induction solution is formulated as Algorithm 2.

Algorithm 2 ilpsm $(\overline {B}, E^+, E^-)$

If the set of positive examples is not empty, $\mathit{PPE}(E^+)$ in line (4) and $\mathit{PNE}(\overline {E},E^+)$ in line (6) respectively ensure that condition (G1) and condition (G2) in Definition 3.1 are achieved. By Proposition 3.4, negative example satisfying ${\mathcal{T}}_{\overline {B} \sqcup \overline {H}}(\overline {e}) \not \sqsubseteq \overline {e}$ does not require further consideration. Thus, such negative examples are eliminated in line (6). When $E^+ \neq \emptyset$ , condition (G2) in Definition 3.1 is guaranteed by either $\mathit{PNE}(E^-,\emptyset )$ in line (9) or $\mathit{PPE}(\overline {G})$ in line (12).

Consider the following example.

We note that, in the definition of $\mathit{PNE}(\overline {E},E^+)$ in Equation (5), the head $x$ of the rule $(x \leftarrow I, \textit {not} \, ({\mathcal{A}} - I), \mu )$ is an arbitrary element in ${\mathcal{A}} - I$ , which is sufficient to guarantee that the output of Algorithm 2 is an induction solution. However, for different choices of the head $x$ , we may come up with different induction solutions.

We note that checking the existence of induction solutions and the subset relation ${\mathcal{T}}_{\overline {B} \sqcup \overline {H}}(\overline {e}) \sqsubseteq \overline {e}$ can be done in polynomial time. Also, the tasks of computing $\mathit{PPE}(E^+)$ , $\mathit{PNE}(\overline {E},E^+)$ are also in polynomial time. The task of line 11 can be achieved by enumerating $\overline G\in \overline {\mathbb A}-E^-$ and checking if ${\mathcal{T}}_{\overline B}(\overline G)\sqsubseteq \overline G$ , which is bounded by $O(k2^m)$ where $m=|\overline {\mathbb A}|$ and $k=|B|+|G|$ (we assume ${\mathcal{T}}_{\overline B}(\overline G)$ is computable in linear time). Therefore, the time complexity of this algorithm is $O(n2^n)$ where $n$ is the size of its input.

While algorithm ilpsm always returns an induction solution if it exists, the solution may not minimal w.r.t. the number of rules in the solution. We note that all rules in $\mathit{PPE}(E^+)$ has are negative (i. e., no positive body atoms), which causes some minimal solutions are missed in the construction process. For instance in Example 3.1, the minimal solution $\overline {H_1}$ will be missed by this algorithm since the positive body of rule $(medA \leftarrow vomiting, \textit {not} \, medB, 1)$ is not empty.

In the rest of this section, we introduce an algorithm that computes a minimal induction solution. This algorithm is completely different from the previous one and we need to first investigate some properties of minimal induction solutions.

Let $\alpha \in \mathcal{Q}$ and $\star \in \{\gt ,\ge , =\}$ . The $\alpha ^\star$ -relevant atoms of a possibilistic interpretation $\overline {I}$ , written as ${RA}^\star (\overline {I},\alpha )$ , is defined by ${RA}^\star (\overline {I},\alpha )=\{ p \mid (p,\beta )\in \overline {I}, \beta \star \alpha \}$ . Note that when $\star$ is the equality sign $=$ , ${RA}^= (\overline {I},\alpha )$ is empty if there is no atom $p$ such that $(p,\beta )\in \overline {I}$ . So, the set ${RA}^=(\overline {I},\alpha )$ is not always be $\{p\}$ .

Given a collection $A$ of sets, a set $B$ is a hitting set of $A$ if $X \cap B \neq \emptyset$ for each $X \in A$ . A hitting set $B$ of $A$ is called minimal if there exists no $C \subset B$ such that $C$ is still a hitting set of $A$ . We use $\mathit{SMHS}(A)$ to denote the set of all minimal hitting sets of $A$ . Now we can define the notion of positive solution spaces in the following definition.

The next propositions shows that $|P\cap S^+(\overline I,\epsilon )|=1$ for each $P\in S^+(I)$ and $\epsilon \in \overline I$ .

Intuitively, $S^+(\overline {I})$ is a collection of all poss-NLPs containing exactly $\vert I \vert$ rules, to which each $\epsilon \in \overline {I}$ provides exactly one rule from $S^+(\overline {I}, \epsilon )$ . Please note that the size of $S^+(\overline I,\epsilon )$ is bounded by $O(k2^n)$ where $n=|{\mathcal{A}}|$ and $k=|{\mathcal{Q}}|$ since there are possibly $2^n$ number of different $\textit {bd}{^+}$ and $\textit {bd}{^-}$ in Equation (6). Thus, the size of $S^+(\overline I)$ is bounded by $O((n2^n)^m)=O(n^m\times 2^{nm})$ where $n=|\overline I|+|{\mathcal{Q}}|$ and $m=|\overline I|$ .

Let us look at some examples of $S^+(\overline {I}, \epsilon )$ and $S^+(\overline {I})$ .

The notion of positive solution space is helpful for guiding the search process for a minimal solution. The below lemma reveals a relation between the least fixpoint of a poss-definite program and the positive solution spaces.

This result can be generalized to general poss-NLPs.

By Definition 4.1, it is straightforward to get $\overline {H} \in S^+(\overline {I})$ in Proposition 4.3. The next step is to check whether $H^I$ is grounded. As each stable model has at least one acyclic support graph (Cabalar and Muñiz Reference Cabalar and Muñiz2023),Footnote ² Proposition 4.4 provides a default method for checking the groundness of $H^I$ via the positive loop in its support graph.

To characterize the impact of negative examples on minimal induction solutions, we introduce the notion of negative solution spaces. Informally, the negative solution space collects all potential rules that prevent $\overline {I} \in E^-$ from being a poss-stable model. While the positive solution space uses a minimal hitting set to ensure the outcome of each possibilistic atom in a poss-stable model, only one rule in the negative solution space (in a solution) can block the undesired poss-stable model. A negative solution space comprises two parts: one part leads to an existing interpretation bounded with an unacceptably higher necessity; and the other leads directly to an unacceptable new interpretation.

Please note that the size of $S^-(\overline I,\epsilon )$ is bounded by $O(k2^n)$ where $n=|{\mathcal{A}}|$ and $k=|{\mathcal{Q}}|$ since there are $2^n$ number of different $\textit {bd}{^+}$ and $\textit {bd}{^-}$ . Thus, the size of $S^-(\overline I)$ is bounded by $O(n2^n)$ where $n=|\overline I|+|{\mathcal{Q}}|$ .

Lemma 4.2 connects the incoherency and the negative solution space. Combining both $S^-(\overline {I})$ and $S^+(\overline {I})$ , we have the following result, which provides a characterization of poss-stable models. Intuitively, $\overline {I}$ being a poss-stable model of $\overline {P}$ if and only if (i) $\overline {P}$ does not contain any superfluous support for $\overline {I}$ , and (ii) $\overline {P}$ provides sufficient support for $\overline {I}$ .

By Theorem 4.1, we have the following corollary.

Proposition 4.5 shows that every rule in a minimal solution must directly affect at least a positive example or a negative example. Besides, the necessity of each rule has a certain degree of freedom within the corresponding solution spaces.

We are now ready to present algorithm ilpsmmin, Algorithm 3, for computing a minimal solution for an induction task (i. e. with a minimum number of rules), if there is a solution. Please note that the algorithm does not directly deal with atoms. The order in which the (positive and negative) examples are processed is not very relevant to its efficiency.

Algorithm 3 ilpsmmin $(\overline {B}, E^+, E^-)$

The algorithm ilpsmmin first collects poss-programs that cover positive examples in $E^+$ in lines 4–8 and then in lines 9–20, the resulting poss-programs are checked to make sure that no negative examples in $E^-$ are covered and if yes, they are expanded and the minimality is guaranteed.

Below is a legend for the variables used in the algorithm.

1. - $\overline {H}$ : The minimal solution returned by the algorithm if it exists.

2. - $norm$ : The number of rules in the minimal solution.

3. - $seeds$ : The set of poss-NLPs supporting the set $E^+$ of positive examples. When $E^+ = \emptyset$ , the $seeds$ comprises $\emptyset$ so that the following analysis of the $E^-$ can go on.

4. - $blacklist$ : Rules in those poss-NLPs that make a positive example uncovered.

5. - $W_i$ : The set of poss-NLPs supporting the $i$ -th positive example.

6. - $whitelist$ : Rules in poss-NLPs that cover a negative example in $\overline {E}$ .

7. - $patches$ : A candidate patch for a preliminary hypothesis $X$ to block the negative examples that are covered.

We briefly explain each step in the algorithm as follows.

1. - Line 1: Checking the existence of an induction solution;

2. - Line 2: Initialize the global variables (especially, $norm$ is set to an upper bound of the rule numbers in induction solutions);

3. - Line 3: According to the negative solution space, find a set of rules that cannot appear in the induction solution and assign the set to the variable $blacklist$ ;

4. - Lines 4–6: Iterate through each positive example $\overline {I}$ in $E^+$ , ensuring that each positive example corresponds to a set $W_i$ of poss-NLPs that covers $\overline {I}$ with the minimum number of rules;

5. - Lines 7–8: If the set of positive examples is not empty, update the $seeds$ by combining poss-programs that cover each positive example into a poss-program that covers the whole $E^+$ ;

6. - Line 9: Iterate through each poss-NLP $X$ in $seeds$ ;

7. - Lines 10–12: If the coverage requirements of the negative examples have been met, then $X - \overline {B}$ is an induction solution; if this induction solution has fewer rules than all other induction solutions that have been examined, set this solution as the new candidate for the minimal solution;

8. - Line 13: If the coverage requirements of the negative examples have not been met, iteratively add rules in $whitelist$ to the candidate solution in lines 14–20;

9. - Line 14: Select negative examples that violate conditions w.r.t. the candidate solution.

10. - Line 15: Based in the negative examples obtained in line 14, collect those rules that can potentially be added to the candidate solution to construct a solution finally.

11. - Line 16: Iterate through the integer $j$ from $1$ to $\vert \overline {E} \vert$ .

12. - Line 17: Pick up only those poss-programs having exactly $j$ rules from the $whitelist$ ;

13. - Line 18: Check only those poss-programs in $patches$ whose number of rules is less than the number of rules of the current candidate (otherwise, it is not minimal even if it can leads to a solution).

14. - Line 19–20: If the condition for $E^-$ in the definition of induction solutions is satisfied, then the current candidate solution is a solution. Moreover, if the number of rules in this solution is less than the value in $norm$ , the new solution is set as the latest candidate for a minimal solution.

15. - Line 21: Return a minimal solution $\overline {H}$ .

Note that the operator $\sqcup$ can not be replaced by $\cup$ in the algorithm. Otherwise, the algorithm may be incorrect for some inputs. To see this, let us look at the following example.

In the following we briefly analyze the time complexity of Algorithm 3. Assume that every elements of $\mathcal{A}$ and $\mathcal{Q}$ have occurred in its input. Let $n=|\overline B|+\sum _{\overline I\in E^+\cup E^-}|I|$ .

1. - Construct $blacklist$ : The search space for constructing $S^-(\overline {I},\epsilon )$ is exponential and thus the task of constructing $blacklist$ is exponential $O(n^22^n)$ since $S^-(\overline I)$ is bounded by $O(n2^n)$ .

2. - The first loop (lines 4-6) is bounded by $O(n^{n+1}2^{n^2})$ , which is thus the upper bound of $|\mathit{seeds}|$ , since $S^+(\overline I)$ is bounded by $O(n^n2^{n^2})$ .

3. - In lines 10 and 19, for each poss-interpretation in $\overline {E}$ , a poss-stable model needs to be computed, which is in exponential time $O(2^n)$ in the worst case since the problem of deciding if a poss-NLP has a poss-stable model is NP-complete (Nicolas et al. Reference Nicolas, Garcia, Stéphan and Lefèvre2006). Note that $|\overline E|$ at line 14 is bounded by $n$ , the size of $\mathit{whitelist}$ at line 15 is bounded by $O(n^22^n)$ and the size of $\mathit{patches}$ at line 17 is bounded by $O(2^{n^22^n})$ . Therefore, the loop (lines 16-20) is bounded by $O(n\times 2^{n^22^n}\times n2^n)=O(n^22^{n^22^n+n})$ . Thus, the loop (lines 9-20) is bounded by $O(n^n2^{n^2}\times n^22^{n^22^n+n})= O(n^{n+2}2^{n^2(2^n+1)+n})$ .

Thus, the overall time complexity of Algorithm 3 is bounded by $O(n^n2^{n^2}+n^{n+2}2^{n^2(2^n+1)+n})=O(n^{n+2}2^{n^2(2^n+1)+n})$ .

Example 4.5 illustrates the main process when given the same induction task as in Example 4.1, allowing us to discern the differences between these two algorithms.

Example 4.5 (Continued from Examples 4.1–4.3). Let $T_{31} = {\langle {\overline {B}, E^+, E^-}\rangle }$ where $\overline {B} = \{ (p \leftarrow q, 0.3), (q \leftarrow \textit {not} \, r, 0.5) \}$ , $E^+ = \{ \{ (r, 0.3) \} \}$ and $E^- = \{ \{ (q, 0.3), (r, 0.5) \}, \{ (p, 0.3), (q, 0.5) \} \}$ . Then we have ${\mathcal{A}} = \{ p,q,r \}$ , ${\mathcal{Q}} = \{ 0.3, 0.5 \}$ and $0.3 \leq 0.5$ . When inputting $T_{31}$ into algorithm ilpsmmin, it can yield the minimal solution $\overline {H_{32}} = \{ (r \leftarrow , 0.3) \}$ according to the analysis as follows.

• • In line (1): The value of $\mathit{Existence}(\overline {B}, E^+, E^-)$ is true , so the algorithm continues;

• • In line (3): According to the analysis in example 4.3 , the value of $S^-(\{ (r, 0.3) \})$ can be determined. For example, $(r \leftarrow , 0.5) \in blacklist$ ;

• • In line (4): Let $\overline {I} = \{ (r, 0.3) \}$ be the first positive example analyzed;

• • In line (6): According to the analysis in example 4.2 , $\{ \{(r \leftarrow \textit {not} \, q, 0.3)\}, \{(r \leftarrow , 0.3)\}, \{(r \leftarrow \textit {not} \, p, \textit {not} \, q, 0.3)\} \} \subseteq W_1$ ;

• • In line (8): $\{ \{(r \leftarrow \textit {not} \, q, 0.3)\}, \{(r \leftarrow , 0.3)\}, \{(r \leftarrow \textit {not} \, p, \textit {not} \, q, 0.3)\}\} \subseteq seeds$ ;

• • In line (9): Assume $\{(r \leftarrow \textit {not} \, q, 0.3)\}$ enters the loop first, that is $X = \{(r \leftarrow \textit {not} \, q, 0.3)\}$ ;

• • In line (10): $E^- \cap \mathit{PSM}(\overline {B} \sqcup X) = \{ \{ (p, 0.3), (q, 0.5) \} \} \neq \emptyset$ , that is the condition in line (10) does not satisfy but the condition in line (13) satisfies, so lines (11-12) will be skipped and the algorithm jumps to lines (14-20);

• • In line (14): Since $\{ (q, 0.3), (r, 0.3) \} \parallel E^+$ , $\overline {E}$ is assigned the value $\{ \{ (p, 0.3), (q, 0.5) \} \}$ ;

• • In line (15): Now only $\{ (p, 0.3), (q, 0.5) \} \in \overline {E}$ , so $\overline {I}$ can only take $\{ (p, 0.3), (q, 0.5) \}$ . Since $\{ (r \leftarrow , 0.5), (p \leftarrow q, 0.5), (r \leftarrow p, 0.3) \} \subseteq S^-(\overline {I})$ and $(r \leftarrow , 0.5) \in blacklist$ , it follows that $\{ (p \leftarrow q, 0.5), (r \leftarrow p, 0.3) \} \subseteq whitelist$ ;

• • In line (16): Now $j$ can only take the value 1, so this loop executes only once;

• • In line (17): $\{ \{(p \leftarrow q, 0.5)\}, \{(r \leftarrow p, 0.3)\} \} \subseteq patches$ ;

• • In line (18): Assume $\{(p \leftarrow q, 0.5)\}$ enters the loop first, that is let $\overline {P} = \{(p \leftarrow q, 0.5)\}$ ;

• • In line (19): Two conditions are both fulfilled;

• • In line (20): $\overline {H}$ is updated from initial $\emptyset$ to $\{ (r \leftarrow \textit {not} \, q, 0.3), (p \leftarrow q, 0.5) \}$ , while $norm$ is updated from initial value to $2$ ;

• • In lines (18-20): Continue looping through the other poss-NLPs in $patches$ , but the condition $\vert X \sqcup \overline {P} - \overline {B} \vert \lt norm$ remains false regardless of how $X$ is assigned afterwards. Therefore, execution jumps back to line (9) since the update operation in line (20) will not execute again until the loop ends;

• • In line (9): Assume $\{(r \leftarrow , 0.3)\}$ consequently enters the loop, that is let $X = \{(r \leftarrow , 0.3)\}$ ;

• • In line (10): Jump to line (11) since $E^- \cap \mathit{PSM}(\overline {B} \sqcup X) = \emptyset$ ;

• • In line (11): Jump to line (12) since $\vert X - \overline {B} \vert = 1 \lt 2$ ;

• • In line (12): $\overline {H}$ is updated to $\{(r \leftarrow , 0.3)\}$ , and $norm$ is updated to $1$ , then jump back to line (9) to continue the subsequent loop;

• • In line (9): Jump to line (21) since the subsequent loop can no longer find a solution whose total number is less than $1$ ;

• • In line (21): Return the minimal solution $\overline {H} = \{(r \leftarrow , 0.3)\}$ ;

It is evident that $\emptyset$ is not a solution. A minimal solution yielded by algorithm ilpsmmin contains only one rule. In contrast, the solution yielded by algorithm ilpsm contains two rules.

To see a more general case, let us revisit Example 3.1 . When $T_1$ is fed to algorithm ilpsmmin, it can return a minimal solution.

\begin{equation*} \overline {H} = \{ (medA \leftarrow vomiting, \textit {not} \, medB, 1) \} . \end{equation*}

This hypothesis means that the specialist absolutely suggests prescribing medicine A if the patient vomits and has not taken medicine B. In contrast, algorithm ilpsm must yield a solution with 10 rules. When $T_1$ is inputted into algorithm ilpsm, $\mathit{PPE}(E^+)$ in line (4) is constructed as the following poss-NLP

\begin{equation*} \left \{ \begin{array}{c} (pregnancy \leftarrow \textit {not} \, medB, 1), (vomiting \leftarrow \textit {not} \, medB, 1), \\ (medA \leftarrow \textit {not} \, medB, 1), (\mathit{relief} \leftarrow \textit {not} \, medB, 0.7), \\ (malnutrition \leftarrow \textit {not} \, medB, 0.7), (pregnancy \leftarrow \textit {not} \, medA, 1), \\ (vomiting \leftarrow \textit {not} \, medA, 1), (medB \leftarrow \textit {not} \, medA, 1), \\ (\mathit{relief} \leftarrow \textit {not} \, medA, 0.6), (malnutrition \leftarrow \textit {not} \, medA, 0.1) \end{array} \right \}. \end{equation*}

After assigning $\mathit{PPE}(E^+)$ to $\overline {H}$ in this step, $\overline {H}$ remains unchanged as $\overline {E} = \emptyset$ and then $\overline {H} \sqcup \mathit{PNE}(\overline {E},E^+) - \overline {B} = \overline {H}$ .

As demonstrated in the example, the algorithm ilpsmmin has a certain degree of randomness such as the variable order of values in line (9). But this does not hinder the algorithm from achieving the desired goal, that is the returned poss-NLP is guaranteed to be a minimal solution of the induction task.

In this section, we have introduced two algorithms ilpsm for computing an induction solution and ilpsmmin for computing a minimal induction solution. The time complexity of algorithms ilpsm (resp. ilpsmmin) is bounded by $O(n2^n)$ (resp. $O(n^{n+2}2^{n^2(2^n+1)+n})$ ).

5 Variants of possibilistic induction tasks

In this section we study three variants of the induction task for possibilistic logic programs under stable models. In the first subsection, we consider two special cases of induction tasks. The first special case is when $\overline {\mathbb{U}} = E^+\cup E^-$ , that is, each atom is either a positive sample or a negative example. We provide a characterization of induction solutions through poss-stable models. The second special case is when input programs are only ordinary NLPs for an induction task for poss-NLPs. In this case, the definition of induction tasks for poss-NLPs collapses to the induction for ordinary NLPs. In the second subsection, we consider the generalization of induction tasks for poss-NLPs by allowing partial interpretations. Interestingly, we show that such a generalized induction task can be equivalently reduced to an induction task as defined in Definition 3.1.

5.2 Induction from partial stable models

We first extend the definition of inductive learning for poss-NLPs by allowing partial interpretation. Recall that a partial interpretation is a pair $\langle {E^{inc}, E^{exc}}\rangle$ of two sets $E^{inc}$ and $E^{exc}$ of atoms where $E^{inc} \cap E^{exc} = \emptyset$ . Intuitively, each atom in $E^{inc}$ is true, each atom in $E^{exc}$ is false, and each element in ${\mathcal{A}} - E^{inc} - E^{exc}$ is unknown. If $E^{inc} \cup E^{exc} = {\mathcal{A}}$ , the partial interpretation $\langle {E^{inc}, E^{exc}}\rangle$ is equivalent to the interpretation $E^{inc}$ . An interpretation $I$ extends a partial interpretation $E = {\langle {E^{inc}, E^{exc}}\rangle }$ , written as $I \propto E$ , if $(E^{inc} \subseteq I) \land (E^{exc} \cap I = \emptyset )$ .

The generalized notion of induction tasks for poss-NLPs is formally defined as follows.

Definition 5.2 (Induction task from partial stable models). An induction task from partial stable models is a tuple $T = {\langle {B, E^+, E^-}\rangle }$ where NLP $B$ is the background knowledge, two sets $E^+$ and $E^-$ of partial interpretations are respectively called the positive and negative examples. A hypothesis $H$ belongs to the set of induction solutions of $T$ , written as $H \in \mathit{ILP_{LPaSM}}(T)$ , if it achieves two aims

(a1) \begin{align} \forall e^+ \in E^+ \exists I \in \mathit{SM}(B \cup H), I \propto e^+, \end{align}

(a2) \begin{align} \forall e^- \in E^- \nexists I \in \mathit{SM}(B \cup H), I \propto e^-. \end{align}

We borrow an Example 5.3 in Law et al. (Reference Law, Russo and Broda2014) to illustrate the above definition.

Example 5.3. Let $T_{43} = {\langle {B, E^+, E^-}\rangle }$ where $E^+ = \{ {\langle {\{ p \}, \emptyset }\rangle }, {\langle {\{ q \}, \{ p \}}\rangle } \}$ , $E^- = \{ {\langle {\{ p,q \}, \emptyset }\rangle } \}$ and $B = \{ q \leftarrow r. \}$ . Then we have ${\mathcal{A}} = \{ p, q, r \}$ , $\mathit{SM}(B \cup H_{1}) = \{ \{ p \}, \{ q, r \} \}$ and $\mathit{SM}(B \cup H_{2}) = \{ p, q, r \}$ where $H_{1} = \{ p \leftarrow \textit {not} \, r. , r \leftarrow \textit {not} \, p. \}$ and $H_{2} = \{ p \leftarrow r. , r \leftarrow . \}$ . It is easy to verify that $\{ p \} \propto {\langle {\{ p \}, \emptyset }\rangle }$ , $\{ q, r \} \propto {\langle {\{ q \}, \{ p \}}\rangle }$ and $\{ p, q, r \} \propto {\langle {\{ p,q \}, \emptyset }\rangle }$ . As a result, $H_{1} \in \mathit{ILP_{LPaSM}}(T_{43})$ and $H_{2} \notin \mathit{ILP_{LPaSM}}(T_{43})$ .

As we can see, the induction task in Definition 5.2 and a LSM induction task reflect the same requirements. The denotation of a partial interpretation $o$ is essentially the set

\begin{equation*} \mathit{de}(o) = \{ I \in 2^{\mathcal{A}} \mid I \propto o \} \end{equation*}

of interpretations. The two induction tasks can be transformed as Proposition 5.3. See Example 5.4 for more details.

Proposition 5.3 (Transformations between induction tasks). Let $T = {\langle {B, O^+, O^-}\rangle }$ be an induction task of NLP from partial stable models. A NLP $H$ such that $H \in \mathit{ILP_{LPaSM}}(T)$ if and only if $H \in \mathit{ILP_{LSM}}(T_1)$ for some $T_1 = {\langle {B, E^+, E^-}\rangle }$ such that

1. (c1) $E^+$ is a minimal hitting set of $\{\mathit{de}(o) \mid o \in O^+ \}$ , written as $E^+ \in \mathit{SMHS}(\{\mathit{de}(o) \mid o \in O^+ \})$ , and

2. (c2) $E^- = \{ J \in \mathit{de}(o) \mid o \in O^- \}$ .

Example 5.4 (Continued from Example 5.3). We have $\{\mathit{de}(o) \mid o \in E^+ \} = \{ \{ \{ p \}, \{ p,q \}, \{ p,r \}, \{ p,q,r \} \}, \{ \{ q \}, \{ q,r \} \} \}$ , $\{ J \in \mathit{de}(o) \mid o \in E^- \} = \{ \{ p,q \}, \{ p,q,r \} \}$ and $\{ \{ p \}, \{ q, r \} \} \in \mathit{SMHS}(\{\mathit{de}(o) \mid o \in O^+ \})$ . As a result, induction task $T_{43}$ can generate a LSM task $T_1 = {\langle {B, \{ \{ p \}, \{ q, r \} \}, \{ \{ p,q \}, \{ p,q,r \} \}}\rangle }$ . It is evident that $H_{1} \in \mathit{ILP_{LSM}}(T_1)$ .

According to Proposition 5.3, an induction task $T = {\langle {B, O^+, O^-}\rangle }$ from partial stable models can be transformed into $\vert \mathit{SMHS}(\{\mathit{de}(o) \mid o \in O^+ \}) \vert$ induction tasks $T_i = {\langle {B, E^+_i, E^-}\rangle }$ where

• • $E^+_i$ is the $i$ -th element in $\mathit{SMHS}(\{\mathit{de}(o) \mid o \in O^+ \})$ , and

• • $E^- = \{ J \in \mathit{de}(o) \mid o \in O^- \}$ .

For convenience, we use $\mathit{trans}(T)$ to denote the set of all corresponding induction tasks $T_i$ deriving from induction task $T$ . A NLP $H$ is a solution of $T$ from partial stable models if and only if there exists $T_i \in \mathit{trans}(T)$ such that $H \in \mathit{ILP_{LSM}}(T_i)$ . Formally, we have the necessary and sufficient condition as Corollary 5.1. Example 5.5 helps to illustrate this conclusion.

Corollary 5.1 (Existence of a NLP solution). Let $T = {\langle {B, O^+, O^-}\rangle }$ be an induction task of NLP from partial stable models. $\mathit{ILP_{LPaSM}}(T) \neq \emptyset$ if and only if there exists $E^+ \in \mathit{SMHS}(\{\mathit{de}(o) \mid o \in O^+ \})$ and $E^- = \{ J \in \mathit{de}(o) \mid o \in O^- \}$ such that

1. (c1) $E^+$ is $\subseteq$ -incomparable, and

2. (c2) $e \models B$ for each $e \in E^+$ , and

3. (c3) ${\mathcal{A}} \notin E^-$ or ${\mathcal{A}} \neq \mathit{lfp}(T_{B^{\mathcal{A}}})$ , and

4. (c4) $E^+ \cap E^- = \emptyset$ .

Example 5.5. Let $T = {\langle {B, O^+, O^-}\rangle }$ where $O^+ = \{ {\langle {\{ p \}, \emptyset }\rangle } \}$ , $O^- = \{ {\langle {\{ p,q \}, \emptyset }\rangle } \}$ and $B = \{ q \leftarrow p. \}$ . Then we have ${\mathcal{A}} = \{ p, q \}$ , $\{\mathit{de}(o) \mid o \in O^+ \} = \{ \{ \{ p \}, \{ p,q \} \} \}$ , $E^- = \{ J \in \mathit{de}(o) \mid o \in O^- \} = \{ \{ p,q \} \}$ and $\mathit{SMHS}(\{\mathit{de}(o) \mid o \in O^+ \}) = \{ \{ \{ p \} \}, \{ \{ p,q \} \} \}$ . As a consequence, $\mathit{trans}(T) = \{ T_1, T_2 \}$ where $T_1 = {\langle {B, E^+_1, E^-}\rangle }$ , $T_2 = {\langle {B, E^+_2, E^-}\rangle } \}$ , $E^+_1 = \{ \{ p \} \}$ and $E^+_2 = \{ \{ p,q \} \}$ . Finally, $\mathit{ILP_{LPaSM}}(T) \neq \emptyset$ since $\{ p \} \in E^+_1, \{ p \} \not \models B$ and $E^+_2 \cap E^- \neq \emptyset$ .

6 Implementation and experiments

We have implemented a prototype for computing minimal solutions for induction tasks based on Algorithm 3. In this section, we first describe technical details of our implementation and report some preliminary experimental results, which show that our prototype significantly outperforms the baseline ilasp4 (Law Reference Law2023) when inducing ordinary NLPs.

As we have seen in the algorithm, a solver for computing poss-stable models will be called. While several algorithms for computing poss-stable models have been proposed in the literature, we have not seen any efficient implementation for computing poss-stable models. For this reason, our implementation runs only for induction tasks in ordinary NLPs. But it can be easily adapted to the case of full poss-NLPs when an efficient solver for computing poss-stable models is available.

The source code of implementation, testing datasets and experimental results are all available on Github.Footnote ³