JSFHPolynomizing Lukasiewicz's Many-Valued Logics by Maple Kahtan H. AlzubaidyIntroduction Polynomizing of Lukasiewicz's Many-Valued Logics 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 by certain reduced polynomials of several variables over the ring 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 This method is applicable when n is a prime or a power of a prime. We shall restrict ourselves to n = 2, 3, 4 only.Numerical representationsPropositional Lukasiewicz's many valued logics can be introduced by the matrix 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Suppose that n is a prime.Then connectives are expressed by the numeric functions.negation(ne) \342\210\274x = n-1- ximplication(im) xLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2MFEnJnJhcnI7RicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSy1JI21pR0YkNihRInlGJ0YvRjIvJSdpdGFsaWNHRjxGNUY4LUYsNjBRIj1GJ0YvRjJGNUY4RjpGPS9GQEY8RkFGQ0ZFRkdGSUZMLUZPNihRJG1pbkYnRi9GMkZSRjVGOC1JKG1mZW5jZWRHRiQ2Jy1GIzYxLUZPNihRIm5GJ0YvRjJGUkY1RjgtRiw2MFEoJm1pbnVzO0YnRi9GMkY1RjhGOkY9RldGQUZDRkVGRy9GSlEsMC4yMjIyMjIyZW1GJy9GTUZhby1JI21uR0YkNidRIjFGJ0YvRjJGNUY4LUYsNjBRIixGJ0YvRjJGNUY4RjovRj5GNEZXRkFGQ0ZFRkcvRkpRJjAuMGVtRicvRk1RLDAuMzMzMzMzM2VtRidGam5GXW9GY29GXW8tRk82KFEieEYnRi9GMkZSRjVGOC1GLDYwUSIrRidGL0YyRjVGOEY6Rj1GV0ZBRkNGRUZHRmBvRmJvRk5GL0YyRjVGOEYvRjJGNUY4Ri9GMkY1Rjg=disjunction(di) xLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2MFElJm9yO0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkstSSNtaUdGJDYoUSJ5RidGL0YyLyUnaXRhbGljR0Y8RjVGOC1GLDYwUSI9RidGL0YyRjVGOEY6Rj0vRkBGPEZBRkNGRUZHL0ZKUSwwLjI3Nzc3NzhlbUYnL0ZNRlktRk82KFEkbWF4RidGL0YyRlJGNUY4LUkobWZlbmNlZEdGJDYnLUYjNiktRk82KFEieEYnRi9GMkZSRjVGOC1GLDYwUSIsRidGL0YyRjVGOEY6L0Y+RjRGV0ZBRkNGRUZHL0ZKUSYwLjBlbUYnL0ZNUSwwLjMzMzMzMzNlbUYnRk5GL0YyRjVGOEYvRjJGNUY4Ri9GMkY1Rjg=conjunction(co) 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For LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2KFEiTEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGOi1JJW1yb3dHRiQ2Jy1JI21uR0YkNidRIjNGJ0YvRjJGOEY7Ri9GMkY4RjsvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Representations by polynomials Let n be a prime. 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 Any polynomial p in 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 ; a,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 binary connectives 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 ; 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To find now the connectivesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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conn2:=proc(ff)local DS2,Eq2;DS2 := [seq(seq([i, j], i = 0 .. 1), j = 0 .. 1)];Eq2 := [seq(p2(u[])-ff(u[]), `in`(u, DS2))];modp(Linear(Eq2, [a, b, c, d]),2);end proc;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 the connectives in 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\342\210\274x = 1+xLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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In terms of polynomial functions 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:=proc(ff)local DS3,Eq3;DS3 := [seq(seq([i, j], i = 0 .. 2), j = 0 .. 2)];Eq3:= [seq(p3(u[])-ff(u[]), `in`(u, DS3))];`mod`(Linear(Eq3, [a, b, c, d,e,f,g,h,k]), 3);end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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Therefore the connectives in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiTEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y3LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GIzYqLUkjbW5HRiQ2J1EiM0YnRjJGNS9GO0Y/Rj1GMkY1RjgvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y3RjpGPS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2MFEifkYnRjJGNUZGRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlUvJSlzdHJldGNoeUdGVS8lKnN5bW1ldHJpY0dGVS8lKGxhcmdlb3BHRlUvJS5tb3ZhYmxlbGltaXRzR0ZVLyUnYWNjZW50R0ZVLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGXm8tRlA2MFEjaW5GJ0YyRjVGRkY9RlNGVkZYRlpGZm5GaG5Gam5GXG9GX29GTy1GLzYoUSZ0ZXJtc0YnRjJGNS9GOUZVRkZGPUZPLUYvNihRI29mRidGMkY1RmdvRkZGPUZPLUYvNihRLHBvbHlub21pYWxzRidGMkY1RmdvRkZGPUZPLUYvNihRJGFyZUYnRjJGNUZnb0ZGRj1GTy1GLzYoUSZnaXZlbkYnRjJGNUZnb0ZGRj1GTy1GUDYwUSNieUYnRjJGNUZGRj1GU0ZWRlhGWkZmbkZobkZqbkZcb0Zfb0ZPRjJGNUZGRj0= \342\210\274x = 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 xLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbW9HRiQ2MFExJkxlZnRSaWdodEFycm93O0YnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkstSSNtaUdGJDYoUSJ5RidGL0YyLyUnaXRhbGljR0Y8RjVGOC1GLDYwUSI9RidGL0YyRjVGOEY6Rj0vRkBGPEZBRkNGRUZHRklGTC1JJW1zdXBHRiQ2JS1GTzYoUSJ4RidGL0YyRlJGNUY4LUYjNiktSSNtbkdGJDYnUSIyRidGL0YyRjVGOEYvRjIvJStleGVjdXRhYmxlR0Y0LyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Y1RjgvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUZZNiVGTkZobkZjby1GLDYwUSIrRidGL0YyRjVGOEY6Rj1GV0ZBRkNGRUZHL0ZKUSwwLjIyMjIyMjJlbUYnL0ZNRlxwRlhGTkZoby1GWTYlLUZPNihRI3h5RidGL0YyRlJGNUY4RmhuRmNvRmhvRmpuLUYsNjBRIn5GJ0YvRjJGNUY4RjpGPUZXRkFGQ0ZFRkcvRkpRJjAuMGVtRicvRk1GZ3BGYHBGaG9Gam5GY3BGZW5GaG9Gam5GY3BGTkZob0ZqbkYvRjJGNUY4In terms of polynomial functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVElcG5lM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInhGJ0YvRjItRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUkjbW5HRiQ2JFEiMkYnRjktRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZqbkZPLUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGVUZNLyUrZXhlY3V0YWJsZUdGPUY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZELUkjbWlHRiQ2JVElcGltM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYkLUYjNiYtRiw2JVEieEYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGOUY5LUY2Ni1RKCZzcmFycjtGJ0Y5RjtGPkZARkJGREZGRkhGZW4vRk5GZm4tSSNtbkdGJDYkUSIyRidGOS1GNjYtUScmc2RvdDtGJ0Y5RjtGPkZARkJGREZGRkhGZW5GX28tSSVtc3VwR0YkNiVGVC1GIzYkRmBvRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRmRvLUZobzYlRmluRmpvRlxwLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GZXBGYG9GZG9GZ29GZG9GaW5GYXBGYG9GZG9GVEZkb0ZfcEZhcEZURmRvRmluRmFwRmBvRmRvRlRGYXBGYG8tRjY2LVEiO0YnRjlGO0ZaRkBGQkZERkZGSEZlbkZNLyUrZXhlY3V0YWJsZUdGPUY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZALUkjbWlHRiQ2JVElcGRpM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEieEYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOS1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRmVuL0ZORmZuLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGZW5GYW8tSSVtc3VwR0YkNiVGVC1GIzYkLUkjbW5HRiQ2JFEiMkYnRjlGOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIRmVuRmFvLUZmbzYlRmluRmhvRl5wLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GanBGZW9GYXBGaW5GZnBGVEZhcEZkcEZmcEZqb0ZhcEZib0ZURmFwRmluRmZwRlRGZnBGaW4tRjY2LVEiO0YnRjlGO0ZaRkBGQkZERkZGSEZlbkZNRlxvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JVElcGNvM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEieEYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOS1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRmVuL0ZORmZuLUkjbW5HRiQ2JFEiMkYnRjktRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIRmVuRmFvLUklbXN1cEdGJDYlRlQtRiM2JEZib0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Zmby1Gam82JUZpbkZccEZecC1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmdwRmJvRmZvRmlvRmZvRmluRmNwRmJvRmZvRlRGZm9GYXBGY3BGVEZmb0Zpbi1GNjYtUSI7RidGOUY7RlpGQEZCRkRGRkZIRmVuRk1GXG9GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZFLUkjbWlHRiQ2JVElcGJpM0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEieEYnRi9GMi1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0Y9RjlGOS1GNjYtUSgmc3JhcnI7RidGOUY7Rj5GQEZCRkRGRkZIRmVuL0ZORmZuLUklbXN1cEdGJDYlRlQtRiM2JC1JI21uR0YkNiRRIjJGJ0Y5RjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZlbkZhby1GY282JUZpbkZlb0ZbcC1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmdwRmJvRl5wRmluRmNwRlRGXnBGYXBGY3BGZ29GXnAtRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZlbkZhb0ZURl5wRmluRmNwRmdvRl5wRlRGY3BGZ29GXnBGaW5GY3BGZ28tRjY2LVEiO0YnRjlGO0ZaRkBGQkZERkZGSEZlbkZNRlxvRjk= Checking tautologies The following procedure based on Groebner basis can compute compound proposition as a reduced polynomial in 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 The proposition is a tautolog if the result is 1 for 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 In the following procedure ev (evaluate). p = 2, 3 primes V is the list of variables x,y,z,... q is the proposition as a polynomial in x, y, z, ...LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRKUdyb2VibmVyRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC8lK2V4ZWN1dGFibGVHRkVGPQ==ev:=proc(p,V,q)local n,W,K,B;n:=nops(V);W:=[seq(V[i],i=1..n)];K:=[seq(x^p-x,x in W)];B:=Basis(K,tdeg(seq(V[i],i=1..n)));modp(NormalForm(q,B,tdeg(seq(V[i],i=1..n))),p);end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiTEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPS1GIzYqLUkjbW5HRiQ2J1EiMkYnRjJGNUY7Rj4tSSNtb0dGJDYwUSJ+RidGMkY1RjtGPi8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlpGMkY1LyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGN0Y7Rj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZGLUYvNihRJ2F4aW9tc0YnRjJGNUY4RjtGPkYyRjVGO0Y+1) x\342\206\222(y\342\206\222x)2) (x\342\206\222(y\342\206\222z)\342\206\222((x\342\206\222y)\342\206\222(x\342\206\222z))3) (\342\210\274x\342\206\222\342\210\274y)\342\206\222(y\342\206\222x)To verify the second axiom for example. The other two axioms are similar.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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) x\342\206\222(y\342\206\222x)2) (x\342\206\222y)\342\206\222((y\342\206\222z)\342\206\222(x\342\206\222z)) 3) (\342\210\274x\342\206\222\342\210\274y)\342\206\222(y\342\206\222x)4) ((x\342\206\222\342\210\274x )\342\206\222x)\342\206\222xTo verify the fourth axiom. The rest are similar. 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 4-valued Logic LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2KVEiTEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lKnVuZGVybGluZUdGNy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPy1GIzYqLUkjbW5HRiQ2KFEiNEYnRjJGNUY7Rj1GQEYyRjVGOy8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjdGPUZALyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkY1RjtGPUZAThe truth values of propositional Lukasiewicz's 4- valued logic 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GF(4)=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Minimum and maximum functions on GF(4) are given by the following two proceddures.m:=proc(u,v)local r;r:=min(u,v);r:=eval(subs(t=2,r));subs(2=t,3=t+1,r);end proc;M:=proc(u,v)local r;r:=max(u,v);r:=eval(subs(t=2,r));subs(2=t,3=t+1,r);end proc;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 The order on GF(4) is therefore given by 0 < 1 < t < t+1 . The connectives are given by the following 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Consider the following 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The reduced polynomial when n = 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conn4:=proc(ff)local DS4,Eq4;DS4 := [seq(seq([i, j], `in`(i, L4)), `in`(j, L4))];Eq4:= [seq(p4(u[])-ff(u[]), `in`(u, DS4))];`mod`(Linear(Eq4, CC), 2);end proc;JSFH The 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Connectives in terms of 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 following procedure evaluate compound propositions in terms of reduced polynomials in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiTEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPS1GIzYpLUkjbW5HRiQ2J1EiNEYnRjJGNUY7Rj5GMkY1LyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGN0Y7Rj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNjBRIn5GJ0YyRjVGO0Y+LyUmZmVuY2VHRjovJSpzZXBhcmF0b3JHRjovJSlzdHJldGNoeUdGOi8lKnN5bW1ldHJpY0dGOi8lKGxhcmdlb3BHRjovJS5tb3ZhYmxlbGltaXRzR0Y6LyUnYWNjZW50R0Y6LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGXG8tRk82MFEiLkYnRjJGNUY7Rj5GUkZURlZGWEZaRmZuRmhuRmpuRl1vRk5GMkY1RjtGPg== p = 2 and V, q as above in ev.ev4:=proc(p,V,q)local n,W,K,B;n:=nops(V);W:=[seq(V[i],i=1..n)];K:=[seq(x^(2*p)-x,x in W)];B:=Basis(K,tdeg(seq(V[i],i=1..n)));modp(NormalForm(q,B,tdeg(seq(V[i],i=1..n))),p);end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Checking tautologies in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiTEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y3LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GIzYqLUkjbW5HRiQ2J1EiNEYnRjJGNS9GO0Y/Rj1GMkY1RjgvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y3RjpGPS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjJGNUZGRj0= The above proceduer ev4 can be used to check tautologies in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZGLUklbXN1YkdGJDYlLUkjbWlHRiQ2KFEiTEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y3LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GIzYrLUkjbW5HRiQ2J1EiNEYnRjJGNS9GO0Y/Rj1GMkY1RjgvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y3LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjpGPS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LlEiLkYnRjIvRjtRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZaLyUpc3RyZXRjaHlHRlovJSpzeW1tZXRyaWNHRlovJShsYXJnZW9wR0ZaLyUubW92YWJsZWxpbWl0c0dGWi8lJ2FjY2VudEdGWi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmNvLUZTNjBRIn5GJ0YyRjVGRkY9RlhGZW5GZ25GaW5GW29GXW9GX29GYW9GZG9GZm8tRlM2LkZob0YyRlZGWEZlbkZnbkZpbkZbb0Zdb0Zfb0Zhb0Zkby1GLzYoUSNJZkYnRjJGNS9GOUZaRkZGPUZmby1GLzYoUSR0aGVGJ0YyRjVGXnBGRkY9RmZvLUYvNihRJ3Jlc3VsdEYnRjJGNUZecEZGRj1GZm8tRi82KFEjaXNGJ0YyRjVGXnBGRkY9RmZvRmZvLUYvNihRInRGJ0YyRjVGXnBGRkY9LUZTNjBRIitGJ0YyRjVGRkY9RlhGZW5GZ25GaW5GW29GXW9GX28vRmJvUSwwLjIyMjIyMjJlbUYnL0Zlb0ZfcS1GQzYnUSIxRidGMkY1RkZGPS1GUzYuUSIsRidGMkZWRlgvRmZuRjdGZ25GaW5GW29GXW9GX29GYW8vRmVvUSwwLjMzMzMzMzNlbUYnRmlvLUZTNjBRJXRoZW5GJ0YyRjVGRkY9RlhGZW5GZ25GaW5GW29GXW9GX29GYW9GZG9GZm9GX3BGZm8tRi82KFEscHJvcG9zaXRpb25GJ0YyRjVGXnBGRkY9RmZvRmVwRmZvLUYvNihRImFGJ0YyRjVGXnBGRkY9RmZvLUYvNihRKnRhdXRvbG9neUYnRjJGNUZecEZGRj0tRlM2MEZVRjJGNUZGRj1GWEZlbkZnbkZpbkZbb0Zdb0Zfb0Zhb0Zkb0YyRjVGTEZGRj0= Examples 1) x \342\206\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 2) x\342\206\222(y\342\206\222x)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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 3) x\342\210\250\342\210\274x is not tautology in 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JSFH Remark We can deduce from \342\206\222 the other connectives \342\210\250,\342\210\247,\342\206\224 . For example x\342\210\250y = (x\342\206\222y)\342\206\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 References1) M. Bergmann Many-Valued and Fuzzy Logic Cambridge 2008. 2) Walter Carnielli Polynomial Ring Calculus for Many-Valued Logics Proceedings of the 35th International Symposium on Multiple-Valued Logic (ISMVL\342\200\23105)