Anatol Slissenko

Polytime algorithm for the shortest path in a homotopy class amidst semi-algebraic obstacles in the plane

Given a set of semi-algebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct the shortest path between these points in a given homotopy class. This path is unique and has some canonical form. We use the representation of homotopy classes in a way that is as general as the classical one. It consists in representing generators of a free group which describes the classes of homotopy by disjoint cuts [GS97] homeomorphic to rays. We show that given such a system of generators and a word representing a homotopy class, one can contruct the shortest path of this class in time polynomial in the size of the word and in the size of the representation of the obstacles and the cuts. The homotopy class may also be represented by a path, then the polynomial complexity will depend on the size of the representation of this path. As a technical notion we introduce one particular system of cuts, which we call an extremity basis, that proves to be especially convenient for algorithmic purposes. The considered problem is motivated by robot motion planning and by theoretical questions arising in shortest path approximations in higher dimensions.

On the complexity of partially observed Markov decision processes

Abstract In the paper we consider the complexity of constructing optimal policies (strategies) for some type of partially observed Markov decision processes. This particular case of the classical problem deals with finite stationary processes, and can be represented as constructing optimal strategies to reach target vertices from a starting vertex in a graph with colored vertices and probabilistic deviations from an edge chosen to follow. The colors of the visited vertices is the only information available to a strategy. The complexity of Markov decision in the case of perfect information (bijective coloring of vertices) is known and briefly surveyed at the beginning of the paper. For the unobservable case (all the colors are equal) we give an improvement of the result of Papadimitriou and Tsitsiklis, namely we show that the problem of constructing even a very weak approximation to an optimal strategy is NP-hard. Our main results concern the case of a fixed bound on the multiplicity of coloring, that is a case of partially observed processes where some upper bound on the unobservability is supposed. We show that the problem of finding an optimal strategy is still NP-hard, but polytime approximations are possible. Some relations of our results to the Max-Word Problem are also indicated.

A first order logic for specification of timed algorithms: basic properties and a decidable class

Abstract We consider one aspect of the problem of specification and verification of reactive real-time systems which involve operations and constraints concerning time. Time is continuous what is motivated by specifications of hybrid systems. Our goal is to try to find a framework that is based on applied first order logic that permits to represent the verification problem directly, completely and conservatively (as explained in Introduction), and that is apt to describe interesting decidable classes, maybe showing way to feasible algorithms. To achieve this goal we use a first order timed logic that is an extension of a decidable theory of reals with timed functions. This logic permits, on the one hand, to rewrite directly and completely requirements and, on the other hand, to describe executions of various timed algorithms—here we consider block Gurevich abstract state machines because of their theoretical clarity and sufficient expressive power. Then we describe one decidable class of the verification problem that is based on notions reflecting finiteness properties of systems of control. These notions may be of independent interest, as, in particular, they give a way to describe a limited usage of arithmetics preserving decidability that is not covered by existing model-theoretic approaches. As an example we consider the generalized railroad crossing problem that we analyze in its entirety.

A Logic of Probability with Decidable Model-Checking

A predicate logic of probability, close to logics of probability of Halpern and al., is introduced. Our main result concerns the following model-checking problem: deciding whether a given formula holds on the structure defined by a given Finite Probabilistic Process. We show that this model-checking problem is decidable for a rather large subclass of formulas of a second-order monadic logic of probability. We discuss also the decidability of satisfiability and compare our logic of probability with the probabilistic temporal logic pCTL*.

The Railroad Crossing Problem: Towards Semantics of Timed Algorithms and Their Model Checking in High Level Languages

The goal of this paper is to analyse semantics of algorithms with explicit continuous time with further aim to find approaches to automatize model checking in high level, easily understandable languages. We give here a general notion of timed transition system and its formula representation that are sufficient to deal with some known examples of timed algorithms. We prove that the general semantics gives the same executions as direct, more intuitive interpretations of executions of algorithms. In a way, we try to give a general treatment of considerations of Yu.Gurevich and his co-authors concerning concrete Gurevich machines (called evolving algebras in [Gur95]), in particular, related to Railroad Crossing Problem [GH96]. Besides that we formalize specifications of this problem in a high level language which permits to rewrite directly natural language formulations, and to give a formal proof of correctness of the railroad crossing algorithm using rather a small amount of logical means, and this leads to hypotheses how automatize inference search.

Polytime model checking for timed probabilistic computation tree logic

Abstract. We consider the model checking problem for Timed Probabilistic Computation Tree Logic (TPCTL) introduced by H.A. Hansson and D. Jonsson, and studied in a recent book by H.A. Hansson [Han94]. The semantics of TPCTL is defined in terms of probabilistic transition systems. It is shown in [Han94] that model checking for TPCTL has exponential time complexity, the latter being measured in terms of the size of formula, the size of transition system and the value of explicit time that appears in the formula. Besides that, [Han94] describes some polytime decidable classes, the proofs being rather voluminous. We show, by a short proof, that this model checking problem is polytime decidable in the general case. The proof is essentially based on results on the complexity of Markov decision processes.

On the Complexity of Finite Memory Policies for Markov Decision Processes

We consider some complexity questions concerning a model of uncertainty known as Markov decision processes. Our results concern the problem of constructing optimal policies under a criterion of optimality defined in terms of constraints on the behavior of the process. The constraints are described by regular languages, and the motivation goes from robot motion planning. It is known that, in the case of perfect information, optimal policies under the traditional cost criteria can be found among Markov policies and in polytime. We show, firstly, that for the behavior criterion optimal policies are not Markovian for finite as well as infinite horizon. On the other hand, optimal policies in this case lie in the class of finite memory policies defined in the paper, and can be found in polytime. We remark that in the case of partial information, finite memory policies cannot be optimal in the general situation. Nevertheless, the class of finite memory policies seems to be of interest for probabilistic policies: though probabilistic policies are not better than deterministic ones in the general class of history remembering policies, the former ones can be better in the class of finite memory policies.

Computing Minimum-Link Path in a Homotopy Class amidst Semi-Algebraic Obstacles in the Plane

Given a set of semi-algebraic obstacles in the plane and two points in the same connected component of the complement, the problem is to construct a polygonal path between these points which has the minimum number of segments (links) and the minimum ‘total turn’, that is the sum of absolute values of angles of turns of the consecutive polygon links. We describe an algorithm that solves the problem spending polynomial time to construct one segment of the minimum-link and minimum-turn polygon if to use a modification of real RAMs which permits to handle the solutions of algebraic equations. It is known that the number of segments in such a minimum-link polygon can be exponential as function of the length of the input data or even of the degree of polynomials representing the semi-algebraic set. In fact, we describe how to construct a minimum-link-turn path for a given class of homotopy(whose shortest path has no self-intersections), and provide a rigorous and rather a universal way of reasoning about homotopy classes in contexts related to algorithms. It was previously shown by Heintz-Krick-Slissenko-Solerno that a shortest path in the situation under consideration is semi-algebraic, and an extended real RAM that is able to compute integrals of algebraic functions can find it in polytime.

A Logic of Probability with Decidable Model Checking* Partially supported by French-Israeli Arc-en-ciel/Keshet project No. 30 and No. 15.

A predicate logic of probability, close to the logics of probability of Halpern et al., is introduced. Our main result concerns the following model-checking problem: deciding whether a given formula holds on the structure defined by a given finite probabilistic process. We show that this model-checking problem is decidable for a rather large subclass of formulas of a second-order monadic logic of probability. We discuss also the decidability of satisfiability and compare our logic of probability with the probabilistic temporal logic pCTL*.

Periodicity Based Decidable Classes in a First Order Timed Logic

Abstract We describe a decidable class of formulas in a first order timed logic based on a generalized small model property: if a formula has a model then it has a model composed of a finite number of ultimately repetitive models (“ultimate repetitiveness” is a generalization of “ultimate periodicity”). This class covers a wide range of properties arising in the verification of real-time distributed systems with metric time constraints. An important feature of this class is that it makes easy the description of properties of parametric systems, in particular those with real time parameters, with parametric number of processes, and moreover, properties involving arithmetical operations. Another feature of this class is important for the verification: if a formula is not true (in the context of verification this means that one of the specifications under consideration is erroneous), then our algorithm gives a quantifier-free description of all counter-models of this formula of the complexity involved in the definition of the decidable class. Such counter-models facilitate the detection of errors in the specifications. Earlier we described decidable classes of verification problems based on a small model property. However, the ‘small models’ that we used were either finite, or similar to those that are introduced in this paper, but without the possibility of treatment of systems with a parametric number of processes.