On the complexity of bounded time and precision reachability for piecewise affine systems

Reachability for piecewise affine systems is known to be undecidable, starting from dimension $2$. In this paper we investigate the exact complexity of several decidable variants of reachability and control questions for piecewise affine systems. We show in particular that the region to region bounded time versions leads to $NP$-complete or co-$NP$-complete problems, starting from dimension $2$. We also prove that a bounded precision version leads to $PSPACE$-complete problems

MODEL THEORY AND COMPLEXITY THEORY

Descriptive complexity theory is a branch of complexity theory that views the hardness of a problem in terms of the complexity of expressing it in some logical formalism; among the resources considered are the number of object variables, quantifier depth, type, and alternation, sentences length (finite/infinite), etc. In this field we have studied two problems: (i) expressibility in $\exists SO$ and (ii) the descriptive complexity of finite abelian groups. Inspired by Fagin's result that $NP=\exists SO$, we have developed a partial framework to investigate expressibility inside $\exists SO$ so as to have a finer look into $NP$. The framework uses combinatorics derived from second-order Ehrenfeucht-Fra\"{\i}ss\'{e} games and the notion of game types. Among the results obtained is that for any $k$, divisibility by $k$ is not expressible by an $\exists SO$ sentence where (1) each second-order variable has arity at most $2$, (2) the first-order part has at most $2$ first-order variables, and (3) the first-order part has quantifier depth at most $3$. In the second project we have investigated the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Fra\"{\i}ss\'e games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let $G_1$ and $G_2$ be a pair of non-isomorphic finite abelian groups, and let $m$ be a number that divides one of the two groups' orders. Then the following hold: (1) there exists a first-order sentence $\varphi$ that distinguishes $G_1$ and $G_2$ such that $\varphi$ is existential, has quantifier depth $O(\log{m})$, and has at most 5 variables and (2) if $\varphi$ is a sentence that distinguishes $G_1$ and $G_2$ then $\varphi$ must have quantifier depth $\Omega(\log m)$. In infinitary model theory we have studied abstract elementary classes. We have defined Galois types over arbitrary subsets of the monster (large enough homogeneous model), have defined a simple notion of splitting, and have proved some properties of this notion such as invariance under isomorphism, monotonicity, reflexivity, existence of non-splitting extensions

Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions

Recursive analysis is the most classical approach to model and discuss computations over the reals. It is usually presented using Type 2 or higher order Turing machines. Recently, it has been shown that computability classes of functions computable in recursive analysis can also be defined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub- and sup-classes) over the reals and algebraically defined (sub- and sup-) classes of $\R$-recursive functions à la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of functions over the reals. In particular we provide the first algebraic characterization of polynomial time computable functions over the reals. This framework opens the field of implicit complexity of functions over the reals, and also provide a new reading of some of the existing characterizations at the computability level

On the extension of computable real functions

International audienceWe investigate interrelationships among different notions from mathematical analysis, effective topology, and classical computability theory. Our main object of study is the class of computable functions defined over an interval with the boundary being a left-c.e. real number. We investigate necessary and sufficient conditions under which such functions can be computably extended. It turns out that this depends on the behavior of the function near the boundary as well as on the class of left-c.e. real numbers to which the boundary belongs, that is, how it can be constructed. Of particular interest a class of functions is investigated: sawtooth functions constructed from computable enumerations of c.e. sets

Trans-Sense: Real Time Transportation Schedule Estimation Using Smart Phones

Developing countries suffer from traffic congestion, poorly planned road/rail networks, and lack of access to public transportation facilities. This context results in an increase in fuel consumption, pollution level, monetary losses, massive delays, and less productivity. On the other hand, it has a negative impact on the commuters feelings and moods. Availability of real-time transit information - by providing public transportation vehicles locations using GPS devices - helps in estimating a passenger's waiting time and addressing the above issues. However, such solution is expensive for developing countries. This paper aims at designing and implementing a crowd-sourced mobile phones-based solution to estimate the expected waiting time of a passenger in public transit systems, the prediction of the remaining time to get on/off a vehicle, and to construct a real time public transit schedule. Trans-Sense has been evaluated using real data collected for over 800 hours, on a daily basis, by different Android phones, and using different light rail transit lines at different time spans. The results show that Trans-Sense can achieve an average recall and precision of 95.35% and 90.1%, respectively, in discriminating lightrail stations. Moreover, the empirical distributions governing the different time delays affecting a passenger's total trip time enable predicting the right time of arrival of a passenger to her destination with an accuracy of 91.81%.In addition, the system estimates the stations dimensions with an accuracy of 95.71%.Comment: 8 pages, 11 figures