Parallel communicating flip pushdown automata systems
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  • 1. International Journal of Computer Engineering (IJCET), ISSN 0976 – 6367(Print), International Journal of Computer Engineering and Technologyand Technology (IJCET), ISSN 0976 1, May - June (2010), © IAEME ISSN 0976 – 6375(Online) Volume 1, Number – 6367(Print)ISSN 0976 – 6375(Online) Volume 1 IJCETNumber 1, May - June (2010), pp. 34-45 ©IAEME© IAEME, http://www.iaeme.com/ijcet.html PARALLEL COMMUNICATING FLIP PUSHDOWN AUTOMATA SYSTEMS COMMUNICATING BY STACKS M.Ramakrishnan Research Scholar Anna University of Technology, Coimbatore Coimbatore – 641 047 S.Balasubramanian Research Scholar Anna University of Technology, Coimbatore Coimbatore – 641 047ABSTRACT In this paper, we introduced parallel communicating flip pushdown automatacommunicating by stacks. PCFPA (2) equals the family of recursively enumerablelanguages Also we have proved that. RPCFPA (3) equals the family of recursivelyenumerable languages.Key words: Pushdown automata, Flip pushdown automata, finite automata1. INTRODUCTION A pushdown automaton is one way finite automaton with a separatepushdown store, that is first in first – out storage structure, which is manipulating bypushing and popping. Probably, such machines are best known for capturing the familyof context free languages, which was independently established by Chomsky (Cole1971). Pushdown automata have been extended in various ways. The extension of pushdown automata is recently introduced, is called flippushdown automata (Border et al 1982). A is an ordinary pushdown with the additionalability to flip its pushdown push down during the computation. This allows the machineto push and pop at both ends of the pushdown, therefore, a flip-pushdown is a form of a 34
  • 2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEdequeue storage structure and it becomes an equally power to Turing machines. Adequeue automaton com simulate two pushdowns.DEFINITION 1.1: PARALLEL COMMUNICATING FLIPPUSHDOWN AUTOMATA Parallel communicating flip pushdown automata system is of degree n is aconstruct A = (V, Γ , A1, A2,…..,An,K) where V is the input alphabet, Γ is the alphabetof pushdown symbols, for each 1 ≤ i ≤ n,Ai = (Qi , V , Γ , fi, ∆ i,qi, Zi, Fi ) where Qi is a finite set of states, Γ i is a finite pushdownalphabet , fi is a transition mapping from V∪ {ε } × Γ i to finite subsets of Qi × Γ i* iscalled transition function , ∆ i is a mapping from Qi to 2Qi, qi ∈ Qi , Zi ∈ Γ i , Fi ⊆ Qi bethe set of final states and K ⊆ {K1, K2,……, Kn } ⊆ Γ is the set of query symbols. TheFlip Pushdown automata A1, A2,…..,An are components of the system A. If there exists only one component, Ai, 1 ≤ i ≤ n, such that (r, α) ∈ fi (q, a, A)with α ∈ Γ *, |α|K > 0, for some r, q ∈ Qi, a ∈ V ∪ {∈}, A ∈ Γ , then the system is saidto be centralized and Ai is said to be the master of the system. For the sake of simplicity,whenever a system is centralized its master is the first component. A parallel communicating flip pushdown automata system is schematicallyrepresented in Figure 1.1. As one can see in Figure 1.1, all stacks are connected with each other, eachstack can send its contents to any other stack or receive the contents of any other stack.In the centralized case all stacks send their contents to a distinguished stack. In the caseof the non-returning strategy, every stack preserves its contents after sending a copy of itto another component while in the case of the returning strategy the stack returns to itsinitial contents after communication. 35
  • 3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME Figure 1.1 parallel communicating flips pushdown automata system The configuration or instantaneous of a parallel communicating flip pushdownautomata system is defined as a 3n-tuple (s1, x1, α1, s2, x2, α2… sn, xn, αn)Where for 1 ≤ i ≤ n, si ∈ Qi is the current state of the component Ai, xi ∈ V* is the remaining part of the input word which has not yet been read by Ai, αi ∈ Γ * is the contents of the ith stack, the rightmost symbol being the topmost symbol on the pushdown store If a is in V ∪ {∈}, xi ∈ V* , αi , βi ∈ Γ * and Zi is in Γ , Then ( si , a xi , αi Zi) ├( pi , xi , αi βi ), if the pair (pi , βi ) is in fi (si , a xi , αi ),for the ordinary pushdown transitions and ( si , a xi , αi Zi) ├ (pi , xi , Zi αRi ), if pi is in ∆ i (si ), for the flip pushdown transition or a pushdown reversal transitions. Whenever, thereis a choice between an ordinary pushdown transition or a pushdown reversal of any i,then the pushdown automaton non-deterministically chooses the next move. We do not 36
  • 4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEwant the flip pushdown automaton to move the bottom of pushdown symbol when thepushdown is flipped.We define two variants of transition relations on the set of configurations of A in thefollowing way:1. (s1, x1, B1α1, ..... s2, xn, Bnαn ) ├ ( p1, y1, β1, …, pn, yn, βn)Where Bi ∈ Γ , αi, βi ∈ Γ *, 1 ≤ i ≤ n, iff one of the following two conditions holds: (i) K ∩ {B1, B2, …, Bn} = φ and xi = aiyi, ai ∈ V ∪ {ε}. {pi, β’i) ∈ fi(si, ai, Bi). βi = β’iαi, 1 ≤ i ≤ n, (ii) for all i, 1 ≤ i n such that Bi = Kji and Bji ε Ki βi = Bj αji αi, for all other r, 1 ≤ r ≤ n, βr = Brαr, and yt = xt, pt = st, for all t, 1 ≤ t ≤ n. 2. (s1, x1, B1 α1, …., sn, xn, Bn αn) ├ r (p1, y1, β1, ... pn, yn, βn)Where Bi ∈ Γ , αi, βi ∈ Γ *, 1 ≤ i ≤ n, iff one of the following two conditions holds: (i) K ∩ {B1, B2, …, Bn} = φ and xi = aiyi, ai ∈ V ∪ {ε}, {pi, β’i) ∈ fi(si, ai, Bi). βi = β’iαi, 1 ≤ i ≤ n, (ii) for all 1 ≤ i ≤ n such that Bi = Kj,, and Bji ∉ K, for all other r, 1 ≤ r ≤ n, βr = Brαr, and yt = xt, pt = st, for all t, 1 ≤ t ≤ n. The two transition relations defined above differ when the topmost symbols ofsome stacks are execute a communication step replacing each query symbol with therequested stack contents of the corresponding component. If the topmost symbol of thequeried stack is also a query symbol, then first this query symbol must be replaced withthe contents of the corresponding stack. The top of each communicated stack must be anon-query symbol before its contents can be sent to another component. If this conditioncannot be fulfilled, (a circular query appeared), then the work of the automation system isblocked. After communication, the stack contents of the sending components remainthe same in the case of relation, whereas it becomes the initial pushdown memory symbol 37
  • 5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEin the case of relation. A parallel communicating flip pushdown automata system whosecomputations are based on relation is said to be non-returning; if its computations arebased on relation, it is said to be returning. The language accepted by a parallel communicating flip automata system A asabove is defined asRec(A) = {x ∈ V* | (q1, x, Z1, …, qn, x, Zn) ├* (s1, ε, α1, …, sn, ε, α1, …, sn, ε, αn), with exactly n pushdown reversals , for any αi∈ Γ * and any si ∈ Fi, 1 ≤ i ≤ n},Recr(A) = {x ∈ V* | (q1, x, Z1, …, qn, x, Zn) ├ *r (s1, ε, α1, …, sn, ε, α1, …, sn, ε, αn), with exactly n pushdown reversals , for any αi∈ Γ * and any si ∈ Fi, 1 ≤ i ≤ n}, where ├* and ├ *r , denote the reflexive and transitive closure of ├ and ├ r respectively. In the following we use the notationsrcpcpfa (n) - for returning centralized parallel communicating flip pushdown automata systems of degree n,rpcfpa (n) - for returning parallel communicating flip pushdown automata systems of degree n,rpcfpa (n) - for centralized parallel communicating flip pushdown automata systems of degree n,pcfpa (n) - for parallel communicating flip pushdown automata systems of degree n. If x(n) is a type of automata system, then X(n) is the class of languagesaccepted by pushdown automata systems of type x(n). For example, RCPCFPA (n) is theclass of languages accepted by automata of type rcpcfpa (n) (returning centralizedparallel communicating flip pushdown automata systems of degree n). 38
  • 6. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME The following examples shows to help the better understanding of the notionsdefined above.EXAMPLE1.1 Let us consider the following rpfcpca (4) given by the transition mappings ofits components.f1(q1, ε, Z1) = {(s1, a)}, f1(s1, ε, Z1) = {(p1, Z1)}, f1(p1, ε, Z1) = {(h1, Z1)}, f1(h1, ε, Z1) = {(r1, Z1), (t1, K3)}, f1(t1, ε, a) = {(s1, a)}, f3(q3, ε, Z3) = {(s3, K1)}, f3(s3, ε, a) = {(p3, K2 a)}, f3(p3, ε, a) = {(h3, a)}, f3(h3, ε, a) = {(q3, a)}, f3(h3, ε, Z3) = {(r3, Z3)}, f3(r3, a, Z3) = {(r3, Z3)}, f2(q2, ε, Z2) = {(s2, K1)}, f2(s2, ε, a) = {(p2, a)}, f2(p2, ε, Z2) = {(h2, Z2)}, f2(p2, ε, Z2) = {(h2, Z2)}, f2(r2, a, Z2) = {(r2, Z2)}, f f4(s4, ε, Z4) = {(h4, Z2)}, 4(q4, ε, Z4) = {(s4, Z4)}, f4(h4, ε, Z4) = {(t4, Z4)}, (v4, K3Z4)}, f4(t4, ε, Z4) = {(u4, Z4)}, f4(v4, ε, a) = {(s4, Z4)}, f4(v4, ε, a) = {(p4, a}, f4(p4, a, a = {(p4, ε)}, f4(p4, ε, Z4) = {(r4, Z4)} 39
  • 7. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEThe final states are F1 = {r1}, F2 = {r2}, F3 = {r3}, F4 = {r4}.EXAMPLE1.2 A more intricate way of computation can be observed in the following cpcfpa (2). f1(q1, X, Z1) = {(q1, Z1)} f1(s1, ε, Z2) = {(p1, ε)} f1(p1, X, X) = {(p2, ε)} f1(p2, ε, X) = {(p2, ε)} f1(p2, ε, Z2) = {(p1, ε)} f1(p1, ε, Z1) = {(p1, Z1)} f2(q2, X, Z2) = {(q2, X, Z2)} f1(q1, c, Z1) = {(s1, K2 Z1)} f2(s2, X, Y) = {(q2, XY)} f1(s1, ε, X) = {(s1, K2 X)} f2(q2, c, X) = {(s2, X)} f2(s2, ε, X) = {(s2, ε} f2(s2, ε, Z2) = {(sf, Z2)} f2(sf, ε, Z2) = {(sf, Z2)} f2(sf, X, Z2) = {(sf, Z2)}where X, Y ε {a, b}, and the sets of final states are F1 = {pf}, F2 = {sf}.LEMMA1.11. RCPCFPA(n) ⊆ RPCFPA(n) and CPCFPA (n) ⊆ PCFPA(n) for all n ≥ 1.2. X(1) equals the family of context-free languages and X(n) ⊆ X(n+1) for X ∈ {RCPCFPA, RPCFPA, CPCFPA, PCFPA}, n ≥ 1.2 COMPUTATIONAL POWER We start by showing how non-returning parallel communicating pushdownautomata systems can be simulated with returning systems. 40
  • 8. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMETHEOREM 2.1 PCFPA(n) ⊆ RPCFPA (2n) for all n ≥ 2.PROOF Let A = (a, ∆, A1, A2, ..., An, K) be a pcpa(n) with Ai = (Qi, V, ∆, fi, qi, Zi, Fi),1 ≤ i ≤ n. We construct the rpcpa (2n) A’ = (V, Γ ’, A’1, A1 , A’2, A 2 , …, A’n, A n , K’),Where K’ = {K , K , K , ..., K } ∪ K , K , K , ..., Kn}, and 1 2 3 n 1 2 3 for all 1 ≤ i ≤ n, A’1 = (Qi ∪ {q’| q ∈ Qi}, V, Γ ∪ { K i }, f’i, qi, Zi, Fi},with (1) f’i (q, a, A) = {(r’, x) | (r, x) ∈ fi(q, a, A)} (2) f’i (q’, ε, Zi) = {(q, K i )},where q, r ∈ Qi, a ∈ V ∪ {ε}, A ∈ Γ , and A i = ({ qi }, V, Γ ∪ {Zi, K’1}, f i , qi , Zi { qi }},with (3) f i ( qi , a, Zi } = {( q i , K’1)}, (4) f i ( qi , ε, A} = {( q i , A)},where a ∈ V ∪ {ε}, A ∈ Γ As one can easily see, every component of A has a “satellite” component inA’. Each accepting step in A is simulated by two accepting and two communicationsteps in A’ in the following way: In the first accepting step A’i and A i use the rules (1) and (3), respectively.Now the stacks of all components A’i have the same contents as the corresponding onesin A. Moreover, the current states of A’i are copies of the current states of thecorresponding ones in A. 41
  • 9. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMETHEOREM 2.2 PCFPA (2) equals the family of recursively enumerable languages.PROOF Let A = (Q, V, Γ , f, q, B1, B2, F) be a two-stack flip pushdown automaton.We construct the pcfpa(2) A = (V, Γ ’, A1, A2, {K1, K2})with Γ ’ = Γ ∪ {Zi, [q, a, X, Y, α, β]i |q, α, β) ∈ f(r, a, X, Y), q, r, ∈ Q, a ∈ V ∪ {ε}, X, Y ∈ Γ , 1 ≤ i ≤ 2},and Ai = {Qi, V, Γ ’, A1, A2, {K1, K2})with Γ ’ = Γ ∪ {Zi, [q, a, X, Y, α, β]i | (q, α, β) ∈ f(r, a, X, Y), q, r, ∈ Q, a ∈ V ∪{ε}, X, Y ∈ Γ , 1 ≤ i ≤ 2},and Ai = {Qi, V, Γ ’, fi, qi, Zi, F’), 1 ≤ i ≤ 2.where Qi = {qch, qrec, qw, [q, α], [q, β] | (q, α, β) ∈ f(r, a, X, Y), q, r, ∈ Q, a ∈ V ∪{ε}, X, Y ∈ Γ } ∪ {〈q, X〉 | q ∈ Q, X ∈ Γ } ∪ {q1, q2}, and F’ = {qch, qrec | q ∈ F}.and the transition mappings are defined as follows: (1) f1(q1, ∈, Z1) = {(qch, B1 Z1)}, f1(q2, ∈, Z2) = {(qrec, B2 Z2)}, This is the initial step in which both components change their states and stackcontents in order to begin the simulation. (2) f1(qch, a, X) = {〈〈qch, [r, a, X, Y, α, β] 1) | r, α, β) ∈ f(q, a, X, Y),, Y ∈ Γ }, f2(qrec, ∈, X)= {〈〈q, X〉, K1)}, f1(qrec, ε, X = {〈〈q, X〉, K2〉}, f2(qch, a, Y = {〈〈qch, [r, a, X, Y, α, β]2 | (r, α, β) ∈ f(q, a, X, Y), X ∈ ∆},where a ∈ V ∪ {ε}, X ∈ Γ . 42
  • 10. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME The first automaton, being in a state qch, q ∈ Q, readinga ∈ V ∪ {ε} on its input tape and X from its memory, chooses a possible move of thetwo-stack flip pushdown automaton in the same state, reading the same input symbol andreading X from its first stack. This move is encoded in a flip pushdown memory symboland stored in the memory of the first component. The other component, being in stateqrec, q ∈ Q requests the memory contents of the first component without moving itsreading head. With the second group of transitions of (2) the same process can be done, butwith A2, choosing a possible transition of A to be simulated. In simulation of A1 and A2take turns in choosing the transitions. (3) f1 (qch, ε, [r, α, X, Y, α, β]1) = {(rw, α)}, f1(qw, ε, X) = {(qw, X), (qrec, X)}, f2 (q, Y), α, [r, α, X, Y, α, β]1) = {([r, β], ε)}, f2(|q, α], ε, X) = {([q, α], ε)}, f2(|q, α], ε, Z1) = {([qch, α)}, f2 (qch, ε, [r, α, X, Y, α, β]2) = {(rw, β)}, f2(qw, ε, X) = {(qw, X), (qrec, X)}, f1 (〈q, X〉, α, [r, α, X, Y, α, β]2) = {([r, β], ε)}, f1(|q, α], ε, X) = {([q, α], ε)}, f1(|q, α], ε, Z2) = {(qch, α)}Where q ∈ Q and α ∈ V ∪ {ε}.as the non-returning variants.THEOREM 2.3 RPCFPA (3) equals the family of recursively enumerable languages.PROOF We first show that for every two-stack flip pushdown automaton. Construct an rpcfpa system with three components such that the two devices acceptthe same language.Let A = (Q, V, Γ , f, qo, Z1 Z0, F) be a two-stack flip pushdown automaton. 0, 2 43
  • 11. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEME Let us associate to any transition (r, α, β) ∈ f(q, a, A, B), whereq, r ∈ Q, a ∈ V ∪ {∈}, A, B ∈ Γ , α, β, ∈ Γ * a new symbol[q, a, A, B, r, α, β] and let us denote the set of these symbols by ∆t. Moreover, let Q’ = {q’ q ∈ Q} and Qc = {〈q, x, X〉 | q ∈ Q,x ∈ V ∪ {ε}, X ∈ Γ }. The simulating rpcfpa A is constructed as follows: A = (V, Γ , A, A1, A2, A3, {K1}),Where A = {Z1, Z2, Z3,} ∪ Γ ∪ Γ t, and A1 = (Q ∪ Q’, V, Γ A, f1, q0, Z1, F) A2 = (Q ∪ Qc, V, Γ A, f2, q0, Z1 , F) 0 2 A3 = (Q ∪ Qc, V, Γ A, f3, q0, Z0 , F).We define the transition mappings of the components as follows: (1) f1(q, a, Z1) = {(r’, [q, a, A, B, r, α, β]) | [q, a, A, B, r, α, β] ∈ Γ t}, f2(q, a, A) = {(〈q, a, A〉, K1)}, f3(q, a, B) = {(〈q, a, B〉, K1)},where q ∈ Q, a ∈ V ∪ {ε}, A, B ∈ Γ , (2) f1(r’, ε, Z1)= (r, Z1), f2(〈q, a, A), ε, [q, a, A, B, r, α, β]) = (r, α), f3(〈q, a, B), ε, [q, a, A, B, r, α, β]) = (r, β),where r’ ∈ Q’, [q, a, A, B, r, α, β] ∈ Γ t.3. CONCLUSION We define a property called communication parallel communicating flippushdown automata. Flip pushdown automata are pushdown with the additional ability toflip or reverse its pushdown. We have proved that RPCFPA equals the family ofrecursively enumerable languages. 44
  • 12. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976 – 6367(Print),ISSN 0976 – 6375(Online) Volume 1, Number 1, May - June (2010), © IAEMEREFERERENCES: 1. Martin-Vide, C., Mateescu, A. and Mitrana, V. (2002), Parallel finite automata Systems communicating by states, International Journal of the Foundations of Computing Science 13 (5),pp. 733-749. 2. A.O.Budo , “ Multiprocessor automata ” ,Infor.Process.Lett.25(1977) 257-261. 3. Elena Czeizler and Eugen Czeizler (2006), ‘Parallel Communicating Watson- Crick Automat Systems, Acta Cybernetica 17 ,685-700. 4. Csuhaj – Varju, E. and Dassow, J. (1990), On cooperating / distributed grammar systerms, Journal of Information processing and Cybernetics 26, 49-63. 5. Markus Holzer(2003), ‘Flip pushdown automata: Nondeterminism is better than Determinism, FIG Research Report. 45