Multicounter Machines Research Articles

Nondeterministic multicounter machines and complementation

It is proved that the family of languages recognized by one-way ƒ( n)-time-bounded nondeterministic multicounter machines is not closed under complementation for any polynomial function ƒ(n)⩾n. This solves the open problem of Wagner and Wechsung (1986).

Pushdown automata with reversal-bounded counters

The main result of this paper is that a pushdown automaton M augmented with R(n) reversal-bounded counters can be simulated by a Turing machine in time polynomial in n n 2 + R( n) if M is 2-way, and in time polynomial in n + R( n) if M is 1-way. It follows from this and previous results that for sufficiently large R(n), the addition of a pushdown store to R(n) reversal-bounded multicounter machines has little effect on the computing powers of the machines. The proof of the main result yields three interesting corollaries. First, relaxing the revversal bound from one counter in an R(n) reversal-bounded multicountter machine leads to very little increase in computing power. Second, 1-way pushdown automata augmented with 1-reversal counters accept in linear time and are therefore equivalent to 1-way simple multihead pushdown automata. Third, for every 1-way pushdown automaton, there is a constant c such that every accepting computation on an arbitrary nonempty input w contains a subsequence (of not necessarily consecutive operations) which is also an accepting computation on w and which has length at most c| w|.

An analysis of the nonemptiness problem for classes of reversal-bounded multicounter machines

In this paper, we present an efficient nondeterministic algorithm to decide nonemptiness for reversal-bounded multicounter machines. This algorithm executes in time polynomial in the size of the input and the number of reversals the counters are allowed. Previously the best known upper bound required space logarithmic in the size of the input and linear in the number of reversals. We argue that our algorithm is within some polynomial function of the optimal nondeterministic run time for many classes of these machines. Furthermore, we show that in most cases, the complexity of the nonemptiness problem does not change significantly when the reversal bound is dropped for one of the counters. In addition, we explore the changes in the complexity of the nonemptiness problem for these classes as different parameters are fixed or are given in either unary or binary. Our results yield as corollaries the answers to several unanswered questions regarding the disjointness, equivalence, and containment problems for reversal-bounded multicounter machines. Finally, we use our results to solve several problems regarding deadlock detection and unboundedness detection for systems of communicating finite state machines.

Reversal-bounded nondeterministic multicounter machines and complementation

It is proved that the family of languages recognized by one-way real-time nondeterministic multicounter machines with constant number of counter reversals is not closed under complementation. This solves an open problem of Wagner and Wechsung (1986).

On the decidability of some problems about rational subsets of free partially commutative monoids

Let I = A ∪ B be a partially commutative alphabet such that two letters commute iff one of them belongs to A and the other one belongs to B. Let M = A ∗ × B ∗ denote the free partially commutative monoid generated by I. We consider the following six problems for rational (given by regular expressions) subsets, X, Y of M: (Q1): X∪ Y=0? (Q2): X⊆ Y? (Q3): X= Y? (Q4): X= M? (Q5): M− X finite? (Q6): X is recognized? It is known (see (Berstel, 1979)) that all these problems are undecidable if Card A > 1 and Card B > 1, and they are decidable if Card A = Card B = 1 (Card U denotes the cardinality of U). It was conjectured (see (Choffrut, 1986, p. 79)) that these problems are decidable in the remaining cases, where Card A = 1 and Card B > 1. In this paper we show that if Card A = 1 and Card B > 1, then the problem (Q1) is decidable, and problems (Q2)–(Q6) are undecidable. Our paper is an application of results concerning reversal-bounded, nondeterministic, multicounter machines and nondeterministic, general sequential machines.

On the power of alternation in automata theory

We shall deal with the following three questions concerning the power of alternation in automata theory: (1) What is the simplest kind of device for which alternation adds computational power? (2) What are the simplest devices (according to the language family accepted by them) such that the alternating version of these devices is as powerful as Turing machines? (3) Can the number of branchings (universal configurations) in the computations of alternating devices be bounded by a function of the input word length without loss of computational power? We give a partial answer to Questions 1 and 2, i.e., we find the simplest known devices (multihead simple finite automata and one-way blind multicounter machines, respectively) having the required properties according to alternation. Besides this, considering one-way multicounter machines whose counter contents are bounded by the input word length, we find a new characterization of P. For one-way alternating (simple) multihead finite automata we show that the number of universal configurations in computations cannot be bounded by o( n log 2 n ) 1 2 ( o( n log 2 n ) ), where n is the input word length.

Alternating multicounter machines with constant number of reversals

Alternating multicounter machines with constant number of reversals

An Optimal Simulation of Counter Machines: The ACM Case

An Augmented Counter Machine (ACM) is a multicounter machine, with initially nonzero counters allowed, and the additional one-step instruction “set counter i to the value of counter j”, for any pair of counters i and j. Each ACM can be real-time simulated by an oblivious one-head tape unit using the information-theoretical storage optimum.

An Optimal Simulation of Counter Machines

Each multicounter machine can be simulated by an oblivious one-head tape unit in real-time, using logarithmic space. The solution uses redundant symmetric number representation and implicit recursion. It represents a new positional representation for (vectors of) the integers.

Extending regular expressions with iterated shuffle

It is shown that every finite expression which uses the operations union, product, Kleene star, and iterated shuffle in any order, starting with finite sets, defines a language which can be recognized non-deterministically by some multicounter machine in quasirealtime. It is known that this family is in general not closed with respect to iterated shuffle. As a consequence of the characterization each such language is in NSPACE(log n) and thus in P. However, if P ≠ NP, then also neither P nor NSPACE(log n) are closed under iterated shuffle. The proof uses the new concept of so-called shuffle schemes and a number of results on algebraic language theory.

On efficient simulations of multicounter machines

An oblivious 1-tape Turing machine can simulate a multicounter machine on-line in linear time and logarithmic space. This leads to a linear cost combinational logic network implementing the first n steps of a multicounter machine and also to a linear time/logarithmic space on-line simulation by an oblivious logarithmic cost RAM. An oblivious log* n -head tape unit can simulate the first n steps of a multicounter machine in real-time, which leads to a linear cost combinational logic network with a constant data rate.

2DST mappings on languages and related problems

Let l be a family of languages effectively closed under inverse homomorphism and intersection with regular sets and such that the languages have effectively constructible semilinear Parikh maps. We show that there is an algorithm to decide given a language L in L and a language R accepted by a one-way nondeterministic multicounter machine, where each counter makes exactly one reversal, whether L∩ R is empty. This result has many applications. In particular, it can be used to show that there is an algorithm to decide given a language L in L and two-way deterministic sequential transducers (2DST's) S 1 and S 2 whether S 1 and S 2 are equivalent on L.

Unary multiple equality sets: The languages of rational matrices

Unary multiple equality sets are equality sets of finite sets of homomorphisms, which map into the free monoids over single letter alphabets. These languages can equally well be defined by rational matrices. Unary multiple equality sets have extraordinary language theoretic properties. They coincide with the commutative multiple equality sets, each unary multiple equality set is precisely classified by the rank of its defining matrices, and all fundamental decision questions are polynomially decidable. The least trio containing all unary multiple equality sets equals the class of languages accepted by blind multicounter machines and by simple multihead finite automata. Finally crossing-free unary multiple equality sets are introduced, which vary from unary multiple equality sets in a similar way as the one-sided Dyck set varies from the two-sided Dyck set.

The complexity of decision problems for finite-turn multicounter machines

We exhibit a large class of machines with polynomial time decidable containment and equivalence problems. The machines in the class accept more than the regular sets. We know of no other class (different from the finite-state acceptors) for which the containment and equivalence problems have been shown polynomially decidable. We also discuss the complexity of other decision problems.

The complexity of the equivalence problem for two characterizations of presburger sets

It is shown that the class of relations (functions) definable by Presburger formulas is exactly the class of relations (functions) computable by reversal-bounded multicounter machines. An upper bound of 2 c(N/log N) 4 on the deterministic time complexity of the equivalence problem for such machines is established ( c is a positive constant and N the sum of the sizes of the machines). In fact, it is proved that the inequivalence problem is NP-complete. It is also shown that the equivalence problem for semilinear sets (these sets are exactly the Presburger relations) is decidable in deterministic time 2 2 cN 2 (where N is the sum of the sizes of the descriptions of the semilinear sets). The bounds obtained here are smaller than the best known deterministic upper bound of 2 2 cN2 for deciding equivalence of Presburger formulas.

Three-way two-dimensional multicounter automata

In this paper, we introduce a three-way two-dimensional multicounter automaton which can be considered as an extension of an on-line multicounter machine to two dimensions, and investigate some of its properties. We first investigate the difference between the accepting powers of three-way and four-way two-dimensional multicounter automata, and between the accepting powers of nondeterministic and deterministic three-way two-dimensional multicounter automata. We then show that hierarchies can be obtained by varying the number of counters or the amount of space allowed, for classes of sets accepted by three-way two-dimensional multicounter automata. We finally examine closure properties under several operations. Input tapes in this paper are restricted to square ones.

On-line n-bounded multicounter automata

In this paper, we investigate several properties of on-line n-bounded multicounter machines. An on-line n-bounded multicounter machine is an on-line multicounter machine in space n which can test whether or not the contents of each counter is equal to the input length. It is shown that 1. (1) there is a set accepted by a deterministic on-line n-bounded 1-counter machine but not by any deterministic on-line multicounter machine in space n, 2. (2) there is a set accepted by a nondeterministic on-line n-bounded 1-counter machine but not by any deterministic on-line n-bounded multicounter machine, 3. (3) there is a set accepted by a deterministic on-line n-bounded ( k + 1)-counter machine ( k ⩾ 1) but not by any nondeterministic on-line n-bounded k-counter machine, and 4. (4) the class of sets accepted by deterministic on-line n-bounded k-counter machines ( k ⩾ 1) is not closed under union, intersection, concatenation, Kleene closure, reversal, or e-free homomorphism.

Remarks on blind and partially blind one-way multicounter machines

We consider one-way nondeterministic machines which have counters allowed to hold positive or negative integers and which accept by final state with all counters zero. Such machines are called blind if their action depends on state and input alone and not on the counter configuration. They are partially blind if they block when any counter is negative (i.e., only nonnegative counter contents are permissible) but do not know whether or not any of the counters contain zero. Blind multicounter machines are equivalent in power to the reversal bounded multicounter machines of Baker and Book [1], and for both blind and reversal bounded multicounter machines, the quasirealtime family is as powerful as the full family. The family of languages accepted by blind multicounter machines is the least intersection closed semiAFL containing { a n b n | n⩾0} and also the least intersection closed semiAFL containing the two-sided Dyck set on one letter. Blind multicounter machines are strictly less powerful than quasirealtime partially blind multicounter machines. Quasirealtime partially blind multicounter machines accept the family of computation state sequences or Petri net languages which is equal to the least intersection closed semiAFL containing the one-sided Dyck set on one letter but is not a principal semiAFL. For partially blind multicounter machines, as opposed to blind machines, linear time is more powerful than quasirealtime. Assuming that the reachability problem for vector addition systems is decidable [16], partially blind multicounter machines accept only recursive sets and do not accept even {a nb n|n⩾0 ∗ , and quasirealtime partially blind multicounter machines are less powerful than general quasirealtime multicounter machines.

Reversal-Bounded Multicounter Machines and Their Decision Problems

article Free AccessReversal-Bounded Multicounter Machines and Their Decision Problems Author: Oscar H. Ibarra Department of Computer Science, University of Minnesota, Minneapolis, MN Department of Computer Science, University of Minnesota, Minneapolis, MNView Profile Authors Info & Claims Journal of the ACMVolume 25Issue 1Jan. 1978 pp 116–133https://doi.org/10.1145/322047.322058Published:01 January 1978Publication History 360citation936DownloadsMetricsTotal Citations360Total Downloads936Last 12 Months103Last 6 weeks22 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Remarks on the complexity of nondeterministic counter languages

For classes of languages accepted in polynomial time by multicounter machines, various trade-offs in computing power obtain among the number of counters, the amount of time, and the amount of space required in all cases, deterministic and nondeterministic, on-line and off-line. Hierarchies can be obtained in all cases by varying the number of counters or the amount of time allowed. In the on-line case, nondeterministic counter machines are always more powerful than deterministic counter machines for the same number of counters and the same polynomial time bound. Relationships with open problems are explored.