Qualitative Logics and Equivalences for Probabilistic Systems

We investigate logics and equivalence relations that capture the qualitative behavior of Markov Decision Processes (MDPs). We present Qualitative Randomized CTL (QRCTL): formulas of this logic can express the fact that certain temporal properties hold over all paths, or with probability 0 or 1, but they do not distinguish among intermediate probability values. We present a symbolic, polynomial time model-checking algorithm for QRCTL on MDPs. The logic QRCTL induces an equivalence relation over states of an MDP that we call qualitative equivalence: informally, two states are qualitatively equivalent if the sets of formulas that hold with probability 0 or 1 at the two states are the same. We show that for finite alternating MDPs, where nondeterministic and probabilistic choices occur in different states, qualitative equivalence coincides with alternating bisimulation, and can thus be computed via efficient partition-refinement algorithms. On the other hand, in non-alternating MDPs the equivalence relations cannot be computed via partition-refinement algorithms, but rather, they require non-local computation. Finally, we consider QRCTL*, that extends QRCTL with nested temporal operators in the same manner in which CTL* extends CTL. We show that QRCTL and QRCTL* induce the same qualitative equivalence on alternating MDPs, while on non-alternating MDPs, the equivalence arising from QRCTL* can be strictly finer. We also provide a full characterization of the relation between qualitative equivalence, bisimulation, and alternating bisimulation, according to whether the MDPs are finite, and to whether their transition relations are finitely-branching.Comment: The paper is accepted for LMC

Game Refinement Relations and Metrics

We consider two-player games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal, for example, reach a target state, the question of winning is thus a probabilistic one: what is the maximal probability of winning from a given state? On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state similarity. We introduce equivalences and metrics for two-player game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative mu-calculus. The quantitative mu-calculus can express a large set of goals, including reachability, safety, and omega-regular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation

State Space Reduction For Parity Automata

Exact minimization of ?-automata is a difficult problem and heuristic algorithms are a subject of current research. We propose several new approaches to reduce the state space of deterministic parity automata. These are based on extracting information from structures within the automaton, such as strongly connected components, coloring of the states, and equivalence classes of given relations, to determine states that can safely be merged. We also establish a framework to generalize the notion of quotient automata and uniformly describe such algorithms. The description of these procedures consists of a theoretical analysis as well as data collected from experiments

Interface Simulation Distances

The classical (boolean) notion of refinement for behavioral interfaces of system components is the alternating refinement preorder. In this paper, we define a distance for interfaces, called interface simulation distance. It makes the alternating refinement preorder quantitative by, intuitively, tolerating errors (while counting them) in the alternating simulation game. We show that the interface simulation distance satisfies the triangle inequality, that the distance between two interfaces does not increase under parallel composition with a third interface, and that the distance between two interfaces can be bounded from above and below by distances between abstractions of the two interfaces. We illustrate the framework, and the properties of the distances under composition of interfaces, with two case studies.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Invariant theory in exterior algebras and Amitsur-Levitzki type theorems

This article discusses invariant theories in some exterior algebras, which are closely related to Amitsur-Levitzki type theorems. First we consider the exterior algebra on the vector space of square matrices of size $n$, and look at the invariants under conjugations. We see that the algebra of these invariants is isomorphic to the exterior algebra on an $n$-dimensional vector space. Moreover we give a Cayley-Hamilton type theorem for these invariants (the anticommutative version of the Cayley-Hamilton theorem). This Cayley-Hamilton type theorem can also be regarded as a refinement of the Amitsur-Levitzki theorem. We discuss two more Amitsur-Levitzki type theorems related to invariant theories in exterior algebras. One is a famous Amitsur-Levitzki type theorem due to Kostant and Rowen, and this is related to $O(V)$-invariants in $\Lambda(\Lambda_2(V))$. The other is a new Amitsur-Levitzki type theorem, and this is related to $GL(V)$-invariants in $\Lambda(\Lambda_2(V) \oplus S_2(V^*))$.Comment: 18 pages; minor revision; to appear in Adv. Mat