A Translation of Weighted LTL Formulas to Weighted B{\" u}chi Automata over {\(\omega\)}-valuation Monoids

In this paper we introduce a weighted LTL over product omega-valuation monoids that satisfy specific properties. We also introduce weighted generalized Buchi automata with epsilon-transitions, as well as weighted Buchi automata with epsilon-transitions over product omega-valuation monoids and prove that these two models are expressively equivalent and also equivalent to weighted Buchi automata already introduced in the literature. We prove that every formula of a syntactic fragment of our logic can be effectively translated to a weighted generalized Buchi automaton with epsilon-transitions. For generalized product omega-valuation monoids that satisfy specific properties we define a weighted LTL, weighted generalized Buchi automata with epsilon-transitions, and weighted Buchi automata with epsilon-transitions, and we prove the aforementioned results for generalized product omega-valuation monoids as well. The translation of weighted LTL formulas to weighted generalized Buchi automata with epsilon-transitions is now obtained for a restricted syntactical fragment of the logic.