A NOTE ON ENTROPY OF LOGIC

Transcription

1 Yugoslav Journal of Operations Research 7 07), Nuber 3, DOI: 0.98/YJOR5050B A NOTE ON ENTROPY OF LOGIC Marija BORIČIĆ Faculty of Organizational Sciences, University of Belgrade, Serbia arija.boricic@fon.bg.ac.rs Received: October 05 / Accepted: April 06 Abstract: We propose an entropy based classification of propositional calculi. Our ethod can be applied to finite valued propositional ics and then, extended asyptotically to infinite valued ics. In this paper we consider a classification depending on the nuber of truth values of a ic and not on the nuber of its designated values. Furtherore, we believe that alost the sae approach can be useful in classification of finite algebras. Keywords: Many valued Propositional Logics, Lindenbau Tarski Algebra, Partition, Entropy. MSC: 03B50, 03B05, 94A7, 37A35.. INTRODUCTION We present one way of ical systes classification based on their entropies see [] and [3]). The concept of generalized Shannons entropy, entropy of a partition and the ical syste represented by its Linednbau Tarski algebra, ake it possible to define the entropy of a any valued propositional ic, and then to extend it asyptotically to infinite valued ics. Our finite easure of uncertainty H of a finite valued ic onotonically increases with the growth of truth values nuber. This easure is sensitive to both the nuber of truth values of a finite valued ic and the nuber of its designated true) values see [] and [3]). In this paper we deal with a classification depending only on the nuber of truth values.. LINDEBAUM TARSKI ALGEBRA Let us keep in ind the following two well known facts. The first is related to the nuber n of utually non equivalent forulae over the finite set of propositional letters {p,..., p n } in the classical two valued ic. The second one

2 386 Marija Boričić / A Note on Entropy of Logic is that Nishiura has shown that in case of Heyting s propositional ic, an infinite valued ic, even in case when the set of propositional forulae is built up over the set of a single propositional letter, there exist countably any utually nonequivalent forulae see []). These exaples show the essential difference between finite valued and infinite valued ics fro the stand point of our intentions. Naely, our ai is to consider possibilities for defining probabilistic easure over partitions of propositional forulae set, denoted by For, defined by the corresponding Linednbau Tarski algebra. By a partition of a nonepty set X we ean any finite or denuerable collection A i ) of nonepty subsets of X such that i, j)i j A i A j = ) and i A i = X. Partition of the set For of propositional forulae is defined on the basis of an equivalence relation L, related to a propositional ic L, by the following condition A L B iff both consequences A B and B A are derivable in L, for any A, B For. This equivalence relation L divides the set For on non epty utually disjoint sets and fors a quotient algebra For / L, usually called the Lindenbau Tarski algebra of L. If by For n we denote a subset of For built up over a finite set of propositional letters {p,..., p n } and a usual list of propositional connectives,, and, then, in case of an valued propositional ic L, the corresponding quotient algebra For n / L will consist of at ost n eleents. 3. ENTROPY OF PARTITIONS OF For A natural generalization of Shannon s entropy, appearing in Measure theory, is defined over a easurable partition α = {A i i I} of a space X, equipped with a easure µ, such that i)i I µa i ) 0), i, j)i I j I i j A i A j = ) and µx \ i I A i ) = 0. In this context, the entropy is defined as follows: Hα) = µa i ) µa i ) i I with the usual convention that µa i ) µa i ) = 0, for µa i ) = 0, by definition, having in ind that li x 0+ x x = 0. Our central proble is how to define a easure over a finite faily of sets consequently extendable to a denuerable faily, in order to get a finite philosophically well founded and ically justified entropy of partition. Let us describe the basic idea and the construction. More accurately, the proble is to define a easure µ over the set For n / L and to extend it into For / L, obtaining the finite entropy for For / L. As we stated in M. Boričić 03, 04), the easures distributed uniforly or binoially do not give satisfiable results. Naely, even in the case of classical two valued propositional ic, neither unifor, nor binoial probability distribution do not give a finite entropy. If we suppose that the easure µa i ) of the class A i is uniforly distributed, eaning that µa i ) =, then, by ), the corresponding entropies HL n ) and HL n ) over partitions of sets For n and For, respectively, are: HL n ) = n ln and HL ) = li n HL n ) = +, where HL n ) and HL ) denote entropies of two valued ic L over the sets

3 Marija Boričić / A Note on Entropy of Logic 387 For n and For, respectively. Alternatively, if we suppose that these easures are binoially distributed, eaning that ) n pa i ) = i n then, by ) and the known asyptotic relation: HL n ) ln eπ n ) as n see [3]), we also conclude that HL ) = li n HL n ) = +. Here we will present, according to [], a definition enabling a good possibility for classification of finite valued propositional ics on the basis of a finite entropy of a countable partition of For. In order to give a siple and clear definition, we will consider here the case when an valued ic has only one designated value, eaning that only one value of designates the truth. The case of valued ic with k =,,..., designated values is considered in [] and [3]. Let L be an valued ic with one designated value, and L n its part built up over a set consisting of n propositional letters only. By HL n ) and HL ) we denote entropies of L n and L, respectively. Let µa i ) = ) i for i =,,..., n and µa i ) = ) i for i = n. Lea. HL ) = ) ) Proof. Using the forula for a geoetric series, following essentially fro [5], i.e. fro the fact that n k= kz k = z n + )zn + nz n+ z) which is provable, for exaple, by atheatical induction on n, for M = n,

4 388 Marija Boričić / A Note on Entropy of Logic and using ), we calculate: HL n ) = M µa i ) µa i ) i= = ) M = ) M ) ) ) M ) M and finally, we find: ) )) ) M ) ) M ) M ) M ) ) M ) M + M ) HL ) = li n HL n ) = ) )7pt Using this Lea we justified the definition of entropy of valued ic L. Consequently, we find see []) that: HL ) Siple onotonicity analysis of the function f x) = x x x ) x ) leads us to the following conclusion: Lea. For any two valued and n valued ics L and L n, if n, then HL ) HL n ). 4. ENTROPY OF SOME KNOWN LOGICS Here we ention soe features of the well known finite valued ics, give their entropies and consider entropies of infinite valued propositional ics. First of all, we note that the classical propositional ic has the entropy less than or equal to, and that both Lukasiewicz s see [8], [9], [4] and []) and Kleene s see [6], [7] and []) three valued ics, with one designated value,

5 Marija Boričić / A Note on Entropy of Logic 389 have the entropies less than or equal to Belnap s four valued ic see []), with one designated value, has the entropy less than or equal to Let us consider the sequence H + E of finite valued extensions of Heyting s propositional ic H by axio scheata E : A i A j ) i<j for 3, where A B is an abbreviation for A B) B A), introduced by McKay see [0]), presents a strictly descending sequence H + E of ) valued ics, with one designated value, interediate between H and classical two valued ic L see [4])), i.e. H... H + E + H + E... H + E 4 H + E 3 = L having the following property: li H + E ) = + H + E ) = H see [0]), gives us the reason to consider an asyptotic approxiation of the entropy of Heyting propositional ic, as well. For the entropy of H + E, we have: HH + E ) ) ) 3 Acknowledgeent: The author thanks the anonyous referees of YUJOR and the ebers of the Seinar for Probabilistic Logic of Matheatical Institute of SANU, Belgrade, particularly her teachers N. Ikodinović, Z. Marković, Z. Ognjanović, A. Perović and M. Rašković, for aking very valuable suggestions and helpful coents regarding earlier versions of this paper. This research was supported in part by the Ministry of Science and Technoy of Serbia, grant nuber REFERENCES [] Belnap, N. D., A useful four valued ic, in J M. Dunn, G. Epstein eds.), Modern Uses of Multiple Valued Logic, D. Reidel, Dordrecht, 977, [] Boričić, M., On entropy of a ical syste, Journal of Multiple Valued Logic and Soft Coputing, 5 6) 03) [3] Boričić, M., On entropy of a propositional ic, Bulletin of Sybolic Logic, 0 ) 04) 5, Abstract, Logic Colloquiu, 03. [4] Hosoi, T., On interediate ics, Journal of the Faculty of Science, University of Tokyo, section I, 4 967) 93 3, and 6 969). Review by C. G. McKay, J. Sybolic Logic 36 97) 39). [5] Jonquiére, A., Note sur la série n= xn n s, Bulletin de la Société Mathéatique de France, 7 889) 4-5. [6] Kleene, S. C., On notation for ordinal nubers, Journal of Sybolic Logic, 3 938) [7] Kleene, S. C., Introduction to Metaatheatics, North-Holland, Asterda, 95.

6 390 Marija Boričić / A Note on Entropy of Logic [8] Lukasiewicz, J., O ice trójwartosćiowej, Ruch filozoficzny 5 90) English translation: On three-valued ic, in: L. Borkowski ed.), J. Lukasiewicz, Selected Works, North Holland, Asterda, 970, ) [9] Lukasiewicz, J., Tarski, A., Untersuchungen über den Aussagenkalkül, Coptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III 3 930) Translated in: S. R. Givant and R. N. McKenzie eds.) A. Tarski, Collected Papers, Vol., Birkhäuser, Bazel, 986, [0] McKay, C. G., On finite ics, Indagationes Matheaticae, ) Review by T. Uezawa, J. Sybolic Logic, 36 97) 330.) [] Nishiura, I., On forulas of one variable in intuitionistic propositional calculus, Journal of Sybolic Logic, 5 960) [] Priest, G., An Introduction to Non Classical Logic, Cabridge University Press, New York, 00. Second edition 008.) [3] Shepp, L., and Olkin, I., Entropy of the su of independent bernoulli rando variables and of the ultinoial distribution, in: J. Gani V. K. Rohatgi eds.) Contributions to Probability,, Acadeic Press, New York, 98, [4] Wajsberg, M., Aksjoatyzacja trójwartściowego rachunku zdań, Coptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Cl. III 3 93) English translation: Axioatization of the three-valued propositional calculus, 3-9, in: S. J. Sura ed.), Mordechaj Wajsberg. Logical Works, Polish Acadey of Sciences, Ossolineu, Wroclaw 977.)