Small(purely) Catalytic P systems simulating register machineAuthored bypetr sosik,Miroslav langer

PRESENTED BY

MUDHIGONDA,VENKAT SAI SHARATH CHANDRA

FOR THE COURSE

CS5813 FORMAL LANGUAGES

11/22/2016

Table of Contents

Introduction Definitions P systems simulation in different modes P systems Generating a Non-Semilineaar set Conclusion

INTRODUCTION

P System is an abstract computing model inspired by the behavior of membranes in living cells, active transport of molecules,catalyzed reactions etc.

Catalyst is a substance that increases the rate of a chemical reaction without itself undergoing any permanent chemical change.

Catalyst cannot change during the reaction and they cannot travel between regions

Difference between Small and purely catalytic Systems

Small Catalytic P Systems Small Catalytic P System consists of

difference natural behaviors including catalytic reactions

Two catalyst are required to prove them as Turing –computable

Purely Catalytic P Systems Purely catalytic P system only involves

catalyzed reactions

Three catalyst are required to prove them as Turing-Computable

What is Register Machine?

In mathematical logic and theoretical computer science a register machine is a generic class of abstract machines used in a manner similar to a Turing machine. All the models are Turing equivalent.

In contrast to the tape and head used by a Turing machine, the model uses multiple, uniquely addressed registers, each of which holds a single positive integer.

Two catalyst are enough to simulate any register machine with two decrementable register which , in turn can compute any Turing-Computable function.

Basic idea

P systems in different computing modes can only compute semilinear sets Non-Semilinear set is set of integers or tuple of integers constructed in such a way

that they do not follow any sequence(like arthmetic progression) The P system with catalyst proved to have computational completeness but there

was no example of cataysts generating a non- Semilinear set

Definitions

By N we denote the set of all nonnegative integers, NRE denotes the class of all recursively enumerable sets of non negative integers

PsRE the class of all recursively enumerable vectors of nonnegative integers. For a finite set of symbolsV, by V∗

we denote the set of all strings consisting of letters from V, including the empty string.

Following the notation , for integers m ≥2and k, lsuch that 1 ≤k, l ≤m, we define

Definitions

A register machine is a tuple M=(m,H,lo,lh,I), where:

• m is the number of registers;• H is the set of labels of instructions;

• lo is the label of the initial instruction;

• lh is the label of the final (halting) instruction;

• I is the program, i.e., a set of instructions labeled in a one-to-one manner by the elements of H.

Register Instructions

The instruction types are:• (ADD(r),i,j)– add 1 to the contents of register r and choose non-deterministically

to continue with the instruction labeled i or j ; if the machine is deterministic, then the instruction adopts the form (ADD(r),i);

• (SUB(r),i,j)– if the contents of the register r is nonzero, then subtract 1 from it and continue with instruction i, else continue with instruction j;

• HALT– halt the machine.

Catalytic P system

Acatalytic P systemof degree m ≥1is a construct

• =(O,C,μ,w1,...,wm,R1,...,Rm,i0)• where:

• O is the alphabet of objects;

• C ⊆ Ois the alphabet of catalysts;

• μ is a membrane structure of degree m with membranes labeled in a one-to-one manner with the natural numbers 1, 2, ..., m;

• w1, ..., wm ∈ O are strings representing multisets of objects initially present in the ∗ m regions of μ;

• Ri, 1 ≤i ≤m, are finite sets of evolution rules over O associated with the regions 1, 2, ..., mof μ; these evolution rules are of the forms ca →cv or a →v, where c is a catalyst, a is an object from O −C, and vis a string from ((O −C) ×{here, out, in}) ;∗

• I0 {0, 1, ..., m} indicates the output region of .∈ 𝝅

Catalytic P Systems in Generating mode

Here # is called trap symbol

# is used to when two objects are in same state and need a catalyst to complete the reaction

# is also used when an wrong step is chosen non deterministically such that the process never halts

Rules for trap symbol

Rules for add instruction

Rules for sub instruction

Simulation of ADD instruction

An ADD-instruction j :(ADD(r), k, l) ∈ I is simulated by the no n-deterministic choice of one of the rules

c1p̃j→c1pkd2or

c1p̃j→c1pld2or

Simultaneously, the rule c2d→c2is executed in order to keep the catalyst c2busy. If this catalyst instead acts in the rule c2o2→c2d2 then in the next step there are

two objects d2and at least one is subject to the rule d2→#, introducing the trap symbol #.

simulation of SUB instructions

Simulation of HALT instruction

The instruction lh:(HALT)is simulated without using any special objects or rules of the P system. We simply modify each previously described rule producing the symbol plhby replacing this object at the right-hand side of the rule by d1. When such a rule is executed, the P systems finally halts with both simulated registers r1and r2being empty

Result Cataltyic P systems halt these instructions after following rules |R| ≤ 3nA + 6nS + 11 for catalytic Psystems, |R| ≤ 3nA + 6nS + 13 for purely catalyticP systems • where nA and nS is the number of instructions ADD and SUB, respectively

Catalytic P Systems in Computing mode

In this mode there are ‘r’ number of catalysts which can decrement with associated r registers

Here ADD instructions are simulated deterministically

j : (ADD(r),k) I∈ SUB instructions are guessed non-

determinstically

Simulation of SUB instructions When the wrong step is chosen

non-deterministically # is introduced

The correct steps of simulation results in halting of instructions else the computation cannot produce any result

If there are any undesired rules catalysts are kept busy in other auxiliary rules

Result

Catalytic P system can achieve its result in following rules |R| ≤ 2nA+6nS+5m+1. By using generalized register machine we can reduce the result to |R| ≤ 6nS + 5m +1 for catalytic P systems, |R| ≤ 6nS + 6m + 1 forpurely catalytic P systems. In similar manner Catalytic P systems in accepting mode can be achieved in

following rules |R| ≤ 2nA+6nS+5m+1.

Catalytic P systems generating Semilinear sets Consider a non-deterministic register

machine with three registers generating the set {2n −2n | n ≥2}. The machine starts with all registers empty and it runs the following program which stores the result in register 3

Catalytic P systems generating Semilinear sets

The register machine program described works as follows:

• The contents of register 1 is emptied and duplicated to register 2

• then the contents of register 2 is copied to register 1 and added to register 3 .

• Finally, the only non-deterministic instruction ADD labeled 7 increments register 1 and it decides non-deterministically whether the computation continues or whether it halts.

• The whole cycle is repeated until a jump to the instruction HALT is chosen non-deterministically.

GENERALISED REGISTER MACHINE INSTRUCTIONS

we can express the program given above with generalized SUB-instructions

1: (SUB(1),ADD(2)ADD(2) 1,2)2: (SUB(2),ADD(1)ADD(3)2,ADD(1)3)3: HALT

CONCLUSION

The paper was hard to understand in the beginning. So, I have started reading the relevant papers to get idea of P systems

This paper proves Universal Completeness of P systems by computing non-semilinear sets

The author utilizes catalyst which are equal to no off register machines in computing mode but the paper states that they use only two catalysts