Knowledge Representation, Coalgebraically Dirk Pattinson, Imperial College London (based on joint work with Rajeev Gorè, Clemens Kupke and Lutz Schröder) Oxford, October 2010
Back in Tudor England . . . 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” October 30, 2010 1
What do we know? ♥ 20% • Mary Boleyn was Henry’s Mistress time between 1520 and 1526 (approx.) • suppose that there’s a 20 % chance that Henry Carey is a royal bastard October 30, 2010 2
What’s a good model? Quantitative Uncertainty about offspring: { , , , . . . } 0 . 2 { , , , , . . . } 0 . 2 0 . 6 { , . . . } , Models are probability distributions over sets of successors : W → D ( P ( W )) October 30, 2010 3
What’s a good model? Certainty about amorous affairs: Madge Shelton Elizabeth Blount Models are relations : W → P ( W ) Combinations of both facets: W → D ( P ( W )) × P ( W ) � �� � � �� � offspring affairs October 30, 2010 4
Logical Language Syntax based on • a set P of propositional variables (concepts) like king • a set N of nominals (individual variables) like henry • a set Λ of modal operators (like “had an affair with”) Formulas H (Λ) ∋ A, B ::= x | A ∧ B | A ∨ B | ♥ ( A 1 , . . . , A n ) | ¯ ♥ ( A 1 , . . . , A n ) | @ i A where x ∈ P ∪ ¯ P ∪ N ∪ ¯ N , i ∈ N and ♥ ∈ Λ is n -ary. Interpretation. • nominals denote singletons, @ i moves evaluation context to point i • modalities “scan successors” for relevant properties item M, w | = A for some w ∈ W — A is satisfiable in M October 30, 2010 5
How do we interpret this? Tudor Example. Λ = { ✸ } ∪ { L p | p ∈ Q ∩ [0 , 1] } Models are triples M = ( W, σ, π ) where σ : W → TW = D ( P ( W )) × P ( W ) π : P ∪ N → P ( W ) and are such that π ( n ) is a singleton for all n ∈ N . Modalities (where � A � M = { w ∈ W | M, w | = A } is the truth-set of A ) M, w | = ✸ A ⇐ ⇒ σ ( w ) ∈ { ( µ, S ) ∈ TW | S ∩ � A � M � = ∅} ⇒ σ ( w ) ∈ { ( µ, S ) ∈ TW | µ ( { S ′ ⊆ W | S ′ ∩ � A � M � = ∅} ) ≥ p } M, w | = L p A ⇐ Satisfaction Operators and Variables (where x ∈ P ∪ N ) M, w | = @ i A ⇐ ⇒ M, π ( i ) | M, w | = x ⇐ ⇒ w ∈ π ( x ) = A October 30, 2010 6
Enter Coalgebra . . . Models in the Tudor example: M = ( W, σ, π ) where σ : W → TW = P ( D ( W )) × P ( W ) Modalities in the Tudor Example: M, w | = ✸ A ⇐ ⇒ σ ( w ) ∈ { ( µ, S ) ∈ TW | S ∩ � A � M � = ∅} ⇒ σ ( w ) ∈ { ( µ, S ) ∈ TW | µ ( { S ′ ⊆ W | S ′ ∩ � A � M � = ∅} ) ≥ p } M, w | = L p A ⇐ Coalgebraic Models. M = ( W, σ, π ) where σ : W → TW Coalgebraic Modalities. M, w | = ♥ ( A ) ⇐ ⇒ σ ( w ) ∈ � ♥ � W ( � A � M ) and � ♥ � is a predicate lifting , i.e. a natural family of mappings of type � ♥ � X : P ( X ) → P ( TX ) October 30, 2010 7
Coalgebraic Setup Given. • a collection Λ of modal operators • a functor T : Set → Set inducing T -models ( C, γ : C → TC ) • an interpretation � ♥ � : P → P ◦ T of every ♥ ∈ Λ Problem. Decide whether A ∈ H (Λ) is satisfiable in Mod (Ξ) , for Ξ ⊆ H (Λ) . Moreover, what’s the complexity and what’s a feasible algorithm? Examples. • the Tudors (as we’ve seen before) • and many more by varying T and Λ (graded, multi-agent, conditional etc.) • also quantum phenomena? October 30, 2010 8
Enter Tableau Methods . . . Basic Ingredient. One-Step (Tableau) Rules ♥ 1 A 1 , . . . , ♥ n A n , ¬♠ 1 B 1 , . . . , ¬♠ k B k Σ 1 . . . Σ k where Σ 1 , . . . , Σ k ⊆ {± A 1 , . . . , ± B k } . Basic Soundness / Completeness If Γ = ♥ 1 A 1 , . . . , ♥ n A n , ¬♠ 1 B 1 , . . . , ¬♠ k B k and τ ( A i ) , τ ( B j ) ⊆ X then � Γ � � = ∅ ⇐ ⇒ � Γ i � � = ∅ Γ 0 Γ k with Γ 0 ⊆ Γ and some 1 ≤ i ≤ n where for some rule Γ 1 ... • � ♥ A � = � ♥ � X ( τ ( A )) • � Σ 1 , Σ 2 � = � Σ 1 � ∩ � Σ 2 � • � A � = τ ( A ) (These rulesets are known for many examples.) October 30, 2010 9
Basic Tableaux Basic Result. Suppose that A ∈ H (Λ) does not contain any nominals or @ -operators. Then A unsatisfiable in Coalg ( T ) ⇐ ⇒ A has closed tableau where tableaux are constructed from one-step rules and propositional rules Γ , A ∧ B Γ , ¬ ( A ∧ B ) Γ , A, ¬ A Γ , A, B Γ , ¬ A Γ , ¬ B 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 Complexity. P SPACE for tractable rule sets October 30, 2010 10
Global Assumption Tableaux Basic Result. Suppose that A ∈ H (Λ) and Θ ⊆ H (Λ) do not contain any nominals or @ -operators. Then A unsatisfiable in Mod (Θ) ⇐ ⇒ A, Θ has closed Θ - tableau where Θ -tableaux are constructed from propositional and augmented one-step rules ♥ 1 A 1 , . . . , ♥ n A n , ¬♠ 1 B 1 , . . . , ¬♠ k B k Σ 1 , Θ . . . Σ k , Θ 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. Complexity. E XPTIME for tractable rule sets. October 30, 2010 11
Enter Nominals and Satisfaction Operators . . . Task. For A ∈ H (Λ) and Θ ⊆ H (Λ) , decide whether A satisfiable in Mod (Θ) . Complexity Theorist’s Take. That’s easy . . . • for each nominal i , guess its theory K i in the subformulas of A, Θ • use Θ -tableau to check satisfiability of A, Θ replacing @ i A by ⊤ if A ∈ K i and by ⊥ , otherwise • additionally check satisfiability of K i . Proof Idea. Satisfiability entails that K i as required exist Complexity. E XPTIME as there are only exponentially many possible guesses. Practitioner’s Complaint. Guessing K i is ‘too much work’. October 30, 2010 12
Coalgebraic Tableaux in Action Basic Idea. Satisfiability as a game played by ∃ (claiming sat) and ∀ • every rule application (challenge of ∀ ) must have a satisfiable conclusion ( ∃ ) • rule application may uncover @ -prefixed formulas that need to be propagated • propagated @ -formulas can be challenged by ∀ Example 1. ∃ shows satisfiability ✸ @ i p, ✸ ( ✸ @ i q ∨ ✸ A ) @ i p, @ i q @ i p ✸ @ i q ∨ ✸ A i, p, q, @ i p, @ i q ✸ @ i q @ i q October 30, 2010 13
Coalgebraic Tableaux in Action Basic Idea. Satisfiability as a game played by ∃ (claiming sat) and ∀ • every rule application (challenge of ∀ ) must have a satisfiable conclusion ( ∃ ) • rule application may uncover @ -prefixed formulas that need to be propagated • propagated @ -formulas can be challenged by ∀ Example 2. ∀ shows unsatisfiability ✸ @ i ⊥ , ✸ A @ i ⊥ , • @ i ⊥ i, ⊥ , @ i ⊥ Note. In both examples, wins were announced without fully unfolding the tableau. October 30, 2010 14
Conclusions Complexity. For coalgebraic logics, satisfiability is EXPTIME-decidable over a (finite) set of global assumptions • this has been known before, but we have a new proof Practicability. The decision procedure is purely syntax driven • amenable to ‘usual’ optimisations for tableau-bases algorithms • room for heuristics Novelty. Even for (description) logics with relational semantics, our algorithm appears to be new? Still Missing (but not for long): implementation and experiments. October 30, 2010 15
Technicalities: Caching Graphs Basic Entities. • Sequents are finite sets of formulas • @ -constraints are finite sets of @ -prefixed formulas possibly containing • [ • signifies incomplete information ] Expansion Rules • for sequents : propositional logic + modal rules (logic specific, but known) • for @ -constraints : Υ i, Υ / @ i , Υ \ {•} where i ∈ N (Υ) is a nominal occurring in Υ and Υ / @ i = { A | @ i A ∈ Υ } . [ Υ is needed as further constraints may be uncovered by modal unfolding ] October 30, 2010 16
Expansion and Propagation in a Nutshell Sequent Expansion → S • select unexpanded sequent ( X ) and create/reuse nodes for all applicable rules • create associated @ -constraint containing just • • mark sequent as unknown ( U ) Constraint Expansion → C . • select unexpanded constraint ( T ) and create/reuse nodes for all applicable rules • mark constraint as done ( D ) Constraint Propagation → P • percolate constraints through the graph using greatest-fixpoint construction Position Update → U . • Update winning positions for ∀ inductively : add sequents that are unsat • Update winning positions for ∃ dually (using coinduction ) October 30, 2010 17
Properties of Ensuing Algorithm Correctness. Given that the modal rules are sound and complete, sequents marked ‘satisfiable’ are indeed satisfiable (dually for unsat). Proof. If a sequent is marked ‘sat’, player ∃ has a winning strategy in a game played on sequents, which induces a satisfying model. Termination. If modal rules are decidable, every execution terminates and all sequents will be marked as either sat or unsat. Proof. Determinacy of the associated two-player game. Complexity. If modal rules are EXPTIME decidable, every execution of the algorithm will terminate in EXPTIME. Proof. The number of sequents that appear in the game is exponential in the size of the root sequent. Non-Determinism leaves room for heuristics. October 30, 2010 18
Recommend
More recommend
