On axiomatization of fuzzy logic

SISY 2007 • 5th International Symposium on Intelligent Systems and Informatics • 24-25 August, 2007 Subotica, Serbia

On axiomatization of fuzzy logic Aleksandar Perović

Maja Jovanović

Aleksandar Jovanović

Group for intelligent systems Email: [email protected]

Group for intelligent systems

Group for intelligent systems Email: [email protected]

Abstract—During the last two decades, Group for intelligent systems at Mathematical faculty in Belgrade developed several theorem provers for different kind of formal systems. Lately, we turn our attention to fuzzy logic and development of the corresponding theorem prover. The first step is to find suitable axiomatization, i.e. the formalization of fuzzy logic that is sound, complete and decidable. Here we will discuss first order approach to the axiomatization of Łukasiewicz logic and Product logic.

I. I NTRODUCTION The fuzzy logic emerges in mid sixties of XX century in order to mathematically capture the notion of uncertain reasoning, i.e. the reasoning about notions with inherited fuzzyness, such as being tall, young, fat, bald etc. Similarly to probability logic, in fuzzy logic we have the real valued truth, i.e. the truth value of certain statement can be any real number in the interval [0, 1]. The semantics of fuzzy logic is based on T -norms as a new kind of “and” and “or” operators. In that way, simple “logical” computations are possible even without any formal (in the strict logical sense) framework. However, any attempt of automatization of such reasoning requires deeper understanding of the heart of the mater, which inevitably leads to the formalization and clear distinction between syntax and semantics. At the very beginning of the process of axiomatization, one encounters the following difficulty: which T -norm to chose? Over forty years of application taught that the list of possible meaningful T -norms is inexhaustible. Since the one of the key features of such axiomatization should be decidability, we will restrict our selves to finite list of such norms, based on Łukasiewicz and product norm. In particular, we will consider the following connectives: unary connectives ¬L ¬Π M O

binary connectives ⇒L ⇒Π ¯ ⊕ ª ∧ ∨ ≡ &

(1)

As it is well known (see[1]), the corresponding evaluation function e is defined as follows:

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e(¬L α) = 1½− e(α) 1 , e(α) = 0 • e(¬Π α) = ½ 0 , e(α) > 0 1 , e(α) = 1 • e(M α) = ½ 0 , e(α) < 1 1 , e(α) > 0 • e(Oα) = 0 , e(α) = 0 • e(α ⇒L β) = min(1 − e(α) + e(β), 1) ( 1 , e(α) > e(β) • e(α ⇒Π β) = e(β) , e(α) > e(β) e(α) • e(α ¯ β) = e(α) · e(β) • e(α ⊕ β) = min(1, e(α) + e(β)) • e(α ª β) = max(0, e(α) − e(β)) • e(α ∧ β) = min(e(α), e(β)) • e(α ∨ β) = max(e(α), e(β)) • e(α&β) = max(0, e(α) + e(β) − 1) • e(α ≡ β) = 1 − |e(α) − e(β)| The rest of the paper is organized as follows: in Section 2 we define a first-order translation of the propositional fuzzy logic based on the connectives listed in (1) and prove that the obtained first order theory is consistent. In section 3 we prove that the theory developed in Section 2 is decidable. We conclude in Section 4. •

II. T HE TLΠ THEORY Let P be a recursive countable set of propositional letters. By F orP we will denote the set of all propositional formulas built over P by means of the connectives listed in (1). For each α ∈ F orP we introduce a new constant symbol Cα . Here Cα should be understood as the weight of α. TLΠ is a first order theory of the language L = {+, ·, 6, 0, 1} ∪ {Cα | α ∈ F orP }, with the following axioms: 1) Axioms of the theory of real closed fields. 2) 0 6 Cα 6 1, α ∈ F orP . 3) Cα + C¬L α = 1. 4) (Cα = 0 ∧ C¬Π α = 1) ∨ (Cα > 0 ∧ C¬Π α = 0). 5) (CMα = 1 ∧ Cα = 1) ∨ (CMα = 0 ∧ Cα < 1). 6) (COα = 1 ∧ Cα > 0) ∨ (COα = 0 ∧ Cα = 0). 7) Cα⇒L β = min(1 − Cα + Cβ , 1). We remind the reader that “ min ”, “ max ”, “| |” and “ − ” are definable in the language of the ordered fields. In this context, those are purely syntactical objects. 8) (Cα⇒Π β = 1 ∧ Cα 6 Cβ ) ∨ (Cα⇒Π β · Cα = Cβ ∧ Cβ < Cα ).

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A. Perović, M. Jovanović, A. Jovanović • On Axiomatization of Fuzzy Logic

9) 10) 11) 12) 13) 14) 15)

Cα¯β = Cα · Cβ . Cα⊕β = min(1, Cα + Cβ ). Cαªβ = max(0, Cα − Cβ ). Cα&β = max(0, Cα + Cβ − 1). Cα∧β = min(Cα , Cβ ). Cα∨β = max(Cα , Cβ ). Cα≡β = 1 − |Cα − Cβ |.

Before we prove the consistency of TLΠ , we notice that TLΠ actually represents a kind embedding of the logic ŁΠ (see [1]) into the theory of real closed fields. This particular approach is similar to the one described in [2]. Next we will indicate how can be proved consistency of TLΠ . Due to Compactness theorem for the first order logic, it is sufficient to prove that each finite subset of TLΠ has a model. Suppose that S is an arbitrary finite subset of TLΠ . Then, there are only finitely many constant symbols Cα , say Cα1 , . . . , Cαm , that appear in axioms of S. Let p1 , . . . , pn be all propositional letters that have at least one appearance in any of formulas α1 , . . . , αm . We construct the model of S as follows: let M be an arbitrary real closed field (here we identify the model M with its universe). Furthermore, let ξ1 , . . . , ξn ∈ M be such that

if and only if RCF ` ∃x1 . . . ∃xn (

CpMi = ξi , i = 1, . . . , n. Similarly to the evaluation function e, we define the interpretation of every Cα(p1 ,...,pn ) in M . For instance, CpM1 ¯p2 = ξ1 ·M ξ2 . If α ∈ F orP is not built over p1 , . . . , pn , let CαM = 0M . It is easy to see that hM, CαM iα∈F orP |= S. Hence, TLΠ is consistent. III. D ECIDABILITY OF TLΠ In this section we will discuss decidability of introduced theory. Namely, we can prove the following two facts: • •

TLΠ admits the quantifier elimination. ϕ is an arbitrary sentence of L, then the satisfiability of ϕ is decidable.

The key idea is to show that TLΠ is interpretable in the theory of real closed fields (RCF). In order to do so, for each α(p1 , . . . , pn ) ∈ F orP , we will define the n-ary function symbol Fα recursively as follows: • • •

Fpi (x1 , . . . , xn ) = xi , i = 1, . . . , n. F¬L α (¯ x) = 1 − Fα (¯ x). Fα¯β (¯ x) = Fα (¯ x) · Fβ (¯ x). The case of other connectives we leave to the reader.

Clearly, each Fα is definable in RCF. Then, it is easy to see that TLΠ ` ϕ(Cα1 (p) ¯ , . . . , Cαm (p) ¯ )

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0 6 xi

6 1 ∧ ϕ(Fα

1

(¯ x), . . . , Fαm (¯ x))).

i=1

Since RCF admits quantifier elimination, and since RCF is a sub-theory of TLΠ , we have the first part of Theorem. To conclude the proof, we only need to notice that satisfiability of any sentence of L can be equivalently reduced to the satisfiability of certain Σ0 sentence of RCF. As it is well known, the latter is decidable. The above mentioned decision procedure goes as follows: Input: An arbitrary sentence ϕ of L. Output: YES, if ϕ is satisfiable; otherwise NO. Step 1 Using appropriate Fα s, equivalently transform ϕ into sentence ϕ∗ of RCF. Step 2 Apply the quantifier elimination procedure for RCF to obtain a Σ0 sentence φ of RCF such that RCF ` ϕ∗ ⇔ φ. Step 3 Apply the decision procedure for the satisfiability of Σ0 sentences of RCF. The output of that procedure is by definition the output of the whole procedure.

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