Annotated Probabilistic Temporal Logic Paulo Shakarian, Austin Parker, Gerardo Simari, V.S. Subrahmanian The semantics of most logics of time and probability is given via a probability distribution over “threads” where a thread is a structure specifying what will be true at different points in time (in the future). When assessing the probabilities of statements such as “Event a will occur within 5 units of time of event b”, there are many different semantics possible, even when assessing the truth of this statement within a single thread. We introduce the syntax of annotated probabilistic temporal (APT) logic programs and axiomatically introduce the key notion of a frequency func- tion (for the first time) to capture different types of intra-thread reasoning, and then provide a semantics for intra-thread and inter-thread reasoning in APT logic programs parameterized by such frequency functions. We develop a comprehensive set of complexity results for consistency checking and entailment in APT logic programs, together with sound and complete algorithms to check consistency and entailment. The basic algorithms use linear programming, but we then show how to substantially (and correctly) reduce the sizes of these linear programs to yield better computational properties. We describe a real world application we are developing using APT logic programs. Categories and Subject Descriptors: I.2.4 [Knowledge Representation Formalisms and Methods]: Temporal Logic; I.2.3 [Deduction and Theorem Proving]: Probabilistic Rea- soning General Terms: Algorithms, Languages Additional Key Words and Phrases: Probabilistic and Temporal Reasoning, Threads, Frequency Functions, Imprecise Probabilities 1. INTRODUCTION There are numerous applications where we need to make statements of the form “Formula G becomes true with 50 − 60% probability 5 time units after formula F became true.” We now give four examples of how such statements might be applied. Stock Market Prediction. There is ample evidence [Fujiwara et al. 2008] that re- ports in newspapers and blogs [De Choudhury et al. 2008] have an impact on stock market prices. For instance, major investment banks invest a lot of time, effort and money attempting to learn predictors of future stock prices by analyzing a variety of indicators together with historical data about the values of these indicators. As we will show later in Figure 1, we may wish to write rules such as “The probability that the stock of company C will drop by 10% at time (T +2) is over 70% if at time T , there is a news report of a rumor of an SEC investigation of the company and (at time T ) there is a projected earnings increase of 10%. It is clear that such rules Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior specific permission and/or a fee. c  2009 ACM 1529-3785/2009/0700-0001 $5.00 ACM Transactions on Computational Logic, Vol. V, No. N, August 2009, Pages 1–42.