Queue Memory Automata by Adrian ATANASIU Faculty of Mathematics, Bucharest University Str. Academiei 14, sector 1 70109 Bucharest, Romania E-mail:
[email protected]
Abstract. The automata with queue memory (QM A) are constructed in a similar way as the pushdown automata; however, the new type of memory of QM A augments their generative capacity to L0 . The pushdown (P DA) and finite automata can be regarded as particular cases of QM A.
AMS CLASSIFICATIONS: 03D05. C.R. CATEGORIES; F.4.3.
The definition of queue memory automata is very similar to that of pushdown automata; the difference concerns the type of memory: F IF O (First In First Out) instead of LIF O (Last In First Out). Other types of extensions for pushdown automata are proposed in [2], [3] and [4]. Definition 1 A queue memory automata (QM A) is a sextuple M = (Q, Σ, Γ, δ, q0 , Z0 ), where the notations are the same as for pushdown automata ([1], pp. 167 - 176). For the definition of the language accepted, we use the triples (q, α, β), where q ∈ Q is a state, α ∈ Σ∗ is a pushdown list with left top, β ∈ Γ∗ is a queue with the top in the left hand and the positions for memorization in the right hand. One defines the relation ` between such triples as follows: (q, aα, xβ) ` (p, α, βγ) if and only if (p, γ) ∈ δ(q, a, x), where p, q ∈ Q, a ∈ Σ ∪ {λ} (λ – the empty sentence), x ∈ Γ, α ∈ Σ∗ , β, γ ∈ Γ∗ . If card(δ(q, a, x)) ≤ 1 for all q ∈ Q, a ∈ Σ ∪ {λ}, x ∈ Γ, the QM A M is deterministic. Otherwise, M is a nondeterministic QM A. ∗
By denoting ` the reflexive and transitive closure of `, the language accepted by a QM A can be defined as follows: ∗
LQM A (M ) = {w | w ∈ Σ∗ , (q0 , w, Z0 ) ` (q, λ, λ)} 1
Example 1 The language L1 = {an bn cn | n ≥ 1} is accepted by deterministic QM A M = (Q, Σ, Γ, δ, a0 , Z0 ) where Q = {q0 , q1 , q2 }, Σ = {a, b, c}, Γ = {Z0 , A, B}, δ(q0 , a, Z0 ) = (q0 , A), δ(q0 , b, A) = (q1 , B), δ(q1 , c, B) = (q2 , λ), δ(q0 , a, A) = (q0 , AA), δ(q1 , b, A) = (q1 , B), δ(q2 , c, B) = (q2 , λ). Instead of proving the equality LQM A (M ) = L1 , we prefer to show how the automaton works in a peculiar case: (q0 , a2 b2 c2 , Z0 ) ` (q0 , ab2 c2 , A) ` (q0 , b2 c2 , AA) ` (q1 , bc2 , AB) ` (q1 , c2 , BB) ` (q2 , c, B) ` (q2 , λ, λ) Example 2 The nondeterministic QM A defined by Q = {q0 , q1 }, Σ = {a}, Γ = {Z0 , A} and δ(q0 , a, Z0 ) = {(q0 , Z0 AA), (q1 , λ)}, δ(q0 , a, A) = {(q0 , A)}, δ(q1 , a, A) = {(q1 , λ)} 2 accepts the language L2 = {an | n ≥ 1}. For example, (q0 , a9 , Z0 ) ` (q0 , a8 , Z0 AA) ` (q0 , a7 , AAZ0 AA) ` ∗
(q0 , a6 , AZ0 AAA) ` (q0 , a5 Z0 A5 ) ` (q0 , a4 , Z0 A4 ) ` (q0 , λ, λ). Example 3 n For the language L3 = {a2 | n ≥ 1}, the next QM A can be constructed(only the mapping δ will be detailed): δ(q0 , a, Z0 ) = {(q1 , Z0 A)}, δ(q0 , λ, a) = {(q0 , A)}, δ(q2 , λ, B) = {(q2 , B)}, δ(q1 , a, Z0 ) = {(q0 , Z0 ), (q0 , B)}, δ(q0 , λ, B) = {(q2 , B), (q4 , B)}, δ(q1 , λ, A) = {(q1 , A)}, δ(q2 , λ, A) = {(q3 , λ)}, δ(q3 , λ, A) = {(q2 , A)}, δ(q3 , λ, B) = {(q4 , λ)}. Proposition 1 Any P DA can be simulated by a QM A. Proof: Let us consider a pushdown automaton Mp = (Qp , Σp , Γp , δp , q0 , Z0 ). We shall construct the QM A Mc = (Qc , Σc , Γc , δc , q0 , Z0 ) as follows: Σc = Σp , Qc = Qp ∪ {qc }, Γc = Γp ∪ {Zi0 | qi ∈ Qp }, (qc 6∈ Qp , Zi0 6∈ Γp ). For every (qj , α) ∈ δp (qi , a, x) one defines δc (qi , a, x) = {(qc , Zj0 , α)}, δc (qc , λ, Zj0 ) = {(qj , λ)}. Finally, (δc , λ, x) = {(qc , x)}, ∀x ∈ Γp . The equality L(Mp ) = LQM A (Mc ) is immediate. We remark that Mc is deterministic if and only if Mp is deterministic. 2 Corollary 1 Any finite automaton (F A) can be simulated by a queue memory automaton. Proposition 2 L is a regular language if and only if it is accepted by a QM A with queue length at most 1. Proof: ”=⇒”: Obvious. ”⇐=”: Let us consider a QM A M = (Q, Σ, Γ, δ, q0 , Z0 ) where the mapping δ is defined as follows: (p, y) ∈ δ(q, a, x) with y ∈ Γ ∪ {λ} 2
The use of a rule with y = λ will empty the queue, so this rule have to be applied only once, at the last step. We shall construct the regular grammar G = (VN , VT , S, P ) where VN = Q × Γ, VT = Σ, S = [q0 , Z0 ] and [q, x] −→ a[p, y] if and only if (p, y) ∈ δ(q, a, x). The equality L(G) = LQM A (M ) is immediate. 2 Remark 1 It seems that the class of languages accepted by queue memory automata with card(Q) = 1 is L3 , but this assertion is not true. Indeed, for example, the QM A with δ defined as follows: δ(q0 , a, Z0 ) = (q0 , Z1 Z1 ), δ(q0 , b, Z1 ) = (q0 , Z0 Z0 ), δ(q0 , a, Z1 ) = (q0 , λ), δ(q0 , b, Z0 ) = (q0 , λ) accepts the language n n n n L = {ab2 a4 b8 . . . a2 b2 |n ≥ 0} ∪ {ab2 . . . b2 a2 | n ≥ 0}. Let Lq be the class of languages accepted by queue memory automata. We have Theorem 1 Lq = L0 . Proof: Let G = (VN , VT , S, P ) be a grammar of type 0 with L(G) = L. We shall divide the set of rules P into two sets P1 ∪ P2 , where P1 = {i : x1 x2 . . . xni −→ y1 y2 . . . yk | ni ≥ 0, k ≥ 0, 1 ≤ i ≤ p} (the productions which are not context free are labelled) and P2 = P \ P1 . The QM A M = (Q, Σ, Γ, δ, q0 , Z0 ) is build as follows: Σ = VT ,
Q = {q0 , q1 , q2 } ∪
p [
{qir | 1 ≤ r ≤ ni − 1},
Γ = VN ∪ VT ∪ {Z0 , ·}.
i=1
For every rule i : x1 x2 . . . xni −→ y1 y2 . . . yk ∈ P1 we define δ(q0 , λ, x1 ) = {(q0 , x1 ), (qi1 , λ)} 1 ≤ r ≤ ni − 2 δ(qir , λ, xr+1 ) = {(qir+1 , λ)} ni −1 δ(qi , λ, xni ) = {(q1 , y1 y2 , . . . yk )} For every production x −→∈ P2 , Further on one define
δ(q0 , λ, x) = {(q0 , x), (q1 , α)}
δ(q0 , λ, Z0 ) = {(q0 , S·)}, δ(q0 , λ, a) = {(q0 , a)}, ∀a ∈ VT δ(q1 , λ, ·) = {(q0 , ·)}, δ(q0 , λ, ·) = {(q2 , λ)}, δ(q2 , a, a) = {(q2 , λ)}, δ(q1 , λ, x) = {(q1 , x)} ∀a ∈ VT , x ∈ Γ \ {·} The equality LQM A (M ) = L results from the following facts: • If no production is applied, then the automaton reaches the state q0 . • The application of a production passes the automaton in a state q1 . Now, the queue is turned around until the delimitator ’·’ reaches the top (the goal is the realization of the same sentential form as in grammar derivation). So, the derivation in G is simulated step by step in M . • If the automaton reaches the state q0 with ’·’ in the queue top, the no rule has to be applied in the last turning round. So, in QM A memory there is a terminal word. Now, ’·’ is erased and the automaton passes in the state q2 . 3
• The state q2 reduces the word from memory, character by character, ticking it with the input. That is, the recognition part of the QM A automaton. • The 0 - type grammars with left-derivations do not cover the class of L0 languages. Accordingly, this restriction was avoided in the queue automata by (q0 , u) ∈ δ(q0 , λ, u), u ∈ {x, x1 }. The inclusion Lq ⊆ L0 is obtained by Church’s thesis.
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References [1] A.V. Aho, J.D. Ullman - The Theory of Parsing, Translation and Compiling, Prentice Hall Englewood Cliffs N.J. 1972, 1973. [2] M.D. Davis, E.J. Weyuker - Computability, Complexity and Languages, Academic press Inc. 1983. [3] J. Hopcroft, A.V. Ulmann - The Formal languages and their relation to automata, Addison Wesley Reading, Mass. 1969. [4] - Automata, Languages and Programming, Proceedings of a symposium IRIA, July 3-7 (1972), North Holland Publ. Co, 1973.
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