Models of Non-standard Computation

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

Citation preview

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

Models of

Non-standard

Computation

A. Syropoulos

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Presentation Outline

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?

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?

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?

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?

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?

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 outto be at least as solvable!

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 outto be at least as solvable!

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 outto be at least as solvable!

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 outto be at least as solvable!

Models of

Non-standard

Computation

A. Syropoulos

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....

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....

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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 outto be at least as solvable!

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 outto be at least as solvable!

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.

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.

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.

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.

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.

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…

P systems are not a non-standard model of computation.

Models of

Non-standard

Computation

A. Syropoulos

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.....

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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.

Models of

Non-standard

Computation

A. Syropoulos

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.....

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....

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.....

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....

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.....

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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.

Models of

Non-standard

Computation

A. Syropoulos

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.....

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....

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.....

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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.

Models of

Non-standard

Computation

A. Syropoulos

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.....

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....

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....

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.....

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....

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.....

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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.

Models of

Non-standard

Computation

A. Syropoulos

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.

....

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.....

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....

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....

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....

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.....

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....

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.....

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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.

Models of

Non-standard

Computation

A. Syropoulos

.....

.

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.....

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....

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....

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....

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.....

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....

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.....

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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.

Models of

Non-standard

Computation

A. Syropoulos

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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˝.)

Models of

Non-standard

Computation

A. Syropoulos

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.....

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.....

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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˝.)

Models of

Non-standard

Computation

A. Syropoulos

.....

.

....

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....

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.....

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....

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....

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.....

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....

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....

.

....

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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˝.)

Models of

Non-standard

Computation

A. Syropoulos

.....

.

....

.

....

.

.....

.

....

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....

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....

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.....

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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˝.)

Models of

Non-standard

Computation

A. Syropoulos

.....

.

....

.

....

.

.....

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....

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....

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....

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.....

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....

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....

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.....

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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˝.)

Models of

Non-standard

Computation

A. Syropoulos

.....

.

....

.

....

.

.....

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....

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....

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....

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.....

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....

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....

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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˝.)

Models of

Non-standard

Computation

A. Syropoulos

.....

.

....

.

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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 arequadruples 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.

Models of

Non-standard

Computation

A. Syropoulos

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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 arequadruples 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.

Models of

Non-standard

Computation

A. Syropoulos

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....

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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 arequadruples 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.

Models of

Non-standard

Computation

A. Syropoulos

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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)lsuch 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.

Models of

Non-standard

Computation

A. Syropoulos

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.....

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....

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....

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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)lsuch 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.

Models of

Non-standard

Computation

A. Syropoulos

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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).

Models of

Non-standard

Computation

A. Syropoulos

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....

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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).

Models of

Non-standard

Computation

A. Syropoulos

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.....

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....

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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).

Models of

Non-standard

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.

Models of

Non-standard

Computation

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Paraconsistent P Systems

A P system with inconsistent rewriting rules.

It is possible to define fuzzy PP systems.

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.

Models of

Non-standard

Computation

A. Syropoulos

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....

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....

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....

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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.

Models of

Non-standard

Computation

A. Syropoulos

.....

.

....

.

....

.

.....

.

....

.

....

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....

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.....

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.

....

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.....

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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.

Models of

Non-standard

Computation

A. Syropoulos

.....

.

....

.

....

.

.....

.

....

.

....

.

....

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.....

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....

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....

.

....

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.....

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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.

Models of

Non-standard

Computation

A. Syropoulos

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....

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....

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Finally…

Thank you very much for your attention.

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