Belief Revision and Progression of KBs in the Epistemic Situation Calculus

Transcription

1 Belief Revision and Progression of KBs in the Epistemic Situation Calculus Christoph Schwering 1 Gerhard Lakemeyer 1 Maurice Pagnucco 2 1 RWTH Aachen University, Germany 2 University of New South Wales, Australia

2 2 / 10 Actions and Beliefs Robot is holding a box, does not know what is in it, but 1. believes it is not fragile and not metallic 2. considers fragility more plausible than it being metallic 3. knows it is not broken yet

3 2 / 10 Actions and Beliefs Robot is holding a box, does not know what is in it, but 1. believes it is not fragile and not metallic 2. considers fragility more plausible than it being metallic 3. knows it is not broken yet Drop!

4 2 / 10 Actions and Beliefs Robot is holding a box, does not know what is in it, but 1. believes it is not fragile and not metallic 2. considers fragility more plausible than it being metallic 3. knows it is not broken yet Clink!

5 2 / 10 Actions and Beliefs Robot is holding a box, does not know what is in it, but 1. believes it is not fragile and not metallic 2. considers fragility more plausible than it being metallic 3. knows it is not broken yet How to reason about this?

6 2 / 10 Actions and Beliefs Robot is holding a box, does not know what is in it, but 1. believes it is not fragile and not metallic 2. considers fragility more plausible than it being metallic 3. knows it is not broken yet If 1 3 is all it believed initially, what is all it believes now? Disclaimer: talk includes improvements over paper (journal version in preparation)

7 3 / 10 Logic for Actions and Beliefs First-order logic with modalities: α holds after action A [A]α α holds forever α if φ held, ψ would hold B(φ ψ) Bψ all we believe is φ i ψ i O{φ 1 ψ 1,, φ m ψ m } before forgetting P, O P { }

8 3 / 10 Logic for Actions and Beliefs First-order logic with modalities: α holds after action A [A]α α holds forever α if φ held, ψ would hold B(φ ψ) Bψ all we believe is φ i ψ i O{φ 1 ψ 1,, φ m ψ m } before forgetting P, O P { } Semantics: worlds specify initial and future truth of fluents w = B []B

9 3 / 10 Logic for Actions and Beliefs First-order logic with modalities: α holds after action A [A]α α holds forever α if φ held, ψ would hold B(φ ψ) Bψ all we believe is φ i ψ i O{φ 1 ψ 1,, φ m ψ m } before forgetting P, O P { } Semantics: possible worlds ranked by plausibilities e = B B []B B e = [][]BB (due to revision)

10 4 / 10 All We Believe Only-believing uniquely determines belief structure Related to Levesque s only-knowing, Pearl s Z-Ordering Subsumes only-knowing α by conditional α Theorem: Unique-Model Property O{φ 1 ψ 1,, φ m ψ m } has unique model if φ i, ψ i are obj. Theorem generalizes for O P {φ 1 ψ 1,, φ m ψ m }

11 All We Believe Only-believing uniquely determines belief structure Related to Levesque s only-knowing, Pearl s Z-Ordering Subsumes only-knowing α by conditional α Theorem: Unique-Model Property O{φ 1 ψ 1,, φ m ψ m } has unique model if φ i, ψ i are obj. Usually all we believe is a Basic Action Theory (BAT) with initial beliefs Σ bel knowledge about dynamics physical effect (successor-state axioms due to Reiter): a. [a]b a = F B epistemic effect (action A leads to revision by IF (A)): a. IF (a) (a = B M ) 4 / 10

12 All We Believe O{ F M, F M M, B, dynamic axioms} believes it is not fragile and not metallic considers fragility more plausible than metallic knows it is not broken yet B F M B M B 4 / 10

13 Progression of an Epistemic State 5 / 10 B F M B M B Initially e = B B

14 Progression of an Epistemic State 5 / 10 B F M (B F ) M (B F ) After progression still e = B B

15 Progression of an Epistemic State 5 / 10 B F M (B F ) M (B F ) Natural revision by B M promotes B-worlds to the top

16 Progression of an Epistemic State 5 / 10 B F M (B F ) M (B F ) After revision (e ) (B M ) = BB

17 Progression of an Epistemic State 5 / 10 B F M (B F ) M (B F ) After progression e = BB

18 Progression of a BAT by a Physical Action 6 / 10 Similar to Lin and Reiter s progression Let A have no epistemic effect Let F be fluents of BAT with axioms [a]f ( x) γ F Let P be new predicates Beliefs after doing A Σ bel A = Σ bel F P { ( x.f ( x) γ F a F A P) F F} Substitute P for F to capture pre-a beliefs Assert x.f ( x) γ F a F A P to set post-a beliefs

19 7 / 10 Progression of a BAT by an Epistemic Action Let A have no physical effect Progression Σ bel A = Σ bel IF (A) Let = {φ ψ Σ bel OΣ bel = B(α φ ψ)} Let P be a new predicate Beliefs after promoting the most-plausible α-worlds Σ bel α = { P } { (P α) } { (φ P ψ) φ ψ } {φ P ψ φ ψ Σ bel }

20 P -worlds are the most plausible worlds P -worlds represent promoted α-worlds P -worlds represent original belief structure 7 / 10 Progression of a BAT by an Epistemic Action Let A have no physical effect Progression Σ bel A = Σ bel IF (A) Let = {φ ψ Σ bel OΣ bel = B(α φ ψ)} Let P be a new predicate Beliefs after promoting the most-plausible α-worlds Σ bel α = { P } { (P α) } { (φ P ψ) φ ψ } {φ P ψ φ ψ Σ bel }

21 8 / 10 Progression of a BAT Briefly: BAT progression matches semantic progression Theorem: Progression #1 = OΣ [A]O P {P } (Σ A) Roughly: If all we believe is Σ, then all we believe after A is Σ A. Theorem: Progression #2 = OΣ [A]α iff = O P {P } (Σ A) α Roughly: Σ A entails the same beliefs as Σ has after A.

22 9 / 10 Belief Revision Postulates Alchourron Gärdenfors Makinson (AGM) hold Darwiche Pearl (DP) hold with a little restriction on DP2 Original DP2 is violated because we cannot recover from an inconsistent state Nayak Pagnucco Peppas (NPP) violated because the order matters in natural revision

23 10 / 10 Conclusion and Ongoing/Future Work Situation calculus plus natural revision Belief progression using only-believing Other revision schemes, e.g., lexicographic Projection by regression Elimination of (nested) beliefs similar to our AAAI-15 paper When is progression first-order-definable? Feasible subclass Implementation based on Lakemeyer & Levesque, KR-14

24 Appendix

25 All We Believe Believe it is not fragile and not metallic F M Fragility is more plausible than metallic F M M Know that it is not broken B Know dynamic axioms (omitted) 1. ( F M ) (F M M ) (B ) B F M 2. (F M M ) (B ) B M 3. (B ) B 2 / 2