Computational aspects of reasoning about action

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Computational aspects of reasoning about action
Jérôme Lang (IRIT, Toulouse)
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Ontic vs. epistemic actions
Purely ontic (physical) actions - may change
the state of the world - do not bring feedback
to the agent (what you foresee is what
you get)
Purely epistemic actions (sensing actions) -
meant to provide new information - do not change
the state of the world (only the
agents beliefs)
binary tests???????test (?) returns the truth
value of ?
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Ontic vs. epistemic actions
Complex actions have both ontic and epistemic
effects (changes in the world
feedback) Example toss a coin (and observe the
outcome) However any complex action can be
written as a purely ontic action followed
by a purely epistemic action
Without loss of generality partition between
purely ontic and purely epistemic actions
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Ontic (physical) actions meant to change the
state of the world (physical) action
transition system automaton
?? eat an apple

deterministic action deterministic automaton
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????eat a fruit
nondeterministic action nondeterministic
automaton
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stochastic action stochastic automatpn
1/6
1/6
1/6

1/6
1/6
1/6
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combinatorial explosion n fruits, maximum p
items of each (p1)n possible states
temporal aspect (p1)n.T possible
trajectories incomplete knowledge 2(p1) n
-1 possible belief states (in the simplest
uncertainty model) how can action effects be
represented in a concise way?
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(No Transcript)
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Compact expression of ontic action effects
Actions representation in propositional logic
FL a,b,c,d, finite set of fluents
(propositional variables) S 2FL set of states

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Compact expression of ontic action effects
An elementary action language STRIPS
eff (??? ??causes li , i 1 .. q ( li
literals) ? li , i 1 .. q
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Compact expression of ontic action effects
An elementary action representation language
STRIPS
Why elementary?
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Compact expression of ontic action effects
1. conditional effects
eff (??? if prei then ??causes li , i 1 ..
q
prei propositional fornula li literal
filter (s, ?)???? li s ?????prei
assumed consistent

s
eff (??????s filter (s, ?)?
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Compact expression of ontic action effects
eff (??? if p ? q then ??causes ? p,
if p ? q then ??causes p, (if T
then) ??causes q
?
?p q
?
?
?
?p ?q
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Compact expression of ontic action effects
 Compilation  of an action
FLt at, bt,
FL a, b,
FLt1 at1, bt1,
? ( f ) ? prei f li
For all f ? FL
?- ( f ) ? (? f )
f is true at time t1 if there is an
applicable effect making it true or f was
true at time t and no applicable effect makes it
false
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Compact expression of ontic action effects
eff (??? if p ? q then ??causes ? p,
if ? p ? q then ??causes p, (if T
then) ??causes q
?
p q
?p q
?
?
?
p ?q
?p ?q
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Compact expression of ontic action effects
eff (??? if p ? q then ??causes ? p,
if ? p ? q then ??causes p, (if T
then) ??causes q
?
p q
?p q
?
?
?
pt1 ? ((? pt ? qt ) ? (pt ? ? (pt ? qt ))
p ?q
?p ?q
??
qt1 ?
?
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Compact expression of ontic action effects
eff (??? if p ? q then ??causes ? p,
if ? p ? q then ??causes p, (if T
then) ??causes q
? (p) ? p ? q
?- (p) ? (? p) p ? q
?
?
? (q)
p q
?p q
?- (q) ?
?
?
?
??
qt1 ? (pt1 ? (pt ? ? qt ))
p ?q
?p ?q

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Compact expression of ontic action effects
2. causal rules
static laws (instantaneous) causality
outside ? ? umbrella ? rain causes ? dry
action laws (dynamic causality)
eff (go-out)? go-out?causes outside
outside t1 umbrellat1 ? umbrella t raint1 ?
raint dryt1 ? dryt ? ? (outside t 1 ? ?
umbrellat1 ? raint1 )
?go-out
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Ontic actions progression
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Ontic actions progression
?
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Ontic actions progression
prog(???)
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Ontic actions regression
Regression (deterministic actions)
?
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Ontic actions regression
Regression (deterministic actions)
reg (???)
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t dryt ? umbrellat
?t ? ?go-out ? outsidet1 ? umbrellat
? umbrellat1 ? (raint ? raint1 )
? dryt ? dryt1
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t dryt ? umbrellat
?t ? ?go-out ? outsidet1 ? umbrellat
? umbrellat1 ? (raint ? raint1 )
? dryt ? dryt1
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
projection on FLt1
?t dryt ? umbrellat
?t ? ?go-out ? outsidet1 ? umbrellat
? umbrellat1 ? (raint ? raint1 )
? dryt ? dryt1
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
projection on FLt1
?t dryt ? umbrellat
?t ? ?go-out ? outsidet1 ? umbrellat
? umbrellat1 ? (raint ? raint1 )
? dryt ? dryt1
outsidet1 ? umbrellat1 ? dryt1
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projecting on FLt1 forgetting the symbols in
FLt
Forget (Ø , ?) ?
Forget (x , ??)
??x
?
Forget (X ? x , ??) Forget (x, Forget (X,
??))
?t ? ?go-out ? outsidet1 ? umbrellat
? umbrellat1 ? (raint ? raint1 )
? dryt ? dryt1
outsidet1 ? umbrellat ?
umbrellat1 ? (raint ? raint1 )
? dryt ? dryt1
forget outsidet
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projecting on FLt1 forgetting the symbols in
FLt
Forget (Ø , ?) ?
Forget (x , ??)
??x
?
Forget (X ? x , ??) Forget (x, Forget (X,
??))
outsidet1 ? umbrellat ?
umbrellat1 ? (raint ? raint1 )
? dryt ? dryt1
outsidet1 ? umbrellat1 ?
(raint ? raint1 ) ? dryt ? dryt1
forget umbrellat
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projecting on FLt1 forgetting the symbols in
FLt
Forget (Ø , ?) ?
Forget (x , ??)
??x
?
Forget (X ? x , ??) Forget (x, Forget (X,
??))
outsidet1 ? umbrellat1 ?
(raint ? raint1 ) ? dryt ? dryt1
outsidet1 ? umbrellat1 ?
? dryt ? dryt1
?
forget raint
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projecting on FLt1 forgetting the symbols in
FLt
Forget (Ø , ?) ?
Forget (x , ??)
??x
?
Forget (X ? x , ??) Forget (x, Forget (X,
??))
outsidet1 ? umbrellat1 ?
dryt1
outsidet1 ? umbrellat1 ?
dryt ? dryt1
?
forget dryt
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Progression
?t dryt ? umbrellat
projection on FLt1
? outsidet1 ? umbrellat ?
umbrellat1 ? (raint ? raint1 )
? dryt ? dryt1
outsidet1 ? umbrellat1 ? dryt1
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t dryt ? ? umbrellat
?t ? ?go-out ? outsidet1 ? ?
umbrellat ? ? umbrellat1 ? (raint ?
raint1 ) ? dryt ? (dryt1 ? ? raint1 )
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t dryt ? ? umbrellat
outsidet1 ? ? umbrellat ? ? umbrellat1 ? (raint
? raint1 ) ? dryt ? (dryt1 ? ? raint1 )
?t ? ?go-out
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t dryt ? ? umbrellat
projection
outsidet1 ? ? umbrellat ? ? umbrellat1 ? (raint
? raint1 ) ? dryt ? (dryt1 ? ? raint1 )
outsidet1 ? ? umbrellat1 ? (dryt1 ? ?
raint1 )
?t ? ?go-out
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t ? umbrellat
outsidet1 ? ? umbrellat ? ? umbrellat1 ? (raint
? raint1 ) ? (dryt1 ? dryt ? ? raint1 )
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t ? umbrellat
outsidet1 ? ? umbrellat ? ? umbrellat1 ? (raint
? raint1 ) ? (dryt1 ? dryt ) ? (dryt1 ? ?
raint1 ) ? (dryt ? ? raint1 ? dryt1 )
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Progression
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t ? umbrellat
projection
outsidet1 ? ? umbrellat ? ? umbrellat1 ? (raint
? raint1 ) ? (dryt1 ? dryt ) ? (dryt1 ? ?
raint1 ) ? (dryt ? ? raint1 ? dryt1 )
outsidet1 ? ? umbrellat1

? (dryt1 ? ? raint1 )
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Progression complexity
?t ????
iff
?t 1
PROGRESSION is coNP-complete
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Regression (deterministic actions)
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t1 dryt1 ? raint1
?t1 ? ?go-out ? outsidet1 ?
umbrellat ? umbrellat1 ? raint ?
raint1 ? dryt1 ? dryt ? ?
(outsidet1 ? ? umbrellat1 ? raint1 )


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Regression (deterministic actions)
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t1 dryt1 ? raint1
?t1 ? ?go-out ? outsidet1 ?
umbrellat ? umbrellat1 ? raint ?
raint1 ? dryt1 ? dryt ? umbrellat1


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Regression (deterministic actions)
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t1 dryt1 ? raint1
?t1 ? ?go-out ? outsidet1 ?
umbrellat ? umbrellat1 ? raint ?
raint1 ? dryt1 ? dryt


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Regression (deterministic actions)
outsidet1 umbrellat1 ? umbrellat raint1 ?
raint dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?go-out
?t1 dryt1 ? raint1
projection on FLt
?t1 ? ?go-out ? outsidet1 ?
umbrellat ? umbrellat1 ? raint ?
raint1 ? dryt1 ? dryt


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Compact expression of ontic action effects
3. nondeterminism
two possible ways or representating
nondeterminism
disjunctive effects
disjunction of effects
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Compact expression of ontic action effects
3. non-déterminisme
two possible ways or representating
nondeterminism
toss-coin causes heads or?causes ? heads
? heads
heads
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Compact expression of ontic action effects
3. nondeterminism
two possible ways or representating
nondeterminism
????make (up1 ? up2) true
??causes up1 ? up2 up1 ? up2 causes light up1 ? ?
up2 causes ? light
?up1 ?up2
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Progression (nondeterministic actions)
?
?
?
?
?
t
t1
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Progression (nondeterministic actions)
?
prog(???)
?
?
?
?
t
t1
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Regression (nondeterministic actions) two forms
Weak regression given - a belief state ?t1 at
time t1 - an action ??performed between t et
t1 which beliefs can we infer about the state of
the world at time t ?
Strong regression given - a goal ?t1 - an
action ? what are the states of the world in
which performing ??guarantees to reach ?t1 ?
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Weak regression (nondeterministic actions)
?
?
?
?
?
t
t1
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Weak regression (nondeterministic actions)
?
t
t1
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Strong regression (nondeterministic actions)
?
?
?
?
?
t
t1
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eff (go-out)? go-out?cause outside
eff (wait)? wait?causes rain or causes ?
rain
outside ? ? umbrella ? rain causes ? dry
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait
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Weak regression (nondeterministic actions)
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait
?t1 outsidet1 ? dryt1
?t1 ? ?go-out ? outsidet1 ? outsidet
? (umbrellat ? umbrellat1 ) ? dryt1 ?
dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )




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Weak regression (nondeterministic actions)
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait
?t1 outsidet1 ? dryt1
?t1 ? ?go-out ? outsidet1 ? outsidet
? (umbrellat ? umbrellat1 ) ? dryt1 ?
dryt ? (umbrellat1 ? ? raint1 )




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Weak regression (nondeterministic actions)
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait
?t1 outsidet1 ? dryt1
?t1 ? ?go-out ? outsidet1 ? outsidet
? (umbrellat ? umbrellat1 ) ? dryt1
? dryt ? (umbrellat ? ? raint1 )




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Weak regression (nondeterministic actions)
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait
?t1 outsidet1 ? dryt1
projection on FLt
?t1 ? ?attendre ? outsidet1 ? outsidet
? (umbrellat ? umbrellat1 ) ?
dryt1 ? dryt ? (umbrellat ? ?
raint1 )





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Strong regression (nondeterministic actions)
Models of StrongReg(a, y)







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Strong regression (nondeterministic actions)
? (conjunction of literals) minimal success
condition (MSC)
(2) there is no ? ? ??such that prog (?? ?)




ABDUCTION
StrongReg (?, ?) ? ? ?t ?t MSC for ?????
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Strong regression (nondeterministic actions)
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait
?t1 outsidet1 ? dryt1
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Strong regression (nondeterministic actions)
?t1 dryt1 ? raint1
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait


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Strong regression (nondeterministic actions)
?t1 dryt1 ? raint1
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?attendre
minimal success conditions dryt ? outsidet ?
umbrellat


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Strong regression (nondeterministic actions)
?t1 dryt1 ? raint1
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait
minimal success conditions dryt ? outsidet ?
umbrellat


StrongReg (?, wait) dry ? outside ? umbrella
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Strong regression (nondeterministic actions)
?t1 dryt1 ? raint1
outsidet1 ? outsidet umbrellat1 ?
umbrellat dryt1 ? dryt ? ? (outsidet1 ? ?
umbrellat1 ? raint1 )
?wait
minimal success conditions dryt ? outsidet ?
umbrellat


StrongReg (?, wait) dry ? outside ? umbrella
? WeakReg (?, wait) dry ? outside
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Weak regression complexity
Weakreg (?,?)
?t 1 ????
iff
?t
WEAK REGRESSION is coNP-complete
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Strong regression complexity
StrongReg (?,?)
STRONG REGRESSION is -complete
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ontic vs. epistemic actions
meant to bring new information to the
agent leave the state of the world unchanged
meant to change the state of the world
no feedback
Any complex action can be decomposed in a
(purely) ontic action followed by a (purely)
epistemic action
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Belief states

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Belief states

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Epistemic actions progression
?? test (?)
?
???
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Epistemic actions progression
?? test (?)
?
?
?
???
?
???
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Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
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Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
G
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Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
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Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
??
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Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
77
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
78
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
79
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
80
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
81
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
82
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
83
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
84
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
85
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
86
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
87
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
88
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
89
Epistemic actions strong regression
?
???
? test (?) s1,s3, s2,s4??
?
s1
s3
? ?
s2
s4
90
Partial observability and epistemic actions in
logic making explicit the distinction between
facts and knowledge/beliefs

epistemic/doxastic logics
91
Propositional epistemic logic S5
Language LK,PS
PS finite set of propositional symbols
f ? F formula of
LK,PS
if ????
K ???? ????????????????

K ?? I know ?
? ????? is true?
? denoted by ?
??objective iff ??is modality-free
92
Propositional doxastic logic KD45
Language LK,PS
PS finite set of propositional symbols
f ? F formula of
LK,PS
if ????
B ???? ????????????????

B ?? I believe ?
? ????? is true?
? denoted by ?
??objective iff ??is modality-free
93
Propositional epistemic logic S5
epistemic atom
K (? a ? b)
epistemic formula Boolean combination of
epistemic atoms
K (? a ? b) ? (? K c ? ? K c)
but not c ? K (? a ? b)

positive epistemic formula epistemic formula
without any occurrence of ? K
K (? a ? b) ? (K c ? K ? c)
94
Propositional epistemic logic S5 axiomatics
axiom schemata
A0. all tautologies of propositional calculus
K ????K (??? ?) ? K ?
K.
distribution
K ? ? ?
T.
correct beliefs
inference rules
from ??and ????????? ? infer ?
MP.
from ?? infer K ???????necessitation
N.
95
Propositional doxastic logic KD45 axiomatics
axiom schemata
A0. all tautologies of propositional calculus
inference rules
from ?? infer K ???????necessitation
N.
96
Propositional S5 semantics
M ? S, val, s? S nonempty subset of
states s ? S (actual state) val S ? (PS ?
true, false) valuation function
iff val (s) (p) true
p
for p ? PS
(M, s)
K ?
??)
iff (? s ? M ) (s
(M, s)
and
iff
(M, s)
?
?????
(M, s)
iff
or
?????
(M, s)
iff
????
(M, s)
iff
??
M
(M, s)
?
97
Propositional KD45 semantics
M ? S, s, val? S nonempty subset of states
s ? actual state, not necessarily in S val
(S? s) ? (PS ? true, false)
iff val (s) (p) true
p
for p ? PS
(M, s)
B ?
??)
iff (? s ? M ) (s
(M, s)
and
iff
(M, s)
?
?????
(M, s)
iff
or
?????
(M, s)
iff
????
(M, s)
iff
??
M
98
Propositional epistemic logic S5
Depth of a formula depth (p) 0 for p ?
PS depth (? ?????depth (? ??? depth
(?????????max (depth (????depth (???? depth (K
?????????depth (???
A fundamental property for all ??? LK,PS there
exists a ? ? LK,PS such that
????? and depth (?) ? 1
S5
K ?
without loss of generality, only objective
formulas in the scope of K
K ?
Similar result holds in KD45
99
Propositional epistemic logic S5
Important tautologies and non-tautologies
K ???? ????? K ?????K ???
S5
K ???? ????? K ????K ???
S5
? (K ??? K ? ???
S5
100
Propositional S5 decidability
Proposition 1 there is an algorithm that,
given a model M, a state s ? M and a formula ?
? LK,PS , determines in time O(M.?) whether
(M,s) ? Proposition 2 if ? ? LK,PS
is satisfiable in S5 then ? is satisfiable in a
model with at most ? states Corollary
satisfiability in S5 is decidable
S5
Similar result holds in KD45
101
Propositional S5 complexity
Proposition 2 if ? ? LK,PS is satisfiable in S5
then ? is satisfiable in a model with at most
? states Corollary satisfiability in S5
is in NP satisfiability in S5 is NP-complete
Similar result holds in KD45
102
S5 and reasoning about action / planning
off-line reasoning (plan generation)
????test(?)
K ??? K ??
complex knowledge state
103
Simple knowledge state (SKS) epistemic atom
Complex knowledge state (CKS) positive
epistemic formula
Any CKS can be equivalently written as a
disjunction of SKS
????K ?1 ? ?K ?n
K a ? ( K b ? K ?b ) ??K (a ? b) ? K (a ? ?b )
104
??CKS
??action (ontic or epistemic)

Progression of a CKS by an action

Prog (????) strongest formula known to hold
after ??is performed, given that
??holds initially
Goal regression of a CKS by an action
Reg (????) weakest formula whose
progression by ??entails ??
105
Ontic actions in S5 progression
??ontic action
classical progression
ProgO (????) is a CKS
ProgO CKS AO ? CKS
106
Ontic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegO (????) K reg (?1????)????????K reg
(?n????)?
RegO CKS AO ? CKS
Computing regression compute minimal
success conditions abduction
107
Epistemic actions in S5 progression
Epistemic action
??
o1, , op possible observations
such that (o1 ? ? op) tautology
ProgE CKS AE ? CKS
Binary tests??????test (?) returns the truth
value of ?
Prog E (???test (??)) ??? ( K ? ??K ? ??)
108
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
? test (?)??
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
RegE CKS ACTO ? CKS
109
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
110
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
RegE (????) K ((u ? v ? v ) ? ((?u ? ?v) ?
v ) ) ? K ((u ? v ? v ) ? ((?u ?
?v) ? ?v ) ) ? K ((u ? v ? ?v ) ? ((?u ? ?v) ?
v ) ) ? K ((u ? v ? ?v ) ? ((?u ? ?v) ? ?v ) )
111
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
RegE (????) K ((u ? v ? v ) ? ((?u ? ?v) ?
v ) ) ? K ((u ? v ? v ) ? ((?u ?
?v) ? ?v ) ) ? K ((u ? v ? ?v ) ? ((?u ? ?v) ?
v ) ) ? K ((u ? v ? ?v ) ? ((?u ? ?v) ? ?v ) )
K v
112
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
RegE (????) K v ? K ((u ? v ?
v ) ? ((?u ? ?v) ? ?v ) ) ? K ((u ? v ? ?v ) ?
((?u ? ?v) ? v ) ) ? K ((u ? v ? ?v ) ? ((?u ?
?v) ? ?v ) )
K?v
113
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
RegE (????) K v ? K ((u ? v
? v ) ? ((?u ? ?v) ? ?v ) ) ? K ((u ? v ? ?v )
? ((?u ? ?v) ? v ) ) ? K?v
v ? u
?
114
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
RegE (????) K v ? K (v ?
u) ? K ((u ? v ? ?v ) ? ((?u ? ?v) ? v ) ) ?
K?v
v
u ? ?v
115
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
RegE (????) K v ? K (v ?
u) ? K (v ? ? u) ? K?v
116
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
RegE (????) K v ? K (v ?
u) ? K (v ? ? u) ? K?v
117
Epistemic actions in S5 regression
????K ??????????K ?n
complex knowledge state
RegE (????) ? K ((?????i )???(? ?????j ))?
i,j ??1 .. n?
????K v ???K ? v
? test (?)? test (u ? v)
RegE (????) K v ? K (v ? u)
118
Plans
? (empty plan) is a plan for any action ??AO ?
AE , ????is a plan if ? and ? are plans then ?
? is a plan if ? and ? are plans and ? a
complex epistemic state then if ??then ? else
? is a plan
119
Progression of a CKS by a plan
Prog(?? ?????? Prog(?? ???????prog(??
??? Prog(?? (? ?)??? Prog( Prog(? ? ????
??? Prog(?? if ??then ? else ?)
Prog(?? ?)
Prog(?? ?)
sinon?
120
Planning problem
P ? K ?init??? AO ? AE , ???
simple knowledge state
goal complex knowledge state
? is a valid plan for P iff Prog (?? ???
??
121
Plan verification example
P ? K ?init??? AO ? AE , ???
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
122
Plan verification example
P ? K ?init??? AO ? AE , ???
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
???????if K (u ? v) then ??else????????
123
Plan verification example
P ? K ?init??? AO ? AE , ???
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
???????if K (u ? v) then ??else?????????
124
Plan verification example
P ? K ?init??? AO ? AE , ???
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
???????if K (u ? v) then ??else???????
??
valid for P
?
125
Plan generation
P ? K ?init??? AO ? AE , ????K G1 ? ? K Gn ?
?? ???
repeat
choose an action?????AO ? AE ?
?? Reg (?????) ? ?
or ? can no longer be changed
K ?init
??
until
126
Plan generation example
P ? K ?init??? AO ? AO , ????K G1 ? ? K Gn ?
?? ???
? ???????
??
repeat
choose an action?????AO ? AE ?
?? Reg (?????) ? ?

• ???? ? Reg (?????) ?? ????????

or ? can no longer be changed
K ?init
??
until
127
Plan generation example
P ? K ?init??? AO ? AO , ????K G1 ? ? K Gn ?
?? ???
? ???????
??
repeat
choose an action?????AO ? AE ?
?? Reg (?????) ? ?

• ???? ? Reg (?????) ???????

or ? can no longer be changed
K ?init
??
until
128
Plan generation example
P ? K ?init??? AO ? AO , ????K G1 ? ? K Gn ?
?? ???
? K G1 ???????? , ? K Gn ???????
??
repeat
choose an action?????AO ? AE ?
?? Reg (?????) ? ?

• ???? ? K ?i ????????Reg (?????) ?i ,

? ? K Fj ???????in ??s.t. Fj?
?
?i

or ? can no longer be changed
K ?init
??
until
129
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????


?? K v ???K ? v
? K v????????? K ? v??????
??


??
130
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
?? K v ???K ? v
? K v????????? K ? v??????
??
??
Reg (???? ) K v ? K (v ? u)
Regression by ?
131
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
?? K v ???K ? v
? K v????????? K ? v??????
??
??
Reg (???? ) K v ? K (v ? u)
Regression by ?
?? K v ???K ? v ? K (v ? u)
132
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
?? K v ???K ? v
? K v????????? K ? v??????
??
??
Reg (???? ) K v ? K (v ? u)
Regression by ?
?? K v ???K ? v ? K (v ? u) ?? K v ???K (v ? u)
133
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
?? K v ???K (v ? u)
? K v????????? K ? v?????? ? K (v ? u)?????
??
??
Reg (???? ) K v ? K (v ? u)
Regression by ?
?? K v ???K (v ? u)
134
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
? K v????????? K ? v?????? ? K (v ?
u)???????? ? K (v ? ? u)???????
?? K v ???K (v ? u)
??
??
Regression by ?
Reg (???? ) K v ? K (v ? ? u)
?? K v ? K (v ? u) ? K (v ? ? u)
135
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
?? K v ???K (v ? u)
? K v????????? K ? v?????? ? K (v ?
u)???????? ? K (v ? ? u)?????? ?????K?????????
??
??
Regression by ?
?
Reg (???? ) K
?
?? K K ?init
STOP
?
136
?init
?
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
? K v????????? K ? v????????? K (v ?
u)???????? ? K (v ? ? u)??????????K??????????
??
??
?
137
?
?init
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
? K v????????? K ? v????????? K (v ?
u)???????? ? K (v ? ? u)??????????K??????????
??
??
?
138
?
?init
AO ? ? switch (u)
AE ???? ? test (u ? v) ?? test
(u ? v)?
K v ???K ? v
????
? K v????????? K ? v????????? K (v ?
u)???????? ? K (v ? ? u)??????????K??????????
??
??
?
???????? if K (u ? v)? then ? else ?? ?
139
knowledge-based programs (Fagin-Halpern-Moses-Var
di, 95)
involve deduction tasks at execution time
???????? if K (u ? v)? then ? else ?? ?
140
Bibliography
141
Bibliography

• H. Levesque, What is planning in the presence
  of sensing? AAAI-96
• R. Scherl and H. Levesque. Knowledge, action,
  and the frame
• problem. Artificial Intelligence Journal 144,
  2003
• C. Baral T. Son, Formalizing sensing actions
  A transition-bsed
• approach. Artificial Intelligence Journal
  125, 2001
• G. de Giacomo and R. Rosati, Minimal knowledge
  approach to
• reasoning about action and sensing,
  Electronic Transactions on
• Artificial Intelligence, 1999
• - A. Herzig, J. Lang, D. Longin and T. Polacsek.
  A logic for planning
• under partial observability. AAAI-00.

142
Bibliography

• R. Fagin, J. Halpern, Y. Moses and M. Vardi.
  Reasoning about
• knowledge, MIT Press, 1995