Reversal-bounded Counters Research Articles

Language Acceptors with a Pushdown: Characterizations and Complexity

We study one-way nondeterministic pushdown automata ([Formula: see text]), optionally with reversal-bounded counters. Finite-turn pushdown automata are pushdown automata with a bound on the number of switches between pushing and popping. We give new characterizations for finite-turn pushdown automata, and for finite-turn pushdown automata augmented with reversal-bounded counters. The first is in terms of multi-tape nondeterministic finite automata ([Formula: see text]), and the second is in terms of multi-tape [Formula: see text] with reversal-bounded counters. We then use the characterizations to determine the complexity of the languages defined by these automata. In particular, we show that languages accepted by finite-turn [Formula: see text] augmented with reversal-bounded counters are in [Formula: see text]. For the non-finite-turn case, the languages are in [Formula: see text] and in [Formula: see text]. We also look at the space complexity of languages accepted by two-way machines. In particular, we show that every language accepted by a two-way [Formula: see text] with reversal-bounded counters that makes a polynomial (resp., exponential) number of input head reversals is in [Formula: see text] (resp., [Formula: see text]). This remains true if the pushdown can flip its contents a bounded number of times.

Semilinearity of Families of Languages

Techniques are developed for creating new and general language families of only semilinear languages, and for showing families only contain semilinear languages. It is shown that for language families [Formula: see text] that are semilinear full trios, the smallest full AFL containing [Formula: see text] that is also closed under intersection with languages in [Formula: see text] (where [Formula: see text] is the family of languages accepted by [Formula: see text]s augmented with reversal-bounded counters), is also semilinear. If these closure properties are effective, this also immediately implies decidability of membership, emptiness, and infiniteness for these general families. From the general techniques, new grammar systems are given that are extensions of well-known families of semilinear full trios, whereby it is implied that these extensions must only describe semilinear languages. This also implies positive decidability properties for the new systems. Some characterizations of the new families are also given.

On families of full trios containing counter machine languages

We look at nondeterministic finite automata augmented with multiple reversal-bounded counters where, during an accepting computation, the behavior of the counters is specified by some fixed pattern. These patterns can serve as a useful “bridge” to other important automata and grammar models in the theoretical computer science literature, thereby helping in their study. Various pattern behaviors are considered, together with characterizations and comparisons. For example, one such pattern defines exactly the smallest full trio containing all the bounded semilinear languages. Another pattern defines the smallest full trio containing all the bounded context-free languages. The “bridging” to other families is then applied, e.g. to certain Turing machine restrictions, as well as other families. Certain general decidability properties are also studied using this framework.

State grammars with stores

State grammars are context-free grammars where the productions have states associated with them, and a production can only be applied to a nonterminal if the current state matches the state in the production. Once states are added to grammars, it is natural to add various stores, similar to machine models. With such extensions, productions can only be applied if both the state and the value read from each store matches between the current sentential form and the production. Here, generative capacity results are presented for different derivation modes, with and without additional stores. In particular, with the standard derivation relation, it is shown that adding reversal-bounded counters does not increase the capacity, and states are enough. Also, state grammars with reversal-bounded counters that operate using leftmost derivations are shown to coincide with languages accepted by one-way machines with a pushdown and reversal-bounded counters, and these are surprisingly shown to be strictly weaker than state grammars with the standard derivation relation (and no counters). The complexity of the emptiness problem involving state grammars with reversal-bounded counters is also studied.

On store languages and applications

The store language of a machine of some arbitrary type is the set of all store configurations (state plus store contents but not input) that can appear in an accepting computation. New algorithms and characterizations of store languages are obtained, such as the result that any nondeterministic pushdown automaton augmented with reversal-bounded counters, where the pushdown can “flip” its contents a bounded number of times, can be accepted by a machine with only reversal-bounded counters. Connections are made between store languages and several model checking and reachability problems, such as accepting the set of all predecessor and successor configurations of a machine from a given (regular) set of configurations. For a variety of different machine models often containing multiple parallel data stores, these sets can be accepted with a machine model that is simpler than the original model itself. Store languages are key to showing these properties.

Grammatical characterizations of NPDAs and VPDAs with counters

We give a characterization of NPDAs with reversal-bounded counters (NPCMs) in terms of context-free grammars with monotonic counters. We show that the grammar characterization can be used to give simple proofs of previously known results such as the semilinearity of the Parikh map of any language accepted by an NPCM. We prove a Chomsky–Schutzenberger-like theorem: A language L is accepted by an NPCM if and only if there is a k≥1 and an alphabet Σ containing at least k distinguished symbols, p1,...,pk, such that L=h(D∩Ek(R)) for some homomorphism h, Dyck language D⊆Σ⁎, and regular set R⊆Σ⁎, where Ek(R)={w|w∈R,|w|p1=⋯=|w|pk}. We also give characterizations of other machine models, such as visibly pushdown automata with reversal-bounded counters (VPCMs). We then investigate the complexity of some decision problems concerning these grammatical models. Finally, we introduce other grammatical models equivalent to NPCM.

Decidable models of integer-manipulating programs with recursive parallelism

We study safety verification for multithreaded programs with recursive parallelism (i.e. unbounded thread creation and recursion) as well as unbounded integer variables. Since the threads in each program configuration are structured in a hierarchical fashion, our model is state-extended ground-tree rewrite systems equipped with shared unbounded integer counters that can be incremented, decremented, and compared against an integer constant. Since the model is Turing-complete, we propose a decidable underapproximation. First, using a restriction similar to context-bounding, we underapproximate the global control by a weak global control (i.e. DAGs possibly with self-loops), thereby limiting the number of synchronisations between different threads. Second, we bound the number of reversals between non-decrementing and non-incrementing modes of the counters. Under this restriction, we show that reachability becomes NP-complete. In fact, it is poly-time reducible to satisfaction over existential Presburger formulas, which allows one to tap into highly optimised SMT solvers. Our decidable approximation strictly generalises known decidable models including (i) weakly-synchronised ground-tree rewrite systems, and (ii) synchronisation/reversal-bounded concurrent pushdown systems with counters. Finally, we show that, when equipped with reversal-bounded counters, relaxing the weak control restriction by the notion of senescence results in undecidability.

Variations of checking stack automata: Obtaining unexpected decidability properties

We introduce a model of one-way language acceptors (a variant of a checking stack automaton) and show the following decidability properties:1.The deterministic version has a decidable membership problem but has an undecidable emptiness problem.2.The nondeterministic version has an undecidable membership problem and emptiness problem. There are many models of accepting devices for which there is no difference with these problems between deterministic and nondeterministic versions, i.e., the membership problem for both versions are either decidable or undecidable, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first one-way model to exhibit properties (1) and (2). We define another family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. We also know of no other model with this property in the literature. We also investigate decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.). Surprisingly, two-way deterministic machines with multiple checking stacks and multiple reversal-bounded counters are shown to have a decidable membership problem, a very general model with this property.

Insertion operations on deterministic reversal-bounded counter machines

Several insertion operations are studied applied to languages accepted by one-way and two-way deterministic reversal-bounded multicounter machines. These operations are defined by the ideals obtained from relations such as the prefix, infix, suffix, and outfix relations, as well as operations defined from inverses of a type of deterministic transducer with reversal-bounded counters attached. The question of whether the resulting languages can always be accepted by deterministic machines with the same number (or larger number) of input-turns (resp., counters, counter-reversals, etc.) is investigated.

Deletion operations on deterministic families of automata

It is shown that one-way deterministic reversal-bounded multicounter languages are closed under right quotient with languages from many language families; even those defined by nondeterministic machines such as the context-free languages. Also, it is shown that when starting with one-way deterministic machines with one counter that makes only one reversal, taking the left quotient with languages from many different language families — again including those defined by nondeterministic machines such as the context-free languages — yields only one-way deterministic reversal-bounded multicounter languages. These results are surprising given the nondeterministic nature of the deletion. However, if there are two more reversals on the counter, or a second 1-reversal-bounded counter, taking the left quotient (or even just the suffix operation) yields languages that can neither be accepted by deterministic reversal-bounded multi-counter machines, nor by 2-way deterministic machines with one reversal-bounded counter.

The effect of end-markers on counter machines and commutativity

Restrictions of reversal-bounded multicounter machines are studied; in particular, those that cannot subtract from any counter until it has reached the end of the input. It is proven that this does not alter the languages accepted when the machines are nondeterministic. When the machines are deterministic, the languages (denoted by eDCM) are shown to coincide with those accepted by deterministic Parikh automata, but are strictly contained in the class of languages accepted by machines without this condition. It then follows that all commutative semilinear languages are in this restricted class. A number of decidability and complexity properties are shown, such as the ability to test, given a deterministic pushdown automaton (even if augmented by a fixed number of reversal-bounded counters), whether it is commutative. Lastly, this deterministic family, eDCM, is shown to be the smallest family of languages closed under commutative closure, right quotient with regular languages and inverse deterministic finite transductions.

Semilinear Sets and Counter Machines: a Brief Survey

Semilinear sets are one of the most important concepts in theoretical computer science, as illustrated by the fact that the set of nonnegative integer solutions to any system of Diophantine equations is semilinear. Parikh's theorem enables us to represent any semilinear set as a pushdown automaton PDA. We summarize recent results on the descriptional complexity of conversions among different representations of a semilinear set: as a vector set conventional, a finite automaton FA, a PDA, etc.. We also discuss semilinearity-preserving operations like union, intersection, and complement. We use Parikh's theorem to enlarge the class of finite-state machines that can represent semilinear sets. In particular, we give a simpler proof of a known result that characterizes semilinear sets in terms of machines with reversal-bounded counters. We then investigate the power of such a machine with only one counter in the context of a long-standing conjecture about repetition on words.

Affine Parikh automata

The Parikh finite word automaton (PA) was introduced and studied in 2003 by Klaedtke and Rues. Natural variants of the PA arise from viewing a PA equivalently as an automaton that keeps a count of its transitions and semilinearly constrains their numbers. Here we adopt this view and define the affine PA , that extends the PA by having each transition induce an affine transformation on the PA registers, and the PA on letters , that restricts the PA by forcing any two transitions on the same letter to affect the registers equally. Then we report on the expressiveness, closure, and decidability properties of such PA variants. We note that deterministic PA are strictly weaker than deterministic reversal-bounded counter machines.

On the reachability set of automaton counter machines

Properties of automaton counter machines are considered. The set of reachability states of any automaton one-counter machine is proved to be a semilinear set. An algorithm for constructing this set is described. In addition, the reachability sets of any reversal-bounded automaton counter machine and any flat automaton counter machine are also semilinear.

Model checking memoryful linear-time logics over one-counter automata

We study complexity of the model-checking problems for LTL with registers (also known as freeze LTL and written LTL ↓ ) and for first-order logic with data equality tests (written FO ( ∼ , < , + 1 ) ) over one-counter automata. We consider several classes of one-counter automata (mainly deterministic vs. nondeterministic) and several logical fragments (restriction on the number of registers or variables and on the use of propositional variables for control states). The logics have the ability to store a counter value and to test it later against the current counter value. We show that model checking LTL ↓ and FO ( ∼ , < , + 1 ) over deterministic one-counter automata is PSpace-complete with infinite and finite accepting runs. By contrast, we prove that model checking LTL ↓ in which the until operator U is restricted to the eventually F over nondeterministic one-counter automata is Σ 1 1 -complete [resp. Σ 1 0 -complete] in the infinitary [resp. finitary] case even if only one register is used and with no propositional variable. As a corollary of our proof, this also holds for FO ( ∼ , < , + 1 ) restricted to two variables (written FO 2 ( ∼ , < , + 1 ) ). This makes a difference with respect to the facts that several verification problems for one-counter automata are known to be decidable with relatively low complexity, and that finitary satisfiability problems for LTL ↓ and FO 2 ( ∼ , < , + 1 ) are decidable. Our results pave the way for model checking memoryful (linear-time) logics over other classes of operational models, such as reversal-bounded counter machines.

ON COUNTER MACHINES, REACHABILITY PROBLEMS, AND DIOPHANTINE EQUATIONS

We give a brief survey of results that use “reversal-bounded counters” in studying reachability problems for various classes of transition systems. We also discuss the connection between the decidability of reachability in counter machines and the solvability of certain Diophantine equations.

On two-way nondeterministic finite automata with one reversal-bounded counter

We show that the emptiness problem for two-way nondeterministic finite automata augmented with one reversal-bounded counter (i.e., the counter alternates between nondecreasing and nonincreasing modes for a fixed number of times) operating on bounded languages (i.e., subsets of w 1 * … w k * for some nonnull words w 1 , … , w k ) is decidable, resolving an open problem. The proof is a rather involved reduction to the solution of a special class of Diophantine systems of degree 2 via a class of programs called two-phase programs.

Eliminating the storage tape in reachability constructions

A discrete pushdown timed automaton is a pushdown machine with integer-valued clocks. It has been shown recently that the binary reachability of a discrete pushdown timed automaton can be accepted by a two-tape pushdown acceptor with reversal-bounded counters. We improve this result by showing that the stack can be eliminated from the acceptor, i.e., the binary reachability can be accepted by a two-tape finite-state acceptor with reversal-bounded counters. We also obtain similar results for other machine models. Our results can be used to verify certain properties concerning these machines that were not verifiable before using previous techniques. For example, we are able to formulate a subset of Presburger LTL that is decidable for satisfiability checking with respect to these machines. We also discuss the “boundedness problem” for reachability sets. Finally, we explain how the storage tape elimination technique can be applied to machines with real-valued clocks.

Automatic verification of recursive procedures with one integer parameter

Context-free processes (BPA) have been used for dataflow analysis in recursive procedures with applications in optimizing compilers (Proceedings of FOSSaCS’99, Lecture Notes in Computer Science, Vol. 1578, Springer, Berlin, 1999, pp. 14–30). We introduce a more refined model called BPA(Z) that can model not only recursive dependencies, but also the passing of an integer parameter to a subroutine. Moreover, this parameter can be tested against conditions expressible in Presburger arithmetic. This new and more expressive model can still be analyzed automatically. We define Z-input 1-CM, a new class of 1-counter machines (cm) that take integer numbers as input, to describe sets of configurations of BPA(Z). We show that the Post∗ (the set of successors) of a set of BPA(Z)-configurations described by a Z-input 1-CM can be effectively constructed. The Pre∗ (set of predecessors) of a regular set can be effectively constructed as well. However, the Pre∗ of a set described by a Z-input 1-CM cannot be represented by a Z-input 1-CM, in general, and has an undecidable membership problem. Then we develop a new temporal logic based on reversal-bounded counter machines (i.e. machines which use counters such that the change between increasing and decreasing mode of each counter is bounded (J. Assoc. Comput. Mach. 25 (1978) 116) that can be used to describe properties of BPA(Z) and show that the model-checking problem is decidable.

THE EXISTENCE OF ω-CHAINS FOR TRANSITIVE MIXED LINEAR RELATIONS AND ITS APPLICATIONS

We show that it is decidable whether a transitive mixed linear relation has an ω-chain. Using this result, we study a number of liveness verification problems for generalized timed automata within a unified framework. More precisely, we prove that (1) the mixed linear liveness problem for a timed automaton with dense clocks, reversal-bounded counters, and a free counter is decidable, and (2) the Presburger liveness problem for a timed automaton with discrete clocks, reversal-bounded counters, and a pushdown stack is decidable.