On Enumerating All the Minimal Models for Particular

CNF Formula Classes

Yakoub Salhi

CRIL-CNRS, Universit

e d’Artois, F-62307 Lens Cedex, France

Keywords:

Minimal Models, SAT, Enumeration Problem, Computational Complexity.

Abstract:

In this work, we propose approaches for enumerating all the minimal models for two particular classes of CNF

formulæ. The ﬁrst class is that of PN formulæ which are deﬁned as CNF formulæ where each clause is either

positive or negative, whereas the second class is that of PH formulæ in which each clause is either positive

or a Horn clause. We ﬁrst provide an approach for enumerating all the minimal models in the case of PN

formulæ that is based on the use of an algorithm for generating the minimal transversals of a hypergraph. We

also propose a SAT-based encoding for solving the same problem. Then, we provide a characterization of the

minimal models in the case of PH formulæ, which allows us to use our approaches in the case of PN formulæ

for solving the problem of minimal model enumeration for PH formulæ. Finally, we describe an application

in datamining of the problem of enumerating the minimal models in the case of PN formulæ.

1 INTRODUCTION

The problem of minimal model enumeration is based

on preferring models that assign false to the proposi-

tional variables (see, e.g., (Ben-Eliyahu and Dechter,

1996; Niemel

a, 1996)). This problem has impor-

tant applications in AI, such as propositional circum-

scription (McCarthy, 1980), minimal diagnosis (Re-

iter, 1987) and logic programming (Bidoit and Froide-

vaux, 1987; Gelfond and Lifschitz, 1988).

It is noteworthy that the problem of checking

whether a model is minimal in the general case

is coNP-complete (Cadoli, 1992b). Different algo-

rithms have been proposed in the literature for com-

puting minimal models. The algorithms that have

been proposed in (Ben-Eliyahu and Dechter, 1996)

exploit tractable classes. For instance, knowing that

any Horn formula admits a single minimal model,

which can be found in linear time using the procedure

of unit propagation, the main idea consists in instan-

tiating as few variables as possible so that the remain-

ing formula is a Horn formula. In addition, an in-

cremental algorithm for enumerating all the minimal

models has been proposed in (Ben-Eliyahu-Zohary,

2005). Other algorithms and approaches for the prob-

lem of enumerating minimal models and some re-

lated problems are proposed in (Reiter, 1987; Avin

and Ben-Eliyahu-Zohary, 2001; Angiulli et al., 2014;

Ben-Eliyahu-Zohary et al., 2017).

In this work, we mainly propose approaches to

enumerate all the minimal models for two particular

classes of CNF formulæ. The ﬁrst class corresponds

to CNF formulæ where each clause is either positive

or negative, called PN formulæ. The second class is

that of CNF formulæ where each clause is either pos-

itive or a Horn clause, called PH formulæ. Clearly,

the class of PH formulæ is a generalization of that

of PN formulæ. From the computational complexity

point of view, we show that the problem of checking

whether a model is minimal is tractable in the case

of PN formulæ, while it is coNP-complete in the case

of PH formulæ. To the best of our knowledge, the

formula classes of PN formulæ and PH formulæ have

not been studied in the case of the problem of minimal

model enumeration.

We provide two methods to enumerate the min-

imal models in the case of PN formulæ. The ﬁrst

one consists in using the problem of enumerating the

minimal transversals of a hypergraph to obtain all the

minimal models that are contained in a model. Note

that we provide an interesting property that allows us

to reduce the search space in solving the latter prob-

lem. Indeed, we provide in this property the part

shared between all the minimal models that are con-

tained in a given model. Our second method consists

in a SAT-based encoding. The main idea consists in

using a formula expressing the fact that the truth value

true is assigned to a variable if and only if there is a

Salhi, Y.

On Enumerating All the Minimal Models for Particular CNF Formula Classes.

DOI: 10.5220/0007257104030410

In Proceedings of the 11th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2019), pages 403-410

ISBN: 978-989-758-350-6

403

positive clause which is satisﬁed only by this assign-

ment. Since checking whether a Boolean interpreta-

tion is a model of a CNF formula or not is a linear

time task, our encoding can be seen as a proof that the

problem of checking whether a model is minimal or

not is tractable in the case of PN formulæ.

Our approach in the case of PH formulæ is based

on enumerating the minimal models of its greatest PN

subformula (the subset of all the positive and negative

clauses). In fact, we provide a simple characterization

of the minimal models in the case of PH formulæ that

uses the procedure of unit propagation and the mini-

mal models of the greatest PN subformulæ. Using this

characterization, we propose an algorithm that uses

our SAT-based encoding in the case of PN formulæ.

2 PRELIMINARIES

2.1 Propositional Satisﬁability and

Minimal Models

A CNF formula is a conjunction of clauses where a

clause is a disjunction of literals. A literal is a propo-

sitional variable (positive literal) or its negation (neg-

ative literal). A positive clause (resp. negative clause)

is a clause that contains only positive (resp. negative)

literals. In addition, a Horn clause is a clause that

contains at most one positive literal. We denote by

Var(φ) (resp. Lit(φ)) the set of propositional vari-

ables (resp. literals) occurring in the formula φ. A

CNF formula is binary if each clause contains at most

two literals. Propositional variables are denoted by

the lowercase letters p, q, r, and CNF formulæ by the

Greek letters φ, ψ, χ.

A CNF formula can also be seen as a set of clauses

and a clause as a set of literals. Thus, a CNF formula

of the form (l

∨ · ·· ∨ l

) ∧ ··· ∧ (l

∨ · ·· ∨ l

) can

be also represented in this work by the set of clauses

{(l

∨ ·· · ∨ l

),. .. ,(l

∨ ·· · ∨ l

)} and the set of sets

of literals {{l

,. ..,l

},. ..,{l

,. ..,l

}}.

A Boolean interpretation of a formula φ is an as-

signment that associates truth values in {0,1} to the

propositional variables in Var(φ), where 0 stands for

false and 1 stands for true. It is extended to formulæ

as usual. A model of a formula is a Boolean inter-

pretation satisfying this formula, i.e., an interpreta-

tion making this formula true. We use Models(φ) to

denote the set of models of φ. The problem of de-

termining if there exists a model that satisﬁes a given

propositional formula (|Models(φ)| ≥ 1), abbreviated

as SAT, is one of the most studied NP-complete prob-

lems.

Given a set of literals L, we use φ

to denote the

formula obtained from φ by removing every clauses

containing a literal in L and by removing all the

complement literals of those in L. More precisely,

= {c ∈ φ | ∀l ∈ L,l /∈ c & l /∈ c} ∪ {c \ {l} | c ∈

φ & l ∈ L & l ∈ c}.

For convenience purposes, we represent the

Boolean interpretations as sets of variables. More pre-

cisely, the set m represents the Boolean interpretation

that associates 1 to the variables in m and 0 to the

variables in Var(φ) \ m. Further, we use (m)

−

to de-

note the set of negative literals that are true by m, i.e.,

(m)

−

= {¬p | p ∈ Var(φ) \ m}.

Deﬁnition 2.1 (Minimal Model). Let φ be a CNF for-

mula and m a model of φ. Then, m is a minimal model

of φ iff there is no model m

of φ such that m

is a

proper subset of m (m

⊂ m).

The problem of checking whether a model of a

CNF formula is minimal is coNP-complete (Cadoli,

1992a).

We use MinModels(φ) to denote the set of min-

imal models of φ. Moreover, given a model m of

φ, we use MinMS(φ,m) to denote the set of minimal

models that are contained in m, i.e., MinMS(φ,m) =

MinModels(φ) ∩ {m

∈ Models(φ) | m

⊆ m}.

2.2 PN and PH formulæ

We here describe two classes of CNF formulæ,

namely PN (for Positive and Negative) and PH (for

Positive and Horn) formulæ. In this work, we de-

ﬁne different methods for enumerating all the mini-

mal models in the case of these classes.

Deﬁnition 2.2 (PN formula). A PN formula φ is a

CNF formula where each clause is either positive or

negative.

We use Pos(φ) (resp. Neg(φ)) to denote the set of

positive (resp. negative) clauses of φ.

One can easily show that each CNF formula can

be linearly transformed into an equisatisﬁable PN for-

mula. Indeed, we only have to use fresh proposi-

tional variables for renaming negative literals in the

mixed clauses. As a consequence, the problem of

checking the satisﬁability of a PN formula is also NP-

complete. Note that this property has been pointed

out in (Boumaraﬁ et al., 2017).

Deﬁnition 2.3 (PH formula). A PH formula φ is a

CNF formula where each clause is either positive or

Horn.

We use Pos(φ) (resp. Horn(φ)) to denote the set

of positive (resp. Horn) clauses of φ.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

404

2.3 Hypergraphs and Minimal

Transversals

A hypergraph H is an ordered pair (V,E) where V is a

non empty set of vertices and E ⊆ P (V ) \

0 is a set of

hyperedges, where P(V ) denotes the power set of V ,

i.e., its set of subsets. In other words, the hypergraph

notion is a generalization of that of undirected graph.

Further, a transversal of H is a set of vertices s ⊆ V

that intersects every hyperedge of H, i.e., e ∩ s 6=

for every e ∈ E. A transversal of H is minimal if it

is not a proper superset of any other transversal of

H, i.e., s

6⊂ s for every transversal s

of H. Given a

hypergraph H, we use MT (H) to denote the set of all

the minimal transversals of H.

Figure 1: A Hypergraph.

Consider, for instance, the hypergraph described

in Figure 1, the set s = {3,5} is a minimal transversal

since it intersects every hyperedge. The set {2, 4,5}

is the only other minimal transversal.

In the literature, there are several methods for gen-

erating the minimal transversals of a hypergraph (see

eg (Fredman and Khachiyan, 1996; Khachiyan et al.,

2005; Khachiyan et al., 2007; H

ebert et al., 2007)).

In particular, the authors of (H

ebert et al., 2007) have

shown how efﬁcient datamining techniques, such as

the levelwise framework, can be used for generating

the minimal transversals. In this work, we use an al-

gorithm for generating the minimal transversals as a

black box in our methods for generating the minimal

models.

3 COMPUTATIONAL

COMPLEXITY

In this section, we provide computational complexity

results for the problem of checking whether a model is

minimal in the case of PN formulæ and PH formulæ.

Theorem 3.1. The problem of checking whether a

model of a PN formula is minimal or not is in P.

Proof. Using the SAT-based encoding deﬁned in Sec-

tion 5, we know that the problem of checking whether

a model of a PN formula is minimal or not is in P. In-

deed, it is well known that the problem of checking

whether a Boolean interpretation is a model of a for-

mulæ can be solved in linear time.

Let G = (V, E) be an undirected graph such that V

is the set of its vertices and E is the set of its edges.

A 3-coloring of G corresponds to a partition of V into

3 subsets P = {V

} so that no two vertices in

a same subset are adjacent, and V

, V

and V

re-

fer to three different colors c

, c

and c

respectively.

It is well known that deciding whether an undirected

graph admits a 3-coloring is an NP-complete prob-

lem. In the following theorem, we show that the com-

plement of the 3-coloring problem can be transformed

into the problem of checking whether a model of PH

formula is minimal.

Theorem 3.2. The problem of checking whether a

model of PH formula is minimal is coNP-complete.

Proof. Using the fact that the same problem in the

general case is coNP-complete, we know that it is

also in coNP in the case of PH formulæ. In order

to show the coNP-completeness, we reduce the com-

plement of the 3-coloring problem to the problem of

checking whether a model of a PH formula is mini-

mal. Let G = (V, E) be an undirected graph such that

V = {1, ... ,n}. We associate to each vertex i ∈ V

and each color c ∈ {r,g,b} a propositional variable

denoted p

. We also use an additional fresh variable

denoted s. Then, the encoding is deﬁned as the con-

junction of the following formulæ:

i∈1..n

∨ p

) (1)

i∈1..n

[(p

∨ ¬p

)∧

(¬p

∨ p

∨ ¬p

) ∧ (¬p

∨ ¬p

∨ p

)]

(2)

i∈1..n−1

[¬p

∨ ¬p

∨ (p

i+1

∧ p

i+1

∧ p

i+1

)]

∧[¬p

∨ ¬p

∨ (p

∧ p

)]

(3)

i< j,{i, j}∈E,c∈{r,g,b}

(¬p

∨ ¬p

∨ s) (4)

(¬s ∨ p

) ∧ (¬s ∨ p

) (5)

(1) is used to express that each vertex has at least

one color, and (2) expresses that if a vertex has more

than one color, then it has the three considered col-

ors. (3) is used to add the fact that if there is a vertex

that has three colors, then every other vertex has also

three colors. (4) means that s is true if there are two

adjacent vertices that have the same color. Finally, (5)

On Enumerating All the Minimal Models for Particular CNF Formula Classes

405

expresses that the fact that s is true implies that the

vertex 1 has three colors. In fact, we show that our

encoding captures the complement of the 3-coloring

problem by showing that there is no 3-coloring iff the

model that associates three colors to every vertex is a

minimal model.

Using the fact that each formula of the form ¬p

∨

·· · ∨¬p

∨(q

∧· ·· ∧q

) is equivalent to the conjunc-

tion of Horn clauses (¬p

∨ ·· · ∨ ¬p

∨ q

) ∧ ·· · ∧

(¬p

∨ ·· · ∨ ¬p

∨ q

), we know that the encoding

= (1) ∧ (2) ∧ (3) ∧ (4) ∧ (5) corresponds to a PH

formula.

Knowing that each clause in our encoding con-

tains at least one positive literal, we have m =

{s, p

, p

,. .., p

, p

} is a model of this encod-

ing. In this context, the proof of coNP-completeness

comes from the fact that m is a minimal model of our

encoding iff G does not admit a 3-coloring. Using

(1) ∧ (2), we know that each model of φ

satisﬁes

either a unique positive literal or the three literals in

, p

} for every i ∈ 1..n. In addition, using (3),

we know that if a model of φ

satisﬁes the three pos-

itive literals {p

, p

} for some i ∈ 1..n, then this

model satisﬁes all the positive literals in {p

, p

}

for every j ∈ 1..n. In other words, every model of φ

different from m associates a single color to each ver-

tex of G. (4) ∧ (5) is used to force every model of

to satisfy all the literals in {s, p

, p

} if it asso-

ciates the same color to two adjacent vertices. Thus,

every model of φ

which is different from m (and con-

sequently contained in m) corresponds to a 3-coloring

of G. Conversely, every 3-coloring of G can be eas-

ily used for deﬁning a model of φ

different from m.

Thus, m is not a minimal model of φ

if and only if G

admits a 3-coloring.

4 PN FORMULÆ AND MINIMAL

MODELS

We here provide interesting properties that allow us to

reduce the search space for solving in the case of PN

formulæthe problem of enumerating all the minimal

models that are contained in a given model. We also

show how the problem of listing the minimal transver-

sals can be directly used for enumerating these mini-

mal models. Then, we use this result for deﬁning an

algorithm for enumerating all the minimal models of

a PN formula.

Deﬁnition 4.1. Let φ be a PN formula, m a model

of φ and n ∈ N. Then, we deﬁne π(φ,m) and ν(φ, m)

as follows: π(φ,m) = {p ∈ Var(φ) | p ∈ φ

|(m)

−

} and

ν(φ,m) = {¬p | p ∈Var(φ) and ∀c ∈ ψ w. p ∈ c, ∃c

∈

ψ : c

⊆ c \ {p}}, where ψ = φ

|(m)

−

∪π(φ,m)

Note that φ

|(m)

−

∪π(φ,m)

contains only positive

clauses, since φ is a PN formula and all the negative

clauses are satisﬁed by satisfying the literals in (m)

−

Proposition 4.1. Given a PN formula φ and a model

m of φ, (m)

−

⊆ ν(φ,m) holds.

Proof. This property is a direct consequence of the

fact that there is no clause in φ

|(m)

−

∪π(φ,m)

that con-

tains a variable occurring in (m)

−

One can easily see that π(φ, m) is included in ev-

ery model which is contained in m, since this set cor-

responds to unit clauses occurring in φ after assign-

ing the truth value 0 to the variables in Var(φ) \ m.

Moreover, from the deﬁnition of ν(φ,m), the for-

mula obtained by assigning 0 to the variables in S =

{p ∈ Var(φ) | ¬p ∈ ν(φ,m)} is equisatisﬁable to ψ =

|(m)

−

∪π(φ,m)

. As a consequence, we get m

∩S =

0 for

every minimal model m

⊆ m.

Let us recall that a positive monotone formula is

deﬁned as a CNF formula that contains only positive

clauses.

Proposition 4.2. Let φ be a positive monotone for-

mula s.t., for every clause c ∈ φ, c contains at least

two literals. Then, for every positive literal p ∈

Lit(φ), if ∃c ∈ φ s.t. p ∈ c and c

6⊆ c for every c

∈ φ,

then there exists a minimal model m of φ s.t. p ∈ m

Proof. Let p a literal in Lit(φ) and c ∈ φ s.t. p ∈ c and

6⊆ c for every c

∈ φ. Let us consider the formula

ψ = φ

|{¬q|q∈c\{p}}

. Clearly, ψ does not contain the

empty clause since there is no clause c

∈ φ s.t. c

⊆

c \ {p}. Knowing that all the models of ψ contain the

literal p since it is a unit clause in ψ, we deduce that

there exists a minimal model of φ that contains p.

The following theorem allows us to reduce the

search space for solving the problem of enumerating

all the minimal models in the case of PN formulæ.

Theorem 4.1. Let φ be a PN formula and m

a model of φ. Then, the following properties

are satisﬁed: (i)

∈MinMS(φ,m)

= π(φ,m) and

(ii)

∈MinMS(φ,m)

)

−

= ν(φ,m).

Proof. We here use S (resp. S

) to denote the set

∈MinMS(φ,m)

(resp.

∈MinMS(φ,m)

)

−

). We

only consider the part ⊇, the other being similar. For

this part, it sufﬁces to show the two following proper-

ties: (i) π(φ,m) ⊆ S and (ii) ν(φ,m) ⊆ S

. Clearly,

all the minimal models in MinMS(φ, m) satisfy the

literals in (m)

−

. Moreover, all the models of φ

|(m)

−

includes all the unit clause occurring in φ

|(m)

−

. As

a consequence, π(φ,m) ⊆ S holds. Further, let p

be a propositional variable s.t. ∀c ∈ Pos(φ

|(m)

−

∪π(φ,m)

)

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

406

Algorithm 1: An algorithm for computing the minimal mod-

els of a PN formula.

1: procedure MINMODSPN(φ)

2: r ←

3: ψ ← φ

4: while SAT(ψ) do

5: tmp ← MinMS(φ,m)  m is a model of ψ

6: ψ ← ψ ∧

∈tmp

7: r ← r ∪ tmp

8: return r

with p ∈ c, ∃c

∈ Pos(φ

|(m)

−

∪π(φ,m)

) s.t. c

⊆ c \ {p}.

Then, for every model m

of φ

|(m)

−

∪π(φ,m)

, m

\ {p}

satisﬁes also φ

|(m)

−

∪π(φ,m)

. Thus, ν(φ,m) ⊆ S

holds.

Given a positive monotone formula φ, we use H

to denote a hypergraph associated to φ and deﬁned as

the ordered pair (Lit(φ), φ). Clearly, using the deﬁni-

tion of H

and the notion of minimal transversal, we

get the following property.

Proposition 4.3. Let φ be a positive monotone CNF

formula. Then, we have MinModels(φ) = MT (H

Thus, using Theorem 4.1, we obtain a simple char-

acterization of the minimal models that are contained

in a given model in the case of PN formulæ. Indeed,

the following property shows that enumerating these

minimal models is reduced to enumerating the mini-

mal transversals of a hypergraph.

Proposition 4.4. Let φ be a PN formula and m a

model of φ. Then, we have MinMS(φ,m) = {s ∪

π(φ,m) | s ∈ MT (H

)} where ψ = φ

|π(φ,m)∪ν(φ,m)

Algorithm 1 shows how the method described in

Proposition 4.4 can be simply used for enumerat-

ing all the minimal models of a PN formula. In-

deed, given a PN formula φ, MINMODSPN(φ) uses

MinMS(φ,m) to get all the minimal models that are

contained in m in every step of the while loop (see

Line 5). The soundness of Algorithm 1 is a direct

consequence of Proposition 4.4. It is worth mention-

ing that we add in Line 6 the negations of the found

minimal models to avoid ﬁnding the same minimal

model more than once.

As a side note, connections between minimal

transversals and minimal models have already been

made in the literature (e.g. (Reiter, 1987)). More-

over, the graph structure has already been used for

computing minimal models through the notion of de-

pendency graph (Ben-Eliyahu-Zohary, 2000; Ben-

Eliyahu-Zohary, 2005; Angiulli et al., 2014).

Algorithm 2: An algorithm for computing a minimal models

of a PN formula contained in a given model.

1: procedure SOLVEMMP(φ, m)

2: r ← m

3: for p ∈ m do

4: if MOD(r \ {p},φ) then

5: r ← r \ {p}

6: return r

5 A SAT-BASED APPROACH FOR

ENUMERATING MINIMAL

MODELS

In this section, we provide a SAT-based encoding for

enumerating all the minimal models of a PN formula.

the main idea consists in using a formula expressing

the fact that the truth value 1 is assigned to a proposi-

tional variable if and only if there is a positive clause

which is satisﬁed only by this assignment.

The following proposition provides a characteri-

zation of the minimal models in the case of PN for-

mulæ.

Proposition 5.1. Given a PN formula φ and a model

m of φ, m is a minimal model of φ iff for all p ∈ m,

there exists a clause c ∈ Pos(φ) s.t. p ∈ c and

c \ {p} ⊆ Var(φ)\ m.

Proof.

Part ⇒. Let m be a minimal model of φ. Assume that

there exists p ∈ m s.t. for every clause c in Pos(φ)

containing p, we have c \ {p} 6⊆ Var(φ) \ m. Thus,

m \ {p} satisﬁes all the clauses containing p, and con-

sequently, it satisﬁes φ. Therefore, we get a contra-

diction since m is a minimal model of φ.

Part ⇐. Let m be a model of φ s.t., for all p ∈ m,

there exists a clause c ∈ Pos(φ) s.t. p ∈ c and c\{p} ⊆

Var(φ)\m. Then, for all p ∈ m, φ∧(

q∈Var(φ)\m

¬q)∧

¬p is unsatisﬁable. As a consequence, m is a minimal

model of φ.

Now, using the characterization provided in

Proposition 5.1, we propose a SAT-based encoding

for enumerating all the minimal models of a PN for-

mula. Given a PN formula φ, we use E

(φ) to de-

note the following SAT encoding: φ ∧ (

p∈Var(φ)

p →

c∈Pos(φ),q∈c

¬(c \ {q})).

Let us recall that any propositional formula can

be translated to CNF using Tseitin’s linear encod-

ing (Tseitin, 1968). To translate our encoding to

CNF, one can associate to each subformula of the

form c \ {p} a fresh propositional variable r with the

clauses (¬r ∨ c \ {p}) ∧

q∈c\{p}

(r ∨ ¬q) that repre-

sent the equivalence r ↔ c \ {p}.

On Enumerating All the Minimal Models for Particular CNF Formula Classes

407

Theorem 5.1 (Soundness). Let φ be a PN formula.

Then, Models(E

(φ)) = MinModels(φ) holds.

Proof. The soundness of the SAT encoding E

(φ)

is a consequence of Porposition 5.1. Indeed, the

subformula (

p∈Var(φ)

p →

c∈Pos(φ),p∈c

c \ {p}) al-

lows us to get the property that for all p ∈ m,

there exists a clause c ∈ Pos(φ) s.t. p ∈ c and

{¬q | q ∈ c \ {p}} ⊆ m.

As mentioned in our proof of Theorem 3.1, this

encoding shows that the problem of checking whether

a model is minimal (called minimal model checking

problem) in the case of PN formulæ is tractable, since

checking whether a Boolean interpretation is a model

of a CNF formula or not can be performed in linear

time.

Let us now consider a problem related to minimal

model computation, called model minimization prob-

lem (Angiulli et al., 2014).

Deﬁnition 5.1 (Model Minimization Problem).

Given a propositional formula φ and a model m of

φ, compute a minimal model m

for φ contained in the

model m.

The model minimization problem is P

NP[O(log(n))]

hard in the general case where n is the number of

variables (Angiulli et al., 2014). This means that this

problem is as hard as a problem that can be solved us-

ing a polynomial deterministic algorithm which uses

O(log(n)) calls of an NP oracle.

Theorem 5.2. The model minimization problem is

tractable in the case of PN formulæ.

Proof. We here show that Algorithm 2 is sound.

In this algorithm the procedure MOD(r \ {p},φ) in

Line 4 is used for checking whether r \{p} is a model

of φ or not. Clearly, the returned Boolean interpreta-

tion r is a model of φ and contained in m. Assume

now that this returned interpretation is not a minimal

model of φ. Then using Proposition 5.1, there exists

p ∈ r s.t. c \ {p} ∩ r 6=

0 for every clause c contain-

ing the literal p. Thus, for all m

with r ⊆ m

, we get

c\{p}∩m 6=

0 for every clause c containing the literal

p. As a consequence, for all m

with r ⊆ m

, m

\ {p}

is a model of φ. Therefore, p /∈ r holds and we get a

contradiction.

6 MINIMAL MODEL

ENUMERATION FOR PH

FORMULÆ

In this section, we propose an approach for enumerat-

ing all the minimal models in the case of PH formulæ.

Algorithm 3: An algorithm for computing the minimal mod-

els of a PH formula.

1: procedure MINMODSPH(φ)

2: r ←

3: ψ ← E

(Pos(φ) ∪ Neg(φ))

4: while SAT(ψ) do  m is a minimal model of

Pos(φ) ∪ Neg(φ)

5: m

← m ∪ pup(M(φ)

)

6: if (∀c ∈ Neg(φ), {p | ¬p ∈ c} 6⊆ m

) then

7: r ← r ∪ {m

}

8: ψ ← ψ ∧ m

9: return r

This approach is based on enumerating the minimal

models of its greatest PN subformula (the subformula

consisting of all the positive and negative clauses),

which means that our methods for enumerating the

minimal models in the case of PN formulæ can be

used in the case of PH formulæ.

Unit propagation is a procedure consisting in ap-

plying the following two rules for every unit clause l

occurring in the CNF formula: (i) every clause con-

taining the literal l, except the unit clause l, is re-

moved; and (ii) all the occurrences of l are removed.

For instance, applying unit propagation to the for-

mula {¬p ∨ ¬q, p, p ∨ r,q ∨ r} produce the formula

{¬q, p,r}. Indeed, using the two rules with the lit-

eral p produces the unit clause ¬q and removes p ∨ r,

and then, we get the literal r from the clause q ∨ r by

propagating ¬q. It is worth noting that an efﬁcient im-

plementation of unit propagation in linear time is pro-

vided in (Crawford and Auton, 1993). In this work,

we use pup(φ) to denote the set of positive literals

that are propagated in the procedure of unit propaga-

tion. Considering again the previous example, we get

pup({¬p ∨ ¬q, p, p ∨ r, q ∨ r}) = {p,r}.

Given a PH formula φ, we use M(φ) to denote the

set of clauses φ \ (Pos(φ) ∪ Neg(φ)). Note that each

clause in M(φ) contains at least one negative literal

and exactly one positive literal.

Theorem 6.1. Let φ be a PH formula. Then,

MinModels(φ) = {m ∪ pup(M(φ)

) | m ∈

MinModels(Pos(φ) ∪ Neg(φ)) & ∀c ∈ Neg(φ),{p |

¬p ∈ c} 6⊆ m ∪ pup(M(φ)

)} holds.

Proof. We use in this proof S to denote the

set {m ∪ pup(M(φ)

) | m ∈ MinModels(Pos(φ) ∪

Neg(φ)) & ∀c ∈ Neg(φ),{p | ¬p ∈ c} 6⊆ m ∪

pup(M(φ)

)}.

Part ⊆. Let m be a minimal model in MinModels(φ).

Then, we know that there exists a minimal model

of Pos(φ) ∪ Neg(φ) s.t. m

⊆ m since Pos(φ) ∪

Neg(φ) is a subformula of φ. Clearly, we have

pup(M(φ)

)) ⊆ m \ m

. Moreover, knowing that

∀c ∈ Neg(φ),{p | ¬p ∈ c} 6⊆ m since m satisﬁes φ,

{p | ¬p ∈ c} 6⊆ m

∪ pup(M(φ)

) holds for every

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

408

c ∈ Neg(φ). As a consequence, using the fact that m

is a minimal model of φ, we get m = m

∪ pup(φ

Thus, m belongs to the set S.

Part ⊇. Let m = m

∪ pup(M(φ)

) an element of

S. Then, using the property ∀c ∈ Neg(φ),{p | ¬p ∈

c} 6⊆ m and the fact that m

satisﬁes ψ = Pos(φ) ∪

Neg(φ), we know that m is a model of φ. Further,

for all p ∈ m

, φ ∧

q/∈m

¬q ∧ ¬p is unsatisﬁable since

∈ MinModels(ψ). We know also that for all p ∈

pup(M(φ)

), φ∧

q/∈m

¬q ∧¬p is unsatisﬁable since

all the elements of pup(M(φ)

) are logical conse-

quences of φ

. Thus, m is a minimal model of φ.

Theorem 6.1 shows how a method for enumerat-

ing the minimal models of a PN formula can be used

to deﬁne a method in the case of PH formulæ. For

instance, Algorithm 3, shows how our SAT-based en-

coding proposed in Section 5 can be used for gener-

ating all the minimal models of a PH formula. In-

deed, given a PH formula φ, MINMODSPH(φ) uses

the SAT-based encoding E

(Pos(φ) ∪ Neg(φ)) for

generating the minimal models of Pos(φ)∪ Neg(φ).

It is noteworthy that, using the fact that each bi-

nary CNF formula is also a PH formula, we get the

following property.

Corollary 6.1. Let φ be a binary CNF for-

mula. MinModels(φ) = {m ∪ pup(M(φ)

) | m ∈

MinModels(Pos(φ) ∪ Neg(φ)) & ∀c ∈ Neg(φ),{p |

¬p ∈ c} 6⊆ m ∪ pup(M(φ)

)} holds.

7 A MODELING EXAMPLE

We here describe a modeling example showing that

the problem of enumerating the minimal models in

the case of PN formulæ can be used for solving the

datamining problem of enumerating the minimal gen-

erators (Zaki, 2004).

Let T be a set of items. From now on, we as-

sume T ﬁxed. A transaction over T is an ordered

pair (tid,I) where tid ∈ N is its identiﬁer and I ⊆ T

is an itemset. A transaction database is a ﬁnite set of

transactions where each identiﬁer refers to only one

transaction. Given an itemset J, we say that (tid,I)

supports J if we have J ⊆ I. Further, the cover of an

itemset J in a database D, denoted C(J, D), is deﬁned

as follows: C(J,D) = {tid | (tid,I) ∈ D & J ⊆ I}.

Deﬁnition 7.1 (Closed Itemset). Let D be a transac-

tion database and I an itemset. Then, I is closed if, for

every itemset J with I ⊂ J, C (J,D) ⊂ C (I,D) holds.

Deﬁnition 7.2 (Minimal Generator). Let D be a

transaction database and I an itemset. Then, I is a

minimal generator if, for every itemset J with J ⊂ I,

C(I,D) ⊂ C (J,D) holds.

Table 1: Transaction database D.

tid itemset

1 a,b,c, d,e

2 a,b,c, f

3 a,b, c, m

4 a,c,d, f , j

5 j,l

6 e, f

Note that for every minimal generator I, there

exists a single closed itemset J such that C(I,D) =

C(J, D), and we say that I generates J.

Consider the transaction database D described in

Table 1. In this database, {a,b,c} and {a, c} are

closed itemsets and {a,b} and {a} are minimal gen-

erators that generate {a,b, c} and {a,c} respectively.

To deﬁne the encoding of the problem of gen-

erating all the minimal generators, we associate to

each item a a variable denoted p

which is used to

represent the fact that a is in the minimal genera-

tor. Further, for all i ∈ 1..n, the variable q

is used

to represent the fact that the itemset appears in the

transaction t

. The encoding M G(D) is

i=1

(¬q

↔

a/∈T \t

). Clearly, each subformula of the form

¬q

↔

a/∈t

¬p

is equivalent to the following PN

formula: (q

∨

a/∈T \t

) ∧

a/∈T \t

(¬p

∨ ¬q

It is important to note that the encoding M G(D)

is similar to that proposed in (Jabbour et al., 2017) for

generating the frequent itemsets. Furthermore, a SAT-

based encoding for generating the minimal generators

has been proposed in (Boudane et al., 2017).

Proposition 7.1. Let D = {(1,t

),. .. ,(n,t

)} be a

transaction database over the set of items T . Then, m

is a minimal model of M G(D) iff I

= {a | p

∈ m}

is a minimal generator in D such that C (I

,D) = {i ∈

1..n | q

∈ m}.

Proof. We only consider the only if part, the other

part being similar. Let m be a minimal model of

M G(D). Then, C (I

,D) = {i ∈ 1..n | m(q

) = 1}

holds. Assume that I

is not a minimal generator.

Then there exists a ∈ I

such that C(I

,D) = C (I

{a},D). Then, m\{p

} is a model of M G(D). Thus,

we get a contradiction since m is a minimal model of

M G(D). Therefore, I

is a minimal generator in the

transaction database D.

8 CONCLUSION AND

PERSPECTIVES

We proposed approaches for generating all the mini-

mal models in two particular classes of CNF formulæ.

The ﬁrst class is that of PN formulæ which are deﬁned

as CNF formulæ where each clause is either positive

On Enumerating All the Minimal Models for Particular CNF Formula Classes

409

or negative, whereas the second class is that of PH for-

mulæ in which each clause is either positive or a Horn

clause. The ﬁrst contribution consists in showing that

the problem of determining whether a model is mini-

mal is tractable in the case of PN formulæ, whereas it

is coNP-complete in the case of PH formulæ. Then,

we introduced our ﬁrst approach for enumerating all

the minimal models of a PN formula, which is based

on the use of an algorithm for generating the minimal

transversals. We also proposed a SAT-based encoding

for enumerating all the minimal models of a PN for-

mula. Next, we provided an interesting characteriza-

tion of the minimal models in the case of PH formulæ

that allows us to use our approaches in the case of PN

formulæ for enumerating the minimal models for the

PH formulæ. Finally, we described a simple modeling

example in datamining.

As a future work, we intend to implement and

evaluate the proposed methods for generating the

minimal models. We also plan to use similar ap-

proaches for other formula classes.

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