Describing weighted safety with weighted LTL over product $\omega$-valuation monoids

TL;DR

This paper introduces a framework for describing weighted safety properties using weighted LTL over product $ ext{ extsterling}$-valuation monoids, providing algorithms to verify automaton behaviors against formulas.

Contribution

It defines $k$-safe infinitary series over product $ ext{ extsterling}$-valuation monoids and develops algorithms for automaton-formula equivalence in this context.

Findings

Defines $k$-safe infinitary series over product $ ext{ extsterling}$-valuation monoids.

Provides algorithms for automaton and formula behavior equivalence.

Establishes syntactic fragments of weighted LTL with $k$-safety properties.

Abstract

We define the notion of $k$-safe infinitary series over idempotent ordered totally generalized product $\omega $-valuation monoids that satisfy specific properties. For each element $k$ of the underlying structure (different from the neutral elements of the additive, and the multiplicative operation) we determine two syntactic fragments of the weighted $LTL$ with the property that the semantics of the formulas in these fragments are $k$ -safe infinitary series. For specific idempotent ordered totally generalized product $\omega $-valuation monoids we provide algorithms that given a weighted B\"{u}chi automaton and a weighted $LTL$ formula in these fragments, decide whether the behavior of the automaton coincides with the semantics of the formula.