1. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineComputing under VaguenessApostolos SyropoulosXanthi, Greeceasyropoulos@yahoo.comSecond international workshopNew Worlds of Computation 2011LIFO, University of Orlans (France)Monday, May 23, 2011. . . . . .

2. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Outline1. VaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)sets2. Fuzzy Turing Machines3. Fuzzy P Systems4. Fuzzy Chemical Abstract Machine 3. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness? 4. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases. 5. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?. 6. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term. 7. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term.The adjective tall is a typical example of a vague term. 8. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term.The adjective tall is a typical example of a vague term.Examples 9. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term.The adjective tall is a typical example of a vague term.ExamplesSerena is 1,75 m tall, is she tall? 10. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term.The adjective tall is a typical example of a vague term.ExamplesSerena is 1,75 m tall, is she tall? 11. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term.The adjective tall is a typical example of a vague term.ExamplesSerena is 1,75 m tall, is she tall?Is Betelgeuse a huge star? 12. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term.The adjective tall is a typical example of a vague term.ExamplesSerena is 1,75 m tall, is she tall?Is Betelgeuse a huge star?If you cut one head off of a two headed man, have youdecapitated him? 13. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term.The adjective tall is a typical example of a vague term.ExamplesSerena is 1,75 m tall, is she tall?Is Betelgeuse a huge star?If you cut one head off of a two headed man, have youdecapitated him?Eubulides of Miletus: The Sorites Paradox. 14. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .What is Vagueness?It is widely accepted that a term is vague to the extent that ithas borderline cases.What is a borderline case?.A case in which it seems impossible either to apply or not toapply a vague term.The adjective tall is a typical example of a vague term.ExamplesSerena is 1,75 m tall, is she tall?Is Betelgeuse a huge star?If you cut one head off of a two headed man, have youdecapitated him?Eubulides of Miletus: The Sorites Paradox.The old puzzle about bald people. 15. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Defining Vagueness 16. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Defining VaguenessBertrand Russell: Per contra, a representation is vague whenthe relation of the representing system to the representedsystem is not one-one, but one-many. 17. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Defining VaguenessBertrand Russell: Per contra, a representation is vague whenthe relation of the representing system to the representedsystem is not one-one, but one-many.Example: A small-scale map is usually vaguer than a large-scalemap. 18. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Defining VaguenessBertrand Russell: Per contra, a representation is vague whenthe relation of the representing system to the representedsystem is not one-one, but one-many.Example: A small-scale map is usually vaguer than a large-scalemap.Charles Sanders Peirce: A proposition is vague when there arepossible states of things concerning which it is intrinsicallyuncertain whether, had they been contemplated by thespeaker, he would have regarded them as excluded orallowed by the proposition. By intrinsically uncertain wemean not uncertain in consequence of any ignorance of theinterpreter, but because the speakers habits of language wereindeterminate. 19. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Defining VaguenessBertrand Russell: Per contra, a representation is vague whenthe relation of the representing system to the representedsystem is not one-one, but one-many.Example: A small-scale map is usually vaguer than a large-scalemap.Charles Sanders Peirce: A proposition is vague when there arepossible states of things concerning which it is intrinsicallyuncertain whether, had they been contemplated by thespeaker, he would have regarded them as excluded orallowed by the proposition. By intrinsically uncertain wemean not uncertain in consequence of any ignorance of theinterpreter, but because the speakers habits of language wereindeterminate.Max Black demonstrated this definition using the word chair. 20. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineVagueness, Generality, and Ambiguity. . . . . . 21. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineVagueness, Generality, and AmbiguityVagueness, ambibuity, and generality are entirely differentnotions.. . . . . . 22. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineVagueness, Generality, and AmbiguityVagueness, ambibuity, and generality are entirely differentnotions.A term of phrase is ambiguous if it has at least two specificmeanings that make sense in context.. . . . . . 23. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineVagueness, Generality, and AmbiguityVagueness, ambibuity, and generality are entirely differentnotions.A term of phrase is ambiguous if it has at least two specificmeanings that make sense in context.Example: He ate the cookies on the couch.. . . . . . 24. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineVagueness, Generality, and AmbiguityVagueness, ambibuity, and generality are entirely differentnotions.A term of phrase is ambiguous if it has at least two specificmeanings that make sense in context.Example: He ate the cookies on the couch.Bertrand Russell: A proposition involving a generalconcepte.g., This is a manwill be verified by a number offacts, such as This being Brown or Jones or Robinson.. . . . . . 25. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineVagueness, Generality, and AmbiguityVagueness, ambibuity, and generality are entirely differentnotions.A term of phrase is ambiguous if it has at least two specificmeanings that make sense in context.Example: He ate the cookies on the couch.Bertrand Russell: A proposition involving a generalconcepte.g., This is a manwill be verified by a number offacts, such as This being Brown or Jones or Robinson.Example: Again consider the word chair.. . . . . . 26. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineDifferent Expressions of Vagueness. . . . . . 27. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineDifferent Expressions of VaguenessMany-valued Logics and Fuzzy Set Theory. . . . . . 28. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineDifferent Expressions of VaguenessMany-valued Logics and Fuzzy Set TheorySupervaluationism. . . . . . 29. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineDifferent Expressions of VaguenessMany-valued Logics and Fuzzy Set TheorySupervaluationismContextualism. . . . . . 30. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineMany-valued Logics and Fuzzy Set Theory. . . . . . 31. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineMany-valued Logics and Fuzzy Set TheoryAristotle in (De Interpretatione) discussed theproblem of future contingencies (i.e., what is the truth value ofa proposition about a future event?).. . . . . . 32. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineMany-valued Logics and Fuzzy Set TheoryAristotle in (De Interpretatione) discussed theproblem of future contingencies (i.e., what is the truth value ofa proposition about a future event?).Can someone tell us what is the truth value of the propositionThe Higgs boson exists?. . . . . . 33. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineMany-valued Logics and Fuzzy Set TheoryAristotle in (De Interpretatione) discussed theproblem of future contingencies (i.e., what is the truth value ofa proposition about a future event?).Can someone tell us what is the truth value of the propositionThe Higgs boson exists?Jan ukasiewicz defined a three-valued logic to deal with futurecontingencies.. . . . . . 34. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineMany-valued Logics and Fuzzy Set TheoryAristotle in (De Interpretatione) discussed theproblem of future contingencies (i.e., what is the truth value ofa proposition about a future event?).Can someone tell us what is the truth value of the propositionThe Higgs boson exists?Jan ukasiewicz defined a three-valued logic to deal with futurecontingencies.Emil Leon Post introduced the formulation of truth values,where .. . . . . . 35. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineMany-valued Logics and Fuzzy Set TheoryAristotle in (De Interpretatione) discussed theproblem of future contingencies (i.e., what is the truth value ofa proposition about a future event?).Can someone tell us what is the truth value of the propositionThe Higgs boson exists?Jan ukasiewicz defined a three-valued logic to deal with futurecontingencies.Emil Leon Post introduced the formulation of truth values,where .C.C. Chang proposed MV-algebras, that is, an algebraic modelof many-valued logics.. . . . . . 36. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineMany-valued Logics and Fuzzy Set TheoryAristotle in (De Interpretatione) discussed theproblem of future contingencies (i.e., what is the truth value ofa proposition about a future event?).Can someone tell us what is the truth value of the propositionThe Higgs boson exists?Jan ukasiewicz defined a three-valued logic to deal with futurecontingencies.Emil Leon Post introduced the formulation of truth values,where .C.C. Chang proposed MV-algebras, that is, an algebraic modelof many-valued logics.In 1965, Lotfi A. Zadeh introduced his fuzzy set theory andits accompanying fuzzy logic.. . . . . . 37. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Supervaluationism 38. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .SupervaluationismSupervaluationism is a special kind of classical indeterminacytheory. 39. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .SupervaluationismSupervaluationism is a special kind of classical indeterminacytheory.Each predicate is associated with two non-overlapping sets+(), the deterninate extension, and (), the determinateanti-extension. 40. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .SupervaluationismSupervaluationism is a special kind of classical indeterminacytheory.Each predicate is associated with two non-overlapping sets+(), the deterninate extension, and (), the determinateanti-extension.The candidate extension of are sets that extend +()and are disjoint from (). 41. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .SupervaluationismSupervaluationism is a special kind of classical indeterminacytheory.Each predicate is associated with two non-overlapping sets+(), the deterninate extension, and (), the determinateanti-extension.The candidate extension of are sets that extend +()and are disjoint from ().Truth is defined relative to each choice of candidateextensions. 42. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .SupervaluationismSupervaluationism is a special kind of classical indeterminacytheory.Each predicate is associated with two non-overlapping sets+(), the deterninate extension, and (), the determinateanti-extension.The candidate extension of are sets that extend +()and are disjoint from ().Truth is defined relative to each choice of candidateextensions.A sentense is super-true if it comes out true realtive to eachchoice of candidate extensions for its predicates. 43. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Contextualism 44. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .ContextualismThe correct application of a vague predicate varies with context. 45. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .ContextualismThe correct application of a vague predicate varies with context.Example: A person may be tall relative to American men butshort relative to NBA players. 46. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .ContextualismThe correct application of a vague predicate varies with context.Example: A person may be tall relative to American men butshort relative to NBA players.Epistemological contextualism maintains the general idea thatwhether one knows is somehow relative to context. 47. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy (Sub)sets 48. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy (Sub)setsFuzzy set theory is based on the idea that elements maybelong to sets to a degree 49. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy (Sub)setsFuzzy set theory is based on the idea that elements maybelong to sets to a degreeGiven a universe , a fuzzy subset of is characterized bya function , = [, ], where () = means that belongs to with degreee equal to . 50. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy (Sub)setsFuzzy set theory is based on the idea that elements maybelong to sets to a degreeGiven a universe , a fuzzy subset of is characterized bya function , = [, ], where () = means that belongs to with degreee equal to .Example: One can easily build a fuzzy subset of the tallstudents of a class. 51. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy (Sub)setsFuzzy set theory is based on the idea that elements maybelong to sets to a degreeGiven a universe , a fuzzy subset of is characterized bya function , = [, ], where () = means that belongs to with degreee equal to .Example: One can easily build a fuzzy subset of the tallstudents of a class.When () = we say that belongs to with degree equalto not that does not belong to ! 52. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy (Sub)setsFuzzy set theory is based on the idea that elements maybelong to sets to a degreeGiven a universe , a fuzzy subset of is characterized bya function , = [, ], where () = means that belongs to with degreee equal to .Example: One can easily build a fuzzy subset of the tallstudents of a class.When () = we say that belongs to with degree equalto not that does not belong to !A consequence of this is that one cannot build a topos of allfuzzy subsets. 53. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineOperations between Fuzzy Subsets. . . . . . 54. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineOperations between Fuzzy SubsetsAssume that and are two fuzzy subsets of . Then. . . . . . 55. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineOperations between Fuzzy SubsetsAssume that and are two fuzzy subsets of . Then. . . . . .their union is( )() = (), (); 56. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineOperations between Fuzzy SubsetsAssume that and are two fuzzy subsets of . Then. . . . . .their union is( )() = (), ();their intersection is( )() = (), (); 57. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineOperations between Fuzzy SubsetsAssume that and are two fuzzy subsets of . Then. . . . . .their union is( )() = (), ();their intersection is( )() = (), ();the complement of is () for ; 58. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineOperations between Fuzzy SubsetsAssume that and are two fuzzy subsets of . Then. . . . . .their union is( )() = (), ();their intersection is( )() = (), ();the complement of is () for ; is a subset of and write iff() () and; 59. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineOperations between Fuzzy SubsetsAssume that and are two fuzzy subsets of . Then. . . . . .their union is( )() = (), ();their intersection is( )() = (), ();the complement of is () for ; is a subset of and write iff() () and;the scalar cardinality of is|| = (). 60. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Set Theory vs. Probability Theory. . . . . . 61. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Set Theory vs. Probability TheoryLets transform the statementSerena is tall to a degree of0,70 into the Serena is 70% tall!. . . . . . 62. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Set Theory vs. Probability TheoryLets transform the statementSerena is tall to a degree of0,70 into the Serena is 70% tall!So is fuzzy sets just an elementary mistake of logic?. . . . . . 63. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Set Theory vs. Probability TheoryLets transform the statementSerena is tall to a degree of0,70 into the Serena is 70% tall!So is fuzzy sets just an elementary mistake of logic?The statement Serena is 70% tall can be either true or false!. . . . . . 64. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Set Theory vs. Probability TheoryLets transform the statementSerena is tall to a degree of0,70 into the Serena is 70% tall!So is fuzzy sets just an elementary mistake of logic?The statement Serena is 70% tall can be either true or false!he statement Serena is tall to a degree of 0,70 says thatSerena is tall and the truth value of this statement is 0,70!. . . . . . 65. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Set Theory vs. Probability TheoryLets transform the statementSerena is tall to a degree of0,70 into the Serena is 70% tall!So is fuzzy sets just an elementary mistake of logic?The statement Serena is 70% tall can be either true or false!he statement Serena is tall to a degree of 0,70 says thatSerena is tall and the truth value of this statement is 0,70!How can we express in probability theory judgments liketomorrow will be a warm day or there will be a strongearthquake in the near future, etc.?. . . . . . 66. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Set Theory vs. Probability TheoryLets transform the statementSerena is tall to a degree of0,70 into the Serena is 70% tall!So is fuzzy sets just an elementary mistake of logic?The statement Serena is 70% tall can be either true or false!he statement Serena is tall to a degree of 0,70 says thatSerena is tall and the truth value of this statement is 0,70!How can we express in probability theory judgments liketomorrow will be a warm day or there will be a strongearthquake in the near future, etc.?It it possible to perform estimations that involve fuzzyprobabilities expressed as likely, unlikely, not very likely, etc.?. . . . . . 67. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Set Theory vs. Probability TheoryLets transform the statementSerena is tall to a degree of0,70 into the Serena is 70% tall!So is fuzzy sets just an elementary mistake of logic?The statement Serena is 70% tall can be either true or false!he statement Serena is tall to a degree of 0,70 says thatSerena is tall and the truth value of this statement is 0,70!How can we express in probability theory judgments liketomorrow will be a warm day or there will be a strongearthquake in the near future, etc.?It it possible to perform estimations that involve fuzzyprobabilities expressed as likely, unlikely, not very likely, etc.?Bart Kosko advocates the idea that fuzziness is a basiccharacteristic of our cosmos! (I agree with Kosko!). . . . . . 68. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineZadehs Fuzzy Algorithms and Turing Machines. . . . . . 69. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineZadehs Fuzzy Algorithms and Turing MachinesIn 1968, Zadeh informally introduced the notions of fuzzyalgorithm and fuzzy Turing machine.. . . . . . 70. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineZadehs Fuzzy Algorithms and Turing MachinesIn 1968, Zadeh informally introduced the notions of fuzzyalgorithm and fuzzy Turing machine.A fuzzy algorithm contains fuzzy commands.. . . . . . 71. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineZadehs Fuzzy Algorithms and Turing MachinesIn 1968, Zadeh informally introduced the notions of fuzzyalgorithm and fuzzy Turing machine.A fuzzy algorithm contains fuzzy commands.A fuzzy command uses fuzzy sets: Make approximatelyequal to , if is approximately equal to .. . . . . . 72. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineZadehs Fuzzy Algorithms and Turing MachinesIn 1968, Zadeh informally introduced the notions of fuzzyalgorithm and fuzzy Turing machine.A fuzzy algorithm contains fuzzy commands.A fuzzy command uses fuzzy sets: Make approximatelyequal to , if is approximately equal to .Zadeh argued that everyday activities can be viewed as fuzzyalgorithms, but Carol E. Cleland argued that recipes arealgorithms!!!. . . . . . 73. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineZadehs Fuzzy Algorithms and Turing MachinesIn 1968, Zadeh informally introduced the notions of fuzzyalgorithm and fuzzy Turing machine.A fuzzy algorithm contains fuzzy commands.A fuzzy command uses fuzzy sets: Make approximatelyequal to , if is approximately equal to .Zadeh argued that everyday activities can be viewed as fuzzyalgorithms, but Carol E. Cleland argued that recipes arealgorithms!!!A fuzzy Turing machine is Turing machine where+ = (, ) is associated with a feasability degree.. . . . . . 74. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy Algorithms 75. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy AlgorithmsShi-Kuo Chang used a finite-state machine and a fuzzymachine to define the execution of fuzzy algorithms. 76. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy AlgorithmsShi-Kuo Chang used a finite-state machine and a fuzzymachine to define the execution of fuzzy algorithms.Kokichi Tanaka and Masaharu Mizumoto defined an extendedfuzzy machine, which is based on a generalization of Changsmachine. 77. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy AlgorithmsShi-Kuo Chang used a finite-state machine and a fuzzymachine to define the execution of fuzzy algorithms.Kokichi Tanaka and Masaharu Mizumoto defined an extendedfuzzy machine, which is based on a generalization of Changsmachine.Eugene S. Santos reformulated fuzzy programs using DanaScotts work and defined W-machines that are able to executethem. 78. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy AlgorithmsShi-Kuo Chang used a finite-state machine and a fuzzymachine to define the execution of fuzzy algorithms.Kokichi Tanaka and Masaharu Mizumoto defined an extendedfuzzy machine, which is based on a generalization of Changsmachine.Eugene S. Santos reformulated fuzzy programs using DanaScotts work and defined W-machines that are able to executethem.The formulations of Chang and Tanaka-Mizumoto are specialcases of Santoss formulation. 79. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing Machines 80. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing MachinesA Santos type fuzzy Turing machine is a septuple(, , , , , , ) where: 81. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing MachinesA Santos type fuzzy Turing machine is a septuple(, , , , , , ) where: represents a finite non-empty set of input symbols; 82. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing MachinesA Santos type fuzzy Turing machine is a septuple(, , , , , , ) where: represents a finite non-empty set of input symbols; denotes a finite non-empty set of states such that = ; 83. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing MachinesA Santos type fuzzy Turing machine is a septuple(, , , , , , ) where: represents a finite non-empty set of input symbols; denotes a finite non-empty set of states such that = ;, are symbols designating the initial and final state,respectively; 84. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing MachinesA Santos type fuzzy Turing machine is a septuple(, , , , , , ) where: represents a finite non-empty set of input symbols; denotes a finite non-empty set of states such that = ;, are symbols designating the initial and final state,respectively; ( ) ( ( {, , })) is the next-move relation; 85. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing MachinesA Santos type fuzzy Turing machine is a septuple(, , , , , , ) where: represents a finite non-empty set of input symbols; denotes a finite non-empty set of states such that = ;, are symbols designating the initial and final state,respectively; ( ) ( ( {, , })) is the next-move relation; is the semiring (, , ); 86. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing MachinesA Santos type fuzzy Turing machine is a septuple(, , , , , , ) where: represents a finite non-empty set of input symbols; denotes a finite non-empty set of states such that = ;, are symbols designating the initial and final state,respectively; ( ) ( ( {, , })) is the next-move relation; is the semiring (, , ); ( ) ( ( {, , })) is a W-function thatassigns a degree of certainty to each machine transition. 87. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Fuzzy Turing MachinesA Santos type fuzzy Turing machine is a septuple(, , , , , , ) where: represents a finite non-empty set of input symbols; denotes a finite non-empty set of states such that = ;, are symbols designating the initial and final state,respectively; ( ) ( ( {, , })) is the next-move relation; is the semiring (, , ); ( ) ( ( {, , })) is a W-function thatassigns a degree of certainty to each machine transition.Nowadays, we use t-norms and t-conorms instead of semirings. 88. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineResults Concerning Fuzzy Turing Machines. . . . . . 89. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineResults Concerning Fuzzy Turing MachinesThere is no universal fuzzy Turing machine!. . . . . . 90. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineResults Concerning Fuzzy Turing MachinesThere is no universal fuzzy Turing machine!Selim Akl had argued that the concept of a universal computercannot be realized.. . . . . . 91. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineResults Concerning Fuzzy Turing MachinesThere is no universal fuzzy Turing machine!Selim Akl had argued that the concept of a universal computercannot be realized.Ji Wiedermann had shown that fuzzy languages accepted bythese machines with a computable t-norm correspond exactly tothe union of recursively enumerable languages and. . . . . .their complements. 92. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineResults Concerning Fuzzy Turing MachinesThere is no universal fuzzy Turing machine!Selim Akl had argued that the concept of a universal computercannot be realized.Ji Wiedermann had shown that fuzzy languages accepted bythese machines with a computable t-norm correspond exactly tothe union of recursively enumerable languages andtheir complements.Benjamn Callejas Bedregal and Santiago Figueira argue thatthey have found flaws in Wiedermann proof though they didnot refute his original argument.. . . . . . 93. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineResults Concerning Fuzzy Turing MachinesThere is no universal fuzzy Turing machine!Selim Akl had argued that the concept of a universal computercannot be realized.Ji Wiedermann had shown that fuzzy languages accepted bythese machines with a computable t-norm correspond exactly tothe union of recursively enumerable languages andtheir complements.Benjamn Callejas Bedregal and Santiago Figueira argue thatthey have found flaws in Wiedermann proof though they didnot refute his original argument.Later on Yongming Li showed that Wiedermanns results is truefor -fuzzy Turing machines.. . . . . . 94. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems in a Nutshell 95. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems in a NutshellA model of computation mimicking the way cells functionwhich was proposed by Gheorghe Pun. 96. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems in a NutshellA model of computation mimicking the way cells functionwhich was proposed by Gheorghe Pun.A compartment is space that is surrounded by a porousmembrane. 97. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems in a NutshellA model of computation mimicking the way cells functionwhich was proposed by Gheorghe Pun.A compartment is space that is surrounded by a porousmembrane.Nested compartments make a P system. Each compartment isa multiset. 98. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems in a NutshellA model of computation mimicking the way cells functionwhich was proposed by Gheorghe Pun.A compartment is space that is surrounded by a porousmembrane.Nested compartments make a P system. Each compartment isa multiset.Data flow from one compartment to another according to somemultiset processing rules. 99. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems in a NutshellA model of computation mimicking the way cells functionwhich was proposed by Gheorghe Pun.A compartment is space that is surrounded by a porousmembrane.Nested compartments make a P system. Each compartment isa multiset.Data flow from one compartment to another according to somemultiset processing rules.Computation stops when no rule can be applied. 100. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems in a NutshellA model of computation mimicking the way cells functionwhich was proposed by Gheorghe Pun.A compartment is space that is surrounded by a porousmembrane.Nested compartments make a P system. Each compartment isa multiset.Data flow from one compartment to another according to somemultiset processing rules.Computation stops when no rule can be applied.The result of the computation is equal to the cardinality of themultiset contained in a special compartment called the outputcompartment. 101. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems in a NutshellA model of computation mimicking the way cells functionwhich was proposed by Gheorghe Pun.A compartment is space that is surrounded by a porousmembrane.Nested compartments make a P system. Each compartment isa multiset.Data flow from one compartment to another according to somemultiset processing rules.Computation stops when no rule can be applied.The result of the computation is equal to the cardinality of themultiset contained in a special compartment called the outputcompartment.Processing takes place in discrete steps but rules are applied inparallel. 102. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems with Fuzzy Data 103. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems with Fuzzy DataA P system with fuzzy data is a n-tuple(, , (), , (), , , , )where 104. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems with Fuzzy DataA P system with fuzzy data is a n-tuple(, , (), , (), , , , )where is the alphabet of objects; 105. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems with Fuzzy DataA P system with fuzzy data is a n-tuple(, , (), , (), , , , )where is the alphabet of objects; is the membrane structure; 106. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems with Fuzzy DataA P system with fuzzy data is a n-tuple(, , (), , (), , , , )where is the alphabet of objects; is the membrane structure;() , , represent multi-fuzzy sets over associated with the region surrounded by membrane ; 107. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems with Fuzzy DataA P system with fuzzy data is a n-tuple(, , (), , (), , , , )where is the alphabet of objects; is the membrane structure;() , , represent multi-fuzzy sets over associated with the region surrounded by membrane ;, , are finite sets of multiset rewriting rules of theform , and , where = ,and = {, } {| }. 108. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .P Systems with Fuzzy DataA P system with fuzzy data is a n-tuple(, , (), , (), , , , )where is the alphabet of objects; is the membrane structure;() , , represent multi-fuzzy sets over associated with the region surrounded by membrane ;, , are finite sets of multiset rewriting rules of theform , and , where = ,and = {, } {| }. {, , , } is the label of a compartment called the outputmembrane. 109. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy P Systems 110. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy P SystemsA P system is fuzzy when it uses fuzzy multisets, fuzzy rewriterules or both. 111. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy P SystemsA P system is fuzzy when it uses fuzzy multisets, fuzzy rewriterules or both.Fuzzy rewrite rules have attached a feasability degree. 112. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy P SystemsA P system is fuzzy when it uses fuzzy multisets, fuzzy rewriterules or both.Fuzzy rewrite rules have attached a feasability degree.P systems that process fuzzy multisets can compute realnumbers, therefore are more expressive than Turing machines. 113. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy P SystemsA P system is fuzzy when it uses fuzzy multisets, fuzzy rewriterules or both.Fuzzy rewrite rules have attached a feasability degree.P systems that process fuzzy multisets can compute realnumbers, therefore are more expressive than Turing machines.Conjecture P systems with fuzzy processing rules that processfuzzy data should have at least the computational power offuzzy Turing machines. 114. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy P SystemsA P system is fuzzy when it uses fuzzy multisets, fuzzy rewriterules or both.Fuzzy rewrite rules have attached a feasability degree.P systems that process fuzzy multisets can compute realnumbers, therefore are more expressive than Turing machines.Conjecture P systems with fuzzy processing rules that processfuzzy data should have at least the computational power offuzzy Turing machines.By replaceing fuzzy multisets with -fuzzy multisets, we get-fuzzy P systems. 115. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .More on Fuzzy P SystemsA P system is fuzzy when it uses fuzzy multisets, fuzzy rewriterules or both.Fuzzy rewrite rules have attached a feasability degree.P systems that process fuzzy multisets can compute realnumbers, therefore are more expressive than Turing machines.Conjecture P systems with fuzzy processing rules that processfuzzy data should have at least the computational power offuzzy Turing machines.By replaceing fuzzy multisets with -fuzzy multisets, we get-fuzzy P systems.Analogously, -fuzzy P systems should be as powerful as-fuzzy Turing machines. Maybe, they are more powerful. 116. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Similar Processes 117. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Similar ProcessesTypically, programs are viewed as types and processes as tokens. 118. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Similar ProcessesTypically, programs are viewed as types and processes as tokens.But, for example, different people that use a particular webbrowser view different web pages and have different numbers oftabs open at any given moment. 119. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Similar ProcessesTypically, programs are viewed as types and processes as tokens.But, for example, different people that use a particular webbrowser view different web pages and have different numbers oftabs open at any given moment.Thus, we cannot talk about processes that are identical, but wecan surely talk about processes that are similar. 120. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Similar ProcessesTypically, programs are viewed as types and processes as tokens.But, for example, different people that use a particular webbrowser view different web pages and have different numbers oftabs open at any given moment.Thus, we cannot talk about processes that are identical, but wecan surely talk about processes that are similar.How can we compare processes? 121. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Similar ProcessesTypically, programs are viewed as types and processes as tokens.But, for example, different people that use a particular webbrowser view different web pages and have different numbers oftabs open at any given moment.Thus, we cannot talk about processes that are identical, but wecan surely talk about processes that are similar.How can we compare processes?Assume that is an executable program for somecomputational platform (operating system, CPU, etc.). Thenan archetypal process of is the program running withminimum resources. 122. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Similar ProcessesTypically, programs are viewed as types and processes as tokens.But, for example, different people that use a particular webbrowser view different web pages and have different numbers oftabs open at any given moment.Thus, we cannot talk about processes that are identical, but wecan surely talk about processes that are similar.How can we compare processes?Assume that is an executable program for somecomputational platform (operating system, CPU, etc.). Thenan archetypal process of is the program running withminimum resources.The term minimum resources means that the archetypalprocess consumes the least possible resources (e.g., memory,CPU time, etc.). 123. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Comparing Processes 124. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Comparing ProcessesAssume that and are two processes initiated from thesame binary . 125. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Comparing ProcessesAssume that and are two processes initiated from thesame binary .Also assume that is an archetypal process and that is amethod that measures the similarity degree of some process to . 126. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Comparing ProcessesAssume that and are two processes initiated from thesame binary .Also assume that is an archetypal process and that is amethod that measures the similarity degree of some process to .That is, () = means that is similar with to a degreeequal to . 127. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Comparing ProcessesAssume that and are two processes initiated from thesame binary .Also assume that is an archetypal process and that is amethod that measures the similarity degree of some process to .That is, () = means that is similar with to a degreeequal to .If (, ) denotes the degree to which the two processes and are similar, then(, ) = () () . 128. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Labeled Transition Systems. . . . . . 129. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Labeled Transition SystemsA fuzzy labeled transition system over a crisp set of actions is a pair (, ) consisting of. . . . . . 130. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Labeled Transition SystemsA fuzzy labeled transition system over a crisp set of actions is a pair (, ) consisting ofa set of states;. . . . . . 131. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Labeled Transition SystemsA fuzzy labeled transition system over a crisp set of actions is a pair (, ) consisting ofa set of states;a fuzzy relation (, , ) called the fuzzy transition relation.. . . . . . 132. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineFuzzy Labeled Transition SystemsA fuzzy labeled transition system over a crisp set of actions is a pair (, ) consisting ofa set of states;a fuzzy relation (, , ) called the fuzzy transition relation.If the membership degree of (, , ) is , denotes thatthe plausibility degree to go from state to state by action is .. . . . . . 133. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Strong Fuzzy Simulation 134. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Strong Fuzzy SimulationLet (, ) be an FLTS. 135. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Strong Fuzzy SimulationLet (, ) be an FLTS.Let is a fuzzy binary relation over . 136. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Strong Fuzzy SimulationLet (, ) be an FLTS.Let is a fuzzy binary relation over . is a strong fuzzy simulation over (, ) with simulationdegree if, whenever (, ) if , then there exists such that , , and (, ) (, ). 137. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineThe Fuzzy Chemical Abstract machine. . . . . . 138. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineThe Fuzzy Chemical Abstract machine. . . . . .()=,, and ()=,, are archetypal molecules, then, , , , ,is an ideal fuzzy reaction rule; 139. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineThe Fuzzy Chemical Abstract machine. . . . . .()=,, and ()=,, are archetypal molecules, then, , , , ,is an ideal fuzzy reaction rule;Let be an instance of to degree (). Then[, , ][, , ]is feasible with feasibility degree equal to if{(), , ()} . 140. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineThe Fuzzy Chemical Abstract machine. . . . . .()=,, and ()=,, are archetypal molecules, then, , , , ,is an ideal fuzzy reaction rule;Let be an instance of to degree (). Then[, , ][, , ]is feasible with feasibility degree equal to if{(), , ()} .If some reacting molecules may be able to yield differentmolecules, then the reaction rule with the highest feasibilitydegree is really applicable. 141. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineThe Fuzzy Chemical Abstract machine. . . . . . 142. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineThe Fuzzy Chemical Abstract machine. . . . . . () 143. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineThe Fuzzy Chemical Abstract machine. . . . . . () Let () = () , then (), ()[] [ ] 144. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract MachineThe Fuzzy Chemical Abstract machine. . . . . . () Let () = () , then (), ()[] [ ] (), (())[()][()] 145. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Finale! 146. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Finale!I have briefly presented 147. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Finale!I have briefly presentedThe notion of vagueness. 148. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Finale!I have briefly presentedThe notion of vagueness.Fuzzy Turing Machines. 149. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Finale!I have briefly presentedThe notion of vagueness.Fuzzy Turing Machines.Fuzzy P Systems. 150. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Finale!I have briefly presentedThe notion of vagueness.Fuzzy Turing Machines.Fuzzy P Systems.The Fuzzy Chemical Abstract Machine. 151. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Finale!I have briefly presentedThe notion of vagueness.Fuzzy Turing Machines.Fuzzy P Systems.The Fuzzy Chemical Abstract Machine.More about fuzzy computability in my forthcoming book! 152. Computingunder VaguenessAp. SyropoulosVaguenessGeneral IdeasMany-valued LogicsSupervaluationismContextualismFuzzy (Sub)setsFuzzy TuringMachinesFuzzy P SystemsFuzzy ChemicalAbstract Machine. . . . . .Finale!I have briefly presentedThe notion of vagueness.Fuzzy Turing Machines.Fuzzy P Systems.The Fuzzy Chemical Abstract Machine.More about fuzzy computability in my forthcoming book!Thank you so much for your attention!