44 research outputs found
Synchronizing Data Words for Register Automata
Register automata (RAs) are finite automata extended with a finite set of
registers to store and compare data from an infinite domain. We study the
concept of synchronizing data words in RAs: does there exist a data word that
sends all states of the RA to a single state?
For deterministic RAs with k registers (k-DRAs), we prove that inputting data
words with 2k+1 distinct data from the infinite data domain is sufficient to
synchronize. We show that the synchronization problem for DRAs is in general
PSPACE-complete, and it is NLOGSPACE-complete for 1-DRAs. For nondeterministic
RAs (NRAs), we show that Ackermann(n) distinct data (where n is the size of the
RA) might be necessary to synchronize. The synchronization problem for NRAs is
in general undecidable, however, we establish Ackermann-completeness of the
problem for 1-NRAs.
Another main result is the NEXPTIME-completeness of the length-bounded
synchronization problem for NRAs, where a bound on the length of the
synchronizing data word, written in binary, is given. A variant of this last
construction allows to prove that the length-bounded universality problem for
NRAs is co-NEXPTIME-complete
Infinite Synchronizing Words for Probabilistic Automata (Erratum)
In [1], we introduced the weakly synchronizing languages for probabilistic
automata. In this report, we show that the emptiness problem of weakly
synchronizing languages for probabilistic automata is undecidable. This implies
that the decidability result of [1-3] for the emptiness problem of weakly
synchronizing language is incorrect.Comment: 5 pages, 3 figure
Revisiting Reachability in Timed Automata
We revisit a fundamental result in real-time verification, namely that the
binary reachability relation between configurations of a given timed automaton
is definable in linear arithmetic over the integers and reals. In this paper we
give a new and simpler proof of this result, building on the well-known
reachability analysis of timed automata involving difference bound matrices.
Using this new proof, we give an exponential-space procedure for model checking
the reachability fragment of the logic parametric TCTL. Finally we show that
the latter problem is NEXPTIME-hard
On Restricted Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is the problem of decomposing a given
nonnegative matrix into a product of a nonnegative matrix and a nonnegative matrix . Restricted NMF
requires in addition that the column spaces of and coincide. Finding
the minimal inner dimension is known to be NP-hard, both for NMF and
restricted NMF. We show that restricted NMF is closely related to a question
about the nature of minimal probabilistic automata, posed by Paz in his seminal
1971 textbook. We use this connection to answer Paz's question negatively, thus
falsifying a positive answer claimed in 1974. Furthermore, we investigate
whether a rational matrix always has a restricted NMF of minimal inner
dimension whose factors and are also rational. We show that this holds
for matrices of rank at most and we exhibit a rank- matrix for which
and require irrational entries.Comment: Full version of an ICALP'16 pape
Controlling a Random Population is EXPTIME-hard
Bertrand et al. [1] (LMCS 2019) describe two-player zero-sum games in which one player tries to achieve a reachability objective in games (on the same finite arena) simultaneously by broadcasting actions, and where the opponent has full control of resolving non-deterministic choices. They show EXPTIME completeness for the question if such games can be won for every number of games. We consider the almost-sure variant in which the opponent randomizes their actions, and where the player tries to achieve the reachability objective eventually with probability one. The lower bound construction in [1] does not directly carry over to this randomized setting. In this note we show EXPTIME hardness for the almost-sure problem by reduction from Countdown Games
On the Complexity of Value Iteration
Value iteration is a fundamental algorithm for solving Markov Decision Processes (MDPs). It computes the maximal n-step payoff by iterating n times a recurrence equation which is naturally associated to the MDP. At the same time, value iteration provides a policy for the MDP that is optimal on a given finite horizon n. In this paper, we settle the computational complexity of value iteration. We show that, given a horizon n in binary and an MDP, computing an optimal policy is EXPTIME-complete, thus resolving an open problem that goes back to the seminal 1987 paper on the complexity of MDPs by Papadimitriou and Tsitsiklis. To obtain this main result, we develop several stepping stones that yield results of an independent interest. For instance, we show that it is EXPTIME-complete to compute the n-fold iteration (with n in binary) of a function given by a straight-line program over the integers with max and + as operators. We also provide new complexity results for the bounded halting problem in linear-update counter machines
Synchronizing automata over nested words
We extend the concept of a synchronizing word from deterministic finite-state automata (DFA) to nested word automata (NWA): A well-matched nested word is called synchronizing if it resets the control state of any configuration, i. e., takes the NWA from all control states to a single control state.
We show that although the shortest synchronizing word for an NWA, if it exists, can be (at most) exponential in the size of the NWA, the existence of such a word can still be decided in polynomial time. As our main contribution, we show that deciding the existence of a short synchronizing word (of at most given length) becomes PSPACE-complete (as opposed to NP-complete for DFA). The upper bound
makes a connection to pebble games and Strahler numbers, and the lower bound goes via small-cost synchronizing words for DFA, an intermediate problem that we also show PSPACE-complete. We also characterize the complexity of a number of related problems, using the observation that the intersection nonemptiness problem for NWA
is EXP-complete
Costs and Rewards in Priced Timed Automata
We consider Pareto analysis of reachable states of multi-priced timed automata (MPTA): timed automata equipped with multiple observers that keep track of costs (to be minimised) and rewards (to be maximised) along a computation. Each observer has a constant non-negative derivative which may depend on the location of the MPTA.
We study the Pareto Domination Problem, which asks whether it is possible to reach a target location via a run in which the accumulated costs and rewards Pareto dominate a given objective vector. We show that this problem is undecidable in general, but decidable for MPTA with at most three observers. For MPTA whose observers are all costs or all rewards, we show that the Pareto Domination Problem is PSPACE-complete. We also consider an epsilon-approximate Pareto Domination Problem that is decidable without restricting the number and types of observers.
We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert\u27s 10^{th} Problem, while decidability for three observers is shown by a translation to a fragment of arithmetic involving quadratic forms