valued events led Mundici to introduce and axiomatise the notion of states ... tions with respect to Borel probability measures on S. Kühr and Mundici.
Stefano Aguzzoli Brunella Gerla
Probability Measures in the Logic of Nilpotent Minimum
Abstract. We axiomatize the notion of state over finitely generated free NM-algebras, the Lindenbaum algebras of pure Nilpotent Minimum logic. We show that states over the free n-generated NM-algebra NM n exactly correspond to integrals of elements of NM n with respect to Borel probability measures. Keywords: Nilpotent Minimum logic, probability measure, state, G¨ odel logic.
1.
Introduction
The first investigations of probability measures over non-classical manyvalued events led Mundici to introduce and axiomatise the notion of states over MV-algebras [15]. Later on, Kroupa [13] and Panti [16] have shown that, whenever an MV-algebra A is an algebra of functions from some set S ⊆ [0, 1]n to [0, 1], states over A are exactly the integrals of these functions with respect to Borel probability measures on S. K¨ uhr and Mundici in [14] extended the notion of state to all algebraic systems of continuous functions over [0, 1]. In [6] it has been shown that probability measures can be defined for logics whose algebraic semantics is not given by algebras of continuous functions, axiomatising the analogous of the notion of state for the Lindenbaum-Tarski algebras of G¨ odel propositional logic. In this work we show that probability measures can be defined also for logics based on non-continuous t-norms, by solving the problem for the logic of Nilpotent Minimum. The Nilpotent Minimum t-norm was introduced by Fodor [10] as the first example of a left-continuous but non-continuous t-norm, and it is defined, for every x, y ∈ [0, 1], by min{x, y} if x + y > 1 xy = 0 otherwise. The propositional logic of Nilpotent Minimum, NM for short, is a schematic extension of the monoidal t-norm based logic MTL, introduced by Esteva
Special Issue: Algebra and Probability in Many-Valued Reasoning Edited by Ioana Leu¸stean and Vincenzo Marra
Studia Logica (2010) 94: 151–176 DOI: 10.1007/s11225-010-9228-8
© Springer 2010
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and Godo in [9]. We recall the reader that MTL and its extensions are based on the language having as primitive the binary connectives , →, and ∧, and the constant ⊥. In each fixed standard [0, 1]-semantics, is the (left-continuous) t-norm, → is its residuum, ∧ is the minimum, and ⊥ is 0. Usually derived connectives are x ∨ y = ((x → y) → y) ∧ ((y → x) → x), the negation ¬x = x → ⊥, and the constant = ¬⊥. In each [0, 1]-semantics ∨ is maximum and is the constant 1. Throughout the paper we shall identify connectives with the operations that interpret them in the algebraic semantics. Dealing with the Nilpotent Minimum standard [0, 1]-semantics mostly, we deem that no confusion should arise. The logic NM is obtained from MTL by adding the following axiom schemes: (WNM) ¬(ϕ ψ) ∨ ((ϕ ∧ ψ) → (ϕ ψ)) , (INV) ¬¬ϕ → ϕ . The Lindenbaum algebras of the logic NM constitute the variety of NMalgebras, which in turn is generated by the standard NM-algebra [0, 1] = [0, 1], , →, ∧, 0 , where is the Nilpotent Minimum t-norm and 1 if x ≤ y x→y= max{1 − x, y} otherwise. Note that the negation ¬x = x → 0 in the standard NM-algebra is given by the standard involutive negation ¬x = 1 − x. It follows that all De Morgan laws involving ∧ and ∨ hold. Nilpotent Minimum t-norm arises as the Jenei’s rotation [12] of G¨ odel t-norm ∧ (we recall that x ∧ y = min{x, y} and its residuum is given by x →∧ y = 1 if x ≤ y, x →∧ y = y otherwise). This fact often allows to transfer results concerning G¨ odel logic to NM. In particular in this work we shall use the definition of normal forms for NM formulas given in [3], and the techniques used to develop the spectral duality for finite G¨ odel algebras in [8] and for finite NM-algebras1 in [1], to adapt the notion of finitely additive probability measure given in [6] for G¨ odel logic to the case of the logic of Nilpotent Minimum. Denote NM n the free NM-algebra over n many free generators. Since the variety of NM-algebras is generated by the standard NM-algebra, it follows that elements of NM n are real–valued functions f : [0, 1]n → [0, 1], called NM-functions. In this paper we shall axiomatise states over NM n as functions s : NM n → [0, 1] satisfying certain requirements (see Def. 4.1). 1
Both the varieties of NM-algebras and of G¨ odel algebras are locally finite. Hence, in these varieties the classes of finitely presented, of finitely generated, and of finite algebras coincide.
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We shall show that states play the role of finitely additive probability measures in the sense stated in our main theorem: Theorem. The following hold. 1. If s : NM n → [0, 1] is a state, there exists a Borel probability measure µ on [0, 1]n such that f dµ = s(f ) , for every f ∈ NM n . (1) [0,1]n
2. Vice versa, for any Borel probability measure µ on [0, 1]n , the function s : NM n → [0, 1] defined by (1) is a state. In words, fixing a state on NM n precisely amounts to integrating NMfunctions of n variables with respect to an appropriate Borel probability measure µ on [0, 1]n .
2.
NM-algebras, G¨ odel algebras, G¨ odel hoops
In this section we collect and recall some of the results connecting NMalgebras and G¨ odel algebras that appear in the literature. No new results are then introduced in this section, but we need this recapitulation in order to present in a unified notation all notions we shall use in the sequel. If an algebra A has a specified chosen set of generators {g1 , . . . , gn }, and ϕ(x1 , . . . , xn ) is a term (or formula) in the language of A, we shall denote ϕA the interpretation of ϕ in A determined by mapping the variable xi to the generator gi , for all i ∈ {1, . . . , n}. As [0, 1] is generic for the variety of NM-algebras, the free n-generated [0,1]n
NM-algebra NM n is isomorphic to the subalgebra of [0, 1] generated n : (t1 , . . . , tn ) → ti . by the projection functions xNM i Generalised G¨ odel algebras, also called G¨ odel hoops, are the ⊥-free subreducts of G¨ odel algebras. An NM-algebra A is directly indecomposable if and only if it is isomorphic to the connected or disconnected rotation of a uniquely determined generalised G¨ odel algebra G(A) (see [7, 1]). In particular, NM n is isomorphic to the direct product Aλ (2) NM n ∼ = λ∈{−1,0,1}n
where each Aλ is such that G(Aλ ) is isomorphic to the free k-generated G¨ odel hoop, where k = |{i ∈ {1, . . . , n} | λ(i) = 0}|, and Aλ is the dis-
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connected rotation of G(Aλ ) if and only if k = n.2 The free k-generated odel hoop of G¨ odel hoop G(Aλ ) can be realised as the subhoop of the G¨ all functions f : (1/2, 1]k → (1/2, 1] generated by the projections functions (t1 , . . . , tk ) → tj . Let pλ : Aλ → NM n be defined for every a ∈ Aλ as pλ (a) = b where b ∈ NM n is determined by πλ (b) = a and πλ (b) = Aλ for λ = λ (we denote by πλ the canonical projection mapping λ∈{−1,0,1}n Aλ onto Aλ ). We call an element a of an NM-algebra positive if a > ¬a. The definition odel hoop to the subhoop of rotation shows that G(Aλ ) is isomorphic as a G¨ of Aλ formed by all its positive elements. Further, Equation (2) implies that pλ (G(Aλ )) is a subhoop of NM n isomorphic to G(Aλ ). We define N (λ) ⊆ [0, 1]n as the set of all points (t1 , . . . , tn ) such that ti = 1/2 if λ(i) = 0; ti > 1/2 > 1 − ti if λ(i) = 1; ti < 1/2 < 1 − ti if λ(i) = −1. It is obvious that {N (λ) | λ ∈ {−1, 0, 1}n } forms a partition of [0, 1]n . Equation (2) allows us to represent each G(Aλ ) as the algebra of restrictions of positive (with respect to Aλ ) functions of NM n to the domain n N (λ): the free generators of G(Aλ ) can be chosen as the restriction of xNM i NM n for each i such that λ(i) = 1, and as the restriction of (¬xi ) for each i such that λ(i) = −1. We conclude that for each element f ∈ NM n and each λ ∈ {−1, 0, 1}n , if f is not constantly 1/2 over N (λ) then there exists a uniquely determined odel hoop G(Aλ ) such that f either element fλ : N (λ) → (1/2, 1] of the free G¨ coincides with fλ or with ¬fλ over N (λ). We can then write f (t1 , . . . , tn ) =
(pλ (fλ∗ ) ∧ σλ )(t1 , . . . , tn ) ,
(3)
λ∈{−1,0,1}n
where for each λ, the function fλ∗ is either fλ or ¬fλ or the constant 1/2 and σλ = ¬pλ (⊥Aλ ) (notice πλ (σλ ) = Aλ and πλ (σλ ) = ⊥Aλ for λ = λ). Compare with [3, Def.6.14,Thm.6.15],[17, Thm.5.1]. Since each fλ is an element of a free G¨odel hoop, we can adapt to Nilpotent Minimum logic several notions developed for G¨odel logic, as for instance minterms and disjunctive normal forms. We are going to make these notions precise in the following section. 2
Throughout the paper we shall identify n-tuples (c1 , c2 , . . . , cn ) ∈ S n and functions c : {1, . . . , n} → S, for all sets S. Hence we take the liberty of denoting the ith element of such an n-tuple by ci as well as c(i), using each time the notation we find most convenient. |S| denotes the cardinality of S.
Probability Measures in the Logic of Nilpotent Minimum
3.
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Functional representation, positive minterms, disjunctive normal form
From now on, we shall identify formulas (terms) with the functions interpreting them in NM n . For instance, the ith projection function (t1 , . . . , tn ) → ti will be denoted simply xi . For any poset (P, ≤) and x ∈ P , we let ↑ x = {y ∈ P | x ≤ y} and ↓ x = {y ∈ P | y ≤ x}. We recall that a partition subsets of A such that of a set A is a collection {A1 , . . . , Ah } of nonempty Ai ∩ Aj = ∅ for all i = j ∈ {1, . . . , h} and hi=1 Ai = A. Each element Ai is called a block of the partition. An ordered partition of π is a partition equipped with a total order ≤π among its blocks A1 , . . . , Ah . We display such an ordered partition π as π = A1
