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Computing with P SystemsA Syropoulos, S Doumanis and KT Sotiriades

Greek Molecular Computing Group

Xanthi, Greece

E-mail: gmcg@araneous.com

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2/15

Recursive Functions & P Systems

Recursive Functions have been encoded as P systems

by Romero-Jiménez and Pérez-Jiménez.

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2/15

Recursive Functions & P Systems

Recursive Functions have been encoded as P systems

by Romero-Jiménez and Pérez-Jiménez.

Here we will present an alternative encoding based on

“abacus” computing.

M.I.T

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2/15

Recursive Functions & P Systems

Recursive Functions have been encoded as P systems

by Romero-Jiménez and Pérez-Jiménez.

Here we will present an alternative encoding based on

“abacus” computing.

For our encoding it was necessary to introduce new

rewriting rules and to make use of a more “liberal”

membrane structures.

M.I.T

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2/15

Recursive Functions & P Systems

Recursive Functions have been encoded as P systems

by Romero-Jiménez and Pérez-Jiménez.

Here we will present an alternative encoding based on

“abacus” computing.

For our encoding it was necessary to introduce new

rewriting rules and to make use of a more “liberal”

membrane structures.

Our encoding provides a “fresh” look at P systems.

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3/15

The Basic Functions

The three basic functions.

the zero function z(x1, . . . , xn) = 0,

the successor function S(x) = x+ 1, and

the identity function Uni (x1, . . . , xn) = xi, 1 ≤ i ≤ n.

M.I.T

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3/15

The Basic Functions

The three basic functions.

the zero function z(x1, . . . , xn) = 0,

the successor function S(x) = x+ 1, and

the identity function Uni (x1, . . . , xn) = xi, 1 ≤ i ≤ n.

M.I.T

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3/15

The Basic Functions

The three basic functions.

the zero function z(x1, . . . , xn) = 0,

the successor function S(x) = x+ 1, and

the identity function Uni (x1, . . . , xn) = xi, 1 ≤ i ≤ n.

M.I.T

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3/15

The Basic Functions

The three basic functions.

the zero function z(x1, . . . , xn) = 0,

the successor function S(x) = x+ 1, and

the identity function Uni (x1, . . . , xn) = xi, 1 ≤ i ≤ n.

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4/15

Encoding the Zero Function

We assume that each basic function is encoded as a

P system with two membranes.

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4/15

Encoding the Zero Function

We assume that each basic function is encoded as a

P system with two membranes.

This function simply discards its arguments and

returns zero. For each argument xi of the function we

pick up an object αi and place xi copies of it in the

outter compartment.

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4/15

Encoding the Zero Function

We assume that each basic function is encoded as a

P system with two membranes.

This function simply discards its arguments and

returns zero. For each argument xi of the function we

pick up an object αi and place xi copies of it in the

outter compartment.

Next, we associate to each object αi a multiset rewrite

rule of the form

αi → ε

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5/15

Encoding the Zero Function (Cont.)

This is a new kind of rule that simply annihilates all

objects. We can imagine that there is a pipe that is

used to throw the αi’s into the environment.

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5/15

Encoding the Zero Function (Cont.)

This is a new kind of rule that simply annihilates all

objects. We can imagine that there is a pipe that is

used to throw the αi’s into the environment.

This rule can be considered to implement a form of

“catharsis” of a compartment.

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5/15

Encoding the Zero Function (Cont.)

This is a new kind of rule that simply annihilates all

objects. We can imagine that there is a pipe that is

used to throw the αi’s into the environment.

This rule can be considered to implement a form of

“catharsis” of a compartment.

Thus, the outcome of the computation of this P system

is the number zero as all objects are actually thrown to

the environment.

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6/15

Encoding the successor function

Problem: An empty P system encoding the successor

function.

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6/15

Encoding the successor function

Problem: An empty P system encoding the successor

function.

This system must place an object to the target

compartment and stop.

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6/15

Encoding the successor function

Problem: An empty P system encoding the successor

function.

This system must place an object to the target

compartment and stop.

As there is no rule that can be apllied to an empty

compartment we are obliged to introduce a new one:

ε → (α, ini)

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7/15

Encoding the successor function (Cont.)

The rule ε → (α, ini) is applicable only if there are no

objects in a given compartment.

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Encoding the successor function (Cont.)

The rule ε → (α, ini) is applicable only if there are no

objects in a given compartment.

This rule just places an object α to compartment i.

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7/15

Encoding the successor function (Cont.)

The rule ε → (α, ini) is applicable only if there are no

objects in a given compartment.

This rule just places an object α to compartment i.

The rationale behind this rule is that at any moment a

cell can absorb matter from its environment that, in

turn, will be consumed.

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7/15

Encoding the successor function (Cont.)

The rule ε → (α, ini) is applicable only if there are no

objects in a given compartment.

This rule just places an object α to compartment i.

The rationale behind this rule is that at any moment a

cell can absorb matter from its environment that, in

turn, will be consumed.

This new rule makes the encoding of the successor

function trivial.

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8/15

The identity function

This function is similar to the zero function with one

difference: it discards all of its arguments but one.

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8/15

The identity function

This function is similar to the zero function with one

difference: it discards all of its arguments but one.

For each xi we place xi copies of αi to the outter

membrane

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8/15

The identity function

This function is similar to the zero function with one

difference: it discards all of its arguments but one.

For each xi we place xi copies of αi to the outter

membrane

Assume that the identity function returns its jth

argument, then we associate with αj the following rule:

αj → (αj, in2)

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8/15

The identity function

This function is similar to the zero function with one

difference: it discards all of its arguments but one.

For each xi we place xi copies of αi to the outter

membrane

Assume that the identity function returns its jth

argument, then we associate with αj the following rule:

αj → (αj, in2)

The remaing objects are associated with a catharsis

rule: αi → ε, i 6= j

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9/15

The Three Processes

Composition f is a function of m arguments and each

g1, . . . , gm is a function of n arguments, then their

composition is defined as follows:

h(x1, . . . , xn) = f(g1(x1, . . . , xn), . . . , gm(x1, . . . , xn))

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9/15

The Three Processes

Composition f is a function of m arguments and each

g1, . . . , gm is a function of n arguments, then their

composition is defined as follows:

h(x1, . . . , xn) = f(g1(x1, . . . , xn), . . . , gm(x1, . . . , xn))

Primitive Recursion A function h of k + 1 arguments is

definable by (primitive) recursion from functions f and g,

with k and k + 2 arguments, respectively, if it is defined as

follows:

h(x1, . . . , xk, 0) = f(x1, . . . , xk)

h(x1, . . . , xk, S(m)) = g(x1, . . . , xk,m, h(x1, . . . , xk,m))

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The Three Processes (Cont.)

Minimalization The operation of minimalizationassociates with each total function f of k + 1arguments the function h of k arguments. Givena tuple (x1, . . . , xk), the value of h(x1, . . . , xk) isthe least value of xk+1, if one such exists, forwhich f(x1, . . . , xk, xk+1) = 0. If no such xk+1

exists, then its value is undefined.

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11/15

Encoding the composition process

αx1βx2γx3

✛✚

✘✙

✄✂ �✁g1 ✛

✚✘✙

✄✂ �✁g2

✛✚

✘✙

f

❈❈❈❈❈❈

❈❈❈❈❈❈

✁✁✁✁✁✁

✁✁✁✁

✁✁

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12/15

Encoding primitive recursion

αxβy

f(x, y) g(x, 0, f(x))

. . .

g(. . .)

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Encoding minimalization function

αxβy

1

f(x, y)

2 3

4

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The rules

R1 = {a → (a, to2)(⊕, to3),

b → (b, to2)(⊗, to3))}

R2 = {# → (#, to3)}

R3 = {# → (#, to4),

# → (b, to1),

⊕ → (a, to1),

⊗ → (b, to1)}

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That’s all!

Thank you very much for your attention!

M.I.T