Temporal Extensions to Defeasible Logic

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Temporal Extensions to Defeasible Logic
Guido Governatori1, Paolo Terenziani2 1
University of Quuensland, Brisbane,
Australia 2Dipartimento di Informatica, UPO,
Alessandria, Italy
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Introduction

• 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

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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

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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)

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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

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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
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Temporal Rules

• a1d1, ., andn ?d bdb
• ed e is an event whose duration is exactly d
  (d?1)
• a1d1, ., andn are the causes. They can start
  at different points in time
• bdb is the effect
• d is the exact delay between causes and effects

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Temporal Rules

• a1d1, ., andn ?d bdb
• SCHEMA OF RULES
• d is the delay between the beginning of the last
  cause and the beginning of the effect
• d is the delay between the ending point of the
  last cause and the beginning of the effect (here
  finite causes only)

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Temporal Rules

• TRIGGERING CONDITIONS (intuition)
• We must be able to prove each ai for for exactly
  di consecutive time points, i.e.,
  ?i?t0,t1,.,tdi, tdi1 consecutive time points
  such that we can prove ai at points t1,.,tdi and
  we cannot prove it at t0 and tdi1
• Let tmax the last time when the latest cause can
  be proved
• b can be proven for exactly db instants starting
  from time tmaxd

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Example
F a_at_0, b_at_5, c_at_5 r1 a1 ?10 d10 r2 b1 ?7
?d5 r3 c1 ?8 d5 r3 ? r2
0 ... 5 ... 10 11 12 13 ..
17 18 19
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Proof Conditions for _at_

• If ?p_at_t P(n1) then
• ?p_at_t ? P(1..n) or
• (i) -?p_at_t ? P(1..n) and
• (ii) ?r?Rsdp \ either r persists or r is
  ?-applicable at t and
• (iii) ?s?Rp either
• - s is ?-discarded at t or
• - if s is (t-t)-effective,then ?v?Rsdp\ v
  defeats s at t

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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)

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Causation

• a1ta?d btb
• Backward causation 0gttad
• One-shot causation 0?tad and 0lttbd
• Continuous causation 0?tad and 0 ? tbd
• Mutually sustaining causation 0tad and 0
  tbd
• Culminated event causation 0 ? d

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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)