The document discusses the concept of vagueness and its applications in computing. It defines vagueness as arising from terms that have borderline cases where it is unclear whether the term applies or not, like the term "tall" and the height of 1.75m. The document outlines different theories of vagueness, such as supervaluationism, contextualism, and fuzzy set theory. It also discusses how vagueness has been incorporated into computational models through fuzzy Turing machines, fuzzy P systems, and fuzzy chemical abstract machines.

1. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Computing under Vagueness
Apostolos Syropoulos
Xanthi, Greece
asyropoulos@yahoo.com
Second international workshop
New Worlds of Computation 2011
LIFO, University of Orléans (France)
Monday, May 23, 2011
. . . . . .

2. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Outline
1. Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
2. Fuzzy Turing Machines
3. Fuzzy P Systems
4. Fuzzy Chemical Abstract Machine

3. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?

4. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.

5. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.

6. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.

7. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.

8. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples

9. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?

10. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?

11. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena is 1,75 m tall, is she tall?
Is Betelgeuse a huge star?

12. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena 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 you
decapitated him?

13. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena 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 you
decapitated him?
Eubulides of Miletus: The Sorites Paradox.

14. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
What is Vagueness?
It is widely accepted that a term is vague to the extent that it
has borderline cases.
What is a borderline case?.
A case in which it seems impossible either to apply or not to
apply a vague term.
The adjective “tall” is a typical example of a vague term.
Examples
Serena 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 you
decapitated him?
Eubulides of Miletus: The Sorites Paradox.
The old puzzle about bald people.

15. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Defining Vagueness

16. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Defining Vagueness
Bertrand Russell: Per contra, a representation is vague when
the relation of the representing system to the represented
system is not one-one, but one-many.

17. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Defining Vagueness
Bertrand Russell: Per contra, a representation is vague when
the relation of the representing system to the represented
system is not one-one, but one-many.
Example: A small-scale map is usually vaguer than a large-scale
map.

18. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Defining Vagueness
Bertrand Russell: Per contra, a representation is vague when
the relation of the representing system to the represented
system is not one-one, but one-many.
Example: A small-scale map is usually vaguer than a large-scale
map.
Charles Sanders Peirce: A proposition is vague when there are
possible states of things concerning which it is intrinsically
uncertain whether, had they been contemplated by the
speaker, he would have regarded them as excluded or
allowed by the proposition. By intrinsically uncertain we
mean not uncertain in consequence of any ignorance of the
interpreter, but because the speaker’s habits of language were
indeterminate.

19. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Defining Vagueness
Bertrand Russell: Per contra, a representation is vague when
the relation of the representing system to the represented
system is not one-one, but one-many.
Example: A small-scale map is usually vaguer than a large-scale
map.
Charles Sanders Peirce: A proposition is vague when there are
possible states of things concerning which it is intrinsically
uncertain whether, had they been contemplated by the
speaker, he would have regarded them as excluded or
allowed by the proposition. By intrinsically uncertain we
mean not uncertain in consequence of any ignorance of the
interpreter, but because the speaker’s habits of language were
indeterminate.
Max Black demonstrated this definition using the word chair.

20. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Vagueness, Generality, and Ambiguity
. . . . . .

21. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
. . . . . .

22. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
A term of phrase is ambiguous if it has at least two specific
meanings that make sense in context.
. . . . . .

23. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
A term of phrase is ambiguous if it has at least two specific
meanings that make sense in context.
Example: He ate the cookies on the couch.
. . . . . .

24. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
A term of phrase is ambiguous if it has at least two specific
meanings that make sense in context.
Example: He ate the cookies on the couch.
Bertrand Russell: “A proposition involving a general
concept—e.g., ‘This is a man’—will be verified by a number of
facts, such as ‘This’ being Brown or Jones or Robinson.”
. . . . . .

25. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Vagueness, Generality, and Ambiguity
Vagueness, ambibuity, and generality are entirely different
notions.
A term of phrase is ambiguous if it has at least two specific
meanings that make sense in context.
Example: He ate the cookies on the couch.
Bertrand Russell: “A proposition involving a general
concept—e.g., ‘This is a man’—will be verified by a number of
facts, such as ‘This’ being Brown or Jones or Robinson.”
Example: Again consider the word chair.
. . . . . .

26. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Different Expressions of Vagueness
. . . . . .

27. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Different Expressions of Vagueness
Many-valued Logics and Fuzzy Set Theory
. . . . . .

28. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Different Expressions of Vagueness
Many-valued Logics and Fuzzy Set Theory
Supervaluationism
. . . . . .

29. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Different Expressions of Vagueness
Many-valued Logics and Fuzzy Set Theory
Supervaluationism
Contextualism
. . . . . .

30. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Many-valued Logics and Fuzzy Set Theory
. . . . . .

31. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
. . . . . .

32. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
. . . . . .

33. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
Jan Łukasiewicz defined a three-valued logic to deal with future
contingencies.
. . . . . .

34. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
Jan Łukasiewicz defined a three-valued logic to deal with future
contingencies.
Emil Leon Post introduced the formulation of 푛 truth values,
where 푛 ≥ ﷡.
. . . . . .

35. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
Jan Łukasiewicz defined a three-valued logic to deal with future
contingencies.
Emil Leon Post introduced the formulation of 푛 truth values,
where 푛 ≥ ﷡.
C.C. Chang proposed MV-algebras, that is, an algebraic model
of many-valued logics.
. . . . . .

36. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Many-valued Logics and Fuzzy Set Theory
Aristotle in Περὶ ἑρμηνεῖας (De Interpretatione) discussed the
problem of future contingencies (i.e., what is the truth value of
a proposition about a future event?).
Can someone tell us what is the truth value of the proposition
“The Higgs boson exists?”
Jan Łukasiewicz defined a three-valued logic to deal with future
contingencies.
Emil Leon Post introduced the formulation of 푛 truth values,
where 푛 ≥ ﷡.
C.C. Chang proposed MV-algebras, that is, an algebraic model
of many-valued logics.
In 1965, Lotfi A. Zadeh introduced his fuzzy set theory and
its accompanying fuzzy logic.
. . . . . .

37. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Supervaluationism

38. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Supervaluationism
Supervaluationism is a special kind of classical indeterminacy
theory.

39. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Supervaluationism
Supervaluationism is a special kind of classical indeterminacy
theory.
Each predicate 푃 is associated with two non-overlapping sets
퐸+(푃), the deterninate extension, and 퐸−(푃), the determinate
anti-extension.

40. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Supervaluationism
Supervaluationism is a special kind of classical indeterminacy
theory.
Each predicate 푃 is associated with two non-overlapping sets
퐸+(푃), the deterninate extension, and 퐸−(푃), the determinate
anti-extension.
The candidate extension of 푃 are sets 푋 that extend 퐸+(푃)
and are disjoint from 퐸−(푃).

41. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Supervaluationism
Supervaluationism is a special kind of classical indeterminacy
theory.
Each predicate 푃 is associated with two non-overlapping sets
퐸+(푃), the deterninate extension, and 퐸−(푃), the determinate
anti-extension.
The candidate extension of 푃 are sets 푋 that extend 퐸+(푃)
and are disjoint from 퐸−(푃).
Truth is defined relative to each choice of candidate
extensions.

42. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Supervaluationism
Supervaluationism is a special kind of classical indeterminacy
theory.
Each predicate 푃 is associated with two non-overlapping sets
퐸+(푃), the deterninate extension, and 퐸−(푃), the determinate
anti-extension.
The candidate extension of 푃 are sets 푋 that extend 퐸+(푃)
and are disjoint from 퐸−(푃).
Truth is defined relative to each choice of candidate
extensions.
A sentense is “super-true” if it comes out true realtive to each
choice of candidate extensions for its predicates.

43. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Contextualism

44. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Contextualism
The correct application of a vague predicate varies with context.

45. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Contextualism
The correct application of a vague predicate varies with context.
Example: A person may be tall relative to American men but
short relative to NBA players.

46. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Contextualism
The correct application of a vague predicate varies with context.
Example: A person may be tall relative to American men but
short relative to NBA players.
Epistemological contextualism maintains the general idea that
whether one knows is somehow relative to context.

47. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy (Sub)sets

48. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to sets to a degree

49. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to sets to a degree
Given a universe 푋, a fuzzy subset 퐴 of 푋 is characterized by
a function 퐴 ∶ 푋 → ﹚, ﹚ = [﷟, ﷠], where 퐴(푥) = 푖 means that 푥
belongs to 퐴 with degreee equal to 푖.

50. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to sets to a degree
Given a universe 푋, a fuzzy subset 퐴 of 푋 is characterized by
a function 퐴 ∶ 푋 → ﹚, ﹚ = [﷟, ﷠], where 퐴(푥) = 푖 means that 푥
belongs to 퐴 with degreee equal to 푖.
Example: One can easily build a fuzzy subset of the tall
students of a class.

51. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to sets to a degree
Given a universe 푋, a fuzzy subset 퐴 of 푋 is characterized by
a function 퐴 ∶ 푋 → ﹚, ﹚ = [﷟, ﷠], where 퐴(푥) = 푖 means that 푥
belongs to 퐴 with degreee equal to 푖.
Example: One can easily build a fuzzy subset of the tall
students of a class.
When 퐴(푥) = ﷟ we say that 푥 belongs to 퐴 with degree equal
to ﷟ not that 푥 does not belong to 퐴!

52. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy (Sub)sets
Fuzzy set theory is based on the idea that elements may
belong to sets to a degree
Given a universe 푋, a fuzzy subset 퐴 of 푋 is characterized by
a function 퐴 ∶ 푋 → ﹚, ﹚ = [﷟, ﷠], where 퐴(푥) = 푖 means that 푥
belongs to 퐴 with degreee equal to 푖.
Example: One can easily build a fuzzy subset of the tall
students of a class.
When 퐴(푥) = ﷟ we say that 푥 belongs to 퐴 with degree equal
to ﷟ not that 푥 does not belong to 퐴!
A consequence of this is that one cannot build a topos of all
fuzzy subsets.

53. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Operations between Fuzzy Subsets
. . . . . .

54. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Operations between Fuzzy Subsets
Assume that 퐴 and 퐵 are two fuzzy subsets of 푋. Then
. . . . . .

55. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Operations between Fuzzy Subsets
Assume that 퐴 and 퐵 are two fuzzy subsets of 푋. Then
. . . . . .
their union is
(퐴 ∪ 퐵)(푥) = ︌︀︗퐴(푥), 퐵(푥)�;

56. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Operations between Fuzzy Subsets
Assume that 퐴 and 퐵 are two fuzzy subsets of 푋. Then
. . . . . .
their union is
(퐴 ∪ 퐵)(푥) = ︌︀︗퐴(푥), 퐵(푥)�;
their intersection is
(퐴 ∩ 퐵)(푥) = ︌︈︍퐴(푥), 퐵(푥)�;

57. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Operations between Fuzzy Subsets
Assume that 퐴 and 퐵 are two fuzzy subsets of 푋. Then
. . . . . .
their union is
(퐴 ∪ 퐵)(푥) = ︌︀︗퐴(푥), 퐵(푥)�;
their intersection is
(퐴 ∩ 퐵)(푥) = ︌︈︍퐴(푥), 퐵(푥)�;
the complement of 퐴 is
﷠ − 퐴(푥) for 푥 ∈ 푋;

58. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Operations between Fuzzy Subsets
Assume that 퐴 and 퐵 are two fuzzy subsets of 푋. Then
. . . . . .
their union is
(퐴 ∪ 퐵)(푥) = ︌︀︗퐴(푥), 퐵(푥)�;
their intersection is
(퐴 ∩ 퐵)(푥) = ︌︈︍퐴(푥), 퐵(푥)�;
the complement of 퐴 is
﷠ − 퐴(푥) for 푥 ∈ 푋;
퐴 is a subset of 퐵 and write 퐴 ⊆ 퐵 iff
퐴(푥) ≤ 퐵(푋) ∀푥 ∈ 푋 and;

59. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Operations between Fuzzy Subsets
Assume that 퐴 and 퐵 are two fuzzy subsets of 푋. Then
. . . . . .
their union is
(퐴 ∪ 퐵)(푥) = ︌︀︗퐴(푥), 퐵(푥)�;
their intersection is
(퐴 ∩ 퐵)(푥) = ︌︈︍퐴(푥), 퐵(푥)�;
the complement of 퐴 is
﷠ − 퐴(푥) for 푥 ∈ 푋;
퐴 is a subset of 퐵 and write 퐴 ⊆ 퐵 iff
퐴(푥) ≤ 퐵(푋) ∀푥 ∈ 푋 and;
the scalar cardinality of 퐴 is
|퐴| = ﾝ
푥∈푋
퐴(푋).

60. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Set Theory vs. Probability Theory
. . . . . .

61. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Set Theory vs. Probability Theory
Let’s transform the statement“Serena is tall to a degree of
0,70” into the “Serena is 70% tall!”
. . . . . .

62. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Set Theory vs. Probability Theory
Let’s transform the statement“Serena is tall to a degree of
0,70” into the “Serena is 70% tall!”
So is fuzzy sets just an “elementary mistake of logic?”
. . . . . .

63. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Set Theory vs. Probability Theory
Let’s transform the statement“Serena is tall to a degree of
0,70” 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. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Set Theory vs. Probability Theory
Let’s transform the statement“Serena is tall to a degree of
0,70” 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!
Τhe statement “Serena is tall to a degree of 0,70” says that
Serena is tall and the truth value of this statement is 0,70!
. . . . . .

65. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Set Theory vs. Probability Theory
Let’s transform the statement“Serena is tall to a degree of
0,70” 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!
Τhe statement “Serena is tall to a degree of 0,70” says that
Serena is tall and the truth value of this statement is 0,70!
How can we express in probability theory judgments like
tomorrow will be a warm day or there will be a strong
earthquake in the near future, etc.?
. . . . . .

66. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Set Theory vs. Probability Theory
Let’s transform the statement“Serena is tall to a degree of
0,70” 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!
Τhe statement “Serena is tall to a degree of 0,70” says that
Serena is tall and the truth value of this statement is 0,70!
How can we express in probability theory judgments like
tomorrow will be a warm day or there will be a strong
earthquake in the near future, etc.?
It it possible to perform estimations that involve fuzzy
probabilities expressed as likely, unlikely, not very likely, etc.?
. . . . . .

67. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Set Theory vs. Probability Theory
Let’s transform the statement“Serena is tall to a degree of
0,70” 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!
Τhe statement “Serena is tall to a degree of 0,70” says that
Serena is tall and the truth value of this statement is 0,70!
How can we express in probability theory judgments like
tomorrow will be a warm day or there will be a strong
earthquake in the near future, etc.?
It it possible to perform estimations that involve fuzzy
probabilities expressed as likely, unlikely, not very likely, etc.?
Bart Kosko advocates the idea that fuzziness is a basic
characteristic of our cosmos! (I agree with Kosko!)
. . . . . .

68. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Zadeh’s Fuzzy Algorithms and Turing Machines
. . . . . .

69. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Zadeh’s Fuzzy Algorithms and Turing Machines
In 1968, Zadeh informally introduced the notions of fuzzy
algorithm and fuzzy Turing machine.
. . . . . .

70. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Zadeh’s Fuzzy Algorithms and Turing Machines
In 1968, Zadeh informally introduced the notions of fuzzy
algorithm and fuzzy Turing machine.
A fuzzy algorithm contains fuzzy commands.
. . . . . .

71. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Zadeh’s Fuzzy Algorithms and Turing Machines
In 1968, Zadeh informally introduced the notions of fuzzy
algorithm and fuzzy Turing machine.
A fuzzy algorithm contains fuzzy commands.
A fuzzy command uses fuzzy sets: Make 푦 approximately
equal to ﷠﷟, if 푥 is approximately equal to ﷤.
. . . . . .

72. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Zadeh’s Fuzzy Algorithms and Turing Machines
In 1968, Zadeh informally introduced the notions of fuzzy
algorithm and fuzzy Turing machine.
A fuzzy algorithm contains fuzzy commands.
A fuzzy command uses fuzzy sets: Make 푦 approximately
equal to ﷠﷟, if 푥 is approximately equal to ﷤.
Zadeh argued that everyday activities can be viewed as fuzzy
algorithms, but Carol E. Cleland argued that recipes are
algorithms!!!
. . . . . .

73. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Zadeh’s Fuzzy Algorithms and Turing Machines
In 1968, Zadeh informally introduced the notions of fuzzy
algorithm and fuzzy Turing machine.
A fuzzy algorithm contains fuzzy commands.
A fuzzy command uses fuzzy sets: Make 푦 approximately
equal to ﷠﷟, if 푥 is approximately equal to ﷤.
Zadeh argued that everyday activities can be viewed as fuzzy
algorithms, but Carol E. Cleland argued that recipes are
algorithms!!!
A fuzzy Turing machine is Turing machine where
푞푛+ = 푓(푞푛, 푠푛) is associated with a feasability degree.
. . . . . .

74. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy Algorithms

75. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy Algorithms
Shi-Kuo Chang used a finite-state machine and a fuzzy
machine to define the execution of fuzzy algorithms.

76. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy Algorithms
Shi-Kuo Chang used a finite-state machine and a fuzzy
machine to define the execution of fuzzy algorithms.
Kokichi Tanaka and Masaharu Mizumoto defined an extended
fuzzy machine, which is based on a generalization of Chang’s
machine.

77. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy Algorithms
Shi-Kuo Chang used a finite-state machine and a fuzzy
machine to define the execution of fuzzy algorithms.
Kokichi Tanaka and Masaharu Mizumoto defined an extended
fuzzy machine, which is based on a generalization of Chang’s
machine.
Eugene S. Santos reformulated fuzzy programs using Dana
Scott’s work and defined W-machines that are able to execute
them.

78. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy Algorithms
Shi-Kuo Chang used a finite-state machine and a fuzzy
machine to define the execution of fuzzy algorithms.
Kokichi Tanaka and Masaharu Mizumoto defined an extended
fuzzy machine, which is based on a generalization of Chang’s
machine.
Eugene S. Santos reformulated fuzzy programs using Dana
Scott’s work and defined W-machines that are able to execute
them.
The formulations of Chang and Tanaka-Mizumoto are special
cases of Santos’s formulation.

79. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines

80. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines
A Santos type fuzzy Turing machine is a septuple
(푆, 푄, 푞푖, 푞푓 , 훿, 푊, 훿푊 ) where:

81. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines
A Santos type fuzzy Turing machine is a septuple
(푆, 푄, 푞푖, 푞푓 , 훿, 푊, 훿푊 ) where:
푆 represents a finite non-empty set of input symbols;

82. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines
A Santos type fuzzy Turing machine is a septuple
(푆, 푄, 푞푖, 푞푓 , 훿, 푊, 훿푊 ) where:
푆 represents a finite non-empty set of input symbols;
푄 denotes a finite non-empty set of states such that 푆 ∩ 푄 = ∅;

83. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines
A Santos type fuzzy Turing machine is a septuple
(푆, 푄, 푞푖, 푞푓 , 훿, 푊, 훿푊 ) where:
푆 represents a finite non-empty set of input symbols;
푄 denotes a finite non-empty set of states such that 푆 ∩ 푄 = ∅;
푞푖, 푞푓 ∈ 푄 are symbols designating the initial and final state,
respectively;

84. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines
A Santos type fuzzy Turing machine is a septuple
(푆, 푄, 푞푖, 푞푓 , 훿, 푊, 훿푊 ) where:
푆 represents a finite non-empty set of input symbols;
푄 denotes a finite non-empty set of states such that 푆 ∩ 푄 = ∅;
푞푖, 푞푓 ∈ 푄 are symbols designating the initial and final state,
respectively;
훿 ⊂ (푄 × 푆) × (푄 × (푆 × {퐿, 푁, 푅})) is the next-move relation;

85. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines
A Santos type fuzzy Turing machine is a septuple
(푆, 푄, 푞푖, 푞푓 , 훿, 푊, 훿푊 ) where:
푆 represents a finite non-empty set of input symbols;
푄 denotes a finite non-empty set of states such that 푆 ∩ 푄 = ∅;
푞푖, 푞푓 ∈ 푄 are symbols designating the initial and final state,
respectively;
훿 ⊂ (푄 × 푆) × (푄 × (푆 × {퐿, 푁, 푅})) is the next-move relation;
푊 is the semiring (푊, ∧, ∨);

86. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines
A Santos type fuzzy Turing machine is a septuple
(푆, 푄, 푞푖, 푞푓 , 훿, 푊, 훿푊 ) where:
푆 represents a finite non-empty set of input symbols;
푄 denotes a finite non-empty set of states such that 푆 ∩ 푄 = ∅;
푞푖, 푞푓 ∈ 푄 are symbols designating the initial and final state,
respectively;
훿 ⊂ (푄 × 푆) × (푄 × (푆 × {퐿, 푁, 푅})) is the next-move relation;
푊 is the semiring (푊, ∧, ∨);
훿푊 ∶ (푄 × 푆) × (푄 × (푆 × {퐿, 푁, 푅})) → 푊 is a W-function that
assigns a degree of certainty to each machine transition.

87. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Fuzzy Turing Machines
A Santos type fuzzy Turing machine is a septuple
(푆, 푄, 푞푖, 푞푓 , 훿, 푊, 훿푊 ) where:
푆 represents a finite non-empty set of input symbols;
푄 denotes a finite non-empty set of states such that 푆 ∩ 푄 = ∅;
푞푖, 푞푓 ∈ 푄 are symbols designating the initial and final state,
respectively;
훿 ⊂ (푄 × 푆) × (푄 × (푆 × {퐿, 푁, 푅})) is the next-move relation;
푊 is the semiring (푊, ∧, ∨);
훿푊 ∶ (푄 × 푆) × (푄 × (푆 × {퐿, 푁, 푅})) → 푊 is a W-function that
assigns a degree of certainty to each machine transition.
Nowadays, we use t-norms and t-conorms instead of semirings.

88. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Results Concerning Fuzzy Turing Machines
. . . . . .

89. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Results Concerning Fuzzy Turing Machines
There is no universal fuzzy Turing machine!
. . . . . .

90. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Results Concerning Fuzzy Turing Machines
There is no universal fuzzy Turing machine!
Selim Akl had argued that the concept of a universal computer
cannot be realized.
. . . . . .

91. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Results Concerning Fuzzy Turing Machines
﷪
﷪﷩﷩There is no universal fuzzy Turing machine!
Selim Akl had argued that the concept of a universal computer
cannot be realized.
Jiří Wiedermann had shown that fuzzy languages accepted by
these machines with a computable t-norm correspond exactly to
the union ︼∪ ︻of recursively enumerable languages and
. . . . . .
their complements.

92. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Results Concerning Fuzzy Turing Machines
﷪
﷪﷩﷩There is no universal fuzzy Turing machine!
Selim Akl had argued that the concept of a universal computer
cannot be realized.
Jiří Wiedermann had shown that fuzzy languages accepted by
these machines with a computable t-norm correspond exactly to
the union ︼∪ ︻of recursively enumerable languages and
their complements.
Benjamín Callejas Bedregal and Santiago Figueira argue that
they have found flaws in Wiedermann proof though they did
not refute his original argument.
. . . . . .

93. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Results Concerning Fuzzy Turing Machines
﷪
﷪﷩﷩There is no universal fuzzy Turing machine!
Selim Akl had argued that the concept of a universal computer
cannot be realized.
Jiří Wiedermann had shown that fuzzy languages accepted by
these machines with a computable t-norm correspond exactly to
the union ︼∪ ︻of recursively enumerable languages and
their complements.
Benjamín Callejas Bedregal and Santiago Figueira argue that
they have found flaws in Wiedermann proof though they did
not refute his original argument.
Later on Yongming Li showed that Wiedermann’s results is true
for 퐿-fuzzy Turing machines.
. . . . . .

94. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems in a Nutshell

95. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems in a Nutshell
A model of computation mimicking the way cells function
which was proposed by Gheorghe Păun.

96. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems in a Nutshell
A model of computation mimicking the way cells function
which was proposed by Gheorghe Păun.
A compartment is space that is surrounded by a porous
membrane.

97. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems in a Nutshell
A model of computation mimicking the way cells function
which was proposed by Gheorghe Păun.
A compartment is space that is surrounded by a porous
membrane.
Nested compartments make a P system. Each compartment is
a multiset.

98. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems in a Nutshell
A model of computation mimicking the way cells function
which was proposed by Gheorghe Păun.
A compartment is space that is surrounded by a porous
membrane.
Nested compartments make a P system. Each compartment is
a multiset.
Data flow from one compartment to another according to some
multiset processing rules.

99. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems in a Nutshell
A model of computation mimicking the way cells function
which was proposed by Gheorghe Păun.
A compartment is space that is surrounded by a porous
membrane.
Nested compartments make a P system. Each compartment is
a multiset.
Data flow from one compartment to another according to some
multiset processing rules.
Computation stops when no rule can be applied.

100. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems in a Nutshell
A model of computation mimicking the way cells function
which was proposed by Gheorghe Păun.
A compartment is space that is surrounded by a porous
membrane.
Nested compartments make a P system. Each compartment is
a multiset.
Data flow from one compartment to another according to some
multiset processing rules.
Computation stops when no rule can be applied.
The result of the computation is equal to the cardinality of the
multiset contained in a special compartment called the output
compartment.

101. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems in a Nutshell
A model of computation mimicking the way cells function
which was proposed by Gheorghe Păun.
A compartment is space that is surrounded by a porous
membrane.
Nested compartments make a P system. Each compartment is
a multiset.
Data flow from one compartment to another according to some
multiset processing rules.
Computation stops when no rule can be applied.
The result of the computation is equal to the cardinality of the
multiset contained in a special compartment called the output
compartment.
Processing takes place in discrete steps but rules are applied in
parallel.

102. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems with Fuzzy Data

103. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems with Fuzzy Data
A P system with fuzzy data is a n-tuple
(푂, 휇, 푤(﷪), … , 푤(푚), 푅﷪, … , 푅푚, 푖﷩)
where

104. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems with Fuzzy Data
A P system with fuzzy data is a n-tuple
(푂, 휇, 푤(﷪), … , 푤(푚), 푅﷪, … , 푅푚, 푖﷩)
where
푂 is the alphabet of objects;

105. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems with Fuzzy Data
A P system with fuzzy data is a n-tuple
(푂, 휇, 푤(﷪), … , 푤(푚), 푅﷪, … , 푅푚, 푖﷩)
where
푂 is the alphabet of objects;
휇 is the membrane structure;

106. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems with Fuzzy Data
A P system with fuzzy data is a n-tuple
(푂, 휇, 푤(﷪), … , 푤(푚), 푅﷪, … , 푅푚, 푖﷩)
where
푂 is the alphabet of objects;
휇 is the membrane structure;
푤(푖) ∶ 푂 → ℕ × ﹚, ﷠ ≤ 푖 ≤ 푚, represent multi-fuzzy sets over 푂
associated with the region surrounded by membrane 푖;

107. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems with Fuzzy Data
A P system with fuzzy data is a n-tuple
(푂, 휇, 푤(﷪), … , 푤(푚), 푅﷪, … , 푅푚, 푖﷩)
where
푂 is the alphabet of objects;
휇 is the membrane structure;
푤(푖) ∶ 푂 → ℕ × ﹚, ﷠ ≤ 푖 ≤ 푚, represent multi-fuzzy sets over 푂
associated with the region surrounded by membrane 푖;
푅푖, ﷠ ≤ 푖 ≤ 푚, are finite sets of multiset rewriting rules of the
form 푢 → 푣, 푢 ∈ 푂∗ and 푣 ∈ 푂∗
ﹿ﹬ﹽ, where 푂ﹿ﹬ﹽ = 푂 × ﹥﹒﹣,
and ﹥﹒﹣ = {︇︄︑︄, ︎︔︓} ∪ {︈︍푗|﷠ ≤ 푗 ≤ 푚}.

108. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
P Systems with Fuzzy Data
A P system with fuzzy data is a n-tuple
(푂, 휇, 푤(﷪), … , 푤(푚), 푅﷪, … , 푅푚, 푖﷩)
where
푂 is the alphabet of objects;
휇 is the membrane structure;
푤(푖) ∶ 푂 → ℕ × ﹚, ﷠ ≤ 푖 ≤ 푚, represent multi-fuzzy sets over 푂
associated with the region surrounded by membrane 푖;
푅푖, ﷠ ≤ 푖 ≤ 푚, are finite sets of multiset rewriting rules of the
form 푢 → 푣, 푢 ∈ 푂∗ and 푣 ∈ 푂∗
ﹿ﹬ﹽ, where 푂ﹿ﹬ﹽ = 푂 × ﹥﹒﹣,
and ﹥﹒﹣ = {︇︄︑︄, ︎︔︓} ∪ {︈︍푗|﷠ ≤ 푗 ≤ 푚}.
푖﷩ ∈ {﷠, ﷡, … , 푚} is the label of a compartment called the output
membrane.

109. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy P Systems

110. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy P Systems
A P system is fuzzy when it uses fuzzy multisets, fuzzy rewrite
rules or both.

111. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy P Systems
A P system is fuzzy when it uses fuzzy multisets, fuzzy rewrite
rules or both.
Fuzzy rewrite rules have attached a feasability degree.

112. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy P Systems
A P system is fuzzy when it uses fuzzy multisets, fuzzy rewrite
rules or both.
Fuzzy rewrite rules have attached a feasability degree.
P systems that process fuzzy multisets can compute real
numbers, therefore are more expressive than Turing machines.

113. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy P Systems
A P system is fuzzy when it uses fuzzy multisets, fuzzy rewrite
rules or both.
Fuzzy rewrite rules have attached a feasability degree.
P systems that process fuzzy multisets can compute real
numbers, therefore are more expressive than Turing machines.
Conjecture P systems with fuzzy processing rules that process
fuzzy data should have at least the computational power of
fuzzy Turing machines.

114. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy P Systems
A P system is fuzzy when it uses fuzzy multisets, fuzzy rewrite
rules or both.
Fuzzy rewrite rules have attached a feasability degree.
P systems that process fuzzy multisets can compute real
numbers, therefore are more expressive than Turing machines.
Conjecture P systems with fuzzy processing rules that process
fuzzy data should have at least the computational power of
fuzzy Turing machines.
By replaceing fuzzy multisets with 퐿-fuzzy multisets, we get
퐿-fuzzy P systems.

115. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
More on Fuzzy P Systems
A P system is fuzzy when it uses fuzzy multisets, fuzzy rewrite
rules or both.
Fuzzy rewrite rules have attached a feasability degree.
P systems that process fuzzy multisets can compute real
numbers, therefore are more expressive than Turing machines.
Conjecture P systems with fuzzy processing rules that process
fuzzy data should have at least the computational power of
fuzzy Turing machines.
By replaceing fuzzy multisets with 퐿-fuzzy multisets, we get
퐿-fuzzy P systems.
Analogously, 퐿-fuzzy P systems should be as powerful as
퐿-fuzzy Turing machines. Maybe, they are more powerful.

116. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Similar Processes

117. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Similar Processes
Typically, programs are viewed as types and processes as tokens.

118. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Similar Processes
Typically, programs are viewed as types and processes as tokens.
But, for example, different people that use a particular web
browser view different web pages and have different numbers of
tabs open at any given moment.

119. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Similar Processes
Typically, programs are viewed as types and processes as tokens.
But, for example, different people that use a particular web
browser view different web pages and have different numbers of
tabs open at any given moment.
Thus, we cannot talk about processes that are identical, but we
can surely talk about processes that are similar.

120. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Similar Processes
Typically, programs are viewed as types and processes as tokens.
But, for example, different people that use a particular web
browser view different web pages and have different numbers of
tabs open at any given moment.
Thus, we cannot talk about processes that are identical, but we
can surely talk about processes that are similar.
How can we compare processes?

121. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Similar Processes
Typically, programs are viewed as types and processes as tokens.
But, for example, different people that use a particular web
browser view different web pages and have different numbers of
tabs open at any given moment.
Thus, we cannot talk about processes that are identical, but we
can surely talk about processes that are similar.
How can we compare processes?
Assume that 푝 is an executable program for some
computational platform (operating system, CPU, etc.). Then
an archetypal process 휋 of 푝 is the program 푝 running with
minimum resources.

122. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Similar Processes
Typically, programs are viewed as types and processes as tokens.
But, for example, different people that use a particular web
browser view different web pages and have different numbers of
tabs open at any given moment.
Thus, we cannot talk about processes that are identical, but we
can surely talk about processes that are similar.
How can we compare processes?
Assume that 푝 is an executable program for some
computational platform (operating system, CPU, etc.). Then
an archetypal process 휋 of 푝 is the program 푝 running with
minimum resources.
The term minimum resources means that the archetypal
process consumes the least possible resources (e.g., memory,
CPU time, etc.).

123. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Comparing Processes

124. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Comparing Processes
Assume that 푝﷪ and 푝﷫ are two processes initiated from the
same binary 푝.

125. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Comparing Processes
Assume that 푝﷪ and 푝﷫ are two processes initiated from the
same binary 푝.
Also assume that 휋 is an archetypal process and that 훿휋 is a
method that measures the similarity degree of some process 푝푖
to 휋.

126. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Comparing Processes
Assume that 푝﷪ and 푝﷫ are two processes initiated from the
same binary 푝.
Also assume that 휋 is an archetypal process and that 훿휋 is a
method that measures the similarity degree of some process 푝푖
to 휋.
That is, 훿휋(푝) = 푖 means that 푝 is similar with 휋푖 to a degree
equal to 푖.

127. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Comparing Processes
Assume that 푝﷪ and 푝﷫ are two processes initiated from the
same binary 푝.
Also assume that 휋 is an archetypal process and that 훿휋 is a
method that measures the similarity degree of some process 푝푖
to 휋.
That is, 훿휋(푝) = 푖 means that 푝 is similar with 휋푖 to a degree
equal to 푖.
If ︷휋(푝﷪, 푝﷫) denotes the degree to which the two processes 푝﷪
and 푝﷫ are similar, then
︷휋(푝﷪, 푝﷫) = ﷠− ￗ 훿휋(푝﷪) − 훿휋(푝﷫) ￗ .

128. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Labeled Transition Systems
. . . . . .

129. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Labeled Transition Systems
A fuzzy labeled transition system over a crisp set of actions 풜
is a pair (풬, 풯) consisting of
. . . . . .

130. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Labeled Transition Systems
A fuzzy labeled transition system over a crisp set of actions 풜
is a pair (풬, 풯) consisting of
a set 풬 of states;
. . . . . .

131. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Labeled Transition Systems
A fuzzy labeled transition system over a crisp set of actions 풜
is a pair (풬, 풯) consisting of
a set 풬 of states;
a fuzzy relation 풯(풬, 풜, 풬) called the fuzzy transition relation.
. . . . . .

132. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
Fuzzy Labeled Transition Systems
A fuzzy labeled transition system over a crisp set of actions 풜
is a pair (풬, 풯) consisting of
a set 풬 of states;
a fuzzy relation 풯(풬, 풜, 풬) called the fuzzy transition relation.
If the membership degree of (푞, 훼, 푞′) is 푑, 푞 훼→푑
푞′ denotes that
the plausibility degree to go from state 푞 to state 푞′ by action
훼 is 푑.
. . . . . .

133. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Strong Fuzzy Simulation

134. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Strong Fuzzy Simulation
Let (풬, 풯) be an FLTS.

135. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Strong Fuzzy Simulation
Let (풬, 풯) be an FLTS.
Let 풮 is a fuzzy binary relation over 풬 .

136. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Strong Fuzzy Simulation
Let (풬, 풯) be an FLTS.
Let 풮 is a fuzzy binary relation over 풬 .
풮 is a strong fuzzy simulation over (풬, 풯) with simulation
degree 푠 if, whenever 풮(푝, 푞) ≥ 푠 if 푝 훼→푑
﷪
푝′, then there exists
푞′ ∈ 풬 such that 푞 훼→푑
﷫
푞′, 푑﷫ ≥ 푑﷪, and 풮(푝′, 푞′) ≥ 풮(푝, 푞).

137. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
The Fuzzy Chemical Abstract machine
. . . . . .

138. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
The Fuzzy Chemical Abstract machine
. . . . . .
(푚푖)푖=﷪,…,푘 and (푚′푗
)푗=﷪,…,푙 are archetypal molecules, then
푚﷪, … , 푚푘 →휆
푚′
﷪, … , 푚′푙
,
is an ideal fuzzy reaction rule;

139. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
The Fuzzy Chemical Abstract machine
. . . . . .
(푚푖)푖=﷪,…,푘 and (푚′푗
)푗=﷪,…,푙 are archetypal molecules, then
푚﷪, … , 푚푘 →휆
푚′
﷪, … , 푚′푙
,
is an ideal fuzzy reaction rule;
Let 푀푖 be an instance of 푚푖 to degree 휆휋(푀푖). Then
[푀﷪, … , 푀푘]→휆
[푀′
﷪, … , 푀′푙
]
is feasible with feasibility degree equal to 휆 if
︌︈︍{훿휋(푀﷪), … , 훿휋(푀푘)} ≥ 휆.

140. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
The Fuzzy Chemical Abstract machine
. . . . . .
(푚푖)푖=﷪,…,푘 and (푚′푗
)푗=﷪,…,푙 are archetypal molecules, then
푚﷪, … , 푚푘 →휆
푚′
﷪, … , 푚′푙
,
is an ideal fuzzy reaction rule;
Let 푀푖 be an instance of 푚푖 to degree 휆휋(푀푖). Then
[푀﷪, … , 푀푘]→휆
[푀′
﷪, … , 푀′푙
]
is feasible with feasibility degree equal to 휆 if
︌︈︍{훿휋(푀﷪), … , 훿휋(푀푘)} ≥ 휆.
If some reacting molecules may be able to yield different
molecules, then the reaction rule with the highest feasibility
degree is really applicable.

141. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
The Fuzzy Chemical Abstract machine
. . . . . .

142. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
The Fuzzy Chemical Abstract machine
. . . . . .
￵푆﷪ →휆
휆
→푆﷫￸ ￵∀푚푖 ∈ 푆﷬ ∶ 훿휋(푚푖) ≥ 휆￸
푆﷪ ⊎ 푆﷬ 푆﷫ ⊎ 푆﷬

143. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
The Fuzzy Chemical Abstract machine
. . . . . .
￵푆﷪ →휆
휆
→푆﷫￸ ￵∀푚푖 ∈ 푆﷬ ∶ 훿휋(푚푖) ≥ 휆￸
푆﷪ ⊎ 푆﷬ 푆﷫ ⊎ 푆﷬
Let ︷훿(푆) = ︌︈︍훿휋(푝) ￗ 푝 ∈ 푆, then
휆
↔휆 ≤ ︌︈︍︷훿(푆), 훿휋(푚)
[푚] ⊎ 푆[푚 ⊲ 푆]

144. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
The Fuzzy Chemical Abstract machine
. . . . . .
￵푆﷪ →휆
휆
→푆﷫￸ ￵∀푚푖 ∈ 푆﷬ ∶ 훿휋(푚푖) ≥ 휆￸
푆﷪ ⊎ 푆﷬ 푆﷫ ⊎ 푆﷬
Let ︷훿(푆) = ︌︈︍훿휋(푝) ￗ 푝 ∈ 푆, then
휆
↔휆 ≤ ︌︈︍︷훿(푆), 훿휋(푚)
[푚] ⊎ 푆[푚 ⊲ 푆]
￵푆→휆
푆′￸ ￵휆 ≤ ︌︈︍︷훿(푆), 훿휋(퐶())￸
[퐶(푆)]→휆
[퐶(푆′)]

145. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Finale!

146. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Finale!
I have briefly presented

147. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Finale!
I have briefly presented
The notion of vagueness.

148. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Finale!
I have briefly presented
The notion of vagueness.
Fuzzy Turing Machines.

149. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Finale!
I have briefly presented
The notion of vagueness.
Fuzzy Turing Machines.
Fuzzy P Systems.

150. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Finale!
I have briefly presented
The notion of vagueness.
Fuzzy Turing Machines.
Fuzzy P Systems.
The Fuzzy Chemical Abstract Machine.

151. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Finale!
I have briefly presented
The notion of vagueness.
Fuzzy Turing Machines.
Fuzzy P Systems.
The Fuzzy Chemical Abstract Machine.
More about fuzzy computability in my forthcoming book!

152. Computing
under Vagueness
Ap. Syropoulos
Vagueness
General Ideas
Many-valued Logics
Supervaluationism
Contextualism
Fuzzy (Sub)sets
Fuzzy Turing
Machines
Fuzzy P Systems
Fuzzy Chemical
Abstract Machine
. . . . . .
Finale!
I have briefly presented
The 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!