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Representation Theorems and Automated Theorem Proving in Certain Classes of Non-Classical Logics extended abstract

Viorica Sofronie-Stokkermans1 Abstract. The main goal of this paper is to present a method for translation to clause form in nitely-valued logics having as algebras of truth values distributive lattices with certain types of operators. The method uses the Priestley dual of the algebra of truth values. We illustrate these general ideas by several examples, and show that the general complexity can be further improved by using the structure of particular algebras of truth values. We then show that these ideas are actually much more general: we further develop one of our previous ideas where we showed that Priestley duality is useful in better understanding the link between algebraic and Kripke-style models for certain non-classical logics, and give several examples.

1 Introduction

Ecient reasoning on incomplete, vague and imprecise knowledge requires the development of ecient many-valued theorem provers. This paper can be seen as a presentation of our ongoing work towards exploiting recent theoretical results concerning properties of algebraic models for some logics useful for modeling imprecise knowledge, in order to improve the eciency of automated theorem proving in these logics. Since many non-classical logics that occur in a natural way in practical applications can be proved to be sound and complete with respect to certain classes of distributive lattices with operators, in this paper we will focus on this kind of logics. One of the main advantages of distributive lattices (with well-behaved operators) is that they usually have good and economical representation theorems. Our method uses the Priestley representation for distributive lattices with operators. The ideas behind it are very natural, even if the algebraic notions used may at rst sight seem to be rather involved. This is why we illustrate the main ideas by examples. The main sources of inspiration for our work are the method for automated theorem proving by many-valued resolution of Baaz and Fermuller [5], and the results due to Hahnle [15, 16] on the one hand; and the work of Goldblatt [12] on Priestleytype dualities for distributive algebras with operators on the other hand. The main contributions of this paper are the following: First, we illustrate, by means of several examples (all having 1 Max-Planck-Institut fur Informatik, D-66123 Saarbrucken, Ger-

many email: [email protected]

c 1998 V. Sofronie-Stokkermans ECAI 98. Workshop MVL for AI Applications

possible applications in dealing with uncertain information or with inconsistencies in databases) a method for clause generation and, ultimately, automated theorem proving in nitelyvalued logics, that uses the Priestley dual of the algebra of truth value ( rst presented in [37]). This method leads in many important cases to a reduction of the number of clauses compared to methods using the algebra of truth values. Second, we show that this method is actually much more general, and can be applied to wider classes of possibly in nite-valued non-classical logics. We rst discuss logics that are sound and complete with respect to varieties of distributive lattices with operators, and then mention examples of logics where the algebraic models are lattices, not necessarily distributive. The paper is structured as follows. We start by giving some background information in Section 2. In Section 3 we brie y present some generalities on Priestley duality for distributive lattices with operators, and then describe a method for translation to clause form based on the Priestley dual of the algebra of truth values. We also show that Priestley duality helps in establishing links between algebraic and Kripke-style models for certain non-classical logics. The translation to clause form can be regarded as a translation to rst-order logic; thus, classical tools for automated theorem proving can be applied to the resulting sets of clauses. Therefore we do not insist on speci c methods for automated theorem proving (for more results in this direction we refer for instance to [37]). In Section 4 we illustrate these ideas by means of several examples: logics based on nite p-lattices (modeling inconsistencies in databases in the presence of some distinguished error values), Pmn -logics (containing among other classes also the fragment Efde of relevance logic), and logics based on implicative algebras (containing as subclasses for instance the Lukasiewicz logics). Then some not necessarily nitely-valued non-classical logics based on classes of lattices are brie y presented. We end by mentioning some related approaches and by presenting plans for future work.

2 Preliminaries In what follows we assume known standard notions, such as the notion of partially-ordered set and lattice, as well as the notions of order- lter and order-ideal in partially-ordered sets, meet- and join-irreducible elements, and lter and ideal (respectively prime lter and prime ideal) in lattices. For the

de nitions and further informations we refer to [8]. Given a partially-ordered set (P; ), by O(P ) we will denote the set of order- lters of P ; for every i 2 P , we will use the following notations: " i = fj 2 P j j ig, and # i = fj 2 P j i j g. In particular, we would like to point out that all ideals (resp. lters) in nite distributive lattices are principal (i.e. every ideal is of the form # i and every lter is of the form " i for some element i in the lattice). Moreover, an ideal (resp. lter) in a nite distributive lattice is prime if and only if it is of the form # i (resp. " i) for some meet-irreducible (resp. join-irreducible) element i of the lattice. For the basic notions of universal algebra needed in what follows, we refer for instance to [6]. For every signature and every arity function a : ! N a -algebra is a structure (A; fA g2 ), where for every 2 , A : Aa() ! A. If the signature is assumed to be known we will sometimes use the notation A for the -algebra (A; fA g2 ). The free algebra over a given signature freely generated by a set X will be called the algebra of terms in X over and will be denoted Term (X ). Given a set X and two terms t1 ; t2 2 Term (X ), and a -algebra A, we say that A satis es the identity t1 = a t2 (denoted in what follows by A j= t1 = t2 ) if for every assignment f : X ! A, f (t1 ) = f (t2 ), where f : Term (X ) ! A is the unique morphism of -algebras that extends f . A class K of algebras satis es the identity t1 = t2 (denoted a K j= t1 = t2 ) if every algebra in K satis es t1 = t2 . A variety is a class of algebras which is closed under homomorphic images, subalgebras and direct products. Birkho proved that a class of algebras is a variety if and only if it is the class of all algebras that satisfy a set of identities. For a given class K of algebras, the variety generated by K is the class of all algebras that are homomorphic images of a subalgebra of a product of elements in K. The variety generated by K will be denoted V (K) (or, sometimes, due to the closure properties mentioned before, HSP (K)). Given a variety V , if V is generated by one a single algebra A (i.e. V = V (A)) then K j= t1 = t2 if and only a if A j= t1 = t2 . In what follows we mention logics which are sound and complete with respect to certain varieties of algebras. Roughly speaking, a logic is described by a pair L = (Fma(L); `L ), where Fma(L) is the set of formulae in the logic and `L is a relation between sets of formulae and formulae called provability relation. Usually (for simplicity we only consider here the propositional case), Fma(L) is constructed starting from a set Var of propositional variables and a set of logical connectives. In this case Fma(L) can be regarded as a -algebra (actually the free -algebra freely generated by Var). If the set of logical connectives is considered xed, instead of Fma(L) we will use the notation Fma(Var). In general `L is introduced by de ning a notion of proof, starting from a set of formulae called axioms, and a ( nite) set f(r1 ); : : : ; (rn )g of rules of inference. A map h : Var ! A, where A is a -algebra is called a valuation in A. From the universality property of Fma(L) it follows that for every -algebra A and every valuation h : Var ! A there is a unique homomorphism of -algebras h : Fma(L) ! A that extends h. In what follows we will call a logic L with consequence relation `L sound with respect to a class of K of algebras, if

Workshop on Many-Valued Logic for AI Applications 2

a

for every formula 2 Fma(L), if `L then K j= = 1; the logic L will be called complete with respect to K if if for every a formula 2 Fma(L), K j= = 1 implies `L .

3 Priestley Duality and Translation to Clause Form

We start by presenting some results on Priestley representation for distributive lattices with operators. For more details we refer to [8, 12, 37, 38]. The Priestley representation theorem [34] states that every distributive lattice A is isomorphic to the lattice of clopen order lters of the ordered topological space having as points the prime lters of A, ordered by inclusion, and the topology generated by the sets of the form Xa = fF j F prime lter, a 2 F g and their complements as a subbasis (if A is nite, this topology is discrete). In what follows, the partially ordered set of all prime lters of A, ordered by inclusion, and endowed with the topology mentioned above will be denoted D(A) (we will refer to it as the dual of A). Given an ordered topological space X = (X; ; ), its lattice of clopen order lters will be denoted by E (X ). Thus, the Priestley representation theorem states that every distributive lattice A is isomorphic to E (D(A)). A dual equivalence exists between the category D01 of bounded distributive lattices and a category P of Priestley spaces (compact, totally order-disconnected ordered topological spaces). This duality was extended to bounded distributive lattices with operators (dlo) [12, 7, 37, 38]. In what follows we will consider the following operators on bounded distributive lattices: lattice homomorphisms and antimorphisms, join- and meet-hemimorphisms (maps of arbitrary arity, preserving nite joins resp. meets in every argument), and join- resp. meetantihemimorphisms (maps of arbitrary arity, mapping nite meets in joins (resp. joins in meets) in every argument). These additional operators on a dlo A induce functions resp. relations on D(A), which, again, de ne corresponding operators on both E (D(A)) and O(D(A)). For instance, with every lattice homomorphism hA : A ! A, one can associate an order-preserving map D(hA ) : D(A) ! D(A) de ned for every prime lter F by D(hA )(F ) = fa 2 A j h(a) 2 F g = h?A1 (F ); for every lattice antimorphism kA : A ! A, one can associate an order-reversing map D(kA ) : D(A) ! D(A) de ned for every prime lter F by D(kA )(F ) = fa 2 A j k(a) 62 F g = AnkA?1 (F ); for every join hemimorphism fA : An ! A (resp. join antihemimorphism fA0 , meet hemimorphism gA or meet antihemimorphism gA0 ), D(A) can be endowed with an increasing relation D(fA ) D(A)n+1 (resp. D(fA0 ), or decreasing relation D(gA ) resp. D(gA0 )) de ned for every prime lters F1 ; : : : ; Fn ; Fn+1 respectively by: D(fA )(F1 ; : : : ; Fn ; Fn+1 ) i 8a1 ; : : : ; an (a1 2 F1 ; : : : ; an 2 Fn implies fA (a1 ; : : : ; an ) 2 Fn+1 );

D(fA0 )(F1 ; : : : ; Fn ; Fn+1 ) i 8a1 ; : : : ; a0 n (a1 62 F1 ; : : : ; an 62 Fn ; implies fA (a1 ; : : : ; an ) 2 Fn+1 );

V. Sofronie-Stokkermans

{ a, b, c, e}

1

{ a, c, b} f

g d

c

a D(~)

e

a

b

c

b

e

{ b, e, a} { a, b}

{ a, c}

{ b, e}

{ a}

{ b}

0 A

E(D(A))

D(A)

Figure 1.

D(gA )(F1 ; : : : ; Fn ; Fn+1 ) i 8a1 ; : : : ; an (gA (a1 ; : : : ; an ) 2 Fn+1 implies 9i 2 f1; : : : ; ng with ai 2 Fi ); D(gA0 )(F1 ; : : : ; Fn ; Fn+1 ) i 8a1 ; : : : ; an (gA0 (a1 ; : : : ; an ) 2 Fn+1 implies 9i 2 f1; : : : ; ng with ai 62 Fi ): Conversely, the Priestley spaces corresponding to these distributive lattices with operators are endowed with continuous order-preserving maps (Lh), continuous order-reversing maps (La), increasing relations with certain additional properties (Jh; Ja), and decreasing relations with additional properties (Mh; Ma). For every such relational Priestley space X , the corresponding operators on E (X ) are de ned in a canonical way as follows: For every continuous, order-preserving map HX : X ! X , E (HX ) : E (X ) ! E (X ) is de ned for every clopen order lter U by E (HX )(U ) = fx 2 X j HX (x) 2 U g = HX?1 (U ); for every continuous, order-reversing map KX : X ! X , E (KX )(U ) = fx 2 X j KX (x) 62 U g = X nKX?1 (U ): For every increasing relation RX X n+1 corresponding to the set Jh (resp. RX0 corresponding to the set Ja, or for every descreasing relation QX corresponding to Mh or resp. Q0X corresponding to Ma), E (RX ) : E (X )n ! E (X ) (resp. E (RX0 ); E (QX ), resp. E (Q0X )) are de ned by:

E (RX )(U1 ; : : : ; Un ) = fx 2 X j 9x1 2 U1 ; : : : ; xn 2 Un s.t. RX (x1 ; : : : ; xn ; x)g; E (RX0 )(U1 ; : : : ; Un ) = fx 2 X j 9x1 62 U1 ; : : : ; xn 62 Un s.t. RX0 (x1 ; : : : ; xn ; x)g; E (QX )(U1 ; : : : ; Un ) = fx 2 X j 8x1 2 X; : : : ; xn 2 X (if QX (x1 ; : : : ; xn ; x) then 9i; xi 2 Ui )g; E (Q0X )(U1 ; : : : ; Un ) = fx 2 X j 8x1 2 X; : : : ; xn 2 X (if Q0X (x1 ; : : : ; xn ; x) then 9i; xi 62 Ui )g: With these notations, one can show that for every dlo A, there exists an isomorphism (of dlo) A : A ' E (D(A)).


Example 1

Moreover, for every suitable relational Priestley space X there exists a homeomorphism X between X and D(E (X )). The constructions D and E described before induce functors between a category of distributive lattices with operators and a suitable category of relational Priestley spaces, and they actually de ne a dual equivalence between these two categories (for further details on this topic we refer for instance to [12, 38]). We illustrate the main ideas by one example: Example 1 Let A be the nite lattice represented in Figure 1, with an additional operator (a lattice antimorphism) : A ! A de ned by: 0 = 1; 1 = 0; a = f; f = a; b = g; g = b; c = c; e = e; d = d: The Priestley dual D(A) of A, also represented in Figure 1, has as elements the principal lters generated by the join-irreducible elements of A (with the order given by the inclusion), and, additionally, a map D() induced by , de ned by D()(F ) = An ?1 (F ): Thus, D()(" a) =" e; D()(" e) =" a; D()(" b) =" c, and D()(" c) =" b: D() induces a map E (D()) on E (D(A)) by E (D())(U ) = D(A)nD()?1 (U ): It can be seen that A and E (D(A)) are isomorphic. The extension of the Priestley representation to dlo makes it possible, for certain nitely-valued logics, to express their semantics by using instead of the algebra of truth values, its Priestley dual D(A). The space D(A) is an ordered space on which operations and relations associated to the operators on A are de ned in a standard way as shown before. If A is nite then the topology on D(A) is discrete, hence E (D(A)) = O(D(A)). We now explain in more detail the main ideas of a translation to clause form based on this semantics. The details can be found for instance in [37].

3.1 Finitely-Valued Logics

Let L be a propositional logic with operators in the following classes: f_; ^g (binary), Lh; La (unary), Jh; Ja; Mh; Ma (arbitrary arity), satisfying the following properties: (P1) L is sound and complete with respect to a variety of algebras VL , such that the variety VL is generated by A, where A is a nite algebra.


(P2) The algebras in VL are distributive lattices with opera-

tors. The operators corresponding to connectives in Lh; La are lattice homomorphisms resp. antimorphisms, those corresponding to connectives in Jh; Mh are join- and meethemimorphisms, and those corresponding to connectives in Ja; Ma are join- resp. meet-antihemimorphisms. We will denote by = f_; ^; 0; 1g[ Lh [ La [ Jh [ Ja [ Mh [ Ma, the signature of the algebras in VL . The additional operators correspond to versions of non-classical negation, or to certain operators modeling knowledge, belief, necessity, or possibility, that generalize similar operators from modal logics. We now show that the Priestley representation theorem for A helps us to de ne a satis ability relation with respect to the relational space D(A). The Priestley dual D(A) of A can be seen as a nite Kripke-style frame, where evaluations at every possible world are 2-valued, and meaning functions are hereditary, i.e. if a formula is true at 2 D(A) then it is true at every with . Thus, all meaning functions in D(A) are required to have as values order- lters (for details about the reasons leading to this assumption see [37, 38]). Let Var be a countably in nite set of propositional variables. Let be a formula in the language of L and variables Var, 2 Fma(Var). Let m : Var ! O(D(A)) be a meaning function. Var

m

A?1 A

/ O(DO (A)) 7A PPP n n n PPP nn n PPP n n PPP m nnn PP( nnn A?1 m /

o

(1)

Fma(Var)

where Dom 6= ; is the domain of interpretation, d : X ! Dom, M (f ) : Domn ! Dom for every function symbol of arity n, and M (R) : Domn ! O(D(A)) for every predicate symbol of arity n. Interpretations in O(D(A)) extend to formulae in a similar way as interpretations in A: M = (Dom; M; d) de nes a map vM : Fma(Var) ! O(D(A)). Notions as validity and satis ability can then be de ned by using the Priestley dual D(A) of A: is satis ed by M at x 2 D(A) if x 2 vM (); is valid i it is satis ed by every interpretation M at every x 2 D(A). The main idea of our approach to automated theorem proving is to use signed literals, where the signs are \possible worlds", i.e. elements of D(A), corresponding to prime lters of truth values. Given an interpretation M = (D; M; d) we will now consider signed formulae (in classical logic) of the type:

t for \ is true at " in M (i.e. 2 vM ()), f for \ is false at " in M (i.e. 62 vM ()), where is a \possible world", i.e. an element of D(A).

De nition 2 (Literal, Clause, Signed CNF) Let L be an atomic formula and 2 D(A) be a \possible world". Then t f

L is a positive literal (with sign ) and L is a negative literal (with sign ). A set of (positive or negative) signed literals is called a (signed) clause. A formula in signed conjunctive normal form (CNF) is a nite set of signed clauses. We require that the clauses in a formula have disjoint variables.

De nition 1 Let m : Fma(Var) ! O(D(A)) be de ned by m = A A?1 m. We de ne: r (1) D(A) j=m;x i x 2 m(), A positive literal Lt (resp. a negative literal Lf ) is r (2) D(A) j=m i m() = D(A); satis able if for some interpretation M of L in O(D(A)), L r r (3) D(A) j= i D(A) j=m for every m : Var ! O(D(A)): is true (resp. false) in M at . A signed clause is satis able if at least one of its literals is satis able. A formula in signed conjunctive normal form is satis able if all clauses in are Proposition 1 Let L be a logic satisfying the conditions (P 1) simultaneously satis able by the same interpretation. and (P 2). Then for every propositional formula in the logic r In order to de ne an automated theorem proving proceL, L ` i D(A) j= : dure, we need a method for translation into clause form, and Example 2 Consider a logic sound and complete with respect a resolution procedure (which has to be proved correct and to the nite lattice of Example 1. Let = p, where p is complete). In [37] we give a structure-preserving transformaa propositional variable, and let m : Var ! O(D(A)) be a tion method to clause form in rst-order logic given in [9], see also [5] and [15]. The main idea of structure-preserving meaning function. Then: methods for transformation to clause form is to introduce for r D(A) j=m;x p i x 2 m( p) = D(A)nD()?1 (m(p)) every non-atomic subformula of a new atomic formula of the form P (x), where P is a new predicate symbol and x i x 62 D()?1 (m(p)) are the free variables in . i D()(x) 62 m(p) r The next result is a consequence of the fact that a formula i D(A) 6j=m;D()(x) p. is valid i for no interpretation is false at some world Similar considerations can be made in the case of rst- 2 D(A) (and, thus, at some minimal world), taking into order logics, by following the ideas in [5] for de ning rst-order account the renamings mentioned before. many-valued logics based on A, and using the fact that if A is a nite dlo then A ' O(D(A)). We obtain a concept of inter- Proposition 2 The formula is valid if and only if there pretation based on D(A) instead of A as a triple (Dom; M; d), exists no interpretation M = (D; M; d) of L in O(D(A)) such Workshop on Many-Valued Logic for AI Applications 4


that 8 > > > > > > > > > > > > > > > > > > > > > > >
> > > > > ( P (x1 ; : : : ; xn )t _ (Qx)P 1 (x; x1 ; : : : ; xn )f )^ > > > ^ ( P (x1 ; : : : ; xn )f _ (Qx)P 1 (x; x1 ; : : : ; xn )t ) > > > > in M; for every 2 D(A) > >

> > > > > for all subformulae = (Qx) 1 (x; x1 ; : : : ; xn ) of and > > > : every instantiation in D of the free variables x1 ; : : : ; xn

of :

The rules for eliminating _, ^, the lattice homomorphisms and antimorphisms, and the quanti ers are schematically presented in what follows (SL is an arbitrary signed literal, and L; L1 ; L2 are atomic formulae): (_)

(^)

SL; Lt1 ; Lt2

SL _ (L1 _ L2 )f

SL _ (L1 ^ L2 )t

SL; Lt1 ; SL; Lt2

SL; Lf1 ; SL; Lf2

SL; Lf1 ; Lf2

SL; D(hA )() L

t

SL; D(hA )() L

SL _ k(L)t

(La)

(8)

(9)

SL _ (8x)L(x; y)t SL; L(x; y)t

f

SL; L(fs (y); y)t

SL; D(kA )() Lt

SL _ (8x)L(x; y)f

SL _ (9x)L(x; y)t

SL; L(fs (y); y)f

SL _ (9x)L(x; y)f

SL; L(x; y)f

V SL; 1 Lf (1 ; ) 2 D(3)

SL _ 2(L)t

SL _ 2(L)f

SL; 1 Lf ; (1 ; ) 2 D(2)

From Proposition 2 and the subsequent rules it follows that for every formula a nite set of signed clauses can be constructed such that is valid i is unsatis able. The next proposition gives an estimate of the number of clauses generated from a given formula by using Proposition 2 and the transformation rules above.

Proposition 3 Let n be the number of elements of D(A), and l the number of subformulae of . If the formula only contains the connectives ^; _ and lattice morphisms or antimorphisms, then the number of clauses generated from is O(ln). If in addition contains unary join- or meet-hemimorphisms, then the number of clauses generated from is O(ln2 ).

For comparison, we would like to point out that existing methods using the algebra of truth values such as [5] would generate O(ljAj2 ) clauses for formulae containing at least one binary operator. However, a naive application of our method may lead to the generation of exponentially many clauses in the case of operators in Jh [ Ja [ Mh [ Ma with arity higher than 1. The number of clauses can be further reduced by introducing quanti cation over the elements of D(A); this approach will be described in future work.

3.2 Other Types of Non-Classical Logics

SL _ k(L)f

SL; D(kA )() Lf

SL _ h(L)f

SL _ h(L)t

(Lh)

SL _ (L1 ^ L2 )f

V SL; 1 Lt (1 ; ) 2 D(2)

in M; for every 2 D(A)

SL _ (L1 _ L2 )t

SL _ 3(L)f

SL _ 3(L)t

(3)

where fs are suitable Skolem functions. We illustrate the rules for join- and meet-hemimorphisms by means of two simple examples: unary join-hemimorphisms 3 : A ! A and unary meet-hemimorphisms 2 : A ! A (similar to the modal operators for \possibility" and \necessity").


In [38] we explain how, by using the Priestley duality for distributive lattices with operators, one can better express the links between the algebraic and Kripke-models for nonclassical logics based on distributive lattices with operators, and, that, in certain situations, starting from a logic which is sound and complete with respect to a variety of distributive lattices, one can determine a class of Kripke-style models for that logic, and a satis ability relation with respect to this class of models, by using only algebraic tools. The idea goes back to the work of Jonsson and Tarski [20, 21]. Let L be a logic which is sound and complete with respect to a variety VL of algebras with a distributive lattice reduct and operators in the classes discussed in Section 3. Assume that the Priestley duality for bounded distributive lattices induces a dual equivalence between VL and a suitable class VL Sp of Priestley spaces endowed with additional functions and relations associated with the operators in L dierent from the operations _ and ^. The elements of VL Sp are usually characterized by a set Top of conditions involving the topology, and a set NonTop of conditions which do not involve the topology. Let RL be the class of all relational structures of the same similarity type as the elements in VL Sp. Let KL be the subclass of RL consisting of all structures that satisfy all the conditions in NonTop. For every such structure (X; ; fRgR2Rel ),


the relations in Rel usually induce corresponding operations on O(X ) in a canonical way, thus inducing a -algebra structure [37, 38]. The elements in KL can be seen as Kripkestyle frames for L. A de nition of satis ability very similar to that in De nition 1 can be given: the meaning functions are supposed to be hereditary, and for every m : Var ! O(X ), r X j=m;x i x 2 m(), where m is the unique extension of m to formulae2. The following result follows then easily:

Theorem 1 If for every X 2 KL , O(X ) 2 VL, then L is sound and complete with respect to the class KL . Other representation theorems can be used as well, with similar results. A detailed presentation of these results is beyond the scope of the present paper. In Section 4.4 we brie y mention some examples in which representation theorems such as, for instance, the Priestley representation theorem, can be used for establishing a link between the algebraic and Kripke models of logics. (However, we do not restrict ourselves to the distributive case, nor to nitely-valued logics.) The existence of classes of Kripke models with respect to which a logic is sound and complete is important because of several reasons. On the one hand, the semantics of the operators of the logic (via the canonical de nition of the operations on O(X )) provides a method for translation to clause form (even to classical logic) of a given formula, especially if the class of Kripke models has good properties e.g. is rst-order de nable (then these properties can be additionally taken into account; one obtains this way rst-order classical formulae which may be checked by a theorem prover). On the other hand, Orlowska and other authors have devised relational deductive systems for many logics which have sound and complete Kripke semantics (cf. e.g. [31, 32]).

3.3 Theorem Proving The translation to clause form for nitely-valued logics whose algebra of truth values is a distributive lattice can be seen as a translation to classical rst-order logic; thus general methods for automated theorem proving in classical logic can be applied to the sets of clauses obtained as explained in Section 3. To these clauses additional classes of clauses should be added, expressing, on the one hand, the fact that the meaning functions are hereditary, and, on the other hand, if necessary, the properties of the set of possible worlds. Similar considerations also apply to logics that are not nitely-valued if the properties of the corresponding classes of Kripke models can be described by rst-order formulae. In the nitely-valued case also many-valued versions of resolution can be applied, e.g. the signed version of hyperresolution presented in [37] extending Hahnle's signed hyperresolution for regular logics.

4 Examples In what follows we will illustrate the method for translation to clause form presented above by means of several examples: logics based on nite p-lattices, Pmn -logics, and n-valued

Lukasiewicz logics. These examples bring new elements compared with the results presented in Section 3: additional operators, such as pseudocomplementation or implication, which are not in the classes mentioned before, are considered as well; the main ideas in the clause transformation process remain however the same. Other examples, such as SHn-logics (useful in modeling posets of interacting agents, as noticed by Iturrioz) or SHKn-logics, are presented in detail in [37]. We end by mentioning some examples of logics which are not nitely-valued.

4.1 Logics Based on Finite Distributive p-Lattices

A distributive p-lattice is an algebra (A; _; ^; ; 0; 1) of type (2; 2; 1; 0; 0) such that (A; _; ^; 0; 1) is a distributive lattice with 0 and 1, and is an operation of pseudocomplementation, i.e. a = maxfb 2 A j a ^ b = 0g: This class of algebras has been studied, e.g. by Priestley [33]. It is known that the distributive p-lattices form a variety B! , and that all the subvarieties of B! form a chain B?1 B0 : : : Bn : : : B! , where B?1 is the trivial variety, B0 the variety of Boolean algebras, and each variety Bn , with n 2 is generated by Pn = 2n 1 (the n-th power of the 2-element chain to which a new top element is adjoined). Applications. Finitely-valued logics based on the algebra Pn = 2n 1 of truth values can be used for modeling inconsistent information in databases. Consider a database where every item can be assigned a value in a set of values V having n + 1 elements; n of these values are considered as acceptable values, and one is a special \alarm value", showing that something is wrong with the speci cation of the corresponding item in the database. When performing repeated queries in the database, asking for the value of a certain item, one may obtain contradictory answers. The elements of Pn in 2n stand for all possible combinations of values that can be obtained this way. The additional 1 adjoined as top element stands for the situation when the answer to one of the queries was the \alarm value" (it is supposed that no answer can be excluded from appearing in the database if this alarm value is encountered). Clause form transformation. Concerning deduction in logics based on the algebra Pn , we make the following remarks concerning the use of the Priestley duality: it is known that in particular, every algebra Pn = 2n 1 is a nite Heyting algebra. Its Priestley space is the space Vn , represented in Figure 2. Thus, for every n, Pn is isomorphic to the lattice of order lters of Vn , where for every U O(Vn ), U = X n # U . 1 OOOO 2 @@

:::

OOO @@ OOO @@ OOO@@ O

?

n ? 1 lll n x l xx lllll x xxllll x x l lxll

2 Further considerations lead to the replacement of the condition

that is a partial order with the condition that is re exive and transitive. Then O(X ) can be seen as the family of all increasing subsets of X w.r.t. this relation.


Figure 2.

Vn


We can thus apply the method in Section 3 to this type of nitely-valued logics. The rules for translation to clause form for _; ^ are those in Section 3. Additionally, we have the following rules for the pseudocomplementation: Lemma 4 For every meaning function m, 2 m( ) i 8 ; 62 m(), i.e. the following hold: (; t) ()t i 8 ; f ; (; f ) ()f i 9 ; t : We have therefore the following additional rules for clause form transformation: SL n_ (L )t o V (; t) SL; Lf

(; f )

f n SL _ (L ) o SL; Lt ;

Complexity. Due to the particular form of Vn , for every x 2 Vn nf?g there is exactly one y 2 Vn with y x. For x =?

there are exactly n + 1 such elements. Thus, the number of clauses generated from a formula by the procedure described in Section 3 for the logic based on Pn is 1 + 3(k1 + k2 )(n + 1) + k3 ((n + 2) + 2n), where k1 ; k2 ; k3 are the numbers of occurrences of _; ^; resp. in . The method of Baaz and Fermuller, in its most general form, generates in this case (k1 + k2 )(2n + 1)2 + k3 (2n + 1) clauses (similar results are also obtained with other methods using the algebra of truth values).

4.2

Pmn -logics

Pmn -logics are logics which are sound and complete with respect to the variety of Pmn -algebras. Pmn -algebras are distributive lattices with 0 and 1, endowed with a unary operation f satisfying f (0) = 1, f (1) = 0, the De Morgan laws with respect to _ and ^, and the additional conditions [40, 10]: f m (x) = f n (x) 8x 2 L if m ? n is even; m f (x) _ f n (x) = 1

8x 2 L f m (x) ^ f n (x) = 0 8x 2 L if m ? n is odd. Applications. The P20 -logic is the fragment of relevance logic Efed considered by Anderson and Belnap [2]. This logic is sound and complete with respect to the variety P20 of De Morgan algebras. The nite Kripke model corresponding to the generator of this variety has two incomparable elements. Kleene logics are sound and complete with respect to the subvariety K of the variety P20 , consisting of those De Morgan algebras that additionally satisfy the Kleene law (a^ a) (b_ b): The nite Kripke model corresponding to the 3-element generator of this variety K3 is a chain with two elements. We also mention logics sound and complete with respect to the variety of Stone algebras (algebras in P21 that satisfy the identity x ^ x = 0; every Stone algebra is a p-lattice, and the variety of Stone algebras is the coincides with the variety B1 described in Section 4.1). Clause Form Transformation. Urquhart [40] shows that the Priestley duality induces a dual equivalence between the


category of Pmn -algebras and a corresponding category of Priestley spaces endowed with an operation g that satis es the identity gm = gn . For m > n 0, let Smn be the structure (X; ; ; g) de ned by X = f0; 1; : : : ; m ? 1g, with the discrete topology and order, and with g(k) = k + 1 for k < m ? 1 and g(m ? 1) = n, as shown in Figure 3. Now let Lmn be the dual algebra of the space Smn (the lattice of all subsets of Smn , with f (Y ) = Smn ng?1 (Y )). The variety Pmn is generated by Lmn (which is a subdirectly irreducible algebra). The method for translation to clause form presented in Section 3 can also be applied in this case. 0

1

2

n-1

n

n+1

m-1

n+2

m-2

Figure 3.

m-3

Smn

Complexity. The complexity of the clause-form translation

of a formula of length l in this case (expressed by the number of clauses generated) is O(lm). By using approaches based on the algebra of truth values O(l22m ) clauses would be generated.

4.3 Logics Based on Implicative Lattices

Note that the remarks in Section 3 are a simpli cation; the results actually apply more generally. In this section we present a class of logics sound and complete with respect to a class of lattices endowed with an operation which is not in the classes Lh [ La [ Jh [ Mh [ Ja [ Ma, namely implicative lattices. An implicative lattice is an algebraic structure (A; _; ^; !), such that (A; _; ^) is a distributive lattice and ! is a binary operation satisfying the following equations: (IL1) x ! (y ^ y0 ) = (x ! y) ^ (x ! y0 ) (IL2) (x _ x0 ) ! y = (x ! y) ^ (x0 ! y) (IL3) x ! (y _ y0 ) = (x ! y) _ (x ! y0 ) (IL4) (x ^ x0 ) ! y = (x ! y) _ (x0 ! y) Note that ! is neither a join-hemimorphism nor a meethemimorphism (it actually is a lattice-homomorphism in the second argument and a lattice antimorphism in the rst argument). Examples of implicative lattices are provided by algebraic structures such as Boolean algebras, Wajsberg algebras, Ockham algebras, and lattice-ordered groups. Priestley representation theorems for implicative lattices have been studied by Martnez in [24]. The dual D (A) of an implicative algebra A is equal to D(A) [ f;; Ag, where D(A) is the Priestley space associated to A, on which an additional binary operation ', induced by the implication !, is de ned by: [ '(F1 ; F2 ) = fy j x ! y 2 F2 g: x2F1

In [24] a dual equivalence is established between the category of implicative lattices and a category of implicative spaces


(Priestley spaces with endpoints, additionally endowed with a binary relation having certain properties). In particular, it is shown that the lattice (O (D (A)); [; \; )) of proper order lters of D (A) is isomorphic to (A; _; ^; !) as an implicative lattice, where ) is de ned by:

U )V = 4.3.1

\

F 2U

fF 0 j '(F; F 0 ) 2 V g:

(!; f ) " i (L1 ! L2 )f i 9j : " j Lt1 ^ " (i + j ? 1) Lf2

Lukasiewicz logics

The Lukasiewicz logic of order n, LUKn is the logic with connectives _; ^; ! that is sound and complete with respect to the algebra Ln = (f0; n?1 1 ; : : : ; nn??12 ; 1g; _; ^; ; !); where for every x; y 2 f0; n?1 1 ; : : : ; nn??12 ; 1g we have x ^ y = min(x; y), x _ y = max(x;y), x = 1 ? x, and

x!y=

1 if x y 1 ? (x ? y) if x y :

It can be seen that Ln = (Ln ; _; ^; !) is an implicative lattice. By the Priestley duality for implicative algebras the dual space of Ln is the space of all prime lters (plus ; and Ln ), namely the ordered set D (Ln ) = f;; " 1; " nn??12 ; : : : ; " 1 2 n?1 ; " n?1 ; Ln g ordered by inclusion. The implication ! on Ln induces, according to the duality theory of implicative lat 2 tices [24] a binary S map ' : D (Ln ) ! D (Ln ), de ned by '(F1 ; F2 ) = x2F1 fy j x ! y 2 F2 g: One can check that '(;; " j ) = '(" j; ;) = ;, and '(" i; " j ) =" (i + j ? 1) (by convention, we assume that if k is negative, then " k = Ln ). Conversely, the lattice O (D (Ln )) of proper order- lters of D (Ln ) becomes an implicative lattice isomorphic to Ln , where the implication is de ned by

U )V =

\

p2U

r

D (Ln ) 6j=m;"i 1 ! 2 i " i 62 m(1 ! 2 ) i (9j such that " j 2 m(1 ) and " (i + j ? 1) 62 m(2 )) We thus obtain the following rules: V (!; t) " i (L1 ! L2 )t i j " j Lf1 _ " (i + j ? 1) Lt2

fq j '(p; q) 2 V g:

Clause form transformation. The theoretical considerations in Section 3 show that (D (Ln ); ; ') can be regarded as nite Kripke-style frames for the Lukasiewicz logic. Models are then pairs (D (Ln ); m), where m : Var ! P (D (Ln )) is a meaning function, required to be proper and hereditary (i.e. to take values in O (D (Ln ))). A satis ability relation can be de ned in a canonical way, as explained in De nition 1, by r

D (Ln ) j=m;x i x 2 m(): r

r

Thus, D (Ln ) j=m i m() = D (Ln ); and D (Ln ) j= r i for every meaning function m; D (Ln ) j=m ; where m is the unique (0; 1; _; ^; !)-homomorphism that extends m to formulae. We can then use the de nitions of the implicative algebra operations induced on O(D (Ln )) by the operations on D (Ln ), inTparticular the fact that m(1 ! 2 ) = m(1 ) ) m(2 ) = "j2m(1 ) f" i j" (i + j ? 1) 2 m(2 )g: r

D (Ln ) j=m;"i 1 ! 2 i " i 2 m(1 ! 2 ) i (8j; " j 2 m(1 ) implies " (i + j ? 1) 2 m(2 ))


Exploiting the isomorphism between Ln and O (D (Ln )), we can read " i Lt as L i, and # i Lt as L i (w.r.t. a certain valuation in Ln ). We then have " i Lf i # (i ? n?1 1 ) Lt and vice-versa. We thus obtain a further transformation of the rules above, similar to the translation to mixed integer programming given in [14], by using the fact that 1 ! 2 i i (9j : 1 j and 2 i + j ? 1): (!; t)

" i (L1 ! L2 )t 9j : # j Lt1 ^ " (i + j ? 1) Lt2

(!; f )

" i (L1 ! L2 )f 9j : " j Lt1 ^ # (i ? n?1 1 + j ? 1) Lt2

In further work we would like to see whether similar results can be established also for more general classes of (e.g. nite) implicative algebras (the dual spaces are in certain interesting cases ordered semigroups, or monoids).

4.4 Other Types of Non-Classical Logics In this section we brie y mention some examples in which representation theorems such as, for instance, the Priestley representation theorem, can be used for establishing a link between the algebraic and Kripke models of logics. Logics based on p-lattices. By the duality theorem for plattices (cf. e.g. [18]) and Theorem 1 it follows that the logics based on p-lattices are sound and complete w.r.t. the class of all Kripke models (X; R), where R is a partial order on X (further considerations may be used to show soundness and completeness w.r.t. the class of the Kripke models (X; R), where R X X is re exive and transitive). Logics based on Ockham algebras. Ockham algebras are distributive lattices with 0, 1, and one unary lattice antimorphism . Ockham algebras are models for logics endowed with a negation operator that satis es the De Morgan laws. In particular all Pmn -algebras are Ockham algebras. A Priestley duality for Ockham algebras has been studied in [40]. The spaces corresponding to the Ockham algebras are Priestley spaces endowed with a unary, continuous, order-reversing map g. Thus, the class of spaces de ned by the family of all non-topological properties of these Priestley duals is the class of all ordered spaces endowed with an order-reversing unary map. It turns out that for every such space, its lattice of order- lters is an Ockham algebra, and thus, by Theorem 1, it follows that the logics which are sound and complete with respect to the variety of Ockham algebras have as Kripke-style models ordered spaces (X; ; g) endowed with an order-reversing map (the


condition that is a partial order can be relaxed to the condition that is re exive and transitive). Relevance logics. Urquhart [41] gave an algebraic semantics for relevance logics in terms of distributive lattices with additional operators (called in what follows relevant algebras). Moreover, he proved that a Priestley duality holds for this type of algebras. It turns out that the class of ordered spaces de ned by the family of all those properties of the Priestley duals of relevant algebras which do not involve topological notions satis es the conditions of Theorem 1, i.e. it has the property that for every such space X , O(X ), with operations de ned in an appropriate way, is a relevant algebra. Thus, by Theorem 1, they provide a family of Kripke-style models for relevant logics. This family of Kripke models coincides with the Kripke models for a version of relevance logics studied by Routley and Meyer (cf. [35]). Theorem 1 shows that soundness and completeness with respect to this class of Kripkestyle models can be proved by completely algebraical means, by using the Priestley duality for relevant algebras and the soundness and completeness with respect to the class of relevant algebras. Quantum logics. A similar result can be established for quantum logics: starting from a Priestley-style representation theorem for orthomodular lattices due to Iturrioz [19] we obtain a class of Kripke-style models for quantum logics (similar models are introduced by Goldblatt in [11], cf. also [13]). One of the properties in the de nition of this class of Kripkestyle models is expressed by a second-order formula; actually Goldblatt proved that there is no rst-order characterization of orthomodularity for orthoframes [13]. We also refer to the work of Allwein [1] who introduced a Kripke-style semantics (3-valued in every possible world) for linear logics starting from Urquhart's representation theorem for non-distributive lattices [39]. These results seem to be only the beginning of a promising eld of research that we would like to explore in future work. We expect that they will enable us to use recent results in the algebraic investigation of various classes of distributive lattices with operators for obtaining Kripke models for some logics with interesting practical applications. We think that this would gradually make automated theorem proving in more classes of logics possible and ecient.

5 Related Work

The work reported here was inspired by the papers of Baaz and Fermuller [3, 4, 5] and Hahnle [15] on resolution-based theorem proving in nite-valued quanti cational rst-order logics on the one hand, and by the existing representation theorems for distributive lattices with operators [34, 40, 10, 12, 7, 24, 18] on the other hand. The goal of our approach is to exploit the ner structure of the algebra of truth values of a given logic, and, when this structure allows it, to improve the eciency of the automated theorem proving procedure both by reducing the number of clauses in the clause form of a formula, and by re ning the resolution procedure. Among the methods exploiting the structure of the algebra of truth values, we would like to mention the following: Orlowska and Morgan [28, 26] give theorem proving systems for m-valued Post logics, and [29, 30] for algorithmic logics.


Salzer [36] investigates the problem of nding optimal axiomatizations for operators and distribution quanti ers in nitelyvalued rst-order logics based on semi-lattices. Hahnle [17] derives tableau-style axiomatizations of distribution quanti ers by using e.g. Birkho's representation theorem for nite distributive lattices. The signed normal forms obtained by applying this method contain as signs both order-ideals and order- lters. In [17], propositional connectives are not taken into account. In our approach we explicitly request that the many-valued logics considered are based on distributive lattices, and we obtained a uniform way of expressing signed clauses, the signs being pairs consisting of an order- lter and one of the values t (true) or f (false). Approaches to automated proving in paraconsistent [22] or annotated logics [23, 25] also have certain similarities with our own approach (in the case of nitely-valued logics), especially concerning the form of literals (in our approach the signs are of the form " i with i join-irreducible and arise in a natural way from theoretical considerations); moreover, although a version of signed resolution has been proved correct for paraconsistent logics [22, 23], we are not aware of any paper using a version of signed hyperresolution, as in [37]. It seems that known methods from modal logic occur as particular cases of more general concepts such as those considered here (see also [12]).

6 Conclusions

In this paper we presented the use of Priestley representation theorem (in general, of representation theorems) to automated theorem proving in non-classical logics. We considered both nitely-valued logics and other non-classical logics, not necessarily nitely-valued. We showed that the Priestley duality for distributive lattices with quanti ers provides a theoretical tool for better understanding the link between algebraic and Kripke-style models. In the nitely-valued case the de nitions of the functions and relations associated with the operations on the algebra of truth values A immediately suggest a notion of satis ability with respect to the Kripke dual D(A) of A, as well as a method for translation to clause form. We gave several examples in order to illustrate how this translation to clause form takes place. We then showed that similar ideas can be applied to logics which are not necessarily nitely-valued. Situations when the algebra of truth values is not a distributive lattice, such as in the case of quantum logics are also mentioned; we also mention existing related approaches to giving Kripke-style models for linear logics. There remain many things we would like to do in future work. First of all we would like to improve our method for proving in nitely-valued logics. One of the possibilities would be to use a notion of polarity in the clause generation process, such as done in [15]. We are also interested in better exploiting the more general results presented in Section 3.2. The link between algebraic and Kripke-style models presented there is only a starting point towards a systematic way of translating formulae into classical two-valued logics (or, in the case of linear logic, to a special kind of three-valued logics, cf. [1]). Until now we did not study problems such as the existence of decision procedure for these very general classes of propositional logics; in future work we would like to investigate (when possible) whether methods similar to the ones used in certain


modal logics (for instance by Ohlbach cf. e.g. [27]) would yield decision procedures for some more general logics, having e.g. the nite model property. Acknowledgements. This research has been supported partially by Esprit brp 6471 (Medlar II) at RISC-Linz, and partially by a postdoctoral fellowship at MPII. We gratefully acknowledge the support by Cost Action 15 mvl. Many thanks to M. Baaz, H. Ganzinger, R. Hahnle, L. Iturrioz, H. de Nivelle, H.J. Ohlbach, E. Orlowska, J. Pfalzgraf, H. Priestley, and K. Stokkermans for their papers and for stimulating discussions.

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