Further Remarks on Trace Languages in P Systems with Symport/Antiport

Further Remarks on Trace Languages inP Systems with Symport/Antiport

Guangwu Liu1,2 Mihai Ionescu1

1Research Group on Mathematical LinguisticsRovira i Virgili University

Pl. Imperial Tarraco 1, 43005 Tarragona, [email protected], [email protected]

2Department of Control Science and EngineeringHuazhong University of Science and Technology

Wuhan 430074, Hubei, P.R. China

Workshop on Membrane Computing 7, 2006Leiden - The Netherlands

G. Liu, M. Ionescu () Trace Languages WMC7 2006 1 / 23

Outline

1 HistoryP Systems with Symport/AntiportConsider the Trace of Certain Objects

2 Trace Languages in S/A P Systems with Sets of ObjectsDefinitionResults

3 Decreasing the Number of Membranes. ProposalsSeveral TravelersInverse MorphismChanging Labels

History P Systems with Symport/Antiport

Outline

History P Systems with Symport/Antiport

P Systems with Symport/Antiport. Definition.

Π = (V , µ, w1, . . . , wm, E , R1, . . . , Rm, io),

where:1 V is the alphabet of chemicals (objects);2 µ is a membrane structure with m membranes; m ≥ 1 is the

degree of the system;3 w1, . . . , wm are strings over V - the multisets of objects;4 E - the environment;5 R1, . . . , Rm are finite sets of rules of the forms (x , in), (x , out), and

(x , out ; y , in), for x , y ∈ V ∗;6 io ∈ {1, . . . , m} - the output membrane.

History Consider the Trace of Certain Objects

Outline

The idea.

Instead of counting the number of objectsItineraries of a certain object through membranes1

the result -> a coding of the string of labels of the visitedmembranes

1M.Ionescu, C.Martín-Vide, Gh.Paun: P systems with symport/antiport rules: Thetraces of objects.Grammars, 5, 2 (2002), 65–70.

The idea.

Definition.

Π = (V , t , T , h, µ, w1, . . . , wm, E , R1, . . . , Rm)

V , µ, w1, . . . , wm, E , R1, . . . , Rm are as above;t ∈ V (a distinguished object,“the traveler");T is an alphabet;h : {1, 2, . . . , m} → T ∪ {λ} is a weak coding.

The traveler is present in exactly one copy in the system -|w1 . . . wm|t = 1, t /∈ E .

Definition.

Π = (V , t , T , h, µ, w1, . . . , wm, E , R1, . . . , Rm)

Definition.

Π = (V , t , T , h, µ, w1, . . . , wm, E , R1, . . . , Rm)

Definition.

Π = (V , t , T , h, µ, w1, . . . , wm, E , R1, . . . , Rm)

Definition.

Π = (V , t , T , h, µ, w1, . . . , wm, E , R1, . . . , Rm)

The Computation.

σ = C1C2 . . . Ck , k ≥ 1 - halting computationC1 = (w1, . . . , wm, λ) - initial configuration

Ci = (z(i)1 , . . . , z(i)

m , z(i)e ) - configuration at step i , 1 ≤ i ≤ k

If |z(i)j |t = 1 for some 1 ≤ j ≤ m we write Ci(t) = j

the trace of t in the computation σ:

trace(t , σ) = C1(t)C2(t) . . . Ck (t)

computation σ generates the string h(trace(t , σ))

the language generated by Π is L(Π) = {h(trace(t , σ)) | σ is ahalting computation in Π}

The Computation.

Ci = (z(i)1 , . . . , z(i)

trace(t , σ) = C1(t)C2(t) . . . Ck (t)

The Computation.

Ci = (z(i)1 , . . . , z(i)

trace(t , σ) = C1(t)C2(t) . . . Ck (t)

The Computation.

Ci = (z(i)1 , . . . , z(i)

trace(t , σ) = C1(t)C2(t) . . . Ck (t)

The Computation.

Ci = (z(i)1 , . . . , z(i)

trace(t , σ) = C1(t)C2(t) . . . Ck (t)

The Computation.

Ci = (z(i)1 , . . . , z(i)

trace(t , σ) = C1(t)C2(t) . . . Ck (t)

Currently Best Results.

RE = LTP∗(sym0, anti2)2

lRE = lLTPl+1(sym0, anti2)lRE = lLTPl+1(sym3, anti0)lRE = lLTPl+2(sym2, anti0)

2P. Frisco, H.J. Hoogeboom: P systems with symport/antiport simulating counterautomata, Acta Informatica, 41(2-3), 2004, 145–170.

Trace Languages in S/A P Systems with Sets of Objects Definition

Outline

Definition I.

Π = (V , t , µ, w1, . . . , wm, R1, . . . , Rm),

1 V – alphabet of chemicals (objects);2 t ∈ V – the traveler; |w1 · · ·wm|t = 1;3 µ – membrane structure with m membranes;4 wi – strings representing the sets of objects present in the regions

of µ, 1 ≤ i ≤ m;5 Ri – set of symport and antiport rules for membrane i ;

(x , in), (x , out) and (x , out ; y , in), for x , y ∈ V ∗, 1 ≤ i ≤ m.The trace: encoded as a string over {a1, a2, . . . , am} marking everymembrane visited by t .

Definition I.

Π = (V , t , µ, w1, . . . , wm, R1, . . . , Rm),

Definition I.

Π = (V , t , µ, w1, . . . , wm, R1, . . . , Rm),

Definition I.

Π = (V , t , µ, w1, . . . , wm, R1, . . . , Rm),

Definition I.

Π = (V , t , µ, w1, . . . , wm, R1, . . . , Rm),

Definition II.

Two cases of marking the trace:only when the traveler enters a membrane:

LTPsetm (symp, antiq, in)

both when the traveler enters and exits a membrane (we collecttwice the label of the membrane t visits):

LTPsetm (symp, antiq, in/out)

Definition II.

Two cases of marking the trace:only when the traveler enters a membrane:

LTPsetm (symp, antiq, in)

both when the traveler enters and exits a membrane (we collecttwice the label of the membrane t visits):

LTPsetm (symp, antiq, in/out)

Trace Languages in S/A P Systems with Sets of Objects Results

Outline

Results I.

Theorem

LTPset∗ (sym∗, anti∗, in) = REG.

A → aiB ∈ R

step R0 Ri1 (fi , out ; cAt , in)

2 (d , out ; fi , in) (At , out)3 (fiB′, out ; A, in) (c, out ; d , in)

4 (d , out ; fi , in)

5 (B, out ; dB′, in)

degree of the systems and the weight of rules:

mREG = LTPsetm+2(sym2, anti3, in)

Results II.

Theorem

LTPset∗ (sym∗, anti∗, in/out) ( REG.

Strict inclusion. L = {a2a3, a2a2a3}

Results II.

Theorem

Strict inclusion. L = {a2a3, a2a2a3}'

Results II.

Theorem

Results II.

Theorem

Results II.

Theorem

Results II.

Theorem

Strict inclusion. L = {a2a3, a2a2a3} a2'

Results II.

Theorem

Strict inclusion. L = {a2a3, a2a2a3} a2a2'

Results II.

Theorem

Strict inclusion. L = {a2a3, a2a2a3} a2a2a3'

Decreasing the Number of Membranes. Proposals Several Travelers

Outline

T = {t1, t2, . . . , tk} - k travelers1, 2, . . . , m - membranes⇒ alphabet of trace symbols: k ·m symbols ai,j ,1 ≤ i ≤ k , 1 ≤ j ≤ mσ = C1C2 . . . Ck , k ≥ 1 halting computationCi(T ) = {ai,j | ti is in membrane j}trace(T , σ) = {w1w2 . . . wk | wi ∈ V ∗,ΨV (wi) = ΨV (Ci(T ))}, i.econcatenation of permutations of stringstrace language: LT (Π) = {h(trace(T , σ)) | σ is a haltingcomputation in Π}

Remark: k travelers, m membranes ⇒ LT (Π) can be on k ·m symbols

Decreasing the Number of Membranes. Proposals Inverse Morphism

Outline

h : V ∗ → U∗,h−1 : U∗ → 2V∗

; h−1(y) = {x ∈ V ∗ | h(x) = y}, y ∈ U∗

Example

V = {a1, a2, . . . , am}U = {0, 1}h(ai) = 0i1, 1 ≤ i ≤ m !Injective, hence card(h−1(y)) = 1 foreach y ∈ h(V ∗)

⇒ for any language L ⊆ V ∗ -> L = h−1(h(L))

L ∈ mFL ⇒ h(L) ∈ 2FLbecause mREG = LTPset

m+2(sym2, anti3, in) ⇒h(L) ∈ 2LTPset

4 (sym2, anti3, in)

Proposition: Every L ∈ mREG can be written in the form L = h−1(L′),for L′ ∈ 2LTPset

4 (sym2, anti3, in).

h : V ∗ → U∗,h−1 : U∗ → 2V∗

; h−1(y) = {x ∈ V ∗ | h(x) = y}, y ∈ U∗

Example

4 (sym2, anti3, in)

h : V ∗ → U∗,h−1 : U∗ → 2V∗

; h−1(y) = {x ∈ V ∗ | h(x) = y}, y ∈ U∗

Example

4 (sym2, anti3, in)

h : V ∗ → U∗,h−1 : U∗ → 2V∗

; h−1(y) = {x ∈ V ∗ | h(x) = y}, y ∈ U∗

Example

4 (sym2, anti3, in)

h : V ∗ → U∗,h−1 : U∗ → 2V∗

; h−1(y) = {x ∈ V ∗ | h(x) = y}, y ∈ U∗

Example

4 (sym2, anti3, in)

h : V ∗ → U∗,h−1 : U∗ → 2V∗

; h−1(y) = {x ∈ V ∗ | h(x) = y}, y ∈ U∗

Example

4 (sym2, anti3, in)

h : V ∗ → U∗,h−1 : U∗ → 2V∗

; h−1(y) = {x ∈ V ∗ | h(x) = y}, y ∈ U∗

Example

4 (sym2, anti3, in)

h : V ∗ → U∗,h−1 : U∗ → 2V∗

; h−1(y) = {x ∈ V ∗ | h(x) = y}, y ∈ U∗

Example

4 (sym2, anti3, in)

Decreasing the Number of Membranes. Proposals Changing Labels

Outline

Rewrite:(x , in): x [ ] i → [x ] i(x , out): [x ] i → [ ] ix(x , out ; y , in): y [x ] i → [y ] ixGeneralize:x [ ] i → [x ] j[x ] i → [ ] jxy [x ] i → [y ] jx⇒ conflict with the new labelsSolutions to avoid conflict:

sequential use of the rulesuse only (i , j)-coherent set of rules, all passing from i to jallow only to certain rules to change the labels

Conclusions

P systems with sets of objects. Computational power does not gobeyond REG.Three ideas to break the infinite hierarchy provoked by cardinalityof the alphabet of languages - no. of membranes.

Conclusions

The end.

THANK YOU!

Further Remarks on Trace Languages in P Systems with Symport/Antiport

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