On the Progression of Knowledge in the Situation Calculus

Transcription

1 On the Progression of Knowledge in the Situation Calculus Yongmei Liu Department of Computer Science Sun Yat-sen University Guangzhou, China Joint work with Ximing Wen KRW-2012, Guiyang June 17, 2012 Y. Liu Progression of Knowledge 1 / 24

2 Litmus test example [Moore 1985] The initial KB: {acid, red, K( red)} Y. Liu Progression of Knowledge 2 / 24

3 Litmus test example [Moore 1985] The initial KB: {acid, red, K( red)} How should we update the KB? Intuitively, the new KB should entail K(acid) Y. Liu Progression of Knowledge 2 / 24

4 Main problem How to update the knowledge base (KB) after an action is executed? The KB includes both objective and subjective knowledge 1 Whether the new KB is definable in first-order modal logic? 2 Whether the new KB is finite? 3 Whether the new KB is computable? Y. Liu Progression of Knowledge 3 / 24

5 Our main work 1 Definition of knowledge progression 2 Progression wrt. physical actions this reduces to forgetting in first-order modal logic progression wrt local-effect physical actions 3 Progression wrt. sensing actions 4 Extension to the multi-agent case Y. Liu Progression of Knowledge 4 / 24

6 Situation calculus with knowledge Sensing actions of the form sense ψ ( x) A special fluent K(s, s), meaning that situation s is accessible from situation s. Knowing φ at situation s is represented as: Knows(φ, s) def = s.k(s, s) φ[s ]. Y. Liu Progression of Knowledge 5 / 24

7 Basic action theories with knowledge D = Σ K Init D una D ap D ss D S0, where D ss : successor state axioms (SSA), e.g., red(do(a, s)) a = dip acid(s) red(s), acid(do(a, s)) acid(s) a dilute D includes the SSA for the K fluent [Scherl & Levesque 2003]: K(s, do(a, s)) s.k(s, s) s = do(a, s ) x[a = senseψ ( x) ψ( x, s ) ψ( x, s)]. A consequence of the SSA for K is that the agent knows her successor state axioms for ordinary fluents D S0 : the initial KB, a finite set of sentences about S 0, e.g. acid(s 0 ), Knows( red, S 0 ) Y. Liu Progression of Knowledge 6 / 24

8 Progression [Lin & Reiter 1997] An intuitive definition Let α be a ground action, and let S α denote do(α, S 0 ). A progression of D S0 wrt α is a set of sentences D Sα s.t. 1 Just as D S0 is about S 0 only, D Sα should be about S α only; 2 the old theory D and the new theory D = (D D S0 ) D Sα should be equivalent wrt the possible futures of S α. Y. Liu Progression of Knowledge 7 / 24

9 Progression A formal definition (A variant from [Lin & Reiter 1997]) α D Sα is a progression of D S0 wrt α, written D S0 DSα, if 1 D Sα is about S α only, D = D Sα, and 2 for every M = D, there is M = D such that M Sα M. How to define M Sα M in the presence of knowledge? We adapt the concept of bisimulation from modal logic Y. Liu Progression of Knowledge 8 / 24

10 Bisimulation 2.2 Bisimulations 65 he converse of (ii): if wzw 0 and R 0 w 0 v 0, then there exists v (in M) such hat vzv 0 and Rwv (the back condition). is a bisimulation linking two states w in M and w 0 in M 0 we say that w re bisimilar, and we write Z : M;w $ M 0 ;w 0. If there is a bisimulation that Z : M;w $ M 0 ;w 0, we sometimes write M;w $ M 0 ;w 0 ; likewise, is some bisimulation between M and M 0, we write M $ M 0. a M Sα M if there is a relation Z between their situations s.t. the denotations of S α are Z-related, and whenever w Z w, M 1 w and w agree on all ordinary fluents; f Definition 2.16 pictorially. Suppose we know that wzw 0 and Rwv (the solid arrow in M and the Z- e bottom of the diagram display this information). Then the forth condition t it is always possible to find a v 0 that completes the square (this is shown ashed arrow in M 0 and the dotted Z-link at the top of the diagram). Note metry between the back and forth clauses: to visualize the back clause, eflect the picture through its vertical axis. vq Z qv The forth Figure condition: 2.3 shows the content for all of the v accessible forth (by K) from w, there is v accessible from w s.t. v Z v ; 3 The back condition: for all v accessible from w, there is v accessible from w s.t. v Z v. wq Z Fig The forth condition. qw 0 M 0 Proposition. If M Sα M, then for any formula φ about S α and the futures of S α, M = φ iff M = φ. t, bisimulations are a relational generalization of boundedy. morphisms: Liu Progression we of Knowledge 9 / 24

11 Outline 1 Definition of knowledge progression 2 Progression wrt. physical actions this reduces to forgetting in first-order modal logic progression wrt local-effect physical actions 3 Progression wrt. sensing actions 4 Extension to the multi-agent case Y. Liu Progression of Knowledge 10 / 24

12 Progression wrt physical actions The modal formula Kφ means knowing φ. Theorem Let D S0 be φ[s 0 ] where φ is in FO modal logic, and α a physical action. Let kforget(φ KDss[α, S 0 ], F ) ψ, where ψ is in α second-order modal logic. Then D S0 ψ[sα ]. D ss [α, S 0 ] denotes the instantiation of D ss wrt α and S 0 ψ denotes the result obtained from ψ by replacing F ( t, S 0 ) with F ( t) and F ( t, S α ) with F ( t) So for physical actions, progression of knowledge reduces to forgetting predicates in first-order modal logic. Y. Liu Progression of Knowledge 11 / 24

13 Forgetting in propositional modal logic [Zhang & Zhou 2009] studied forgetting in propositional S5 They defined forgetting based on the notion of bisimulation Their definition coincides with the semantic definition of formula V φ in [French 2005] and the notion of uniform interpolation in [Ghilardi et al. 2006] A result in [Ghilardi et al. 2006] shows that propositional S5 is closed under forgetting. Y. Liu Progression of Knowledge 12 / 24

14 Forgetting atoms in first-order modal logic Definition φ is in -DNF if it is of the form x(φ 1... φ n ), where each φ i is of the form α Kβ i x imγ i, called an extended term, where α, β, and γ i s are all objective, and Mγ abbreviates for K γ, and means possibly γ -DNF goes beyond formulas without quantifying-in Proposition. kforget(α Kβ i x imγ i, µ) forget(α β, µ) Kforget(β, µ) i x imforget(γ i β, µ). Theorem Let φ be in -DNF. Then the result of forgetting an atom in φ is definable in first-order modal logic and computable. Y. Liu Progression of Knowledge 13 / 24

15 Forgetting predicates in first-order modal logic Theorem If a sentence φ entails knowing that the truth values of predicates P and Q are different at only a finite number of certain instances, then forgetting Q in φ can be achieved by forgetting the Q atoms of these instances in φ and then replacing Q by P in the result. Y. Liu Progression of Knowledge 14 / 24

16 Progression wrt local-effect physical actions Local-effect actions: if an action A( c) changes the truth value of an atom F ( a, s), then a is contained in c. For action A( c), let Ω = {F ( a) a is contained in c }. Theorem Let α be a local-effect physical action. Let D S0 be φ[s 0 ] where φ is in -DNF. Then there is ψ in FO modal logic s.t. kforget(φ KD ss[ω], Ω) ψ, and D S0 α ψ( F / F )[S α ]. In this case, progression of knowledge is definable in first-order modal logic. Y. Liu Progression of Knowledge 15 / 24

17 Example: Progression wrt dip D S0 = {acid(s 0 ), K( red, S 0 )}. S 1 = do(dip, S 0 ). Ω = {red}, D ss[ω] includes: red acid red kf orget(φ, red), where φ = acid K red K(red acid red) D S1 = {acid(s 1 ), red(s 1 ), K(acid red, S 1 )} Y. Liu Progression of Knowledge 16 / 24

18 Outline 1 Definition of knowledge progression 2 Progression wrt. physical actions 3 Progression wrt. sensing actions 4 Extension to the multi-agent case Y. Liu Progression of Knowledge 17 / 24

19 Progression wrt sensing actions The modal formula W φ abbreviates for Kφ K φ Theorem Assume that D S0 does not contain negative knowledge. Let α be sense ψ ( c), where ψ( x) is an objective formula. α Then D S0 DS0 (S 0 /S α ) {W ψ( c)[s α ]}. A formula does not contain negative knowledge if no knowledge atom appears in the scope of an odd number of negation operators. In the general case, without the assumption, we do not yet know how to do progression wrt sensing actions. Y. Liu Progression of Knowledge 18 / 24

20 Example: Progression wrt sense D S1 = {acid(s 1 ), red(s 1 ), K(acid red, S 1 )} S 2 = do(sense red, S 1 ) D S2 = {acid(s 2 ), red(s 2 ), K(red acid, S 2 ), W (red, S 2 )} D S2 {K(red, S 2 ), K(acid, S 2 )} Y. Liu Progression of Knowledge 19 / 24

21 Outline 1 Definition of knowledge progression 2 Progression wrt. physical actions 3 Progression wrt. sensing actions 4 Extension to the multi-agent case Y. Liu Progression of Knowledge 20 / 24

22 Extension to the multi-agent case: Initial results The modal formula Cφ means that φ is common knowledge. By introducing a second-order axiom describing the transitive closure of the union of the K i fluent, we can represent common knowledge in the SitCalc. We only consider public actions whose occurrence is common knowledge. Y. Liu Progression of Knowledge 21 / 24

23 Extension to the multi-agent case: Initial results Theorem Let D S0 be φ[s 0 ] where φ is in FO modal logic, and α a public physical action. Let kforget(φ CDss[α, S 0 ], F ) ψ, where ψ is α in SO modal logic. Then D S0 ψ[sα ]. Theorem Assume that D S0 does not contain negative knowledge. Let ψ( x) be an objective formula, and α a public sensing action sense i,ψ ( c). α Then D S0 DS0 (S 0 /S α ) {CW i ψ( c)[s α ]}. Y. Liu Progression of Knowledge 22 / 24

24 Conclusions 1 Adapted the concept of bisimulation and extended Lin & Reiter s notion of progression to accommodate knowledge 2 Showed for physical actions, progression of knowledge reduces to forgetting predicates in FO modal logic 3 Showed for local-effect physical actions, when the initial KB is in -DNF, prog n of knowledge is definable in FO modal logic 4 Showed for sensing actions, when the initial KB does not contain negative knowledge, prog n can be simply achieved 5 Initial results on extension to the multi-agent case Y. Liu Progression of Knowledge 23 / 24

25 Future work 1 An in-depth study of forgetting in FO modal logic 2 Progression wrt sensing actions in the general case 3 Progression in the general multi-agent case Y. Liu Progression of Knowledge 24 / 24