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, 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


Example


Action:• listen(radio)

Direct Effect of “listen(radio)”

noise(radio)


ok(radio)

plugged-in(radio)

ok(power)

Indirect Effects of “listen(radio)”

noise(radio)Action:• listen(radio)

on(radio)


Determine whether “ok(power)”

? ok(power)


Action:• turn_on(radio)


? ok(power)on(radio)


Action:• turn_on(radio)• listen(radio)

? ok(power)on(radio)


noise(radio)


on(radio)

ok(radio)

plugged-in(radio)

ok(power)


noise(radio)Action:• turn_on(radio)• listen(radio)



on(radio)

?? ()ok(radio)

?? ()plugged-in(radio)

?? ()ok(power)

… silence ...

… ...

… ... …...

...…... …

noise(radio)Action:• turn_on(radio)• listen(radio)


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.


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


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.


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


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.


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


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




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]


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.


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) . . .


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


Contributions



Effect of test outcomes on different hypothesis spaces




Contributions



• Effect of test outcome on different hypothesis spaces

Complex tests as Golog procedures



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.


Contributions





Verification and generation of complex tests


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]







Summary

Theory of testing for deterministic, partially observable dynamical systems that exploits the relationship between objects/properties in the world to infer unobservable properties.


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

Documents

Sheila McIlraith, Knowledge Systems Lab, Stanford University AAAI’00 08/2000 What Sensing Tells Us: Towards a Formal Theory of Testing for Dynamical Systems