Soft Comput (2009) 13:1089–1097 DOI 10.1007/s00500-008-0382-0
ORIGINAL PAPER
Computational complexities of axiomatic extensions of monoidal t-norm based logic Moataz Saleh El-Zekey Æ Wafik Boulos Lotfallah Æ Nehad Nashaat Morsi
Published online: 11 November 2008 Ó Springer-Verlag 2008
Abstract We study the computational complexity of some axiomatic extensions of the monoidal t-norm based logic (MTL), namely NM corresponding to the logic of the socalled nilpotent minimum t-norm (due to Fodor in Fuzzy Sets Syst 69:141–156, 1995); and SMTL corresponding to leftcontinuous strict t-norms, introduced by Esteva (and others) (Fuzzy Sets Syst 132(1):107–112, 2002; 136(3):263–282, 2003). In particular, we show that the sets of 1-satisfiable and positively satisfiable formulae of both NM and SMTL are NP-complete, while the set of 1-tautologies of NM and the set of positive tautologies of both NM and SMTL are co-NPcomplete. The set of 1-tautologies of SMTL is only shown to be co-NP-hard, and it remains open if this set is in co-NP. Also, some results on the relations between these sets are obtained. We point out that results about 1-satisfiability and 1-tautology for NM are already well-known. However, in this paper, those results are proved in different ways.
W. B. Lotfallah is on leave from Cairo University. M. S. El-Zekey Department of Basic Sciences, Benha High Institute of Technology, Benha, Egypt e-mail:
[email protected] W. B. Lotfallah (&) Department of Mathematics, The German University in Cairo, Cairo, Egypt e-mail:
[email protected] N. N. Morsi Department of Basic Sciences, Arab Academy for Science, Technology and Maritime Transport, Cairo, Egypt
1 Introduction Questions about the computational complexity of fuzzy propositional calculi have been recently studied, and important results have been obtained for those logics based on continuous t-norms, namely Ha´jek basic logic BL, Łukasiewicz logic Ł, Go¨del logic G and product logic P (see the monograph, Ha´jek 1998, for a complete survey on these logical formalisms). For the computational complexity of propositional logics Ł, G, and P, complete information is already stated in Ha´jek (1998). On the other hand, Baaz et al. (2002) shows that the set of standard BL-tautologies (1-tautologies or identically 1-true formulae) to be co-NP-complete, while positive tautologies as well as satisfiable formulae are studied in Ha´jek (2003). Also, see Aguzzoli et al. (2005) for a complete survey on the complexity results concerning Ha´jek basic logic BL and for several of its schematic extensions. Getting results for the monoidal t-norm based logic (MTL) analogous to the ones obtained for BL seems quite a complex task, since the structure of left-continuous t-norms is not yet totally known (see Vetterlein 2007 for really nice improvements about that topic). However, we can do some steps forward by considering the computational complexity of some particular schematic extensions of MTL which do exhibit positive results. In particular we shall study the computational complexity of the two extensions NM and SMTL corresponding to the logic of the so-called nilpotent minimum t-norm (due to Fodor 1995) and left-continuous strict t-norms, introduced independently in Ha´jek (2002) and Jenei and Montagna (2003). We show that the sets of 1-satisfiable and positively satisfiable formulae of both NM and SMTL are NP-complete, while the set of 1-tautologies of NM and the set of positive tautologies of both NM and
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SMTL are co-NP-complete. The set of 1-tautologies of SMTL is shown to be co-NP-hard, and it remains open if this set is in co-NP. Also, some results on the relations between these sets are obtained. We point out that results about 1-satisfiability and 1-tautology for NM are already well-known (see, for instance, Aguzzoli and Gerla 2006; Esteva et al. 2008a, 2008b; Gispert 2003). However, note that the proof given here differs from those given in the above cited works. Thus there are no problem in recognizing the originality of the theorems presented in the present paper. Before studying the computational complexity of the two extensions NM and SMTL, let us present an overview of some fundamental results, both logical and algebraic, on leftcontinuous t-norms and their residua that will be needed in this work. 1.1 Basic notions A triangular norm (see Schweizer and Sklar 1983) T is a binary operation on [0, 1] that is associative, commutative and monotone in both arguments, and has 1 as an identity element. If T is a t-norm, the residuum I of T is defined as: for all a; b 2 ½0; 1 Iða; bÞ ¼ supfc 2 ½0; 1 : Tða; cÞ bg: For our purposes it is interesting to note the following basic well-known results on t-norms and their residua on [0, 1] (see for instance Esteva and Godo 2001; Ha´jek 1998): 1.
Left-continuity is the necessary and sufficient condition for a t-norm T and its residuum I (R-implication, for short) to verify the residuation property:
c Iða; bÞ
iff Tðc; aÞ b:
In that case (T, I) is called a residuated pair (see Pavelka 1979). A corresponding negation operation can also be defined by putting nðaÞ ¼ Iða; 0Þ: 2. A residuated implication satisfies the following prelinearity property: maxðIða; bÞ; Iðb; aÞÞ ¼ 1: 3.
Given a residuated pair (T, I); max is definable from min by the equation,
maxða; bÞ ¼ minðIðIða; bÞ; bÞ; IðIðb; aÞ; aÞÞ: A t-norm has no zero-divisor (x [ 0 is a zero-divisor if there exists y [ 0 such that Tðx; yÞ ¼ 0) iff the corresponding negation is ( 1; if x ¼ 0 nðxÞ ¼ Iðx; 0Þ ¼ ð1Þ 0; otherwise 4.
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called Go¨del negation, or equivalently iff the pseudocomplementation condition minðx; nðxÞÞ ¼ 0 holds. A tnorm without zero-divisors is called a strict t-norm. Actually, in the literature of t-norms, the class of strict tnorms usually refers to the class of Archimedean t-norms (continuous t-norms that do not have idempotents except 0 and 1) with no zero divisors, that is, those t-norms isomorphic to the product t-norm. In this paper, following Esteva et al. (2003), we use strictness as equivalent to having no zero divisors, i.e. which verify for each x, y in [0, 1]: T(x, y) = 0 iff (x = 0 or y = 0). Example 1.1 The nilpotent minimum (Fodor 1995) is a left-continuous, but not continuous, t-norm. It and its residuum are defined by ( 0; xþy 1 Tðx; yÞ ¼ ; minðx; yÞ; x þ y [ 1 ( 1; x z Iðx; zÞ ¼ : maxf1 x; zg; x [ z The negation operation corresponding to nilpotent minimum t-norm is nðxÞ ¼ 1 x: MTL-algebra (Esteva and Godo 2001) is a bounded residuated lattice ðP; ^; _; T; I; 0; 1Þ; where ^ and _ are the lattice meet and join operations and (T, I) is a residuated pair, satisfying the prelinearity condition, 8a; c 2 P: Iða; cÞ _ Iðc; aÞ ¼ 1:
ðprelinearity conditionÞ
Monoidal t-norm based logic (Esteva and Godo 2001), MTL for short, is a logic whose algebraic counterpart is the class of MTL-algebras. The language of MTL is built in the usual way from a denumerable set of propositional symbols using the primitive connectives ; !; ^ and the truth constant 0: As definable connectives, it has the negation and the double implication: :s is s ! 0; n !s is ðn ! sÞ ðs ! nÞ: Also, in MTL, _ becomes definable in terms of ; ! and ^ as n _ s isððn ! sÞ ! sÞ ^ ððs ! nÞ ! nÞ: An axiomatic system of MTL can be found in Esteva and Godo (2001). The deduction rule of MTL is modus ponens, i.e. from n and n ! s derive s: Let K ¼ ðP; ^; _; T; I; 0; 1Þ be an MTL-algebra. An evaluation of propositional variables is a mapping e assigning to each propositional variable p its truth value eðpÞ 2 P: This extends uniquely in the obvious way to an evaluation eK of all propositional formulae using the operations on K as truth functions.
Computational complexities of axiomatic extensions
Completeness of the logic MTL with respect to MTLalgebras and linearly ordered MTL algebras holds (see Esteva and Godo 2001). Furthermore, standard completeness for MTL (i.e. completeness w.r.t. the class of left-continuous t-norm based [0, 1] structures) has been proved by Jenei and Montagana (see Jenei and Montagna 2002). Therefore, since MTL-structures in the unit interval [0, 1] are defined by left-continuous t-norms, MTL can be properly called the logic of left-continuous t-norms and their residua, hence MTL is the most general residuated fuzzy logic related to t-norms on the unit interval [0, 1] (recall that a t-norm has residuum if and only if it is left continuous). The logical system MTL is directly related to Ha´jek basic fuzzy logic BL (Ha´jek 1998), the fuzzy logic capturing the tautologies of continuous t-norms and their residua (Cignoli et al. 2000). The difference with MTL is due to the fact that in BL: (a) the divisibility axiom ðnðn ! sÞÞ !ðn ^ sÞ holds since it is a tautology for those continuous t-norms; and (b), as a consequence of (a), the min-conjunction ^ is definable in BL (n ^ s is n ðn ! sÞ), and need not to be introduced as a further primitive connective. For a comprehensive study of basic logic, see Ha´jek (1998) and Ho¨hle (1992, 1994, 1995a, 1995b). Łukasiewicz fuzzy logic Ł, Go¨del fuzzy logic G, and product fuzzy logic P are obtained as extensions of BL by one or two additional axioms. Moreover Boolean propositional calculus (Boole) can be introduced as schematic extension of BL as Boole ¼ BL þ ðf_:fÞ (Ha´jek 1998). [For historical comments see Ha´jek (1998, Chap. 10); we point out that the notion of a t-norm has been originated outside logic (Schweizer and Sklar 1983).] For a clear overview of t-norm based logical systems, the reader is referred to Gottwald’s (2001) work. 1.2 SAT and TAUT problems in a many-valued setting Let SATBoole be the set of all satisfiable formulas f in the Boolean propositional calculus, i.e. there is a {0, 1}-evaluation e of propositional variables of f (i.e. e(p) = 0 or e(p) = 1) such that eBoole ðfÞ ¼ 1 ðeBoole ðfÞ is the value of f given by the {0, 1}-evaluation e of propositional variables of f using Boolean truth tables). Theorem 1.1 (Papadimitriou 1994, Cook’s theorem) SATBoole is NP-complete. Remark 1.1 Let TAUTBoole be the set of all tautologies of Boolean propositional logic. Clearly, TAUTBoole is co-NPcomplete since f 2 TAUTBoole
iff :f 62 SATBoole :
With a many-valued logic L; or a L-algebra K; one might wonder about the definition of the SAT and TAUT
problems, as the classical dichotomy is no longer at hand. For a fixed semantics given by an algebra K; it makes sense to distinguish the following sets of formulae (cf. Ha´jek 1998). In all cases f stands for propositional formulae in the MTL-language and eK runs over evaluations in K: TAUTK 1 ¼ ffj for all eK ; eK ðfÞ ¼ 1g; TAUTK pos ¼ ffj for all eK ; eK ðfÞ [ 0g; SATK 1 ¼ ffj for some eK ; eK ðfÞ ¼ 1g; SATK pos ¼ ffj for some eK ; eK ðfÞ [ 0g: These sets are referred to as 1-tautologies, positive tautologies, 1-satisfiable formulae and positively satisfiable formulae of K: Following Aguzzoli et al. (2005) and Baaz et al. (2002), for a class of algebras of the same type, one may generalize:
Unlike in classical logic, for a many-valued semantics there need not be a simple relationship between its TAUT and SAT problems. The 1-tautologies will probably be of most interest; however, the other sets present themselves to be investigated as well.
2 Some schematic extensions of the monoidal t-norm logic: an overview Some genuine axiomatic extensions of MTL have been considered in Esteva and Godo (2001) and Esteva et al. (2002) where issues of standard completeness are analyzed. Two of the possible schematics extensions are the so-called nilpotent minimum logic (NM) and strict monoidal t-norm logic (SMTL). We first consider the extension NM corresponding to a class of left-continuous t-norms called nilpotent minimum t-norms. Then, we consider the extension SMTL capturing the logic of left-continuous strict t-norms, i.e. those having the Go¨del negation as the associated negation. 2.1 The logic of nilpotent minimum Nilpotent minimums are left-continuous t-norms that were introduced by Fodor (1995). Given an involutive negation n on [0, 1], the corresponding nilpotent minimum t-norm Tn is defined as
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( Tn ðx; yÞ ¼
0;
if x nðyÞ;
minðx; yÞ;
otherwise.
ð2Þ
The corresponding residuated implication is given by ( 1; if x y; ð3Þ In ðx; yÞ ¼ maxðnðxÞ; yÞ; otherwise. It is straightforward to see that In ðx; 0Þ ¼ nðxÞ; so the involution n is actually the negation corresponding to Tn. Nilpotent minimum has been discovered not by chance. There is a study on contrapositive symmetry of fuzzy implications (Fodor 1995). A particular case of those investigations yielded nilpotent minimum. It is the standard example of a left-continuous t-norm in the literature. In Cignoli et al. (2002), a subclass of left-continuous t-norms was studied, which are definable by an arbitrary continuous t-norm T (rather than min) and a weak (i.e. non necessarily involutive) negation n, thus generalizing the construction of the nilpotent minimum t-norms. Studies on properties of fuzzy logics based on left-continuous t-norms, and especially on the nilpotent minimum have started only recently; see Pei (2003, 2002) and Wang et al. (2004, 2005) along this line. The logic of nilpotent minimum (NM from now on), introduced by Esteva and Godo (2001), is the schematic extension of MTL resulting when we add the following two axioms: ðWNMÞððf sÞ ! 0Þ _ ððf ^ sÞ ! ðf sÞÞ: ðINVÞ::f ! f: An NM-algebra is a structure ðL; ^; _; Tn ; In ; 0; 1Þ which is an MTL-algebra and satisfies the equations corresponding to (WNM) and (INV) axioms, that is In ðTn ða; bÞ; 0Þ _ In ða ^ b; Tn ða; bÞÞ ¼ 1;
nðnðaÞÞ ¼ a:
The decomposition of NM-algebras into subdirect products of linearly ordered ones is proved as usual. Hence, completeness of the logic NM with respect to NM-algebras and linearly ordered NM algebras also holds (see Esteva and Godo 2001). The NM-algebra in [0, 1] defined by taking nðaÞ ¼ 1 a is called the standard NM-algebra (see Example 1.1). Following Ha´jek’s (2003) denotation, we will denote by ½0; 1NM the standard NM-algebra. It is also proved in Esteva and Godo (2001) that all nilpotent minima on [0, 1] are isomorphic, which implies the following standard completeness theorem of NM logic. Theorem 2.1 (Esteva and Godo 2001, standard completeness) NM proves a formula f if and only if f is a tautology with respect to the standard NM-algebra ½0; 1NM :
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Theorem 2.1 says that the set of tautologies over all NM-algebras equal to the set of tautologies over one particular algebra, which is the standard NM-algebra ½0; 1NM : The nilpotent minimum was slightly extended in Esteva and Godo (2001) by allowing a weak negation (a negation that satisfies the inequality a nðnðaÞÞ) instead of a strong one in the construction. Based on this extension, MTL were studied also in Esteva and Godo (2001), together with the involutive case (IMTL). Properties and applications of the nilpotent minimum t-norm based implication (called R0 implication there) were published in Pei (2002). Its richness is due to the fact that it is both an S-implication and an R-implication at the same time, and thus advantageous features of both classes are combined. 2.2 The logic SMTL Apart from the logic NM, another interesting extension of MTL, namely the logic SMTL, has been considered in Esteva et al. (2002). The logic SMTL is an extension of MTL by adding the axiom n ^ :n !0: The corresponding semantical models are the MTL-algebras satisfying the equation x ^ nðxÞ ¼ 0 (Pseudo-complementation axiom). Of course both Product and Go¨del logics are extensions of SMTL. SMTL is proved to be standard complete (Esteva et al. 2002) as MTL, i.e. complete w.r.t. the class of left-continuous strict t-norm based [0, 1] structures. Hence SMTL captures the logic of left-continuous strict t-norms or equivalently left-continuous t-norms without zero divisors, i.e. those having the Go¨del negation (1) as the associated negation. Actually, it is easy to find examples (see Esteva et al. 2002) of SMTL-algebras in [0, 1] that are not BL-algebras, or in other words, to find left-continuous t-norms T, which are not continuous, satisfying the axiom x ^ Iðx; 0Þ ¼ 0 where I is the residuum of T. One example is already shown in Ha´jek (2003), here we show a simpler one (Esteva et al. 2002). Namely, for 0 \ a \ 1 take the following left-continuous t-norm: ( a; if x; y a and x þ y 1 þ a Ta ðx; yÞ ¼ minðx; yÞ; otherwise ð4Þ which is basically the ordinal sum of a Go¨del t-norm and of a nilpotent minimum. It is easy to check that Ia ðx; 0Þ ¼ supfz 2 ½0; 1jTa ðx; zÞ 0g is 1 if x = 0 and is 0 otherwise, and hence the axiom x ^ Ia ðx; 0Þ ¼ 0 is satisfied where Ia is the residuum of Ta.
Computational complexities of axiomatic extensions
3 Computational complexity of NM
the values 0; 1). Thus, the proof of the lemma runs as follows:
Since NM is a standard complete logic (see Theorem 2.1) whose corresponding variety is singly generated by the standard NM-algebra K ¼ ½0; 1NM ; we shall set TAUTNM ¼ TAUTK 1 1 : The same kind of notation will be NM applied to the other problems (i.e. TAUTNM and pos ; SAT1 NM SATpos ) in our classification. Moreover, an evaluation eK will be denoted by eNM ðeNM ðfÞ is the value of f given by the evaluation e of propositional variables (by elements of [0, 1]) and by the standard algebra of truth functions ½0; 1NM :) Recall that in NM (since the negation is involutive), eNM ðfÞ ¼ 1 iff eNM ð:fÞ ¼ 0; likewise eNM ðfÞ ¼ 0 iff eNM ð:fÞ ¼ 1: Thus we have the following lemma:
iff there is an evaluation e of propositional f# 2 SATNM 1 variables of f such that eNM ðf# Þ ¼ 1 iff there is an evaluation e of propositional variables of f such that for all i ¼ 1; . . .; m; eNM ðpi _ :pi Þ ¼ 1 and eNM ðfÞ ¼ 1 iff e is a f0; 1g-evaluation of propositional variables of f such that eNM ðfÞ ¼ 1 iff f 2 SATBoole : (
Lemma 3.1
Lemma 3.4 Let e be an evaluation of propositional variables of an arbitrary formula f in NM logic (by elements of ½0; 1) such that eðpÞ 6¼ 12 for all propositional variables occurring in f: Then there is a f0; 1g-evaluation e0 of the propositional variables of f which satisfies e0NM ðfÞ ¼ 1
1 iff eNM ðfÞ [ ; 2
iff :f 62 SATNM (i) f 2 TAUTNM 1 pos ; NM (ii) f 2 TAUTpos iff :f 62 SATNM 1 :
and
Lemma 3.2
Proof For all propositional variables occurring in f; let e0 ðpÞ ¼ 1 iff eðpÞ [ 12 and e0 ðpÞ ¼ 0 iff eðpÞ\12: Hence, the proof of the lemma goes by induction on the construction of f: h
Boole
SATNM 1
e0NM ðfÞ ¼ 0
SATNM pos ;
(i) SAT NM Boole (ii) TAUTNM TAUT : 1 pos TAUT Proof Non-strict inclusions are obvious from the definitions; strictness is showed by simple examples. Indeed, NM ðp ^ :pÞ 2 SATNM pos SAT1 ;
where p1 ; . . .; pm are the propositional variables in f: Then
ðp _ :pÞ ðp _ :pÞ 2 TAUTBoole TAUTNM pos ;
f 2 SATBoole
NM ðp _ :pÞ 2 TAUTNM pos TAUT1 :
( The above two lemmas are analogous to the ones obtained for Łukasiewicz logic in Ha´jek (1998). This is due to the fact that NM and Łukasiewicz logics are quite close. In both, the negation is involutive and the implication can be defined as in the classical case from negation and conjunction as :ðn :sÞ: For each formula f in NM logic, let f# be
ðp1 _ :p1 Þ ðpm _ :pm Þ f where p1 ; . . .; pm are the propositional variables in f: Then f 2 SATBoole
For each formula f in NM logic, let f## be
ðp1 _ :p1 Þ2 ðpm _ :pm Þ2 f2
Boole ; ðp ! :pÞ ð:p ! pÞ 2 SATNM 1 SAT
Lemma 3.3 the formula
Lemma 3.5 the formula
1 iff eNM ðfÞ\ : 2
iff f# 2 SATNM 1 :
Proof To prove the lemma observe first that for all evaluation e, eNM ðp_:pÞ ¼ 1 is equivalent to e(p) = 1 or e(p) = 0. Also, for any f0; 1g-evaluation e of propositional variables of f (i.e. eðpi Þ ¼ 1 or eðpi Þ ¼ 0; i ¼ 1; . . .; m), eBoole ðfÞ ¼ eNM ðfÞ (i.e. they coincide for
iff f## 2 SATNM pos :
Proof To prove the lemma observe first that eNM ððp_:pÞ2 Þ [ 0 is equivalent to eNM ðp _ :pÞ [ 12; which in turn is equivalent to eðpÞ 6¼ 12: Now, suppose f## 2 SATNM pos : Then there is an evaluation e of p1 ; . . .; pm (variables of f) such that eNM ðf## Þ [ 0: This implies that eNM ððpi _:pi Þ2 Þ [ 0 for all i ¼ 1; . . .; m and eNM ðf2 Þ [ 0; which is equivalent to eðpi Þ 6¼ 12 for all i ¼ 1; . . .; m and eðfÞ [ 12: Thus, by Lemma 3.4, there is a f0; 1g-evaluation e0 of the propositional variables of f which satisfies e0NM ðfÞ ¼ 1: Thus, f 2 SATBoole : This proves that if f## 2 SATNM pos then f 2 SATBoole : The converse is immediate. ( Lemma 3.6 Let f be an arbitrary formula in NM logic having m propositional variables p1 ; . . .; pm : Then the following two statements are equivalent: There is an evaluation e0 of the propositional variables of f (by elements of ½0; 1) such that e0NM ðfÞ ¼ 1 ðor e0NM ðfÞ [ 0Þ: (ii) There is an evaluation e of the propositional variables of f taking only values in the set of
(i)
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M. S. El-Zekey et al. 1 2m þ 3 elements Dm ¼ f0; 2mþ2 ; . . .; 2mþ1 2mþ2; 1g such that eNM ðfÞ ¼ 1 (or eNM ðfÞ [ 0).
Proof Suppose (i) holds. Let n be the involutive negation associated with the standard NM-algebra. Then the set Z ¼ fe0 ðp1 Þ _ nðe0 ðp1 ÞÞ; . . .; e0 ðpm Þ _ nðe0 ðpm ÞÞg is contained in the subinterval ½12; 1: Hence, there is an order isomorphism f of ½12; 1 onto ½12; 1 taking Z into Dm : We extend f into an order isomorphism on ½0; 1 by setting for all t 2 ½0; 12: f ðtÞ ¼ 1 f ðnðtÞÞ: Thus, it is easy to prove that f is an isomorphism of the standard NM-algebra ½0; 1NM sending the set fe0 ðp1 Þ; . . .; e0 ðpm Þ; nðe0 ðp1 ÞÞ; . . .; nðe0 ðpm ÞÞg into Dm : Thus, defining eðpi Þ ¼ f ðe0 ðpi ÞÞ for all propositional variables occurring in f; eNM ðfÞ ¼ f ðe0NM ðfÞÞ ¼ 1 (or [ 0). This proves (ii). The converse is immediate. ( Theorem 3.1
NP-completeness of SAT1 and SATpos for the slightly more general logical system obtained by adding rational truth constants to NM, and hence the NP-completeness of NM follows as a corollary.
4 Computational complexity of SMTL The standard semantics of SMTL is formed by the class of all SMTL-algebras ½0; 1T ¼ ð½0; 1; ^; _; T; IÞ where T is a left-continuous strict t-norm with its residuum I, and ^; _ are min and max on ½0; 1: Since SMTL is standard complete (Esteva et al. 2002) (i.e. complete w.r.t. the class of all SMTL-algebras K ¼ ½0; 1T ), we shall set
SATNM pos is NP-complete.
Proof Hardness: Let for every formula f; f## be the formula obtained from f as in Lemma 3.5. Clearly, f## is computed from f in polynomial time. Hence, Lemma 3.5 proves that SATBoole is polynomially reducible to SATNM Pos and thus, SATNM pos is NP-hard. Membership: By Lemma 3.6, f 2 SATNM pos iff there is an evaluation e of p1 ; . . .; pmn (variables of f) taking o only val1 ues in the set Dm ¼ 0; 2mþ2 ; . . .; 2mþ1 ; 1 such that 2mþ2 eNM ðfÞ [ 0: Thus an algorithm showing f 2 SATNM pos guesses an evaluation e; eðpi Þ 2 Dm ; and computes the truth value. This shows that SATNM ( pos is in NP. Theorem 3.1 and Lemma 3.1(i) render the following corollary: Corollary 3.1
TAUTNM is co-NP-complete. 1
Theorem 3.2
is NP-complete. SATNM 1
Proof Let for every formula f; f# be the formula obtained from f as in Lemma 3.3. Clearly, f# is computed from f in polynomial time. Hence, Lemma 3.3 proves that SATBoole is polynomially reducible to SATNM and thus, SATNM is NP1 1 NM hard. Membership (i.e. SAT1 is in NP): Similar to the proof of the membership in Theorem 3.1. (
The same kind of notation will be applied to the other SMTL problems (i.e. TAUTSMTL and SATSMTL pos ; SAT1 pos ) in our classification. Moreover, for SMTL-algebra K ¼ ½0; 1T ; an evaluation eK will be denoted by eT : Here eT ðfÞ is the value of f given by the evaluation e of propositional variables (by elements of ½0; 1) and by the standard algebra of truth functions ½0; 1T : Recall that both Product and Go¨del logics are extensions of SMTL and all these logics are characterized by having the Go¨del negation as the associated negation. In Ha´jek (1998), the study of the computational complexity of Product and Go¨del logics depends mainly on this (i.e. their associated negations are Go¨del negation or equivalently their corresponding t-norms are without zero divisors). Here, in our study of the complexity of SMTL, we use and modify the apparatus used in Ha´jek (1998, 6.2.2–6.2.9). We have to start with the following definition: Definition 4.1 Let p1 ; . . .; pm be propositional variables and J fp1 ; . . .; pm g: J is understood as the set of variables evaluated by 0: We define, for each formula f in SMTL built from p1 ; . . .; pm ; its translation fJ as follows: 0J ¼ 0;
Theorem 3.2 with Lemma 3.1(ii) render the following corollary: Corollary 3.2
TAUTNM pos is co-NP-complete.
We point out that there are results about NM in literature (see, for instance, Aguzzoli and Gerla 2006, Esteva et al. 2008a, 2008b and Gispert 2003) proving the NP-completeness of SATNM and SATNM 1 pos : However, note that the proof given here differs from those given in the above cited works. In particular, Esteva et al. (2008b) have showed the
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( pJi
¼
1J ¼ 1; x
0; if pi 2 J pi ; otherwise
;
8 > < 1; J ðf ! sÞ ¼ 0; > : J f ! sJ ;
if fJ ¼ 0 if fJ 6¼ 0 and sJ ¼ 0 ; otherwise
Computational complexities of axiomatic extensions
( J
ðf sÞ ¼ ( ðf ^ sÞJ ¼
0;
if fJ ¼ 0 or sJ ¼ 0
fJ s J ;
otherwise
0;
if fJ ¼ 0 or sJ ¼ 0
fJ ^ s J ;
otherwise
;
:
Recall that, in SMTL, _ and : are definable in terms of !; ^ and 0 as n _ s is ððn ! sÞ ! sÞ ^ ððs ! nÞ ! nÞ; :s is s ! 0: Hence, by Definition 4.1 it is direct to see that ( 0; if fJ ¼ 0 and sJ ¼ 0 J ðf _ sÞ ¼ ; J J f _ s ; otherwise ( 0; if fJ 6¼ 0 J : ð:fÞ ¼ 1; otherwise
This proves that (in the case that f is f1 f2 ) if fJ is not 0 then eT ðfJ Þ 6¼ 0: The converse is immediate. Similarly for the other cases. 2-ii. The induction hypothesis states that for all subformulae s of f; eT ðsÞ ¼ eT ðsJ Þ: Suppose f is f1 ! f2 : Thus, if fJ1 6¼ 0 and fJ2 ¼ 0 then fJ ¼ 0 (by Definition 4.1), hence eT ðfJ Þ ¼ eT ð0Þ ¼ 0: On the other hand eT ðfÞ ¼ eT ðf1 ! f2 Þ ¼ IðeT ðf1 Þ; eT ðf2 ÞÞ ¼ IðeT ðfJ1 Þ; eT ðfJ2 ÞÞ ðby induction hypothesisÞ ¼ IðeT ðfJ Þ; 0Þ ðby the assumption fJ ¼ 0Þ ¼ 0; 1
since eT ðfJ1 Þ 6¼ 0 [by the assumption fJ1 6¼ 0 and the part (2-i)] and I is the residuum of the left-continuous strict t-norm T. Hence, eT ðfJ Þ ¼ 0 ¼ eT ðfÞ: Otherwise [i.e. if not (fJ1 6¼ 0 and fJ2 ¼ 0)], eT ðfJ Þ ¼ eT ððf1 ! f2 ÞJ Þ
Lemma 4.1 For each formula f in SMTL built from p1 ; . . .; pm ; and each set J fp1 ; . . .; pm g; the following hold: 1. 2.
or fJ is a formula not containing 0: Either fJ is 0 For each left-continuous strict t-norm T and for each evaluation e such that eðpi Þ ¼ 0 iff pi 2 J; the following hold: (i) eT ðfJ Þ ¼ 0 iff fJ is 0; (ii) eT ðfÞ ¼ eT ðfJ Þ:
¼ eT ðfJ1 ! fJ2 Þ ðby Definition 4:1Þ ¼ IðeT ðfJ1 Þ; eT ðfJ2 ÞÞ ¼ IðeT ðf1 Þ; eT ðf2 ÞÞ ðby the induction hypothesisÞ ¼ eT ðf1 ! f2 Þ ¼ eT ðfÞ: This proves that (in the case that f is f1 ! f2 ), eT ðfJ Þ ¼ eT ðfÞ: Similarly for the other cases. ( Recall that SATBoole is the set of all formulae f satisfiable in Boolean (two-valued) logic. Similarly for TAUTBoole :
Proof
Lemma 4.2
1.
1. 2. 3. 4.
This follows directly from Definition 4.1 and by induction on the construction of f; e.g. if f is f1 f2 ; then by induction hypothesis either fJ1 (or fJ2 ) is 0 or fJ1 (and fJ2 ) is a formula not containing 0: Thus by Definition 4.1 either fJ ¼ ðf1 f2 ÞJ is 0 or fJ ¼ ðf1 f2 ÞJ ¼ fJ1 fJ2 is a formula not containing 0: Similarly for the other cases. 2. Let T be a left-continuous strict t-norm and e be an evaluation such that eðpi Þ ¼ 0 iff pi 2 J: Then if f is pj ; j 2 1; . . .; m; then (by Definition 4.1) eðpj Þ ¼ eðpJj Þ and eðpJj Þ ¼ 0 iff pJj is 0: We now use the induction on the construction of f: 2-i. The induction hypothesis states that for all subformulae s of f; eT ðsJ Þ ¼ 0 iff sJ is 0: Suppose f is f1 f2 : Thus, if fJ is not 0 then by part (1) fJ is a formula not containing 0; hence both fJ1 and fJ2 are not 0: Thus, by induction hypothesis, eT ðfJ1 Þ 6¼ 0 and eT ðfJ2 Þ 6¼ 0: Then we get eT ðfJ Þ ¼ eT ððf1 f2 ÞJ Þ ¼ eT ðfJ1 fJ2 Þ ¼
TðeT ðfJ1 Þ;
eT ðfJ2 ÞÞ
6¼ 0:
2
The following are equivalent:
f 2 SATSMTL pos ; for some J; fJ is not 0; f 2 SATBoole ; f 2 SATSMTL : 1
Proof Evidently, (3) implies (4) and (4) implies (1). (1) implies (2): Suppose f 2 SATSMTL pos : Then for a leftcontinuous strict t-norm T; an evaluation e exists such that for J ¼ fpi jeðpi Þ ¼ 0g we get [by Lemma 4.1(2-ii)] eT ðfJ Þ ¼ eT ðfÞ [ 0 and hence fJ is not 0; by Lemma 4.1(2-i). (2) implies (3): Now assume (2) and let e be a f0; 1gevaluation such that eðpi Þ ¼ 0 for pi 2 J and eðpi Þ ¼ 1 otherwise. Since fJ is not 0 we get [by Lemma 4.1(2-i)], for each left-continuous strict t-norm T; eT ðfJ Þ 6¼ 0 and hence eT ðfJ Þ ¼ 1: Then, by Lemma 4.1(2-ii), eT ðfÞ ¼ 1 and hence eBoole ðfÞ ¼ 1 (since e is a f0; 1g-evaluation and both of eBoole and eT coincide for values 0; 1). Thus f 2 SATBoole : This completes the proof. ( Boole ; SATSMTL Corollary 4.1 The sets SATSMTL 1 pos ; SAT are all equal; hence they are NP-complete.
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Lemma 4.3 The set TAUTSMTL is closed under modus pos ponens, that is if n 2 TAUTSMTL and ðn ! sÞ 2 TAUTSMTL pos pos SMTL then s 2 TAUTpos : Proof For each left-continuous strict t-norm T and for each evaluation e; TðeT ðnÞ; eT ðn ! sÞÞ eT ðsÞ: Thus if we assume eT ðnÞ [ 0 and eT ðn ! sÞ [ 0 then TðeT ðnÞ; eT ðn ! sÞÞ [ 0 (since T has no zero divisors), which implies eT ðsÞ [ 0: This gives the result. ( The local deduction theorem is valid in MTL (Esteva and Godo 2001) and hence it is valid in SMTL, i.e. given a theory C and formulae n; s in SMTL, then C[fng ‘SMTL s iff there is an m such that C ‘SMTL nm ! s (where nm is n n; m factors). Lemma 4.4 TAUTSMTL ¼ TAUTBoole ; thus TAUTSMTL pos pos is co-NP-complete. TAUTBoole : On the other Proof Clearly, TAUTSMTL pos hand, for all evaluations e; one has eSMTL ðf_:fÞ ¼ 1 iff eSMTL ðfÞ ¼ 1 or eSMTL ðfÞ ¼ 0 (by the fact that, the negation in SMTL is Go¨del negation). Hence, if f 2 TAUTBoole then f_:f SMTL f; and hence f_:f ‘SMTL f: Thus by the local deduction theorem, ‘SMTL ðf_:fÞm ! f: Hence ½ðf_:fÞm ! f 2 TAUTSMTL : 1 Moreover, it is clear that f_:f 2 TAUTSMTL pos : Thus ðf_:fÞm 2 TAUTSMTL (since has no zero divisors in pos SMTL). Hence, by Lemma 4.3, we get f 2 TAUTSMTL pos : This proves that TAUTBoole TAUTSMTL and completes pos the proof. ( SMTL SMTL Note that f 2 TAUTpos iff :f 62 SATpos : Lemma 4.5 For each f; let f:: be the formula resulting from f by replacing each propositional variable pi by its double negation ::pi : Then f 2 TAUTBoole iff f:: 2 TAUTSMTL : 1 Proof For each [0, 1]-evaluation e, let e0 be an evaluation such that e0 ðpi Þ ¼ nðnðeðpi ÞÞÞ; clearly e0 ðpi Þ is 0 or 1 (since n is Go¨del negation). We now claim that for each left-continuous strict t-norm T; eT ðf:: Þ ¼ e0T ðfÞ: The proof of this claim is easy by induction on the construction of f as follows: It is clear that the induction step holds by the definition of e0 : The induction hypothesis is that for all subformulae s of f; e0T ðsÞ ¼ eT ðs:: Þ: Suppose f is f1 f2 : Thus e0T ðfÞ ¼ e0T ðf1 f2 Þ ¼ Tðe0T ðf1 Þ; e0T ðf2 ÞÞ :: ¼ TðeT ðf:: 1 Þ; eT ðf2 ÞÞ ðby induction hypothesisÞ :: ¼ eT ðf:: 1 f2 Þ
¼ eT ðf:: Þ:
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Similarly for the other cases.
Now, suppose f 2 TAUTBoole : Let e be any [0, 1]-evaluation of propositional variables of f: Then e0 ðpi Þ ¼ nðnðeðpi ÞÞÞ is a {0,1}-evaluation and eT ðf:: Þ ¼ e0T ðfÞ ¼ 1: Thus, f:: 2 TAUTSMTL : 1 On the other hand, suppose f:: 2 TAUTSMTL : Let e be 1 any {0,1}-evaluation of propositional variables of f: Then e0 ðpi Þ ¼ nðnðeðpi ÞÞÞ ¼ eðpi Þ and hence, eT ðfÞ ¼ e0T ðfÞ ¼ eT ðf:: Þ ¼ 1: Thus, f 2 TAUTBoole : ( Lemma 4.5 proves that TAUTBoole is polynomially reducible to TAUTSMTL : Thus, we have the following 1 corollary: Corollary 4.2
TAUTSMTL is co-NP-hard. 1
Corollary 4.3 Let K stands for any particular SMTLalgebra ½0; 1T where T is a left-continuous strict t-norm. Then the following hold: K Boole (i) SATK ; is NP-complete. 1 ¼ SATpos ¼ SAT K Boole (ii) TAUTpos ¼ TAUT ; is co-NP-complete. (iii) TAUTK 1 is co-NP-hard.
Proof (i)
Direct by the following obvious inclusions SMTL SATBoole SATK ¼ SATBoole (by Corol1 SAT1 lary 4.1). Similarly for SATK pos :
(ii) TAUTBoole ¼ TAUTSMTL (by Lemma 4.4) pos (iii) Direct by Corollary 4.2 and by the fact that TAUTSMTL TAUTK ( 1 1: Þ has only The set of 1-tautologies of SMTL (TAUTSMTL 1 been proved in Corollary 4.2 to be co-NP-hard. It remains open if the set TAUTSMTL is also in co-NP. We can solve 1 the problem positively for some particular left-continuous strict t-norms as what was done in Ha´jek (1998), where Ha´jek proved that the set of 1-tautologies of Go¨del and product logics (as particular cases of strict t-norms) are pairwise distinct and are co-NP-complete. This is illustrated in the following example. Example 4.1 Consider the SMTL-algebra K ¼ ½0; 1Ta ; where Ta is the left-continuous strict t-norm given by (4). Based on the embeddability of the standard NM-algebra ½0; 1NM in the SMTL-algebra ½0; 1Ta ; it is easy to see that TAUTK 1 (the set of 1-tautologies in the SMTL-algebra K ¼ ½0; 1Ta ) is co-NP-complete. For the argument of this result, it suffices to notice that f 62 TAUTK 1 if and only if there exist J (recall the mapping fJ from Definition 4.1) such that
Computational complexities of axiomatic extensions
^
! J
pi ^ :ðf Þ
2 SATNM pos
pi 62J
where p1 ; . . .; pm be the propositional variables of f and J fp1 ; . . .; pm g: This reduces polynomially the compleNM ment of TAUTK 1 to SATpos ; hence by Theorem 3.1, we get the result. We finally note that a careful inspection of the NP- and co-NP-hardness proofs of Theorems 3.1, 3.2, and Corollary 4.2, reveals that the polynomial time reductions used there can actually be done in logarithmic space with linearly bounded output, and are even first order reductions (see Immerman 1999, Papadimitriou 1994, for definitions of these terms). Thus, all of our completeness and hardness results are still valid via logarithmic space reductions, and first order reductions. Acknowledgments We thank the three learned referees for their apt and expansive advice, which was instrumental in correcting and improving this article.
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