Reachability Problems 

Abstract: Interval Markov chains extend classical Markov chains with the possibility to describe transition probabilities using intervals, rather than exact values. While the standard formulation of interval Markov chains features closed intervals, previous work has considered also open interval Markov chains, in which the intervals can also be open or half-open. In this paper we focus on qualitative reachability problems for open interval Markov chains, which consider whether the optimal (maximum or minimum) probability with which a certain set of states can be reached is equal to 0 or 1. We present polynomial-time algorithms for these problems for both of the standard semantics of interval Markov chains. Our methods do not rely on the closure of open intervals, in contrast to previous approaches for open interval Markov chains, and can characterise situations in which probability 0 or 1 can be attained not exactly but arbitrarily closely.

Abstract: An automaton is history-deterministic (HD) if one can safely resolve its non-deterministic choices on the fly. In a recent paper, Henzinger, Lehtinen and Totzke studied this in the context of Timed Automata [9], where it was conjectured that the class of timed $$\omega $$ -languages recognised by HD-timed automata strictly extends that of deterministic ones. We provide a proof for this fact.

Abstract: . In this paper, we consider a variant of the classical algorithmic problem of checking whether a given word v is a subsequence of another word w . More precisely, we consider the problem of deciding, given a number p (defining a range-bound) and two words v and w , whether there exists a factor w [ i : i + p − 1] (or, in other words, a range of length p ) of w having v as subsequence (i. e., v occurs as a subsequence in the bounded range w [ i : i + p − 1] ). We give matching upper and lower quadratic bounds for the time complexity of this problem. Further, we consider a series of algorithmic problems in this setting, in which, for given integers k , p and a word w , we analyse the set p - Subseq k ( w ) of all words of length k which occur as subsequence of some factor of length p of w . Among these, we consider the k -universality problem, the k -equivalence problem, as well as problems related to absent subsequences. Surprisingly, unlike the case of the classical model of subsequences in words where such problems have efficient solutions in general, we show that most of these problems become intractable in the new setting when subsequences in bounded ranges are considered. Fi-nally, we provide an example of how some of our results can be applied to subsequence matching problems for circular