Annotated Probabilistic Temporal Logic

Abstract

Annotated Probabilistic Temporal (APT) logic programs support building applications where we wish to reason about statements of the form "Formula G becomes true with a probability in the range [L, U] within (or in exactly) t time units after formula F became true." In this paper, we present a sound, but incomplete fixpoint operator that can be used to check consistency and entailment in APT logic programs. We present the first implementation of APT-logic programs and evaluate both its compute time and convergence on a suite of 23 ground APT-logic programs that were automatically learned from two real-world data sets. In both cases, the APT-logic programs contained up to 1,000 ground rules. In one data set, entailment problems were solved on average in under 0.1 seconds per ground rule, while in the other, it took up to 1.3 seconds per ground rule. Consistency was also checked in a reasonable amount of time. When discussing entailment of APT-logic formulas, convergence of the fixpoint operator refers to (U − L) being below a certain threshold. We show that on virtually all of the 23 automatically generated APT-logic programs, convergence was quick-often in just 2-3 iterations of the fixpoint operator. Thus, our implementation is a practical first step towards checking consistency and entailment in temporal probabilistic logics without independence or Markovian assumptions.

Event correlation and forecasting over high-dimensional streaming sensor data

“…Though, most of the times, this is not sufficient for expressing temporal correlations among the observed patterns. In order to represent uncertain data and time dependencies, probabilistic temporal logic (PTL) programming paradigms have been proposed [29] [30] [31] [32]. Such approaches extend the syntax and the semantics of probabilistic logic programs by terms of allowing for reasoning about point probabilities over time intervals through the use of probabilistic temporal rules.…”

“…Additionally, PTL programs allow for the use of integrity constraints that should not be violated across the execution time of the logic program. Specifically, two types of integrity constraints can be considered [29]: block-size constraints (BLK) are used to state that an atom A cannot be consecutively true for more than a number of times that is defined as an integer value within the constraint expression; similarly, occurrence constraints (OCC) are proposed to state that an atom A must stand true for a number of times that is within a range of minimum and maximum number of times value.…”

A Probabilistic Logic of Cyber Deception

“…In this paper we use part of the annotated probabilistic logic introduced in [8]. The computational aspect of this logic is related to linear program formulations that are sometimes very hard to compute.…”

Cyber Attribution: An Argumentation-Based Approach

“…In this paper, we assume that information is properly extracted from a set of historic data and hence consistent; (recall that inconsistent information can only be handled in the AM, not the EM). A consistent knowledge base could also be obtained as a result of curation by experts, such that all inconsistencies were removed -see [10,17] for algorithms for learning rules of this type.…”

“…One way to represent this type of information in InCA is by replacing the EM with a probabilistic temporal logic (i.e. [8,4,17,20]). However, even though this would be a relatively straightforward adjustment to the framework, it leads to several interesting questions, specifically:…”