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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 McIlraithKnowledge Systems LabDept. Computer Science
Stanford University
Richard ScherlDept. Computer Science
New Jersey Inst. of Technology
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 hypothesesHow do we select actions to reduce the hypothesis space?
Approach:Provide a theory of testing for dynamical systemsin 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]
S0
...
do(turn_on(radio), S0)
do(unplug(radio), do(turn_on(radio), S0))
do(unplug(radio), S0)
......
Sorted First-Order Language:
Situations: e.g., S0, do(turn_on(radio), S0)
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 theframe and ramification problems for knowledge and sensing*
sensing actions & knowledge
Knowledge fluent, Successor state axiom “solution”
to the frame problem for knowledgeand 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(a3, s)do(a1,s)
.........
. . .
S0
do(a3,S0)do(a1,S0)
.........
s’
do(a3, s’)do(a1, 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)
. . .
KK
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 iffD 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 Hi, Hj HYP such that outcome of test (I,a) refutes either Hi or Hj no matter what the outcome.
If Hi = Hj, (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
S0
do(a3,S0)do(a1,S0)
.........
Golog [Levesque et al, 97]• sequencing• if-then-else• while-do• nondeterministic choiceetc.
Proc CHECKBATTERY
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 CHECKFUSES);
if Knows( AB(FUSES) then METERCHECKBATTERY
else (FIXFUSES; CHECKBATTERY))endProcOBSERVE: 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 theoremproving in the initial situation.
Verification: We can automatically verify certain properties of arestricted 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 initialsituation. (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