Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 What Sensing Tells Us: Towards a Formal Theory of Testing for Dynamical Systems Sheila McIlraith Knowledge Systems Lab Dept. Computer Science Stanford University Richard Scherl Dept. Computer Science New Jersey Inst. of Technology

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Example

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Action: listen(radio) Direct Effect of “ listen(radio) ” noise(radio)

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 ok(radio) plugged-in(radio) ok(power) Indirect Effects of “ listen(radio) ” noise(radio) Action: listen(radio) on(radio)

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Determine whether “ ok(power) ” ? ok(power)

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Action: turn_on(radio) Determine whether “ ok(power) ” ? ok(power) on(radio)

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Action: turn_on(radio) listen(radio) ? ok(power) on(radio) Determine whether “ ok(power) ” noise(radio)

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 on(radio) ok(radio) plugged-in(radio) ok(power) Determine whether “ ok(power) ” noise(radio) Action: turn_on(radio) listen(radio)

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Determine whether “ ok(power) ” on(radio) ?? (   )ok(radio) ?? (   )plugged-in(radio) ?? (   )ok(power) … silence... …... …... … …   noise(radio) Action: turn_on(radio) listen(radio)

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Problem and Approach Problem: Given an axiomatization of a deterministic, partially observable dynamical system with sensing actions state constraints (relationships between properties/objects in the world). and a set of unobservable hypotheses How do we select actions to reduce the hypothesis space? Approach: Provide a theory of testing for dynamical systems in the situation calculus.

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Contributions  “Solution” to the ramification problem for sensing actions Characterization of tests, and the effect of test outcomes Effect of test outcomes on different hypothesis spaces Complex tests as Golog procedures Verification and generation of complex tests

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Situation Calculus [McCarthy, 68], [Reiter, 92] S0S0... do(turn_on(radio), S 0 ) do(unplug(radio), do(turn_on(radio), S 0 )) do(unplug(radio), S 0 )... Sorted First-Order Language: Situations: e.g., S 0, do(turn_on(radio), S 0 ) Parameterized Actions: e.g., turn_on(radio) Fluents: e.g., on(radio) Etc.

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Situation Calculus Axiomatizations Knowledge fluent, successor state axiom “solution” to the frame and ramification problems for knowledge and sensing* sensing actions & knowledge Knowledge fluent, Successor state axiom “solution” to the frame problem for knowledge and sensing [Scherl & Levesque, 93] * sometimes Successor state axiom “solution” to the frame and ramification problems * [Lin&Reiter, 94],[McIlraith,97] state constraints causality & completeness assumptions Situation Calculus [McCarthy,68] Successor state axiom “solution” to the frame problem [Reiter, 92] completion & causality assumptions

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Ramification Problem for Sensing Actions Theorem (informally stated): Our representation addresses the frame and ramification problems for world-altering and sensing actions. Using this representation the agent knows the indirect effects of both its world-altering and sensing actions.

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Knowledge Fluent/Accessibility Relation K s do(a 3, s)do(a 1,s)... S0S0 do(a 3,S 0 )do(a 1,S 0 )... s’ do(a 3, s’)do(a 1, s’)... Knows( ,s)   s’ K(s’,s)   (s’) Knows(on(radio),s)   s’ K(s’,s)  on(radio,s’) Kwhether( , s)  Knows( ,s)  Knows(   ,s) Kwhether(on(radio), s)  Knows(on(radio), s)  Knows(   on(radio), s)... K K Knowledge Fluent K(s’,s) K... K

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Contributions “Solution” to the ramification problem for sensing actions  Characterization of tests, and the effect of test outcomes Effect of test outcomes on different hypothesis spaces Complex tests as Golog procedures Verification and generation of complex tests

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Definition of a Test Simple Test: A simple test is a pair (I,a) where I, the initial conditions, is a conjunction of literals, and a is a binary sense action. E.g., (on(radio), listen(radio)) [McIlraith & Reiter, 92] [McIlraith, 94]

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Tests for Hypothesis Spaces Car Domain Example [Idiots Guide to Car Repair] (1) ab(battery,s)  on(radio,s)    noise(radio,s) (2) ab(radio,s)    noise(radio,s) (3) sparking  sparks(s) (4) sparks(s)  gas_leak(s)  explosion(s) (5)   explosion(s)... Test of Hypothesis Space HYP: A test (I,a) is a test for hypothesis space HYP in situation s iff D  I  Poss(a,s)  H(s) is satisfiable for every H  HYP. E.g., Hyp = {gas_leak(s), ab(battery,s), ab(spark_plugs,s), empty(tank,s)} test (sparking, check_sparking(spark_plugs)) is not a test for hypothesis space HYP.

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Confirmation and Refutation Confirmation and Refutation: The outcome  of test (I,a) confirms H  HYP iff D  I  Poss(a,s)  Knows(H  ,s) The outcome  of test (I,a) refutes H  HYP iff D  I  Poss(a,s)  Knows(H    ,s) E.g., Hyp = {gas_leak(s), ab(battery,s), ab(spark_plugs,s), empty(tank,s)} test = (on(radio), listen(radio)) outcome noise(radio,s) refutes hypothesis ab(battery,s). outcome   noise(radio,s) confirms hypothesis ab(battery,s). Car Domain Example (repeated) (1) ab(battery,s)  on(radio,s)    noise(radio,s) (2) ab(radio,s)    noise(radio,s)...

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Discriminating Tests Discriminating Test: A test (I,a) is a discriminating test for hypothesis space HYP iff D  I  Poss(a,s)  H(s) is satisfiable for every H  HYP, and There exists H i, H j  HYP such that outcome  of test (I,a) refutes either H i or H j no matter what the outcome. If H i =   H j, (I,a) is an individual discriminating test. E.g., Hyp = {gas_leak(s), ab(battery,s), ab(spark_plugs,s), empty(tank,s)} test (true, check_empty(tank)) is an individual discriminating test. Other Tests: relevant test constraining test

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Contributions “Solution” to the ramification problem for sensing actions Characterization of tests, and the effect of test outcomes  Effect of test outcomes on different hypothesis spaces Complex tests as Golog procedures Verification and generation of complex tests

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Contributions “Solution” to the ramification problem for sensing actions Characterization of tests, and the effect of test outcomes Effect of test outcome on different hypothesis spaces  Complex tests as Golog procedures Verification and generation of complex tests

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Complex Tests as Golog Procedures S0S0 do(a 3,S 0 )do(a 1,S 0 )... Golog [Levesque et al, 97] sequencing if-then-else while-do nondeterministic choice etc. Proc C HECK B ATTERY TURN_ON(RADIO); LISTEN(RADIO); if  Kwhether(AB(BATTERY) then (TURN_ON(LIGHTS); LOOK(LIGHTS)); if  Kwhether(AB(BATTERY) then (if  Kwhether(AB(FUSES) then CHECK F USES); if Knows(  AB(FUSES) then METER C HECK B ATTERY else (FIX F USES; C HECK B ATTERY)) endProc OBSERVE: Complex tests can have side-effects on the world.

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Contributions “Solution” to the ramification problem for sensing actions Characterization of tests, and the effect of test outcomes Effect of test outcome on different hypothesis spaces Complex tests as Golog procedures  Verification and generation of complex tests

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Test Verification and Generation Theorem (informally stated): Regression rewriting reduces the verification problem to theorem proving in the initial situation. Verification: We can automatically verify certain properties of a restricted class of complex tests, e.g., Proving Verifies that the procedure  H  HYP Kwhether(H,s) reduces the hypothesis space HYP  H  HYP Knows(  H,s) is a discriminating test for HYP Generation: We can automatically generate an even more restricted class of complex tests that satisfy particular properties, e.g., Kwhether(ab(battery),s) in a brute-force manner by searching through the space of conditional plans, followed by regression and theorem proving in the initial situation. (not efficient!) [Lesperance, 94]

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 “Solution” to the ramification problem for sensing actions Characterization of tests, and the effect of test outcomes Effect of test outcome on different hypothesis spaces Complex tests as Golog procedures Verification and generation of complex tests Summary Theory of testing for deterministic, partially observable dynamical systems that exploits the relationship between objects/properties in the world to infer unobservable properties.

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 Testing [Roth, 80], [Larrabee,92], [Shirley & Davis, 83], [McIlraith & Reiter, 92], [McIlraith 94], etc. Knowledge and Sensing [Moore, 85], [Etzioni et al., 92], [Scherl & Levesque, 93], [Lesperance, 94], [Golden & Weld,96], [Baral & Son, 98], [Funge, 98], [Weld et al., 98], [Lakemeyer, 99], [de Giacomo & Levesque, 99a,99b], [Lesperance & Ng, 00], [Reiter, 00, 00a], etc. Assimilation of Observations [Shanahan, 96,96a], [McIlraith,97,98], [Baral et al., 00], [Son, 00] Related Probabilistic Approaches (e.g., POMDPs) [Smallwood & Sondik, 73], [Horwitz, 88], [Littman,96], etc. Related Work