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

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

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 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 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)

TRIGGERING CONDITIONS (intuition): 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, 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 Let tmax the last time when the latest cause can be proved 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 11 12 13 …….. 17 18 19 a b c r2 +d +d r2 terminates r1 +d r1 r1 +d r3  r2 +d r3  r2 +d r3 +d

Proof Conditions for +@ If +p@t = P(n+1) then + p@t  P(1..n) or (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 “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)