A talk given at The Science and Philosophy of Unconventional Computing Workshop, which took place in Cambridge, UK, March 2009.

1. Models of
Non-standard
Computation
A. Syropoulos
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. Models of Non-standard Computation
Apostolos Syropoulos1
1Greek Molecular Computing Group
Xanthi, Greece
The Science and Philosophy of Unconventional Computing

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Presentation Outline

3. Models of
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Non-standard Computing

4. Models of
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Computation
A. Syropoulos
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Non-standard Computing
Non-standard is something that is varying from or not adhering
to the standard.

5. Models of
Non-standard
Computation
A. Syropoulos
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Non-standard Computing
Non-standard is something that is varying from or not adhering
to the standard.
Use non-standard ideas to build new computing devices.

6. Models of
Non-standard
Computation
A. Syropoulos
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Non-standard Computing
Non-standard is something that is varying from or not adhering
to the standard.
Use non-standard ideas to build new computing devices.
Paraconsistency and Fuzziness are non-standard ideas.

7. Models of
Non-standard
Computation
A. Syropoulos
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Non-standard Computing
Non-standard is something that is varying from or not adhering
to the standard.
Use non-standard ideas to build new computing devices.
Paraconsistency and Fuzziness are non-standard ideas.
Why bother with non-standard ideas?

8. Models of
Non-standard
Computation
A. Syropoulos
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Paraconsist Logic(s)

9. Models of
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Computation
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Paraconsist Logic(s)
Paraconsistency: the idea that contradictions makes sense.

10. Models of
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Computation
A. Syropoulos
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Paraconsist Logic(s)
Paraconsistency: the idea that contradictions makes sense.
A set of statements is incosistent if it contains both some
statement A and its negation A.

11. Models of
Non-standard
Computation
A. Syropoulos
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Paraconsist Logic(s)
Paraconsistency: the idea that contradictions makes sense.
A set of statements is incosistent if it contains both some
statement A and its negation A.
A logic is called paraconsistent if from an incosistent set of
statements one cannot prove all statements.

12. Models of
Non-standard
Computation
A. Syropoulos
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Paraconsist Logic(s)
Paraconsistency: the idea that contradictions makes sense.
A set of statements is incosistent if it contains both some
statement A and its negation A.
A logic is called paraconsistent if from an incosistent set of
statements one cannot prove all statements.
Example: the wave–particle duality is a form of inconsistency in
nature.

13. Models of
Non-standard
Computation
A. Syropoulos
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Paraconsist Logic(s)
Paraconsistency: the idea that contradictions makes sense.
A set of statements is incosistent if it contains both some
statement A and its negation A.
A logic is called paraconsistent if from an incosistent set of
statements one cannot prove all statements.
Example: the wave–particle duality is a form of inconsistency in
nature.
Manuel Bremer has speculated that for paraconsistent Turing
machines All problems (like decidability of Firts Order Logic)
that are at least as hard as the Halting Problem may turn out
to be at least as solvable!

14. Models of
Non-standard
Computation
A. Syropoulos
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What is Fuzziness

15. Models of
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Computation
A. Syropoulos
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What is Fuzziness
Fuzzy logic is a precise system of reasoning, deduction and
computation in which the objects of discourse and analysis are
associated with information which is, or is allowed to be,
imprecise, uncertain, incomplete, unreliable, partially true or
partially possible (Zadeh, BISC-group mailing list).

16. Models of
Non-standard
Computation
A. Syropoulos
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What is Fuzziness
Fuzzy logic is a precise system of reasoning, deduction and
computation in which the objects of discourse and analysis are
associated with information which is, or is allowed to be,
imprecise, uncertain, incomplete, unreliable, partially true or
partially possible (Zadeh, BISC-group mailing list).
Fuzzy logic and fuzzy set theory is very popular among
engineers!

17. Models of
Non-standard
Computation
A. Syropoulos
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What is Fuzziness
Fuzzy logic is a precise system of reasoning, deduction and
computation in which the objects of discourse and analysis are
associated with information which is, or is allowed to be,
imprecise, uncertain, incomplete, unreliable, partially true or
partially possible (Zadeh, BISC-group mailing list).
Fuzzy logic and fuzzy set theory is very popular among
engineers!
A fuzzy (sub)set A of a crisp set X is characterized by a
function
A X [0, 1],
where A(x) = i means that x belongs to A with degree i.

18. Models of
Non-standard
Computation
A. Syropoulos
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What is Fuzziness
Fuzzy logic is a precise system of reasoning, deduction and
computation in which the objects of discourse and analysis are
associated with information which is, or is allowed to be,
imprecise, uncertain, incomplete, unreliable, partially true or
partially possible (Zadeh, BISC-group mailing list).
Fuzzy logic and fuzzy set theory is very popular among
engineers!
A fuzzy (sub)set A of a crisp set X is characterized by a
function
A X [0, 1],
where A(x) = i means that x belongs to A with degree i.
Real world example of fuzzy set: scales of gray.

19. Models of
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Computation
A. Syropoulos
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Non-Standard Turing Machines

20. Models of
Non-standard
Computation
A. Syropoulos
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Non-Standard Turing Machines
Juan C. Agudelo and Andrés Sicard introduced paraconsistent
Turing machines.

21. Models of
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Computation
A. Syropoulos
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Non-Standard Turing Machines
Juan C. Agudelo and Andrés Sicard introduced paraconsistent
Turing machines.
Juan C. Agudelo and Walter Carnielli elaborated the theory by
showing that their PTMs are a model of quantum computation.

22. Models of
Non-standard
Computation
A. Syropoulos
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Non-Standard Turing Machines
Juan C. Agudelo and Andrés Sicard introduced paraconsistent
Turing machines.
Juan C. Agudelo and Walter Carnielli elaborated the theory by
showing that their PTMs are a model of quantum computation.
Eugene S. Santos defined fuzzy Turing Machines.

23. Models of
Non-standard
Computation
A. Syropoulos
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Non-Standard Turing Machines
Juan C. Agudelo and Andrés Sicard introduced paraconsistent
Turing machines.
Juan C. Agudelo and Walter Carnielli elaborated the theory by
showing that their PTMs are a model of quantum computation.
Eugene S. Santos defined fuzzy Turing Machines.
Jiří Wiedermann showed that fuzzy Turing machines have
hypercomputational powers.

24. Models of
Non-standard
Computation
A. Syropoulos
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Non-Standard Turing Machines
Juan C. Agudelo and Andrés Sicard introduced paraconsistent
Turing machines.
Juan C. Agudelo and Walter Carnielli elaborated the theory by
showing that their PTMs are a model of quantum computation.
Eugene S. Santos defined fuzzy Turing Machines.
Jiří Wiedermann showed that fuzzy Turing machines have
hypercomputational powers.
Benjamím Callejas Bedregal and Santiago Figueira question
Wiedermann’s results…

25. Models of
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Computation
A. Syropoulos
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Non-Standard Turing Machines
Juan C. Agudelo and Andrés Sicard introduced paraconsistent
Turing machines.
Juan C. Agudelo and Walter Carnielli elaborated the theory by
showing that their PTMs are a model of quantum computation.
Eugene S. Santos defined fuzzy Turing Machines.
Jiří Wiedermann showed that fuzzy Turing machines have
hypercomputational powers.
Benjamím Callejas Bedregal and Santiago Figueira question
Wiedermann’s results…
P systems are not a non-standard model of computation.

26. Models of
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Computation
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Paraconsistent Turing machines

27. Models of
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Computation
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Paraconsistent Turing machines
Cells do not hold a single symbol but instead they hold a
multiset of symbols.

28. Models of
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Paraconsistent Turing machines
Cells do not hold a single symbol but instead they hold a
multiset of symbols.
Cells that are supposed to be initially empty hold empty sets.

29. Models of
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Computation
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Paraconsistent Turing machines
Cells do not hold a single symbol but instead they hold a
multiset of symbols.
Cells that are supposed to be initially empty hold empty sets.
Each quadruple is associated with a plausibility degree, which is
number between zero and one.

30. Models of
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Computation
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Paraconsistent Turing machines
Cells do not hold a single symbol but instead they hold a
multiset of symbols.
Cells that are supposed to be initially empty hold empty sets.
Each quadruple is associated with a plausibility degree, which is
number between zero and one.
The consistency restriction is relaxed.

31. Models of
Non-standard
Computation
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Paraconsistent Turing machines
Cells do not hold a single symbol but instead they hold a
multiset of symbols.
Cells that are supposed to be initially empty hold empty sets.
Each quadruple is associated with a plausibility degree, which is
number between zero and one.
The consistency restriction is relaxed.
Qc, the “current state,” is a set of ordinary states ( initially
Qc = {q0}.)

32. Models of
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Computation
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How PTMs Operate?

33. Models of
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How PTMs Operate?
If there is only one quadruple qi Sj Sk ql such that qi Qc and
Sj Cc (Cc contens of current cell), then
Cc (Cc {Sj}) {Sk} and Qc (Qc {qi}) {ql}.

34. Models of
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Computation
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How PTMs Operate?
If there is only one quadruple qi Sj Sk ql such that qi Qc and
Sj Cc (Cc contens of current cell), then
Cc (Cc {Sj}) {Sk} and Qc (Qc {qi}) {ql}.
If the only quadruple for which qi Qc and Sj Cc are
quadruples of the form qi Sj L ql or qi Sj R ql, then
Qc (Qc {qi}) {ql},
and the scanning head moves left or right, respectively.

35. Models of
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Computation
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How PTMs Operate?
If there are several quadruples qi Sj Ski
qlj
such that qi Qc and
Sj Cc, then
Cc (Cc {Sj}) {Sk1
, , Skm
}
and
Qc (Qc {qi}) {ql1
, , qln
}.

36. Models of
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Computation
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How PTMs Operate?
If there are several quadruples qi Sj Ski
qlj
such that qi Qc and
Sj Cc, then
Cc (Cc {Sj}) {Sk1
, , Skm
}
and
Qc (Qc {qi}) {ql1
, , qln
}.
If there are several entirely different quadruples q(i)
i S(i)
j S(i)
k q(i)
l
such that q(i)
i Qc and S(j)
j Cc, then
Cc (Cc {Sj}) {S(i)
k } and Qc (Qc {qi}) {q(i)
l },
provided that the corresponding ith quadruple has the highest
plausibility degree.

37. Models of
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Computation
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Fuzzy P Systems
A P system with only fuzzy data is a construct
FD = (O, , w(1), , w(m), R1, , Rm, i0, )

38. Models of
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Computation
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Fuzzy P Systems
A P system with only fuzzy data is a construct
FD = (O, , w(1), , w(m), R1, , Rm, i0, )
Extend systems by adding fuzzy rewrite rules (it was proposed,
but not seriously considered…).

39. Models of
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Computation
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Fuzzy P Systems
A P system with only fuzzy data is a construct
FD = (O, , w(1), , w(m), R1, , Rm, i0, )
Extend systems by adding fuzzy rewrite rules (it was proposed,
but not seriously considered…).
Rules should be of the form
12 n 12 m,
Such a rule is feasible iff (i) for all i = 1, , n, where (i) is a
similarity degree (the idea is borrowed from the description of
fuzzy chemical abstract machines).

40. Models of
Non-standard
Computation
A. Syropoulos
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Paraconsistent P Systems
A P system with inconsistent rewriting rules.

41. Models of
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Computation
A. Syropoulos
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Paraconsistent P Systems
A P system with inconsistent rewriting rules.
It is possible to define fuzzy PP systems.

42. Models of
Non-standard
Computation
A. Syropoulos
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Speculations about the Computational Power

43. Models of
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Computation
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Speculations about the Computational Power
Fuzzy paraconsistent Turing machines are as powerful as fuzzy
Turing machines.

44. Models of
Non-standard
Computation
A. Syropoulos
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Speculations about the Computational Power
Fuzzy paraconsistent Turing machines are as powerful as fuzzy
Turing machines.
The same applies to fuzzy P systems.

45. Models of
Non-standard
Computation
A. Syropoulos
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Speculations about the Computational Power
Fuzzy paraconsistent Turing machines are as powerful as fuzzy
Turing machines.
The same applies to fuzzy P systems.
PP systems is something that has to be studied further.

46. Models of
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Computation
A. Syropoulos
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Finally…
Thank you very much for your attention.