Uncertain Interval Algebra via Fuzzy/Probabilistic Modeling Keyvan Mir Mohammad Sadeghi and Ben Goertzel Abstract—A novel approach to uncertain temporal inference is presented. Allen Interval Algebra is extended to fuzzy time-intervals via representing the latter as trapeziums with distinct beginning, middle and end. An uncertain version of the Interval Algebra composition table is developed via running a computer simulation in which a large number of fuzzy time-intervals are drawn from an assumed probability distribution. I. I NTRODUCTION T EMPORAL inference is a critical part of human commonsense reasoning, and important to many practical AI applications as well. Further, temporal inference appears to have unique aspects relative to general-purpose inference. Time is a deeply fundamental aspect of the ex- perience of humans and other animals, making it highly plausible that evolution has provided the human brain with special mechanisms for drawing conclusions about time (e.g. for drawing conclusion about the tem- poral aspects of events, based on the temporal aspects of other events) [8]. Similarly, from the standpoint of developing artificial reasoning systems, it seems very reasonable to consider the augmentation of general- purpose inference mechanisms with specialized tools for handing temporality. The present work arose in the context of the Probabilistic Logic Networks (PLN) [2] uncertain in- ference system, which synthesizes probabilistic and fuzzy reasoning in the context of highly general logical inference, and is designed to provide a declarative reasoning component to the OpenCog integrative cog- nitive architecture [3], [4]. PLN is, in principle, capable of carrying out temporal inference as a special conse- quence of its general reasoning methods. In practice, however, this strategy is not very computationally effi- cient or intuitive, resulting in lengthy reasoning chains for inferences that are humanly intuitively simple. Thus it seems appropriate to consider augmenting PLN with a specialized temporal reasoning component – such as the one described here, based on an uncertain version of Allen interval algebra. While the work proposed here has its roots in PLN, however, its potential applicability is much more Keyvan Sadeghi (email: keyvan@opencog.org) and Ben Goertzel (e-mail: ben@ goertzel.org) are with the OpenCog Foundation, 35516 Aspen Court, Rehoboth DE, 19971 USA general. What we have done is to take the Allen Interval Algebra, a standard and time-tested formalism for reasoning about crisp time intervals, and extend it to enable judgments about fuzzy time intervals. The approach is probabilistic as well as fuzzy, in the sense that probabilistic methods are used to estimate the appropriate inference rules for combining fuzzy temporal relationships. The result is a robust, extensible framework for reasoning about uncertain intervals, incorporating fuzzy and probabilistic aspects into a common conceptual and mathematical understanding. Prior work extending Allen Interval Algebra to handle uncertainty has mainly been purely fuzzy set theoretic in nature, meaning that the results are not optimal for integration into probabilistic reasoning frameworks like PLN. Although, some of these fuzzy approaches have advantages in terms of preserving the exact transitivity relations possessed by the crisp Allen algebra [10]. On the other hand, prior work on proba- bilistic extensions of Allen algebra have involved fairly complex computational problems such as probabilistic constraint satisfaction [6]; or else have been overly simplistic, e.g. viewing probabilistic intervals in terms of their endpoints [9]. Here we draw our fundamental Trapezium-based model of uncertain temporal events from the fuzzy Allen algebra literature, but carry out calculations utilizing these Trapeziums according to probabilistic mathematics, hence in some ways get- ting the best of both worlds – a simple conceptual foundation using fuzzy concepts, but a set of resultant values that are harmonious with probabilistic inference systems. Naturally this approach may not be the most appropriate for all purposes, but we have found it valuable in our own work and feel it may be more generally useful as well. II. ALLEN I NTERVAL ALGEBRA A common and fairly effective way to approach temporal reasoning is to represent events as intervals. One might at first think to deal with individual points of time as primitive entities, but time-points turn out to be problematic from a variety of perspectives. Most human commonsense reasoning about time is more easily formulated in terms of intervals. A time-interval captures some portion of time within which certain features (identified as characteriz- ing an event) hold. It is represented as [a, b] where a is 2014 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) July 6-11, 2014, Beijing, China 978-1-4799-2072-3/14/$31.00 ©2014 IEEE 591