Almost Sure Degrees of Truth and Finite Model Theory of Łukasiewicz Fuzzy Logic

Transcription

1 Almos Sure Degrees of Truh and Finie odel Theory of Łukasiewicz Fuzzy Logic Rober Kosik Insiue of Informaion Business, Vienna Universiy of Economics and Business Adminisraion, Wirschafsuniversiä Wien, Augasse 2-6, 1090 Wien, Ausria, Europe (Corresponding Auhor) and Chrisian Fermüller Insiue of Compuer Languages, Vienna Universiy of Technology Technische Universiä Wien, Favoriensraße 9, A-1040 Wien, Ausria, Europe Absrac- An imporan resul of finie model heory are he zero-one laws, which esablish ha many ypes of logical formulas hold for eiher almos all or almos no finie srucures. We sudy he generalizaion of he zero-one law of firs order classical logic o many-valued Łukasiewicz logic. Łukasiewicz logic is a basic fuzzy logic, where ruh degrees are real numbers 0 1. For he special case, where he number of ruh values is a power of 2, we prove ha o each formula φ in firs order Łukasiewicz logic here is a (unique) ruh value such ha φ has ruh degree almos surely in he sense of finie model heory. Keywords- almos sure, finie model heory, fuzzy logic, Łukasiewicz logic, zeroone law unary connecive, and he ruh consan. Formulas are buil in he usual way from an infinie se of proposiional variables. In firs order Łukasiewicz logic, we have in addiion o he connecives also quanifiers and. We will consider firs-order logic wih (crisp, i.e., wo-valued) equaliy = and unary, binary, ernary, ec. relaion symbols, bu wihou funcion symbols or individual consans. Synacical noions are defined as in he classical firs order case. B. V Algebras V-algebras are for Łukasiewicz logic wha Boolean algebras are for classical logic [1]. There is a naural V-algebra on he se T = [0, 1] of ruh degrees. We define a consan, a unary operaion and binary operaions,,, &,, on he uni inerval [0, 1] as follows: I. ŁUKASIEWICZ LOGIC According o Zadeh fuzzy logic in he narrow sense is symbolic logic wih a comparaive noion of ruh developed fully in he spiri of classical logic (synax, semanics, deducion, model heory). One example is Łukasiewicz logic, which is welldeveloped mahemaically. We sar by recalling he synax, semanics and algebra of Łukasiewicz logics. A. Basic Synax Proposiional Łukasiewicz logic uses he connecives,,, &, and, he Noe ha all operaors can be defined using and he ruh value. For example, φ can be defined as φ. By changing he se T of ruh values o we ge finie V algebras, which correspond o finie-valued Łukasiewicz logics. Inernaional Conference on IT o Celebrae S. Charmonman's 72nd Birhday, arch 2009, Thailand 20.1

2 Rober Kosik and Chrisian Fermüller C. odel Theory Le σ be a (finie) se of relaion symbols, wih an ariy aached o each relaion symbol. We call σ he signaure. A model wih signaure σ is given by a nonempy se, called he domain, ogeher wih inerpreaions R of all relaion symbols. If R is a k-ary relaion symbol, hen R is a map from k ino T. [0, 1] is defined by inducion for all closed -formulas in he naural way: We le φ be he infimum over φ Μ, aken over all fuzzy models. We say ha a closed formula φ is valid iff φ = 1, i.e., iff φ Μ = 1 for all models. I is easy o see ha formulas valid in infinie Łukasiewicz logic are also valid in all finie logics. II. ZERO-ONE LAWS Finie model heory is a subfield of model heory ha focuses on properies of logical languages, such as firs-order logic, over finie srucures, such as finie groups, graphs, daabases, and mos absrac machines. I focuses in paricular on connecions beween logical languages and compuaion, and is closely associaed wih discree mahemaics, complexiy heory, and daabase heory[3]. An imporan resul of finie model heory are he zero-one laws, which esablish ha many ypes of logical formulas hold for eiher almos all or almos no finie srucures. A. Zero-One Law for Firs Order Classical Logic A model is called finie, if is domain is a finie se. If = {1,...,m} we call a labelled finie model wih m elemens. Le φ be a closed formula in firs oder classical logic wih equaliy (wihou consans and wihou funcion symbols). Define μ m (φ) o be he fracion of labeled models wih m elemens ha saisfy he formula φ Here he hash operaor denoes cardinaliy. We say he senence φ is almos surely rue if is asympoic probabiliy is 1, ha is, if μ m (φ) converges o 1 as m goes o infiniy. Theorem 1 (Classical Zero-One-Law): Eiher φ is rue almos surely or φ is rue almos surely. A proof can be found in [3]. A quick and very readable inroducion is given in [2]. B. Generalizaion o Łukasiewicz Logic We will give he generalizaion of he classical zero-one-law o Łukasiewicz logic. This is our main heorem. Theorem 2 (Almos Sure Degree of Truh): Le φ be a closed formula in firs order manyvalued Łukasiewicz logic wih equaliy (wihou consans and wihou funcion symbols). Then here is a (unique) ruh value such ha φ Μ = is almos surely rue. Our proof relies on an encoding of manyvalued models ino classical models and a compaible ranslaion of many-valued formulas ino wo-valued formulas. An absrac and much more general form of his encoding is given in [4]. The encoding of models is given in Secion IV. The ranslaion is described in Secion V. The classical zero-one-law is applicable o ranslaed formulas. By looking a he ranslaion of several formulas in parallel we derive a proof of he main heorem in Secion VI. III. NORAL FORS These are synacical ools we will use laer for he ask of formula ranslaion as described in Equaion (5). Special Issue of he Inernaional Journal of he Compuer, he Inerne and anagemen, Vol.17 No. SP1, arch,

3 P Almos Sure Degrees of Truh and Finie odel Theory of Łukasiewicz Fuzzy Logic A. Exisence of Prenex Forms The following equivalences ("quanifier shifs") known from classical logic are also valid in infinie Łukasiewicz logic. Here, by definiion, he bound variable x does no occur free in A: bijecive, if he number of ruh values n is a power of 2. A. Basic case: one elemen, a single relaion Consider a many-valued model wih a single elemen and signaure {R Ł } (a single relaion symbol). A model Ł in n-valued logic is given by a single assignmen (where he do denoes he elemen): Using hese equivalences we can rewrie any formula as (Q x )P( x ) where Q is a chain of quanifiers (Q i x i ) and P is quanifier free. B. Disjuncive Normal Form Le φ be a closed formula of firs order n- valued Łukasiewicz logic. By he prenex normal form heorem we have Here P( x ) is free of quanifiers, i.e., purely proposiional. We look a he aomic expressions {PPi } occurring in P( x ), hey are relaional expressions of he form R(y 1,..., y k) for a k-ary relaional symbol R and free variables y i. Le N denoe he number of aoms. An assignmen A is a map from he aoms {P i } ino he se of ruh values. Here equaliy has o be reaed as a classical ({0, 1}-valued) relaion. By assigning all possible ruh values o he aoms and calculaing he value of P( x ) we form he ruh able for P( x ). From he ruh able we can exrac all assignmens A which give he ruh value. Le L denoe he number of hese assignmens. We ge: Here he variable PPi runs over all aoms. The variables ij denoe ruh values and depend on on he righ. In he classical case our consrucion corresponds o he disjuncive normal form. IV. CLASSICAL ENCODING OF ANY-VALUED ODELS In his secion we define an encoding (denoed by ilde ) of many valued models as classical models. The encoding is The aim is o give a wo-valued represenaion. The simple idea is o encode a many-valued assignmen by a se of wovalued assignmens: We wrie j as a binary number and define he classical model ~ where a i {0, 1} are he digis of j in base 2. The signaure ~ σ of he classical model consiss of l clone relaion symbols R ~ i. Wih his signaure, here are 2 l differen oneelemen models. On he oher hand using l digis all numbers 0 j (2 l 1) can be represened (leading zeros have o be allowed). Hence if he number of ruh values is a power of 2, i.e., n = 2 l, we ge a bijecion beween he se { Ł } of many-valued models and he se { ~ } of classical models. B. any Elemens, any Relaions So far, our encoding has been defined only in he case of a single elemen and a single relaion. However, exension is rivial: for any k-ary relaion symbol R and any k-uple of elemens apply exacly he same encoding, see Equaion (3) and Equaion (4). One cavea: as defined, our mapping is based on he bijecion beween numbers j, and heir binary represenaion as l-uples Inernaional Conference on IT o Celebrae S. Charmonman's 72nd Birhday, arch 2009, Thailand 20.3

4 Rober Kosik and Chrisian Fermüller Of course here are oher bijecions beween hose wo ses. I is possible o apply hem o differen aoms and combine hem. However, aoms wih free variables can only be compaibly ranslaed (see ahead for Equaion (8)), if always he same bijecion is used. I.e., he encoding of R(.) = may only depend on, bu no on he argumen (.) of R. V. COPATIBLE TRANSLATION OF ANY-VALUED FORULAS In Secion IV-A we have given a bijecive encoding of many-valued models Ł as classical models ~. The aim of his secion is o give a ranslaion of a many-valued formula ϕ o a formula ~ ϕ of firs order classical logic, which is compaible wih he encoding in he following sense: For all Ł and for all ruh values We wrie ~ ϕ o indicae ha he ranslaion of ϕ depends on he ruh degree and on he chosen bijecion beween models (bu no on a single model Ł!). In words: If ϕ has degree in Ł, hen is ranslaion ~ ϕ rue in ~. Else, if ϕ does no have degree in Ł, hen is ranslaion ~ ϕ is false in ~. A. Translaion of Aoms Firs, noe ha equaliy is a crisp relaion, i is no many-valued and he ranslaion is sraighforward, as boh ses of models share he same domain (in case = 0 negaion has o be used on he righ). We now wan o map many-valued aomic formulas o classical formulas in a compaible way, i.e.: is side rue. By consrucion he corresponding classical model ~ has assignmens where he a i are he binary digis of j (possibly including leading zeros). We wan o find a formula R ~ (.) which is only rue for hese assignmens. I is easy o see ha fullfills hese requiremens. In summary we have ( basic ranslaion"): wih a i {0, 1} he firs l digis of j in base 2, see Equaion (3). (Remark: we could find a compaible ranslaion of aoms for any bijecion beween ruh values and uples.) By he consrucion in Subsecion IV-B, he encoding of aoms only depends on he ruh value. Then we have Equaion (7) universally: Here all relaions are k-ary and x is shorhand for a lis of k free and disinc variables. B. Compaible Translaion of Formulas I is possible o reduce he ranslaion of full formulas o he aomic case by he use of equivalence ransformaions in Łukasiewicz logic. Transforming ϕ o prenex form we ge where Q is a (, )-chain. The ransformaion now goes by disincion of cases (formula sars wih or ). Firs case: ( χ ) ϕ( χ) =. This is equivalen o: j We have: for each ruh value =, here n 1 is (up o isomorphism) exacly one singleelemen model Ł which makes he lef hand Special Issue of he Inernaional Journal of he Compuer, he Inerne and anagemen, Vol.17 No. SP1, arch,

5 Almos Sure Degrees of Truh and Finie odel Theory of Łukasiewicz Fuzzy Logic Second case: analogy: ( χ ) ϕ( χ) =. We ge in We can move quanifiers ouside by repeaing hese wo seps and finally end up wih a formula which has expressions P(χ ) = as is building blocks. These expressions can be synacically ranslaed by firs bringing hem o generalized disjuncive form according o Equaion (2) and hen by compaible ranslaion of he aoms as defined in Equaion (8). VI. THE ALOST SURE DEGREE OF TRUTH -THEORE So far we have consruced an encoding of models and a ranslaion wih he following properies: Assume ha he number of ruh values n is a power of 2. Then for each ruh value and each formula ϕ of firs order n- valued Łukasiewicz logic we can find a formula ~ ϕ of classical firs order logic, such ha for all labelled models Ł : Here Ł is a many-valued model, while ~ is a classical model. They are linked by a binary encoding of ruh values on an aomic level. This encoding is inverible. They have he same domain bu differen signaures (and differen logic). For each cardinaliy of labels m, he number of models Ł is equal o he number of models ~. I is easy o see ha ~ μ m ( ϕ ) (see Equaion (1)) is he proporion of labelled manyvalued models wih ruh value, where m is he number of labels. We have Exacly one elemen ou of he se of equaions has o be rue, oherwise here is a conradicion wih he sum in Equaion (9). This proves our main heorem, Theorem 2. Hence every formula of Łukasiewicz logic has an almos sure degree of ruh. For n = 2 his is he classical zero-one-law. ACKNOWLEDGEENT Rober Kosik was parially suppored by he FWF (Ausrian Science Foundaion) Projec no. P16563-N14 and Projec no. I143-G15. Chrisian Fermüller was parially suppored by he FWF (Ausrian Science Foundaion) Projec no. I143-G15. LITERATUR [1] Per Hajek. eamahemaics of Fuzzy Logic. Springer Verlag, [2] Yuri Gurevich. Zero-One Laws. Bullein of he European Associaion for Theoreical Compuer Science, No. 51, Feb. 1991, [3] Heinz-Dieer Ebbinghaus und Jörg Flum. Finie odel Theory. Springer Verlag, [4] ahias Baaz and Chrisian Fermüller. Auomaed Deducion for any-valued Logics. Handbook of Auomaed Reasoning, Elsevier Science Publishers, 2001, Applying he classical zero-one heorem o he formula ~ ϕ, we have for each := i Inernaional Conference on IT o Celebrae S. Charmonman's 72nd Birhday, arch 2009, Thailand 20.5