BACK TO THE FUTURE On the State of the Art in Default Reasoning - PowerPoint PPT Presentation

BACK TO THE FUTURE On the State of the Art in Default Reasoning Emil Weydert Individual and Collective Reasoning Group Lab for Intelligent and Adaptive Systems University of Luxembourg Dagstuhl 2015, May 26-29

What to expect • On default reasoning ... • ... using ranking measures • Beyond plausibility maximization • System JZ - a canonical ranking construction • Why not Maximum Entropy ranking-style?

Reasoning with defaults Nonmonotonic inference: Φ | ∼ ψ but Φ ∪ Ψ �| ∼ ψ Examples: inductive, abductive, context-dependent, prioritized, legal reasoning, inconsistency elimination - and in particular: Default reasoning: nonmonotonic reasoning with defaults Defaults: weak implicational assertions allowing exceptions ϕ ❀ ψ : if ϕ then by default ψ , ϕ plausibly/normally implies ψ , ϕ is a prima facie reason for ψ , ... Early default formalisms: ... followed by many more ... - Default logic [Reiter 79] - Conditional logics [Delgrande 87, Kraus, Lehmann, Magidor 90] - Defeasible argumentation [Pollock 87, Loui 87] Because: some desiderata are surprisingly hard to realize!

Default inference - a general perspective 3 Lessons about defaults: - Defaults are essentially object-level entities - The monotonic logic of defaults may be of secondary importance - Defaults get their meaning in the context of defeasible inference Basic default language: over some classical logic ( L, ⊢ ) L ( ։ , ❀ ) = { ϕ ։ ψ, ϕ ❀ ψ | ϕ, ψ ∈ L } ϕ ։ ψ strict, ϕ ❀ ψ defeasible conditional Default Inference: Σ ∪ ∆ | ∼ ψ or Σ | ∼ ∆ ψ Fact base Σ ⊆ L , default base ∆ ⊆ L ( ։ , ❀ ), ψ ∈ L . e.g. { E } ∪ { E ։ B, B ❀ F, E ❀ ¬ F } | ∼ ¬ F , and ... �| ∼ F

Some basic requirements - Strict Modus Ponens for ։ and | ∼ - Defeasible Modus Ponens for ❀ and | ∼ : – Σ ∪ { ϕ, ϕ ❀ ψ } | ∼ ψ if Σ ∪ { ϕ } � ⊢ ¬ ψ – Σ ∪ { ϕ, ¬ ψ } ∪ { ϕ ❀ ψ } ∪ ∆ �| ∼ F is possible - Local exchange of logically equivalent conditionals - Boolean invariance - only the boolean structure is relevant - Irrelevance of independent new info X : Σ ∪ ∆ ( ∪ X ) | ∼ ψ - Preferentiality for | ∼ ∆ (has preferred model theory) [KLM 90], fails e.g. for Reiter’s default logic To implement these principles, to suitably handle independence → semantics using reasonable conditional plausibility valuations

Ranking semantics Ranking measures: plausibility + independence + valuation Math. generalizes Spohn’s ranking functions [Spohn 88, Wey 94] Coarse-graine, quasi-probabilistic implausibility valuations → to model iterated revision, graded plain belief, independence Real-valued ranking measures: enough for default reasoning R : Prop L → ([0 , ∞ ] , 0 , ∞ , + , ≤ ) with R ( T ) = 0, R ( F ) = ∞ , R ( A ∪ B ) = min ≤ { R ( A ) , R ( B ) } , R ( A | B ) = R ( A ∩ B ) − R ( B ) R 0 : uniform ranking measure, ∞ − ∞ := ∞ Ranking epistemology: R ( ¬ ϕ ) = degree of belief in ϕ Monotonic semantics for L ( ։ , ❀ ) : model sets Mod rk (∆) • R ∈ Mod rk ( { ϕ ❀ ψ } ) iff R ( ϕ ∧ ψ ) + 1 ≤ R ( ϕ ∧ ¬ ψ ) • R ∈ Mod rk ( { ϕ ։ ψ } ) iff R ( ϕ ∧ ¬ ψ ) = ∞

Ranking-based default inference Idea: take preferred ranking models Pref (∆) ⊆ Mod rk (∆) ∼ Pref ψ iff { ϕ 0 , ..., ϕ n } ∪ ∆ | Pref (∆) ⊆ Mod rk ( ϕ 0 ∧ ... ∧ ϕ n ❀ ψ ) - System P [KLM 90]: Pref P (∆) = Mod rk (∆), weakest - System Z [Pearl 90]: Pref Z (∆) = { Min ≤ ( Mod rk (∆)) } But: no irrelevance of independent info, failing inheritance: ∼ Z ψ for independent ϕ, ψ {¬ ϕ } ∪ { T ❀ ϕ, T ❀ ψ } �|

System J Epistemic construction strategy: [Wey 96] Idea: focus on those R ∈ Mod rk (∆) which can be obtained by iterated Spohn-revision with ϕ i → ψ i if ∆ = { ϕ i ❀ ψ i | i ≤ n } Pref J (∆) = { R 0 ⋆ a 0 ( ϕ 0 → ψ 0 ) ... ⋆ a n ( ϕ n → ψ n ) | = rk ∆ | 0 ≤ a i } = { R 0 + s 0 ( ϕ 0 ∧ ¬ ψ 0 ) + . . . + s n ( ϕ n ∧ ¬ ψ n ) | = rk ∆ | 0 ≤ s i } Yes: basic inference principles, exceptional inheritance, benchmarks But: still too weak, unjustified s i , no plausibility maximization

System JZ ∼ : Pref ∗ (∆) = { R ∗ Strongest constructible | ∆ } ⊆ Pref J (∆) Flagship: canonical JZ-ranking construction (see System Z): System JZ : based on canonical JZ-model R jz ∆ [Wey 98,03] • Maximizing plausibility: Hierarchical bottom-up construc- tion targeting for each ( ϕ i ∧ ¬ ψ i ) the smallest possible rank • Minimizing local shifting: Lexicographically minimize first the longest/costliest shifting moves aimed at a given rank • Justifiable constructibility: Proper shifting only if the corresponding ranking constraint is not oversatisfied Example: b: bird, s: small, f: canfly ∼ jz s • { b, ¬ f } ∪ { b ❀ s, b ❀ f } | ∼ jz s • { b, ¬ f } ∪ { b ❀ s, b ❀ f, b ∧ ¬ s ❀ ¬ f } �|

System ME Entropy maximization: well-justified probabilistic inference method minimizing information bias. Convex constraints → unique P me ME applicable to default reasoning [Goldsmidt et al. 93, Bacchus et al. 96, Wey 96, 98, Kern-Isberner 98, Bourne, Parsons 99 ...] Idea: ϕ ❀ ψ translated by P ( ψ | ϕ ) ≥ 1 − ε (0 < ε infinitesimal) The resulting constraint set ∆ P ε is closed and convex → a unique nonstandard maximum-entropy-model P me ∆ P ε This gives - independently from the choice of ε - a canonical justifi- ably constructible ranking measure model R me ∆ with R me ∆ ( A ) = standard ( log ε ( P me ε ( A ))) ∆ P ∆ = R jz Fact: For irredundant default bases ∆, we have R me ∆

Conclusions Problems with ME: - No nice algorithms for more general scenarios - No intuitive canonical ranking construction - Representation dependence (not robust w.r.t. small changes) A problem with JZ? ∼ jz ∼ jz - Mod rk (∆) = Mod rk (∆ ′ ) with | ∆ � = | ∆ ′ is possible ∼ jz → the ranking semantic content of ∆ does not characterize | ∆ But: this is implied by major desiderata for default reasoning! Conclusion: The practical issues and theoretical problems suggest that the direct implementation of ME at the ranking level doesn’t provide an adequate solution for default reasoning However, System JZ, or a close relative, may actually take its place in the ranking context

Some references Wey 94 General belief measures. Proc. of UAI 94 . Morgan Kauf- mann, 1994. Wey 95 Defaults and infinitesimals. Defeasible inference by non- archimedean entropy maximization. Proc. of UAI 95 . Morgan Kaufmann, 1995. Wey 96 System J - Revision entailment. Proc. of FAPR 96 . Springer, 1996. Wey 98 System JZ - How to build a canonical ranking model of a default knowledge base. Proc. of KR 98 . Morgan Kaufmann, 1998. Wey 03 System JLZ - Rational default reasoning by minimal rank- ing constructions. Journal of Applied Logic 1(3-4):273308. Else- vier, 2003.