Metric Boolean Algebras and an Application To Propositional Logic

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1 Global Journal of Science Frontier Research Volume 11 Issue 5 Version 1.0 August 2011 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) ISSN: Metric Boolean Algebras and an Application To Propositional Logic By Li FU, Guojun Wang Qinghai Nationalities University, China Abstract - Let B be a Boolean algebra and Ω be the set of all homomorphisms from B into D, and μ be a probability measure on Ω. We introduce the concepts of sizes of elements of B and similarity degrees of pairs of elements of B by means of μ, and then define a metric on B. As an application, we propose a kind of approximate reasoning theory for propositional logic. Keywords : Boolean algebra, probability measure, size, similarity degree, approximate reasoning, propositional logic. GJSFR Classification : 03G05, 54E35, 60A10 Strictly as per the compliance and regulations of: 2011.Li FU, Guojun Wang.This is a research/review paper, distributed under the terms of the Creative Commons Attribution- Noncommercial 3.0 Unported License permitting all non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

2 Metric Boolean Algebras and an Application To Propositional Logic Li FU α, Guojun Wang Ω Abstract - Let B be a Boolean algebra and Ω be the set of all homomorphisms from B into D, and µ be a probability measure on Ω. We introduce the concepts of sizes of elements of B and similarity degrees of pairs of elements of B by means of µ, and then define a metric on B. As an application, we propose a kind of approximate reasoning theory for propositional logic. Keywords : Boolean algebra, probability measure, size, similarity degree, approximate reasoning, propositional logic. I. PRELIMINARIES L et L be a distributive lattice and D be the lattice {0,1}. Assume Ω is the set of all homomorphism from L to D, then f Ω, f is order-preserving, i.e., a, b L, a b if and only if f(a) f(b). In fact, if a, b L, a b, then there exists an f Ω, such that f(a) = 1 and f(b) = 0 ( see [4]). Therefore, if we think of elements of L as functions from Ω to D, i.e., a L defining a : Ω {0, 1}, a(f) =f(a)(f Ω), then the partial Ω order or L will have a representation by means of the partial order among function of 2.This fact is also true for Boolean algebras. We use the same symbol Ω to denote the set of all (Boolean) homomorphisms from a Boolean algebra B into D. Proposition 1 : Suppose that B is a Boolean algebra, D = {0, 1}, and Ω is the set of all homomorphism from L to D, then a, b B, a b if and only if f Ω, f(a) f(b). Throughout this paper, assume that B is a Boolean algebra and a probability measure µ is given on Ω, we will introduce the concepts of sizes of elements of B and similarity degrees between elements pairs of elements of B by means of µ and proposition 1, and finally define a metric ρ on B therefrom. Especially, this can be done the Lindenbaum algebra B L = F (S)/, where F (S) is the set consisting of all logic propositions and is the congruence relation of logically equivalence[6], and Ω is the set consisting of all valuations of F(S) of which a probability measure can naturally be introduced. Finally, a kind of approximate reasoning theory can be developed on F(S) because there have been distances among propositions. II. METRIC BOOLEAN ALGEBRAS Suppose that µ is a probability measure on Ω Definition 1 : Define τ : B [0, 1] as follows: τ(a) = µ({f Ω a(f) = 1}), a B then τ(a) is called the size of a with respect to µ, or briefly, size of a if no confusion arises. The following proposition is obvious. Proposition 2 : (i) 0 τ(a) 1, a B; (ii) τ(1 B ) = 1, τ(0 B) = 0, where 1 B and 0 B are the greatest element and the least element of B respectively; (iii) τ(a ) = 1 τ(a), a B (iv) If a b, then τ(a) τ(b), a, b B August Global Journal of Science Frontier Research Volume XI Issue vvvvvvvvvvvvvvv ersion I 1vvvvvvvvvvvvvvV * Foundation item : QingHai Province 135 program. Biographyical : FUli, Female, Doctor, Research field : Non- determinacy Reasoning, @ 163.com. Author α : College of Mathematics and Statistics, Qinghai Nationalities University, Xining , P.R.China. Author Ω : Shanxi Normal University, Xi'an,710062, P.R.China.

3 Journal of Science Frontier Resear ch Volume XI Issue vvvvvvvvvvvvvvv Version I 2Global August Example 1 : Consider the Boolean algebra P(X), where X is non-empty finite set, and P(X ) is the power set of X. It is easy to verify that a mapping f : P(X) D is a homomorphism iff there exists an (unique) element x of X such that f 1 (1) = {A X x A}. Hence, Ω = X. Assume that µ is the evenly distributed probability measure on Ω, then the size of an element A of P(X) is the cardinal of A over X, i.e., τ(a) = A / X (A P(X)). Proposition 3 : τ(a b) = τ(a) + τ(b) τ(a b), a, b B. Assume that a b = a b(a, b B) (see 5), then we have: Proposition 4 : τ(a) α, τ(a b) β, then τ(b) α + β 1, where α, β [0, 1]. Proposition 5 : If τ(a b) α, τ(b c) β, then τ(a c) α + β 1. Definition 2 : Define η : B B [0, 1] as follows: η(a, b) = τ(a b) (b a), a, b B. then η(a, b) is called the similarity degree between a and b, and η the similar relation on B with respect to µ. Example 2 : Consider the Boolean algebra P(X) where X is any non-empty set, and µ is any probability measure on Ω. Then (1) η(a, X A) = 0,(A P(X)). In fact, A (X A) = (X A) (X A) = X A, (X A) A = A A = A Hence, η(a, X A) = τ((x A) A) = τ( ) = 0. (2) η(a, B) + η(a, X B ) = 1. In fact, let G = ((X A) B) ((X B) A), H = ((X A) (X B)) (B A). then it is routine to verify that G H = X, G H =. Hence, η(a, B) + η(a, X B) = µ(g) + µ(h) = 1. Proposition 6 : η(a, b) = µ({f Ω f(a) = f(b)}), a, b B. Proposition 7 : (i) η(a, b ) = 1 if and only if a = b. (ii) η(a, a ) = 0 (iii) η(a, b) + η(a, b ) = 1. (iv) η(a, c) η(a, b) + η(b, c) 1. Proposition 8 : Suppose that η(x, a) α, η(y, b) β then η(x y, a b) α + β 1. It follow from proposition 7 that the function ρ defined below is a metric on B. Definition 3 : Define ρ : B B as follows: ρ(a, b) = 1 η(a, b), a, b B. Then ρ is a metric on B and ( B, ρ) is called the metric Boolean algebra with respect to µ. Theorem 1 : Let ( B, ρ) be the metric Boolean algebra, then (i) All the operations,,, are uniformly continuous. (ii) If ( B, ρ) is a complete metric space, then B is a ω complete lattice. Proof : (i) By proposition 7, it follows that ρ (a, b ) = 1 η (a, b ) = η (a, b) = 1 η (a, b) = ρ (a, b). If ρ (a, b) ε, then ρ(a, b ) ε, where ε is any given positive number. This proves the continuity of the operation : B B. Suppose that ρ(x, a) ε, ρ(y, b) ε, then η(x, a) > 1 ε, η(y, b) > 1 ε, and it follows proposition 8, having η (x y, a b) (1 ε)+(1 ε) 1 = 1 2ε, hence ρ (x y, a b) 2ε. This proves that the operation : B B B is uniformly continuous. Since a b = a b, a b = (a b ), hence the operations and also uniformly continuous. (ii) Suppose that ( B, ρ) is a complete metric space, and = {a 1, a 2, } B. Let b n = n i=1 a i, then b 1, b 2, is an increasing sequence and hence τ (b 1 ), τ (b 2 ), is an increase sequence in [0,1] and is therefore Cauchy. Note that b n b m = b n b m = 1 B, and b m b n = 0 B whenever b n b m, then ρ (b m, b n ) = 1 η(b m, b n ) = 1 τ((b m b n ) (b n b m )) = 1 τ(b m b n ) = 1 τ(b m b n ) = 1 [τ(b m) + τ(b n ) τ(b m b n )] = 1 τ(b m) τ(b n ) = τ(b m ) τ(b n ). hence b 1, b 2, is a Cauchy sequence in ( B, ρ) and converges to an element c of B. Fix an n we have from b n = b n b n+k and the continuity of that

4 b n = lim k (b n b n+k ) = b n lim k b n+k b n c, hence b n c and c is an upper bound of Σ = {b 1, b 2, }. On the other hand, let e be any upper bound of Σ, then b n = b n e and hence c = lim n b n = lim n (b n e) = (lim n b n ) e = c e. This means that c e, hence c = sup Σ. It is clear that sup = supσ = c. We can prove that inf exists in a similar way, hence B is ω complete lattice. III. AN APPLICATION Consider the set of all abstract formulas in propositional logic. Let S = { p 1, p 2, } be a countable set and F(S) be free algebra of type (, ) generated by S where and are unary and binary operations respectively. Homomorphisms from F(S) to D = {0, 1} called valuations of F(S), the set is consisting of valuations will be denoted Ω. Assume that A, B are formulas (propositions) of F (S), A and B are logically equivalent if ν Ω, ν(a) = ν(b). It is well known that the logical equivalence relation is a congruence relation on F(S) with respect to (, ), and the quotient algebra F(S) / = is a Boolean algebra and called the Lindenbaum algebra[6], and denoted B L. Let X = n=1 X n, where X n = {0, 1}, and µ n be the evenly distributed probability measure on X n, i.e., µ( ) = 0, µ (X n ) = 1, and µ n ({0}) = µ n ({1}) = 1 2, (n = 1, 2, ), and let µ be the infinite product of µ 1, µ 2, on X (see[3]). Since F (S) is the free algebra generated by S, a valuation ν : F (S) {0, 1} is completely decided by its restriction ν S. Since ν S = {ν (p 1 ), ν (p 2 ), } can be thought of as a point of X, there is a bijection between Ω and X and hence the measure µ on X can be transplanted on Ω. We use the same symbol µ to denote the measure on Ω, i.e., µ(σ) = µ({(x 1, x 2, ) X ν Σ, x n = ν(p n ), n = 1, 2, })Σ Ω. Assume that A, B F(S ) and A B and ν Ω, then ν (A) = ν (B) hence ν induces a unique Boolean homomorphism ν : B L = F (S)/ {0, 1} defined by ν (a) = ν(a), where a is the congruence class containing A (a B L ). In the following ν will be simplified as ν and Ω can be thought of as the set consisting of all Boolean homomorphisms B L to {0, 1}. Therefore there is a metric ρ on the Lindenbaum algebra B L and ( B L, ρ) will be called the metric Lindenbaum algebra. Denote the congruence class B L containing A by [A], where A F (S) and define d : F (S) F (S) [0, 1] as follows: d(a, B) = ρ([a], [B]), then d is a pesudo-metric on F(S). Now that there exists the concept of distances among formulas, an approximate reasoning theory can naturally be developed on F(S). We give in the following only a short sketch. Definition 4 : Let T be the set consisting of all theorems in F (S) (see[2]) and A F (S ). If d(a, T) < ε, then A is called a theorem with error ε, and denoted ( ε) A. Moreover, let D(Γ) be the set of all Γ conclusions, i.e., D(Γ) = {A F (S) Γ A}. If inf{h(d(γ), T ) Γ A} < ε, where H(Σ, ) is Hausdorff distance between Σ and (Σ, ) are nonempty subsets of F (S), for a general definition of the Hausdorff distance between two non-empty, bounded subsets of a metric space without the requirement of closedness see[1], and that of subsets of a pseudo-metric space, see[6]), then A is called ε quasi theorem and denoted (ε)a. Theorem 2 : Suppose that A is any formula of F(S), then A, B F (S), (ε) A if and only if (ε)a Definition 5 : Suppose that Γ F (S), let Div(Γ) = sup {ρ (A, B) A, B D(Γ)}, where sup = 0 is assumed, then Div (Γ) is called the divergence degree of Gamma. Moreover, let Dev (Γ) = H (D (Γ), T ), then Dev (Γ) is called the deviation of Γ. Theorem 3 : Suppose that Γ F (S ), then Div(Γ) = Dev(Γ). Proof : Suppose that Dev(Γ) = α and ε > 0, then it follows from Global Journal of Science Frontier Research Volume XI Issue vvvvvvvvvvvvvvv ersion I 3vvvvvvvvvvvvvvV August 2 011

5 H(D(Γ), T ) = max(sup{d(a, T ) A D(Γ)}, sup{d(t, D(Γ)) T T }) = sup{d(a, T ) A D(Γ)} = α. that there is an A D (Γ) such that d (A, T ) α ε. Choose any T from T, then A, T D(Γ) and we have Div(Γ) d(a, T ) α ε Since ε is arbitrary, we have Div(Γ) α = Dev(Γ). Conversely, we have Journal of Science Frontier Resear Volume XI Issue vvvvvvvvvvvvvvv Version I 4Global August Dev(Γ) = H(D(Γ), T ) = max(sup{d(a, T ) A D(Γ)}, sup{d(t, T ) T D(Γ)}) Moreover, assume that A, B D(Γ). Note that [ A] [B] [A], and [ B] [A] [B], we have d (A, B) = Hence, Div(Γ) = sup{d(a, B) A, B D(Γ)} D(Γ). =sup{d(a, T ) A D(Γ)} = sup{1 τ([a]) A D(Γ)}. ρ([a], [B]) = 1 η([a], [B]) = 1 τ(([a] [B]) [B] [A])) 1 τ([a]) τ([b]) = (1 τ([a])) (1 τ([a])). References Références Referencias 1. A.Csaszar, General Topology, Adam Hilger Ltd, Bristol, A.G.Hamilton, Logic for Mathematicians, Cambridge University Press, London, P.R.Halmos, Measure Theory, Springer-Verlag, New York, P.J.Johnstone, Stone Space,Cambridge University Press, London, H.Rasiowa,R.Sikorski, The Mathematics of Metamathematics, Panstwowe Wydawnicto Naukowe,Warszwa, G.J.Wang, Non - classical Mathematicical Logis and Approximate Reasoning, Science in China Press, Bejing (in Chinese), ch