Tree languages defined in first-order logic with one quantifier alternation

International audienceWe study tree languages that can be defined in Δ2. These are tree languages definable by a first-order formula whose quantifier prefix is ∃ * ∀ * , and simultaneously by a first-order formula whose quantifier prefix is ∀ * ∃ *. For the quantifier free part we consider two signatures, either the descendant relation alone or together with the lexicographical order relation on nodes. We provide an effective characterization of tree and forest languages definable in Δ2. This characterization is in terms of algebraic equations. Over words, the class of word languages definable in Δ2 forms a robust class, which was given an effective algebraic characterization by Pin and Weil [11]

Mixing Probabilistic and non-Probabilistic Objectives in Markov Decision Processes

In this paper, we consider algorithms to decide the existence of strategies in MDPs for Boolean combinations of objectives. These objectives are omega-regular properties that need to be enforced either surely, almost surely, existentially, or with non-zero probability. In this setting, relevant strategies are randomized infinite memory strategies: both infinite memory and randomization may be needed to play optimally. We provide algorithms to solve the general case of Boolean combinations and we also investigate relevant subcases. We further report on complexity bounds for these problems.Comment: Paper accepted to LICS 2020 - Full versio

Characterizing EF and EX tree logics

AbstractThe expressive power of temporal branching time logics that use the modalities EX and EF is described. Forbidden pattern characterizations are given for tree languages definable in three logics: EX, EF and EX+EF. The characterizations give algorithms for the definability problem in the respective logics that are polynomial in the size of a deterministic tree automaton representing the language

Wreath Products of Forest Algebras, with Applications to Tree Logics

Abstract—We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics CTL and EF, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics. I