Hromkovič, Juraj

Ďuriš, Pavol

Institute of Mathematics AS CR, v. v. i.

KybernetikaPavol Ďuriš; Juraj HromkovičZerotesting bounded one-way multicounter machinesKybernetika, Vol. 23 (1987), No. 1, 13--18Persistent URL: http://dml.cz/dmlcz/124775Terms of use:© Institute of Information Theory and Automation AS CR, 1987Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czKYBERNETIKA- VOLUME 23 (1987), NUMBER 1 ZEROTESTING BOUNDED ONE-WAY MULTICOUNTER MACHINES PAVOL ĎURIŠ, JURAJ HROMKOVIČ One-way multicounter machines with bounds on the number of reversals and zerotests in accepting computations are studied. The bounds are considered as functions of the length of input words. The first hieararchy results for nonconstant bounds on the number of zerotests are obtained. Further results relate the reversal complexity and zerotest complexity again in relation to nondeterminism, time, and the number of counters as additional complexity par­ameters. 1. INTRODUCTION AND DEFINITIONS We shall study the hierarchy of zerotesting bounded one-way multicounter machines and their relation to reversal bounded one-way multicounter machines in this paper. We consider a multicounter machine as a multipushdown machine whose pushdown stores have a single-letter alphabet. Informally (the formal definition can be found in [4, 5]), a one-way multicounter machine consists of a finite state control, a one-way reading head which reads the input from the input tape, and a finite number of counters. We can regard a counter as an arithmetic register containing an integer which can be positive or zero. In one step, a one-way multicounter machine may increase or decrease a counter by 1. The action or the choice of actions of the machine is determined by the input symbol currently scanned, the state of the machine, and the sign of each counter: positive or zero. The machine starts with all counters empty and accepts if it reaches a final state with all counters empty. The class of one-way multicounter machines working in quasirealtime (for each machine there exists such a constant d that the length of each part of any computation, in which the reading head is stationary, is bounded by d) will be denoted by QR-COUNTER, the deterministic version by QR-DCOUNTER. The class one-way multicounter machines without any restriction will be denoted by COUNTER, the deterministic version by DCOUNTER. We shall study the reversal and zerotesting bounded versions of these machines, 13 where the number of reversals is bounded by a function of the input word length. The constant-reversal bounded computations were studied in [5, 7], the /-reversal bounded computations, for an increasing function/, were studied in [1, 2, 8, 10, 11]. The zerotesting bounded multicounter machines have been so far considered for constant bound only [12]. For increasing functions f(n) = o(g(n)), where g(n) = o(log2 n), and one-way deterministic quasirealtime multicounter machines, we prove the existence of strong hierarchy according to zerotest bounds / and g. Further, we show that there is a lan-guage L' recognized by a one-way deterministic quasirealtime multicounter machine with one zerotest which can be accepted by no one-way nondeterministic multi-counter machine with f(n) = o(n) reversal number bound. This result relates the reversal complexity and the zerotest complexity in relation to nondeterminism and time as additional complexity parameters. Now, let us defirie formally the reversal, and zerotest complexity measures. Let J5?(M) be the class of languages accepted by the machines in a class M. Let A be a multicounter machine from the class M and L(A) be the language accepted by A. Let / be a real function defined on natural numbers. Then LRf(A) denotes the set of all words in L(A) for which there is an accepting computation containing at most f(n) reversals (i.e. changes from increasing to decreasing contents of a counter or vice versa), where n is the length of the input word. Let F be the class of all functions q from natural numbers to positive real numbers such that for all natural numbers n:f(n) ^ q(n). Then we define the classes of languages &Rf(M) = \){LRf(B)} and 2>(M - R(f)) = \\ £Rq(M). BBM qsF Now, we shall introduce the zerotesting restriction. Let / be a function from natural numbers to positive real numbers, and let A be a multicounter machine from a class M. Then LZf(A) is the set of all words in L(A) for which there is some accepting computation having its zerotest number at most f(n) (i.e. the machine A empties the counters at most f(n) times in the computation), where n is the length of the input word. Let F be the class of all functions q from natural numbers to positive real numbers such that for all natural n:f(n) g q(n). We define the following classes of languages <ezs(M) = U {Lzf(B)} and if(M - Z(f)) = \J <?Zq(M) . BeM qeF In what follows we shall often consider computations in which a multicounter machine reads a group of identical symbols whose number is greater than the number of states. Clearly, there has to be a state q which will be entered twice (or more) in different configurations in this part of computation. If these two occurrences of the state q are adjacent (no further state q and no two equal states different from q occurs inbetween) we say that this part of the computation is a cycle with state characteristic q, reading head characteristic — the number of symbols over which the reading head moves to the right in this cycle, and counter characteristic, for 14 each counter, which is the difference between the counter contents at the beginning and at the end of the cycle. Thus, the counter characteristic can be positive, if the machine increases the contents of the counter in the cycle, it can be negative if the counter contents is decreased in this part of the computation, and obviously it can be zero. We call attention to the fact that if a cycle occurs in a computation of a one-way deterministic multicounter machine reading a group of identical symbols it follows that this cycle will be executed repeatedly until the reading head reaches some different symbols (i.e. until the reading head reads through the whole group of identical symbols) or a counter reaches zero. Let s be the number of states of a multicounter machine A and k be the number of A counters. Then we can bound the number of all cycles with different character-istics by s(s + 1) (2s + If . Now, we introduce the following notation. Let d be a real number. Then {d} is the smallest natural number k such that d g k, and [d] is the largest natural number m such that d ^ m. Le t / , and g be functions defined on natural numbers. The fact that lira f(n)jg(n) = 0 will be denoted by/(n) = o(g(n)). 2. RESULTS Using modifications of the proof technique developed in [3, 7, 10] a hieararchy of one-way quasirealtime deterministic multicounter machines according to zerotest bound is esatblished and a relation between reversal and zerotest complexity measures is shown. Theorem 1. Let / and g be increasing functions from naturals to positive reals such that g(n) j> f(n) for all natural n, f(n) = o(g(n)) and g(n) = o(log2 n). Then £e(QR-DCOUNTER-Z(f)) £ J?(QR-DCOUNTER-Z(g)). Proof. Let us consider the language Lg = {utu2 ... up\ p ^ g(\ulu2 ... up\) and M. = a"'fo"' for all i = l,...,p, where nt > 0}. Clearly, this language can be accepted by a QR-COUNTER machine B with one counter such that LZg(B) = Lg. Now, we shall show by contradiction that the language Lg does not belong to ^(QR-DCOUNTER-Z(f)). Let h be a function from natural numbers to positive real numbers such that h(n) ^ f(n) and let there exist a QR-COUNTER machine A such that Lzh(A) = Lg. Let A have s states and let the length of each part of any computation, in which the reading head is stationary, be bounded by a constant c. We shall consider the word x = attlb"lan2bn2... a"»<»>£,"»(••) 15 i - 1 where Mx >. s + 1, n; k 2sc £ n ; for i = 1, . . . , g(n). Since a(n) = o(log2 n) we J = I can assume that such a word exists and belongs to Lg. Since f(n) = o(g(n)) there exists such subword y = a">b"> ...a"kb"k of the word x that k — j is greater than the number of all cycles of A with different characteristics, and the machine A empties no counter during the computation on y. It is easy to see that, for i = j , j + 1, ..., k, the counter characteristics of all cycles used in the computation on a"' must be equal or greater than 0. It follows from the fact that if one of the counter characteristics of a cycle is negative in the computation on a"r, for some j S r S k, then the corresponding counter must be emptied in the computation on a"r, what is the consequence of the assumption r - l n,. ^ 2sc Y, nj . J = I Since k — j is greater than the number of all cycles of A with different charac-teristics there exist such two numbers d, z: j <. d < z ^ k that the machine A works on a"d in a cycle sx and on a"" in a cycle s2, where st and s2 have the same characteristics. Let m be the reading head characteristic of these cycles. Clearly, we can assume m > 0 because no quasirealtime computation of a deterministic multicounter machine can involve a cycle with the reading head characteristic 0. It can be simply seen that if A accepts the word x = vtyv2 then A must accept the word x' = vxa">b">... a"d+mb"d... an'~mbn* ... a"kb""v2 what proves our assertion because x' does not belong to Lg, and if w — w1s1w2s2w3 is the accepting computation on x then w' = w1s1slw2w3 is the accepting computa-tion of A on x'. • Now, we show that quasirealtime, determinism, and one zerotest can be sufficient for the recognition of a language, for which nondeterminism, unbounded time, and f(n) = o(n) reversals do not suffice. Theorem 2. Let f(n) = o(n) be a function from naturals to positive reals. Then <e(QR-DCOUNTER-Z(l)) - £?(COUNTER-R(f)) 4= 0. Proof. Let us consider the language L' = {w | w in {a, b}* such that Jka(w) = = #fo(w) andfor allx, j>in {a, b} + , w = xyimplies #o(x) > # &(x)}. This Ian uage can be accepted by a QR-DCOUNTER machine B' with one counter. The machine B' increases (decreases) the contents of its counter for all a(b) on the input tape. If B' computing on an input word will empty its counter, and is not scanning the last symbol of the input word then B' does not accept. If B' will empty its counter and is scanning the last symbol on the input word B' accepts. Now, we shall prove by contradiction that L' does not belong to 3?(COUNTER-16 -R(f)). Let h be a function from natural numbers to positive real numbers such that h(n) :» f(n). Let us assume that there exists a COUNTER machine B such that LRh(B) = L. Let E have s states, k counters, and the number of all cycles with different characteristics be bounded by a constant d. We shall consider the input word x = a(as+ i&»+»)«»+«<-+ WW) ambm+i in L', where c > d. Sincej(n) = o(n), for a sufficiently large n, there is a nonnegative integer m such that the length of x is n = 2 + 2(s + 1) c(fc + l) {f(n)} + 2m. So, there exists a subword XX = (a*+l^+l)*(*+l>(*+l) of x such that B computing on xt reversals no counter. It is easy to see that in the same way as in the proof of Theorem 1 of [10] we can construct a word x' which does not belong to 11 but which is accepted by B. Obviously x' will be constructed from x in such way that we first take some amount of a's of a n a s + 1 of x, and then pump a group of a's of x, what breaks the prefix property of x. Corollary 1. Let f(n) = o(n) be a function defined on natural numbers. Then ^(QR-COUNTER-R(f)-Z(l)) «-: if(gR-CoUiVTER-Z(l)) ^(COUNTER-R(f)-Z(l)) £ g(COUNTER-Z(\)). We call attention to the fact that the hierarchy results formulated in Corrolary 1 can be formulated for deterministic machines, and machines with different time restrictions too. Concluding this paper we give some open problems concerning zerotesting multi-counter machines. Open Problems. Let f(n) g g(n) be functions such that f(n) = o(g(n)). What is the relation between: 1. &(DCOUNTER-Z(g)) and £C(DCOUNTER-Z(f)) 2. &(QR-DCOUNTER-Z(g)) and ^(QR-DCOUNTER-Z(f)) for g(n) £ log2 n 3. ^(QR-DCOUNTER-Zff)) and £C(DCOUNTER-Z(f)) 4. £C(DCOUNTER-Z(f)) and &(COUNTER-Z(f)) 5. Se(QR-DCOUNTER-Z(f)) and &(QR-COUNTER-Z(f))l We note that Problems 1, 2, 3 can be formulated for nondeterministic machines too. We conjecture that we shall be able to solve Problems 3, 5, and Problem 2 for g(n) S n. On the other hand we have no idea which can help to solve the additional problems. (Received August 5, 1985.) 17 R E F E R E N C E S [1] T. Chan: Reversal complexity of counter machines. In: Proc. IEEE Symposium on Theory of Computing 1981, IEEE, New York, pp. 146—157. [2] P. Buris and Z. Galil: On reversal-bounded counter machines and on pushdown automata with a bound on the size of the pushdown store. Inform, and Control 54 (1982), 3, 217—227. [3] P. Duris and J. Hromkovic: One-way simple multihead finite automata are not closed under concatenation. Theoret. Comput. Sci. 27 (1983), 121—125. [4] S. Ginsburg: Algebraic and Automata — Theoretic Properties of Formal Languages. North-Holland Publ. Comp., Amsterdam 1975. [5] S. A. Greibach: Remarks on blind and partially blind one-way multicounter machines. Theoret. Comput. Sci. 7 (1978), 311-324. [6] M. Hack: Petri Net Languages, Computation Structures. Group Memo 124, Project MAC, MIT, 1975. [7] J. Hromkovic: Closure properties of the family of languages recognized by one-way two-head deterministic finite state automata. In: Mathematical Foundations of Computer Science 1981 — Proc. 10th Symposium, Strbske plesp, Czechoslovakia, August 31 — September 4, 1981 (J. Gruska, M. Chytil,eds.). (Lecture Notes in Computer Science 118.) Springer-Verlag Berlin-Heidelberg-New York 1981, pp. 304-313. [8] J. Hromkovic: Hierarchy of reversal and zerotesting bounded multicounter machines. In: Mathematical Foundations of Computer Science 1984— Proc. 11th Symposium, Prague, Czechoslovakia, September 3—7, 1984 (M. P. Chytil, V. Koubek, eds.). (Lecture Notes in Computer Science 176.) Springer-Verlag, Berlin—Heidelberg—New York—Tokyo 1984, pp. 312-321. [9] J. Hromkovic: Reversal bounded multicounter machines. Computers and Artificial Intel-ligence 4 (1985), 4, 361-366. [10] J. Hromkovic: Hierarchy of reversal bounded one-way multicounter machines. Kybernetika 22(1986), 2, 200-206. [11] O. H. Ibarra: Reversal-bounded multicounter machines and their decision problems. J. Assoc. Comput. Mach. 25(1978), 116-133. [12] M. Jantzen: On zerotesting-bounded multicounter machines. In: Proc. 4th GI Conference, 1979, Springer-Verlag, Berl in-Heidelberg-New York, pp. 158-169. RNDr. Pavol Duris, Vypodtove stredisko Slovenskej Akademie Vied (Computer Centre — Slovak Academy of Sciences), 842 35 Bratislava. Czechoslovakia. RNDr. Juraj Hromkovic, Katedra teoretickej kybernetiky a matematickej informatiky Univerzity Komenskeho (Department of Theoretical Cybernetics and Mathematical Informatics — Comenius University), 842 15 Bratislava. Czechoslovakia. 18 

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Zerotesting bounded one-way multicounter machines

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Zerotesting bounded one-way multicounter machines

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