Abstract
We consider Pareto analysis of multi-priced timed automata (MPTA) having multiple observers recording costs (to be minimised) and rewards (to be maximised) along a computation.We study the Pareto Domination Problem, which asks whether it is possible to reach a target location such that the accumulated costs and rewards Pareto dominate a given vector. We show that this problem is undecidable in general, but decidable for MPTA with at most three observers. We show the problem to be PSPACE-complete for MPTA recording only costs or only rewards. We also consider an approximate Pareto Domination that is decidable in exponential time with no restrictions on types and number of observers.We develop connections between MPTA and Diophantine equations. Undecidability of the Pareto Domination Problem is shown by reduction from Hilbert's 10th Problem, while decidability for three observers entails translation to a decidable fragment of arithmetic involving quadratic forms.
Highlights
We consider Pareto analysis of reachable states of multi-priced timed automata (MPTA): timed automata equipped with multiple observers that keep track of costs and rewards along a computation
For MPTA whose observers are all costs or all rewards, we show that the Pareto Domination Problem is PSPACE-complete
Multi Priced Timed Automata (MPTA) [5, 7, 8, 10, 17, 18, 19] extend priced timed automata [2, 3, 4, 6, 16] with multiple observers that capture the accumulation of costs and rewards along a computation
Summary
Multi Priced Timed Automata (MPTA) [5, 7, 8, 10, 17, 18, 19] extend priced timed automata [2, 3, 4, 6, 16] with multiple observers that capture the accumulation of costs and rewards along a computation. This extension allows to model multi-objective optimization problems beyond the scope of timed automata [1]. MPTA lie at the frontier between timed automata (for which reachability is decidable [1]) and linear hybrid automata (for which reachability is undecidable [13]). The observers exhibit richer dynamics than the clocks of timed automata by not being confined to unit slope in locations, but may neither be queried nor reset while
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