Download Dipartimento di Informatica

Temporal Extensions to
Defeasible Logic
Guido Governatori1, Paolo Terenziani2
1 University
of Quuensland, Brisbane, Australia
2Dipartimento
di Informatica, UPO, Alessandria, Italy
Introduction
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Defeasible conclusions  nonmonotonic logic
Trade-off: expressiveness vs comp. complexity
Defeasible Logic [Nute,94]: a linear logic
Several applications:
legal reasoning
contracts and agent negotiations
Semantic Web
Defeasible Logic
• Facts (predicate; e.g., penguin(Tweedy))
• Strict Rules A1..An  B (classical rules)
• Defeasible Rules A1..An B (rules that can be defeated by
contrary evidence; e.g., “birds usually fly”)
• Defeaters A1..An  B (rules to prevent derivation of
conclusions; “e.g., if something is heavy it might not fly”)
• Priorities between rules
• “skeptical” nonmonotonic logic: it does not support
contraddictory conclusions
Provability in DL
Let D be a Theory
• +q (q is definitely provable in D, i.e., using only facts
and strict rules)
• -q (we proved that q is not definitely provable in D)
• +q (q is defeasibly provable in D)
• - q (we proved that q is not defeasibly provable in D)
Derivability
A conclusion p is derivable when
• p is a fact
• there is an applicable strict or defeasible rule for p, and
all the rules for  p are discarded (i.e., proved not to
be applicable), or
every applicable rule for  p is weaker than an
applicable strict or defeasible ruple for p
Temporal Extensions
Explicit representation of time need to cope with large parts
of reality (e.g., causation)
durative actions
delays
Trade-off between expressiveness and computational
complexity
GOAL: temporal extension to DL retaining LINEAR
complexity
Temporal Rules
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a1:d1, …., an:dn d b:db
e:d e is an event whose duration is exactly d (d1)
a1:d1, …., an:dn are the “causes”. They can start at
different points in time
b:db is the effect
d is the exact delay between causes and effects
Temporal Rules
a1:d1, …., an:dn d b:db
SCHEMA OF RULES
(1) d is the delay between the beginning of the last cause
and the beginning of the effect
(2) d is the delay between the ending point of the last cause
and the beginning of the effect (here finite causes only)
Temporal Rules
TRIGGERING CONDITIONS (intuition):
(1) We must be able to prove each ai for for exactly di
consecutive time points, i.e., it0,t1,….,tdi, tdi+1
consecutive time points such that we can prove ai at
points t1,….,tdi and we cannot prove it at t0 and tdi+1
(2) Let tmax the last time when the latest cause can be proved
(3) b can be proven for exactly db instants starting from time
tmax+d
Example
F = {a@0, b@5, c@5}
r1: a:1 10 d:10; r2: b:1 7 d:5; r3: c:1 8 d:5; r3  r2
0 …... 5 .….. 10
a
b
c
11
12
r1
r1
r2
+d
+d
r2 terminates r1
13 …….. 17
r 3  r2
+d +d
+d
r3  r2
+d
18
r3
+d
19
Proof Conditions for +@
If +p@t = P(n+1) then
(1) + p@t  P(1..n) or
(2) (i) -~p@t  P(1..n) and
(ii) rRsd[p] \ either r persists or r is -applicable at t
and
(iii) sR[~p] either
- s is -discarded at t or
- if s is (t-t’)-effective,then vRsd[p]\ v defeats s at t’
Complexity
THEOREM 1
Let D be a temporalized defeasible theory without
backward causation. Then the extension of D from time
t0 to t (i.e., the set of all consequences of D derivable
from t0 to t) can be computed in time linear to the size of
the theory, i.e., O(|Prop||R|  t)
Causation
a1:tad b:tb
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“Backward” causation: 0>ta+d
“One-shot” causation: 0ta+d and 0<tb+d
“Continuous” causation: 0ta+d and 0  tb+d
“Mutually sustaining” causation: 0=ta+d and 0 = tb+d
“Culminated event” causation: 0  d
Conclusions & Future Work
TEMPORAL EXTENSION TO DL
- increased expressiveness
- retaining linear complexity
FUTURE
-Complexity of theories with backward causation
-Type of events (e.g., states vs accomplishments vs processes)