Temporal Extensions to Defeasible Logic Guido Governatori 1

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

Introduction • • Defeasible conclusions nonmonotonic logic Trade-off: expressiveness vs 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 A 1. . An B (classical rules) • Defeasible Rules A 1. . An B (rules that can be defeated by contrary evidence; e. g. , “birds usually fly”) • Defeaters A 1. . 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 • • a 1: d 1, …. , an: dn d b: db e: d e is an event whose duration is exactly d (d 1) a 1: d 1, …. , 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 a 1: d 1, …. , 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 exactly di consecutive time points, i. e. , i t 0, t 1, …. , tdi+1 consecutive time points such that we can prove ai at points t 1, …. , tdi and we cannot prove it at t 0 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} r 1: a: 1 10 d: 10; r 2: b: 1 7 d: 5; r 3: c: 1 8 d: 5; r 3 r 2 0 …. . . 5. …. . 10 a b c 11 12 r 1 r 2 r 3 r 2 + d + d + d r 2 terminates r 1 13 ……. . 17 18 r 3 + 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 t 0 to t (i. e. , the set of all consequences of D derivable from t 0 to t) can be computed in time linear to the size of theory, i. e. , O(|Prop| |R| t)

Causation a 1: 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)