Dung T. Huynh

Finite-automaton aperiodicity is PSPACE-complete

Abstract In this paper, we solve an open problem raised by Stern (1985) — “Is finite-automaton aperiodicity PSPACE-complete?” — by providing an affirmative answer. We also characterize the exact complexity of two other problems considered by Stern: (1) dot-depth-one language recognition and (2) piecewise testable language recognition. We show that these two problems are logspace- complete for NL (the class of languages accepted by nondeterministic logspace-bounded Turing machines.

Some complexity bounds for problems concerning finite and 2-dimensional vector addition systems with states

Abstract In this paper, we analyse the complexity of the reachability, containment, and equivalence problems for two classes of vector addition systems with states (VASSs): finite VASSs and 2-dimensional VASSs. Both of these classes are known to have effectively computable semilinear reachibility sets (SLSs). By giving upper bounds on the sizes of the SLS representations, we achieve upper bounds on each of the aforementioned problems. In the case of the finite VASSs, the SLS representation is simply a listing of the reachability set; therefore, we derive a bound on the norm of any reachable vector based on the dimension, number of states, and amount of increment caused by any move in the VASS. The bound we derive shows an improvement of two levels in the primitive recursive hierarchy over results previously obtained by McAloon (1984), thus answering a question posed by Clote (1986). We then show this bound to be optimal. We feel that the techniques we use in deriving our upper bounds represent an original approach to the problem, and since they yield improvements over previous results, we feel these techniques may have applications to other problems. In the case of 2-dimensional VASSs, we analyse an algorithm given by Hopcroft and Pansiot (1979) that generates an SLS representation of the reachability set. Specifically, we show that the algorithm operates in 2 2 cln nondeterministic time, where l is the length of the binary representation of the largest integer in the VASS, n is the number of transitions, and c is some fixed constant. We also give examples for which this algorithm will take 2 2 dln nondeterministic time for some positive constant d . Finally, we give a method of determinizing the algorithm in such a way that it requires no more than 2 2 cln deterministic time. From this upper bound and special properties of the generated SLSs, we derive upper bounds of Dtime (2 2 cln ) for the three problems mentioned above.

The parallel complexity of finite-state automata problems

The goal of this paper is to study the exact complexity of several important problems concerning finite-state automata and to classify the degrees of ambiguity of nondeterministic finite-state automata. Our results are as follows: (1) Minimization of deterministic finite automata is NC^1-complete for NL. (2) Testing whether the degree of ambiguity of a nondeterministic finite automaton is exponential, or polynomial, or bounded is NC^1-complete for NL. (3) Checking whether a given nondeterministic finite automaton is unambiguous or k-ambiguous is NC^1-complete for NL, where k is some fixed constant. (4) The bounded nonuniversality problem for nondeterministic finite automata (which is the problem of deciding whether L(M) @? @S^@?^n @S^@?^n for a given nondeterministic finite automaton M and a unary integer n) is log-space complete for NP. (5) The bounded nonuniversality problem for unambiguous finite automata is in DET (the class of problems NC^1-reducible to computing the determinants of integer matrices), and for deterministic finite automata, it is NC^1-complete for NL. (6) The inequivalence problems for unambiguous and k-ambiguous finite automata are both in DET, where k is some fixed constant.

A note on almost-everywhere-complex sets and separating deterministic-time-complexity classes

Abstract For each time bound T: {input strings} → {natural numbers} that is some machines exact running time, there is a {0, 1}-valued function fT that can be computed within time proportional to T, but that cannot be computed within any time bound T′ that is infinitely often significantly smaller than T ( T′ ≠ Ω(T) , typically). Equivalently, every algorithm to compute fT requires time T′ on almost every input if T′ is almost everywhere significantly smaller than T (T′ = o(T), typically).

Deciding bisimilarity of normed context-free processes is in σ p 2

Existing decision algorithms for bisimulation equivalence for normed context-free processes require at least exponential time. We develop a Σp2 (a subclass of PSPACE) algorithm for deciding bisimulation equivalence for normed context-free processes.

On Deciding Readiness and Failure Equivalences for Processes

In this paper, we study the complexity of deciding readiness and failure equivalences for finite state processes and recursively defined processes specified by normed context-free grammars (CFGs) in Greibach normal form (GNF). The results are as follows: (1) Readiness and failure equivalences for processes specified by normed GNF CFGs are both undecidable. For this class of processes, the regularity problem with respect to failure or readiness equivalence is also undecidable. Moreover, all these undecidability results hold even for locally unary processes. In the unary case, these problems become decidable. In fact, they are ?p2-complete, We also show that with respect to bisimulation equivalence, the regularity for processes specified by normed GNF CFGs is NL-complete. (2) Readiness and failure equivalences for finite state processes are PSPACE-complete. This holds even for locally unary finite state processes. These two equivalences are co-NP-complete for unary finite state processes. Further, for acyclic finite state processes, readiness and failure equivalences are co-NP-complete and they are NL-complete in the unary case. (3) For finite tree processes, we show that finite trace, readiness, and failure equivalences are all L-complete. Further, the results remain true for the unary case. Our results provide a complete characterization of the computational complexity of deciding readiness and failure equivalences for several important classes of processes.

Some observations about the randomness of hard problems

In this note we investigate some connections between hard languages and random languages. We show that there exist languages that are both hard and random. We also show that every EXPTIME-hard language is polynomial-time weakly random.

Minimum latency data aggregation in the physical interference model

Data aggregation has been the focus of many researchers as one of the most important applications in Wireless Sensor Networks. A main issue of data aggregation is how to construct efficient schedules by which data can be aggregated without any interference. The problem of constructing minimum latency data aggregation schedules (MLAS) has been extensively studied in the literature although most of existing works use the graph-based interference model. In this paper, we study the MLAS problem in the more realistic physical model known as signal-to-interference-noise-ratio (SINR). We first derive an @Wlogn approximation lower bound for the MLAS problem in the metric SINR model. We also prove the NP-hardness of the decision version of MLAS in the geometric SINR model. This is a significant contribution as these results have not been obtained before for the SINR model. In addition, we propose two constant factor approximation algorithms whose latency is bounded by O(@D+R) for the dual power model, where @D is the maximum node degree of a network and R is the network radius. Finally we study the performance of the algorithms through simulation.

The parallel complexity of coarsest set partition problems

Abstract In this paper we investigate two different versions of coarsest set partition problems. They are (1) single-function coarsest set partition, and (2) multi-function coarsest set partition. We classify the parallel complexity of these two problems and present for them several parallel algorithms. We also note that the single-relation and multi-relation coarsest set partition problems are both P-complete.

On a complexity hierarchy between L and NL

Abstract This paper attempts to explain the complexity of the unary 0–1 knapsack problem which lies between L and NL. We introduce a new complexity class of languages log-space reducible to languages accepted by the family of one-way one-turn Nondeterministic auxiliary counter machines whose auxiliary worktapes are O( log n ) bounded. This complexity class is denoted by LOG(1-1-NAuxCM( log n )). We show that the modified unary knapsnack problem with bandwidth 2O( log n ) is log-space complete for LOG(1-1-NAuxCM( log n )). By varying the space bound on the auxiliary worktape, we obtain a hierarchy of complexity classes between L and NL.