A Data-Transparent Probabilistic Model of Temporal Propositional Abstraction

Definition (Model distribution)! Let be a function that maps each data point to the corresponding model, i.e., valuation in propositional logic.! ! Namely, each data point supports a single model deterministically. m:Data ×Models p(mt|d1:t,m1:t−1,x1:t−1 1:I) Def.2 ={1"if"mt=m(dt) 0"otherwise Prop.2 =p(mt|dt) FUNDAMENTAL PROBLEMS Data scarcity, e.g., zero frequency No answer is straightforward as the robot has never been in Room 3.! Data smoothing works only when probabilistic models are simple enough.! Huge hypothesis spaces The three time-series data best match the given locations. This leads to the prediction of Room 10. This requires a 4th-order Markov chain with parameters.! ! Poor data transparency The prediction of Room 10 cannot be grounded in . Learning is the process of exploiting data to adjust the parameters of probabilistic models, whereas reasoning is the process of using the parameters, not the data itself, to make predictions. d2-4 9⋯104 p(Xt|Xt−4:t−1) = p(xt 1|xt−4:t−1 1)⋮p(xt 10 |xt−4:t−1 1) ⋱ p(xt 1|xt−4:t−1 10 )⋮p(xt 10 |xt−4:t−1 10 ) d2-4 MOTIVATING EXAMPLE We want to predict the location of a robot moving around the building with ten rooms shown below.! Only information we can use is the twelve timeseries data, , collected by the robot.! Question The robot moved through Rooms 2, 3, and 8. Which room is the robot likely to be two time steps later — Room 4 or Room 10?! Existing solutions! Probabilistic models, e.g., Markov chains and# hidden Markov models d1,d2, . . . , d12 A Data-Transparent Probabilistic Model of Temporal Propositional Abstraction Hiroyuki Kido Cardiff University, Park Place,"Cardiff, CF10 3AT, United Kingdom Presented at The Twelfth Annual Conference on Advances in Cognitive Systems,! Atlanta, Georgia, USA, 13-15 October 2025 ABSTRACT Standard probabilistic models face fundamental challenges such as data scarcity, a large hypothesis space, and poor data transparency. To address these challenges, we propose a novel probabilistic model of data-driven temporal propositional reasoning. Unlike conventional probabilistic models where data is a product of domain knowledge encoded in the probabilistic model, we explore the reverse direction where domain knowledge is a product of data encoded in the probabilistic model. This more data-driven perspective suggests no distinction between maximum likelihood parameter learning and temporal propositional reasoning. We show that our probabilistic model is equivalent to a highest-order, i.e., full-memory, Markov chain, and it can also be viewed as a hidden Markov model requiring no distinction between hidden and observable variables. We discuss that limits provide a natural and mathematically rigorous way to handle data scarcity, including the zero-frequency problem. We also discuss that a probability distribution over data generated by our probabilistic model helps data transparency by revealing influential data used in predictions. The reproducibility of this theoretical work is fully demonstrated by the included proofs. OUR SOLUTION Reasoning is the process of using data for predictions (see ↓ and →). 3 ? Definition (Data distribution)! Let be a function that maps each data point to the one at the next time step.! ! Namely, data changes deterministically. n:Data ×Data p(dt|d1:t−1,m1:t−1,x1:t−1 1:I) Def.1 = 1 |Data|"if"t= 1 1if"t≈1"and"dt=n(dt−1) 0otherwise Prop.1 =p(dt|dt−1) Definition (Knowledge distribution)! Let be the truth value of in the model and .! ! Namely, the truth values of formulas obey the semantics of propositional logic. [[Xi]]mt Xi mt μ∈[0.5,1] p(xt i|d1:t,m1:t,x1:t−1 1:I,xt 1:i−1) Def.3 ={μ"if"xt i= [[Xi]]mt 1−μ"otherwise Prop.3 =p(xt i|mt) EVALUATIONS A discrete-time, discrete-space localisation problem in a grid. A robot retains the last ten visited locations, and it moves randomly while avoiding those in memory. When all adjacent accessible locations are stored, it remains in place. 7⋯7 matrix 104⋯10 Room at Time ! given Rooms for the past 4 time steps t Room 10 at Time ! given Room 1 for the past 4 time steps t Data d1-12 x1 2,x2 3,x3 8 Inputs/Output x5 10 Reasoning Probabilistic model X5 X4 X3 X2 X1 x1 2,x2 3,x3 8 Inputs/Output x5 10 Learning Reasoning Data d1-12 2nd-order Markov chain Data at Time 1 Models (i.e., valuations in propositional logic) at Time 1 Truth values of propositional formula at Time 1 X1 Equiv. → Poor data transparency Probability distributions over data ground predictions in data. In fact, the prediction of Room 10 at Time 5 is grounded in at Time 1. d2 Theoretical result Given , our solution (TA: temporal abstraction) is equivalent to Markov chains (MC) with maximum likelihood parameters. μ= 1 Data scarcity Given , TA outperforms MC with simple Laplace smoothing, which assigns an equal probability to zero frequency events. μ≈1 Huge hypothesis spaces TA has essentially linear complexity due to the deterministic data trajectory and deterministic data– model support relation. Abstraction (i.e.! selective ignorance) p(D1:T,M1:T,X1:T 1:I)Thm.1 = T ∏ t=1 [p(Dt|Dt−1)p(Mt|Dt) I ∏ i=1 p(Xt i|Mt)] p(X1:T 1:I)Thm.2 =1 K K ∑ k=1 T ∏ t=1 I ∏ i=1 p(Xt i|m(nt−1(dk)))