649 research outputs found
Computing Optimal Coverability Costs in Priced Timed Petri Nets
We consider timed Petri nets, i.e., unbounded Petri nets where each token
carries a real-valued clock. Transition arcs are labeled with time intervals,
which specify constraints on the ages of tokens. Our cost model assigns token
storage costs per time unit to places, and firing costs to transitions. We
study the cost to reach a given control-state. In general, a cost-optimal run
may not exist. However, we show that the infimum of the costs is computable.Comment: 26 pages. Contribution to LICS 201
Decisive Markov Chains
We consider qualitative and quantitative verification problems for
infinite-state Markov chains. We call a Markov chain decisive w.r.t. a given
set of target states F if it almost certainly eventually reaches either F or a
state from which F can no longer be reached. While all finite Markov chains are
trivially decisive (for every set F), this also holds for many classes of
infinite Markov chains. Infinite Markov chains which contain a finite attractor
are decisive w.r.t. every set F. In particular, this holds for probabilistic
lossy channel systems (PLCS). Furthermore, all globally coarse Markov chains
are decisive. This class includes probabilistic vector addition systems (PVASS)
and probabilistic noisy Turing machines (PNTM). We consider both safety and
liveness problems for decisive Markov chains, i.e., the probabilities that a
given set of states F is eventually reached or reached infinitely often,
respectively. 1. We express the qualitative problems in abstract terms for
decisive Markov chains, and show an almost complete picture of its decidability
for PLCS, PVASS and PNTM. 2. We also show that the path enumeration algorithm
of Iyer and Narasimha terminates for decisive Markov chains and can thus be
used to solve the approximate quantitative safety problem. A modified variant
of this algorithm solves the approximate quantitative liveness problem. 3.
Finally, we show that the exact probability of (repeatedly) reaching F cannot
be effectively expressed (in a uniform way) in Tarski-algebra for either PLCS,
PVASS or (P)NTM.Comment: 32 pages, 0 figure
Dense-Timed Petri Nets: Checking Zenoness, Token liveness and Boundedness
We consider Dense-Timed Petri Nets (TPN), an extension of Petri nets in which
each token is equipped with a real-valued clock and where the semantics is lazy
(i.e., enabled transitions need not fire; time can pass and disable
transitions). We consider the following verification problems for TPNs. (i)
Zenoness: whether there exists a zeno-computation from a given marking, i.e.,
an infinite computation which takes only a finite amount of time. We show
decidability of zenoness for TPNs, thus solving an open problem from [Escrig et
al.]. Furthermore, the related question if there exist arbitrarily fast
computations from a given marking is also decidable. On the other hand,
universal zenoness, i.e., the question if all infinite computations from a
given marking are zeno, is undecidable. (ii) Token liveness: whether a token is
alive in a marking, i.e., whether there is a computation from the marking which
eventually consumes the token. We show decidability of the problem by reducing
it to the coverability problem, which is decidable for TPNs. (iii) Boundedness:
whether the size of the reachable markings is bounded. We consider two versions
of the problem; namely semantic boundedness where only live tokens are taken
into consideration in the markings, and syntactic boundedness where also dead
tokens are considered. We show undecidability of semantic boundedness, while we
prove that syntactic boundedness is decidable through an extension of the
Karp-Miller algorithm.Comment: 61 pages, 18 figure
Multipebble Simulations for Alternating Automata - (Extended Abstract)
Abstract. We study generalized simulation relations for alternating BĂŒchi automata (ABA), as well as alternating finite automata. Having multiple pebbles allows the Duplicator to âhedge her bets â and delay decisions in the simulation game, thus yielding a coarser simulation relation. We define (k1, k2)-simulations, with k1/k2 pebbles on the left/right, respectively. This generalizes previous work on ordinary simulation (i.e., (1, 1)-simulation) for nondeterministic BĂŒchi automata (NBA) in [3] and ABA in [4], and (1, k)-simulation for NBA in [2]. We consider direct, delayed and fair simulations. In each case, the (k1, k2)simulations induce a complete lattice of simulations where (1,1)- and (n, n)simulations are the bottom and top element (if the automaton has n states), respectively, and the order is strict. For any fixed k1, k2, the (k1, k2)-simulation implies (Ï-)language inclusion and can be computed in polynomial time. Furthermore, quotienting an ABA w.r.t. (1, n)-delayed simulation preserves its language. Finally, multipebble simulations yield new insights into the Miyano-Hayashi construction [10] on ABA.
Strategy Complexity of Point Payoff, Mean Payoff and Total Payoff Objectives in Countable MDPs
We study countably infinite Markov decision processes (MDPs) with real-valued
transition rewards. Every infinite run induces the following sequences of
payoffs: 1. Point payoff (the sequence of directly seen transition rewards), 2.
Mean payoff (the sequence of the sums of all rewards so far, divided by the
number of steps), and 3. Total payoff (the sequence of the sums of all rewards
so far). For each payoff type, the objective is to maximize the probability
that the is non-negative. We establish the complete picture of the
strategy complexity of these objectives, i.e., how much memory is necessary and
sufficient for -optimal (resp. optimal) strategies. Some cases can
be won with memoryless deterministic strategies, while others require a step
counter, a reward counter, or both.Comment: Revised and extended journal version of results presented at the
CONCUR 2021 conference. For a special issue in the arxiv overlay journal LMCS
(https://lmcs.episciences.org). This is not a duplicate of arXiv:2107.03287
(the conference version), but the significantly changed journal version for
LMCS (which uses arXiv as a backend
Approximating the Value of Energy-Parity Objectives in Simple Stochastic Games
We consider simple stochastic games G with energy-parity objectives, a combination of quantitative rewards with a qualitative parity condition. The Maximizer tries to avoid running out of energy while simultaneously satisfying a parity condition.
We present an algorithm to approximate the value of a given configuration in 2-NEXPTIME. Moreover, ?-optimal strategies for either player require at most O(2-EXP(|G|)?log(1/?)) memory modes
Strategy Complexity of Mean Payoff, Total Payoff and Point Payoff Objectives in Countable MDPs
We study countably infinite Markov decision processes (MDPs) with real-valued
transition rewards. Every infinite run induces the following sequences of
payoffs: 1. Point payoff (the sequence of directly seen transition rewards), 2.
Total payoff (the sequence of the sums of all rewards so far), and 3. Mean
payoff. For each payoff type, the objective is to maximize the probability that
the is non-negative. We establish the complete picture of the
strategy complexity of these objectives, i.e., how much memory is necessary and
sufficient for -optimal (resp. optimal) strategies. Some cases can
be won with memoryless deterministic strategies, while others require a step
counter, a reward counter, or both.Comment: Full version of a conference paper at CONCUR 2021. 41 page
Strategy Complexity of Threshold Payoff with Applications to Optimal Expected Payoff
We study countably infinite Markov decision processes (MDPs) with transition
rewards. The (resp. ) threshold objective is to maximize the
probability that the (resp. ) of the infinite sequence of
transition rewards is non-negative. We establish the complete picture of the
strategy complexity of these objectives, i.e., the upper and lower bounds on
the memory required by -optimal (resp. optimal) strategies. We
then apply these results to solve two open problems from [Sudderth, Decisions
in Economics and Finance, 2020] about the strategy complexity of optimal
strategies for the expected (resp. ) payoff.Comment: 53 page
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