A Distributed Logical Filter for Connected Row Convex Constraints - PowerPoint PPT Presentation

A Distributed Logical Filter for Connected Row Convex Constraints T. K. Satish Kumar Hong Xu Zheng Tang Anoop Kumar Craig Milo Rogers Craig A. Knoblock tkskwork@gmail.com, hongx@usc.edu, {zhengtan, anoopk, rogers, knoblock}@isi.edu November 6, 2017 Information Sciences Institute, University of Southern California The 29th IEEE International Conference on Tools with Artifjcial Intelligence (ICTAI 2017) Boston, Massachusetts, the United States of America

Executive Summary The Kalman fjlter and its distributed variants are successful methods in state estimation in stochastic models. We develop the analogues in domains described using constraints. Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 1 / 19

Agenda What is Filtering and What is Its Motivation The Connected Row Convex (CRC) Filter Distributed Connected Row Convex (CRC) Filter Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 2 / 19

Agenda What is Filtering and What is Its Motivation The Connected Row Convex (CRC) Filter Distributed Connected Row Convex (CRC) Filter

Motivation Navigation system (Zarchan et al. 2015) Image by Hervé Cozanet (CC BY-SA 3.0). Retreived from: https://commons.wikimedia.org/wiki/File: Navigation_system_on_a_merchant_ship.jpg Econometrics (Schneider 1988) Image retrieved from: https://i.ytimg.com/vi/vEP4RIOKuE4/hqdefault.jpg Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 3 / 19

Filtering In a partially observable or uncertain environment, an agent often needs to maintain its belief state (a representation of its knowledge about the world) based on • What are the beliefs at previous time steps? • What does the agent observe at the current time step? Filtering denotes any method whereby an agent updates its belief state. Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 4 / 19

Example: The Kalman Filter Kalman fjlter (Kalman 1960) A Distributed Logical Filter for Connected Row Convex Constraints Kumar et al. (Information Sciences Institute, USC) by Petteri Aimonen (CC0). Retreived from: https://commons.wikimedia.org/wiki/File:Basic_concept_of_Kalman_filtering.svg 5 / 19 Prediction step Prior knowledge Based on e.g. of state physical model Next timestep Update step Measurements Compare prediction to measurements Output estimate of state

Agenda What is Filtering and What is Its Motivation The Connected Row Convex (CRC) Filter Distributed Connected Row Convex (CRC) Filter

Logical Filter A logical fjlter applies to domains that are described using logical et al. 2006), a logical fjlter that uses CRC constraints. Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 6 / 19 formulae or constraints. Here, we are interested in the connected row convex (CRC) fjlter (Kumar

Motivation: Multi-Robot Localization by James McLurkin. Retreived from: https://people.csail.mit.edu/jamesm/project-MultiRobotSystemsEngineering.php Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 7 / 19

• Find an assignment a of values to these variables such that all Constraint Satisfaction Problems (CSPs) constraints allow it. • Known to be NP-hard. Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 8 / 19 • N variables X = { X 1 , X 2 , . . . , X N } . • Each variable X i has a discrete-valued domain D ( X i ) . • M constraints C = { C 1 , C 2 , . . . , C M } . • Each constraint C i specifjes allowed and disallowed assignments of values to a subset S ( C i ) of the variables.

A Filter Based on Constraints: Framework • Observations at t are modeled as A Distributed Logical Filter for Connected Row Convex Constraints Kumar et al. (Information Sciences Institute, USC) t (with Markovian assumption). consistent extension to variables at assignments of values to variables state is defjned by all allowed modeled as the constraints between constraints on the variables at t . 9 / 19 • Transitions from t to t + 1 are Time = 0 Time = t Time = t+1 X 1 X 1 t X 1 t+1 0 variables at t and t + 1. X 2 X 2 t X 2 t+1 0 • At each time step t + 1, the belief X N X N t+1 X N 0 t at t + 1 that satisfy observation observations at 0 observations at t observations at t+1 constraints at t + 1, and have a

A Filter Based on Constraints: Framework But… A Distributed Logical Filter for Connected Row Convex Constraints Kumar et al. (Information Sciences Institute, USC) Connected Row Convex (CRC) Constraints Solution: We’d like to have compact information. time step requires looking further back. a consistent extension to the previous in general, determining the existence of 10 / 19 Time = 0 Time = t Time = t+1 X 1 X 1 t+1 X 1 0 t X 2 X 2 t X 2 t+1 0 X N X N t+1 X N 0 t observations at 0 observations at t observations at t+1

The Connected Row Convex (CRC) Constraint ‘1’: Allowed assignment A Distributed Logical Filter for Connected Row Convex Constraints Kumar et al. (Information Sciences Institute, USC) intersect or are consecutive after removing empty rows/columns CRC constraint: Row convex + The ‘1’s in any two successive rows/columns Row convex constraint: All ‘1’s in each row are consecutive ‘0’: Disallowed assignment 11 / 19 X j X j d j1 d j2 d j3 d j4 d j5 d j1 d j2 d j3 d j4 d j5 X i X i 1 0 0 0 1 0 0 0 d i1 0 d i1 0 0 0 0 0 d i2 1 1 0 d i2 1 0 0 0 1 1 1 d i3 0 1 1 d i3 0 0 1 0 0 d i4 0 0 1 0 d i4 0 0 1 0 d i5 0 1 0 0 d i5 0 1 0 0 0 0 (a) ✓ (b) ✗

The Connected Row Convex (CRC) Constraint • Path consistency: For any consistent assignment of values to any two variable X k . • After enforcing path consistency on constraint networks with only CRC constraints, all constraints are still CRC. This is not true for row convex constraints. Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 12 / 19 variables X i and X j , there exists a consistent extension to any other

The Connected Row Convex (CRC) Constraint CSP search tree Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 13 / 19 () (X 1 = d 11 ) (X 1 = d 11 , X 2 = d 22 ) (X 1 = d 11 , X 2 = d 22 , X 3 = d 31 ) (X 1 = d 11 , X 2 = d 22 , X 3 = d 31 , X 4 = d 43 ) (X 1 = d 11 , X 2 = d 22 , X 3 = d 31 , X 4 = d 43 , X 5 = ?)

The Connected Row Convex (CRC) Constraint Row convexity implies global consistency in path consistent constraint networks A Distributed Logical Filter for Connected Row Convex Constraints Kumar et al. (Information Sciences Institute, USC) 14 / 19 X 5 X 5 d 51 d 52 d 53 d 54 d 55 d 51 d 52 d 53 d 54 d 55 0 0 1 1 1 X 1 = d 11 X 1 = d 11 X 2 = d 22 X 2 = d 22 0 0 1 1 1 X 3 = d 31 X 3 = d 31 0 1 1 1 0 X 4 = d 43 1 X 4 = d 43 0 0 0 1

The Connected Row Convex (CRC) Filter If all constraints are CRC, enforcing A Distributed Logical Filter for Connected Row Convex Constraints Kumar et al. (Information Sciences Institute, USC) consistent assignments. constraints contain all information of leads to new CRC constraints path consistency between every two 15 / 19 Time = t-1 Time = t Time = t+1 consecutive time steps t and t + 1 X 1 X 1 t-1 X 1 t t+1 X 2 t-1 X 2 t X 2 t+1 between variables at t + 1. These CRC X N t-1 X N t X N t+1 observations at t-1 observations at t observations at t+1

Example: Multi-Robot Localization (Kumar et al. 2006) (b) Each robot estimate its A Distributed Logical Filter for Connected Row Convex Constraints Kumar et al. (Information Sciences Institute, USC) (Kumar et al. 2006, Fig. 18) (c) The constraints are CRC. distances from other robots. 16 / 19 movement. (a) Each robot estimate its own Y – X = U K Y Y – X = L (X i t+1 , Y i t+1 ) a 2 0 X H H a 1 (X i t , Y i t )

Agenda What is Filtering and What is Its Motivation The Connected Row Convex (CRC) Filter Distributed Connected Row Convex (CRC) Filter

The Distributed Kalman Filter The distributed version of the Kalman fjlter has been successful in state estimation in wireless sensor networks (Rao et al. 1993), including large scale systems (Olfati-Saber 2007). What about a distributed CRC fjlter? Kumar et al. (Information Sciences Institute, USC) A Distributed Logical Filter for Connected Row Convex Constraints 17 / 19

Distributed Connected Row Convex (CRC) Filter • Each agent is in charge of a A Distributed Logical Filter for Connected Row Convex Constraints Kumar et al. (Information Sciences Institute, USC) the success of the CRC fjlter. consistency algorithms is key to • Improving distributed path distributed path consistency. • The system evolves using variables. constraints involving those subset of variables, and all 18 / 19 S 1 = {X 1 , X 2 } n 3 n 1 S 3 = {X 1 , X 3 , X 4 } n 5 S 5 = {X 3 , X 4 , X 5 } n 2 X 3 0 X 3 t X 3 t+1 X 4 0 X 4 X 4 t+1 t S 2 = {X 2 , X 4 } X 5 X 5 0 X 5 t t+1 n 4 S 4 = {X 4 , X 6 } X 4 X 4 X 4 t+1 0 t X 6 X 6 0 X 6 t t+1