Water computing: A P system variant Alec Henderson 1 , Radu Nicolescu - - PowerPoint PPT Presentation

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Water computing: A P system variant Alec Henderson 1 , Radu Nicolescu 1 , Michael J. Dinneen 1 , TN Chan 2 , Hendrik Happe 3 , and Thomas Hinze 3 1 School of Computer Science The University of Auckland, Auckland, New Zealand 2 Compucon New Zealand,

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Water computing: A P system variant

Alec Henderson1, Radu Nicolescu1, Michael J. Dinneen1, TN Chan2, Hendrik Happe3, and Thomas Hinze3

1School of Computer Science

The University of Auckland, Auckland, New Zealand

2Compucon New Zealand, Auckland, New Zealand 3Department of Bioinformatics

Friedrich Schiller University of Jena, Jena, Germany

September 10, 2020

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Outline of presentation

1

Introduction

2

Previous Work

3

Control tanks

4

Example

5

Turing Completeness

6

A restricted cP system

7

Conclusion

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SLIDE 3

The power of water

Coromandel, New Zealand The magical sound,

  • f the cascading water,

natural beauty

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Water Integrator

First model built in 1936, in USSR; modular model in 1941, standard unified units in 1949-1955

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Water Integrator

First model built in 1936, in USSR; modular model in 1941, standard unified units in 1949-1955 Used to solve inhomogeneous differential equations with applications such as: solving construction issues in the sands

  • f Central Asia and in

permafrost and in studying the temperature regime of the Antarctic ice sheet

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Water Integrator

First model built in 1936, in USSR; modular model in 1941, standard unified units in 1949-1955 Used to solve inhomogeneous differential equations with applications such as: solving construction issues in the sands

  • f Central Asia and in

permafrost and in studying the temperature regime of the Antarctic ice sheet Only surpassed by digital computers in the 1980’s.

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SLIDE 7

MONIAC (Monetary National Income Analogue Computer)

MONIAC (Monetary National Income Analogue Computer) also known as the Phillips Hydraulic Computer and the Financephalograph

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MONIAC (Monetary National Income Analogue Computer)

MONIAC (Monetary National Income Analogue Computer) also known as the Phillips Hydraulic Computer and the Financephalograph First built in 1949 by New Zealand economist Bill Phillips to model the UK economy.

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MONIAC (Monetary National Income Analogue Computer)

MONIAC (Monetary National Income Analogue Computer) also known as the Phillips Hydraulic Computer and the Financephalograph First built in 1949 by New Zealand economist Bill Phillips to model the UK economy. Built as a teaching aid it was discovered that it was also an effective economic simulator.

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SLIDE 10

Previous Work1

No centre of control. Water flows if and only if all valves on a pipe are

  • pen.

Water flows between tanks concurrently.

1Thomas Hinze et al. “Membrane computing with water”. In: J. Membr. Comput.

2.2 (2020), pp. 121–136.

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SLIDE 11

Open Problems

How can functions be stacked without a combinatorial explosion of the number of valves?

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SLIDE 12

Open Problems

How can functions be stacked without a combinatorial explosion of the number of valves? How can termination of the system be detected?

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SLIDE 13

Open Problems

How can functions be stacked without a combinatorial explosion of the number of valves? How can termination of the system be detected? How to reset the system?

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SLIDE 14

Open Problems

How can functions be stacked without a combinatorial explosion of the number of valves? How can termination of the system be detected? How to reset the system? Is the system Turing complete?

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SLIDE 15

Open Problems

How can functions be stacked without a combinatorial explosion of the number of valves? How can termination of the system be detected? How to reset the system? Is the system Turing complete? We solve these problems by introducing a set of control tanks.

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Control tanks

x1 x′ 1 xn x′ n

... ... y1, ..., ym = f(x1, ..., xn)

y1 ym

...

y′ 1 y′ m

...

A control tank for each input and output.

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Control tanks

x1 x′ 1 xn x′ n

... ... y1, ..., ym = f(x1, ..., xn)

y1 ym

...

y′ 1 y′ m

...

A control tank for each input and output. Start the computation

  • nce all control tanks are

filled. An output is ready when it’s control tank is full.

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Subtraction

x x′ y′ z z′ y q′ = 1 q′ = 1 y = 0 y = 0 q′ = 1 q′ = 1 q′ = 1 q′ x′ = 1 y′ = 1 x = 0 y = 0 x′ = 0 y′ = 0

z = x − y

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Subtraction

x x′ y′ z z′ y q′ = 1 q′ = 1 y = 0 y = 0 q′ = 1 q′ = 1 q′ = 1 q′ x′ = 1 y′ = 1 x = 0 y = 0 x′ = 0 y′ = 0

z = x − y

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Subtraction

x x′ y′ z z′ y q′ = 1 q′ = 1 y = 0 y =′ 0 q′ = 1 q′ = 1 q′ = 1 q′ x′ = 1 y′ = 1 x = 0 y = 0 x′ = 0 y′ = 0

z = x − y

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Subtraction

x x′ y′ z z′ y q′ = 1 q′ = 1 y = 0 y = 0 q′ = 1 q′ = 1 q′ = 1 q′ x′ = 1 y′ = 1 x = 0 y = 0 x′ = 0 y′ = 0

z = x − y

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Subtraction

x x′ y′ z z′ y q′ = 1 q′ = 1 y = 0 y = 0 q′ = 1 q′ = 1 q′ = 1 q′ x′ = 1 y′ = 1 x = 0 y = 0 x′ = 0 y′ = 0

z = x − y

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SLIDE 23

Subtraction

x x′ y′ z y q′ = 1 q′ = 1 y = 0 y = 0 q′ = 1 q′ = 1 q′ = 1 q′ x′ = 1 y′ = 1 x = 0 y = 0 x′ = 0 y′ = 0

z = x − y

z′ Water computing: A P system variant 14 / 29

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Primitive Recursion

To prove our system can construct all unary primitive recursive functions we use the following base functions and closure operators2: Successor function: S(x) = x + 1 Subtraction function: B(x, y) = x − y Composition operator: C(h, g)(x) = h(g(x)) Difference operator: D(f , g)(x) = f (x) − g(x) Primitive recursion operator: P(f )(0) = 0, P(x + 1) = f (P(x)) To assist in the proof we also construct two copy functions: inplace i(x) and destructive c(x).

2Cristian Calude and Lila Sˆ

  • antean. “On a theorem of G¨

unter Asser”. In: Mathematical Logic Quarterly 36.2 (1990), pp. 143–147.

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Destructive copy x1 = x2 = x

x x′ x1 q′ = 1 q′ = 1 x2 q′ x′ = 1 x′ 1 x = 0 q′ = 1 x = 0 x′ 2 q′ = 0 x = 0

x1 = x2 = x

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Inplace copy x1 = x

x x′ q′ x′ = 1 a x1 x′ 1 x = 0 q′ = 1 x′ = 0 x = 0 q′ = 1 x′ = 1 q′ = 1 a = 0 x′ = 0

x1 = x

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Successor function S(x) = x + 1

x x′ q′ = 1 x′ = 1 x = 0 x′ = 0 q′ = 1

z = x + 1

q′ z′ z

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Composition operator C(h, g)(x) = h(g(x))

x x′

y = g(x)

y y′

z = f(y)

z z′

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Difference operator D(f , g)(x) = f (x) − g(x)

x1 x′ 1

u = g(x)

u u′

z = B(u, v) = u − v

x x′ z z′ x2 x′ 2

v = f(x)

v v′

c(x) : x1 = x2 = x

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Primitive recursion operator P(f )(0) = 0, P(x + 1) = f (P(x))

x x′ q′ = 1 u

v = f(u)

v v′ x′ = 0 q′ = 1 u′ x = 0 x = 0 v = 0 y y′ x = 0 x = 0 v′ = 1 v = 0 d′ x = 0 v = 0 v′ = 0 d′ = 0 v′ = 0 x′ = 0 x = 0 u′ = 0 q′ x′ = 1 u′ = 0

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SLIDE 31

To prove Turing completeness we require that our system can construct the unary primitive recursive functions as well as3: Addition function: A(x, y) = x + y µ operator: µy(f )(x, y) = miny{f (x, y) = 0}

3Julia Robinson. “General recursive functions”. In: Proceedings of the american

mathematical society 1.6 (1950), pp. 703–718.

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Addition function A(x, y) = x + y

x y′ z z′ x′ y q′ = 1 q′ = 1 q′ = 1 q′ = 1 q′ x′ = 1 y′ = 1 y = 0 x = 0 x′ = 0 y′ = 0

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µ operator µy(f )(x, y) = miny{f (x, y) = 0}

x x′ y′ q′ = 1 u u′ v v′ w w′

w = f(u, v)

x1 q′ = 1 x′ 1 x = 0

u = i(x) = x1

y y′

v = i(y) = y

q′ x′ = 1 w′ = 1 w = 0 y w′ = 1 w = 0 y = 0 w = 0 y = 0 d′ w = 0 d′ = 0 w = 0 d′ = 0 w = 0 d′ = 0 w = 0 d′ = 0 w = 0 q′ = 1 x = 0 x′ = 0 w′ = 0 w′ = 1 w = 0 x1 = 0

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A restricted cP system

cP systems subcells act as data storage the same as our water tanks.

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A restricted cP system

cP systems subcells act as data storage the same as our water tanks. cP systems have a set of rules for changing the content in the

  • subcells. Similarly we have a set of valves and pipes.

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A restricted cP system

cP systems subcells act as data storage the same as our water tanks. cP systems have a set of rules for changing the content in the

  • subcells. Similarly we have a set of valves and pipes.

cP systems are able to create and consume subcells whereas in our system we cannot create and consume tanks.

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A restricted cP system

cP systems subcells act as data storage the same as our water tanks. cP systems have a set of rules for changing the content in the

  • subcells. Similarly we have a set of valves and pipes.

cP systems are able to create and consume subcells whereas in our system we cannot create and consume tanks. Using the similarities we are able to construct cP system-like rules which don’t contain creating or consuming of subcells.

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Rules for subtraction

x x′ y′ z z′ y q′ = 1 q′ = 1 y = 0 y = 0 q′ = 1 q′ = 1 q′ = 1 q′ x′ = 1 y′ = 1 x = 0 y = 0 x′ = 0 y′ = 0

z = x − y s1 q() → s2 q(1) | cx (1) cy (1) (1) s2 cx (1) → s2 cx () | q(1) (2) s2 cy (1) → s2 cy () | q(1) (3) s2 vx (X1) → s2 vx (X) | q(1) vy ( 1) (4) s2 vy (Y 1) → s2 vy (Y ) | q(1) (5) s2 vx (X1) vz (Z) → s2 vx (X) vz (Z1) | q(1) vy () (6) s2 q(1) cz () → s3 q() cz (1) | cx () cy () (7) Water computing: A P system variant 26 / 29

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Conclusion

We have proven that our water tank system is Turing complete, via construction of µ-recursive functions.

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Conclusion

We have proven that our water tank system is Turing complete, via construction of µ-recursive functions. We have demonstrated how termination can be detected, as well as how to combine different functions without an exponential explosion

  • f the number of valves.

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Conclusion

We have proven that our water tank system is Turing complete, via construction of µ-recursive functions. We have demonstrated how termination can be detected, as well as how to combine different functions without an exponential explosion

  • f the number of valves.

We have given a brief description on how our water tank system is a restricted version of cP systems.

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Future Work

Being able to run a function without all of the controls being filled.

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Future Work

Being able to run a function without all of the controls being filled. Solving practical problems with the system.

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Future Work

Being able to run a function without all of the controls being filled. Solving practical problems with the system. Using the system to model biological systems.

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Future Work

Being able to run a function without all of the controls being filled. Solving practical problems with the system. Using the system to model biological systems. Constructing a programmable universal water computer.

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Questions

Coromandel New Zealand Thank you for listening.

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Questions

Coromandel New Zealand Thank you for listening. Any questions?

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