The paper considers various formalisms based on Automata, Temporal Logic and Regular Expressions for specifying queries over sequences. Unlike traditional binary semantics, the paper presents a similarity based semantics for thse formalisms. More specifically, a distance measure in the range [0,1] is associated with a sequence, query pair denoting how closely the sequence satisfies the query. These measures are defined using a spectrum of normed vector distance measures. Various distance measures based on the syntax and the traditional semantics of the query are presented. Efficient algorithms for computing these distance measure are presented. These algorithms can be employed for retrieval of sequence from a database that closely satisfy a given.
Since the appearance of a preliminary version of this paper @cite_19 , other non-probabilistic quantitative versions of temporal logic have been proposed in @cite_23 @cite_13 . Both these works consider infinite computations and branching time temporal logics. The similarity measure they give, for the linear time fragment of their logic, corresponds to the infinite norm among the vector distance functions. On the contrary, we consider formalism and logics on finite sequences and give similarity based measures that use a spectrum vector of distance measures. We also present methods fo computing similarity values of a database sequence with respect to queries given in the different formalisms.
@cite_19 @cite_13 @cite_23
2339807279 2049399166 2140028191 2086092403
This paper introduces a framework for inference of timed temporal logic properties from data. The dataset is given as a finite set of pairs of finite-time system traces and labels, where the labels indicate whether the traces exhibit some desired behavior (e.g., a ship traveling along a safe route). We propose a decision-tree based approach for learning signal temporal logic classifiers. The method produces binary decision trees that represent the inferred formulae. Each node of the tree contains a test associated with the satisfaction of a simple formula, optimally tuned from a predefined finite set of primitives. Optimality is assessed using heuristic impurity measures, which capture how well the current primitive splits the data with respect to the traces' labels. We propose extensions of the usual impurity measures from machine learning literature to handle classification of system traces by leveraging upon the robustness degree concept. The proposed incremental construction procedure greatly improves the execution time and the accuracy compared to existing algorithms. We present two case studies that illustrate the usefulness and the computational advantages of the algorithms. The first is an anomaly detection problem in a maritime environment. The second is a fault detection problem in an automotive powertrain system. In this paper, we consider the robust interpretation of Metric Temporal Logic (MTL) formulas over signals that take values in metric spaces. For such signals, which are generated by systems whose states are equipped with non-trivial metrics, for example continuous or hybrid, robustness is not only natural, but also a critical measure of system performance. Thus, we propose multi-valued semantics for MTL formulas, which capture not only the usual Boolean satisfiability of the formula, but also topological information regarding the distance, @e, from unsatisfiability. We prove that any other signal that remains @e-close to the initial one also satisfies the same MTL specification under the usual Boolean semantics. Finally, our framework is applied to the problem of testing formulas of two fragments of MTL, namely Metric Interval Temporal Logic (MITL) and closed Metric Temporal Logic (clMTL), over continuous-time signals using only discrete-time analysis. The motivating idea behind our approach is that if the continuous-time signal fulfills certain conditions and the discrete-time signal robustly satisfies the temporal logic specification, then the corresponding continuous-time signal should also satisfy the same temporal logic specification. A unifying framework for the study of real-time logics is developed. In analogy to the untimed case, the underlying classical theory of timed state sequences is identified, it is shown to be nonelementarily decidable, and its complexity and expressiveness are used as a point of reference. Two orthogonal extensions of PTL (timed propositional temporal logic and metric temporal logic) that inherit its appeal are defined: they capture elementary, yet expressively complete, fragments of the theory of timed state sequences, and thus are excellent candidates for practical real-time specification languages. This paper presents an inference algorithm that can discover temporal logic properties of a system from data. Our algorithm operates on finite time system trajectories that are labeled according to whether or not they demonstrate some desirable system properties (e.g. "the car successfully stops before hitting an obstruction"). A temporal logic formula that can discriminate between the desirable behaviors and the undesirable ones is constructed. The formulae also indicate possible causes for each set of behaviors (e.g. "If the speed of the car is greater than 15 m s within 0.5s of brake application, the obstruction will be struck") which can be used to tune designs or to perform on-line monitoring to ensure the desired behavior. We introduce reactive parameter signal temporal logic (rPSTL), a fragment of parameter signal temporal logic (PSTL) that is expressive enough to capture causal, spatial, and temporal relationships in data. We define a partial order over the set of rPSTL formulae that is based on language inclusion. This order enables a directed search over this set, i.e. given a candidate rPSTL formula that does not adequately match the observed data, we can automatically construct a formula that will fit the data at least as well. Two case studies, one involving a cattle herding scenario and one involving a stochastic hybrid gene circuit model, are presented to illustrate our approach.