Knowledge Representation, Coalgebraically Dirk Pattinson, Imperial - - PowerPoint PPT Presentation
Knowledge Representation, Coalgebraically Dirk Pattinson, Imperial College London (based on joint work with Rajeev Gor, Clemens Kupke and Lutz Schrder) Oxford, October 2010 Back in Tudor England . . . Henry VIII Henry Carey Mary Boleyn
Dirk Pattinson, Imperial College London (based on joint work with Rajeev Gorè, Clemens Kupke and Lutz Schröder) Oxford, October 2010
Henry VIII Henry Carey Mary Boleyn
“There has been speculation that Mary’s two children, Catherine and Henry, were fathered by Henry, but this has never been proven”
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20%
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Quantitative Uncertainty about offspring:
0.2 0.2 0.6
Models are probability distributions over sets of successors: W → D(P(W))
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Certainty about amorous affairs: Madge Shelton Elizabeth Blount Models are relations: W → P(W) Combinations of both facets:
affairs
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Syntax based on
Formulas
where x ∈ P ∪ ¯
Interpretation.
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Tudor Example. Λ = {✸} ∪ {Lp | p ∈ Q ∩ [0, 1]} Models are triples M = (W, σ, π) where
and
are such that π(n) is a singleton for all n ∈ N. Modalities (where AM = {w ∈ W | M, w |
Satisfaction Operators and Variables (where x ∈ P ∪ N)
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Models in the Tudor example: M = (W, σ, π) where
Modalities in the Tudor Example:
Coalgebraic Models. M = (W, σ, π) where σ : W → TW Coalgebraic Modalities.
and ♥ is a predicate lifting, i.e. a natural family of mappings of type
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Given.
Moreover, what’s the complexity and what’s a feasible algorithm? Examples.
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Basic Ingredient. One-Step (Tableau) Rules
where Σ1, . . . , Σk ⊆ {±A1, . . . , ±Bk}. Basic Soundness / Completeness If Γ = ♥1A1, . . . , ♥nAn, ¬♠1B1, . . . , ¬♠kBk and τ(Ai), τ(Bj) ⊆ X then
for some rule
Γ0 Γ1 ... Γk with Γ0 ⊆ Γ and some 1 ≤ i ≤ n where
(These rulesets are known for many examples.)
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Basic Result. Suppose that A ∈ H(Λ) does not contain any nominals or
where tableaux are constructed from one-step rules and propositional rules
and Coalg(T) is the class of all T -coalgebras (C, γ : C → TC). Proof Idea. Construct satisfying model ‘step-by-step’ by basic completeness
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Basic Result. Suppose that A ∈ H(Λ) and Θ ⊆ H(Λ) do not contain any nominals or @-operators. Then
where Θ-tableaux are constructed from propositional and augmented one-step rules
and Mod(Θ) is the class of all T -coalgebras (C, γ : C → TC) validating all formulas in Θ. Proof Idea. Augmented rules force validity of Θ at all constructed states.
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Complexity Theorist’s Take. That’s easy . . .
and by ⊥, otherwise
Proof Idea. Satisfiability entails that Ki as required exist
Practitioner’s Complaint. Guessing Ki is ‘too much work’.
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Basic Idea. Satisfiability as a game played by ∃ (claiming sat) and ∀
Example 1. ∃ shows satisfiability
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Basic Idea. Satisfiability as a game played by ∃ (claiming sat) and ∀
Example 2. ∀ shows unsatisfiability
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(finite) set of global assumptions
appears to be new? Still Missing (but not for long): implementation and experiments.
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Basic Entities.
[ • signifies incomplete information ] Expansion Rules
where i ∈ N(Υ) is a nominal occurring in Υ and Υ/@i = {A | @iA ∈ Υ}. [ Υ is needed as further constraints may be uncovered by modal unfolding ]
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Sequent Expansion →S
Constraint Expansion →C.
Constraint Propagation →P
Position Update →U .
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marked ‘satisfiable’ are indeed satisfiable (dually for unsat).
sequents will be marked as either sat or unsat.
algorithm will terminate in EXPTIME.
the root sequent. Non-Determinism leaves room for heuristics.
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