On a powerful class of non-universalP systems with active membranes

Antonio E. Porreca Alberto Leporati Claudio Zandron

Dipartimento di Informatica, Sistemistica e ComunicazioneUniversità degli Studi di Milano-Bicocca, Italy

14th Developments in Language TheoryLondon, Ontario, 19 August 2010

Introduction

I Membrane systems (P systems) are devices inspiredby structure and functioning of biological cellsintroduced by G. Paun in 1998

I They can be used as language recognisersand most variants are computationally universal

I We show a variant of P system that decides exactlythe languages decided by Turing machines workingin tetrational (iterated exponential) time and space

2/18

Introduction

I Membrane systems (P systems) are devices inspiredby structure and functioning of biological cellsintroduced by G. Paun in 1998

I They can be used as language recognisersand most variants are computationally universal

I We show a variant of P system that decides exactlythe languages decided by Turing machines workingin tetrational (iterated exponential) time and space

2/18

Introduction

I Membrane systems (P systems) are devices inspiredby structure and functioning of biological cellsintroduced by G. Paun in 1998

I They can be used as language recognisersand most variants are computationally universal

I We show a variant of P system that decides exactlythe languages decided by Turing machines workingin tetrational (iterated exponential) time and space

2/18

Outline

I Description of our variant of P systemsI Languages decidable in tetrational timeI How much space can P systems use? (upper bound)I Simulating tetrational-space TMs (lower bound)I Conclusions and open problems

3/18

The membrane structure of P systemsI Membranes divide the cell into regionsI Membranes have fixed label and changeable electrical charge

h0

h1

h2

h3 h4

+

− 0

+

0

4/18

The membrane structure of P systemsI Membranes divide the cell into regionsI Membranes have fixed label and changeable electrical charge

h0

h1

h2

h3 h4

h0

h1 h2

h3 h4

+

− 0

+

0

[ [ ]+h1[ [ ]−h3

[ ]0h4]+h2

]0h0

0

+

−

+

0

4/18

Contents of the membranes

Each membrane contains a multiset of objects(symbols from an alphabet Γ) representing molecules

h0

h1

h2

+ +0

aaabbc

bcc

abc

[aaabbc [bcc]+h1[abc]+h2

]0h0

5/18

Evolution rules

I Each pair (label, charge) has a set of associated rules

I For instance, the rule [a]0h → [ ]+h b can be applied toa membrane having label h and charge 0

I The charge represents a kind of state of the membrane,controlling which set of rules can be applied

I We only use three kinds of rule in this paper:I Send-out communication rules: [a]αh → [ ]βh bI Send-in communication rules: a [ ]αh → [b]βhI Nonelementary division rules

6/18

Evolution rules

I Each pair (label, charge) has a set of associated rulesI For instance, the rule [a]0h → [ ]+h b can be applied to

a membrane having label h and charge 0

I The charge represents a kind of state of the membrane,controlling which set of rules can be applied

I We only use three kinds of rule in this paper:I Send-out communication rules: [a]αh → [ ]βh bI Send-in communication rules: a [ ]αh → [b]βhI Nonelementary division rules

6/18

Evolution rules

I Each pair (label, charge) has a set of associated rulesI For instance, the rule [a]0h → [ ]+h b can be applied to

a membrane having label h and charge 0I The charge represents a kind of state of the membrane,

controlling which set of rules can be applied

I We only use three kinds of rule in this paper:I Send-out communication rules: [a]αh → [ ]βh bI Send-in communication rules: a [ ]αh → [b]βhI Nonelementary division rules

6/18

Evolution rules

I Each pair (label, charge) has a set of associated rulesI For instance, the rule [a]0h → [ ]+h b can be applied to

a membrane having label h and charge 0I The charge represents a kind of state of the membrane,

controlling which set of rules can be appliedI We only use three kinds of rule in this paper:

I Send-out communication rules: [a]αh → [ ]βh bI Send-in communication rules: a [ ]αh → [b]βhI Nonelementary division rules

6/18

Evolution rules

I Each pair (label, charge) has a set of associated rulesI For instance, the rule [a]0h → [ ]+h b can be applied to

a membrane having label h and charge 0I The charge represents a kind of state of the membrane,

controlling which set of rules can be appliedI We only use three kinds of rule in this paper:

I Send-out communication rules: [a]αh → [ ]βh b

I Send-in communication rules: a [ ]αh → [b]βhI Nonelementary division rules

6/18

Evolution rules

I Each pair (label, charge) has a set of associated rulesI For instance, the rule [a]0h → [ ]+h b can be applied to

a membrane having label h and charge 0I The charge represents a kind of state of the membrane,

controlling which set of rules can be appliedI We only use three kinds of rule in this paper:

I Send-out communication rules: [a]αh → [ ]βh bI Send-in communication rules: a [ ]αh → [b]βh

I Nonelementary division rules

6/18

Evolution rules

I Each pair (label, charge) has a set of associated rulesI For instance, the rule [a]0h → [ ]+h b can be applied to

a membrane having label h and charge 0I The charge represents a kind of state of the membrane,

controlling which set of rules can be appliedI We only use three kinds of rule in this paper:

I Send-out communication rules: [a]αh → [ ]βh bI Send-in communication rules: a [ ]αh → [b]βhI Nonelementary division rules

6/18

Nonelementary divisionI A nonelementary division rule separates

positive and negative children membranes

I The neutral children membranes are duplicated insteadI For instance: consider [ [ ]+h1

[ ]−h2]0h → [ [ ]0h1

]+h [ [ ]0h2]−h

7/18

Nonelementary divisionI A nonelementary division rule separates

positive and negative children membranesI The neutral children membranes are duplicated instead

I For instance: consider [ [ ]+h1[ ]−h2

]0h → [ [ ]0h1]+h [ [ ]0h2

]−h

7/18

Nonelementary divisionI A nonelementary division rule separates

positive and negative children membranesI The neutral children membranes are duplicated insteadI For instance: consider [ [ ]+h1

[ ]−h2]0h → [ [ ]0h1

]+h [ [ ]0h2]−h

h1

+

bc

h3

0

aaa

h2

−

h

0

7/18

Nonelementary divisionI A nonelementary division rule separates

positive and negative children membranesI The neutral children membranes are duplicated insteadI For instance: consider [ [ ]+h1

[ ]−h2]0h → [ [ ]0h1

]+h [ [ ]0h2]−h

h1

+

bc

h3

0

aaa

h2

−

h

0

7/18

Nonelementary divisionI A nonelementary division rule separates

positive and negative children membranesI The neutral children membranes are duplicated insteadI For instance: consider [ [ ]+h1

[ ]−h2]0h → [ [ ]0h1

]+h [ [ ]0h2]−h

h1

0

bc

h3

0

aaa

h2

0

h3

0

aaa

h h

−+

7/18

Recogniser P systems

I To decide a language L, we map each string x to a P system Πx

I Πx decides x ∈ L by emitting an object yes or nofrom the outermost membrane and halting

I The map x 7→ Πx is computed by a polytime Turing machine M

M

0 10

x ∈ Σ?

yes

no

aab

Πx = M (x)

a

8/18

Recogniser P systems

I To decide a language L, we map each string x to a P system Πx

I Πx decides x ∈ L by emitting an object yes or nofrom the outermost membrane and halting

I The map x 7→ Πx is computed by a polytime Turing machine M

M

0 10

x ∈ Σ?

yes

no

aab

Πx = M (x)

a

8/18

Recogniser P systems

I To decide a language L, we map each string x to a P system Πx

I Πx decides x ∈ L by emitting an object yes or nofrom the outermost membrane and halting

I The map x 7→ Πx is computed by a polytime Turing machine M

M

0 10

x ∈ Σ?

yes

no

aab

Πx = M (x)

a

8/18

Recogniser P systems

I To decide a language L, we map each string x to a P system Πx

I Πx decides x ∈ L by emitting an object yes or nofrom the outermost membrane and halting

I The map x 7→ Πx is computed by a polytime Turing machine M

M

0 10

x ∈ Σ?

yes

no

aab

Πx = M (x)

a

8/18

The complexity class PTETRA

Denote by n2 the tetration (iterated exponentiation) operation

n2 = 222···

2︸ ︷︷ ︸n times

DefinitionPTETRA is the class of languages decided by Turing machinesoperating in “tetrational time” p(n)2 for some polynomial p

PropositionPTETRA coincides with tetrational space, and is closedunder complement, union, intersection, and polytime reductions

9/18

The complexity class PTETRA

Denote by n2 the tetration (iterated exponentiation) operation

n2 = 222···

2︸ ︷︷ ︸n times

DefinitionPTETRA is the class of languages decided by Turing machinesoperating in “tetrational time” p(n)2 for some polynomial p

PropositionPTETRA coincides with tetrational space, and is closedunder complement, union, intersection, and polytime reductions

9/18

The complexity class PTETRA

Denote by n2 the tetration (iterated exponentiation) operation

n2 = 222···

2︸ ︷︷ ︸n times

DefinitionPTETRA is the class of languages decided by Turing machinesoperating in “tetrational time” p(n)2 for some polynomial p

PropositionPTETRA coincides with tetrational space, and is closedunder complement, union, intersection, and polytime reductions

9/18

How large is PTETRA?

RE

R

PR

PTETRA

ELEMENTARY

EXP

P

All inclusions in this Venn diagram are proper

10/18

Space required to simulate P systems

FactA P system can be simulated by a TM using the same amountof space (number of membranes and objects) moduloa polynomial overhead

I The number of objects in our P systems only increasesvia nonelementary membrane division

I We need to count the maximum number of membranesduring the computation. . .

I . . . keeping in mind that the initial number is m = poly(n)

11/18

Space required to simulate P systems

FactA P system can be simulated by a TM using the same amountof space (number of membranes and objects) moduloa polynomial overhead

I The number of objects in our P systems only increasesvia nonelementary membrane division

I We need to count the maximum number of membranesduring the computation. . .

I . . . keeping in mind that the initial number is m = poly(n)

11/18

Space required to simulate P systems

FactA P system can be simulated by a TM using the same amountof space (number of membranes and objects) moduloa polynomial overhead

I The number of objects in our P systems only increasesvia nonelementary membrane division

I We need to count the maximum number of membranesduring the computation. . .

I . . . keeping in mind that the initial number is m = poly(n)

11/18

Space required to simulate P systems

FactA P system can be simulated by a TM using the same amountof space (number of membranes and objects) moduloa polynomial overhead

I The number of objects in our P systems only increasesvia nonelementary membrane division

I We need to count the maximum number of membranesduring the computation. . .

I . . . keeping in mind that the initial number is m = poly(n)

11/18

Nonelementary division stops in a finite number of steps

+ −

12/18

Nonelementary division stops in a finite number of steps

+ − + +− −

12/18

Nonelementary division stops in a finite number of steps

+ − + + + + + +− − − − − −

12/18

Nonelementary division stops in a finite number of steps

+ − + + + + + +− − − − − −

no further divisionis possible here

12/18

Applying division to all levels

PropositionSuppose we apply all possible division rules to the completem-ary tree of m levels, and that the rules are applied level-by-level,bottom-up (this is the worst case). Then the final number of nodesis bounded by O(m2)2

PropositionA family of P system {Πx : x ∈ Σ?} can be simulatedby a Turing machine in space p(n)2 · p(n) for some polynomial p

TheoremThe class of languages decidable by this variant of P systems isa subset of PTETRA

13/18

Applying division to all levels

PropositionSuppose we apply all possible division rules to the completem-ary tree of m levels, and that the rules are applied level-by-level,bottom-up (this is the worst case). Then the final number of nodesis bounded by O(m2)2

PropositionA family of P system {Πx : x ∈ Σ?} can be simulatedby a Turing machine in space p(n)2 · p(n) for some polynomial p

TheoremThe class of languages decidable by this variant of P systems isa subset of PTETRA

13/18

Applying division to all levels

PropositionSuppose we apply all possible division rules to the completem-ary tree of m levels, and that the rules are applied level-by-level,bottom-up (this is the worst case). Then the final number of nodesis bounded by O(m2)2

PropositionA family of P system {Πx : x ∈ Σ?} can be simulatedby a Turing machine in space p(n)2 · p(n) for some polynomial p

TheoremThe class of languages decidable by this variant of P systems isa subset of PTETRA

13/18

Can the upper bound be actually achieved?

I By using a few auxiliary membranes and objects. . .I . . . and forcing the divisions to occurr in a bottom-up order

by using the electrical charges. . .I . . . we can produce tetrationally many membranes and,

as a consequence, tetrationally many objects

I How can we exploit this feature?

14/18

Can the upper bound be actually achieved?

I By using a few auxiliary membranes and objects. . .I . . . and forcing the divisions to occurr in a bottom-up order

by using the electrical charges. . .I . . . we can produce tetrationally many membranes and,

as a consequence, tetrationally many objectsI How can we exploit this feature?

14/18

Simulating register (counter) machines

r1 r2 r3 r4 r5

one membraneper register

aaaaaaaa

pi

aaaaaaaaaaaaaaaaaaaa

aa

the number of a’s is thevalue of the register

h

000000

“spare” a’s to performincrement operations

program counter object

15/18

Solving PTETRA problems via P systems

By exploiting the nonelementary division processwe can produce tetrationally many spare a’s

PropositionTetrational-space register machines (hence TMs) can be simulatedby our variant of P system

TheoremThe class of languages decided by our variant of P systemis exactly PTETRA

16/18

Solving PTETRA problems via P systems

By exploiting the nonelementary division processwe can produce tetrationally many spare a’s

PropositionTetrational-space register machines (hence TMs) can be simulatedby our variant of P system

TheoremThe class of languages decided by our variant of P systemis exactly PTETRA

16/18

Solving PTETRA problems via P systems

By exploiting the nonelementary division processwe can produce tetrationally many spare a’s

PropositionTetrational-space register machines (hence TMs) can be simulatedby our variant of P system

TheoremThe class of languages decided by our variant of P systemis exactly PTETRA

16/18

Conclusions and open problems

I We described a variant of P systems that, althoughnon computationally universal, decides a very large classof languages

I Do other nature-inspired computing devices characterisePTETRA or other similarly large classes?

I Can we characterise the primitive recursive languages?

17/18

Conclusions and open problems

I We described a variant of P systems that, althoughnon computationally universal, decides a very large classof languages

I Do other nature-inspired computing devices characterisePTETRA or other similarly large classes?

I Can we characterise the primitive recursive languages?

17/18

Conclusions and open problems

I We described a variant of P systems that, althoughnon computationally universal, decides a very large classof languages

I Do other nature-inspired computing devices characterisePTETRA or other similarly large classes?

I Can we characterise the primitive recursive languages?

17/18

Thanks for your attention!