Computational Complexities of Axiomatic Extensions of Monoidal t-Norm Based Logic Moataz Saleh El-Zekey∗, Wafik Boulos Lotfallah†, and Nehad Nashaat Morsi‡ December 12, 2007
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 so-called nilpotent minimum t-norm (due to Fodor [8]); and SMTL corresponding to left-continuous strict t-norms, introduced by Esteva (and others) in [4] and [5]. 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 1tautologies of NM and the set of positive tautologies of both NM and SMTL are co-NP-complete. 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.
1
Introduction
Questions about computational complexity of fuzzy propositional calculi (Basic Fuzzy Logic BL and the most important stronger logics - L Ã ukasiewicz, G¨odel and product logics L Ã , G, Π respectively) have been studied and positive ∗
Department of Basic Sciences, Benha High Institute of Technology, Benha, Egypt, E-mail Address: m s
[email protected] † Corresponding Author: Department of Mathematics, University of WisconsinMadison, USA, E-mail Address:
[email protected], on leave from the German University in Cairo ‡ Department of Basic Sciences, Arab Academy for Science, Technology & Maritime Transport, Cairo, Egypt
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results have been obtained. For propositional logics L Ã , G, and Π, complete information is already stated in [10]. On the other hand, [1] shows 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 [12]. 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 (we think, there is no real hope that a structural characterization of left-continuous t-norms will ever arise). 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 [8]) and left-continuous strict t-norms, introduced independently in [11] and [19]. 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-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. 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 left-continuous t-norms and their residua that will be needed in this work. A triangular norm (see [24]) T on a [0, 1] is a binary operation on [0, 1] that is associative, commutative and monotone in both arguments, and 1 is an identity element for T . 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 [10]): 1. Left-continuity is the necessary and sufficient condition for a t-norm T and its residuum I (R-implication, for short) defined as I(a, b) = sup{c ∈ [0, 1] : T (a, c) ≤ b}, to verify the residuation property: c ≤ I(a, b) iff T (c, a) ≤ b. In that case (T, I) is called a residuated pair (see [21]). A corresponding negation operation can also be defined by putting n(a) = I(a, 0). 2. A residuated implication satisfies the following prelinearity property: max(I(a, b), I(b, a)) = 1. 2
3. Given a residuated pair (T, I), max is definable from min by the equation: max(a, b) = min(I(I(a, b), b), I(I(b, a), a)). 4. A residuated pair (T, I) satisfies the divisibility condition: min(x, y) = T (x, I(x, y)) iff T is continuous. 5. The negation n defined by a continuous t-norm T is involutive (i.e. n(n(x)) = x) iff the t-norm is isomorphic to L Ã ukasiewicz t-norm. (For the definition of L Ã ukasiewicz t-norm, see Example 1.1.) 6. A t-norm has no zero-divisors iff (∀x, y ∈ [0, 1], T (x, y) = 0 iff x = 0 or y = 0). Such a t-norm is called strict 1 . The following statements are equivalent: (a) T is a strict t-norm. odel negation given (b) The corresponding negation is the so-called G¨ by: ½ 1 if x = 0 (1) n(x) = I(x, 0) = 0 otherwise (c) The pseudo-complementation condition min(x, n(x)) = 0 holds. Example 1.1 [10] The following are three basic continuous t-norms on the unit interval, together with their residuation implications: (i) L Ã ukasiewicz t-norm: T (x, y) = max {0, x + y − 1}, L Ã ukasiewicz implication: I (x, z) = min {1, z + 1 − x}. (ii) G¨odel t-norm: T (x, y) = min {x, ½ y}, 1 if x ≤ z . G¨odel implication: I(x, z) = z if x > z (iii) Product (Goguen) t-norm: T (x,½y) = xy (product of reals) , 1 if x ≤ z Goguen implication: I(x, z) = . z if x > z x 1
Actually, in the literature of t-norms, the class of strict t-norms 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, we follow [6] by using strictness as equivalent to having no zero divisors.
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Example 1.2 The nilpotent minimum [8] is a left-continuous, but not continuous, t-norm. It and its residuum are defined by: ½ 0 if x + y ≤ 1 T (x, y) = , min(x, y) if x + y > 1 ½ 1 if x ≤ z I(x, z) = . max{1 − x, z} if x > z The negation operation corresponding to G¨odel and Product t-norms is G¨odel negation (1), while the negation operation corresponding to L Ã ukasiewicz and nilpotent minimum t-norms is n(x) = 1 − x. The problem of finding an appropriate axiomatization of many-valued logics based on a t-norm has been approached by introducing suitable classes of algebraic structures which approximate well the class of algebraic structures determined by the continuous or by the left continuous t-norms with their associated R-implication functions. In [15], the class of residuated lattices is considered, H´ajek in [10] defines the BL-algebras, and the MTL-algebras are introduced in [4]. In the following we will give the exact definitions of residuated lattice, MTL-algebra and BL-algebra. Definition 1.1 [15] A residuated algebra is an algebra (P, ≤, T, I, 1) , in which (P, ≤) is a poset with a top element 1, and T is a triangular norm on (P, ≤) that has a residuated implication (R-implication) I; that is, I is well-defined by the condition: (Residuation): ∀a, b, c ∈ P , T (a, b) ≤ c iff b ≤ I(a, c). Definition 1.2 [15] A residuated lattice is a residuated algebra whose underlying poset is a bounded lattice. In a residuated lattice on P , the meet and join operations on P will be denoted by ∧ and ∨. Definition 1.3 [4] MTL-algebras are residuated lattices on bounded lattices, which satisfy the prelinearity condition: ∀a, c ∈ P, I (a, c) ∨ I (c, a) = 1. Definition 1.4 [10] A BL-algebra is a residuated lattice which satisfies the following two identities for all x, y ∈ P : (1) x ∧ y = T (x, I(x, y)), (divisibility condition) (2) I(x, y) ∨ I(y, x) = 1. (prelinearity condition)
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Note that, BL-algebras are MTL-algebras which satisfy the divisibility condition. A residuated pair endows the unit interval [0, 1] with the structure of residuated lattice satisfying the prelinearity condition, but where the divisibility condition may fail. Monoidal Logic (ML, for short), introduced by H¨ohle in [15], is a logic whose algebraic counterpart is the class of residuated lattices. The language of Propositional Monoidal Logic is built in the usual way from a denumerable set of propositional symbols using the four primitive connectives ∗,→, ∧ and ∨ and the truth constant ¯0. As definable connectives, it has the negation and the double implication: ¬τ ξ ←→ τ
is is
τ → ¯0, (ξ → τ ) ∗ (τ → ξ).
An axiomatic system of Monoidal Logic can be found in [15]. H¨ohle proved that this logic is complete with respect to the variety of residuated lattices. Monoidal t-norm based Logic [4], MTL for short, is a strengthening of Monoidal logic. Namely, it is the extension of Monoidal logic with the axiom: ((ξ → τ ) → χ) → (((τ → ξ) → χ) → χ) which is equivalent to the usual prelinearity axiom. Therefore, any axiomatic extension of MTL is chain complete. Also, in MTL, ∨ becomes definable in terms of ∗, → and ∧ as: ξ ∨ τ is ((ξ → τ ) → τ ) ∧ ((τ → ξ) → ξ). An axiomatic system of Monoidal t-norm based Logic (MTL) can be found in [4]. The deduction rule of MTL is modus ponens, i.e. from ξ and ξ → τ derive τ . Furthermore, standard completeness for MTL has been recently proved by Jenei and Montagana in [18]. 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 a residuum iff it is left continuous). The logical systems MTL and ML are directly related to H´ajek Basic Fuzzy Logic [10]. It is the fuzzy logic capturing the tautologies of continuous t-norms and their residua. H´ajek defines in [10] the so-called Basic Fuzzy Logic (BL from now on). It is the particular case of many-valued residuated logic introduced to cope with the logic of continuous t-norms and their residua, which has BL-algebras as corresponding algebraic structures. BL differs from MTL in that: 5
1. The divisibility axiom (ξ ∗ (ξ → τ )) ←→ (ξ ∧ τ ) holds in BL, since it is a tautology for those continuous t-norms ; and 2. As a consequence, the min-conjunction ∧ is definable in BL as ξ ∧ τ is ξ ∗ (ξ → τ ), and need not to be introduced as a further primitive connective. For a comprehensive study of basic logic, see H´ajek [10] and H¨ohle ([13]-[16]).) In BL, if we fix the interpretation of conjunction to one of the basic tnorms, we obtain L à ukasiewicz fuzzy logic, G¨odel fuzzy logic, and product fuzzy logic respectively. In [10], it is shown that: • G¨odel fuzzy logic is the schematic extension of BL by the axiom: (G) ξ → ξ ∗ ξ, (contraction axiom) • Product fuzzy logic is the schematic extension of BL by the two axioms: (Π1) ¬¬ζ → ((ξ ∗ ζ → τ ∗ ζ) → (ξ → τ )), (Π2) ξ ∧ ¬ξ → ¯0, • L à ukasiewicz fuzzy logic is the schematic extension of BL by the axiom: (ÃL) ¬¬ξ → ξ. The standard semantics for L à , G, Π is [0, 1] with the corresponding continuous t-norm (and its residuum) denoted by [0, 1]Là , [0, 1]G , [0, 1]Π ; the standard semantics of BL is formed by all algebras [0, 1]T where T is a continuous t-norm, i.e. [0, 1] with T and its residuum. All these logics are proved to be standard complete (cf. [10] and [2]), i.e. complete with respect to the corresponding standard semantics. Another extension of BL is worth mentioning, namely the Strict Basic Fuzzy Logic SBL introduced in [7]. It is the schematic extension of BL by the axiom: ξ∧¬ξ → ¯0. The corresponding semantical models are BL-algebras satisfying the equation x ∧ n(x) = 0 (Pseudo). Hence, SBL-algebras can also be called pseudo-complemented BL-algebras. SBL is proved to be standard complete as BL [2], i.e. complete with respect to the corresponding class of strict continuous t-norms (continuous t-norms without zero divisors) based [0, 1] structures. Hence SBL captures the logic of strict continuous t-norms, i.e. those having G¨odel negation as associated negation (see Equation (1)). 6
For a clear overview of t-norm based logical systems ranging from Monoidal logic to the well-known L à ukasiewicz, Product and G¨odel logics, the reader is referred to Gottwald’s work [9]. In the rest of this introduction we survey, for the reader’s convenience, some known notions and facts on the computational complexities of BL and the most important stronger logics - L à ukasiewicz L à , G¨odel G and Product Π logics. All necessary details for the complexities of BL, L à , G and Π are available in [1], [10] and [12]. Let F be any of the logics L à , G, Π, BL, or Bool(= Boolean Logic). An evaluation of propositional variables (atomic formulae) is a mapping e assigning to each propositional variable p its truth value e(p) ∈ [0, 1]. This extends uniquely in the obvious way to an evaluation eF of all formulae using the operations on the standard semantics of F as truth functions. F For each F we define the set T AU T1F of 1-tautologies of F, T AU Tpos of F F positive tautologies of F and SAT1 , SATpos (1-satisfiable, positively satisfiable) in the obvious way (cf. [10, Definition 6.2.1]), i.e. T AU T1F = {ζ| for all evaluations e, eF (ζ) = 1}, F T AU Tpos = {ζ| for all evaluations e, eF (ζ) > 0}, F SAT1 = {ζ| for some e, eF (ζ) = 1}, F SATpos = {ζ| for some e, eF (ζ) > 0}. Note that for Boolean Logic, we only consider {0, 1}-evaluations e of propositional variables (i.e. e(p) ∈ {0, 1}). Theorem 1.1 [20](Cook’s theorem). Bool SAT1Bool = SATpos is NP-complete. Thus, since ζ ∈ T AU T Bool iff ¬ζ ∈ / SAT Bool , Bool Bool T AU T1 = T AU Tpos is co-NP-complete. For the complexities of L à , G, Π; it is shown in [10] that everything is as expected: All the SAT sets are NP-complete and all the T AU T sets are coNP-complete. See [10, Section 6.2] where one can find additional information on which of these classes are equal and which are not. For BL, we define (obviously) T AU T1BL = {ζ| for each evaluation e and each continuous t-norm T, eT (ζ) = BL (eT (ζ) > 0). 1}, and similarly for T AU Tpos For satisfiability let us define SAT1BL = {ζ| for some evaluation e and some continuous t-norm T, eT (ζ) = BL . 1} and similarly for SATpos The following is proved in [1]: Theorem 1.2 T AU T1BL is co-NP-complete (and obviously different from à , G, Π). T AU T1F for F = L 7
Note that [1] contains a remark on an alternative notion of satisfiability which could be called universal satisfiability: U SAT1BL = {ζ| for each T , there is an e such that eT (ζ) = 1}. It is shown that a formula is universally 1-satisfiable iff it is satisfiable BL in Boolean logic. By the same argument, the set U SATpos of universally positively satisfiable formulae equals the set of formulae satisfiable in Boolean logic (therefore these sets are NP-complete). BL The following theorem gives (expected) results on T AU Tpos , SAT1BL , BL SATpos together with (unexpected) results on the relation to corresponding sets over L Ã ukasiewicz logic. BL L Ã = T AU Tpos (hence co-NP-complete), Theorem 1.3 [12](1) T AU Tpos BL L Ã (2) SAT1 = SAT1 (hence NP-complete), BL L Ã (3) SATpos = SATpos (hence NP-complete).
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Some Schematic Extensions of The Monoidal T-norm Logic: An Overview
Some genuine axiomatic extensions of MTL have been considered in [4] and [5] 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 G¨odel negation as the associated negation.
2.1
The Logic of Nilpotent Minimum
Nilpotent Minimums are left-continuous t-norms that were introduced by Fodor in [8]. Given an involutive negation n on [0, 1], the corresponding nilpotent minimum t-norm Tn is defined as ½ 0 if x ≤ n(y), Tn (x, y) = (2) min(x, y) otherwise. The corresponding residuated implication is given by ½ 1 if x ≤ y, In (x, y) = max(n(x), y) otherwise. 8
(3)
It is straightforward to see that In (x, 0) = n(x), 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 [8]. A particular case of those investigations yielded nilpotent minimum. It is the standard example of a left-continuous t-norm in the literature. In [3], 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 , [22], [23], [25] and [26] along this line. The logic of nilpotent minimum (NM from now on), introduced by Esteva [4], is the schematic extension of MTL resulting when we add the following two axioms: (WNM) ((ζ ∗ τ ) → 0) ∨ ((ζ ∧ τ ) → (ζ ∗ τ )). (INV) ¬¬ζ → ζ. An NM-algebra is a structure (L, ∧, ∨, Tn , In , 0, 1) which is an MTLalgebra and satisfies the equations corresponding to (WNM) and (INV) axioms, that is In (Tn (a, b), 0) ∨ In (a ∧ b, Tn (a, b)) = 1. n(n(a)) = a. The decomposition of NM-algebras into subdirect products of linearly ordered ones is proved as usual. Moreover, since NM is a schematic extension of MTL, completeness of the logic NM with respect to NM-algebras and linearly ordered NM algebras also holds (see [4]). The NM-algebra in [0, 1] defined by taking n(a) = 1 − a is called the standard NM-algebra (see Example 1.2). It is also proved in [4] that all nilpotent minima on [0, 1] are isomorphic, which implies the following standard completeness theorem of Nilpotent Minimum Logic. Theorem 2.1 [4] (Standard completeness). NM proves a formula ζ if and only if ζ is a tautology with respect to the standard nilpotent minimum algebra. The standard completeness Theorem 2.1, above, 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”. The nilpotent minimum was slightly extended in [4] by allowing a weak negation (a negation that satisfies the inequality a ≤ n(n(a))) instead of a strong one in the construction. Based on this extension, monoidal t-norm 9
based logics (MTL) were studied also in [4], together with the involutive case (IMTL). Properties and applications of the nilpotent minimum t–norm based implication (called R0 implication there) were published in [23]. 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 [5], very analogous to what was done in BL. The logic SMTL is to MTL as the logic SBL to BL, that is, it is obtained as an extension of MTL by adding the same characteristic axiom of SBL -(ξ ∧ ¬ξ → 0)- as an extension of BL. The corresponding semantical models are the MTL-algebras satisfying the equation x ∧ n(x) = 0 (Pseudo-complementation axiom). Of course both Product and G¨odel logics are extensions of SBL, and therefore they are extensions of SMTL because SBL is an extension of SMTL. SMTL is proved to be standard complete [5] as MTL, i.e. complete with respect to 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 G¨odel negation as the associated negation (i.e. n(x) = I(x, 0) = 1 if x = 0 otherwise n(x) = 0). It is proved that SMTL cannot collapse with SBL (see [5]). Actually, it is easy to find examples 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 ∧ I(x, 0) = 0 where I is the residuum of T . One example is already shown in [12], here we show a simpler one [5]. Namely, for 0 < a < 1 take the following left-continuous t-norm: ½ a if x, y ≥ a and x + y ≤ 1 + a Ta (x, y) = (4) min(x, y) otherwise which is basically the ordinal sum of a G¨odel t-norm and of a nilpotent minimum. It is easy to check that Ia (x, 0) = sup{z ∈ [0, 1]|Ta (x, z) ≤ 0} is 1 if x = 0 and is 0 otherwise, and hence the axiom x ∧ Ia (x, 0) = 0 is satisfied where Ia is the residuum of Ta .
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3
Computational Complexity of NM
Due to the standard completeness Theorem 2.1, the standard semantics for NM is the standard NM-algebra ([0, 1], ∧, ∨, Tn , In ) defined by taking n(a) = 1 − a, where ∧, ∨ are min and max on [0, 1]. Following H´ajek’s denotation [12], we will denote by [0, 1]N M the standard semantics for NM. NM We define the set T AU T1N M of 1-tautologies of NM, T AU Tpos of positive NM NM tautologies of NM and SAT1 , SATpos (1-satisfiable, positively satisfiable) in the obvious way, e.g. NM = {ζ| for all evaluations e, eN M (ζ) > 0}, T AU Tpos NM SATpos = {ζ| for some e, eN M (ζ) > 0}. Here eN M (ζ) is the value of ζ given by the evaluation e of propositional variables (by elements of [0, 1]) and by the standard algebra of truth functions [0, 1]N M . Similarly for 1 instead of pos. Recall that in NM (since the negation is involutive), eN M (ζ) = 1 iff eN M (¬ζ) = 0; likewise eN M (ζ) = 0 iff eN M (¬ζ) = 1. Thus we have the following lemma: NM Lemma 3.1 (i) ζ ∈ T AU T1N M iff ¬ζ ∈ / SATpos , NM NM (ii) ζ ∈ T AU Tpos iff ¬ζ ∈ / SAT1 . NM Lemma 3.2 (i) SAT Bool ⊂ SAT1N M ⊂ SATpos , NM NM Bool (ii) T AU T1 ⊂ T AU Tpos ⊂ T AU T .
Proof. Non-strict inclusions are obvious from the definitions; strictness is showed by simple examples. Indeed, NM (p ∧ ¬p) ∈ SATpos − SAT1N M , (p → ¬p) ∗ (¬p → p) ∈ SAT1N M − SAT Bool , NM (p ∨ ¬p) ∗ (p ∨ ¬p) ∈ T AU T Bool − T AU Tpos , NM NM (p ∨ ¬p) ∈ T AU Tpos − T AU T1 . The above two lemmas are analogous to the ones obtained for L Ã ukasiewicz logic in [10]. This is due to the fact that NM and L Ã ukasiewicz 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 ¬(ξ ∗ ¬τ ). Lemma 3.3 For each formula ζ in NM logic, let ζ # be the formula (p1 ∨ ¬p1 ) ∗ ... ∗ (pm ∨ ¬pm ) ∗ ζ where p1 , ..., pm are the propositional variables in ζ. Then ζ ∈ SAT Bool iff ζ # ∈ SAT1N M . 11
Proof. To prove the lemma observe first that for all evaluation e, eN M (p∨ ¬p) = 1 is equivalent to e(p) = 1 or e(p) = 0. Also, for any {0, 1}-evaluation e of propositional variables of ζ (i.e. e(pi ) = 1 or e(pi ) = 0, i = 1, ..., m), eBool (ζ) = eN M (ζ) (i.e. they coincide for the values 0, 1). Thus, the proof of the lemma runs as follows: ζ # ∈ SAT1N M iff there is an evaluation e of propositional variables of ζ such that eN M (ζ # ) = 1 iff there is an evaluation e of propositional variables of ζ such that for all i = 1, ..., m, eN M (pi ∨ ¬pi ) = 1 and eN M (ζ) = 1 iff e is a {0, 1}-evaluation of propositional variables of ζ such that eN M (ζ) = 1 iff ζ ∈ SAT Bool . Lemma 3.4 Let e be an evaluation of propositional variables of an arbitrary formula ζ in NM logic (by elements of [0, 1]) such that e(p) 6= 21 for all propositional variables occurring in ζ. Then there is a {0, 1}-evaluation e0 of the propositional variables of ζ which satisfies 1 e0N M (ζ) = 1 iff eN M (ζ) > , 2 and
1 e0N M (ζ) = 0 iff eN M (ζ) < . 2 Proof. For all propositional variables occurring in ζ, let e0 (p) = 1 iff e(p) > 21 and e0 (p) = 0 iff e(p) < 12 . Then, the proof of the lemma is by induction on the construction of ζ and runs as follows: Case 1: ζ = ¬ζ 1 e0N M (ζ) = 1 iff e0N M (¬ζ 1 ) = 1 iff n(e0N M (ζ 1 )) = 1 iff e0N M (ζ 1 ) = 0 iff eN M (ζ 1 ) < 12 (by induction hypothesis) iff n(eN M (ζ 1 )) > 12 iff eN M (¬ζ 1 ) > 12 iff eN M (ζ) > 21 . Also, e0N M (ζ) = 0 iff e0N M (¬ζ 1 ) = 0 iff n(e0N M (ζ 1 )) = 0 iff e0N M (ζ 1 ) = 1 iff eN M (ζ 1 ) > 12 (by induction hypothesis) iff n(eN M (ζ 1 )) < 12 iff eN M (¬ζ 1 ) < 12 iff eN M (ζ) < 21 . Case 2: ζ = ζ 1 ∗ ζ 2 . e0N M (ζ) = 1 iff e0N M (ζ 1 ∗ ζ 2 ) = 1 iff Tn (e0N M (ζ 1 ), e0N M (ζ 2 )) = 1 iff e0N M (ζ 1 ) = 1 and e0N M (ζ 2 ) = 1 iff eN M (ζ 1 ) > 21 and eN M (ζ 2 ) > 21 (by induction hypothesis) iff Tn (eN M (ζ 1 ), eN M (ζ 2 )) > 12 iff eN M (ζ 1 ∗ ζ 2 ) > 12 iff eN M (ζ) > 12 . Also, e0N M (ζ) = 0 iff e0N M (ζ 1 ∗ ζ 2 ) = 0 iff Tn (e0N M (ζ 1 ), e0N M (ζ 2 )) = 0 iff e0N M (ζ 1 ) = 0 or e0N M (ζ 2 ) = 0 iff eN M (ζ 1 ) < 21 or eN M (ζ 2 ) < 21 (by induction hypothesis) iff Tn (eN M (ζ 1 ), eN M (ζ 2 )) < 12 iff eN M (ζ 1 ∗ ζ 2 ) < 12 iff eN M (ζ) < 12 . The proof in the other cases (i.e. using →, ∧, ∨) follows on lines much similar to the proof in case 2. 12
Lemma 3.5 For each formula ζ in NM logic, let ζ ## be the formula (p1 ∨ ¬p1 )2 ∗ ... ∗ (pn ∨ ¬pm )2 ∗ ζ 2 where p1 , ..., pm are the propositional variables in ζ. Then NM . ζ ∈ SAT Bool iff ζ ## ∈ SATpos
Proof. To prove the lemma observe first that eN M ((p ∨ ¬p)2 ) > 0 is equivalent to eN M (p ∨ ¬p) > 21 , which in turn is equivalent to e(p) 6= 12 . Now, NM . Then there is an evaluation e of p1 , ..., pm (variables suppose ζ ## ∈ SATpos ## of ζ) such that eN M (ζ ) > 0. This implies that eN M ((pi ∨ ¬pi )2 ) > 0 for all i = 1, ..., m and eN M (ζ 2 ) > 0, which is equivalent to e(pi ) 6= 12 for all i = 1, ..., m and e(ζ) > 12 . Thus, by Lemma 3.4, there is a {0, 1}-evaluation e0 of the propositional variables of ζ which satisfies e0N M (ζ) = 1. Thus, NM ζ ∈ SAT Bool . This proves that if ζ ## ∈ SATpos then ζ ∈ SAT Bool . The converse is immediate. Lemma 3.6 Let ζ be an arbitrary formula in NM logic and have m propositional variables p1 , ..., pm . Then the following two statements are equivalent: (i) There is an evaluation e0 of the propositional variables of ζ (by elements of [0, 1]) such that e0N M (ζ) = 1 (or e0N M (ζ) > 0). (ii) There is an evaluation e of the propositional variables of ζ taking only 1 2m+1 values in the set of 2m + 3 elements Dm = {0, 2m+2 , ..., 2m+2 , 1} such that eN M (ζ) = 1 (or eN M (ζ) > 0). Proof. Suppose (i) holds. Let n be the involutive negation associated with the standard NM-algebra. Then the set Z = {e0 (p1 )∨n(e0 (p1 )), ..., e0 (pm )∨ n(e0 (pm ))} is contained in the subinterval [ 21 , 1]. Hence, there is an order isomorphism f of [ 12 , 1] onto [ 12 , 1] taking Z into Dm . We extend f into an order isomorphism on [0, 1] by setting for all t ∈ [0, 21 ]: f (t) = 1−f (n(t)). It follows that the isomorphism f satisfies, for all t ∈ [0, 1]: f (n(t)) = n(f (t)). This is because if t ≤ 21 then f (t) = 1 − f (n(t)) and if t ≥ 12 then n(t) ≤ 12 , and so also f (n(t)) = 1 − f (n(n(t))) = 1 − f (t) = n(f (t)). Thus, it is easy to prove that the isomorphism f also satisfies for all a, b ∈ [0, 1]: f (Tn (a, b)) = Tn (f (a), f (b)) and f (In (a, b)) = In (f (a), f (b)). That is f is an isomorphism with respect to the operations of the standard NM-algebra. Hence, f is an isomorphism of the standard NM-algebra [0, 1]N M sending the set {e0 (p1 ), ..., e0 (pm ), n(e0 (p1 )), ..., n(e0 (pm ))} into Dm . Thus, defining e(pi ) = f (e0 (pi )) for all propositional variables occurring in ζ, eN M (ζ) = f (e0N M (ζ)) = 1 (or > 0). This proves (ii). The converse is immediate. 13
M Theorem 3.1 SATN pos is NP-complete.
Proof. Hardness: Let let for every formula ζ, ζ ## be the formula obtained from ζ as in Lemma 3.5. Clearly, ζ ## is computed from ζ in polynomial time. Hence, Lemma 3.5 proves that SAT Bool is polynomially reducible M to SATPNosM and thus, SATN pos is NP-hard. NM Membership: By Lemma 3.6, ζ ∈ SATpos iff there is an evaluation e of p1 , 1 2m+1 ..., pm (variables of ζ) taking only values in the set Dm = {0, 2m+2 , ..., 2m+2 , 1} such that eN M (ζ) > 0. NM Thus an algorithm showing ζ ∈ SATpos guesses an evaluation e, e(pi ) ∈ NM Dm , and computes the truth value. This shows that SATpos is in NP. Theorem 3.1 and Lemma 3.1 (i) render the following corollary: M is co-NP-complete. Corollary 3.1 TAUTN 1 M Theorem 3.2 SATN is NP-complete. 1
Proof. Hardness: Let let for every formula ζ, ζ # be the formula obtained from ζ as in Lemma 3.3. Clearly, ζ # is computed from ζ in polynomial time. Hence, Lemma 3.3 proves that SAT Bool is polynomially reducible to SAT1N M M and thus, SATN is NP-hard. 1 Membership: By Lemma 3.6, ζ ∈ SAT1N M iff there is an evaluation e of p1 , 2m+1 1 , ..., 2m+2 , 1} ..., pm (variables of ζ) taking only values in the set Dm = {0, 2m+2 such that eN M (ζ) = 1. Thus an algorithm showing ζ ∈ SAT1N M guesses an evaluation e, e(pi ) ∈ Dm , and computes the truth value. This shows that SAT1N M is in NP. Theorem 3.2 with Lemma 3.1 (ii) render the following corollary: NM Corollary 3.2 T AU Tpos is co-NP-complete.
4
Computational Complexity of SMTL
The standard semantics of SMTL is formed by all algebras ([0, 1], ∧, ∨, T, I) where T is a left-continuous strict t-norm with its residuum I, and ∧, ∨ are min and max on [0, 1], denoted by [0, 1]T . For SMTL, we define (obviously) SM T L = {ζ| for each evaluation e and each left-continuous strict T AU Tpos t-norm T, eT (ζ) > 0} and similarly for T AU T1SM T L (eT (ζ) = 1). Here eT (ζ) is the value of ζ given by the evaluation e of propositional variables (by elements of [0, 1]) and by the standard algebra of truth functions [0, 1]T . For satisfiability let us define 14
SM T L SATpos = {ζ| for some evaluation e and some left-continuous strict t-norm T, eT (ζ) > 0} and similarly for SAT1SM T L (eT (ζ) = 1). Recall that both Product and G¨odel logics are extensions of SMTL and all these logics are characterized by having the G¨odel negation as the associated negation. H´ajek [10], in studying the computational complexity of Product and G¨odel logics, depends mainly on this (i.e. their associated negations are G¨odel 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 that H´ajek (cf. [10, 6.2.2 - 6.2.9]) used in his study. We have to start with the following definition:
Definition 4.1 Let p1 , ..., pm be propositional variables and J ⊆ {p1 , ..., pm }. J is understood as the set of variables evaluated by 0. We define, for each formula ζ in SMTL built from p1 , ..., pm , its translation ζ J as follows: ¯0J = ¯0, ¯1J = ¯1, ½ ¯0 if pi ∈ J piJ = , p otherwise i J ¯1 if ζ = ¯0 ¯0 if ζ J 6= ¯0 and τ J = ¯0 , (ζ → τ )J = J ζ → τ J otherwise ½ ¯0 if ζ J = ¯0 or τ J = ¯0 (ζ ∗ τ )J = , J ζ ∗ τ J otherwise ½ ¯0 if ζ J = ¯0 or τ J = ¯0 (ζ ∧ τ )J = . ζ J ∧ τ J otherwise Recall that, in SMTL, ∨ and ¬ are definable in terms of →, ∧ and ¯0 as ξ ∨ τ is ((ξ → τ ) → τ ) ∧ ((τ → ξ) → ξ), ¬τ is τ → ¯0. Hence, by Definition 4.1 it is direct to see that ½ ¯0 if ζ J = ¯0 and τ J = ¯0 J (ζ ∨ τ ) = , J ζ ∨ τ J otherwise ½ ¯0 if ζ J 6= ¯0 J . (¬ζ) = ¯1 otherwise
Lemma 4.1 For each formula ζ in SMTL built from p1 , ..., pm , and each set J ⊆ {p1 , ..., pm }, the following hold: (1) Either ζ J is 0¯ or ζ J is a formula not containing ¯0. (2) For each left-continuous strict t-norm T and for each evaluation e such that e(pi ) = 0 iff pi ∈ J, the following hold: 15
(i) eT (ζ J ) = 0 iff ζ J is ¯0, (ii) eT (ζ) = eT (ζ J ). Proof. (1) This follows directly from Definition 4.1 and by induction on the construction of ζ, e.g. if ζ is ζ 1 ∗ζ 2 then by induction hypothesis either ζ J1 (or ζ J2 ) is ¯0 or ζ J1 (and ζ J2 ) is a formula not containing ¯0. Thus by Definition 4.1 either ζ J = (ζ 1 ∗ ζ 2 )J is ¯0 or ζ J = (ζ 1 ∗ ζ 2 )J = ζ J1 ∗ ζ J2 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 e(pi ) = 0 iff pi ∈ J. Then if ζ is pj , j ∈ 1, ..., m, then (by Definition 4.1) e(pj ) = e(pJj ) and e(pJj ) = 0 iff pJj is ¯0. We now use the induction on the construction of ζ. (2-i) The induction hypothesis states that for all subformulae τ of ζ, eT (τ J ) = 0 iff τ J is ¯0. Suppose ζ is ζ 1 ∗ ζ 2 . Thus, if ζ J is not ¯0 then by part (1) ζ J is a formula not containing ¯0, hence both ζ J1 and ζ J2 are not ¯0. Thus, by induction hypothesis, eT (ζ J1 ) 6= 0 and eT (ζ J2 ) 6= 0. Then we get eT (ζ J ) = eT ((ζ 1 ∗ ζ 2 )J ) = eT (ζ J1 ∗ ζ J2 ) = T (eT (ζ J1 ), eT (ζ J2 )) 6= 0. This proves that (in the case that ζ is ζ 1 ∗ζ 2 ) if ζ J is not ¯0 then eT (ζ J ) 6= 0. The converse is immediate. Similarly for the other cases. (2-ii) The induction hypothesis states that for all subformulae τ of ζ, eT (τ ) = eT (τ J ). Suppose ζ is ζ 1 → ζ 2 . Thus, if ζ J1 6= ¯0 and ζ J2 = ¯0 then ζ J = ¯0 (by Definition 4.1), hence eT (ζ J ) = eT (¯0) = 0. On the other hand eT (ζ) = eT (ζ 1 → ζ 2 ) = I(eT (ζ 1 ), eT (ζ 2 )) = I(eT (ζ J1 ), eT (ζ J2 )) (by induction hypothesis) = I(eT (ζ J1 ), 0) (by the assumption ζ J2 = ¯0) = 0, since eT (ζ J1 ) 6= 0 (by the assumption ζ J1 6= ¯0 and the part (2-i)) and I is the residuum of the left-continuous strict t-norm T . Hence, eT (ζ J ) = 0 = eT (ζ). Otherwise (i.e. if not (ζ J1 6= ¯0 and ζ J2 = ¯0)), eT (ζ J ) = eT ((ζ 1 → ζ 2 )J ) = eT (ζ J1 → ζ J2 ) (by Definition 4.1) = I(eT (ζ J1 ), eT (ζ J2 )) = I(eT (ζ 1 ), eT (ζ 2 )) (by the induction hypothesis) = eT (ζ 1 → ζ 2 ) = eT (ζ). This proves that (in the case that ζ is ζ 1 → ζ 2 ), eT (ζ J ) = eT (ζ). Similarly for the other cases. Recall that SAT Bool is the set of all formulae ζ satisfiable in Boolean (two-valued) logic. Similarly for T AU T Bool .
16
Lemma 4.2 The following are equivalent: SM T L (1) ζ ∈ SATpos ; J (2) for some J, ζ is not ¯0; (3) ζ ∈ SAT Bool ; (4) ζ ∈ SAT1SM T L . Proof. Evidently, (3) implies (4) and (4) implies (1). SM T L (1) implies (2): Suppose ζ ∈ SATpos . Then for a left-continuous strict t-norm T , an evaluation e exists such that for J = {pi |e(pi ) = 0} we get (by Lemma 4.1 (2-ii)) eT (ζ J ) = eT (ζ) > 0 and hence ζ J is not ¯0, by Lemma 4.1 (2-i). (2) implies (3): Now assume (2) and let e be a {0, 1}-evaluation such that e(pi ) = 0 for pi ∈ J and e(pi ) = 1 otherwise. Since ζ J is not ¯0 we get (by Lemma 4.1 (2-i)), for each left-continuous strict t-norm T , eT (ζ J ) 6= 0 and hence eT (ζ J ) = 1. Then, by Lemma 4.1 (2-ii), eT (ζ) = 1 and hence eBool (ζ) = 1(since e is a {0, 1}-evaluation and both of eBool and eT coincide for values 0, 1). Thus ζ ∈ SAT Bool . This completes the proof. SM T L Corollary 4.1 The sets SAT1SM T L , SATpos , SAT Bool are all equal; hence they are NP-complete.
Recall that the difference between BL and MTL is due to the fact that in BL the divisibility axiom (ξ ∧ τ ) → (ξ ? (ξ → τ )) (DIV) holds since it is a tautology for those continuous t-norms [4]. The following lemma shows that DIV is a positive tautology for those left-continuous strict t-norms. SM T L Lemma 4.3 DIV ∈ T AU Tpos . SM T L Proof. For a contradiction, assume DIV ∈ / T AU Tpos . Then for a left-continuous strict t-norm, an evaluation e exist such that eT (DIV ) = 0, which implies for each ζ, τ , I(eT (ξ) ∧ eT (τ ), T (eT (ξ), I(eT (ξ), eT (τ )))) = 0 (I is the residuum of T ), which in turn is equivalent to the following two statements: (1) eT (ξ) ∧ eT (τ ) 6= 0, which is equivalent to “eT (ξ) 6= 0 and eT (τ ) 6= 0”, and (2) T (eT (ξ), I(eT (ξ), eT (τ ))) = 0. Since T is a left-continuous strict t-norm, (2) is equivalent to eT (ξ) = 0 (rejected because it contradicts (1)) or I(eT (ξ), eT (τ )) = 0, which is equivalent to eT (ξ) 6= 0 and eT (τ ) = 0, a contradiction (with (1)). Thus, DIV ∈ SM T L . T AU Tpos SM T L is closed under modus ponens, that is if Lemma 4.4 The set T AU Tpos SM T L SM T L SM T L . then τ ∈ T AU Tpos and (ξ → τ ) ∈ T AU Tpos ξ ∈ T AU Tpos
17
Proof. For each left-continuous strict t-norm T and for each evaluation e, T (eT (ξ), eT (ζ → τ )) ≤ eT (τ ). Thus if we assume eT (ξ) > 0 and eT (ξ → τ ) > 0 then T (eT (ξ), eT (ζ → τ )) > 0 (since T has no zero divisors), which implies eT (τ ) > 0. This gives the result. The local deduction theorem is valid in MTL [4] and hence it is valid in SMTL, i.e., given a theory Γ and formulae ξ, τ in SMTL, then Γ ∪ {ξ} ` τ iff there is an m such that Γ ` ξ m → τ (where ξ m is ξ ∗ ... ∗ ξ, m factors). SM T L SM T L Lemma 4.5 T AU Tpos = T AU T Bool , thus T AU Tpos is co-NP-complete. SM T L Proof. Clearly, T AU Tpos ⊆ T AU T Bool . On the other hand, recall that Boolean logic is axiomatized by BL + ζ ∨ ¬ζ [10]. Also, it is well known that BL is equivalent to the system resulting from MTL by adding the divisibility axiom [4]: (DIV) (ζ ∧ τ ) → (ζ ∗ (ζ → τ )). Hence, we can say that Boolean logic is axiomatized by MTL + DIV + ζ ∨ ¬ζ. Now, suppose ζ ∈ T AU T Bool . Then ζ is provable in Boolean logic, and hence (over SMTL) {DIV, ζ ∨ ¬ζ} ` ζ. Thus by the local deduction theorem, SM T L ` (DIV ∗ (ζ ∨ ¬ζ))m → ζ. Hence (1) [(DIV ∗ (ζ ∨ ¬ζ))m → ζ] ∈ T AU T1SM T L . SM T L Moreover, it is clear that ζ ∨ ¬ζ ∈ T AU Tpos and by Lemma 4.3, SM T L DIV ∈ T AU Tpos . Thus SM T L (2) (DIV ∗ (ζ ∨ ¬ζ))m ∈ T AU Tpos (since ∗ has no zero divisors in SM T L SMTL). Hence from (1), (2) and Lemma 4.4 we get ζ ∈ T AU Tpos . This Bool SM T L proves that T AU T ⊆ T AU Tpos and completes the proof. SM T L SM T L Note that ζ ∈ T AU Tpos iff ¬ζ ∈ / SATpos .
Lemma 4.6 For each ζ, let ζ ¬¬ be the formula resulting from ζ by replacing each propositional variable pi by its double negation ¬¬pi . Then ζ ∈ T AU T Bool iff ζ ¬¬ ∈ T AU T1SM T L . Proof. For each [0, 1]-evaluation e, let e0 be an evaluation such that e0 (pi ) = n(n(e(pi ))), clearly e0 (pi ) is 0 or 1 (since n is G¨odel negation). We now claim that for each left-continuous strict t-norm T , eT (ζ ¬¬ ) = e0T (ζ). The proof of this claim is easy by induction on the construction of ζ as follows: It is clear that the induction step holds by the definition of e0 . The induction hypothesis is that for all subformulae τ of ζ, e0T (τ ) = eT (τ ¬¬ ). Suppose ζ is ζ 1 ∗ ζ 2 . Thus e0T (ζ) = e0T (ζ 1 ∗ ζ 2 ) = T (e0T (ζ 1 ), e0T (ζ 2 )) 18
¬¬ = T (eT (ζ ¬¬ 1 ), eT (ζ 2 )) (by induction hypothesis) ¬¬ = eT (ζ ¬¬ 1 ∗ ζ2 ) = eT (ζ ¬¬ ). Similarly for the other cases. Now, suppose ζ ∈ T AU T Bool . Let e be any [0, 1]-evaluation of propositional variables of ζ. Then e0 (pi ) = n(n(e(pi ))) is a {0, 1}-evaluation and eT (ζ ¬¬ ) = e0T (ζ) = 1. Thus, ζ ¬¬ ∈ T AU T1SM T L . On the other hand, suppose ζ ¬¬ ∈ T AU T1SM T L . Let e be any {0, 1}evaluation of propositional variables of ζ. Then e0 (pi ) = n(n(e(pi ))) = e(pi ) and hence, eT (ζ) = e0T (ζ) = eT (ζ ¬¬ ) = 1. Thus, ζ ∈ T AU T Bool . Lemma 4.6 proves that T AU T Bool is polynomially reducible to T AU T1SM T L . Thus, we have the following corollary:
Corollary 4.2 T AU T1SM T L is co-NP-hard. In the following, let F stands for any particular SMTL-algebra [0, 1]T where T is a left-continuous strict t-norm. Then T AU T1F denotes the set F F of 1-tautologies in F and similarly for T AU Tpos , SATpos and SAT1F of positive tautologies in F, positively satisfiable formulae in F and 1-satisfiable formulae in F, respectively. Hence we have the following corollary: F = SAT Bool , is NP-complete. Corollary 4.3 (i) SAT1F = SATpos F Bool (ii) T AU Tpos = T AU T , is co-NP-complete. F (iii) T AU T1 is co-NP-hard.
Proof. (i) Direct by the following obvious inclusions SAT Bool ⊆ SAT1F ⊆ SAT1SM T L = SAT Bool (by Corollary 4.1). F Similarly for SATpos . Bool SM T L (ii) T AU T = T AU Tpos (by lemma 4.5) F ⊆ T AU Tpos ⊆ T AU T Bool (obvious from the definitions). (iii) Direct by Corollary 4.2 and by the fact that T AU T1SM T L ⊆ T AU T1F . The set of 1-tautologies of SMTL (T AU T1SM T L ) has only been proved in Corollary 4.2 to be co-NP-hard. It remains open if the set T AU T1SM T L is also in co-NP. We can solve the problem positively for some particular left-continuous strict t-norms as what was done in [10], where H´ajek proved that the set of 1-tautologies of G¨odel and product logics (as particular cases of strict t-norms) are pairwise distinct and are co-NP-complete. Recall that, in Section 2.2, the t-norm Ta (4) is given as an example of a left-continuous strict t-norm, which is not continuous. In the following, we prove that the set of 1-tautologies of Ta is co-NP-complete. To do this we need the following two lemmas. 19
Lemma 4.7 For each c such that 0 < c < 1, the standard NM-algebra on [0, 1] is isomorphic to the algebra ([c, 1], Tc , Ic , ∧c , ∨c , 0c , 1c ) where, for all c ≤ x, y ≤ 1, ½ c if x + y ≤ 1 + c Tc (x, y) = ½ min(x, y) otherwise 1 if x ≤ y Ic (x, y) = max(1 + c − x, y) otherwise ∧c , ∨c are min and max on [c, 1], 0c = c, 1c = 1. Proof. The isomorphism is fc : [0, 1] → [c, 1], defined by fc (x) = (1 − c)x + c. It is clear that x ≤ y iff fc (x) ≤ fc (y). We compute: x + y ≤ 1 iff fc (x + y) ≤ fc (1) = 1 iff fc (x) + fc (y) ≤ 1 + c. Also 1 − x ≤ y iff fc (1 − x) ≤ fc (y) iff 1 + c − fc (x) ≤ fc (y). The rest is evident. Lemma 4.8 For each a, a ´ ∈ ]0, 1[ such that a 6= a ´, let Ta and Ta´ are the leftcontinuous strict t-norm on [0, 1] given by (4). Then ([0, 1], Ta ) is isomorphic to ([0, 1], Ta´ ). Proof. The ½ a´ isomorphism is f : [0, 1] → [0, 1], defined by x if 0 ≤ x < a a f (x) = . a ´−1 (x − 1) + 1 if a≤x≤1 a−1 It is clear that x ≤ y iff f (x) ≤ f (y) and x ≥ a iff f (x) ≥ a ´. Letting x, y ≥ a, hence x + y − a ≥ a, we compute: x + y ≤ 1 + a iff x + y − a ≤ 1 iff a ´−1 f (x + y − a) ≤ f (1) = 1 iff a−1 [x + y − a − 1] + 1 ≤ 1 iff f (x) + f (y) ≤ 1 + a ´. The rest is evident. Lemma 4.9 Let F stand for the SMTL-algebra [0, 1]Ta , where Ta is the leftcontinuous strict t-norm given by (4). Then T AU T1F is co-NP-complete. Proof. By Corollary 4.3 (iii), it remains to prove that T AU T1F is in co-NP, which in turn is equivalent to the statement saying that the set {ζ| for some e, eTa (ζ) < 1} is in NP. Recall the mapping ζ J from Definition 4.1. The following are equivalent: (1) For some evaluation e of p1 , ..., pm (the variables of ζ), eTa (ζ) < 1. (2) An evaluation e exists such that for J = {pi |e(pi ) = 0}, eTa (ζ J ) < 1 (thanks to Lemma 4.1 (2-ii)). (3) For some J ⊆ {p1 , ..., pm } there exists a positive evaluation e0 (i.e. 0 e (pi ) > 0 for pi occurring in ζ J ) such that e0Ta (ζ J ) < 1 (thanks to Lemma 4.1). (4) For some J and for some 0 < a0 (a0 < e0 (pi ) for all pi occurring in ζ J ), a positive evaluation e00 exists such that e00Ta0 (ζ J ) < 1 (thanks to Lemma 4.8). 20
J (5) For some J, a positive evaluation e# exists such that e# N M (ζ ) < 1 (thanks to Lemma 4.7). # J # (6) For some J, a³ positive evaluation ´ e exists such that eN M (¬ζ ) > 0. V J NM (7) For some J, pi ∈J / pi ∧ ¬(ζ ) ∈ SATpos . Thus, we get the following NP algorithm that decides the membership of the set {ζ| for some e, eTa (ζ) < 1}: 1. Guess J ⊆ {p1 , ..., pm } (in polynomial time). ´ ³V J 2. Construct the formula η = (also in polynomial pi ∈J / pi ∧ ¬(ζ ) time). NM 3. Check that η ∈ SATpos (an NP-problem). This shows that the set {ζ| for some e, eTa (ζ) < 1} is in NP, and thus T AU T1F is in co-NP. 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 [20] and [17] for definitions of these terms). Thus, all of our completeness and hardness results are still valid via logarithmic space reductions, and first order reductions.
References [1] M. Baaz, P. H´ajek, F. Montagna, H. Veith, Complexity of t-tautologies, Annals Pure Appl Logic 113 (1-3) (2001) 3-11. [2] R. Cignoli, F. Esteva, L. Godo, and A. Torrens, Basic fuzzy logic is the logic of continuous t-norms and their residua, Soft Computing 2 (2000) 106-112. [3] R. Cignoli, F. Esteva, L. Godo, and F. Montagna, On a class of leftcontinuous t-norms, Fuzzy Sets and Systems 131 (2002) 283-296. [4] F. Esteva, L. Godo, Monoidal t-norm based logic: towards a logic for left-continuous t-norms, Fuzzy Sets and Systems 124 (2001) 271-288. [5] F. Esteva, J.Gispert, L. Godo and F. Montagna, On the standard and rational completeness of some axiomatic extensions of Monoidal t-norm based logic, Studia Logica 71 (2002) 393-420. [6] F. Esteva, L. Godo, and A. Garc’a-Cerdana, On the hierarchy of t-norm based residuated fuzzy logics, in: E. orlowska, M. Fitting, Eds., Studies 21
in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 2003, pp. 251-272. [7] F. Esteva, L. Godo, P. H´ajek and M. Navara, Residuated fuzzy logics with an involutive negation, Archive for Mathematical Logic 39 (2000) 103-124. [8] J.C. Fodor, Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69 (1995) 141-156. [9] S. Gottwald, A Treatise on Many-Valued Logic, Research Studies Press, Baldock, 2001. [10] P. H´ajek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998. [11] Petr H´ajek, Observations on the monoidal t-norm logic, Fuzzy Sets and Systems 132(1) (2002) 107-112 . [12] P. H´ajek. Basic fuzzy logic and BL-algebras II, Soft Computing 7 (2003)179-183. [13] U. H¨ohle, M-valued sets and sheaves over integral commutative CLmonoids, in S.E. Rodabaugh, E.P. Klement and U. H¨ohle, Eds., Applications of Category Theory to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1992, pp. 32-72. [14] U. H¨ohle, Monoidal logic, in: R. Kruse, J. Gebhardt and R. Palm, Eds., Fuzzy Systems in Computer Science, Vieweg, Wiesbaden, 1994, pp. 233243. [15] U. H¨ohle, Commutative, residuated `-monoids, in: U. H¨ohle and E.P. Klement, Eds., Non-classical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1995, pp. 53-106. [16] U. H¨ohle, Presheaves over GL-monoids, in: U. H¨ohle and E.P. Klement, Eds., Non-classical Logics and Their Applications to Fuzzy Subsets, Kluwer Academic Publishers, Dordrecht, 1995, pp. 127-157. [17] N. Immerman, Descriptive Complexity, Springer Verlag (1999). [18] S. Jenei and F. Montagna, A proof of standard completeness for Esteva and Godo’s logic MTL, Studia Logica 70 (2002) 183–192. [19] S. Jenei, F. Montagna, A general method for constructing leftcontinuous t-norms. Fuzzy Sets and Systems 136(3) (2003) 263-282. 22
[20] C. H. Papadimitriou, Computational Complexity, Addison Wesley (1994). [21] J. Pavelka, On fuzzy logic, Zeitsch. f. Math. Logik, I, II, III: 25 (1979) 45-52, 119-134, 447-464. [22] D. Pei, On equivalent forms of fuzzy logic systems NM and IMTL, Fuzzy Sets and Systems 138 (2003) 187-195. [23] D. Pei, R0 implication: characteristics and applications, Fuzzy Sets and Systems 131 (2002) 297 - 302. [24] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983. [25] S.M.Wang, B.-S.Wang and X.-Y.Wang, A characterization of truthfunctions in the nilpotent minimum logic, Fuzzy Sets and Systems 145 (2004) 253-266. [26] S.M. Wang, B.-S. Wang and F. Ren, NMÃL, a schematic extension of F.Esteva and L.Godos logic MTL, Fuzzy Sets and Systems 149 (2005) 285–295.
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