Weight monitoring with linear temporal logic

Abstract

Many important performance and reliability measures can be formalized as the accumulated values of weight functions. In this paper, we introduce an extension of linear time logic including past (LTL) with new operators that impose constraints on the accumulated weight along path fragments. The fragments are characterized by regular conditions formalized by deterministic finite automata (monitor DFA). This new logic covers properties expressible by several recently proposed formalisms. We study the model-checking problem for weighted transition systems, Markov chains and Markov decision processes with rational weights. While the general problem is undecidable, we provide algorithms and sharp complexity bounds for several sublogics that arise by restricting the monitoring DFA.

Trade-off analysis meets probabilistic model checking

“…As a consequence of Proposition 2 and the undecidability results for the model-checking problem for temporal logics extended by assertions on the accumulated weights presented in [11,14], we get the undecidability of the model-checking problem for LTL extended by atoms of the form ratio ≍ ϑ interpreted over weighted transition systems or weighted Markov chains. Nevertheless, there are decidable algorithmic problems for several ratio constraint patterns.…”

“…Even apparently simple problems, such as the task to compute the probability for a weight invariance (wgt > 0) in a Markov chain, turn out to be hard. Our work on ratios in Markovian models is in the line of a current research trend to extend temporal logics, transition systems and game structures with weight functions, see, e.g., [1,11,12,14,16,17,34,39].…”

“…The latter existential formula can be checked efficiently by standard shortest-path algorithms (e.g., the Bellman-Ford algorithm). Similarly, the almostsure ratio problem for some fixed window size ℓ∈N asking whether is solvable in polynomial time by shortest-path algorithms [11].…”

“…For instance, [2,42] addressed expected ratios or [23] considered long-run ratios when the denominator has the purpose of a counter. Recent undecidability results on temporal logics extended by assertions on the accumulated values of weight functions [11,14] impose serious limitations. These undecidability results are mainly due to the fact that this problem is closely related to MDPs with integer weights rather than non-negative rewards.…”

Maximizing the Conditional Expected Reward for Reaching the Goal

“…Using standard automata-based approaches, our method can easily be generalized to compute maximal conditional expected rewards for regular co-safety conditions (rather than reachability conditions ♦G) and/or where the accumulation of rewards is "controlled" by a deterministic finite automaton as in the logics considered in [16,12] (rather than F ). In this paper, we restricted to MDPs with non-negative integer rewards.…”