COMMODIOUS AXIOMATIZATION OF QUANTIFIERS IN MULTIPLE-VALUED LOGIC REINER HA HNLE Abstract. We provide tools for a concise axiomatization of a broad class of
quanti ers in many-valued logic, so-called distribution quanti ers. Although sound and complete axiomatizations for such quanti ers exist, their size renders them virtually useless for practical purposes. We show that for quanti ers based on nite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and lters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkho 's representation theorem for nite distributive lattices are used to derive tableau-style axiomatizations of distribution quanti ers.
Introduction 1 The aim of this paper is to provide concise axiomatizations of certain quanti ers
in many-valued logic which were introduced by Mostowski (1961) and baptized distribution quanti ers by Carnielli (1987). The task of axiomatizing such quanti ers
has been solved satisfactorily in theory: sound and complete axiomatizations for sequent and tableau calculi are given, for example, in (Rousseau, 1967; Carnielli, 1987; Carnielli, 1991; Zabel, 1993; Zach, 1993; Hahnle, 1994; Baaz & Fermuller, 1995). For the purpose of automated theorem proving, however, this is not enough. There, one needs minimal axiomatizations which, moreover, one must be able to nd in a reasonably ecient way. Until recently, only few systematic solutions for restricted cases existed (Zabel, 1993; Hahnle, 1994) which will be extended in this paper (how, is discussed in detail in Section 6). In this paper we provide axiomatizations in the form of sequent or tableau rules for a subclass of distribution quanti ers. The key observation, due to Zabel (1993), is that the bodies of quanti ed signed formulas, when a Skolem term or universal term is substituted for the quanti ed variable, can be used to characterize upsets and downsets of the set lattice over the set of truth values N. This leads ultimately to schematic and compact tableau rules for quanti ers which are de ned as generalized meet and join if the truth values form a nite, distributive lattice. The paper is organized as follows: in Section 1 we provide some basic de nitions to make the paper self-contained. In Section 2 we brie y explain how tableau rules for quanti ed formulas in many-valued logic can be obtained in general. In Section 3 we prove some results about the correspondence between skolemized quanti ed signed formulas and lters and ideals on the set lattice of truth values. In Section 4 we apply these results to obtain schematic and concise tableau rules for certain quanti ers. This paper deals with the case where the set of truth values N is nite, however, in Section 5 we discuss the in nitely-valued case and argue that Date : May 7, 1997. Key words and phrases. Many-valued logic, in nitely-valued logic, distribution quanti ers, distributive lattices, semantic tableaux, sequent calculi. 1 An extended abstract appeared in the Proceedings of ISMVL 1996 held in Santiago de Compostela/Spain, pp. 118{123, published by IEEE Press, Los Alamitos. 1
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the present results do not easily generalize. We close the paper with a discussion of related and future work. 1. Basic Definitions
1.1. Logic: Syntax. De nition 1. A rst-order signature is a triple hP; F ; i, where P is a non-empty family of predicate symbols, F is a possibly empty family of function symbols disjoint from P, and assigns a non-negative arity to each member of P [ F. Let Term be the set of -terms over object variables Var=fx0; x1; : : : g, and let Term0 be the set of ground terms in Term. Atoms are de ned as: At =fp(t1 ; : : : ; tn )jp2P; (p)=n; ti 2Term g : De nition 2. A rst-order language is a triple L=h; ; i, where is a nite or denumerable set of logical connectives and de nes the arity of each connective. Connectives with arity 0 are called logical constants. is a nite or denumerable set of quanti ers. The set L of L-formulas over is inductively de ned as the smallest set with the following properties: 1. At L . 2. If 1 ; : : : ; m 2L , 2 and ()=m then ( 1 ; : : : ; m )2L . 3. If 2, 2L , and x2Var then (x) 2L .
1.2. Logic: Semantics. De nition 3. The set of truth values N is an arbitrary nite set. jN j denotes the cardinality of N . If L=h; ; i is a rst-order language then we call a triple A=hN; A; Qi, where N is a truth value set, A assigns to each 2 a function A() : N () ! N , and Q assigns to each 2 a function Q() : P + (N) ! N a rst-order matrix for L (we abbreviate 2N ? f;g by P + (N)). Q() is called the distribution function of the quanti er . This generalized notion of a quanti er is due to Rosser & Turquette (1952, Chapter IV). The above, simpli ed, de nition is due to Mostowski (1961). The phrase distribution quanti er for referring to quanti ers of this kind was coined by Carnielli (1987). Example 1. Let N = f0; n?1 1 ; : : : ; nn??21 ; 1g. Then one can de ne generalizations of the classical quanti ers via Q(8) = min, Q(9) = max, where min and max are de ned wrt the natural order of N . In particular, for jN j = 2 one has N = f0; 1g and the mapping:
X Q(8)(X) Q(9)(X) f0g 0 0 f1g 1 1 f0,1g 0 1 In three-valued logic Q(8), Q(9) are extended as follows:
1 1 f 12 g 2 21 1 f0; 2 g 0 2 1 f 12 ; 1g 1 2 f0; 12 ; 1g 0 1 De nition 4. A pair L=hL; Ai consisting of a rst-order language L and a rstorder matrix for L is called jN j-valued rst-order logic.
COMMODIOUS AXIOMATIZATION OF QUANTIFIERS IN MULTIPLE-VALUED LOGIC 3 2 M over is De nition 5. Let L be a rst-order logic. An Herbrand structure 0 ( an interpretation I that maps p2P to functions I(p) : (Term ) p) ! N . It is assumed that contains at least one 0-ary function symbol. For a structure M over and a closed rst-order formula we de ne a valuation function vM : L ! N via: 1. If =p(t1 ; : : : ; tn) then vM (t)=I(p)(t1 ; : : : ; tn). 2. If =( 1 ; : : : ; m ) then vM ( )=A()(vM ( 1); : : : ; vM ( m )). 3. The distribution of (x) is dM ( (x)) = fvM ( (t))jt2Term0 g.3 If =(x) (x) then vM ( )=Q()(dM ( (x))). We de ned a distribution as the range of the function vM ( (x)) : Term0 ! N
and mapped distributions to truth values. It would even have been possible to associate a di erent truth value with each function vM ( (x)) instead of taking merely the distribution (the range) which would have given an even ner grained class of quanti ers. In this case already for two-valued N there are in nitely many new quanti ers.
Example 2. The quanti ers 8, 9 de ned in Example 1 are simply the classical quanti ers for jN j = 2, N = f0; 1g. De nition 6. Let S N . A signed rst-order -formula S is said to be ( rstorder) satis able i there is a structure M over such that vM ( ) 2 S . In this case we say that M is a model of S and write M j= S . S is valid, in symbols j= S , i every -structure is a model of .
1.3. Lattices. In this paper we assume that the set of truth values has a lattice structure. Therefore, we need some de nitions and results from lattice theory as can be found for instance in (Birkho , 1970; Davey & Priestley, 1990). De nition 7. A lattice L is a partially ordered set hN; i such that any two elements of N have a unique supremum (called join, we use the symbol t) and a unique in mum (called meet, we use the symbol u) in N . A lattice can be alternatively de ned with join and meet alone in which case it is explicitly stipulated that they are associative, commutative, idempotent and absorptive. With this in mind we write tfi1 ; : : : ; in g for i1 t(i1 t( (in?1 tin ) )) and similarly with u. Any nite lattice is bounded: there is a (unique) minimal element ? and maximal element > in L. A lattice L is distributive i for all a; b; c 2 L: a u (b t c) = (a u b) t (a u c). A particular kind of lattice we will work with is the Boolean set lattice for N 2N = h2N ; ;; N; \; [i, where \ is set intersection, [ is set union, is set inclusion, and N any set. In our present setting we associate elements of 2N with distributions of truth values. De nition 8. Let L be a lattice and i 2 L. Then "i = fx 2 Lj x ig and #i = fx 2 Lj x ig are called the upset, respectively, the downset of i. The interval between i and j is de ned as [i; j] = fx 2 Lj i x j g. We say an element x 2 L is covered by y 2 L i x < y and for any z x z < y implies x = z . The elements that cover ? are called the atoms of L. Obviously, [i; j] = ("i) \ (#j) holds for all i; j 2 L. De nition 9. Let L be a lattice and I , F sets of elements of L. If
2 We work with Herbrand structures in order to avoid unnecessary technical complications. The result that any satis able -formula is satis able already in a Herbrand structure over some is proved exactly as the classical result, see (Baaz & Fermuller, 1995). 3 Note that the value of d M is always a non-empty set of truth values.
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1. a; b 2 I (a; b 2 F ) imply a t b 2 I (a u b 2 F ) and 2. a 2 L, b 2 I (b 2 F ) and a b (a b) imply a 2 I (a 2 F ) then I is called an ideal (F is called a lter) of L. For each a 2 L, #a is an ideal of L. This particular ideal is called the principal ideal generated by a. Principal lters are de ned dually. A maximal ideal (a maximal lter) is an ideal (a lter) such that the only ideal ( lter) properly containing it is L. In nite lattices, all lters and ideals are principal. De nition 10. Let L be a lattice. An element x 2 L is called meet-irreducible (join-irreducible) if 1. x 6= > (x 6= ?) and 2. x = a u b (x = a t b) implies x = a or x = b for all a; b 2 L. The sets of meet-irreducible and join-irreducible elements of L are denoted with M(L) and J (L), respectively. The following lemmas will be needed later on: Lemma 1. Let L be a distributive lattice, i 2 L. If i 2 M(L) then a1; : : : ; ar 2 L and a1 u u ar i imply ai i for at least one 1 i r. For join-irreducible
elements a dual result holds. Proof. See (Birkho , 1970, Lemma 1, Section III.3) or (Davey & Priestley, 1990,
Lemma 8.16)
Lemma 2. Let L be a nite distributive lattice, i 2 L. Let M be the minimal elements of M(L) \"i and J the maximal elements of J (L) \#i. Then i = uM = tJ (using the convention ufg = > and tfg = ?). Thus each element of a nite dis-
tributive lattice either is the top (bottom) element or it can be uniquely represented as a meet (join) of meet-(join-)irreducible elements. Proof. See (Birkho , 1970, Corollary to Lemma 1, Section III.3) or (Davey & Priest-
ley, 1990, Proof of Theorem 8.17).
2. Standard Axiomatization of Distribution Quantifiers The problem one needs to solve in order to provide sequent or tableau rules4 for signed quanti ed formulas is: given a quanti ed signed formula S (x) (x) we are looking for a formula of the form
_^
i2I j 2Ji
Sij (zij )
(1)
where Sij N and the zij are certain ground terms falling in two categories: either zij is a Skolem constant c, then it must be new relative to the proof in which the formula occurs or zij is a term t, then it stands for an arbitrary ground term. In Lemma prop:fol-rule below we stipulate for (1) that any structure M over can be extended to a structure M over such that S (x) (x) is satis able in M i (1) is satis able in M for every possible substitution of ground terms for zij of universal type. This is nothing else than soundness and completeness of the following sequent/tableau rule which is simply (1) written in a slightly di erent notation: 4 We use ground versions of tableau rules a la Smullyan (1995) as opposed to free variable tableau rules (Fitting, 1996) in order to avoid technical complications. It is completely straightforward to give free variable versions of the results in this paper.
COMMODIOUS AXIOMATIZATION OF QUANTIFIERS IN MULTIPLE-VALUED LOGIC 5
S (x) (x) S11 (z11) Sr1 (zr1 ) (2) .. .. . . S1s1 (z1s1 ) Srs (zrs ) where I = f1; : : : ; rg, Ji = f1; : : : ; si g, and the zij are either cij or t according to their status in (1). Such rules are given in (Rousseau, 1967; Carnielli, 1991; Zach, 1993; Baaz & Fermuller, 1995) for the special case where all occurring signs S, S11 , : : : ,Srs are singleton sets. (Hahnle, 1994) deals with the general case which can easily be obtained from the following lemma: Lemma 3 ((Hahnle, 1994)). Let (Q())?1 (S) = f; 6= I N j Q()(I) 2 Sg. Then a signed quanti ed formula S (x) (x) is satis ed in M over i there is an I=fi1; : : : ; ir g 2 (Q())?1 (S) such that 1. for each ik 2 I , there is a constant term ck not occurring in (x) such that fik g (ck ) and 2. for any ground term t, I (t) are simultaneously satis able in a suitable M over = [ fc1 ; : : : ; cr g. Proof. Consider a model M of S (x) (x). The two conditions mean: (1) there are witnesses t1; : : : ; tr such that all of the truth values fi1; : : : ; ir g = I are taken on by vM ( (t1 )), : : : , vM ( (tr )) in M; as usual M is suitably extended to M such that the interpretation of each ci and ti is equal; (2) no other values than those from I are reached by any vM ( (t)). Thus, by de nition, and since the ci do not occur in (x), v M ((x) (x)) = vM ((x) (x)) 2 S. The other direction is similar. Let (Q())?1 (S) = fI1 ; : : : ; Ir g, Ii = fki1; : : : ; kijI j g. Then set I = f1; : : : ; rg, si = jIij +1, Sij = fkij g, zij = cij for j jIi j and Si(jI j+1) = Ii , zi(jI j+1) = t in (1) and (2) to obtain a sound and complete sequent/tableau rule from the preceding lemma. Example 3. Consider N = f0; 12 ; 1g and 8 as de ned in Example 1. All subsets of N that contain at least one of 0 and 21 are contained in (Q(8))?1 (f0; 21 g), all in r
r
i
i
i
all six sets. Hence the tableau rule according to Lemma 3 is as shown in Figure 1.
The obvious problem with such rules is that they can become rather big. This is unavoidable in general: Zach (1993) shows that for similar characterizations as (1) there exist combinations of quanti ers and signs such that jI j = 2jN j?1 . Hence, the branching factor can be exponential with respect to the cardinality of the set of truth values. On the other hand, Lemma 3 (and the corresponding results in (Rousseau, 1967; Carnielli, 1991; Zach, 1993; Baaz & Fermuller, 1995)) tends to give \fat" rules even when \slim" ones exist. For instance, the rule (3) below is a sound and complete rule for the formula in Example 3 as well and it is considerably smaller than the rule displayed in Figure 1. 3. Instances of Quantified Formulas and Boolean Set Lattices In the following we show that in many relevant cases concise representations of distribution quanti ers can be obtained. The starting point is the observation that signed formulas of the form I (c) and J (t) can be used to characterize certain sets of distributions. The important thing to note is that these sets of distributions play special r^oles within the set lattice 2N . For ; 6= F N we introduce the abbreviation U (F) = fX N j X \ F 6= ;g. With it one can prove the following improved version of Lemma 3 for certain cases:
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f0; 21 g (8x) (x) f0g (c) f0g (c) f0g (c) f0g (c) f 12 g (c) f 21 g (d) f 12 g (d) f 12 g (d) f1g (e) f1g (e) f1g (e) f0g (t1 ) f0; 12 g (t2) N (t3 ) f0; 1g (t4) f 12 g (t5 ) f 21 ; 1g (t6) Figure 1. Tableau rule for f0; 21 g (8x) (x), see Example 3.
Lemma 4. Assume U (F) = (Q())?1 (S). Then any structure M over can be extended to a structure M over such that vM ((x) (x)) 2 S i v M ( (c)) 2
F , where c does not occur in . Proof. Assume vM ((x) (x)) 2 S. Thus the distribution I = dM ( (x)) is in (Q())?1 (S) and, by assumption, also in U (F). By de nition of U (F), I \ F 6= ;, thus consider i 2 I \ F. By i 2 I and the de nition of a distribution there must be a ground term t such that vM ( (t)) = i. We de ne M by setting = [ fcg, I M (p(c)) = IM (p(t)) if c occurs in p and I M (p) = IM (p) otherwise. Then we have v M ( (c)) = v M ( (t)) = vM ( (t)) = i 2 F as desired. Conversely, let M be as above and let i = vM ( (t)) = v M ( (c)) 2 F. By de nition, i 2 dM ( (x)) = I. Hence, by de nition of U (F), I 2 U (F) and by assumption I 2 (Q())?1 (S). Finally by de nition of vM we have vM ((x) (x)) = Q()(dM ( (x))) = Q()(I) 2 S : The single extension containing just F (c) is, of course, considerably shorter than the extensions corresponding to the set of distributions U (F), hence one obtains much simpli ed rules, whenever U (F) = (Q())?1 (S) holds for some F N. For instance, because of U (f0; 21 g) = (Q(8))?1 (f0; 21 g) in Example 3, the rule shown in Figure 1 can be simpli ed to:
f0; 21 g (8x) (x) f0; 12 g (c)
(3)
The usefulness of Lemma 4 comes from the fact that the family of sets U (F) has an interesting structure which occurs frequently: Lemma 5. For any F 2 2N, U (F) = Si2F "fig, in particular, U (fig) = "fig. Proof. Assume X 2 U (F) and S let i be any member of X \ F. Then i 2 F and X 2 "fig. Conversely, let X 2 i2F "fig, say X 2 "fig for i 2 F. Hence, i 2 X and i 2 X \ F, so X 2 U (F) by de nition. A principal lter generated by an atom of 2N is a maximal lter of 2N (a maximal lter on N for short) and in the nite case all maximal lters on N are of this form. Thus, according to the previous lemma, whenever (Q())?1 (S) is a union of maximal lters one may choose its upset representation to obtain a single-extension rule. In particular, whenever (Q())?1 (S) is a maximal lter on N, say "fig, then there is a single-extension rule with exactly one formula, namely fig (c). In the nite case this can be generalized:
COMMODIOUS AXIOMATIZATION OF QUANTIFIERS IN MULTIPLE-VALUED LOGIC 7
Lemma 6. For nite N the principal lter "I of 2N is equal to the intersection of T the maximal lters i2I "fig.
Proof. Observe T that the intersection of two lters again is a lter. It isTobvious
that "I i2I "fig as I 2 "fig for all i 2 I. Conversely, for every J 2 i2I "fig one has J I, thus J 2 "I. (Cf. Ex. 9.2(1) in (Davey & Priestley, 1990).)
In other words, for nite N, whenever (Q())?1 (S) is a (principal) lter of 2N , then there is a single extension rule containing the formulas ffig (ci)j i 2 I g, where "I = (Q())?1 (S). As Sa consequence, whenever we have a representation of (Q())?1 (S) of the form k2K Fk , where the Fk are lters of 2N , then, by repeated application and disjunctive combination of Lemmas 4, 5, and 6, there is a tableau rule for S(x) (x) with jK j extensions. Similarly as signed formulas of the form I (c) characterize distribution quanti ers whose distributions correspond to lters of 2N , signed formulas of the form I (t) characterize distribution quanti ers whose distributions correspond to certain ideals of 2N . To make this idea more precise, we de ne D(I) = fX N j ; = 6 X I g. First we prove an analogon of Lemma 4. Lemma 7. Let (x) (x) be a -formula and M any -structure. Assume D(I) = (Q())?1 (S). Then vM ((x) (x)) 2 S i vM ( (t)) 2 I for all ground terms t. Proof. Assume vM ((x) (x)) 2 S. Thus the distribution J = dM ( (x)) is in (Q())?1 (S) and, by assumption, also in D(I). By de nition of D(I), I J 6= ;. By de nition of a distribution and of J, for an arbitrary term t one has vM ( (t)) 2 J I. Conversely, assume vM ( (t)) 2 I for all ground terms t. By de nition, this entails ; 6= dM ( (x)) I, hence dM ( (x)) 2 D(I) = (Q())?1 (S). From this one immediately concludes vM ((x) (x)) 2 S.
D(I) [f;g is the principal ideal of 2N generated by I, that is #I. In nite lattices every ideal is principal, hence, whenever (Q())?1 (S) [ f;g is an ideal #I of 2N , then there is a single-extension rule with exactly one formula, namely I (t). So the characterization of distributions that are ideals of 2N is even slightly easier than that of lters of 2N .
4. Distribution Quantifiers and the Structure of 2N It could be argued that the case when (Q())?1 (S) can be straightforwardly repesented as a DNF combination of lters and ideals is relatively rare and therefore the results above are not particularly relevant. There are two objections to this argument: rst, as Zabel (1993, Section 1.3.3) points out, even when (Q())?1 (S) has no representation as a DNF combination of lters and ideals, it still can possibly be partitioned into (Q())?1 (S) = I [R such that I has a representation as a DNF combination of lters and ideals. In this case, at least the part of the standard rule (2) that corresponds to I can be simpli ed. Second, in the following we show that many \naturally" de ned quanti ers have a representation as a DNF combination of lters and ideals, in fact as a simple conjunction of certain lters and ideals. Example 4. Consider the truth value set FOUR Q = f?; f; t; >g with Q the ordering indicated in Figure 2(a). De ne a quanti er on FOUR via Q( ) = u, where u is the meet operator on FOUR considered as a lattice. Q Let us look at the set of distributions corresponding to
f?g, ff; ?g, and ff g, respectively. One computes
and the sets of signs
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8
(Q(Q))?1 (f?g) =ff?g; f?;f g; f?; tg; ff;tg; f?; >g; f?;f; tg; f?; f; >g; f?; t; >g; ff;t; >g; FOURg further
(Q(Q))?1 (ff g) =fff g; ff; >gg
and
(Q(Q))?1(f?; f g) =(Q(Q))?1 (ff g) [ (Q(Q))?1 (f?g) :
It turns out that all three sets of distributions can be simply characterized using lters and ideals of the set lattice of FOUR, see Figure 2(b):
(Q(Q))?1 (f?; f g) = "ff g [ "f?g = U (f?; f g) (Q(Q))?1(f?g) = "ff; tg [ "f?g (Q(Q))?1 (ff g) = "ff g \ #ff; >g
This leads to the following rules:
f?; f g (Qx) (x) f?; f g (c)
f?g (Qx) (x) ff g (c) f?g (e) ftg (d)
ff g (Qx) (x) ff g (c) ff; >g (t)
FOUR f?; f; tg f?; f; >g f?; t; >g ff; t; >g f?; f g
> f
t
f?; tg
ff; tg
f?; >g
f?g
ff g
ftg
? (a) FOUR
ff; >g ft; >g f>g
fg (b) Set lattice of FOUR
FOUR with an ordering and the set lattice of FOUR. This motivating example is generalized in the following theorem which proves the existence of concise characterizations of truth value lattice-based quanti ers in general: 5 over a nite set of truth values Theorem 8. Assume L = hN; u; ti isQ a lattice P N . We de ne distribution quanti ers and via: Q(Q) = u (4) P Q( ) = t (5) Figure 2. The truth value set
Then the following holds: 5 We use *i and +i to denote the upset and downset of an element i 2 L to distinguish it clearly from upsets and downsets in 2N .
COMMODIOUS AXIOMATIZATION OF QUANTIFIERS IN MULTIPLE-VALUED LOGIC 9
1. If i 2 M(L) then (Q?1(Q))(fig) = U (fig) \ (D(*i) [ f;g) 2. If L is distributive and i 2 M(L) then (Q?1(Q))(+i) = U (+i) 3. For all i 2 N and distributive L: (Q?1 (Q))(fig) =
\
m2Mi
!
U (+m) \ (D(*i) [ f;g)
where Mi are the minimal elements of M(L) \ *i. 4. For all i 2 N and distributive L: \ U (+m) (Q?1 (Q))(+i) = where Mi is as above. 5. For all i 2 N
m2Mi
(Q?1 (Q))(*i) = D(*i)
6. If i 2 J (L) then (Q?1(P))(fig) = U (fig) \ (D(+i) [ f;g) 7. If L is distributive and i 2 J (L) then (Q?1 (P))(*i) = U (*i) 8. For all i 2 N and distributive L:
0 1 \ P (Q?1 ( ))(fig) = @ U (*j)A \ (D(+i) [ f;g) j 2Ji
where Ji are the maximal elements of J (L) \ +i. 9. For all i 2 N and distributive L: \ (Q?1 (P))(*i) = U (*j) where Ji is as above. 10. For all i 2 N
j 2Ji
(Q?1(P))(+i) = D(+i)
Proof.
ad (1): If i is meet-irreducible, then uX = i implies i 2 X, therefore: (Q?1(Q))(fig) = fX N j i 2 X and X L ig = fX N j fig \ X = 6 ;g \ fX N j X L ig then apply the de nitions: = U (fig) \ fX N j X *ig = U (fig) \ (D(*i) [ f;g) ad (2): Follows from (4) by noting that if i is meet-irreducible then Mi = fig (where Mi is de ned as in case (3)).
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10
> F C
> G
D A
F E
B
? (a) M(NINE )
G D
C A
E B
? (b) +D in NINE
Figure 3. Illustration of case (4) in Theorem 8 and of Example 5.
ad (3): If i is meet-irreducible, then, of course, case (1) applies; if i not meetirreducible, then (Q?1 (Q))(fig) may contain additional distributions, because uX = i implies no longer i 2 X: the distributions X with uX = i, but X >L i are exactly the ones that must be added (for instance, ? = f u t in FOUR).
These distributions are dicult to characterize in general. In the case of distributive lattices, however, the unique meet-representation of i with meet-irreducible elements M = Mi provided by Lemma 2 can be used. Assume X 2 (Q?1(Q))(fig), thus uX = i, hence X L i from which X 2 DT(*i) follows immediately by de nition of D. It remains to show that X 2 m2M U (+m). Let X = fx1; : : : ; xr g with uX = i. For any meetirreducible m in a distributive lattice by Lemma 1 we have that m L i = ufx1; : : : ; xr g implies m L xj for some 1 j r, hence X 2 U (+m) for any meet-irreducible element m L i. Conversely, assume X is in the set de ned by the term on the right-hand side of the equation in case (3). We must show thatT uX = i. By de nition of D, one immediately has uX L i. From X 2 m2M U (+m) one infers that there are fx1; : : : ; xjM jg X such that for each xj there is exactly one mj 2 M with xj L mj . Thus i L uX L ufx1; : : : ; xjM jg L ufm1; : : : ; mjM jg = uM = i : ad (4): Assume XT2 (Q?1(Q))(+i), thus uX L i. Let M = Mi ; we need to show that X 2 m2M U (+m). Let X = fx1; : : : ; xr g with uX L i. For any meet-irreducible m in a distributive lattice by Lemma 1 we have that m L i L ufx1; : : : ; xr g implies m L i L xj for some 1 j r, hence X 2 U (+m) for any meet-irreducible element m L i. T Conversely, assume X 2 m2M U (+m). We must show that uX L i. As in the previous case, there are fx1; : : : ; xjM jg X such that for each xj there is exactly one mj 2 M with xj L mj . Thus uX L ufx1; : : : ; xjM jg L ufm1 ; : : : ; mjM jg = uM = i : ad (5): Let X 2 (Q?1 (Q))(*i), hence uX L i i ; 6= X *i. Thus (Q?1(Q))(*i) = fX N j ; = 6 X *ig from which the result follows by de nition of D. ad (6){(10): By duality.
COMMODIOUS AXIOMATIZATION OF QUANTIFIERS IN MULTIPLE-VALUED LOGIC 11
Observe the strong duality between QPand PQ: cases (6){(10) in the previous theorem can be obtained by substituting for , \join" for \meet", etc. in cases (1){(5). This duality generalizes the well known duality between the classical quanti ers 8 and 9. Of course, duality extends to the associated tableau rules:
Corollary. Cases (1){(10) in the previous theorem give rise to the following sound and complete rule schemata for distribution quanti ers: If i is meet-irreducible: If i is join-irreducible: L distributive (6) L distributive (1) (2) (7) fig (Px) (x) fig (Qx) (x) Q P +i ( x) (x) *i ( x) (x) fig (c) fig (c) *i (t) +i (c) +i (t) *i (c) For all i, M= min(M(L) \ *i)= fm1; : : :; mr g, J= max(J (L) \ +i)=fj1 ; : : :; js g If L is distributive (3) (8) (4) Q fig (Qx) (x) fig (Px) (x) +i ( x) (x) +m1 (c1) *j1 (c1) +m (c ) .. .
+mr (cr ) *i (t)
.. .
*js (cs) +i (t)
1
.. .
1
+mr (cr )
(9)
*j (Px) (x) *j1 (c1 ) .. .
*js (cs )
For all i and arbitrary lattices6L:
(5)
*i (Qx) (x) *i (t)
(10)
+i (Px) (x) +i (t)
Proof. Repeated use and conjunctive combination of Lemmas 4 and 7.
Whenever i is the bottom (top) element of L, a signed formula of the form *i (+i ) is tautological and can be removed in the rules above. Note that all signs occurring in the conclusions are of the form *i, +i or fig. As for all these signs and quanti ers based on distributive lattices rules exist we have a complete inference system for quanti ers induced by nite distributive lattices. As is evident from the proof of Theorem 8, rule schema (2) can be considered to be a special case of (4): if i is meet-irreducible then M = fig, thus (4) collapses to (2). Similarly, (3) collapses to a schema with +i (c) and *i (t) in the conclusion. A second application of (3), where t is chosen to be c, yields *i (c). As fig = *i \+i the result is equivalent to the conclusion of (1). Rule schemata (7), (9) and (6), (8) behave dually. If i = >, then M = ; and (3) has the same conclusion as (5). ThisQremains true even for non-distributive lattices: f>g (Qx) (x) is equivalent to P *> ( x) (x), hence schema (5) applies. Dually, schema (10) can be used for f?g ( x) (x).
Example 5. Two of the rules given in Example 4 are easily seen to be instances of the corollary: FOUR is distributive and f is meet-irreducible so the left and the right rule in Example 4 can be obtained from schemata (2) and (1) respectively. In the case of f?g we have to invoke rule schema (3) to obtain the rule below which in fact is simpler than the rule derived \by hand".
6 It is straightforward to see that in the case of (5) even a lower semi-lattice and for (10) an upper semi-lattice is sucient.
REINER HA HNLE
12
f?g (Qx) (x) f?; f g (c) f?; tg (d) Q As a further example we give the rule for +D and in NINE (cf. Figure 3(b)). The meet-irreducible representation of D is F u G, hence: +D (Qx) (x) +F (c) +G (d)
One should realize that a rule constructed with Lemma 3 would have several
hundred extensions.
Hahnle (1994, Section 5.4) lists rules for the special case when L is a nite chain and signs are either upsets, downsets or singletons. Those rules can be obtained from the Corollary immediately as special cases of (1), (2), (5){(7) and (10), because a chain is distributive and all its elements are meet- and join-irreducible.
We stress that propositional connectives do not occur at all in our results. This means that a many-valued logic needs not be lattice-based in general for our results to apply. It suces that the quanti ers can be de ned with the help of lattices. One may even use a di erent lattice for each quanti er7 given that the lattices are upset/downset-compatible (i.e. an upset wrt to one of the lattices is also an upset or downset wrt the other lattices). 5. Infinitely-Valued Logic It is tempting to try to generalize the results in this paper to in nitely-valued logics. Indeed, in lattice theory many results on the representation of nite distributive lattices have in nite generalizations at least for bounded lattices. Given a complete, possibly in nite, lattice L by setting (6) Q(Q)(X) = X G P Q( )(X) = X (7) F where and denotes supremum, respectively, in mum of X in L, one still obtains well-de ned quanti ers. The following brief consideration shows, however, that a straightforward axiomatization cannot be expected, whenever quanti ers are based on a lattice that has an in nite chain (that is an in nite set of distinct, totally ordered elements). Consider a complete lattice L which, to make things simple, consists of a single in nite chain C plus its in mum i such that i = ( C) 62 C and apply rule schema (2) from the corollary to +i (Qx) (x). It is easy to see that now weQno longer have a sound rule: by de nition of Q, for the distribution C we have Q( )(C) = i 2 +i, but vM ( (c)) 2 C >L i for any c, hence the rule conclusion +i (c) is unsatis able. In order to regain a sound rule distributions such as C must be characterized in the conclusion as well. Given the fact that such distributions are necessarily in nite and their number is not denumerable, this seems hardly possible.
G
F
F
7 This is important for logics based on bilattices (Fitting, 1990) where two di erent lattices occur naturally. As an upset (downset) in one of the lattices that constitute a bilattice is an upset (downset) in the other lattice as well, combination of the rules is no problem.
COMMODIOUS AXIOMATIZATION OF QUANTIFIERS IN MULTIPLE-VALUED LOGIC 13
The situation is, of course, completely consistent with the fact that in nitelyvalued rst-order logics in general are not rst-order axiomatizable (Mostowski, 1961; Scarpellini, 1962; Ragaz, 1981; Baaz et al., 1996). Indeed, the breakdown of the rule for *i (Px) (x) in the in nite case is somewhat reminiscent of the failure of Lemma B in (Hay, 1963). Whether an in nitely-valued rst-order logic is rst-order axiomatizable or not depends not only on the quanti ers, but also on the available propositional connectives. As the latter do not play any r^ole in our analysis, we cannot expect to obtain axiomatizations of rst-order logics based on lattices with in nite chains with our present approach. It is not surprising, therefore, that in Takeuti & Titani's (1984) complete rst-order sequent style axiomatization of in nitely-valued Godel logic with the real unit interval [0; 1] as truth values speci c mixed rstorder/propositional axioms and a (highly indeterministic) propositional rule re ecting density of [0; 1] play an important r^ole. It is not obvious how their system could be reformulated akin to ours. What our approach could handle are in nite distributive lattices without in nite chains. Such lattices are always complete and analoga of Lemmas 1 and 2 hold, see (Birkho , 1970, Section VIII.2) for details. It is unnecessary, however, to discuss these lattices separately, because they are in fact nite. Proposition 9. Every distributive lattice L without in nite chains is nite. Proof. Consider any maximal chain C in L (not to be confused with a chain of maximal length which does not necessarily exist) which is nite by assumption. Assume jC j > 2 (if all chains are at most of length 2, then both L and the result are trivial). An elementary result for distributive lattices8 says that for each a 2 L the mapping fa : x 7! hx u a; x t ai is an embedding of L into +a *a. Choose a 6= >; ? and repeat the construction if necessary until an embedding into 2m is reached, where 2 is the two-element lattice. Because C is nite m is nite, and so L must be nite as well. 6. Conclusion Future Work. In fuzzy logic quanti ers usually are based on the concept of a T-norm, respectively, S-norm, for instance, the T-quanti ers and S-quanti ers of Thiele (1994). Can we apply the methods developed in the present paper to fuzzy quanti ers at least in nitely-valued logic, that is, is it possible to characterize their distributions by lattice-theoretic tools? A rst glance seems promising: Thiele's Tand S-quanti ers are based on a chain, where, of course, each (inner) element is meet- and join-irreducible. This is important for his conditions (TQ1) and (SQ1) to hold. All other conditions stated there are also true for lattice-based quanti ers. A further classes of quanti ers that should be considered in the future are branching (Henkin) quanti ers and generalized n-ary (Lindstrom) quanti ers which are frequently encountered in computational linguistics (Westerstahl, 1989). Another perspective is to exploit that 2N is a Boolean lattice, hence complemented. This should extend the sets of distributions that can be characterized with universal/existential type instances of quanti ed formulas. The complement of +D in NINE shown in Figure 3(b), for instance, is rather easy to represent if one notes that it is the union of two upsets of join-irreducible elements. Finally, as already mentioned on page 12, in a many-valued logic several quanti ers and connectives may be present. Even when each of them is based on lattices, 8
See, for example, (Davey & Priestley, 1990, Ex. 6.12(i)).
14
REINER HA HNLE
the underlying orders might not be upset/downset-compatible. Under which restrictions is it possible to give complete rule systems for such logics that still can exploit the results of this paper? Related Work. Fitting (1990) gives signed tableau rules for the case when the set of truth values forms a certain type of nite bilattice (Ginsberg, 1988), however, as Fitting requires his rules to follow Smullyan's uniform notation (Smullyan, 1995), the class of logics for which they work is restricted. Similar results as Lemmata 4 and 7, but restricted to singleton sets as signs, appear in (Zabel, 1993) as Corollaries 1.3.3 and 1.3.4. Zabel also pointed out that the premisses of those lemmata are \les plus communs dans les logiques polyvalentes"; he did not try, however, to make this statement more precise. Zach (1993, Section 1.7) and Baaz & Fermuller (1995, Example 4.20) give single extension rules for singleton signs and quanti ers induced by certain connectives (the latter were shown in turn to be related to upper semi-lattices over the set of truth values). Hahnle (1994) gives rules for upsets, downsets and singleton signs in the case when N is a nite chain. The most relevant and recent work with respect to our discussion is Salzer's (1996) who describes a minimization algorithm for distribution quanti ers. He proves that optimal dual (CNF-based) rules are produced by his algorithm for arbitrary quanti ers and signs in nitely-valued logics. Salzer gives also general tableau rule schemata for quanti ers based on upper/lower semi-lattices and interval-shaped signs. This result is applicable to a wider class of quanti ers than the results presented here, but the number of rule extensions and signs occurring in Salzer's rules cannot be restricted a priori : up to jN j extensions and all 2jN j ? 2 di erent (nontrivial) signs may occur. The results of the present paper imply, however, that in the case of distributive lattices only upsets and downsets occur in Salzer's rules. Summary. It has been pointed out by Carnielli (1987, p. 488) that, even for singleton signs, it is as yet an unsolved problem to nd minimalrules for distribution quanti ers automatically in a feasible way that is without enumerating all possible rules. Zabel (1993, Section 1.3.3) gave simpli ed singleton signs rules in the case when the distributions to be characterized form a sublattice of a nite Boolean set lattice. This is, of course, equivalent to being a nite, distributive lattice (Davey & Priestley, 1990, Corollary 8.18). We extended Zabel's results to upsets and downsets of the truth value lattice as signs and showed how to systematically construct concise rules with at most one extension (Zabel's rules can have jN j many extensions). For quanti ers based on non-distributive lattices partial results are obtained. As in the rule conclusions again only upsets, downsets and singletons occur, one obtains complete calculi for logics based on distributive lattices. In particular, it is sucient to consider merely 2jN j di erent signs, the upsets and downsets of N, because fig = *i \ +i. Together with the result that there exists a broad class of propositional many-valued logics with the same property (so-called regular logics, see (Hahnle, 1991; Hahnle, 1994)) this implies that there is a substantial class of many-valued rst-order logics which is proof-theoretically not a lot more complex than classical logic. Independently of the applicability of Theorem 8, the results of Section 3 tell which kind of sets of distributions can easily be characterized with \universal" and \existential" instances of signed quanti ed formulas. The proof of Theorem 8 above (and thus the existence of single-extension rules) hinges on the existence of unique representations of lattice elements. As this is a
REFERENCES
15
characteristic property of distributive lattices it sets bounds on the availability of small rules in non-distributive lattices. In short, the results of this paper and of (Salzer, 1996) together constitute a satisfying and exhaustive answer to the question of axiomatizability of lattice-based distribution quanti ers in nitely-valued logic.
Acknowledgements. I am grateful to Peter Schmitt for carefully reading a draft
of this paper and for fruitful discussions. He suggested important improvements, corrected an error in an earlier version of the proof of Theorem 8 and supplied a simple proof of Proposition 9. Bernhard Beckert and Gernot Salzer both read drafts and suggested various improvements. An example due to Salzer led to an improvement of Theorem 8, cases (4) and (9). References Baaz, Matthias, & Fermuller, Christian G. 1995. Resolution-Based Theo-
rem Proving for Many-Valued Logics. Journal of Symbolic Computation, 19(4), 353{391. Baaz, Matthias, Leitsch, Alexander, & Zach, Richard. 1996. Incompleteness of a rst-order Godel logic and some temporal logics of programs. Pages 1{15 of: Kleine Buning, Hans (ed), Selected Papers from Computer Science Logic, CSL'95, Paderborn, Germany. LNCS, vol. 1092. Springer-Verlag. Birkhoff, Garrett. 1970. Lattice Theory. third (new) edn. AMS Colloquium Publications, vol. XXV. American Mathematical Society. Carnielli, Walter A. 1987. Systematization of Finite Many-Valued Logics through the Method of Tableaux. Journal of Symbolic Logic, 52(2), 473{493. Carnielli, Walter A. 1991. On Sequents and Tableaux for Many-Valued Logics. Journal of Non-Classical Logic, 8(1), 59{76. Davey, B. A., & Priestley, H. A. 1990. Introduction to Lattices and Order. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge. Fitting, Melvin C. 1990. Bilattices in Logic Programming. Pages 238{247 of: 20th International Symposium on Multiple-Valued Logic, Charlotte. IEEE Press, Los Alamitos. Fitting, Melvin C. 1996. First-Order Logic and Automated Theorem Proving. Second edn. Springer-Verlag, New York. Ginsberg, Matthew L. 1988. Multi-Valued Logics. Computational Intelligence, 4(3). Hahnle, Reiner. 1991. Uniform Notation of Tableaux Rules for Multiple-Valued Logics. Pages 238{245 of: Proc. International Symposium on Multiple-Valued Logic, Victoria. IEEE Press, Los Alamitos. Hahnle, Reiner. 1994. Automated Deduction in Multiple-Valued Logics. International Series of Monographs on Computer Science, vol. 10. Oxford University Press. Hay, Louise Schmir. 1963. Axiomatization of the in nite-valued predicate calculus. Journal of Symbolic Logic, 28(1), 77{86. Mostowski, A. 1961. Axiomatizability of some many valued predicate calculi. Fundamenta Mathematic, L, 165{190. Ragaz, M. E. 1981. Arithmetische Klassi kation von Formelmengen der unendlichwertigen Logik. Ph.D. thesis, ETH Zurich. Rosser, J. B., & Turquette, A. R. 1952. Many-Valued Logics. Amsterdam: North-Holland. Rousseau, G. 1967. Sequents in many valued logic I. Fundamenta Mathematic, LX, 23{33.
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Institute for Logic, Complexity and Deduction Systems, Dept. of Computer Science, University of Karlsruhe, 76128 Karlsruhe, Germany
E-mail address :
[email protected]
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http://i12www.ira.uka.de/~reiner
