atarazanas.sci.uma.esatarazanas.sci.uma.es/docs/articulos/16494489.pdfannals of mathematics and...

Annals of Mathematics and Artificial Intelligence 42: 369–398, 2004. 2004 Kluwer Academic Publishers. Printed in the Netherlands.

A new algebraic tool for Automatic Theorem Provers ∗

Multisemilattice: A structure to improve the efficiency of proversin temporal logics

P. Cordero, G. Gutiérrez, J. Martínez and I.P. de Guzmán

Dpto. Matemática Aplicada, E.T.S.I. Informática, Universidad de Málaga, 29071, SpainE-mail: [email protected], {gloriagb,javim,guzman}@ctima.uma.es

The concepts of implicates and implicants are widely used in several fields of “AutomatedReasoning”. Particularly, our research group has developed several techniques that allow usto reduce efficiently the size of the input, and therefore the complexity of the problem. Thesetechniques are based on obtaining and using implicit information that is collected in termsof unitary implicates and implicants. Thus, we require efficient algorithms to calculate them.In classical propositional logic it is easy to obtain efficient algorithms to calculate the set ofunitary implicants and implicates of a formula. In temporal logics, contrary to what we see inclassical propositional logic, these sets may contain infinitely many members. Thus, in orderto calculate them in an efficient way, we have to base the calculation on the theoretical studyof how these sets behave. Such a study reveals the need to make a generalization of LatticeTheory, which is very important in “Computational Algebra”. In this paper we introduce themultisemilattice structure as a generalization of the semilattice structure. Such a structure isproposed as a particular type of poset. Subsequently, we offer an equivalent algebraic char-acterization based on non-deterministic operators and with a weakly associative property. Wealso show that from the structure of multisemilattice we can obtain an algebraic character-ization of the multilattice structure. This paper concludes by showing the relevance of themultisemilattice structure in the design of algorithms aimed at calculating unitary implicatesand implicants in temporal logics. Concretely, we show that it is possible to design efficientalgorithms to calculate the unitary implicants/implicates only if the unitary formulae set hasthe multisemilattice structure.

Keywords: automated deduction, theorem provers, implicants, lattice theory, multisemilat-tice, ideals

1. Introduction

TAS1-methodology is a well-established alternative [18,19] to Tableaux-methodsand Resolution in Automated Deduction which has already been developed for classical,multivalued, modal, and temporal logics [1,9,13,14,17,26]. TAS is based on using the

∗ This paper has been partially supported by Spanish DGI projects BFM2000-1054-C02-02, TIC2000-1109,TIC2003-08687-C02-01 and TIC2003-09001-C02-01.

1 Spanish translation of “Transformaciones de Arboles Sintacticos”.

370 P. Cordero et al. / A new algebraic tool for Automatic Theorem Provers

information implicit in the formula to reduce its size by using rewriting rules. Thesereduction rules have a cost which is, at most, polynomial and are used to delay branching(responsible for the exponential cost of the prover) and which is applied in a lazy way.

Although the reduction rules have been developed with TAS methodology in mind,they can be easily used to improve the efficiency of any prover.

The implicit information in the formula, which serves as the basis for the reductiontechniques, is obtained with the lowest possible cost from the unitary implicates andimplicants of the formula.

In temporal logics, contrary to what we see in classical propositional logic, thesets of unitary implicates and implicants can have an infinite cardinality and so, in or-der to manipulate them efficiently, a more thorough theoretical study is required. Thisinvolves an important element of computational algebra, namely Lattice Theory and itsgeneralizations [16].

Our work focuses on temporal logics that are an extension of propositional classicallogics, since these preserve all the properties of propositional logic, but expand theirsemantics to work with temporal matters. The ultimate aim is to design an automatedtheorem prover for a fully expressive temporal logic such as the US logic of Kamp [21]or the LN logic introduced in [2].

With this in mind, the first TAS-based prover was designed for FNext logic [13,14].This logic was chosen because it achieves a good balance between complexity and ap-plicability. This TAS-prover, the TAS-FNext prover, was compared to other proversin [14] with very good results. The set of unitary formulae or literals with “logic impli-cation” has a lattice structure (see figure 1 in section 3) which made it possible to designalgorithms able to calculate the unitary implicates and implicants of the formula, withlinear costs.

When the complexity of the FNext± logic is increased by including past connec-tives, we are faced with an important problem: the partially ordered set (poset) of literalsdoes not have a lattice structure (see figure 2 in section 4). However, the properties ofthis poset allows the design of algorithms able to calculate the unitary implicates andimplicants of the formula, with linear costs as well [11,15]. This fact led us to considerwhat the minimal structure a literal poset should have to make possible the design ofthese kinds of algorithms for temporal or modal logics. In this work we deal with thisissue by introducing the multisemilattice structure.

In a nutshell

– Our group has developed reduction techniques that improve the efficiency of anyprover.

– These techniques are based on an efficient calculus of the unitary implicates/impli-cants of a formula.

– It is not possible to calculate the unitary implicates/implicants of a formula in anefficient way in every logic.

– In this paper we present the multisemilattice structure.

P. Cordero et al. / A new algebraic tool for Automatic Theorem Provers 371

– In logics in which the set of literals has a multisemilattice structure, it is possible tocalculate efficiently the unitary implicates/implicants.

We assume readers are familiar with the basic concepts and results obtained fromposets, semilattices, and lattices. In particular, we use the concept of ideal in a lattice(a sublattice that is lower closed). Equally, we assume that the basics of �-algebras andfreely generated �-algebras are known. Nevertheless, we have included an appendixdiscussing them. We will make use of the following notation:

Notation. Given a poset (A,�), X ⊆ A and a ∈ A, we denote by

– (a] and [a) the sets {x ∈ A | x � a} and {x ∈ A | x � a}, respectively.

– (X] the ideal generated by X, that is, the intersection of all the ideals that contain X.

This paper is structured as follows: In section 2 we introduce a formalization oftemporal propositional logic. However, the temporal propositional logics we are inter-ested in are those that preserve all the laws of classical propositional logic. In particularwe are interested in FNext and FNext±. In section 3 we introduce the concepts of aunitary formula and a literal and in section 4 we study the calculus of unitary implicateand implicant sets in temporal logics. In order to do this, we introduce the concepts ofa restriction of an ideal and a generating set. In this section we outline the problem thatgave rise to the introduction of the new multisemilattice structure described in section 5.This structure generalizes the semilattice structure. From this, in section 6, we introducethe structure of multilattice in a natural way. In section 7 we show the the utility of themultisemilattice in the calculus of unitary implicates/implicants. In section 8 we high-light the utility of the multisemilattice structure in the recursive method we are workingwith. Finally, we include an appendix at the end of the paper with some aspects relatedto abstract algebra.

2. Temporal logics that are extensions of classical logic

In this section, we introduce a formalization of temporal propositional logics asan abstract algebra, and within this context, characterize those which are extensions ofclassical propositional logic.

Given that a logical system is formed by a formal language and a model theoryregarding such a language, temporal propositional logic is defined as a pair (L, I) whereL is a propositional language and I a set of interpretations for such a language.

A propositional language can be defined as an �-algebra freely generated by a setof atoms, V , in the � category, in the following way:

Definition 2.1. Let us consider the following sets:

• An infinite denumerable set V = {p, q, . . . , p1, q1, . . . , pn, qn, . . .} whose elementsare called atoms or atomic fomulae.


• A finite domain of operators � = {op1, . . . , opr} such that �(0) ⊆ V , where �(0)

denotes the set of operators with arity 0. The elements in �(0) are called logicalconstants and the elements in � � �(0) are called propositional connectives.

A propositional language is an �-algebra of the words L = 〈V〉 = (L, op1,

. . . , opr ), where the elements of L are called propositional formulae (see definition A.5in appendix A).

The model theory is given by the interpretation set, such interpretations being de-fined in terms of �-algebras as follows:

Definition 2.2. Let L = 〈V〉 = (L, op1, . . . , opr ) be a propositional language. Let usconsider the following concepts:

• A flow of time is a pair (T ,�) where T is a set of temporal instants and � is a partialorder relation on T .

• Mt = ((T ,�), op1, . . . , opr ) is a temporal matrix in L where (T ,�) is a flow oftime and (2T , op1, . . . , opr) is an algebra similar2 to L.

A temporal interpretation for L is a pair I = (Mt , h) where Mt is a temporal matrix inL and h is a homomorphism from L to (2T , op1, . . . , opr). Briefly, we will say that h isa homomorphism from L to Mt .

Definition 2.3. A temporal propositional logic is a pair (L, I) where L is a propositionallanguage and I a set of temporal interpretations for L.

• A formula φ is said to be true in a temporal interpretation I = (Mt , h) if h(φ) = T .This is denoted by |=I φ.

• A formula φ is said to be valid, and denoted by |= φ, if it is true for every I ∈ I .

• A formula, φ, is said to be satisfiable if for some temporal interpretation I = (Mt , h)

there exists a t ∈ h(φ); in this case, we say that h is a model of φ at t .

• Two formulae φ and ψ are said to be semantically equivalent, which is denoted byφ ≡ ψ , if, for each temporal interpretation I = (Mt , h), we have that h(φ) = h(ψ).

2.1. Classical logic extensions

The types of temporal propositional logics we are interested in are those that pre-serve all the laws of classical propositional logic, and are known as extensions of clas-sical logic. In an informal way, we can say that a temporal propositional logic is anextension of classical propositional logic if, for each instant of time, its restriction is aclassical logic. This idea can be formalized with the following definitions:

2 Two �-algebras are similar if they have the same similarity. That is, the lists made by the arities of theiroperators are coincident. See appendix A.


Definition 2.4. Let A = (A, f1, . . . , fm) an �-algebra. A matrix over A is a tuple(B,C, g1, . . . , gm), where (B, g1, . . . , gm) is an �-algebra similar to A and C ⊂ B.

Definition 2.5. Let us consider a propositional language L = (L, op1, . . . , opr ), a ma-trix M = (M,D, op1, . . . , opr) over L, and I a set of homomorphisms I :L → M.Then:

• L = (L,M, I ) is called a propositional logic.

• M is called the set of semantic values of L.

• The elements of D are called relevant semantics values.

• The elements of I are called interpretations.

Definition 2.6. Let L = (L, I) be a temporal propositional logic, where L = 〈V〉 =(L,G), where G is a set of operators. L is an extension of classical logic if there existsF ⊆ G and a matrix M = ({0, 1}, {1},F), such that, for each t ∈ T , if

(1) L(t) = 〈L〉 = (L,F),

(2) I(t) = {Ih | (Mt , h) ∈ I} where, if t ∈ h(φ) then Ih(φ) = 1 else Ih(φ) = 0

then L(t) = (L(t),M, I(t)) is a classical propositional logic [4].

As we pointed out in the introduction, our aim is to deepen, in a formal way, thestudy of implicate and implicant sets in temporal logics. We will begin by defining thesemantic implication relation.

Definition 2.7. Let L = (L, I) be a temporal propositional logic that is an extensionof classical logic. We define a binary relation, �, in its language, L, as follows: letφ,ψ ∈ L then

φ � ψ if and only if |= φ → ψ.

The relation � is a preorder. Indeed, there are different φ,ψ ∈ L such that φ � ψ

and ψ � φ. For example, φ = (p∨q)∧(p∨¬q) and ψ = p. Therefore, in the quotientset L/≡, the relation � defined as

[φ] � [ψ] if and only if φ � ψ

is a partial-order relation, with a bounded lattice structure with element zero [⊥] andelement one [�]. Furthermore, (L/≡,�) is a Boolean algebra.

Definition 2.8. Let L = (L, I) be a temporal propositional logic, an extension of clas-sical logic, and φ,ψ ∈ L. φ is an implicant of ψ or ψ is an implicate of φ, if φ � ψ .


2.2. FNext and FNext± logics

We illustrate previous definitions by using FNext and FNext± logics. In these, theflow of time is (Z,�), and is thus infinite, discrete and linear. The elements of the propo-sitional variable set, V , will be denoted by the last letters of the alphabet, p, q, r, . . . ,possibly with natural numbers as subscripts. FNext is the extension of classical propo-sitional logic with the following temporal connectives involving the future: ⊕ (read as“tomorrow”), F (read as “sometimes in the future”), and G (read as “always in the fu-ture”).

Definition 2.9. FNext is a temporal propositional logic (LFNext, I) given by:

– LFNext = 〈V〉 = (LFNext,⊥,�,¬,⊕,F,G,→,∨,∧) where the arity list of these op-erators (their similarity) is (0, 0, 1, 1, 1, 1, 2, ∗, ∗), where ∗ indicates that the operatorhas flexible arity.

– I , the interpretation set of the type (Mt , h) where:

• Mt = ((Z,�),∅, Z, c,⊕,F,G,→,∪,∩) where c is the complement operator forsets; ⊕,F,G : 2Z → 2Z are given by ⊕� = {t ∈ Z | t + 1 ∈ �}, F� = {t ∈ Z |[t + 1) ∩ � �= ∅}, G� = {t ∈ Z | [t + 1) ⊆ �} and → : 2Z × 2Z → 2Z given by→(�1, �2) = �c

1 ∪ �2.

• h is any homomorphism from LFNext to Mt .

FNext± is an extension of FNext logic with the following temporal connectivesinvolving the past: � (read as “yesterday”), P (read as “sometime in the past”), and H(read as “always in the past”), analogous to the future operators ⊕, F, G, respectively.

Definition 2.10. FNext± is the temporal propositional logic given by (LFNext±, I)

where:

– LFNext± = 〈V〉 = (LFNext±,⊥,�,¬,⊕,F,G,�,P,H,→,∨,∧) where its similarityis (0, 0, 1, 1, 1, 1, 1, 1, 1, 2, ∗, ∗).

– I , the set of interpretations of the type (Mt , h) where:

• Mt = ((Z,�),∅, Z, c,⊕,F,G,�,P,H,→,∪,∩) where c, ⊕, F, G and → aredefined in the same way as in FNext, and �,P,H : 2Z → 2Z are given by �� ={t ∈ Z | t −1 ∈ �}, P� = {t ∈ Z | (t −1]∩� �= ∅}, H� = {t ∈ Z | (t −1] ⊆ �}.

• h is any homomorphism from LFNext± to Mt .

3. Unitary formulae and literals

In this section we introduce the concept of a unitary formula and from this, theconcept of a literal. As we say in the introduction, the importance of these conceptsis related to the reductions techniques to improve the efficiency of automatic provers.These techniques are based on the information obtained in terms of literals.


Definition 3.1. Let (L, I) be a temporal propositional logic, an extension of classicallogic, where L = 〈V〉 in the (�) category. We define the set of unitary formulae, Luni,as the freely generated word algebra by V in the (�(1)) category.

Therefore, unitary formulae are the constants and the formulae which only haveunary connectives. For example, in FNext these are

LuniFNext = {�,⊥} ∪ {

γ1 . . . γkp | γi ∈ {¬,⊕,F,G}, 1 � i � k, p ∈ V}

and in FNext±Luni

FNext± = {�,⊥} ∪ {γ1 . . . γkp | γi ∈ {¬,⊕,F,G,�,P,H}, 1 � i � k, p ∈ V

}.

Obviously the set of classical literals, V± = {p,¬p | p ∈ V}, is a subset of theset of the unitary formulae. From now on, �p will denote a classical literal in p.

Definition 3.2. Let (L, I) be a temporal propositional logic, an extension of the classi-cal logic, where L = 〈V〉 in the (�) category. We define the set of temporal literals asfollows:

Lit = {[φ] ∈ L/≡ | [φ] ∩ Luni �= ∅}.

Therefore, a temporal literal is a class of L/≡ containing some unitary formula.The equivalence laws in FNext logic allow us to choose a canonical representative

for each temporal literal. In [6] an algorithm with linear cost that allows the calculationof this canonical representative is shown. This allows us to define the FNext literal set,Lit+, up to equivalence, as follows:

Lit+ =⋃

�p∈V±Lit+(�p)

where

Lit+(�p) = {�,⊥} ∪ {FG�p,GF�p} ∪ {⊕n�p,F⊕n �p,G⊕n �p | n ∈ N}.

In the same way, FNext± equivalence laws and the algorithm described in [15]allows us to define the FNext± temporal literal set. The Lit± set is, up to equivalence,the following:

Lit± =⋃

�p∈V±Lit±(�p)

where

Lit±(�p) = {�,⊥} ∪ {FG�p,GF�p,PH�p,HP�p,FP�p,GH�p}∪ {�n�p,F�n �p,G�n �p,P�n �p,H�n �p | n ∈ Z

}and �n�p denotes: ⊕ n. . . ⊕�p if n > 0; � −n. . . ��p if n < 0, and �0�p = �p.


Figure 1. The lattice (Lit+(�p),�) in FNext.

4. Unitary implicates and implicants

Prime implicates and implicants are widely used in several areas of artificial in-telligence. For example, they are used to formally model truth maintenance systems(TMSs) and assumption-based truth maintenance systems (ATMSs) [27]. Similarly,they are useful for circumscription, model-based diagnosis, abduction, and relationaldatabases [12,22,25].

Our research focuses on the field of automated deduction where it is possible to ob-tain good results using the concept of unitary implicants and implicates. The advantageof using unitary implicants and implicates resides in the large amount of informationthey provide with linear cost.

The difficulty of calculating unitary implicates and implicants sets in temporallogic is due to the fact that the sets can have an infinite cardinality. In order to overcomethis problem we must deepen the study of the structure of these sets. In this section weprovide some glimpses into a study of this kind to justify the introduction of the newalgebraic structures. To do so, we take as a starting point the following result, which isa direct consequence of the definitions of ideal and filter in lattices [16].

Proposition 4.1. Let (L, I) be a temporal logic, an extension of classical logic, andφ ∈ L. The set of implicants of φ is the ideal (φ], and the set of implicates of φ is thefilter [φ) in the lattice (L/≡,�). In this way, the sets of implicant literals and implicateliterals are, respectively: (φ] ∩ Lit and [φ) ∩ Lit.

Given that the concepts of implicate and filter are dual concepts of implicant andideal, from now on we will only refer to the latter.

Our aim is to determine a suitable structure for (Lit,�) that would make possiblethe design of efficient algorithms for calculating the sets of implicant literals of a for-


mula. To solve this we need to make an algebraic study of the behavior of intersectionsof the type I ∩ Lit where I is an ideal of (L/≡,�).

Notation. Let (A,�) be a lattice, ∅ �= B ⊆ A and X ⊆ A, we denote by:

• X↓ the lower closure of X, that is,

X↓=⋃x∈X

(x] =⋃x∈X

{y ∈ A | y � x}.

• X↓B , the restriction of X↓ to B, i.e., X↓B= X↓ ∩ B.

• If I is an ideal of A, IB denotes the restriction of I to B. In more specific terms (X]Bdenotes the restriction on B of the ideal generated by X, (X];3 that is, (X]B = (X]∩B

and this is called ideal generated by X in B.

4.1. Restrictions of ideals

We introduce here the concept of an ideal restricted to the subset B of a lattice A

and the basic results that call for the introduction of new structures. In a dual way, thecorresponding results are obtained for filters.

Definition 4.2. Let (A,�) be a lattice and ∅ �= B ⊆ A. X ⊆ B is an ideal restricted toB if it satisfies that (X]B = X.

Example 4.1. Let us consider the lattice (A,�) and the subset B in the following dia-grams.

0��

c��

a��

b

�

�d

�1(A,�)

(B,�)

0 X��

c��

a��

b

�

�1

X is an ideal restricted to B, because

(X] = {0, a, b, c, d},(X]B = (X] ∩ B = {0, a, b, c} = X.

Obviously any restricted ideal X satisfies that X↓B= X. The next example ensuresthat the opposite assertion is not true.

Example 4.2. Let us consider the lattice (A,�), and the subsets B = {0, a, b, c, d} andX = {0, a, b, c}.3 This is the standard notation in the lattice theory.


0�

��a

��

��

��b c e

��

��

��

��

d��

1(A,�)

0

�a

��

��

b c��

��

d

X

(B,�)

Then X↓B= X. However, (X] = B and (X]B = B �= X, and so, X is not an idealrestricted to B.

The following proposition justifies the above definition.

Proposition 4.3. Given a lattice (A,�), ∅ �= B ⊆ A and X ⊆ B. Then:

(1) (X]B = {b ∈ B | there exists a finite X0 ⊆ X such that b �∨

x∈X0x},

(2) X is an ideal restricted to B if and only if there is an ideal I such that X = IB ,

(3) (X]B is the intersection of all the ideals restricted to B that include X.

Proof. (1) is a direct consequence of the results about lattices [16].(2) The necessity is obvious by considering I = (X]. Let us prove the sufficiency:

if X = I ∩ B = IB , we have that X ⊆ I and, since I is an ideal of A, the definitionof (X] ensures that (X] ⊆ I and, therefore, (X]B ⊆ IB = X. On the other hand, fromX ⊆ (X] and X ⊆ B, we have that X ⊆ (X]B . In short, X = (X]B .

(3) is consequence of item (2) and from known results about ideals in lattices. �

Notation. If (A,�) is a lattice and ∅ �= B ⊆ A, IdealsB(A) denotes the set of idealsrestricted to B in A.

We are interested in finding the structure of (IdealsB(A),⊆) and, in order to doso, we begin by analyzing the behavior of restricted ideals when they intersect.

Lemma 4.4. Let (A,�) be a lattice and ∅ �= B ⊆ A. Let X1 and X2 be two idealsrestricted to B. Then, X1 ∩ X2 is an ideal restricted to B.

Proof. Indeed, if X1 = (X1]B and X2 = (X2]B we obviously obtain that X1 ∩ X2 ⊆(X1 ∩ X2]B .

On the other hand, (X1 ∩ X2]B ⊆ (X1]B ∩ (X2]B = X1 ∩ X2. �

The following example shows that the union of restricted ideals is not always arestricted ideal.


Example 4.3. Let us consider the lattice (A,�) and the subset B in the following dia-grams.

0�b

��

��c��

j��

g��

��a

e

h��

��

��d��

��f

i�

1(A, �)

�� e

��

��

0�b�

��c��

g�

��

��f

i�

1(B,�)

�

The subsets X1 = {0, b, c, f, g} and X2 = {0, b, c, e, f, i} of B are ideals restrictedto B. For these subsets X1 ∩ X2 is an ideal restricted to B, but X1 ∪ X2 is not, since

X1 ∪ X2 = {0, b, c, e, f, g, i} � ({0, b, c, e, f, g, i}]B = B.

Example 4.4. Let us consider the lattice (LFNext/≡,�) (see figure 1) and the subsetLit+. For the restricted ideals (⊕p]Lit+ and (F ⊕ p]Lit+ we have that (⊕p]Lit+ ∩(F ⊕ p]Lit+ = {⊥,Gp} which is a restricted ideal. However, (⊕p]Lit+ ∪ (F ⊕ p]Lit+is not a restricted ideal, since Fp = ⊕p ∨ F⊕ p /∈ (⊕p]Lit+ ∪ (F⊕ p]Lit+ .

Using the theorem below, we now deal with the study of the structure of(IdealsB(A),⊆).

Theorem 4.5. Let (A,�) be a lattice and ∅ �= B ⊆ A. Then, (IdealsB(A),⊆) is alattice with

inf(X1, X2) = X1 ∩ X2 and sup(X1, X2) = (X1 ∪ X2]Bwhere X1, X2 ∈ IdealsB(A).

Proof. The equality inf(X1, X2) = X1 ∩ X2 is a consequence of lemma 4.4. To provethat sup(X1, X2) = (X1 ∪ X2]B , it is enough to verify that (X1 ∪ X2]B exists, which isthe case, because from item (3) in proposition 4.3, (X1 ∪ X2]B is the intersection of allthe ideals restricted to B containing X1 ∪ X2, and B = AB is an ideal restricted to B

containing X1 ∪ X2. �

4.2. Generating sets

As we have said in the introduction our goal is to manipulate efficiently the set ofunitary implicates and implicants of a formula. These sets, as we have seen in proposi-tion 4.1 are ideals and filters restricted to Lit. As these sets can have an infinite numberof elements, rather than working with these, our aim is to replace them with the finitesubsets that generate them.

Definition 4.6. Let (A,�) be a lattice, ∅ �= B ⊆ A and X an ideal restricted to B. Aset � ⊆ X generates X if and only if (�]B = X.


Figure 2. The poset (Lit±(�p),�) in FNext±.

Example 4.5. In FNext± logic (see figure 2), the set

X = {⊥,GH�p,FG�p,GF�p,PH�p,HP�p,FP�p} ∪ {�m�p | m ∈ Z}

∪ {G�m �p | m ∈ Z

} ∪ {H�m �p | m ∈ Z

} ∪ {F�m �p | m ∈ Z

}∪ {

P�m �p | m ∈ Z}

is an ideal restricted to Lit±, that has as generating sets, among others, �0 = {FP�p}and any set � = {F �m1 �p,P�m2} ∪ {�m�p | m ∈ Z, m1 � m � m2} for every pairm1,m2 ∈ Z such that m1 � m2.

Our main objective is to make available some recursive method that would allowedus to obtain a generating set of the implicant set of a formula from the ideals generatedby the temporal literals of the formula.

The following examples show the need to put conditions on the poset (Lit,�).


Example 4.6. Let us consider the lattice (LFNext/≡,�) (see figure 1) and the subsetLit+. For the restricted ideals (⊕p]Lit+ and (F⊕ p]Lit+ we have that

(⊕p ∧ F⊕ p]Lit+ = (⊕p]Lit+ ∩ (F⊕ p]Lit+ = {⊥,Gp} = (Gp]Lit+ .

Notice that Gp is the infimum of {⊕p,F⊕ p} in the lattice (Lit+,�).

In FNext± the poset (Lit±,�) does not have a lattice structure. For example, theset {p,⊕p} does not have supremum. However, if we analyze figure 2 we can see that itis possible to replace the infimum with the lower bound maximals since any lower boundis less than or equal to one of such maximals.

Example 4.7. Let us now consider the lattice (LFNext±/≡,�) and the subset Lit± (seefigure 2). For the restricted ideals (�p]Lit± and (F� p]Lit± we have that

(�p ∧ F� p]Lit± = (�p]Lit± ∩ (F� p]Lit±

= {⊥,GHp} ∪ {G�n p | n � 0

} ∪ {H�n p | n � 3

}= (

Gp,H�3 p]Lit± .

Notice that Gp and H�3 p are maximals of the set of the lower bounds of {�p,F� p}in the poset (Lit±,�).

The following example illustrates a restricted ideal without finite generating set.

Example 4.8. Let us consider the lattice (A,�) and the set B = {0, c, d}∪{an | n ∈ N}whose diagrams are:

(A,�)

0�

a0�

a1

.

.

.

b�� c d��

1

(B,�)

0�

a0�

a1

.

.

.

�� c d

Then (c]B ∩ (d]B = {0} ∪ {an | n ∈ N} is an ideal restricted to B and it does nothave any finite generating set.

5. Multisemilattices

In this section, we introduce the new structure that generalizes the semilattice struc-ture and which will allow us to deal with the issue at hand. To do so, we introduce theconcepts of multi-supremum and multi-infimum that are generalizations of the conceptsof supremum and infimum.


5.1. Multi-supremums and multi-infimums

Definition 5.1. Given a poset, (A,�), and B ⊆ A, Bounds↑(B) denotes the set of upperbounds of B, and Bounds↓(B) the set of lower bounds of B.

Then we have two operators Bounds↑, Bounds↓ : 2A → 2A that can be describedas follows:

Bounds↑(B) =⋂b∈B

[b), Bounds↓(B) =⋂b∈B

(b].

Definition 5.2. Let (A,�) be a poset and B ⊆ A. A multi-supremum of B is a minimalelement of Bounds↑(B). We denote by Multi-sup(B) the set of multi-suprema of B.A multi-infimum of B is a maximal element of Bounds↓(B). We denote by Multi-inf(B)

the set of multi-infima of B.

5.2. Ordered multisemilattices

Definition 5.3. An ordered ∨-multisemilattice is a poset, (A,�), such that for everyfinite subset H ⊆ A we have that:

Bounds↑(H) = (Multi-sup(H)

) ↑

Dually, we obtain the definition of ordered ∧-multisemilattice.Notice that we do not demand Bounds↑(H) (similarly, Bounds↓(H)) to be a non-

empty set.

The following proposition, whose proof is immediate, provides us an equivalentdefinition of an ordered multisemilattice.

Proposition 5.4. A poset, (A,�), is an ordered ∨-multisemilattice if and only if forevery finite non-empty subset H of A the following condition is satisfied:

If x ∈ Bounds↑(H), then there is a z ∈ Multi-sup(H) such that z � x.

Dually, a poset, (A,�), is an ordered ∧-multisemilattice if and only if for every finitenon-empty subset H of A the following condition is satisfied:

If x ∈ Bounds↓(H), then there is a z ∈ Multi-inf(H) such that x � z.


Example 5.1. Let A be the poset with the following diagram.

0��

a b��

.

.

.

cn

.

.

.

c1

�c0

�

c

e

�

.

.

.

dn

.

.

.

d1

�d0

�d

This poset is a ∧-multisemilattice, but it is not a ∨-multisemilattice because, ci ∈Bounds↑({a, b}), but there is no z ∈ Multi-sup({a, b}) that satisfies z � ci .

Obviously, every finite poset is an ordered ∨-multisemilattice and an ordered ∧-multisemilattice; every ∨-semilattice is an ordered ∨-multisemilattice and every ∧-semilattice is an ordered ∧-multisemilattice.

5.3. Non-deterministic operators

In order to algebraically characterize this new structure, we need to make use ofoperators in which the images are sets rather than elements of the domain.

These operators will be called non-deterministic operators.

Definition 5.5. Let A be a non-empty set.We define the non-deterministic operator (henceforth, ndo) of arity n in A, any

total application of the type F : An → 2A.We define the non-deterministic operator of flexible arity in A, any total appli-

cation of the type F : A∗ → 2A where A∗ is the universal language on A, that is,A∗ = ⋃

n∈NAn.

We also introduce some other concepts about non-deterministic operators that willbe used later.

Definition 5.6. Let F be a ndo of flexible arity in a set A and ∅ �= B ⊆ A.

• We call the restriction of F to B, denoted by F/B , the ndo in B given by F/B(ω) =F(ω) ∩ B for every ω ∈ B∗.

• We say that F is full if F(α) �= ∅ for all α ∈ A∗.


5.3.1. Basic propertiesThe known properties of deterministic operators can be generalized to non-

deterministic operators as follows:

Definition 5.7. Let F be a ndo of flexible arity in a set A.

1. F is commutative if for all n ∈ N and all x1, . . . , xn ∈ A we have

F(x1 . . . xn) = F(xσ(1) . . . xσ(n)), σ ∈ S([n])where S([n]) denotes the permutation set of n elements.

2. F is associative if for all n ∈ N and all x1, . . . , xn ∈ A

F(x1 . . . xn) = F(F(x1 . . . xn−1)xn

) = F(x1F(x2 . . . xn)

).

3. F is idempotent if for all x ∈ A and all n ∈ N − {0} we have that F(n

x . . . x) = {x}.In particular, F(x) = {x}.

Moreover, as usual,

F(a1 . . . ai−1Xai+1 . . . an) =⋃x∈X

F(a1 . . . ai−1xai+1 . . . an)

for all X ⊆ A. Therefore, F(a1 . . . ai−1∅ai+1 . . . an) = ∅.

5.3.2. New propertiesAs we will see later, it is excessive to impose the associative property on the ndo

for the posets under consideration in this work. We introduce a new property whichis weaker than the associative property. Moreover, for clarity, we give firstly this newproperty for binary ndos and later for ndos with flexible arity.

Definition 5.8. Let F be a binary ndo in A. We say that F is weakly associative if forall x1, x2, x3, z ∈ A we have that:

if F(x1, x2) = {z}, then:

{F

(F(x1, x2), x3

) ⊆ F(x1, F (x2, x3)

),

F(x3, F (x1, x2)

) ⊆ F(F(x3, x1), x2

).

Let F be a ndo with flexible arity in A. We say that F is weakly associative if forevery chain ω = α1α2α3 ∈ A∗ 4 with α2 �= ε it is satisfied that, if F(α2) is a singleton,then:

F(α1F(α2)α3

) =⋂

ω1,ω2,ω3∈A∗ω=ω1ω2ω3

ω2 �=ε

F(ω1F(ω2)ω3

).

4 Where α1, α2, α3 ∈ A∗ and ω is the concatenation (denoted by juxtaposition) of these three chains.


Example 5.2. The binary ndo F in 2U given by

F(A,B) = {A ∪ B,A ∩ B}is not associative, but is weakly associative.

Let (A,�) be a poset. The ndo with flexible arity

F(x1, . . . xn) = Minimals{x1, . . . xn}is not associative. However, it is weakly associative.

The previous example shows two weakly associative but not associative ndos. Thefollowing result shows that when we work with deterministic operators both propertiesare the same thing.

Lemma 5.9. Every weakly associative deterministic5 operator is associative.

To conclude this subsection, we highlight a particular result of interest for the restof the development:

Proposition 5.10. Let F be a ndo in A with flexible arity, weakly associative, and idem-potent. Then, the three following conditions are satisfied:

(1) For all ω ∈ A∗,

F(ω) =⋂

ω=ω1ω2ω3ω2 �=ε

F(ω1F(ω2)ω3

),

(2) For all ω = α1α2α3 ∈ A∗, if F(α2) is an unitary set, F(α2) = {z}, we have thatF(ω) = F(α1zα3),

(3) Given ω ∈ A∗ and z ∈ A, if F(xz) = {z} for all x ∈ ω, then F(ωz) = {z}.

Proof. (1) This is an immediate consequence of the weak associativity and the fact that,for the idempotent property, we have that F(x) = {x} for all x ∈ ω.

(2) Let ω = α1α2α3 ∈ A∗ and F(α2) = {z}. Then,

F(α1zα3)†1=

⋂ω=ω1ω2ω3

ω2 �=ε

F(ω1F(ω2)ω3

) †2⊆ F(ω)†3⊆ F(α1zα3)

where in †1 we use weak associativity; in †2 idempotency and in †3 item (1). Conse-quently, F(ω) = F(α1zα3).

5 In this case, we use deterministic operators to refer to a ndo in which all the images are singletons.


(3) If the length of ω is 1, the result is obvious. Let us now assume that the resultis true for length n. If ω = ω1x1 ∈ A∗ is a chain of length n + 1 then, for all x ∈ ω wehave that F(xz) = {z}, in particular F(x1z) = {z}. Therefore,

F(ωz) = F(ω1x1z)†1= F

(ω1F(x1z)

) = F(ω1z)†2= {z}

where we use item (2) in †1, and the induction hypothesis in †2. �

As a consequence of the previous lemma, we obtain the following result:

Corollary 5.11. Let F be a ndo with flexible arity, weakly associative, commutative,and idempotent in a set A. For all ω ∈ A∗ we have that F(ω) = F(ω′) where ω′ is thechain obtained after eliminating repetitions of elements in ω.

Before presenting the algebraic characterization of ordered multisemilattices, wedefine a new property for the ndo which was driven by the possibility of using it whenthe posets are considered domains of such operators.

Definition 5.12. Let F be a ndo of flexible arity in A. F satisfies the comparabilityproperty if for all ω ∈ A∗ the two following conditions are satisfied:

comp1: if z ∈ F(ω), then F(xz) = {z} for all x ∈ ω,

comp2: if z1, z2 ∈ F(ω) and F(z1z2) = {z1} then z1 = z2.

5.4. Algebraic multisemilattices

Definition 5.13. Let (A,�) be an ordered ∨-multisemilattice. We define in A the ndoof flexible arity

F∨(x1 . . . xn) = Multi-sup({x1, . . . , xn}

).

Dually, given an ordered ∧-multisemilattice, (A,�), we define in A the ndo of flexiblearity

F∧(x1 . . . xn) = Multi-inf({x1, . . . , xn}

).

From the definitions of F∨ and F∧, as an immediate consequence, we obtain thefollowing results:

Proposition 5.14. Let (A,�) be an ordered �-multisemilattice, � ∈ {∧,∨}. Thendo F� satisfies the commutative, idempotent, comparability, and weakly associativeproperties.

Now, we can provide a definition of algebraic multisemilattice.

Definition 5.15. An algebraic multisemilattice, (A, F ), is a set A with a ndo F of flex-ible arity in A, that satisfies the following axioms:


(MS1) Commutative law, (MS3) Idempotency law,(MS2) Weakly associative law, (MS4) Comparability law.

The following theorem ensures the equivalence between the definitionos of ordered∨-multisemilattice and algebraic multisemilattice.

Theorem 5.16.

(i) Let M = (A,�) be an ordered ∨-multisemilattice. Then, (A, F∨) is an algebraicmultisemilattice, denoted by Ma, where:

F∨(x1 . . . xn) = Multi-sup({x1, . . . , xn}

).

(ii) Let M = (A, F ) be an algebraic multisemilattice. The set A with the order relation“x � y if and only if F(xy) = {y}” is an ordered ∨-multisemilattice, denoted byMo∨.

(iii) If M = (A,�) is an ordered ∨-multisemilattice, then (Ma)o∨ = M.

(iv) If M = (A, F ) is an algebraic multisemilattice, then (Mo∨)a = M.

By duality, we obtain the corresponding result for ∧-multisemilattices.

Proof. (i) It is an immediate consequence of proposition 5.14 and (iii) is immediate.(ii) Let us assume that M = (A, F ) is an algebraic multisemilattice. First, let us

see that � is an order relation:

• The idempotency law ensures that � is reflexive.

• The commutative law ensures that � is antisymmetric.

• � is transitive because, for all x, y, z ∈ A, if x � y � z, then F(xy) = {y}and F(yz) = {z} and, from weak associativity, F(F(xy)z) = F(xF(yz)), that is,{z} = F(xz).

Let us prove that, if x ∈ Bounds↑({x1, x2}), then there exists z ∈ A, such thatz � x and z ∈ F(x1x2). Based on the hypothesis F(x1x) = {x} and F(x2x) = {x} and,item (2) of proposition 5.10, we have that F(x1x2x) = {x}. Therefore, according to theproperty of weak associativity, {x} ⊆ F(F(x1x2)x), that is, there exists z ∈ F(x1x2)

such that x ∈ F(zx). Finally, if x ∈ F(zx), we have that {x} †1= F(zxx)†2= F(zx) and

so, z � x where we have made use of comp1 and item (2) of lemma 5.10 in †1, and havemade use of corollary 5.11 in †2.

(iv) Let M = (A, F ) be an algebraic multisemilattice, and (Mo∨)a = (A, F ′∨).First, we will prove that F ⊆ F ′∨.

If z ∈ F(ω), by the comparability law, F(xz) = {z} for all x ∈ ω, and so z ∈Bounds↑

�F(ω). Next, we prove that z is a minimal element of the set Bounds↑

�F(ω).

If z1 ∈ Bounds↑�F

(ω), where z1 � z, we can ensure that z2 ∈ F(ω) exists, such thatz2 � z1 � z. Then, by comparability, z2 = z and, consequently, z ∈ F ′∨(ω).


Finally, we prove that F ′∨ ⊆ F∨, that is, if z is a minimal element in Bounds↑�F

(ω),then z ∈ F(ω). Since z is an upper bound (in respect to the order �F ) of ω, we have thatF(xz) = {z} for all x ∈ ω and, therefore, there exists z1 ∈ F(ω) where z1 � z. So thecomparability law ensures that z1 ∈ Bounds↑

�F(ω) and, as z is a minimal element of this

set, z = z1. �

Example 5.3. Let us consider the poset (A,∨) that we saw in example 5.1. As we sawin that example, (A,∨) is not an ordered ∨-multisemilattice and, therefore, (A, F∨) isnot an algebraic multisemilattice. Indeed, the ndo F∨ is not weakly associative, because

F∨(aF∨(bc)

) = F∨(ac) = {c} �⊆ F∨(F∨(ab)c

) = F∨(ec) = ∅.

5.5. Associative multisemilattices

In this section we prove that the presence of associativity reduces multisemilatticesto semilattices.

Definition 5.17. Let (A, F ) be a multisemilattice. If F is full, then we say that (A, F )

is full. If F has the associative property then we say that (A, F ) is associative.

Obviously, every bounded multisemilattice is full.

Example 5.4. The following diagram corresponds to a full but non-associative ∨-multisemilattice and to a non-full and non-associative ∧-multisemilattice.

x1 x2 x3

�

��

��

��

��

��

x z2

y��

�� z1

Theorem 5.18. Let (A, F ) be a multisemilattice. Then, A is a semilattice if and onlyif F is associative and full. Moreover, if (A, F ) is a bounded multisemilattice, A is asemilattice if and only if, F has the associative property.

Proof. It suffices to prove that a multisemilattice, (A, F ), is associative if, and only if,|F(ω)| � 1 for all ω ∈ A∗.

If |F(ω)| � 1 for all ω ∈ A∗, the proof that the weak associativity implies asso-ciativity is immediate. To prove the inverse, it is enough to see that, if F is associative,F(ab) is a singleton for two arbitrary elements a, b ∈ A.

If x ∈ F(ab), by comp1, F(ax) = F(bx) = {x}, and by the associative property,we have that {x} = F(ax) = F(aF(bx)) = F(F(ab)x). Therefore, for all x′ ∈ F(ab)

we have that {x} = F(x′x) and, by the comp2 property, we have that x = x′. �


6. Multilattices

In this section we introduce the multilattice structure as a natural generalization ofthe lattice structure. That is, given two multisemilattices we will obtain a multilattice us-ing a new property generalizing the absorption property for non-deterministic operators.

Definition 6.1. Let F and G be two ndos of flexible arity in a set A. We say that thepair (F,G) has the absorption property if for all ω ∈ A∗:

• If x ∈ ω, then G(xy) = {x} for all y ∈ F(ω).

• If x ∈ ω, then F(xy) = {x} for all y ∈ G(ω).

From the definition of the absorption property, we obtain the following lemma,which has special relevance for the multilattice structure.

Lemma 6.2. Let F and G be two ndos of flexible arity in a set A that possess the ab-sorption property. Then:

(1) F(ab) = {a} if and only if G(ab) = {b},(2) F and G satisfy the comp1 property.

Proof. (1) is immediate. Let us prove (2): if z ∈ F(ω), according to the absorptionproperty, we have that G(xz) = {x} for all x ∈ ω, and according to item (1), if we againapply the absorption property, we have that F(xz) = {z} for all x ∈ ω. �

We may now make the following definition:

Definition 6.3. Let A be a non-empty set and F and G two ndos of flexible arity in a setA. We say that (A, F,G) is a multilattice if (A, F ) and (A,G) are multisemilattices,and (F,G) satisfies the absorption property.

As a direct consequence of the previous definition, we obtain the following propo-sition which characterizes the multilattice structure.

Proposition 6.4. Given a set A �= ∅ with two ndos F and G of flexible arity in a set A,(A, F,G) is a multilattice if and only if F and G satisfy the following properties:

(M1) Commutative laws, (M2) Weakly associative laws,(M3) Idempotency laws, (M4) Comparability laws,(M5) Absorption law.

The following theorem is a direct consequence of item (1) in lemma 6.2.

Theorem 6.5. Let (A, F∨, F∧) be a multilattice, and �∨ and �∧ be partial order rela-tions given by: “x �∨ y if and only if F∨(xy) = {y}” and “x �∧ y if and only ifF∧(xy) = {x}”. Then �∨=�∧.


As a consequence of this theorem and theorem 5.16, an equivalent multilatticedefinition is the following.

Definition 6.6. A multilattice is a poset, (A,�), such that for every finite subset H ⊆ A

we have that:

Bounds↑(H) = (Multi-sup(H)

)↑, Bounds↓(H) = (Multi-inf(H)

)↓.

Obviously, the finite posets and the lattices are also multilattices.

Example 6.1. In FNext±, (Lit±,�) (see figure 2) is an infinite multilattice and not alattice.

7. The utility of the multisemilattice structure in the calculus of unitaryimplicates/implicants

In this section we show the importance of the multisemilattice structure introducedin this article. It is a very suitable structure for Lit when we need efficient algorithmsthat would allow us to calculate implicates and implicants in a given temporal logic.With this aim in mind, it is even more important to make available a proper algebraiccharacterization of such a structure in terms of operators of flexible arity as shown in thefollowing results.

Theorem 7.1. Let (A,�) be a lattice and ∅ �= B ⊆ A. (B,�) is a ∧-multisemilatticeif and only if, for all x1, x2 ∈ B,(

Multi-inf{x1, x2}]B

= Multi-inf{x1, x2} ↓B= (x1]B ∩ (x2]B.

Proof. Its sufficiency is obvious. We show that it is a necessary condition. From thedefinitions of ∧-multisemilattice and restricted ideal, we have that

(x1]B ∩ (x2]B = Bounds↓B

({x1, x2}) = F∧(x1x2)↓B⊆ (

F∧(x1x2)]B.

On the other hand, if a ∈ (F∧(x1x2)]B , there is a finite set � ⊆ F∧(x1x2) ⊆ (x1]B ∩(x2]Bsuch that a �

∨x∈� x, therefore a ∈ (x1]B ∩ (x2]B . We can then conclude that

(x1]B ∩ (x2]B = F∧(x1x2)↓B⊆ (F∧(x1x2)

]B

⊆ (x1]B ∩ (x2]B. �

The following theorem allows us to obtain generating sets of (X1 ∪ X2]B and X1 ∩X2 from the generating sets of X1 and X2.

Theorem 7.2. Let (A,�) be a lattice, ∅ �= B ⊆ A, where (B,�) is a ∧-multisemilattice and �1, �2 ⊆ B.

(1) (�1 ∪ �2]B = ((�1]B ∪ (�2]B ]B .


(2) If (�1]B = �1↓B and (�2]B = �2↓B , then(F∧(�1�2)

]B

= (�1]B ∩ (�2]B.

Proof. (1) Given that �1 ∪ �2 ⊆ (�1]B ∪ (�2]B , we have that (�1 ∪ �2]B ⊆ ((�1]B ∪(�2]B ]B .

We will now prove the other inclusion. Let a ∈ ((�1]B ∪ (�2]B ]B . From theproposition 4.3 there exists x1, . . . , xr ∈ (�1]B and xr+1, . . . , xs ∈ (�2]B such that

a � x1 ∨ · · · ∨ xr ∨ xr+1 ∨ · · · ∨ xs.

However, from the same proposition, for each xi with 1 � i � r there is a finite�i ⊆ �1 such that xi �

∨x∈�i

x, and for each xj with r < j � s there is a finite�j ⊆ �2 such that xj �

∨x∈�j

x, and so we have that:

a � x1 ∨ · · · ∨ xr ∨ · · · ∨ xs �( ∨

x∈�1

x

)∨ · · · ∨

( ∨x∈�r

x

)∨ · · · ∨

( ∨x∈�s

x

)

where �1 ∪ · · · ∪ �r ∪ · · · ∪ �s ⊆ �1 ∪ �2 and it is finite. Therefore, a ∈ (�1 ∪ �2]B .

(2) (�1]B ∩ (�2]B = ((�1]B ∩ (�2]B

]B

= (�1↓B ∩�2↓B

]B

=(( ⋃

x∈�1

(x]B)

∩( ⋃

y∈�2

(y]B)]

B

=( ⋃

x∈�1y∈�2

((x]B ∩ (y]B

)]B

†=( ⋃

x∈�1y∈�2

F∧(x, y)↓B

]B

= (F∧(�1�2)↓B

]B

= (F∧(�1�2)

]B

where in † we have made use of theorem 7.1. �

Given that the sets of implicate/implicant literals can have an infinite number ofelements, rather than working with these, our aim is to replace them with the finitesubsets that generate them. The main objective is to make available some recursivemethod that would allow us to obtain a generating set of the implicant set of the formulafrom the ideals generated by the temporal literals of the formula.

To make this possible we have introduced, in this paper, the minimal structure to bedemanded of Lit. This minimal structure is a generalization of the semilattice structure,called a multisemilattice, and it is obtained in a natural way.

8. Conclusions

The calculation of prime implicants/implicates of a formulae is useful in situationswhere satisfying models are desired as in circuits design, in error analysis during hard-ware verification and, more recently, in Artificial Intelligence applications (diagnosis,


abductive reasoning, automated theorem proving, compilation of knowledge bases and,more generally, in casual and hypothetical reasoning [12,22,24,25]). So, the design ofefficient methods for the computation of prime implicants/implicates from a logical ex-pression has been topic of research over decades and numerous algorithms have beenproposed in the literature [20,23].

If we focus in the field of automatic proving, such study is useful even if it is re-stricted to the set of unitary implicants/implicates [5–8]. The methodology developedby our research group and named TAS, has already taken its place as a solid alterna-tive to existing works on non-clausal Automatic Deduction. Concretely, the results ob-tained [18,19] are based on the efficient manipulation of these sets and, given a formula,they allow us to obtain efficiently another one equivalent to it and smaller. This is a veryimportant improvement in the efficiency of any prover.

In Propositional Classical Logic, it is easy to calculate recursively the set of unitaryimplicants (implicates) of a formula. The reason is the simplicity of the structure of Lit.The diagram shows that, in this logic, (Lit,�) is a lattice:

⊥��

��

��

��

��

�

�� p ¬p q ¬q r ¬r . . .��

�

��

��

��

��

��

This allows us the following recursive method to calculate the set of unitary impli-cants of a formula. If we denote by UI(ψ) the set of unitary implicants of a formula ψ ,then:

1. UI(�) = {⊥, �} if � ∈ Lit.

2. UI(φ ∨ ψ) ={Lit if p,¬p ∈ UI(φ) ∪ UI(ψ),UI(φ) ∪ UI(ψ) in other case.

3. UI(φ ∧ ψ) = UI(φ) ∩ UI(ψ).

Example 8.1. If we consider the following propositional formula:

ψ = ((p ∨ q) ∧ (p ∨ r)

) ∨ ((q ∨ r) ∧ (¬p ∨ ¬q)

).

Recursively we obtain that:

UI(ψ) = {p}.This information allows us to obtain an equivalent and smaller formula. If an interpreta-tion is a model of p, it is a model of ψ , and if it is not a model of p, it should be a model


of ψ ′, where ψ ′ is obtained from ψ replacing p by ⊥. That is:

ψ ≡ p ∨ [((⊥ ∨ q) ∧ (⊥ ∨ r)

) ∨ ((q ∨ r) ∧ (¬⊥ ∨ ¬q)

)]†≡ p ∨ (q ∧ r) ∨ (

(q ∨ r) ∧ �)††≡ p ∨ (q ∧ r) ∨ q ∨ r

†††≡ p ∨ q ∨ r

where in † we use that ⊥ ∨ ψ ≡ ψ and � ∨ ψ ≡ �, in †† we use that � ∧ ψ ≡ ψ , in††† we repeat this process.

Note that this process has at most polynomial cost.

The greatest obstacle we have found when trying to apply the obtained results forClassical Logic and multivalued logics [10,17–19] to non-classical logics and, in particu-lar, to temporal logics, is the higher complexity of the set of unitary implicants/implicateswith the relation of “logic implication”.

The difficulty of the calculus of the unitary implicates and implicants sets in tem-poral logic is due to the fact that the sets can have an infinite cardinality. For example,in FNext± logic (figure 2), the set of implicants of P⊕ p ∨ F� p is

{⊥,GHp,FGp,GFp,PHp,HPp,FPp} ∪ {�mp | m ∈ Z} ∪ {

G�m p | m ∈ Z}

∪ {H�m p | m ∈ Z

} ∪ {F�m p | m ∈ Z

} ∪ {P�m p | m ∈ Z

}.

However, a minimal generating set exists: {FPp}.As these sets can have an infinite number of elements, rather than working with

these, our aim is to replace them with the finite subsets that generate them. The mainobjective is to make available some recursive method that would allow us to obtaina generating set of the implicant set of the formula from the ideals generated by thetemporal literals of the formula.

Only if (Lit,�) is an ∨-multisemilattice and F∧(a, b) is finite for all a, b ∈ Lit,theorems 7.1 and 7.2 allow us the following recursive method to calculate a generatingset of the set of unitary implicants of a formula that we will denote by UIm(φ):

1. UIm(�) = {�} if � ∈ Lit,

2. UIm(φ ∨ψ) = Maximals(UIm(φ)∪UIm(ψ)∪{�∨k | �∨k ∈ Lit, � ∈ UIm(φ), k ∈UIm(ψ)}),

3. UIm(φ ∧ ψ) = Maximals(F∧(UIm(φ), UIm(ψ))).

Example 8.2. In FNext± (see figure 2), let us consider φ = P ⊕3 p and ψ = F �2 p.We have that:

1. UIm(φ ∨ ψ) = Maximals{P⊕3 p,F�2 p,FPp} = {FPp},2. UIm(φ ∧ ψ) = Maximals{�p, p,⊕p,⊕2p} = {�p, p,⊕p,⊕2p}.


In a nutshell, in order to apply the reductions techniques (or designing a TASprover, or improving the efficiency of any prover) in a new temporal logic that is anextension of classical logic, we must give the following steps:

• To determine if the set of literals has the multisemilattice structure and if the operatorhas a finite set of images.

• In such case, to design an efficient algorithm based on the recursive method we havedescribed. We have made this for the logics Fnext in [8] and FNext± in [11].

Acknowledgements

The authors wish to thank the reviewers for their helpful comments and suggestedimprovements.

Appendix A. Universal algebras

In this section we introduce the concept of �-algebra or abstract algebra thatwe use in the formal definition of a logic. From now on we use the following naturalextension of N:

N = N ∪ {∗}

Definition A.1. A domain of operators is a set � (whose elements are called operators)and a map Ar : � → N where:

• If f ∈ � and Ar(f ) = n ∈ N, f is an operator with arity n.

• If f ∈ � and Ar(f ) = ∗, f is an operator with flexible arity.

• For each ρ ∈ N we define �(ρ) = {f ∈ � | Ar(f ) = ρ}.

To define an �-algebra structure on a set A, we assign an operator on A to eachelement of �. In the following definition, we will use the notation: If we consider twosets A and B, we will denote by Map(A,B) the set of all the maps from A into B.

Definition A.2. Let us consider a set A and a domain of operators �. An �-algebrastructure on A is a family of maps:

()A : �(0) → A,

()A : �(ρ) → Map(Aρ,A

), ρ ∈ N, ρ �= 0

where if ρ = ∗, then A∗ denotes the universal language on A.

That is, a family of maps that apply an element of A to each element of � witharity 0 (that are called operators with arity 0 on A), and to each f ∈ � with Ar(f ) �= 0,


a map from AAr(f ) in A:

fA : AAr(f ) → A.

The set A with this structure is called an �-algebra or abstract algebra over theuniverse A. If we denote the family of operators {fA | f ∈ �} by F , the �-algebra isdeterminated by the pair (A,F).

Given a domain of operators �, (�) denotes the category of the �-algebras.Let A be an �-algebra over the universe A and (f1, . . . , fm) a listing of the ele-

ments of �. In this case, the algebra A is represented by (A, f1A, . . . , fmA).The tuple (Ar(f1A), . . . , Ar(fmA)) is named the similarity of A. The position of

the operators in (A, f1A, . . . , fmA) and the similarity determine the �-algebra structure.Two algebras (A, g1, . . . , gm) and (B, h1, . . . , hm) are similar if they have the same

similarity, that is, if they belong to the same category (�).

Definition A.3. Let A and B be two �-algebras over the universes A and B respectively.B is a subalgebra of A if:

1. B ⊆ A.

2. For each f ∈ �, fB is the restriction of fA to BAr(f ).

If h is a map from A into B, h : A → B, we define the maps:

1. (h, (n). . ., h) : An → Bn given by (h, (n). . . , h)(a1, . . . , an) = (h(a1), . . . , h(an)),

2. h∗ : A∗ → B∗ given by h∗(a1 . . . am) = h(a1) . . . h(am).

Definition A.4. Let us consider two �-algebras A and B over the universes A and B,respectively. A map h : A → B is a homomorphism of algebras, and denoted byh :A → B if:

1. For each f ∈ � with Ar(f ) = 0, h(fA) = fB .

2. For each f ∈ � with Ar(f ) ∈ N and Ar(f ) �= 0 we have that

h(fA(a1, . . . , aAr(f ))

) = fB

(h(a1), . . . , h(aAr(f ))

).

3. For each f ∈ � with Ar(f ) = ∗ we have that

h(fA(a1 . . . am)

) = fB

(h(a1) . . . h(am)

).

A.1. Free generated algebras

The notion of free generated algebras is a basic concept in the formalization ofpropositional logics. A subalgebra B of an algebra A over the universe A is generatedby G ⊂ A, if B is the minor subalgebra of A that contains G.


Definition A.5. Let A be an �-algebra over the universe A and let G ⊂ A such that{fA | f ∈ �(0)} ⊂ G.

We define the following sets:

B0 = G,

Bi+1 = Bi ∪ {f (a1, . . . , aAr(f )) | aj ∈ Bi,Ar(f ) � 1

}∪ {

f (a1 . . . an) | aj ∈ Bi,Ar(f ) = ∗},

B =⋃i∈N

Bi.

We can define an �-algebra structure over B as: for each f ∈ � with Ar(f ) �= 0,fB is the restriction of fA to BAr(f ). We denote this subalgebra by 〈G〉, and we say that〈G〉 is the algebra generated by G or that G is a set of generators of 〈G〉.

The following result establishes that the previous concept is the same that the intu-itive concept given in the introduction of this section:

Proposition A.6. Let A be an �-algebra over the universe A, and let G ⊂ A such that{fA | f ∈ �(0)

} ⊂ G.

Then the subalgebra 〈G〉 is the minor subalgebra of A that contains G.

Definition A.7. We say that an �-algebra over the universe A is freely generated byG ⊂ A in the category (�) if:

1. A = 〈G〉 and

2. If B is another �-algebra over the universe B and h : G → B is a map, then h can beuniquely extended to an homomorphism of algebras h :A → B.

Definition A.8. Let � be a domain of operators, G a set such that G ∩ � = �(0) andlet us consider the set X = G ∪ � ∪ {(, )}. Given the universal language X∗ we definethe �-algebra structure:

• If f ∈ � and Ar(f ) = 0, fU = f .

• If f ∈ � and Ar(f ) = n, fU(γ1, . . . , γn) = f (γ1, . . . , γn).

• If f ∈ � and Ar(f ) = ∗, fU(γ1, . . . , γm) = f (γ1, . . . , γm) for any m ∈ N and anychain γ1, . . . , γm with length m.

We call �-algebra of the words over G to the subalgebra 〈G〉.

Theorem A.9. For any domain of operators � and any set G such that G ∩ � = �(0),the �-algebra of the words over G is a freely generated �-algebra by G in the cate-gory (�).


References

[1] G. Aguilera, M. Ojeda and I.P. de Guzmán, A new reduction-based theorem prover for 3-valued logic,Mathware & Soft Computing 4(2) (1997) 99–127.

[2] A. Burrieza and I.P. de Guzmán, A new algebraic semantic approach and adequate connectives forcomputation with temporal logic over discrete time, Journal of Applied Non-Classical Logics 2(2)(1991) 181–200.

[3] A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Science Publications (Clarendon Press,Oxford, 1997).

[4] P.M. Cohn, Universal Algebra, Mathematics and Its Applications, Vol. 6 (D. Reidel, 1965).[5] P. Cordero, M. Enciso and I.P. de Guzmán, Structure Theorems for Closed Sets of Implicates/Impli-

cants in Temporal Logic, Lecture Notes in Artificial Intelligence, Vol. 1695 (1999).[6] P. Cordero, M. Enciso and I.P. de Guzmán, A temporal negative normal form which preserves impli-

cants and implicates, Journal of Applied Non-Classical Logics 10(3–4) (2000) 243–272.[7] P. Cordero, M. Enciso and I.P. de Guzmán, From the posets of temporal implicates/implicants to a

temporal negative normal form, Reports on Mathematical Logics 36 (2002) 3–48.[8] P. Cordero, M. Enciso and I.P. de Guzmán, Bases for closed sets of implicants and implicates in

temporal logic, Acta Informatica 38 (2002) 599–619.[9] I.P. de Guzmán, M. Ojeda and A. Valverde, Implicates and reduction techniques for temporal logics,

Annals of Mathematics and Artificial Intelligence 27(1–4) (1999) 3–23.[10] I.P. de Guzmán, M. Ojeda and A. Valverde, Reductions for non-clausal theorem proving, Theoretical

Computer Science 266(1–2) (2001) 81–112.[11] I.P. de Guzmán, P. Cordero and M. Enciso, Structure theorems for closed sets of implicates/implicants

in temporal logic, in: Lecture Notes in Artificial Intelligence, Vol. 1695 (1999) pp. 193–207.[12] J. de Kleer, A.K. Mackworth and R. Reiter, Characterizing diagnoses and systems, Artificial Intelli-

gence 56(2–3) (1992) 192–222.[13] M. Enciso and I.P. de Guzmán, A new and complete theorem prover for temporal logic, in: IJCAI’95 –

Workshop on Executable Temporal Logics (August 1995).[14] M. Enciso, I.P. de Guzmán and C. Rossi, Temporal reasoning over linear discrete time, in: Logics in

Artificial Intelligence, Lecture Notes in Artificial Intelligence, Vol. 1126 (Springer, 1996) pp. 303–319.

[15] M. Enciso, P. Cordero and I.P. de Guzmán, From the posets of temporal implicates/implicants to atemporal negative normal form, Reports on Mathematical Logics 36 (2002) 3–48.

[16] G. Grätzer, General Lattice Theory (Birkhäuser, Basel, 1998).[17] G. Gutiérrez, J. Martínez, I.P. de Guzmán, M. Ojeda and A. Valverde, Reduction theorems for boolean

formulas using -trees, Lecture Notes in Artificial Intelligence, Vol. 1919 (2000) pp. 179–192.[18] R. Hähnle, Handbook of Philosophical Logic. Advances in Many-Valued Logics, 2nd edition (Kluwer,

1999).[19] R. Hähnle and G. Escalada, Deduction in many valued logics: a survey, Mathware & Soft Computing

4(2) (1997).[20] P. Jackson and J. Pais, Computing prime implicants, in: Lecture Notes in Artificial Intelligence,

Vol. 449 (Springer, 1990) pp. 543–557.[21] H. Kamp, Tense logic and theory of linear orders, Ph.D. Thesis, University of California, Los Angeles

(1968).[22] A. Kean, The approximation of implicates and explanations, International Journal of Approximate

Reasoning 9 (1993) 97–128.[23] A. Kean and G. Tsiknis, An incremental method for generating prime implicants/implicates, Journal

of Symbolic Computation 9 (1990) 185–206.[24] A. Kogan, T. Ibaraki and K. Makino, Functional dependencies in horn theories, Artificial Intelligence

108(1–30) (1999).


[25] P. Marquis, Extending abduction from propositional to first-order logic, Fundamentals of ArtificialIntelligence Research (1991) 141–155.

[26] M. Ojeda, G. Aguilera, A. Valverde and I.P. de Guzmán, Reductions for non-clausal theorem proving,Theoretical Computer Science 266(1/2) (2001) 81–112.

[27] R. Socher, Optimizing the clausal normal form transformation, Journal of Automated Reasoning 7(3)(1991) 325–336.

atarazanas.sci.uma.esatarazanas.sci.uma.es/docs/articulos/16494489.pdfannals of mathematics and...

Documents