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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
−+
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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
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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
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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
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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
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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
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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
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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
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How large is PTETRA?
RE
R
PR
PTETRA
ELEMENTARY
EXP
P
All inclusions in this Venn diagram are proper
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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)
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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)
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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)
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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)
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Nonelementary division stops in a finite number of steps
+ −
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Nonelementary division stops in a finite number of steps
+ − + +− −
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Nonelementary division stops in a finite number of steps
+ − + + + + + +− − − − − −
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Nonelementary division stops in a finite number of steps
+ − + + + + + +− − − − − −
no further divisionis possible here
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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
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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
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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
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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?
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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?
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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
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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
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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
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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
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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?
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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?
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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?
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Thanks for your attention!