Weighted automata model quantitative aspects of systems like memory or power consumption. Recently, Chatterjee, Doyen, and Henzinger introduced a new kind of weighted automata which compute objectives like the average cost or the long-time peak power consumption. In these automata, operations like average, limit superior, limit inferior, limit average, or discounting are used to assign values to finite or infinite words. In general, these weighted automata are not semiring weighted anymore. Here, we establish a connection between such new kinds of weighted automata and weighted logics. We show that suitable weighted MSO logics and these new weighted automata are expressively equivalent, both for finite and infinite words. The constructions employed are effective, leading to decidability results for the weighted logic formulas considered.

Quantitative aspects of systems like consumption of resources, output of goods, or reliability can be modeled by weighted automata. Recently, objectives like the average cost or the longtime peak power consumption of a system have been modeled by weighted automata which are not semiring weighted anymore. Instead, operations like limit superior, limit average, or discounting are used to determine the behavior of these automata. Here, we introduce a new class of weight structures subsuming a range of these models as well as semirings. Our main result shows that such weighted automata and Kleene-type regular expressions are expressively equivalent both for finite and infinite words.

Since recognizable tree languages are closed under the rational operations, every regular tree expression denotes a recognizable tree language. We provide an alternative proof to this fact that results in smaller tree automata. To this aim, we transfer Antimirov's partial derivatives from regular word expressions to regular tree expressions. For an analysis of the size of the resulting automaton as well as for algorithmic improvements, we also transfer the methods of Champarnaud and Ziadi from words to trees.

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