Models for Many-Valued Probabilistic Reasoning

Journal of Logic and Computation Advance Access published March 18, 2009

Models for Many-Valued Probabilistic Reasoning TOMMASO FLAMINIO and FRANCO MONTAGNA, Department of Mathematics and Computer Science, University of Siena, Pian dei Matellini 44, 53100 Siena, Italy. E-mail: [email protected]; [email protected] Abstract In this article, we compare models for many-valued probabilistic reasoning from the point of view of the sets of satisfiable formulas, positive satisfiable formulas, and tautologies. The results arising from this comparison will be used in the final part of the present article to provide results about the computational complexity for the problem of deciding if a formula belongs to one of the previously discussed sets. Keywords: SMV-algebras, states on MV-algebras, probabilistic Kripke models, PSPACE containment.

1

Introduction

In his book [9], Hàjek introduces a discussion about the connections between fuzzy logic and probability. He points out several differences: for instance, fuzzy logic is a logic for the treatment of vagueness, whereas probabilistic logic is a logic for the treatment of uncertainty. Moreover, the most prominent fuzzy logics are truth functional, whereas probabilistic logic is not. Nevertheless, Hàjek shows that probabilistic logic can be treated inside fuzzy logic, by means of a modality Pr to be interpreted as probably. Then, Flaminio and Godo [6], introduce a logic, called FP(Ł,Ł), in which it is possible to express the probability of fuzzy events. This logic is not algebraizable in the sense of Blok and Pigozzi (cf. [1]), and its semantics is not given by means of universal algebras but rather by means of Kripke models. On the algebraic side, Mundici in [13] introduces states on MV-algebras. These operators can be interpreted as probabilistic measures over many-valued formulas. Moreover, as shown by Mundici in [15], states are also related to subjective probability. Indeed, he shows that a probabilistic assessment over many-valued events can be extended to a state iff there is no Dutch-book for it (cf. Section 2.3 for details). Thus, both states and probabilistic fuzzy logics have to do with probability over fuzzy events. However, so far these two approaches to probability have been investigated separately. In [7, 8], the present authors try to connect them. To this purpose, they introduce an algebraizable logic, called SFP(Ł,Ł), which extends FP(Ł,Ł) (see Section 2.1), and whose equivalent algebraic semantics is constituted by a variety of universal algebras, called SMV-algebras. These algebras are MV-algebras equipped with a unary operation whose properties somehow simulate the properties of states. The purpose of the present article is to connect the semantics of MV-algebras with a state to the semantics given by Kripke models, showing that both semantics are equivalent to a semantics given by a special kind of SMV-algebras, called σ -simple SMV-algebras. As a consequence, we establish some complexity theoretic results for the classes of SFP(Ł,Ł)-formulas which are 1-tautologies or 1-satisfiable (with respect to various kinds of semantics). © The Author, 2009. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected] doi:10.1093/logcom/exp013

2 Models for MV-Probabilistic Reasoning The article is organized as follows. In the next section, we recall the syntax of SFP(Ł,Ł) and we point out the improvements this logic brings, with respect to the weaker logic FP(Ł,Ł). In Section 2, we recall some preliminary definitions about MV-algebras with a state and SMValgebras, and we prove some properties of tensor product of MV-algebras. In Section 3, we introduce various kinds of semantics for SFP(Ł,Ł), namely, the SMV semantics given by general SMV-algebras, the standard semantics given by the so called σ -simple SMValgebras, the state semantics, given by the states on the Lindenbaum algebra of Łukasiewicz logic, and the Kripke semantics, given by Kripke models. For each kind of semantics, we investigate the set of 1-tautologies and the set of 1-satisfiable formulas. For the 1-tautologicity, we prove that the standard semantics, the state semantics and the Kripke semantics are mutually equivalent, and we do not know whether these semantics are equivalent to the SMV semantics or not. For the 1-satisfiability, the situation is more complex: we have two (non-equivalent) kinds of 1-satisfiability, a global one and a local one. The local one is called local 1-satisfiability, and the global one is called just 1-satisfiablity. For 1-satisfiability, we prove that both the Kripke semantics and the standard semantics are equivalent to the SMV semantics, but not to the state semantics; for the local 1-satisfiability, we prove that the state semantics, the Kripke semantics and the standard semantics are mutually equivalent, and we do not know whether they are also equivalent to the SMV semantics or not. Then, in Section 4, we establish some complexity results. In particular, we prove that the set of standard 1-tautologies with respect to either Kripke semantics or the standard semantics or the state semantics, as well as the set of 1-satisfiable, and locally 1-satisfiable formulas with respect to any of the above semantics, are in PSPACE.

2

Preliminaries

2.1 The logic SFP(Ł,Ł) The language of SFP(Ł,Ł) consists of a set Var = {p1 ,p2 ,...} of propositional variables, the constant 0 (for the falsity), the binary connective ⊕, the unary connective ¬ and the unary modality Pr. Formulas are defined by induction as usual: every propositional variable is a formula, 0 is a formula and, whenever ϕ and ψ are formulas, then ϕ ⊕ψ, ¬ϕ and Pr(ϕ) are formulas. The following connectives are definable: ϕ ψ is ¬(¬ϕ ⊕¬ψ) ϕ → ψ is ¬ϕ ⊕ψ ϕ ↔ ψ is (ϕ → ψ)(ψ → ϕ) ϕ ψ is ¬(ϕ → ψ) ϕ ∨ψ is (ϕ → ψ) → ψ ϕ ∧ψ is ¬(¬ϕ ∨¬ψ) We shall henceforth denote by SFP the set of formulas of SFP(Ł,Ł). Axioms are those of Łukasiewicz logic (cf. [9]) and the following shemata for the modality Pr: (1) (2) (3) (4)

Pr(0) ↔ 0 Pr(Pr(ϕ)⊕Pr(ψ)) ↔ Pr(ϕ)⊕Pr(ψ) Pr(ϕ → ψ) → (Pr(ϕ) → Pr(ψ)) Pr(ϕ ⊕ψ) ↔ [Pr(ϕ)⊕Pr(ψ (ϕ ψ))]1

1 In [8] the present authors gave a different axiomatization for SFP(Ł,Ł): the axiom 3 was substituted for Pr(¬ϕ) ↔ ¬Pr(ϕ). The two axiomatizations are equivalent, but this one is preferable because it allows to derive the rule of substitution of equivalents for Pr [from ϕ ↔ ψ, derive Pr(ϕ) ↔ Pr(ψ)] in an easier way.

Models for MV-Probabilistic Reasoning 3 Rules are modus ponens: from ϕ and ϕ → ψ, derive ψ, and necessitation: from ϕ, derive Pr(ϕ). The advantages of dealing with SFP(Ł,Ł) instead of FP(Ł,Ł), are the following: (a) SFP(Ł,Ł) is more expressive than FP(Ł,Ł). Indeed the set of well-founded formulas of FP(Ł,Ł) is a proper subset of SFP, because in FP(Ł,Ł) Pr(ϕ) is a formula iff ϕ is a formula without occurrences of Pr (i.e. it is not allowed to use the modality nested), and a Łukasiewicz connective  is not admitted in contexts like Pr(ϕ)ψ, where ψ is not in the form Pr(γ ). Moreover the algebraic models of SFP(Ł,Ł), the SMV-algebras, allow us to interpret formulas which do not have a natural interpretation in MV-algebras equipped with a state. (b) Whilst we do not know yet if FP(Ł,Ł) is complete with respect to the class of Kripke models, SFP(Ł,Ł) is strongly complete with respect to the class of SMV-algebras (cf. [8, Theorem 4.5]). (c) Whilst the connections between Kripke models and states have not been investigated in details, SMV-algebras are closely related to MV-algebras with a state.

2.2 MV-algebras and tensor product An algebra A = (A,⊕,¬,0,1) of type (2,1,0,0) is said to be an MV-algebra if A satisfies the following equations: (x ⊕y)⊕z = x ⊕(y⊕z) x ⊕y = y⊕x x ⊕0 = x x ⊕1 = 1 ¬1 = 0 and ¬0 = 1 ¬(¬x ⊕y)⊕y = ¬(¬y⊕x)⊕x The class of MV-algebras forms a variety, which will be denoted by MV. In any MV-algebra A, one can define the following operations: x → y = ¬x ⊕y, x y = ¬(x → y), x y = ¬(¬x ⊕¬y), x ↔ y = (x → y)(y → x), x ∨y = (x → y) → y and x ∧y = ¬(¬x ∨¬y). We shall henceforth use the following abbreviations: for every x ∈ A, and every n ∈ N, n nx stands for
x ⊕...⊕x  , and x stands for
x ...x   . n-times n-times

Any MV-algebra A can be equipped with the order relation ≤, defined, for all x,y ∈ A, by x ≤ y iff x → y = 1. An MV-algebra A is said to be linearly ordered (or an MV-chain), provided that the order ≤ is linear. Example 2.1 (1) Consider the real unit interval [0,1] equipped with (Łukasiewicz) operations ⊕ and ¬ defined, for all x,y ∈ [0,1], by x ⊕y = min{1,x +y}, and ¬x = 1−x. The resulting algebra [0,1]MV = ([0,1],⊕,¬,0,1) is an MV-chain. Actually, Chang (cf. [3]) proved that the variety MV is generated by [0,1]MV . [0,1]MV is called the standard MV-algebra.

4 Models for MV-Probabilistic Reasoning (2) Fix a k ∈ N, and let F(k) be the set of all the McNaughton functions (cf. [4]) from the hypercube [0,1]k into [0,1]. In other words, let F(k) be the set of all those functions f : [0,1]k → [0,1] which are continuous and piecewise linear with integer coefficients. The operations ⊕ and ¬ defined on F(k) by (f ⊕g)(x) = min{1,f (x)+g(x)},and ¬f (x) = 1−f (x), make the structure F(k) = (F(k),⊕,¬,0,1) an MV-algebra, where 0 and 1 denote the functions constantly equal to 0 and 1, respectively. Actually, F(k) is the free MV-algebra over k-free generators. Henceforth, F(ω) will denote the free MV-algebra over ω-free generators. Let A be an MV-algebra. Then a non-empty subset f of A is said to be a filter of A iff: (a) 1 ∈ f, (b) if x,y ∈ f, then x y ∈ f, and (c) if x ∈ f and y ≥ x, then y ∈ f. A filter f of an MV-algebra A is said to be proper, if f  = A. A filter m is said to be a maximal filter (or an ultrafilter) if m  = A, and if for any filter f such that f ⊇ m, then either f = A, or f = m. The set of all ultrafilters of an MV-algebra A will be henceforth denoted by M(A), or, when there is no danger of confusion, simply by M. It is well known (see [4] for instance) that the congruences lattice and the filters lattice of any MV-algebra A are mutually isomorphic, via the isomorphism which associates to every congruence θ the filter {x ∈ A | (x,1) ∈ θ }. Definition 2.2 Let A be an MV-algebra. The A is said to be: (i) Simple iff A is non-trivial, and {1} is its only proper filter. (ii) Semisimple iff the intersection of all its maximal filters is {1}. Clearly every simple MV-algebra also is semisimple, but not vice-versa. For instance, the standard MV-algebra [0,1]MV of Example 2.1(1), is simple, whilst the algebra F(ω) of Example 2.1(2), is semisimple but not simple. In fact any simple MV-algebra is, up to isomorphism, an MV-subalgebra of [0,1]MV , hence F(ω), being not linearly ordered, cannot be simple. On the other hand, if A is any MV-algebra and m ∈ M(A), then the quotient A/m is simple (cf. [4, Theorems 2.4.14, 2.5.7]). In [14], Mundici introduces the notion of MV-algebraic tensor product (or simply tensor product) between MV-algebras. Let us recall the main steps of this definition: let A, B and C be MV-algebras. A bimorphism from the direct product A×B of A and B into C, is a map β such that: (i) β(1,1) = 1, and for all a ∈ A, and b ∈ B, β(a,0) = β(0,b) = 0. (ii) For all a,a1 ,a2 ∈ A, and b,b1 ,b2 ∈ B, and for  ∈ {∧,∨}, β(a1 a2 ,b) = β(a1 ,b)β(a2 ,b), and β(a,b1 b2 ) = β(a,b1 )β(a,b2 ). (iii) For all a,a1 ,a2 ∈ A, and for all b,b1 ,b2 ∈ B, if a1 a2 = 0, then β(a1 a2 ,b) = β(a1 ,b) β(a2 ,b) = 0 and β(a1 ⊕a2 ,b) = β(a1 ,b)⊕β(a2 ,b). If b1 b2 = 0, then β(a,b1 b2 ) = β(a,b1 )β(a,b2 ) = 0 and β(a,b1 ⊕b2 ) = β(a,b1 )⊕β(a,b2 ). The tensor product of two MV-algebras A and B, is an MV-algebra A⊗B for which there exists a bimorphism β from A×B into A⊗B fulfilling the following universal property: for every bimorphism β  from A×B into an MV-algebra C, there is a unique homomorphism λ from A⊗B into C such that, for all (a,b) ∈ A×B, λ(β(a,b)) = β  (a,b). The tensor product of two MV-algebras A and B always exists and it is unique up to isomorphism (cf. [14, Theorem 3.2]). We shall henceforth write a⊗b instead of β(a,b). It follows from [14] that the maps a  → a⊗1 and b  → 1⊗b, respectively, are embeddings of A and of B into A⊗B.

Models for MV-Probabilistic Reasoning 5 In [14], Mundici also proves that in general semisimplicity is not preserved by the tensor product: there exists a semisimple MV-algebra A such that A⊗A is no longer semisimple (cf. [14, Theorem 3.3]). The following proposition shows that, in some particular cases, semisimplicity is preserved. Proposition 2.3 If A is a semisimple MV-algebra, then [0,1]MV ⊗A is semisimple. In particular, [0,1]MV ⊗F(ω) is semisimple. Proof. Let α ⊗a ∈ [0,1]MV ⊗A, and assume α ⊗a < 1⊗1. If α < 1, then there is a natural number n such that α n = 0 in [0,1]MV . Therefore, in [0,1]MV ⊗A, (α ⊗a)n ≤ (α ⊗1)n = 0⊗1 = 0. Hence the filter generated by α ⊗a is not proper, whence any maximal filter of [0,1]MV ⊗A entirely consists of elements of the form 1⊗a. Now A is semisimple, whence, if a < 1, there exists an m ∈ M(A) such that a  ∈ m. Therefore, letting m∗ = {1⊗b | b ∈ m}, we have that m∗ is an ultrafilter of [0,1]MV ⊗A such that 1⊗a  ∈ m∗ . It follows that the intersection of all the ultrafilters of [0,1]MV ⊗A is the singleton {1⊗1}, whence [0,1]MV ⊗A is semisimple.
The algebra [0,1]MV ⊗F(ω) will play a central role in this article. We exploit another property of it. Let FR (ω) denote the Lindenbaum algebra of the logic RŁ obtained by extending Łukasiewicz language by a constant α for every α ∈ [0,1], and by adding all the bookkeeping axioms: (α ⊕β) ↔ min{1,α +β} and¬(α) ↔ 1−α. The algebraic counterpart of RŁ is the variety generated byFR (ω), whose members will be henceforth called RŁ-algebras. Proposition 2.4 The algebra FR (ω) is isomorphic to an MV-subalgebra of [0,1]MV ⊗F(ω). Proof. First of all note that F(ω) and [0,1]MV are MV-subalgebras of both FR (ω) and [0,1]MV ⊗ F(ω). Moreover, in [0,1]MV ⊗F(ω) we can interpret every constant α as α ⊗[1], so obtaining an algebra in the signature of FR (ω). By the universal property of free algebras, there is a unique homomorphism h from FR (ω) into [0,1]MV ⊗F(ω) (considered as a model of RŁ, with α interpreted as α ⊗[1]), such that, for each generator [p], h([p]) = 1⊗[p]. Clearly, for each α ∈ [0,1], h([α]) = α ⊗[1]. In order to settle the claim, it is sufficient to show that h is one-one, or equivalently that, if [ψ] < [1], then h([ψ]) < 1⊗[1]. Claim 2.4.1 FR (ω) is semisimple. Proof of Claim 2.4.1. Let [0,1]R MV be the algebra obtained from [0,1]MV by adding a constant α for every α ∈ R, to be interpreted as α. Then [0,1]R MV generates the variety of RŁ-algebras. Therefore, )ω

FR (ω) is the subalgebra of [0,1]MV MV [where ([0,1]MV )ω denotes the direct product of ω copies of [0,1]MV ] generated by the projections, whence FR (ω) is a subdirect product of simple algebras, and hence it is semisimple. Since [ψ] < [1], by the semisimplicity of FR (ω), there is a maximal filter m of FR (ω), such that [ψ]  ∈ m. Let h be the homomorphism associated to m. Then h maps FR (ω) into an algebra isomorphic to [0,1]MV , and h ([ψ]) < 1. Hence, there is a homomorphism h from FR (ω) into [0,1]R MV such that h ([ψ]) < 1. ([0,1]

6 Models for MV-Probabilistic Reasoning Let h1 : FR (ω) → [0,1]MV ⊗F(ω) and h2 : [0,1]MV ⊗F(ω) → [0,1]MV ⊗F(ω) be defined as: h1 : [γ ]  → h ([γ ])⊗[1], and h2 : α ⊗[γ ]  → (α ·h ([γ ]))⊗[1]. Both h1 and h2 are MV-homomorphisms. In fact, let g : [0,1]MV ⊗F(ω) → [0,1]MV be so defined: g(α ⊗[γ ]) = h ([γ ])·α [the definition of g is meaningful because F(ω) is an MV-subalgebra of FR (ω)]. Then g is an MV-homomorphism. Indeed the map β from [0,1]MV ×F(ω) into [0,1]MV such that β(α,[γ ]) = α ·h ([γ ]) is a bimorphism, whence, by the universal property of tensor product, g is that unique homomorphism such that g(α ⊗[γ ]) = β(α,[γ ]). Let i be the embedding of [0,1]MV into [0,1]MV ⊗F(ω), defined by i(α) = α ⊗[1], h1 = i◦h and h2 = i◦g. Then h1 and h2 are MVhomomorphisms. Now let [p] be a generator of FR (ω). Then h1 ([p]) = h ([p])⊗[1] = h2 (1⊗[p]) = h2 (h([p])), whence h1 = h2 ◦h, and hence, being h1 ([ψ]) < 1⊗[1], also h([ψ]) < 1⊗[1].


2.3 States on MV-algebras Let A be an MV-algebra. A state on A (cf. [13]) is a map s : A → [0,1] satisfying the following properties: (i) s(1) = 1. (ii) For all a,b ∈ A, whenever ab = 0, then s(a⊕b) = s(a)+s(b). A state s is said to be faithful if s(x) = 0, implies x = 0. States constitute a generalization of the notion of probability measure on MV-algebras. Moreover, states are related to de Finetti’s coherence criterion: according to de Finetti (cf. [5]), a probabilistic assessment χ : P(ϕ1 ) = α1 ,...,P(ϕn ) = αn of classical events ϕ1 ,...,ϕn is said to be coherent iff there is no system of reversible bets on the events which leads to a win independently on the truth of ϕ1 ,...,ϕn . In other words, the assessment χ is coherent iff for every λ1 ,...,λn ∈ R, there is a valuation V such that n  λi (αi −V (ϕi )) ≥ 0, i=1

The celebrated de Finetti’s Theorem states that an assessment χ is coherent iff it can be extended to a finitely additive measure on the Boolean algebra of all events. In [10, Corollary 4.3], Kühr and Mundici extend de Finetti’s coherence criterion to any (algebraizable, cf. [1]) logic L whose equivalent algebraic semantics is given by the algebraic variety generated by the algebra ([0,1], ), where denotes a set of continuous operations on [0,1]. Hence, their result applies to Łukasiewicz logic, to every extension of Łukasiewicz logic with rational or real constants (as for instance RPL, cf. [9], and to the logic RŁ described in the previous section), and to the PMV+ logic (cf. [11, 12]). Theorem 2.5 [10] Equip the unit interval [0,1] with a set of continuous operations, denote by [0,1] the algebra so obtained, and let V([0,1] ) be the equational class generated by [0,1] . For each algebra A ∈ V([0,1] ), let W be the set of homomorphisms from A into [0,1] . Then, for all A = {a1 ,...,an } ⊆ A, a map χ : A → [0,1] is coherent over A iff χ is extendible to a convex combination of at most n+1 elements of W .

Models for MV-Probabilistic Reasoning 7 In other words, if ϕ1 ,...,ϕn are formulas of L , then an assessment χ : ϕi  → αi (for i = 1,...,n) is coherent iff there exist at most n+1 valuations v1 ,...,vn+1 into [0,1] and n+1 real numbers λ1 ,...,λn+1 , such that n+1 

λi = 1,and for allj ∈ {1,...,n},αj =

i=1

n+1 

λi ·vi (ϕj ).

i=1

2.4 SMV-algebras Definition 2.6 An SMV-algebra (cf. [7, 8]) is an algebra (A,σ ) such that: • A is an MV-algebra, • for all a,b ∈ A, the operation σ satisfies the following equations: (σ 1) σ (0) = 0. (σ 2) σ (¬a) = ¬σ (a). (σ 3) σ (σ (a)⊕σ (b)) = σ (a)⊕σ (b). (σ 4) σ (a⊕b) = σ (a)⊕σ (b(ab)). An SMV-algebra is said to be faithful if it satisfies the quasi equation: σ (a) = 0 implies a = 0. The following proposition collects some properties of SMV-algebras. A proof can be found in [8]. Proposition 2.7 In any SMV-algebra (A,σ ), the following conditions hold: (i) (ii) (iii) (iv) (v)

σ (1) = 1. If a ≤ b, then σ (a) ≤ σ (b). σ (a⊕b) ≤ σ (a)⊕σ (b), and if ab = 0, then σ (a⊕b) = σ (a)⊕σ (b). σ (σ (a)) = σ (a). The image σ (A) of A under σ is the domain of an SMV-subalgebra of (A,σ ), and if (A,σ ) is subdirectly irreducible, then σ (A) is an MV-chain.

In [8] we introduce a way to define, starting from an SMV-algebra (A,σ ), a state on A, and vice-versa, to define an SMV-algebra (T ,σ ) starting from a state on an MV-algebra A, in such a way that A is an MV-subalgebra of T . We recall the main steps of this construction: (1) let (A,σ ) be any SMValgebra and let m be a maximal filter of σ (A). Then σ (A)/m is a simple MV-algebra and it can be embedded into [0,1]MV via an embedding h. The map h◦ηm ◦σ , where ηm denotes the canonical homomorphism of σ (A) into σ (A)/m, is a state on A. (2) Let A be an MV-algebra and let s be a state on A. Let T = [0,1]MV ⊗A, and let σs : T → T be the map defined by σs (α ⊗a) = α ·s(a)⊗1. Then A is an MV-subalgebra of T , and (T ,σs ) is an SMV-algebra. (Further details can be found in [8, Theorem 5.1, Theorem 5.3].) We conclude this section introducing a class of SMV-algebras which will play an important role in the remaining of this article. Definition 2.8 An SMV-algebra (A,σ ) is said to be σ -simple if A is a semisimple MV-algebra, and σ (A) is a simple MV-algebra.

8 Models for MV-Probabilistic Reasoning Example 2.9 Let s be a state on F(ω). Then the SMV-algebra ([0,1]MV ⊗F(ω),σs ), where σs is defined by σs (α ⊗[ψ]) = α ·s([ψ])⊗[1] is σ -simple (the semisimplicity of [0,1]MV ⊗F(ω) follows from Proposition 2.3). Proposition 2.10 Let (A,σ ) be a σ -simple SMV-algebra. Then for every a ∈ A, the equation 1⊗σ (a) = σ (a)⊗1 holds in [0,1]MV ⊗A, where we have identified σ (A) with its isomorphic MV-subalgebra of [0,1]MV . Proof. Assume, by way of contradiction, that for some a ∈ A, 1⊗σ (a)  = σ (a)⊗1. By Proposition 2.3, [0,1]MV ⊗A is semisimple, whence there is a maximal MV-congruence θ which separates 1⊗σ (a) and σ (a)⊗1. The quotient ([0,1]MV ⊗A)/θ is simple, whence it is isomorphic to (and will be identified with) an MV-subalgebra of [0,1]MV . Up to isomorphism, the map h : α ⊗a  → (α ⊗a)/θ , is a homomorphism of [0,1]MV ⊗A into [0,1]MV . Let h be the map on A defined by h (a) = h(a⊗1). The h is a homomorphism, being the composition of h and of the embedding i2 : a  → 1⊗a. Claim 2.10.1 h(α ⊗a) = α ·h (a) = α ·h(a⊗1). Proof of Claim 2.10.1. The claim can be easily proved if α is a rational number. If α is irrational, it follows from the monotonicity of h and ⊗. Now h must be identical on the elements of σ (A), because σ (A) is an MV-subalgebra of [0,1]MV . Therefore, from Claim 2.10.1, we obtain: h(σ (a)⊗1) = 1·h (σ (a)) = σ (a) = σ (a)·h (1) = h(1⊗σ (a)).


3

Comparing the semantics

SFP(Ł,Ł) is the algebraizable logic (in the sense of Blok and Pigozzi, cf. [1]) whose equivalent algebraic semantic is constituted by the variety of SMV-algebras. Thus, in particular the connectives and the constants of SFP(Ł,Ł) are the operations and the constants of SMV-algebras. Moreover, recalling that a valuation of SFP(Ł,Ł) into an SMV-algebra A is a homomorphism from the algebra of formulas of SFP(Ł,Ł) into A, we have that the theorems of SFP(Ł,Ł) are precisely those formulas which are evaluated to 1 by any valuation in any SMV-algebra. The set of SFP(Ł,Ł)-formulas will be henceforth denoted by SFP. Moreover we shall refer to semantics via SMV-algebras as the SMV semantics. Definition 3.1 A formula φ ∈ SFP is said to be (i) an SMV-tautology (φ ∈ SMVTAUT ) iff for every SMV-algebra (A,σ ) and for every valuation v in (A,σ ), v(φ) = 1. (ii) SMV-1-satisfiable (φ ∈ SMV 1SAT ) iff there are an SMV-algebra (A,σ ), and a valuation v in (A,σ ) such that v(φ) = 1. In [8, Theorem 4.5], it is shown that SMVTAUT coincides with the set of theorems of SFP(Ł,Ł). We now introduce other kinds of semantics for formulas in SFP.

Models for MV-Probabilistic Reasoning 9

3.1 The standard semantics While the standard model of MV-algebras is [0,1]MV , we cannot consider this model as a standard representative of SMV-algebras, because the only internal state making [0,1]MV an SMV-algebra is the identity. Our idea is that standard models of SMV-algebras should be MV-algebras of functions on [0,1], with MV-operations defined pointwise, equipped with an internal state σ representing a sort of integral (hence the image of σ should be a subset of [0,1], or equivalently a set of constant [0,1]valued functions). According to this idea, standard SMV-algebras should be precisely the σ -simple SMV-algebras. Thus we interpret formulas in SFP into σ -simple SMV-algebras (A,σ ). From an interpretation into a σ -simple SMV-algebra, we can obtain interpretations in [0,1] combining them with a homomorphism h from A into [0,1]MV . The concept of 1-satisfiable formula splits into two non-equivalent classes, according as we use h or not. Definition 3.2 Let φ ∈ SFP. Then φ is said to be (i) a standard tautology (φ ∈ StdTAUT ) iff for every σ -simple SMV-algebra (A,σ ), and for every valuation v in (A,σ ), v(φ) = 1. (ii) Standard-1-satisfiable (φ ∈ Std1SAT ) iff there is a valuation v into a σ -simple SMV-algebra (A,σ ) such that v(φ) = 1. (iii) Locally standard positively satisfiable (φ ∈ lStdposSAT ) iff there is a valuation v into a σ -simple SMV-algebra (A,σ ) and an MV-homomorphism h from A into [0,1]MV , such that h(v(φ)) > 0. (iv) Locally standard 1-satisfiable (φ ∈ lStd1SAT ) iff there are a valuation v into a σ -simple SMValgebra (A,σ ) and an MV-homomorphism h from A into [0,1]MV , such that h(v(φ)) = 1.

3.2 The state semantics Since states are not internal operations, it is not possible to interpret SMV-formulas directly by means of states. Thus, we first associate to any state s on the countably generated free MV-algebra F(ω) an internal state σs on [0,1]MV ⊗F(ω), defined as in Example 2.9, which makes ([0,1]MV ⊗F(ω),σs ) an SMV-algebra, and then we evaluate SFP-formulas in it. Moreover σs somehow extends s, in the sense that σs (1⊗[φ]) = s([φ])⊗[1]. The most natural valuation in [0,1]MV ⊗F(ω), denoted by us , is inductively defined as follows: (i) (ii) (iii)

us (p) = 1⊗[p] for p atomic; us commutes with all MV-connectives; us (σ (φ)) = σs (us (φ)).

Once again, 1-satisfiability splits into two cases, according as we use a homomorphism h into [0,1] or not. Definition 3.3 Let φ ∈ SFP. Then φ is said to be (i) A state tautology (φ ∈ STTAUT ) iff for every state s on F(ω), us (φ) = 1⊗[1]. (ii) State-1-satisfiable (φ ∈ ST 1SAT ) iff there is a state s on F(ω), such that us (φ) = 1⊗[1]. (iii) Locally state-positively satisfiable (φ ∈ lSTposSAT ) iff there is a state s on F(ω) and a homomorphism h from [0,1]MV ⊗F(ω) into [0,1]MV , such that h(us (φ)) > 0. (iv) Locally state-1-satisfiable (φ ∈ lST 1SAT ) iff there is a state s on F(ω) and a homomorphism h from [0,1]MV ⊗F(ω) into [0,1]MV , such that h(us (φ)) = 1.

10 Models for MV-Probabilistic Reasoning

3.3 The Kripke semantics Let Var be the set of propositional variables in SFP. A probabilistic Kripke model is a triple K = (X,e,µ) where X is a finite or countable set of nodes, e is a map associating  to every pair (x,p) ∈ X ×Var, an element e(x,p) ∈ [0,1] and µ : X → [0,1] is a function such that x∈X µ(x) = 1. For every x ∈ X and every φ ∈ SFP, the truth-value of φ in K at the node x, denoted by φK,x , is inductively defined as follows: if φ is a propositional variable p, then  pK,x = e(x,p); 0K,x = 0; ·K,x commutes with all Łukasiewicz connectives; σ (ψ)K,x = y∈X ψK,y ·µ(y). Notice that, for every ψ ∈ SFP, σ (ψ)K,x does not depend on x, in the sense that, for all x,y ∈ X, σ (ψ)K,x = σ (ψ)K,y . We say that φ is valid in a probabilistic Kripke model K (and we write K |= φ), iff for all x ∈ X, φK,x = 1. Definition 3.4 Let φ ∈ SFP. Then φ is said to be: (i) a Kripke tautology (φ ∈ KRTAUT ) iff for every probabilistic Kripke model K = (X,e,µ), K |= φ. (ii) Kripke 1-satisfiable (φ ∈ KR1SAT ) iff there is a probabilistic Kripke model K such that K |= φ. (iii) Locally Kripke positively satisfiable (φ ∈ lKRposSAT ) iff there are a probabilistic Kripke model K = (X,e,µ) and node x ∈ X, such that φK,x > 0. (iv) Locally Kripke 1-satisfiable (φ ∈ lKR1SAT ) iff there are a probabilistic Kripke model K = (X,e,µ) and a node x ∈ X such that φK,x = 1. Remark 3.5 For MOD ∈ {Std,ST ,KR}, MOD1SAT ⊆ lMOD1SAT . We show that all inclusions are proper. We prove the claim for Std1SAT and lStd1SAT , the proofs for the other cases being similar (notice in particular that, letting p any propositional variable, then p ∈ lST 1SAT , but p  ∈ ST 1SAT ). Let φ = p∧σ (¬p). Clearly φ ∈ Std1SAT . Now let A be the MV-algebra consisting of all functions from [0,1] into [0,1] having only a finite (possibly empty) set of discontinuity points, with MV-operations defined pointwise. Let σ be the Lebesgue integral on A, let f be the characteristic function of {0}, and let h be the MV-homomorphism from A into [0,1]MV defined, for all g ∈ A, by h(g) = g(0). Finally, let v be a valuation in (A,σ ) such that v(p) = f . Then h(v(σ (¬p))) = 1 and h(v(p)) = 1, whence φ ∈ lStd1SAT . Theorem 3.6 Let φ ∈ SFP, and let α ∈ [0,1]. Then the following are equivalent: (i) There are σ -simple SMV-algebra (A,σA ), valuation v into (A,σA ), and homomorphism h : A → [0,1]MV , such that h(v(φ)) = α. (ii) There are state s on F(ω), and homomorphism h : [0,1]MV ⊗F(ω) → [0,1]MV , such that h(us (φ)) = α. (iii) There are probabilistic Kripke model K = (X,e,µ) and a node x ∈ X such that φK,x = α. Proof. (i) ⇒ (ii). Let (A,σA ), h and v be as in (i). We define s : F(ω) → [0,1] letting, for every [ψ] ∈ F(ω), s([ψ]) = σA (v(ψ)). s is a state on F(ω), whence the map σs : [0,1]MV ⊗F(ω) → [0,1]MV ⊗F(ω) defined as in Example 2.9, makes the algebra ([0,1]MV ⊗F(ω),σs ) an SMV-algebra (cf. [8, Theorem 5.3]). Now let σ  : [0,1]MV ⊗A → [0,1]MV ⊗A be a map so defined: for all α ⊗a ∈ [0,1]MV ⊗A, σ  (α ⊗ a) = α ·σA (a). Then ([0,1]MV ⊗A,σ  ) is an SMV-algebra (cf. [8, Theorems 5.2, 5.3]).

Models for MV-Probabilistic Reasoning 11 Consider the map v0 : [0,1]MV ⊗F(ω) → [0,1]MV ⊗A such that, for all α ⊗[ψ] ∈ [0,1]MV ⊗F(ω), v0 (α ⊗[ψ]) = α ⊗v(ψ). Claim 3.6.1 The map v0 is an SMV-homomorphism. Proof of Claim 3.6.1. The map v0 : [0,1]MV ×F(ω) → [0,1]MV ⊗A, defined by v0 (α,[ψ]) = α ⊗v(ψ) is a bimorphism, whence there is a unique homomorphism v0 from [0,1]MV ⊗F(ω) into [0,1]MV ⊗A such that v0 (α,[ψ]) = v0 (α ⊗[ψ]). Then v0 = v0 is an MV-homomorphism. Now we prove that v0 is compatible with σ : v0 (σs (α ⊗[ψ])) = v0 (α ·s([ψ])⊗[1]) = α ·s([ψ])⊗1 = α ·σA (v(ψ))⊗1 = σ  (α ⊗v(ψ)) = σ  (v0 (α ⊗[ψ])). This settles Claim 3.6.1. Claim 3.6.2 For all ψ, v0 ◦us (ψ) = 1⊗v(ψ). Proof of Claim 3.6.2. The proof is by induction on ψ. If ψ is σ -free, then v0 (us (ψ)) = v0 (1⊗[ψ]) = 1⊗v(ψ). If ψ = σ (γ ), then v0 (us (σ (γ ))) = σ  (v0 (us (γ ))). By the inductive hypothesis, v0 (us (γ )) = 1⊗v(γ ), whence (also using Proposition 2.10) σ  (1⊗v(γ )) = σA (v(γ ))⊗1 = 1⊗σA (v(γ )) = 1⊗ v(σ (γ )). Then Claim 3.6.2 holds. Finally let h∗ be the map from [0,1]MV ⊗A into [0,1]MV , defined as follows: for all α ⊗a ∈ [0,1]MV ⊗A, h∗ (α ⊗a) = α ·h(a). From [8, Theorem 5.3], it follows that h∗ is a well-defined MVhomomorphism. Moreover, from Claim 3.6.2, h∗ (v0 (us (φ))) = h∗ (1⊗v(φ)) = h(v(φ)) = α. We have shown that the following diagram commutes. s

u / ([0,1]MV ⊗F(ω),σs ) SFP PP PPP PP1⊗v PPP v0 v PPP (   1⊗IdA / ([0,1]MV ⊗A,σ  ) (A,σA ) PPP PPPh PPP h∗ PPP (  [0,1]MV

(where h and h∗ are MV-homomorphisms). Then (ii) holds. (ii)⇒(i). Let s be a state on F(ω), and let h be a homomorphism from [0,1]MV ⊗F(ω) into [0,1]MV , such that h(us (φ)) = α. Let σs be the internal state on [0,1]MV ⊗F(ω) defined as in the definition of state semantics. Then ([0,1]MV ⊗F(ω),σs ) is a σ -simple SMV-algebra. Taking v = us , we immediately have h(v(φ)) = α. (ii)⇒(iii). Let s be a state on F(ω), and let h be a homomorphism from [0,1]MV ⊗F(ω) into [0,1]MV such that h(us (φ)) = α. For every subformula δ of φ of the form σ (γ ), let αγ = h(us (σ (γ ))). Let us

12 Models for MV-Probabilistic Reasoning write every subformula δ of φ as an MV-combination of (propositional variables and) formulas of the form σ (γ ), and let δ ∗ be the result of substituting in δ each subformula σ (γ ) by αγ . Then δ ∗ (whence, in particular, φ ∗ ) is a formula of RŁ. Let σ (γ1 ),...,σ (γr ) be a list of all subformulas of φ beginning with σ . Then the assessment σ (γi∗ ) = αγi ( i = 1,...,r), is coherent, whence, from Theorem 2.5, there are (at most) r +1 valuations  v1 ,...,vr+1 of RŁ and (at most) r +1 positive real numbers λ1 ,...,λr+1 such that r+1 i=1 λi = 1, and  ∗ ) for all j = 1,...,r +1. αγj = r+1 λ v (γ i=1 i i j Let v be the valuation defined by v(p) = h(us (p)) = h(1⊗[p]), and set X = {v,v1 ,...,vr+1 }. Also define µ : X → [0,1] by µ(vi ) = λi (i = 1,...,r +1), and µ(v) = 0 . Finally let, for every x ∈ X, and for every propositional variable p, e(x,p) = x(p). Then K = (X,e,µ) is a probabilistic Kripke model, such that φK,v = h(us (φ)) = α. (iii)⇒(i). Let K = (X,e,µ) be a probabilistic Kripke model, and let x ∈ X be such that φK,x = α. Let A = [0,1]XMV be the MV-algebra of all functions from X to [0,1]MV with MV-operations defined pointwise. Moreover, for every a = ax | x ∈ X ∈ A, let σA (a) be the function constantly equal to  x∈X ax ·µ(x). Then (A,σA ) is a σ -simple SMV-algebra. Finally, let v be the (A,σA )-valuation defined, for every propositional variable p and for every x ∈ X, by v(p)x = e(x,p). By induction on φ we can see that for every subformula γ of φ and for every x ∈ X, v(γ )x = γ K,x . Let us prove the induction step corresponding to σ : v(σ (ψ))x = σ A (v(ψ))x  = y∈X v(ψ)y ·µ(y)  = y∈X ψK,y ·µ(y) = σ (ψ)K,x . Finally, define h : A → [0,1]MV by h(a) = ax . Then, h is a homomorphism, and h(v(φ)) = w(φ)x = φK,x = α.
Corollary 3.7 Let φ ∈ SFP. Then the following are equivalent: (i) (ii) (iii) (vi) (v) (vi)

φ ∈ StdTAUT . φ ∈ STTAUT . φ ∈ KRTAUT . ¬φ  ∈ lStdposSAT . ¬φ  ∈ lSTposSAT . ¬φ  ∈ lKRposSAT .

Proof. The claim follows from Theorem 3.6, from the fact that φ ∈ MODTAUT iff ¬φ  ∈ lMODposSAT , for MOD being either Std, ST or KR, and from the fact that if either v is a valuation into a σ -simple SMV-algebra (A,σ ) such that v(ψ) < 1, or s is a state on Fω such that us (ψ) < 1, then there is a homomorphism h from A (from [0,1]MV ⊗F(ω), respectively) into [0,1]MV such that h(v(ψ)) < 1 (h(us (ψ)) < 1, respectively).
Corollary 3.8 Let φ ∈ SFP. Then the following are equivalent: (i) φ ∈ lStd1SAT . (ii) φ ∈ lST 1SAT . (iii) φ ∈ lKR1SAT .




Models for MV-Probabilistic Reasoning 13 For 1-satisfiability we have a different result: the state semantics is different from the other three, as e.g. the formula p is not in ST 1SAT , but it clearly belongs to Std1SAT ,SMV 1SAT and KR1SAT . However, SMV semantics coincides with standard semantics and with Kripke semantics as regards to 1-satisfiability. Theorem 3.9 Let φ ∈ SFP. Then the followings are equivalent: (i) φ ∈ Std1SAT . (ii) φ ∈ KR1SAT . (iii) φ ∈ SMV 1SAT . Proof. (i)⇒(ii). Let (A,σ ) be a σ -simple SMV-algebra and let v0 be a valuation in (A,σ ) such that v0 (φ) = 1. Let s be the state on F(ω) defined as follows: for every [γ ] ∈ F(ω), s([γ ]) = σ (v0 (γ )). Up to isomorphism, we can (and we will) identify σ (A) with its isomorphic subalgebra of [0,1]MV . Thus, σ may be regarded as a state on A. Let, for every subformula of φ of the form σ (γ ), αγ = s([γ ]), and let us write every subformula δ of φ as an MV-combination of propositional variables and formulas of the form σ (γ ). Let δ ∗ be the result of substituting in δ each subformula σ (γ ) by αγ . Then δ ∗ is a formula of RŁ. As in the proof of Theorem 3.6, let σ (γ1 ),...,σ (γr ) be a list of all subformulas of φ which begin with σ . Then the assessment consisting of all conditions σ (γi∗ ) = αγi (with i = 1,...,r), is coherent, whence, by Theorem 2.5, there are (at most) r +1 valuations v1 ,...,vr+1 of RŁ in [0,1]MV and (at most) r +1 positive real numbers λ1 ,...,λr+1 such  ∗ that λ1 +...+λr+1 = 1 and αγj = r+1 i=1 λi vi (γj ) for j = 1,...,r. Note that αφ = σ (v0 (φ)) = σ (1) = 1. Therefore, since λi > 0 for i = 1,...,r +1, we obtain v0 (φ ∗ ) = ... = vr+1 (φ ∗ ) =1. Now let X = v0 ,v1 ,...,vr+1 , µ(vi ) = λi for i = 1,...,r +1, and µ(v0 ) = 0. Finally, let for every propositional variable p and for i = 0,...,r +1, e(vi ,p) = vi (p). Then K = (X,e,µ) is a Kripke model. Moreover by induction on δ, we see that for every SFP(Ł,Ł) subformula δ of φ and for j = 0,...,r +1, we have δK,vj = vj (δ ∗ ). The only non-trivial step corresponding to σ is proved as follows: since σ (δ)∗ = αδ , for j = 0,...,r +1 we have: r+1 r+1   σ (δ)K,vj = µ(vi )·δK,vi = λi vi (δ ∗ ) = αδ = vj (δ ∗ ). i=0

Thus, for j = 0,...,r +1, φK,vj = vj

i=0

(φ ∗ ) = 1.

It follows that K |= φ.

(ii)⇒(i). Let K = (X,e,µ) be a Kripke model such that (X,e,µ) |= φ. Let A = [0,1]XMV and let for each a = ay : y ∈ X ∈ [0,1]XMV , σµ (a) be the constant function from X into [0,1], defined, for every y ∈ X, by  (σµ (a))y = µ(z)·az . z∈X

Then (A,σµ ) is a σ -simple SMV-algebra. Finally, let v be the valuation in (A,σµ ) defined, for every propositional variable p and for every y ∈ X, by v(p)y = e(y,p). Then for all y ∈ X we have v(φ)y = φK,y = 1. Hence v(φ) = 1, as desired. (i)⇒(iii) is trivial.

14 Models for MV-Probabilistic Reasoning (iii)⇒(i). Let (A,σ ) be an SMV-algebra, and let v be a valuation into (A,σ ) such that v(φ) = 1. Let m ∈ M(σ (A)), and let n be the filter of A, generated by m. Then n is an MV-filter, closed under σ (i.e., n is a σ -filter in the terminology of [7, 8]). Indeed, if b ∈ n, then there is a σ (a) ∈ m such that σ (a) ≤ b, whence σ (a) ≤ σ (b) (recall that any internal state is monotone and idempotent, Proposition 2.3), and σ (b) ∈ n. Now let (An ,σn ) be the quotient of (A,σ ) modulo n. Then σn (An ) is a simple MValgebra, because it is isomorphic to σ (A) modulo m, which is a maximal filter. Finally, let p denote the intersection of all the maximal filters of An . If a ∈ p, then, for every k ∈ N, k¬(a) ≤ a, whence k¬(σn (a)) ≤ σn (a), and σn (a) ∈ p. Thus p is a σ filter. Moreover if a ∈ p, then σn (a) = 1 because σn (A) is simple. Therefore, the quotient (Ap ,σp ) of (An ,σn ) modulo p is a σ -simple SMV-algebra, which is a quotient of (A,σ ) modulo the composition of the homomorphisms induced by the filters n and p. Call h such a composition. Then h◦v is a valuation into a σ -simple SMV-algebra such that h◦v(φ) = 1.
Corollary 3.10 Let φ ∈ SFP, and let σ (γ1 ),...,σ (γr ) be an enumeration of its subformulas which begin with σ . Then: (1) For every Kripke model K = (X,e,µ) and for every x ∈ X, there exists a Kripke model K  = (X  ,e ,µ ), with |X  | = r +2, and a node x  ∈ X  , such that φK,x = φK  ,x . (2) φ ∈ KR1SAT iff there exists a Kripke model K  with at most r +2 nodes, such that K  |= φ. (3) The following are equivalent: (3a) φ ∈ KR1SAT ; (3b) σ (φ) ∈ lKR1SAT ; (3c) σ (φ) ∈ KR1SAT . (4) φ ∈ KR1SAT iff there exist a σ -simple and faithful SMV-algebra (A,σ ), and a valuation v into (A,σ ), such that v(φ) = 1. Proof. (1) For the non-trivial direction, let K = (X,e,µ) be a Kripke model, and let x be an element of X such that φK,x = α ∈ [0,1]. By Theorem 3.6, (iii)⇒(ii), there exist a state s on F(ω), and an MVhomomorphism h : [0,1]MV ⊗F(ω) → [0,1]MV , such that h(us (φ)) = α. Moreover, inspection on the proof of Theorem 3.6, (ii)⇒(iii), shows that h(us (φ)) = α implies the existence of a Kripke model K  = (X  ,e ,µ ), with at most r +2 nodes, such that for some x  ∈ X  , φK  ,x = α. (2) For the non-trivial direction, by Theorem 3.9, φ ∈ KR1SAT is equivalent to φ ∈ Std1SAT . Inspection on the proof of Theorem 3.9, (i)⇒(ii), shows that φ ∈ Std1SAT , implies the existence of a Kripke model K = (X,e,µ) with at most r +2 nodes, such that for every x ∈ X, e(x,φ) = 1. (3) The direction (3a)⇒(3b) is clear. To prove  (3b)⇒ (3a), let K = (X,e,µ) be a Kripke model such that, for some node y ∈ X, σ (φ)K,y = x∈X µ(x)·φK,x = 1. Let X  = {x ∈ X | µ(x) > 0}, and  let K  = (X  ,e ,µ ) be the Kripke  model resulting from K, restricting the nodes to be in X . Then, for all x  ∈ X, µ (x  ) > 0, and x ∈X  µ (x  )·φK  ,x = 1. This is only possible if φK  ,x = 1 for every x  ∈ X  , whence φ ∈ KR1SAT . Finally note that (3a)⇒(3c) holds because σ (1) = 1 is always satisfied by any state σ , and (3c)⇒(3b) easily follows from Remark 3.5. (4) Let K = (X,e,µ) be a Kripke model such that, for every x ∈ X, φK,x = 1. As in the previous proof of (3b)⇒(3a), let K  = (X  ,e ,µ ) be the Kripke model obtained deleting from X all those nodes xi such that µ(xi ) = 0. Then K  |= φ. Theorem 3.9, (ii)⇒(i), states that K  |= φ implies the

Models for MV-Probabilistic Reasoning 15 existence of a σ -simple SMV-algebra (A,σ ), and a valuation v into (A,σ ), such that v(φ) = 1. Moreover, inspection on the proof of Theorem 3.9 shows that σ (a) = 0 iff a = 0, whence (A,σ ) is faithful.


4

Complexity issues

In this section, we prove a PSPACE containment result for SMV 1SAT and for all the classes MOD1TAUT , MOD1SAT and lMOD1SAT , where MOD ∈ {Std,ST ,KR}. Let φ be in SFP, and let ψ1 ,...,ψt be an enumeration of all its subformulas and, among ψ1 ,...,ψt , let σ (γ1 ),...,σ (γr ) be all those formulas, beginning with σ . j For i = 1,...,t, and j = 1,...,r +2, let xψi and zj be mutually distinct variables, and let φ F be the formula in the language of ordered fields, expressing the following: j

(a) For i = 1,...,t and j = 1,...,r +2, 0 ≤ xψi ≤ 1. (b) For j = 1,...,r +2, zj ≥ 0, and z1 +···+zr+2 = 1. j j j (c) For i,h,k = 1,...,t and j = 1,...,r +2, if ψi = ψh ⊕ψk , then either xψh +xψk ≥ 1 and xψi = 1, or j

j

j

j

j

xψh +xψk < 1 and xψi = xψh +xψk .

j

j

(d) For i,h = 1,...,t and j = 1,...,r +2, if ψi = ¬ψh , then xψi = 1−xψh . (e) For i = 1,...,r and j = 1,...,r +2, j

xψh = i

r+1 

m zm ·xψ , k i

m=1

where hi and ki are such that σ (γi ) = ψhi , and γi = ψki Theorem 4.1 ∗ be an abbreviation for ∃x 1 ...∃x r+2 and ∃z∗ be Let q be such that φ = ψq , and let, for i = 1,...,t, ∃xψ ψi ψi i an abbreviation for ∃z1 ...∃zr+2 . Then: (1) φ ∈ KR1TAUT = Std1TAUT = ST 1TAUT iff the formula  ∗ ∗  F φ TAUT = ∃z∗ ∃xψ φ ∧ ...∃xψ 1 t

r+2 

 xψq < 1 j

j=1

is false in the real-ordered field R. (2) φ ∈ lKR1SAT = lStd1SAT = lST 1SAT iff the formula  ∗ ∗  F φ ∧ ...∃xψ φ lSAT = ∃z∗ ∃xψ 1 t

r+2 

 xψq = 1 j

j=1

is true in the real-ordered field R. Proof. (1) For the left-to-right direction, argue contrapositively. Suppose that φ TAUT is true in the real field.    j Then there is a valuation v in the real field making φ F ∧ r+2 j=1 xψq < 1 true. Let X = x1 ,...,xr+2

16 Models for MV-Probabilistic Reasoning be a set with r +2 elements, and let for j = 1,...,r +2 and for every propositional variable p, j e(xj ,p) = v(xp ) and µ(xj ) = v(zj ). Then K = (X,e,µ) is a Kripke model. We prove that for every j subformula ψh of φ and for j = 1,...,r +2, ψh K,xj = v(xψh ). The proof is by induction on ψh . For ψh atomic, the claim follows from the definition of e, and the steps corresponding to Łukasiewicz connectives are trivial. For the step corresponding to σ , assume ψh = σ (ψk ). From the induction hypothesis and from the definition of φ TAUT , we get: ψh K,xj = σ (ψk )K,xj =

r+1 

µ(xi )ψk K,xi =

i=1 j

r+1 

i i v(zi )v(xψ ) = v(xψ ). k h

i=1

j

Since for some j, v(xφ ) = v(xψq ) < 1, for this j it also holds ψh K,xj < 1, and hence φ ∈ / KR1TAUT . Conversely, let K = (X,e,µ) be a (finite or countable) probabilistic Kripke model with X = {x1 ,...,xn ,...} such that φK,x = α < 1 for some x ∈ X.  }, From Corollary 3.10 (1), there exists a Kripke model K  = (X  ,e ,µ ) with X  = {x1 ,...,xr+2 and a node xl ∈ X  such that φK  ,x = α < 1. Let v be a valuation into the real field R such l

that, for i = 1,...,t and j = 1,...,r +2, v(xψi ) = ψi K  ,x , and v(zj ) = µ (xj ). Then, conditions j

l

l ) < 1, and φ TAUT holds in (a), (b), (c), (d) and (e) preceding Theorem 4.1 hold. Moreover v(xψ q R. This settles part (1). (2) The right-to-left direction is proved essentially by the same argument as in part (1). For the other direction, let K = (X,e,µ) be a Kripke model and let x ∈ X be such that φK,x = 1. From  }, and a Corollary 3.10 (1) there exists a Kripke model K  = (X  ,e ,µ ) with X  = {x1 ,...,xr+2  node xl ∈ X such that φK  ,x = 1. Let v be a valuation into R such that: for i = 1,...,t and j = l   j r+2 j 1,...,r +2, v(xψi ) = ψi K  ,kj , and for j = 1,...,r +2, v(zj ) = µ (kj ). Then, v x = 1 =1 j=1 ψq j

iff there exists a j ∈ {1,...,r +2} such that v(xψq ) = 1 iff there exists a j ∈ {1,...,r +2} such that ψq K  ,kj , and we already know that ψq K  ,kl = φK  ,kl = 1, whence, letting j = l, (2) holds.
Corollary 4.2 The sets KR1TAUT = Std1TAUT = ST 1TAUT , lKR1SAT = lStd1SAT = lST 1SAT and KR1SAT = Std1SAT = SMV 1SAT are in PSPACE. Proof. By Theorem 4.1, checking that a formula φ ∈ SFP is in one of the above sets, reduces to check the truth (or the falsity) of an existential formula of field theory (either φ TAUT , or φ lSAT ), in the real field. Moreover, φ TAUT and φ lSAT can be computed in polynomial time from φ, whence our claim follows from Canny’s Theorem (cf. [2, Theorem 3.3]) stating that the existential theory of the reals can be decided in PSPACE. Finally, Corollary 3.10 (3), φ ∈ KR1SAT iff σ (φ) ∈ lKR1SAT , and σ (φ) can be obviously computed from φ in polynomial time. Thus KR1SAT = ST 1SAT = Std1SAT are in PSPACE.


5

Concluding remarks

In this article, we introduced several semantics for the probabilistic logic SFP(Ł,Ł), namely the SMV-semantics, the standard semantics, the state semantics and the Kripke semantics. For each kind

Models for MV-Probabilistic Reasoning 17 of semantics, we investigated the sets of 1-tautologies, and 1-satisafiable, positively satisfiable and locally-satisfiable formulas. As for 1-tautologies, we proved that those of the standard semantics, state semantics and Kripke semantics coincide, but we do not know yet, if any of them is equivalent to the SMV semantics, whence, since the set of theorems of SFP(Ł,Ł) coincides with the set of SMV-tautologies, the problem left open in [6] to establish the completeness of the weaker logic FP(Ł,Ł) with respect to Kripke semantics, is still open. As for the sets of locally satisfiable formulas we proved that those of standard semantics, state semantics and Kripke semantics coincide, while, as regards 1-satisfiable formulas, those of standard semantics, Kripke semantics and SMV semantics coincide. In the final part of this article, we proved that all the above discussed sets of formulas are in PSPACE. The problem of establishing an NP-containment for them remains open.

Acknowledgements The authors would like to thank the anonymous referee for many valuable suggestions and remarks.

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