A material conditional "P implies Q" has a timeless truth value, but knowing whether it holds does not: every check of the conditional is an act of observation, at finite resolution, with each event separated from its observation by delay, drift, and jitter. This produces a paradox of observed implication -- an internally sound monitor can contradict the timeless truth value in both directions (a spurious falsifier, or a phantom trigger). We make this precise for the temporal implication G(p -> F[<=k] q) and establish a single headline result: the monitorability of the bounded implication is joint in the pair (formula, observation model) -- the observation model being the sampling regime through which the signals are read -- and is not a property of the formula alone. A change of observation model alone can move the property across the safety / cosafety / liveness / coliveness classification. The observation space partitions into a machine-checked taxonomy of eight classes; a monotone ladder of declared observation theories grades which classes become True-attributable.
Observation-Relative Monitorability of Bounded Temporal Implication / Bragetti, D., Bragetti, A., Italiano, G.F.. - (2026 Jan 01), pp. 1-77. [10.5281/zenodo.21700205]
Observation-Relative Monitorability of Bounded Temporal Implication
Davide Bragetti
;Giuseppe Francesco Italiano
2026
Abstract
A material conditional "P implies Q" has a timeless truth value, but knowing whether it holds does not: every check of the conditional is an act of observation, at finite resolution, with each event separated from its observation by delay, drift, and jitter. This produces a paradox of observed implication -- an internally sound monitor can contradict the timeless truth value in both directions (a spurious falsifier, or a phantom trigger). We make this precise for the temporal implication G(p -> F[<=k] q) and establish a single headline result: the monitorability of the bounded implication is joint in the pair (formula, observation model) -- the observation model being the sampling regime through which the signals are read -- and is not a property of the formula alone. A change of observation model alone can move the property across the safety / cosafety / liveness / coliveness classification. The observation space partitions into a machine-checked taxonomy of eight classes; a monotone ladder of declared observation theories grades which classes become True-attributable.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.


